Improved Measurements Of Hadronic-Mass
B → Xc`ν
Determination of |Vcb |
Moments In Decays
and
DISSERTATION
zur Erlangung des akademischen Grades
Doctor rerum naturalium
(Dr. rer. nat.)
vorgelegt
der Fakultät Mathematik und Naturwissenschaften
der Technischen Universität Dresden
von
Diplom-Physiker Jan Erik Sundermann
geboren am 20. November 1976 in Hamburg
Gutachter :
Prof. Dr. Klaus R. Schubert
Prof. Dr. Michael Kobel
Dr. Oliver Buchmüller
Eingereicht am : aa. bb. 2006
Disputation am : aa. bb. 2006
Abstract
Kurzfassung
Contents
1
Introduction
1.1
2
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inclusive Semileptonic
B -Meson Decays and |Vcb |
2.1
The Standard Model of Particle Physics
2.2
Electroweak Interaction and Symmetry Breaking
2
5
. . . . . . . . . . . .
. . . . . . .
5
6
2.3
The CKM Matrix and the Unitary Triangle
. . . . . . . . . .
9
2.4
Heavy Quark Eective Theory . . . . . . . . . . . . . . . . . .
10
2.4.1
The Heavy Quark Limit of QCD
. . . . . . . . . . . .
12
2.4.2
Eective Lagrangian . . . . . . . . . . . . . . . . . . .
14
2.4.3
Non-Perturbative Corrections . . . . . . . . . . . . . .
16
2.5
3
Outline
B -Meson
Inclusive Semileptonic
Decays
. . . . . . . . . . . .
17
. . . . . . . . . . . . . .
18
. . . . . . . . . . . . . . . . . . .
19
2.5.1
The Heavy Quark Expansion
2.5.2
The Kinetic Scheme
The
BABAR Experiment
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
The PEP-II
3.3
The
B -Meson
BABAR Detector
23
23
Factory . . . . . . . . . . . . . . . . . .
23
. . . . . . . . . . . . . . . . . . . . . . .
24
3.3.1
The Silicon Vertex Tracker
. . . . . . . . . . . . . . .
26
3.3.2
The Drift Chamber . . . . . . . . . . . . . . . . . . . .
26
3.3.3
The ƒerenkov Detector
. . . . . . . . . . . . . . . . .
27
3.3.4
The Electromagnetic Calorimeter . . . . . . . . . . . .
28
3.3.5
The Instrumented Flux Return
30
. . . . . . . . . . . . .
4
Outline of the Analysis
33
5
Measurement of Hadronic Mass Moments
37
5.1
Samples of Data and Monte Carlo Simulations . . . . . . . . .
37
5.2
Semi-Exclusive Reconstruction of
5.3
Breco
. . . . . . . . . . . . .
38
5.2.1
Reconstruction Formalism . . . . . . . . . . . . . . . .
38
5.2.2
Kinematic Variables
39
5.2.3
Selection of the Best
Particle Selection for the
. . . . . . . . . . . . . . . . . . .
Breco Candidate
Signal B Meson . .
. . . . . . . . .
41
. . . . . . . . .
43
v
CONTENTS
vi
5.4
5.3.1
Charged Track Selection . . . . . . . . . . . . . . . . .
44
5.3.2
Neutral Candidate Selection Criteria . . . . . . . . . .
47
5.3.3
Particle Identication
. . . . . . . . . . . . . . . . . .
47
Reconstruction of the Hadronic System . . . . . . . . . . . . .
49
5.4.1
Event Selection Criteria
. . . . . . . . . . . . . . . . .
49
5.4.2
Kinematic Fit . . . . . . . . . . . . . . . . . . . . . . .
52
5.4.2.1
Method of Least Squares with Constraints . .
52
5.4.2.2
Linearization . . . . . . . . . . . . . . . . . .
53
5.4.2.3
Solution . . . . . . . . . . . . . . . . . . . . .
54
5.4.2.4
Application to Semileptonic Decays
56
. . . . .
5.5
Comparison of Data and MC Simulations
. . . . . . . . . . .
59
5.6
Background Subtraction . . . . . . . . . . . . . . . . . . . . .
61
Combinatorial
. . . . . . . . . . . .
61
5.6.2
Residual Background . . . . . . . . . . . . . . . . . . .
66
+
and Ds Decays . . .
Background
66
. . . . . . . . . . . . . . . . .
71
Decays
. . . . . . . . . . . .
73
Flavor Anticorrelated
5.6.2.4
τ Decays . . .
J/ψ and ψ(2S)
B 0B 0 Mixing .
. . . . . . . . . . . . . . . . .
73
5.6.2.5
Mis Tagged Events . . . . . . . . . . . . . . .
73
5.6.2.6
Misidentied Hadrons . . . . . . . . . . . . .
74
5.6.2.7
Semileptonic
B → Xu `ν
. . . . . . .
74
Summary of Background Subtraction . . . . . . . . . .
74
5.6.3.1
.
74
. . . . . . . . . . . . . . . . . . . . .
77
Extraction of Moments . . . . . . . . . . . . . . . . . . . . . .
77
5.7.1
. . . . . . . . . . . . . . . . . . . . . . . . .
79
5.6.3
5.6.3.2
5.7.2
Calculation of Background Subtraction Fac-
Overview
Calibration
wi
. . . . . . . . . . . . . . . . . . . . . . . .
79
5.7.2.1
General Construction Principle . . . . . . . .
79
5.7.2.2
Binning of Calibration . . . . . . . . . . . . .
81
5.7.2.3
Linear Dependence . . . . . . . . . . . . . . .
81
5.7.2.4
From the Measured to the Corrected (Calibrated) Masses
5.7.3
Decays
Normalization of Background Distributions
tors
5.8
D
5.6.2.1
5.6.2.2
5.6.2.3
5.7
Breco
5.6.1
Bias Corrections
. . . . . . . . . . . . . . . .
82
. . . . . . . . . . . . . . . . . . . . .
82
5.7.3.1
Bias Correction of Calibration
. . . . . . . .
90
5.7.3.2
Residual Bias Correction
. . . . . . . . . . .
90
5.7.4
Statistical Uncertainties and Correlations
. . . . . . .
93
5.7.5
Verication of the Calibration Procedure . . . . . . . .
96
5.7.5.1
Calibration of Exclusive Signal Decay Modes
96
5.7.5.2
Calibration of Generic MC Simulations
. . .
97
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
CONTENTS
6
vii
Systematic Studies
105
6.1
General Procedure
6.2
Bias of Calibration . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3
Binning of Calibration Curves . . . . . . . . . . . . . . . . . . 106
6.4
Simulation Model of Signal Decays
6.5
. . . . . . . . . . . . . . . . . . . . . . . . 105
. . . . . . . . . . . . . . . 106
6.4.1
Simulation Model Dependence of Calibration
. . . . . 106
6.4.2
Model Dependence of Residual Bias Correction
. . . . 107
Background Subtraction . . . . . . . . . . . . . . . . . . . . . 107
6.5.1
Combinatorial Background Subtraction
6.5.2
Residual Background Subtraction . . . . . . . . . . . . 107
. . . . . . . . 107
6.6
Track Selection Eciency
. . . . . . . . . . . . . . . . . . . . 110
6.7
Photon Selection Eciency
6.8
Hard Photon Emission . . . . . . . . . . . . . . . . . . . . . . 111
6.9
Soft Photons
. . . . . . . . . . . . . . . . . . . 110
. . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.10 Stability Checks . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.10.1
Emiss − |~
pmiss |
Cut Variation
. . . . . . . . . . . . . . 112
6.10.2 Measurements on Disjoined Data Samples . . . . . . . 114
6.11 Summary of Systematic Uncertainties
7
. . . . . . . . . . . . . 117
Interpretation of Results and Extraction of
|Vcb |
125
7.1
Extraction Formalism
7.2
Experimental Input . . . . . . . . . . . . . . . . . . . . . . . . 127
. . . . . . . . . . . . . . . . . . . . . . 125
7.3
Fit Results and Comparison With Other Measurements
. . . 128
A Correlation Matrix
137
Acknowledgements
157
Chapter 1
Introduction
The
Standard Model of Particle Physics
comprises the current knowledge
about the interactions between the smallest constituents of matter, the quarks
and leptons.
Even though the Standard Model is extremely succesfull in
predicting physical observables with high accuracy it seems clear from a
theoretical point of view that the Standard Model has to be extended to
describe physics at higher energies.
However, new physics eects that lie
beyond the descriptiveness of the Standard Model might also be observable
at energies in reach of today's accelerator experiments. For this reason the
Standard Model is subject to several experimental high precision tests that
try to verify the consistency of the theory and search for physical eects beyond that Standard Model. Several of those tests are directly related to the
Cabibbo-Kobayashi-Maskawa
(CKM) matrix that relates the quark's weak
avor eigenstates to their mass eigenstates.
The elements of the CKM matrix are free parameters in the Standard
Model and have to be determined experimentally. A precise measurement of
the individual CKM matrix elements will allow to test the unitarity of the
quark mixing matrix that is an important property of the Standad Model.
This thesis presents a work that allows the determination of the CKM matrix
elements
|Vcb |
together with other fundamental parameters of the Standard
Model, the masses of the
b
and
c
quarks.
One of the most promising approaches for the determination of
|Vcb |
uti-
B
B → Xc `ν .
A theoretical approach that combines properties of the Heavy Quark Eective
Theory (HQET) with an Operator Product Expansion called Heavy Quark
Expansion (HQE) has already proven to be very successfull. The HQE prolizes eective eld theories for the theoretical description of semileptonic
meson decays to hadronic nal states containing a charm quark,
vides reliable predictions of inclusive observables like the total semileptonic
rate
Γ(B → Xc `ν) or
< mnX >.
the moments of the related invariant hadronic mass
distribution
Measuring the moments of the invariant hadronic mass spectrum in
semileptonic
B
meson decays requires the identication of all particles be-
1
Introduction
2
longing to the hadronic system
Xc .
The presented work implements a tech-
nique that fully reconstructs one of the
Υ (4S) → BB ,
B
meson decays in events
A possible semileptonic decay of the re-
meson is identied by its associated lepton. Experimentally the
identication of leptons with low momenta is complicated.
ing a minimum lepton momentum is inevitable.
Thus, requir-
This work presents mea-
surements of the rst to sixth invariant hadronic mass moments,
n = 1 . . . 6, as functions of the
p∗` ≥ 0.8 GeV/c and p∗` ≥ 1.9 GeV/c
with
meson.
e+ e− →
thereby separating its decay products from the hadronic sys-
tem of the semileptonic decays.
maining
B
< mnX >
minimum lepton momentum between
calculated in the restframe of the
B
Measuring moments up to very high minimum lepton momenta,
thereby entering the region near the kinematic endpoint in which the HQE
is expected to break down, allows to evaluate the behavior of the theoretical
prediction in the expected break-down region.
1.1 Outline
The presented work is organized as follows:
• Chapter 2
discusses the physics of
B
meson decays required for the
motivation and interpretations of the presented measurement.
After
a brief introduction to the Standard Model of Particle Physics, electroweak interactions of elementary particles and the CKM quark mixing matrix will be discussed. The two consequent sections cover special
topics concerning the decays of
B
mesons and inclusive semileptonic
decays to hadronic nal states containing a charm quark,
B → Xc `ν .
It
concludes with a discussion of how moments of the invariant hadronic
mass,
< mnX >,
can be used to extract the CKM matrix element
|Vcb |.
• Chapter 3 gives an overview of the BABAR detector located at the
+ −
PEP-II e e storage rings at the Stanford Linear Accelerator Center.
We will give a short overview of the dierent
• Chapter 4
BABAR subdetectors.
contains a brief overview of the applied method to mea-
sure the rst to sixth hadronic mass moments as functions of the minimum lepton momentum in the semileptonic decay. In the employed
technique one
B
meson in events
e+ e− → Υ (4S) → BB
is fully re-
constructed in several hadronic modes, thereby separating its decay
products from the hadronic system of the semileptonic decay.
semileptonic decay of the second
B
The
meson is identied by requiring a
single detected electron. The resolution of the reconstructed hadronic
system suers from the limited acceptance and resolution of the detector. For the extraction of moments we utilize a method that calibrates the measured hadronic masses by applying correction factors
on an event-by-event basis.
1.1 Outline
3
• Chapter 5 gives a detailed description of the performed measurement,
therebye following the brief outline given in the previous chapter. It
starts with an overview of the used data and MC simulation samples
followed by a description of the semi-exclusive reconstruction method
that is utilized for the full reconstruction of hadronic
Then particle selection criteria will be outlined.
B
meson decays.
The reconstruction
of the hadronic system is described in the consequent section followed
by a description of the background subtraction. Finally the extraction
of moments
< mnX >
from the reconstructed hadronic system will be
discussed. The chapter concludes with a presentation of the obtained
results.
• Chapter 6 discusses several potential sources of systematic uncertainties. Described are eects associated with the calibration procedure,
background subtraction, eects stemming from mis modeling in MC
simulations, and concludes with stability tests of the extracted results.
• Chapter 7 the obtained results are interpreted and compared to other
measurements. We use our results together with other measured moments of the lepton energy spectrum in semileptonic
B
extract the CKM matrix element
predicted moments.
meson decays
B → Xs γ to
|Vcb | in a combined t to theoretically
and moments of the photon energy spectrum in decays
Chapter 2
Inclusive Semileptonic
B -Meson
Decays and
|Vcb|
The following chapter will give a brief introduction to theoretical concepts
utilized to describe and interpret the decays of
work.
B
mesons investigated in this
Section 2.1 will give a brief introduction to the Standard Model of
Particle Physics followed by a discussion of the weak interaction part of the
Standard Model and the
Cabibbo-Kobayashi-Maskawa
quark mixing matrix
in sections 2.2 and 2.3.
Theoretical concepts that are of special interest for the description of
the physics of heavy mesons and inclusive measurements of semileptonic
B
meson decays
B → Xc `ν
will be discussed in the sections 2.4 and 2.5.
We will discuss basic aspects of the
Heavy Quark Expansion.
Heavy Quark Eective Theory
and the
There are several review articles covering this eld
of elementary particle physics which are among others [1, 2, 3].
2.1 The Standard Model of Particle Physics
The most comprehensive theoretical framework for the description of interactions between the smallest constituents of matter, the quarks and leptons, is the
Standard Model of Particle Physics.
The Standard Model rep-
resents the best current knowledge about strong, weak, and electromagnetic interactions of elementary particles.
Interactions are described by a
SU (3)C ⊗ SU (2)W ⊗ U (1)Y relativistic quantum eld theory combining the
SU (3)C color charge group, the SU (2)W weak isospin group, and the hypercharge symmetry group U (1)Y .
In the current view all matter is made out of three kinds of elementary
particles: leptons, quarks, and gauge bosons. There are six leptons arranged
into three generations:
νe electron-neutrino
e
electron
νµ
µ
muon-neutr.
muon
ντ tau-neutr.
τ
tau
5
Inclusive Semileptonic B -Meson Decays and |Vcb |
6
Similar, there are six avors of quarks falling into three generations:
c charm-quark
s strange-quark
u up-quark
d down-quark
t
top-quark
b bottom-quark
Furthermore, for every elementary particle there exists a corresponding antiparticle. While the dierent quark and lepton generations are equal regarding their quantum numbers they dier in their masses increasing hierarchically from quark generation to quark generation and from lepton generation
to lepton generation. Neutrinos are known to have masses since the observation of neutrino oscillations in 1998 by the Super-Kamiokande experiment [4].
Since the origin of neutrino masses has not been answered conclusively yet,
that is for example if neutrinos are Majorana or Dirac particles, and the
question of neutrino masses does not aect the theoretical interpretation of
B
meson decays under study in this work, we will consider neutrinos in the
following discussion of electroweak interactions and heavy quark decays as
massless.
Every interaction has its bosonic mediator. Electroweak interactions are
mediated by the photon,
W +, W −,
and
Z 0.
Charged particles interact
electromagnetically, i.e. all quarks and leptons except neutrinos. The weak
interaction couples only to the left-handed quarks and leptons thereby maximally violating parity conservation. The strong interaction is mediated by
eight gluons coupling to the so called color charge of the quarks. Leptons do
not take part in the strong interaction.
2.2 Electroweak Interaction and Symmetry Breaking
Until now three generations of quarks and leptons were observed that can
be arranged in three left-handed quark doublets, six right-handed quark
singulets, three left-handed lepton doublets, and three right-handed lepton
singulets:
quarks:
leptons:
i
u
= iL , uiR , diR
d
L
νLi
i
LL = i , `iR
eL
QiL
(i = 1 . . . 3).
(2.1)
The electroweak interaction part of the standard model has the structure of
a
SU (2)W ⊗ U (1)Y
group which is a combination of the weak
symmetry group and the hypercharge group
U (1)Y .
SU (2)W -
The weak interac-
tion part of the theory couples to the left-handed doublet structure of the
fermions, the weak isospin. The right-handed fermion singlets do not take
part in the weak interaction. The weak interaction is mediated by the three
2.2 Electroweak Interaction and Symmetry Breaking
I
I3
Q
Y
1
2
1
2
+ 12
− 12
0
0
0
0
+ 23
− 13
+ 23
− 13
+ 13
+ 13
+ 43
− 23
1
2
1
2
0
+ 12
− 12
0
0
−1
−1
−1
−1
−2
1
2
1
2
+ 12
− 12
+1
0
+1
+1
ui
ui = u, c, t
i
QL =
,
di L di = d, s, b
uiR = uR , cR , tR
diR = dR , sR , bR
νi
νi = νe , νµ , ντ
i
LL =
,
li L li = e, µ, τ
`iR = eR , µR , τR
+
h
H=
h0
7
Table 2.1: Properties of quark, lepton, and Higgs elds in the electroweak
interaction.
The elds
QiL
and
LiL
are the left-handed quark and lepton
doublets, respectively. Right-handed elds are indicated by a subscript
The elds
H
R.
is the Higgs doublet. The given quantum numbers are the weak
isospin (I ), the third component of the weak isospin (I3 ), the electric charge
(Q), and weak hypercharge,
Y = 2(Q − I3 ).
W3µ . The U (1)Y interaction mediµ
ated by a single abelian gauge eld B couples to the weak hypercharge of
non-abelian gauge elds
W1µ , W2µ ,
and
the fermions. The hypercharge quantum number
and the third component of the weak isospin
I3
Y
is related to the charge
Q
by the Gell-Mann-Nishijima
relation,
Y = 2 (Q − I3 ) .
(2.2)
Table 2.1 summarizes the mentioned quantum numbers of all appearing
elds.
The Lagrangian density describing the dynamics and electroweak interactions of the quark and lepton elds is given by,
L=
X
Ψ̄iL iD
/ L ΨiL +
i=1...3
where
Ψ̄iL
X
i=1...3
1 A Aµν 1
Ψ̄iR iD
/ R ΨiR − Fµν
F
− fµν f µν ,
4
4
is either the left-handed quark doublet
QiL
(2.3)
or the left-handed
i
i
i
i
lepton doublet LL . Ψ̄R are the right-handed quark and lepton elds uR , dR ,
µ
µ
i
or `R . DL are DR are the covariant derivatives of the left- and right-handed
elds,
µ
DL
= ∂ µ + ig1 T A WAµ + ig2 Y B µ
µ
DR
= ∂ µ + ig2 Y B µ
T A = σ A /2
(A = 1 . . . 3),
(2.4)
SU (2)W symmetry group proporA
tional to the Pauli spin matrices σ . The generator of the U (1)Y symmetry
with
the three generators of
Inclusive Semileptonic B -Meson Decays and |Vcb |
8
group is equal to one.
The dynamics of the weak and hypercharge gauge
A (A = 1 . . . 3) and f ,
Fµν
µν
µ
respectively.
B are not the physical elds ob±
0
served in nature. The physical elds W , Z , and A arise from the elds
elds are described by the eld strength tensors
µ
The gauge elds WA and
Wµ
and
B
by performing a linear transformation,
W 1 ∓ iW 2
√
,
2
Z 0 = cos(ΘW ) W 3 − sin(ΘW ) B,
W± =
(2.5)
A = sin(ΘW ) W 3 + sin(ΘW ) B,
and introducing the Weinberg mixing angle
ΘW .
Experimental observations show that the quarks, leptons, and the gauge
bosons
W±
Z0
and
are massive particles.
Up to this point all gauge and
particle elds appearing in the electroweak Lagrangian (cf.
equation 2.3)
are massless, i.e. the Lagrangian contains no mass terms.
Adding Dirac
mass terms like
m2 WAµ WµA
would violate the invariance under local gauge
transformation. In order to obtain gauge invariant mass terms Peter Higgs
introduced in 1964 a doublet of complex scalar elds [5, 6, 7, 8, 9],
H=
h+
,
h0
(2.6)
h+ and h0 are its charged and neutral components, respectively. The
SU (2)W ⊗ U (1)Y symmetry group is spontaneously broken by the specic
where
structure of the vacuum with the Higgs potential,
V (H) = −µ2 H † H + λ2 (H † H)2 .
(2.7)
The Higgs sector of the Standard Model Lagrangian,
LHiggs = (Dµ H)† (Dµ H) − V (H),
is invariant under local gauge transformation. Expanding
uum expectation value,
v 2 = µ2 /2λ2 ,
and eliminating
(2.8)
H around its vach+ and Im h0 by
choosing a convenient gauge,
H=
√v
2
0
,
+ Re h0
(2.9)
leads to massive gauge bosons and a massive Higgs boson. Quarks and leptons get their masses because of their Yukawa coupling to the Higgs doublet,
LYuk = guij ūiR H T QjL − gdij d¯iR H † QjL − g`ij `¯iR H † LjL + h.c.
(2.10)
In every term a right- and a left-handed eld couples to the Higgs eld.
The matrices
g ij
give the strength of the Higgs eld coupling and
is the
2.3 The CKM Matrix and the Unitary Triangle
anti-symmetric matrix,
=
9
01
−1 0 . From equation 2.10 follows for the mass
matrices of the quark- and lepton-elds,
vgu
Mu = √ ,
2
The matrices
M
vgd
Md = √ ,
2
vge
Me = √ .
2
(2.11)
are not necessarily diagonal. By multiplication with unitary
matrices from the left and right they can be diagonalized.
down-type quarks are part of the same
SU (2)
Since up- and
doublet it is not possible to
diagonalize their mass matrices simultaneously. Therefore, transforming the
avor eigenstate elds into their mass eigenstate elds gives,
uL
dL
=
U(u, L)u0L
U(d, L)d0L
0 uL
= U(u, L)
,
V d0L
with the unitary transformation matrices
U(u, L)
(2.12)
U(d, L) for the upd0L are the mass
respectively. The matrix V is
and
0
type and for the down-type elds, respectively. The uL and
eigenstates of the up- and down-type quarks,
the so called
Cabibbo-Kobayashi-Maskawa
(CKM) matrix.
2.3 The CKM Matrix and the Unitary Triangle
The unitary CKM matrix relates the weak avor eigenstates of the quarks
(d,
s, b)
0
to their mass eigenstates (d ,
s0 , b0 ),
  
  0
d
Vud Vus Vub
d
s =  Vcd Vcs Vcb  s0  .
b
Vtd Vts Vtb
b0
(2.13)
As a consequence avor changing charged currents can occur at tree level
with transition rates proportional to
|Vij |2 .
The corresponding CKM ma-
trix element denotes the relative strength of the coupling. Flavor changing
neutral currents at tree level are forbidden in the Standard Model.
The CKM matrix has to be unitary in the standard model. It can be parameterized by three mixing angles and a CP-violating phase. Following the
observation of a hierarchy between the dierent matrix elements Wolfenstein
proposed an expansion of the CKM matrix in terms of the four parameters
λ , A , ρ,
and
η,
dened by the relations,
|Vus |
λ= p
∼ 0.22,
|Vud |2 + |Vus |2
Vcb 2
,
Aλ = λ Vus ∗
Aλ2 (ρ + iη) = Vub
,
(2.14)
Inclusive Semileptonic B -Meson Decays and |Vcb |
10
and
λ
being the expansion parameter.
The CKM matrix written in the
Wolfenstein parameterization is,

2
1 − λ2
λ

2
V =
−λ
1 − λ2
Aλ3 (1 − ρ − iη) −Aλ2

Aλ3 (ρ − iη)

4
 + O(λ ).
Aλ2
1
(2.15)
The unitarity of the CKM matrix imposes the relation,
∗
Vud Vub
Vcd Vcb∗
Vtd Vtb∗
+
+
= 0,
Vcd Vcb∗
Vcd Vcb∗
Vcd Vcb∗
(2.16)
that can be represented in the complex plane as a triangle. There are several experimental constraints on the parameters of this triangle which allow
an overdetermination of the shape of the unitary triangle and thus test the
consistency of the Standard Model. Figure 2.1 illustrates the unitary triangle in the
ρ̄η̄
plane obtained from a t combining several observables [10].
Introducing the parameters
triangle,
ρ̄ + iη̄ =
ρ̄
and
η̄
ensures that the apex of the unitary
∗ /V V ∗ , is phase-convention independent and the
−Vud Vub
cd cb
λ, A, ρ̄, and η̄
|Vcb | which is of
CKM matrix written in terms of
The CKM matrix element
is unitary to all orders in
λ.
special interest for the pre-
sented measurement is currently known to be [11],
|Vcb | = (41.96 ± 0.73) × 10−3 ,
with a relative uncertainty of
1.7%.
It enters in the global CKM t with the
4th power into the relation between
of neutral kaons,
K ,
and
ρ, η .
(2.17)
CP
violation parameter in the system
Experimentally the CKM matrix element
|Vcb | can be determined through the study of semileptonic B meson decays
B → Xc `ν . In the following an overview of the theory involved in the mathematical description of heavy mesons and inclusive semileptonic B meson
decays will be given.
2.4 Heavy Quark Eective Theory
The
Heavy Quark Eective Theory
(HQET) is an eective eld theory that
Quantum Chromo DymQ → ∞. In this limit
can be derived from the theory of strong interaction
namics
(QCD) in the limit of heavy quark masses,
QCD has spin-avor heavy quark symmetry, which has important implications for the properties of hadrons containing a single heavy quark.
eective eld theory is constructed so that only inverse powers of
mQ
The
ap-
pear in the eective Lagrangian, in contrast to the QCD Lagrangian that
has positive powers of
mQ .
2.4 Heavy Quark Eective Theory
11
1.5
lu
exc
excluded area has CL > 0.95
∆md
L>
at C
1
ded
γ
5
0.9
sin2β
∆ms & ∆md
0.5
η
α
α
εK
γ
0
β
V ub/V cb
-0.5
εK
α
-1
CKM
fitter
-1.5
-1
-0.5
sol. w/ cos2β < 0
(excl. at CL > 0.95)
γ
BEAUTY 2006
0
0.5
ρ
1
Figure 2.1: Global t of the unitary triangle in the
1.5
ρ̄η̄
2
plane [10].
Inclusive Semileptonic B -Meson Decays and |Vcb |
12
2.4.1 The Heavy Quark Limit of QCD
The QCD portion of the Standard Model that describes the strong interactions of quarks and gluons is characterized by a renormalized coupling that
depends on the energy scale of the the interaction.
While the mediators
of the electromagnetic force carry no charge and thus do not interact with
each other, the gauge elds of the strong interaction, the gluons, carry color
charge and can interact with each other. The self-interaction of gluons leads
to a modied behavior of the ne structure constant of the strong interaction
as a function of the energy scale
αs (µ) =
µ,
12π
.
(33 − 2Nq ) ln µ2 /Λ2QCD
If the number of accessible quark avors
Nq
(2.18)
is less than
16, αs (µ)
becomes
weak at high energies or low distances. This phenomenon is called asymptotic freedom. The strong interaction ne structure constant, as suggested
in equation 2.18, diverges as
µ → ΛQCD .
The parameter
ΛQCD
is the scale at
which the QCD becomes strongly coupled, perturbation theory breaks down,
and non-perturbative eects become important.
∼ 200 MeV,
Experimentally,
ΛQCD
is
and it sets the scale for non-perturbative strong interaction ef-
fects.
HQ that is composed of a heavy quark Q and light de-
Consider a hadron
grees of freedom, consisting of light quarks and gluons. The Compton wavelength of the heavy quark scales with its mass,
with increasing quark mass.
and decreases
In contrast the light degrees of freedom are
characterized by momenta of order
λl ∼ 1/ΛQCD λQ .
λQ ∼ 1/mQ ,
ΛQCD
and large Compton wavelengths,
Therefore, the light degrees of freedom cannot resolve
properties of the heavy quark other than its conserved gauge quantum numbers. In particular this is the actual value of
mass
mQ
λQ
which corresponds to the
of the heavy quark.
p exchanged by the light degrees of freedom among
ΛQCD and thus nonnature. For these momenta with p mQ the heavy
The typical momenta
each other and with the heavy quark are of order
perturbative in their
quark does not recoil and remains at rest in the rest frame of the hadron
acting as a static source of electric and chromoelectric gauge eld. In the
heavy quark limit,
mQ → ∞,
the dynamics of the light degree of freedom
depend only on the presence of the static gauge eld and remain unchanged
under the exchange of the heavy quark avor or mass.
called
Heavy Quark Flavor Symmetry.
This behavior is
As a consequence the spectrum of
excitations of heavy hadrons is independent of avor and mass of the heavy
quark as depicted in gure 2.2. Heavy quark avor symmetry breaking eects
of rst order in
1/mQ
are proportional to
1/mQa − 1/mQb .
The spin-dependent interactions are proportional to the chromomagnetic
2.4 Heavy Quark Eective Theory
∆3
∆2
∆1
13
Β3
Β2
Β1
C3
∆3
Β0
C2
∆2
m b− mc
C1
∆1
C0
z
s Q=
−12
+ 12
Figure 2.2: Spectroscopy of heavy mesons containing a bottom- or charmquark in the heavy quark limit. The excitation energy
∆i
does not depend
on the mass of the heavy quark. The oset in the energy of the two spectra
is given by the dierence of the two quark masses,
two hadrons with dierent spin quantum numbers,
moment of the quark,
µQ =
mb − mc . The energies of
szQ = ± 12 , is degenerated.
g
.
2mQ
(2.19)
Q has the spin quantum numbers sQ = 21 and szQ = ± 12 . In
the heavy quark limit, mQ → ∞, the light degrees of freedom get insensitive
z
to sQ and the energies of two hadrons with dierent sQ are degenerated (see
gure 2.2). This leads to the Heavy Quark Spin Symmetry. Heavy quark
The heavy quark
spin symmetry breaking does not have to be proportional to the dierence of
1/mQ 's,
since the spin symmetry is broken even for two hadrons containing
heavy quarks with the same mass.
In the heavy quark limit of QCD spin and avor symmetry can be summarized in a bigger heavy quark spin-avor symmetry,
SU (2)spin ⊗ U (NQ )avor → U (2NQ )spin-avor ,
where
NQ
(2.20)
is the number of heavy quarks. In this symmetry the total angular
momentum
Ĵ and the spin of the heavy quark ŜQ are good quantum numbers.
Therefore, the spin of the light degrees of freedom,
Ŝl ≡ Ĵ − ŜQ
is also conserved.
(2.21)
Inclusive Semileptonic B -Meson Decays and |Vcb |
14
Quark
Mass
Content
[ MeV/c2 ]
D+
D∗+
D1+
D2∗+
D0∗+
D10+
cd
1869.3 ± 0.5
2010.0 ± 0.4
2423.4 ± 3.1
2459 ± 4
2403 ± 40
D0
D∗0
D10
D2∗0
D0∗0
D100
cu
1864.5 ± 0.4
2006.7 ± 0.4
2422.3 ± 1.3
2461.1 ± 1.6
2352 ± 50
2427 ± 40
Γ
[ MeV/c2 ]
τ
[ fs]
I(J P )
sl
1040 ± 7
1 −
2 (0 )
1 −
2 (1 )
1 +
2 (1 )
1 +
2 (2 )
1 +
2 (0 )
1 +
2 (1 )
1
2
1
2
3
2
3
2
1
2
1
2
410.1 ± 1.5
1 −
2 (0 )
1 −
2 (1 )
1 +
2 (1 )
1 +
2 (2 )
1 +
2 (0 )
1 +
2 (1 )
1
2
1
2
3
2
3
2
1
2
1
2
0.096 ± 0.022
25 ± 6
29 ± 5
283 ± 40
< 2.1
20.4 ± 1.7
43 ± 4
261 ± 50
384+130
−110
Table 2.2: Properties of heavy charmed mesons containing light
u
or
d
anti-
quarks [12]: total width (Γ), life time (τ ), isospin (I ), total angular momentum (J ), parity (P ), and the spin of the light degrees of freedom (sl ).
c quark1 , carrying the spin
1
quantum number sQ = , and a light antiquark q , either u, d, or s, with the
2
1
1
1
spin sl = . The ground state mesons form a doublet with spin j =
2
2⊗2 =
0 ⊕ 1. If the light antiquark is either a u or d, these mesons are called D and
D∗ if Q is a c quark, or B and B ∗ if Q is a b quark. Accordingly, we call
+
∗+
∗
mesons containing a light s antiquark Ds , Ds , Bs , or Bs .
Heavy mesons are made up of a heavy
b
or
The lowest excited states of heavy mesons are those with a relative orbital
angular momentum,
L = 1, between the light antiquark and the heavy quark.
∗∗
It is called D . The the light degrees of freedom carry the spin quantum
3
1
1
∗
numbers sl =
2 or 2 . If Q is a c quark, mesons with sl = 2 are called D0 and
3
0
∗
D1 , and mesons with sl = 2 form a system of states called D1 and D2 . The
D1 and D2∗ are narrow states while D0∗ and D10 are wide resonances. Table
2.2 summarizes the properties of heavy charmed mesons that contain light
u
or
s
antiquarks.
2.4.2 Eective Lagrangian
The strong interaction of quarks and gluons is characterized in the Standard
Model by a
SU (3)C
symmetry group with the Lagrangian density,
1
LQCD = Ψ̄ (iγ µ Dµ − m) Ψ − Gaµν Gµν
a
4
1
(a = 1 . . . 8),
Top quarks decay too quickly to establish a static chromoelectric eld.
(2.22)
2.4 Heavy Quark Eective Theory
where
Dµ
is the
SU (3)C
covariant derivative,
Dµ = ∂µ + igAaµ Ta
Here, the
λa
Ta = 21 λa
15
a = 1 . . . 8.
are the eight generators of the
the Gell-Mann-Matrices and
Aaµ
(2.23)
SU (3)C
gauge group with
the eight gauge elds of color. The
Gaµν
are the gluon eld strength tensors.
As discussed above, properties of mesons containing a single heavy quark
manifest heavy quark spin-avor symmetry in the limit
mQ → ∞.
Since the
QCD Lagrangian in equation 2.22 does not manifest the spin-avor symmetry it is convenient to reformulate the QCD in an eective eld theory that
provides a controlled expansion about the limit
theory is called
Heavy Quark Eective Theory
mQ → ∞.
The momentum exchange between the heavy quark
grees of freedom is of order
mass shell,
p2Q = m2Q .
ΛQCD mQ
taking
Thus, the momentum
This eective
(HQET).
pµQ
Q
Q
and the light de-
never far away from its
can be decomposed into two
parts,
pµ = mQ v µ + k µ ,
where
mQ v µ
(2.24)
v µ being the four-velocity of the
∼ ΛQCD determines the amount by
is the large on-shell part, with
µ
heavy quark. The small uctuation k
which the quark is o-shell because of its interactions. Since soft interactions
can change
k µ but not v µ in the heavy quark limit, v µ acts as a good quantum
number of the eective QCD Lagrangian.
It is convenient to reformulate the original quark eld
Q(x)
in terms
large and small velocity dependent component elds,
Qv (x) = eimQ v·x P+ Q(x),
Qv (x) = eimQ v·x P− Q(x),
(2.25)
such that,
Q(X) = e−imQ v·x (Qv (x) + Qv (x)) .
The operators
P±
(2.26)
are velocity dependent projection operators dened as,
P± =
1 ± v/
2
that become in the rest frame of heavy quark,
(2.27)
P± = (1±γ 0 )/2, projecting out
the upper and lower two components of the heavy quark spinor. In the limit
mQ → ∞
only the two upper spinor components propagate mixing with the
∼ 1/(2mQ )
∼ 2mQ . Therefore, the lower two components of the spinor
projected out by P− are suppressed with 1/mQ ,
1
1
P+ Q(x) = Q(x) + O
,
P− Q(x) = 0 + O
.
(2.28)
mQ
mQ
two lower components. The mixing occurs with the amplitude
and frequency
While the eld
Qv (x)
in equation 2.26 is mass independent and carries the
heavy quark spin-avor symmetry, the eld
Qv (x)
vanishes in the heavy
Inclusive Semileptonic B -Meson Decays and |Vcb |
16
quark limit. The exponential factor
eimQ v·x
in
Qv (x) and Qv (x) removes the
large on-shell component from the heavy quark momentum.
Starting from the original QCD Lagrangian density in equation 2.22 the
eective HQET Lagrangian at tree level in leading order perturbation theory
follows by substituting equation 2.26 with
Qv (x) = 0:
LHQET = Q̄v (x) (ivµ Dµ ) Qv (x).
(2.29)
The eective Lagrangian in equation 2.29 does not depend on the mass or
spin of the heavy quark and thus incorporates spin-avor symmetry. Nonperturbative corrections proportional to
ΛQCD /mQ that break the spin-avor
symmetry will be introduced in the next section.
2.4.3 Non-Perturbative Corrections
The HQET Lagrangian including
1/mQ
corrections can be derived from the
QCD Lagrangian by substitution with the eective elds in equation 2.26,
LHQET = Q̄v (x) (iv · D) Qv (x) − Q̄v (x) (iv · D + 2mQ ) Qv (x)
(2.30)
+Q̄v (x)iD
/ Qv (x) + Q̄v (x)iD
/ Qv (x).
The eld
Qv (x)
corresponds to an excitation with the mass
2mQ
and can
be integrated out of the theory by application of the classical equation of
motion,
(iD
/ − mQ )Q(x):
(iv · D + 2mQ )Qv (x) = iD
/ ⊥ Qv (x),
(2.31)
µ
D⊥
≡ Dµ − v · D v µ is the perpendicular component
µ
respect to the velocity v . Substituting equation 2.31 back into
Lagrangian in equation 2.30 and expanding in 1/mQ gives,
where
of
Dµ
LHQET = L0 + L1 + O(1/m2b ),
where
L0
with
the eective
(2.32)
is the lowest order Lagrangian as dened in equation 2.29. The
Lagrangian
L1
is of rst order in
L1 = −Q̄v (x)
using the identity
1/mQ .
It is given by,
D
/ 2⊥
σ µν Gµν
Qv (x) − g Q̄v (x)
Qv (x),
2mQ
4mQ
[Dµ , Dν ] = igGµν
and the denition
(2.33)
σµν = i[γµ , γν ]/2.
The
rst term in equation 2.33 is the nonrelativistic heavy quark kinetic energy.
It breaks heavy quark avor symmetry because of the explicit dependence
on
mQ .
The second term describes the magnetic moment interaction of the
heavy quark. It breaks both heavy quark spin and avor symmetry.
Both expressions in equation 2.33 contain operators whose matrix elements will play an important role in the Operator Product Expansion utilized for theoretical description of inclusive semileptonic
This will be discussed in the next section.
B
meson decays.
2.5 Inclusive Semileptonic B -Meson Decays
17
2.5 Inclusive Semileptonic B -Meson Decays
In this section methods will be discussed that allow to extract the CKM
matrix element
|Vcb |
from the total semileptonic decay width
Γ(B → Xc `ν)
calculated in the framework of the Heavy Quark Eective Theory utilizing
an Operator Product Expansion approach.
B -meson decays to hadronic nal states containing a charm
B → Xc `ν , arise from the matrix elements of the weak Hamiltonian
Semileptonic
quark,
density,
4GF
¯ P ν,
HW = √ Vcb cγ µ PL b `γ
| {z } | µ{zL }`
2
µ
J(bc)
where
GF
is the Fermi constant and
jecting out the left-handed eld.
J(`ν)µ
PL = (1 − γ5 )/2
J(`ν)µ ,
and hadronic,
is the operator pro-
The weak Hamiltonian in equation 2.34
depends directly on the CKM matrix element
leptonic,
(2.34)
µ
J(bc)
,
Vcb .
It can be factored into a
current, since leptons do not have any
strong interaction. In contrast to the matrix element of the leptonic current
it is much more dicult to calculate the matrix element of the hadronic
current since the quark level transition is dressed by non-perturbative QCD
eects.
Figure 2.3 shows weak decay diagrams for the semileptonic
meson decays. The hadronic nal state
D∗
Xc
could be for example a resonant
Dπ
or
D∗ π mesons. When calculating observables of the inclusive semileptonic
B
state like a single
decay like
D
b quark and B
or
Γ(B → Xc `ν)
mesons or a multiparticle state comprising
it is possible to revert to the formalism of HQET
and perform a double expansion in powers of
αs
and
ΛQCD /mb
therebye in-
troducing non-perturbative matrix elements rstly appearing at order
1/m2b .
This procedure based on an Operator Product Expansion (OPE) approach
and perturbative QCD is called
Heavy Quark Expansion
(HQE)
The predictions obtained from the HQE for the dierential
B → Xc `ν
semileptonic decay rate cannot be compared directly with experimental measurements in all regions of the phase space.
The HQE provides a reliable
prediction only when the hadronic nal state smoothly averages over several
nal state hadronic masses
mX .
Near the kinematic endpoint only lower-
mass nal hadronic states can contribute and the expansion breaks down.
On the other hand the HQE approach provides reliable predictions of inclusive observables that integrate over a sucient number of exclusive nal
Γ(B → Xc `ν), the moments of the
< mnX >, or the moments of the lepton
states, like the total semileptonic rate
invariant hadronic mass spectrum
energy spectrum
< E`n >.
The HQE also allows the prediction of moments
B → Xs γ .
on |Vcb | they
of the photon energy spectrum in rare decays
moments
<
mnX
>
do not directly depend
Even though the
allow to measure
the dominant non-perturbative matrix elements that cannot be calculated
reliably.
Inclusive Semileptonic B -Meson Decays and |Vcb |
18
ℓ−
(a)
ℓ−
(b)
νℓ
W−
c
b
νℓ
W−
c
b
B
Xc
q
Figure 2.3: Weak decay diagrams for semileptonic
b
quark (a) and
B
meson
(b) decays. The weak Hamiltonian of the heavy meson decay can be factored
into a leptonic and hadronic current, since leptons do not have any strong
interaction.
Perturbative corrections to the moments and total rate can be implemented in miscellaneous schemes using dierent normalization scales.
side the so called
1S scheme
[13] the
kinetic scheme
Be-
[14, 15, 16] by Uraltsev
et al. provides calculations of dierent moments and the total semileptonic
rate. We will use these calculations in the nal interpretation of the measured invariant hadronic mass moments and for the extraction of
|Vcb |.
The next section 2.5.1 will give a short introduction to the HQE followed
by a discussion of its implementation in the kinetic scheme in section 2.5.2.
2.5.1 The Heavy Quark Expansion
The dierential decay rate of the inclusive semileptonic decay in the restframe of the
B
meson after integration over whole the phase space is given
by,
→ Xc `ν)
1 X X |hXc `ν|HW |Bi|2 4
=
δ [pB − (p` + pν ) − pXc ],
dq 2 dE` dEν
4
2mB
dΓ(B
Xc lepton
spins
(2.35)
where
HW
is the weak Hamiltonian as dened in equation 2.34. The kine-
matic variables,
q µ = pµ` + pµν , dene the momentum transfer of the W
boson
transferred to the leptons. The dierential decay rate is calculated by summing over all lepton spin orientations and hadronic nal states
Xc .
Since
the leptonic nal states do not interact strongly with the hadronic system,
the matrix element can be decomposed into leptonic and hadronic matrix
elements,
dΓ(B
→ Xc `ν)
dq 2 dE` dEν
= 2G2F |Vcb |2 Wαβ Lαβ ,
(2.36)
2.5 Inclusive Semileptonic B -Meson Decays
with
Lαβ
the spin summed leptonic tensor and
Wαβ =
X (2π)3
2mB
Xc
×
β
J(bc)
Here
D
Wαβ
19
the hadronic tensor,
δ 4 [pB − q − pXc ]
(2.37)
ED
E
β
Xc (pXc )|J(bc)
|B(pB ) .
†α
B(pB )|J(bc)
|Xc (pXc )
is the left-handed hadronic current as dened in equation 2.34.
The optical theorem of QCD relates the hadronic matrix element to the
imaginary part of the forward scattering amplitude,
Z
h
i E
−iq·x D
1
†α
β
4 e
Im i
d x
B|T J(bc)
(x)J(bc)
(0) |B
π
2mB
1
≡ − Im T αβ ,
π
W αβ = −
where
†α
β
T [J(bc)
(x)J(bc)
(0)]
(2.38)
(2.39)
is the time ordered product of the heavy quark
currents.
The hadronic matrix
T αβ
can be calculated using the Operator Product
Expansion formalism meaning that
T αβ
will be expanded in powers of
1/mb
as a series of local operators of the heavy quark eld. The coecients of the
operators that occur in this expansion can be computed using QCD perturbation theory. The following section will discuss results for total semileptonic
rate and for moments of the invariant hadronic mass distribution obtained
in the kinetic scheme.
2.5.2 The Kinetic Scheme
For the interpretation of the measured invariant hadronic mass moments and
nal extraction of the CKM matrix element
|Vcb |
Expansions calculated in the kinetic scheme.
we will use Heavy Quark
Following up the discussion
in the previous section the total semileptonic rate is calculated up to order
1/m3b
as [15],
G2 m5
ΓSL (B → Xc `ν) = F 3b |Vcb |2 (1 + Aew )Apert (r, µ)
192π
"
ρ3 (µ)+ρ3LS (µ) !
µ2π (µ) − µ2G (µ) + D mb (µ)
× z0 (r) 1 −
2m2b (µ)
− 2(1 − r)4
with
ρ3D (µ)+ρ3LS (µ)
mb (µ)
2
mb (µ)
µ2G (µ) +
r = m2c (µ)/m2b (µ).
(2.40)
#
ρ3D (µ)
+ d(r) 3
+ O(1/m4b ) ,
mb (µ)
The tree level phase space factor
z0 (r) = 1 − 8r + 8r3 − r4 − 12r2 ln r,
z0 (r) is dened as,
(2.41)
Inclusive Semileptonic B -Meson Decays and |Vcb |
20
and the expression
d(r)
is given by,
d(r) = 8 ln r +
The factor
1 + Aew
34 32
32
10
− r − 8r2 + r3 − r4 .
3
3
3
3
(2.42)
accounts for electroweak corrections. It is estimated to
be approximately,
1 + Aew ≈
α MZ
1 + ln
π
mb
2
≈ 1.014.
(2.43)
Apert accounts for perturbative contributions and is estimated
Apert ≈ 0.908. The parameter µ denotes the Wilson normalization
The quantity
to be
scale that separates eects from long- and short-distance dynamics.
chosen to be
It is
µ = 1 GeV.
The expression in equation 2.40 is in leading order equal to the decay
to the free
1/m2b .
b
quark. The leading non-perturbative corrections arise in order
They are controlled by the matrix elements
µ2π
and
µ2G
of the kinetic
and chromomagnetic operators respectively,
µ2π (µ) ≡ −
~
< B|bDb|B
>µ
,
2mB
µ2G (µ) ≡ −
< B|bσ µν Gµν b|B >µ
.
4mB
Both operators have already been derived as corrections of order
1/mb
(2.44)
in the
eective Lagrangian of the HQET (cf. equation 2.33). Corrections of order
1/m3b
are dened by the Darwin and spin-orbital
ρ3D (µ) ≡ −
~ · Eb|B
~
< B|bD
>µ
,
4mB
ρ3LS (µ) ≡
LS
terms,
~ × iD)b|B
~
>µ
< B|b(~σ · E
.
2mB
(2.45)
HQE calculations for other inclusive variables of
B
meson decays rely on the
same set of non-perturbative parameters as the total semileptonic rate. The
moments of the invariant hadronic mass distribution in semileptonic decays
B → Xc `ν , < mnX >, measured in this analysis, have been calculated in [14].
They are given by the expressions,
< mnX > (mb (µ),mc (µ), µ2π (µ), µ2G (µ), ρ3D (µ), ρ3LS (µ); αs ) =
V + B(mb − 4.6 GeV) + C(mc − 1.2 GeV)
+ P (µ2π − 0.4 GeV2 ) + D(ρ3D − 0.1 GeV3 )
(2.46)
+ G(µ2G − 0.35 GeV2 ) + L(ρ3LS + 0.15 GeV3 )
+ S(αs − 0.22),
where the dependence on the HQE parameters has been linearized relative
V , B , C , P , D, G, L, and
GeV depending on the order n
to meaningful reference values. The coecients
S
are of the dimension of proper powers of
2.5 Inclusive Semileptonic B -Meson Decays
21
of the calculated moment. Numerical values have been calculated in [14] as
functions of the minimum lepton momentum
p∗` .
Overall, the HQE in the kinetic scheme up to order
1/m3b
has six free
parameters:
•
•
Leading order:
mb (µ), b
quark mass
mc (µ), c
quark mass
Non-perturbative correction of order
µ2π (µ),
1/m2b :
kinetic expectation value
µ2G (µ), chromomagnetic expectation value
•
Non-perturbative correction of order
ρ3D (µ),
ρ3LS (µ),
1/m3b :
Darwin term
spin-orbital
LS
term
Furthermore, the theoretical calculation of the total semileptonic rate depends on the CKM matrix element
|Vcb |
and will be used for its extraction
in a global t to hadronic mass moments measured in this analysis combined
with other measured moments of the lepton energy spectrum in semilep-
B meson decays and moments of the photon energy spectrum in decay
B → Xs γ . See section 7 for a description of the applied t procedure.
tonic
Chapter 3
The
BABAR Experiment
3.1 Introduction
The
BABAR experiment is located at the Stanford Linear Accelerator Center
(SLAC) in California. Its primary physics goal is the precision measurement
CP -violating asymmetries in the decay of neutral B 0 -
of the time dependent
mesons. Further integral parts of the
BABAR physics program are the precise
measurement of standard model pareters like the angles of the Unitary Triangle
α, β , and γ
or the CKM matrix elements
|Vub | and |Vcb |. Many of the the
10−4 . A high lumi-
involved decays have low branching fractions, typically
nosity as well as high-performace tracking systems, calorimetry, and particle
identication are prerequisite to achieve the formulated goals.
3.2 The PEP-II B -Meson Factory
The PEP-II
B -meson
factory [17] was designed to meet this requirements.
As shown in gure 3.1 it consists of a high energy electron storage ring (HER)
9.0 GeV and a low energy positron storage ring (LER)
with a beam energy of 3.1 GeV resulting in an interaction energy in the center
2
of mass frame of 10.58 GeV/c operating is the mass of the Υ (4S) resonance.
0 0
+ −
The produced Υ (4S) resonance decays almost entirely into B̄ B or B B
with a beam energy of
pairs. Due to the asymmetric conguration of the beam energies their decay
βγ = 0.55 in the laboratory frame resulting in an
the B -meson decay vertices of ∼ 240µm. This feature
the vertices of the two B -mesons, to determine their
products are boosted with
average separation of
allows to reconstruct
relative decay times, and to extract the time dependence of their decay rates.
Between May 1999 and August 2006 the PEP-II collidered delivered an
integrated luminosity of
106
BB
406.28 fb−1
pairs. A peak luminosity of
buches and currents of
2900 mA
corresponding to approximately
12.069 cm−2 s−1
450 ×
was reached with 1722
in the LER as well as
1875 mA
in the HER.
23
The BABAR Experiment
24
PEP II
Low Energy
Ring (LER)
North Damping Ring
BABAR
Detector
Positron Source
Positron Return Line
e-gun
Linac
200 MeV
injector
PEP II
High Energy
Ring (HER)
South Damping Ring
3 km
Figure 3.1: The Stanford Linear Collider and
P EP − II e+ e−
storage rings.
3.3 The BABAR Detector
The
BABAR detector is a general-purpose megnetic spectrometer.
The two
primary objectives are
•
a precise measurement of charged particle trajectories in order to re-
B meson decay vertices with a resolution of 80 µm2 . Together
with a precise measurement of photon energies B -meson decays to sev-
construct
eral nal states can be reconstructed exclusively with low background
and high eciency.
•
the excellent identication of electrons, muons, and kaons with high
eciency and low misidentication rate.
Fig. 3.2 illustrates the layout of the
BABAR detector.
It consists of ve sub-
components. Charged particle tracks can be measured with the silicon vertex
tracker and the drift chamber. The detector of internally reected ƒerenkov
light serves especially for the discrimination of pion and kaon particle types.
Electromagnetic showers of electrons and photons can be measured with the
electromagnetic calorimeter. The
BABAR detector is a magnetic spectrometer
with a solenoidal superconducting coil operating at a eld strength of
1.5 T.
Outside of the solenoidal coil the muon and hadron identication system in
form of the instrumented ux return is placed.
The
BABAR coordinate system to be used throughout this document is
dened as follows:
•
the
z
axis points parallel to the magnetic eld of the solenoid in the
direction of the
e−
beam.
•
the
y
•
the
x axis points horizontally away from the center of the PEP-II rings
•
the point of origineis dened as the nominal interaction points of the
e−
axis points vertical upwards
and
e+
beams.
3.3 The BABAR Detector
Figure 3.2: Layout of the
BABAR detector with its sub-detectors:
25
(1) Silicon
Vertex Tracker (SVT), (2) Drift Chamber (DCH), (3) Detector of Internally
Reected ƒerenkov Light (DIRC), (4) Electromagnetic Calorimeter (EMC,
(5) Solenoid Coil, and (6) Instrumented Flux Return (IFR).
The BABAR Experiment
26
See [18] for a detailed description of the detector and its sub-systems.
A
short overview of the dierent components will be given in the following
sections.
3.3.1 The Silicon Vertex Tracker
The Silicon Vertex Tracker (SVT) provides a tracking system close to the
beam pipe which is appropriate to resolve the decay vertices of the two
B-
mesons. In combination with the drift chamber this allows the reconstruction
of exclusive
B-
and
D-meson
with a high resolution and low background.
The SVT consists of ve concentric layers of double-sided silicon strip
sensors organized in 6, 6, 6, 16, and 18 modules, respectively. The silicon
strips on the opposite sides of each layer are oriented orthogonally to each
other, on one side parallel and on the other transverse to the beam.
The
32 mm to the beam while the
outermost layer has a distance of 114 − 144 mm to the beam. The inner
three layers give a spatial resolution of 10 − 15 µm. Layer four and ve give
a spatial resolution of ≈ 40 µm.
∗
For low energetic charged particles, like pions from D decays, with transverse momenta below pt ∼ 100 MeV/c the SVT is the only appropriate trackinnermost layer is located at a distance of
ing device, since such SVT-only particles do not reach the drift chamber.
The SVT provides additional information about the ionization
dE/dx
with up to ten measurements for every track and a resolution of appoximate
14%.
It can be utilized for particle identication purposes.
3.3.2 The Drift Chamber
For the detection of charged particles with transverse momenta greater than
pt ∼ 100 MeV/c and the measurment of their momenta and angles the multiwire Drift Chamber (DCH) is the main detector. As depicted in gure 3.3
its shape is that of a cylinder with a length of
236 mm, and
370 mm from
809 mm.
an outer radius of
2764 mm,
z
17.2◦ ≤ θ ≤ 152.6◦ .
the interaction point in the direction of the boost along the
axis. Therefore, it has an asymmetric angular coverage of
The DCH is lled with a
typically operating at
cells,
an inner radius of
The chamber center is oset by
11.9 mm
1960 V.
80 : 20
mixture of helium and isobutane and
It is composed of 7104 small hexagonal drift
by approximately
19.0 mm
along the radial and azimuthal
directions, arranged in 40 cylindrical layers. Each drift cell is made up of
20 µm surrounded by six aluminum eld wires with diameters of 80−120 µm. The layone gold-coated tungsten-rhenium sense wire with a diameter of
ers are grouped by four into ten superlayers, with the same wire orientation
and equal numbers of cells in each layer of a superlayer. Spatial resolution
along the beam line is provided by superlayers with dierent stereo angles
varying between
±45 mrad
and
±76 mrad.
The stereo angles alternate be-
tween axial (A) and stereo pairs with positive (U) and negative (V) stereo
3.3 The BABAR Detector
324
1015
27
1749
35
68
469
17.19
202
551
236
973
1618
IP
Figure 3.3: The multi-wire Drift Chamber.
angles, in the order AUVAUVAUVA.
The overall achieved resolution of measured charged track momenta is
dominated by the DCH and followes the linear dependence,
σpt
= (0.13 ± 0.01)% · pt [ GeV/c] + (0.45 ± 0.03)%,
pt
(3.1)
while the position and angular measurements near the interaction point are
dominated by the SVT.
The measurement of the specic energy loss
ticle identication purposes.
The
dE/dx
dE/dx
can be used for par-
is derived from the total charge
deposited in every drift cell with an average resolution of
7.5%.
3.3.3 The ƒerenkov Detector
The Detector of Internally Reected ƒerenkov light (DIRC) is a ƒerenkov
144 bars
made of fused silica each with a length of 4.9 m, a thickness of only 17.25 mm,
and a width of 35 mm arranged in a 12-sided polygonal barrel. They lead
detector designed for charged particle identication. It consists of
into the instrumented imaging region as picture in gure 3.4.
v greater
n = 1.473,
Charged particles traversing the fused silica with a velocity
than the velocity of light
i.e.
v > c/n,
c0
within the radiator of refractive index
emit a cone of light under the ƒerenkov angle,
r
1
cos Θc =
=
nβ
with
β = v/c, m
the particle's mass, and
1+
2
n
p
m
p
,
(3.2)
the particle's momentum.
The emitted photons are guided through multiple total internal reection
to the Stando box. The stando box contains about
6000 l of puried water
The BABAR Experiment
28
PMT + Base
10,752 PMT's
Light Catcher
Purified Water
Standoff
Box
17.25 mm Thickness
(35.00 mm Width)
Bar Box
Track
Trajectory
PMT Surface
Wedge
Mirror
Bar
Window
4.9 m
{
1.17 m
4 x 1.225m Bars
glued end-to-end
{
8-2000
8524A6
Figure 3.4: The detector of Internally Reected ƒerenkov light (DIRC).
with an index of refraction of
n ≈ 1.346 reasonably close to that of the fused
silica and thus minimizing the total internal reection at the silica-water
interface.
The ƒerenkov light entering the imaging region is detected as
cones by an array of
with
29 mm
10572
closely packed photo multuplier tubes (PMTs)
diameter mounted on the toroidal surface on the back.
Together with the momentum provided by the
the angle
Θc
BABAR tracking system
can be used for particle identication.
With a single track
∆Θc = 2.5mrad for the ƒerenkov angle, the DIRC achieves
separation of 4.2σ for particles with momenta of p = 3 GeV/c.
resolution of
pionkaon
a
3.3.4 The Electromagnetic Calorimeter
The Electromagnetic Calorimeter (EMC) is designed to measure electromagnetic showers with a high eciency, as well as angular and energy resolution
over a wide energy range from
20 MeV to 9 GeV.
It is the main detector com-
π 0 or η decays,
above 700 MeV/c. Its
ponent for the measurement of photons, for instance from
and for the identication of electrons with momenta
design is shown in gure 3.5.
The EMC consists of a cylindrical barrel and a conical forward endcap
with an angular coverage of of
cos θ ≤ 0.895
−0.775 ≤ cos θ ≤ 0.962
and of
−0.916 ≤
in the center-of-mass frame.
The barrel consists of 5760 crystals arranged in 48 rings with 120 identical
crystals in each ring. The endcap holds 820 crystals arranged in 8 rings. For
3.3 The BABAR Detector
29
2359
1555
1375
2295
1127
920
External
Support
1801
38.2˚
26.8˚
558
Interaction Point
15.8˚
22.7˚
1-2001
8572A03
1979
Figure 3.5: The Electromagnetic Calorimeter.
the crystal material thallium-doped CsI was chosen which is characterized
by a short radiation length of
3.8 cm.
X0 = 1.85 cm
and a Molière radius of
Rm =
To account for the boost which leads to higher photon energies for
29.6 cm in the
corresponding to 16X0 and
smaller polar angles the crystal lengths increase starting from
32.4 cm in the forward direction,
17.5X0 , respectively. Their cross sectional area is chosen to be comparable
to Rm . Photons from electromagnetic showers developing in the crystals are
backward to
detected with two silicon photodiodes mounted at the rear of each crystal.
Typically an electromagnetic shower spreads over several adjacent crystals forming acluster of energy deposits.
Thus, for the reconstruction of
particles an ecient recognition of those clusters is important.
The cluster recognition algorithm starts from a seed crystal with an energy above
10 MeV.
Surrounding crystals are considered to be part of the
same cluster if their energies exceed a threshold of
tiguous neighbors of crystals with at least
3 MeV.
1 MeV
or if they are con-
Clusters containing more
than one local energy maximum are split into several so called bumps. Using
tracking information obtained by the SVT and DCH, it is decided whether
a cluster or bump is generated by a neutral or charged particle. If a bump
cannot be associated to a charged particle a neutral candidate is created with
the measured energy and shower shape, otherwise the EMC information is
associated to a charged candidate measured in the tracking system.
Due to accumulating damage of beam-generated radiation a frequent
calibration of each individual crystal and clusters has to be performed.
At low energies this taks is performed using a radioactive photon source.
Slow neutrons are used to irradiate Fluorinert resulting in the decay chain
+ n →16 N + α, 16 N →16 O∗ + β , 16 O∗ →16 O + γ . The emitted photons
have an energy of 6.13 MeV. They can be used for an absolute calibration
at this energy scale. For high energies between 3 GeV and 9 GeV the energy
± → e± .
calibration is performed with electrons from Bhabha scattering, e
19 F
The BABAR Experiment
30
Starting from a well dened initial state the kinematics of the nal state depending on the polar angle is known. Calibration is done by comparison to
Monte Carlo simulated Bhabha events. Both calibration methods are combined by a logarithmic interpolation. The overall energy resolution achieved
is,
(2.32 ± 0.30)%
σE
= p
⊕ (1.85 ± 0.12)%.
4
E
E[ GeV]
The
(3.3)
BABAR EMC has an angular resolution of,
σθ = σφ =
3.87 ± 0.07
p
+ 0.00 ± 0.04
E[ GeV]
!
mrad,
corresponding to an angular resolution of approximately
ergies and an angular resolution of about
3 mrad
(3.4)
12 mrad
at low en-
at high photon energies.
3.3.5 The Instrumented Flux Return
The Instrumented Flux Return (IFR) is the main sub-system for the identication of muons and neutral hadrons, like
KL0
and neutrons. It is located
outside the solenoidal coil and uses its steel ux return as muon lter and
hadron absorber. The IFR covers polar angles between
17◦
and
157◦ .
Re-
sistive Plate Chambers (RPCs) are installed as detectors between the segmented steel of the barrel and end door of the ux return (cf. gure 3.6).
The steel is segmented into 18 plates with increasing thickness, from
2 cm for
10 cm for the outermost plate. The total thickness
65 cm in the barrel and 60 cm in the end door.
between the steel plates is 3.5 cm in the inner layers of the
the inner nine plates to
of the steel plates amounts to
The nominal gap
barrel and
3.2 cm
in the endcaps.
elsewhere. There are 19 RPC layers in the barrel and 18
The RPCs are operated in limited streamer mode.
Their
active volumes are lled with a mixture of argon, freon (C2 H2 F4 ) and a few
percent of Isobutane.
65% and 80% with misidentication probabilities between 2% and 4% for pions and between 0.5% and
1% for kaons are achieved for momenta above p = 1.5 GeV/c.
Overall muon identication eciencies between
3.3 The BABAR Detector
Barrel
342 RPC
Modules
31
3200
FW
3200
920
19 Layers
1940
BW
18 Layers
1250
4-2001
8583A3
432 RPC
Modules
End Doors
Figure 3.6: The Instrumented Flux Return is located outside the solenoidal
coil and uses its steel ux return as muon lter and hadron absorber. Resistive Plate Chambers are installed as detectors between the segmented steel
of the barrel (left) and end door (right) of the ux return.
Chapter 4
Outline of the Analysis
Measuring the moments of the invariant hadronic mass spectrum in semileptonic decays
B → Xc `ν
to the hadronic system
the second
B
requires the identication of all particles belonging
Xc .
We employ a technique that fully reconstructs
e+ e− → Υ (4S) → BB
meson in events
in several hadronic de-
cay modes, thereby separating its decay products from the hadronic system
(see section 5.2 for details). Figure 4.1 illustrates the studied event structure
schematically.
Breco
After a fully reconstructed
tonic decay of the second
BSL
candidate has been selected the semilep-
meson in the event is identied by requiring
one identied lepton among the tracks not assigned to the
Breco
candidate.
Afterwards, all remaining charged tracks and neutral candidates are assigned
to the hadronic system
Xc
of the semileptonically decaying
BSL .
Even though it would be preferable for the theoretical interpretation
to measure the moments of the invariant hadronic mass spectrum without
restriction on the momentum of the lepton, the identication of leptons with
low momenta is experimentally complicated.
lepton momentum is inevitable.
Thus, requiring a minimum
We measure the hadronic mass moments
for dierent selection criteria on the minimum lepton momentum between
p∗` ≥ 0.8 GeV/c
and
p∗` ≥ 1.9 GeV/c
calculated in the restframe of the
BSL
meson.
To ensure an optimal reconstruction of the undetected neutrino originating from the semileptonic
B
decay, charged tracks and photons need to
be reconstructed with sucient eciency and quality. Selection criteria are
applied to ensure an ecient selection but suppress misreconsted tracks and
photons like ghost tracks, loopers, and hadronic split-os. Furthermore,
the applied selection criteria are chosen to improve the agreement of data
and MC simulations by avoiding regions with data-MC disagreement. See
the sections 5.3.1 and 5.3.2 for a discussion of the charged track and photon
selection criteria.
The resolution of the reconstructed hadronic system is limited by missing
particles that have not been measured due to the limited acceptance and
33
Outline of the Analysis
34
!#"
Figure 4.1: Illustration of the studied event structure of decays
Breco BSL (Xc `ν).
After a fully reconstructed
lected the semileptonic decay
BSL → Xc `ν
Breco
Υ (4S) →
candidate has been se-
is identied by requiring one
identied lepton among the tracks not assigned to the
Breco
candidate. All
remaining tracks and photons are assigned to the hadronic system
Xc .
resolution of the detector. The missing energy and momentum in the event
is calculated by,
Pmiss = PΥ (4S) − PBreco − PX − P` ,
(4.1)
PΥ (4S) is the four-momentum of the Υ (4S) produced in the e+ e− collision, P` that of the lepton originating from the semileptonic decay, and PX
is the four-momentum of the reconstructed Xc system calculated from the
where
contributing measured particles. The dierence of missing energy and momentum,
Emiss −|~
pmiss |, is equal to zero within the resolution of the detector
if only the neutrino from the semileptonic decay remains unmeasured. To ensure a well reconstructed hadronic system with good resolution the selected
events are restricted to the window
|Emiss − |~
pmiss || < 0.5 GeV. Starting
e+ e− → Υ (4S) → BB , ad-
from a kinematically well dened initial state
ditional knowledge about the kinematics of the semileptonic nal state is
exploited in a kinematic t to improve the overall resolution and reduce the
bias of the measured invariant hadronic mass
mX,reco .
The applied event
selection criteria and the kinematic t are described in sections 5.4.1 and
5.4.2, respectively.
By exploiting the observation that the charge of a primary lepton stem-
B meson is correlated with the known
Breco , a large fraction of background from
0 0
semileptonic decays not originating from B B mix-
ming from a semileptonically decaying
avor of the fully reconstructed
subsequent secondary
35
ing can be suppressed. Remaining sources of background containing rightcharged leptons in the nal state and being experimentally indistinguishable
from true signal decays are subtracted using MC simulations. The shape and
magnitude of combinatorial background stemming from misreconstructed
Breco
candidates is determined directly on data. The performed background
subtraction procedure is described in detail in section 5.6.
Even though the kinematic t already provides an instrument that allows
a reliable measurement of the hadronic system by signicantly reducing the
average bias of the measured
have to be applied.
mX,reco
distribution, additional corrections
For the extraction of moments we utilize a method
that calibrates the measured and kinematically tted
mX,reco
by applying
correction factors on an event-by-event basis. The background is subtracted
by weighting events according to the fraction of signal events expected in the
mX,reco spectrum. The moments of the invariant
n
distribution < mX > of order n = 1 . . . 6 are determined by
corresponding part of the
hadronic mass
calculating the weighted mean,
PNevt
<
mnX
wi (mX )mnX,calib,i
× Ccalib × Ctrue .
PNevt
wi (mX )
i
i=1
>=
(4.2)
wi (mX ) are the mX,reco -dependent background subtraction facmX,calib,i is the corrected mass. Additional corrections have to be
The weights
tors and
applied to account for a small remaining bias. Two dierent bias sources are
distinguished:
1. The nal extraction of moments suers from a bias
Ccalib
that is inher-
ent in the applied calibration method itself.
2. The factor
Ctrue
corrects for changes in the selection eciency intro-
duced by the event selection criteria.
The
Ctrue
also corrects eects
introduced by photons emitted in nal state radiation.
The moment extration formalism is described in detail in section 5.7. The
obtained results are summarized in section 5.8. Systematic studies will be
presented in chapter 6.
Chapter 5
Measurement of Hadronic
Mass Moments
5.1 Samples of Data and Monte Carlo Simulations
This analysis is based on data recorded between January 2000 and July 2004,
210.4 fb−1 collected at the Υ (4S) resonance. It
6
comprises a total of (231.6±2.5)×10 decays of Υ (4S) → BB . The number of
BB pairs is measured by counting hadronic events at the Υ (4S) resonance
and subtracting contributions from non-BB events using an o-resonance
sample recorded 40 MeV below the Υ (4S) resonance [19].
For MC studies three samples of simulated BB decays are used:
Runs 1-4, corresponding to
generator MC for the determination of residual
Both B mesons decay semileptonically into
the decays in all samples are generated with EvtGen
1. We use a sample of
bias correction factors.
B → Xc `ν
where
[20]. The generator MC does neither incorporate a simulation of the
detector response nor the simulation of nal state radiation.
2. The
generic MC
is based on a sample of simulated
BB
possible known nal states. A full simulation of the
based on
GEANT4 [21] is realized.
decays to all
BABAR detector
Furthermore it incorporates QED nal
state radiation simulated with the
PHOTOS
package [22]. The generic
MC simulation is used for background studies and the determination
of bias correction factors.
3. By contrast to the generator and generic MC samples, in the cocktail
B mesons decays generically while the
B decays into one of the hadronic nal states B → D(∗)+ π − ,
B → D(∗)+ ρ− , and B → D(∗)+ a−
1 . It provides a large sample of clean
fully reconstructed B mesons and is used for the construction of the
MC sample only one of the two
other
calibration curves. The detector and nal state simulation is performed
like in the generic MC sample.
37
Measurement of Hadronic Mass Moments
38
NBB
Dataset
(231.6 ± 2.5) × 106
546.50 × 106
544.52 × 106
26.49 × 106
6.28 × 106
23.94 × 106
Data
0 0
Generic MC (B B )
+ −
Generic MC (B B )
Cocktail MC (B
0B 0 )
+ −
Cocktail MC (B B )
Generator MC
Table 5.1:
Summary of datasets.
Given is the overall number of decays
Υ (4S) → BB .
Approximate numbers of simulated
B 0B 0
and
B +B −
decays are summarized
in table 5.1 for all three MC samples.
5.2 Semi-Exclusive Reconstruction of Breco
In the following section the method applied for the full reconstruction of
one of the
B
mesons in the event will be described. It follows a formalism
developed and commonly used within the
BABAR collaboration called semi-
exclusive reconstruction. A more detailed discussion can be found in [23].
5.2.1 Reconstruction Formalism
The semi-exclusive reconstruction implements a procedure to reconstruct
0
+
Breco
→ D(∗)− Y + and Breco
→ D(∗)0 Y + . The hadronic
+ is composed of n π ± , n K ± , n
0
0
system Y
π
K
KS0 KS , and nπ 0 π with ni the
±
±
0
+ −
0
number of π , K , KS → π π , and π → γγ . Starting from a fully
(∗)0
(∗)−
reconstructed D
or D
one charged track is added to form a seed
compatible with the charge of the Breco candidate meant to be reconstructed.
0
Thereupon, the reconstructed seed is combined iteratively with neutral π
0
±
±
and KS candidates, as well as pairs of charged tracks π
and K
to form
+ is
a Breco candidate. The nal composition of the hadronic system Y
hadronic decays
constrained by the following boundary conditions:
nπ
+ nK ≤ 5,
nK 0
≤ 2,
nπ 0
≤ 2.
S
(5.1)
To suppress combinatorial background and to divide single decay channels
into submodes with dierent purity additional cuts are applied on the invariant mass of the
Y+
system depending on its composition.
Table 5.2
summarizes all possible reconstructed combinations adding up to a total
52 modes where the denition of the dierent modes depends also
+ mass.
reconstructed invariant Y
number of
on the
5.2 Semi-Exclusive Reconstruction of Breco
Decay mode
Y+
39
NY +
h
h π0
3h
h 2π 0
3h π 0
3h 2π 0
5h
5h π 0
KS0 h (π 0 )(π 0 )
KS0 h 2π ± (π 0 )
KS0 KS0 X
2
4
7
4
7
6
6
4
6
5
1
52
total
Table 5.2: Summary of reconstructed compositions of the hadronic system
Y +.
The rst column denes the composition of the hadronic
where
h
±
stands either for a reconstructed charged π or
second column gives the number of dened modes.
Y+
system
K ± candidate. The
The denition of the
dierent modes depends also on the invariant hadronic
Y+
mass.
Breco modes. Depending on the
Y + congurations are considered. For
0
− + all modes Y + = 5h π 0 and 3h 2π 0 are left out of
the decays Breco → D Y
∗−
∗− → D 0 π − .
consideration. The D
meson is reconstructed in the channel D
∗−
−
+
Nevertheless, a large fraction of the decays D
→ D π contained in
−
+
−
−
0
the reconstructed decays D
→ K π π π and D− → KS0 π − π 0 can be
Table 5.3 summarizes all reconstructed
reconstructed mode not all possible
recovered by requiring that the dierence between the invariant masses of
the
e.g.
D− π 0 system and the D− system are compatible with a D∗− decay,
0.1366 < ∆m ≡ m(K + π − π − π 0 ) − m(K + π − π − ) < 0.1446 GeV/c2 .
5.2.2 Kinematic Variables
Fully reconstructed
Breco
decays will be described throughout this document
using two kinematic variables, the energy dierence
substituted mass
mES .
∆E
and the energy-
Both variables are constructed to be minimally cor-
related. The energy dierence is
∗
∗
∆E = EB
− Ebeam
,
reco
with
∗
EB
reco
the energy of the reconstructed
Breco
(5.2)
beam energy. Both quantities are calculated in the restframe
beams.
Measured values of
∆E
∗
Ebeam
the
+
−
of the e e
candidate and
are expected to be close to zero within
detector resolution for well reconstructed
Breco
mesons.
Measurement of Hadronic Mass Moments
40
Breco
0
Breco
→ D∗− Y +
+
Breco
→ D∗0 Y +
0
Breco
→
D− Y +
+
Breco
→ D0 Y +
total
Nmodes
decay mode
D∗−
D∗−
D∗−
D∗−
→ D0 π− ,
→ D0 π− ,
→ D0 π− ,
→ D0 π− ,
D∗0
D∗0
D∗0
D∗0
D∗0
D∗0
D∗0
D∗0
→ D0 π 0 , D0 → K + π−
→ D0 π − , D0 → K + π − π 0
→ D0 π − , D0 → K + π − π + π −
→ D0 π − , D0 → KS0 π − π +
→ D0 γ , D0 → K + π −
→ D0 γ , D0 → K + π − π 0
→ D0 γ , D0 → K + π − π + π −
→ D0 γ , D0 → KS0 π − π +
D−
D0
D0
D0
D0
K +π−π−
→ K +π−
→ K +π−π0
→ K +π−π+π−
→ KS0 π − π +
52
52
52
52
52
52
52
52
52
52
52
52
→
→ KS0 π −
−
D → KS0 π − π 0
D− → K + π− π− π0
D− → KS0 π − π + π −
42
→ K +π−
→ K +π−π0
→ K +π−π+π−
→ KS0 π − π +
52
D−
D0
D0
D0
D0
42
43
43
42
52
52
52
1044
Table 5.3: Summary of modes considered in the semi-exclusive
struction.
Breco
recon-
5.2 Semi-Exclusive Reconstruction of Breco
41
The energy-substituted mass,
mES =
q
∗
Ebeam
2
− p~∗2
Breco ,
denes the invariant mass of the reconstructed
the energy dierence
∆E
to
0 MeV.
(5.3)
Breco
candidate constraining
It is calculated in the restframe of
the beams from the beam energy and the momentum
candidate.
B
true
mES
Measured values of
p~∗2
Breco
of the
Breco
are expected to correspond with the
meson mass within detector and storage-ring energy resolution.
As outlined in [24], calculating
mES
B
by PEP-II leads to uctuating mean
This implies an imprecise
Ebeam
using beam energies as reported
meson masses in the
mES
peak.
measurement. Corrections to the PEP-II
energies have been computed for dierent run ranges [25,26]. Applying those
corrections shifts the
mES
peak to the nominal
B
meson mass [12].
5.2.3 Selection of the Best Breco Candidate
Reconstructed
Breco
candidates are preselected by applying cuts on
pending on the composition of the hadronic
Y+
∆E
system. The following
de-
∆E
windows are chosen,
−45 < ∆E < 30 MeV
for
3h,
|∆E| < 45 MeV
for
h
|∆E| < 48 MeV
for
5h
|∆E| < 50 MeV
for
KS0 KS0 X,
−90 < ∆E < 60 MeV
corresponding to
3
the selected mode.
the
Breco
and
KS0 h,
and
KS0 3h,
(5.4)
for all remaining modes including a
standard deviations of the
∆E
π0,
distribution specic to
The mode specic resolutions are determined prior to
candidate selection by tting the
∆E
distributions with a linear
background and a Gaussian signal functions for modes without
state. To avoid biasing the
∆E
distribution towards small values a common
window for all modes containing at least one
cleanest modes. Reconstructed
π 0 in the nal
Breco
π0
is determined from the
candidates are preselected by requiring
mES > 5.20 GeV/c2 .
All possible channels are ranked according to their purity,
P=
Nsignal
,
Nsignal + Nbg
(5.5)
Nsignal the number of signal and Nbg the number of misreconstructed
Breco candidates. After the application of the preselection cuts on ∆E and
mES the Breco candidate with the highest purity is retained.
Figure 5.1 shows the number of reconstructed Breco candidates per event
after the application of the preselection criteria on ∆E and mES . The left
with
Measurement of Hadronic Mass Moments
42
entries
7000
6000
5000
4000
3000
2000
1000
0
3
800
700
600
500
400
300
200
100
0
entries
×10
2
4
6
8
Figure 5.1: Number of
10 12
NB
Breco
×10
3
2
4
6
8
reco
10 12
NB
reco
NBreco , after the appliNBreco distribution for all
subset of modes with P > 28%
candidates per event,
cation of preselection criteria. Plot (a) shows the
Breco
mode. Plot (b) shows the
NBreco
for a
for each individual mode corresponding to a sample with a total purity of
60%
as it will be selected in nal event selection.
NBreco
plot in gure 5.1 shows the
modes. About
In
20%
47%
distribution for all reconstructed
Breco
of all events contain only one reconstructed candidate.
Breco candidate. If
P > 28% for each
with a total purity of 60% as it
nd 64% and 17% of all events
of all events we nd a second reconstructed
the selection of candidates is restricted to modes with
individual mode corresponding to a sample
will be selected in nal event selection, we
with one and two
corresponding
Breco
NBreco
candidates, respectively.
Figure 5.1b shows the
distribution.
Having chosen the best
Breco
candidate a kinematic t is performed [27].
Thereby, masses of intermediate states
D(∗) , KS0 , π 0
are constrained to their
nominal masses [12]. By performing the kinematic t the four-vector of the
reconstructed
Breco
candidate is changed, which is why also the values of the
corresponding kinematic variables
∆E
and
mES
are changed and have to be
recalculated.
The
mES
distribution for the selected
5.2 for dierent subsets of
Breco
Breco
candidates is shown in gure
mES distri28%. Restricting the
and P > 28% (gure
modes. Figure 5.2a shows the
bution for all possible modes with an overall purity of
P > 7% (gure 5.2b)
40% and 60%, respectively. The selecoveral purity of 60% corresponds to that used in
selected modes to those with
5.2c) results in overall purities of
tion criteria leading to an
the nal event selection. Requiring a selected lepton among the tracks not
associated to the
Breco
candidate improves the purity of the selected sam-
ple since semileptonic decays are usually characterized by lower track and
photon multiplicities than hadronic decays reducing the number of random
combinations. We nd an overall purity of
Figure 5.3 shows the
for dierent subsets of
∆E
Breco
78%
as depicted in gure 5.2d.
distributions for the selected
Breco
candidates
modes. While gure 5.3a shows the distribution
5.3 Particle Selection for the Signal B Meson
3
3
400
300
200
100
90000
80000
70000
60000
50000
40000
30000
20000
10000
0
5.2 5.22 5.24 5.26 5.28 5.3
mES [GeV/c2]
entries / 2.5MeV/c2
0
entries / 2.5MeV/c2
×10
entries / 2.5MeV/c2
entries / 2.5MeV/c2
500
43
5.2 5.22 5.24 5.26 5.28 5.3
mES [GeV/c2]
Figure 5.2: Distributions of
mES
200 ×10
180
160
140
120
100
80
60
40
20
0
5.2 5.22 5.24 5.26 5.28 5.3
mES [GeV/c2]
25000
20000
15000
10000
5000
0
for the best
5.2 5.22 5.24 5.26 5.28 5.3
mES [GeV/c2]
Breco
candidate and dierent
criteria on the purity of the selected modes: all modes (a), a subset of modes
P > 7% (b), a subset of modes with P > 28% (c), and a subset of
P > 28% and a selected lepton among the tracks not associated
the Breco candidate (d).
with
modes with
to
for all reconstructed modes, only a subset of modes with
lepton among the tracks not associated to the
Breco
P > 28%
and a
candidate is selected in
gure 5.3b. The edges in both distributions are caused by the preselection
cuts on
∆E .
5.3 Particle Selection for the Signal B Meson
After a fully reconstructed
Breco
candidate has been selected all remaining
charged tracks and neutral candidates are assigned to the decay of the second
BSL
meson in the event. Among those a subset is selected applying selection
criteria that have been developed to improve the agreement of data and
MC simulations by avoiding regions with large data-MC dierences while
retaining a sucient selection eciency [28, 29].
Charged track selection
criteria are summarized in table 5.4 while table 5.5 summarizes all photon
selection criteria.
Measurement of Hadronic Mass Moments
44
×10
3
entries / 2.0MeV/c2
entries / 2.0MeV/c2
350
300
250
200
150
100
50
0
-0.1 -0.05
0
0.05 0.1
mES [GeV/c2]
Figure 5.3: Distributions of
∆E
6000
5000
4000
3000
2000
1000
0
for the best
-0.1 -0.05
Breco
0
0.05 0.1
mES [GeV/c2]
candidate and dierent
criteria on the purity of the selected modes: all modes (a) and a subset of
modes with
to the
Breco
P > 28%
and a selected lepton among the tracks not associated
candidate (b).
5.3.1 Charged Track Selection
Charged tracks are reconstructed using combined information obtained from
the two
BABAR tracking systems:
the DCH and SVT. Therebye, the applied
track selection follows two principal guidelines. To ensure an optimal reconstruction of the unmeasured neutrino originating from the semileptonic
B
decay, tracks need to be reconstructed with sucient eciency and quality.
At the same time it is desirable to suppress misreconstructed additional
tracks like ghosts and loopers that result from misfeatures of the tracking
algorithm. Reconstructed tracks have to fulll the following requirements:
1.
0.41 ≤ θ ≤ 2.54, with θ
the track`s polar angle in the laboratory. Since
a reliable particle identication relies on information provided by the
EMC, tracks are restricted to its acceptance.
2.
p < 10 GeV/c
removes tracks with momenta
p
not compatible with the
beam energies.
3.
|dxy | < 1.5 cm and |dz | < 5 cm, where dxy
and
dz
refer to the distance of
closest approach to the primary vertex in the plane perpendicular and
along the beam axis, respectively. Both cuts aim at a discrimination
against tracks not originating from beam-beam interactions.
4.
pt > 60 MeV/c,
with
pt
the component of the momentum vector trans-
verse to the beam axis. This cut ensures a minimum transverse momentum required for a reliable measurement.
5.
pt < 200 MeV/c.
This cut is only applied to SVT-only tracks, that is
to tracks that have not been measured with the DCH.
5.3 Particle Selection for the Signal B Meson
6. Charged particles with transverse momenta
45
pt . 360 MeV/c
that do
not reach the EMC can loop within the tracking system of the
BABAR
detector. The helix of the corresponding track is usually reconstructed
by the tracking algorithm as several smaller segments, one for every
half-turn of the loop. Half of the reconstructed segments are identied
as positively charged the other half as negatively charged particles.
Looper candidates are identied by selecting pairs of charged tracks
pit < 250 MeV/c, cos θi < 0.2, and |∆pt | ≡ |pit − pjt | < 120 MeV/c,
j
i
where pt and pt are the transverse momenta of the rst and second
with
looper candidate, respectively.
Two cases have to be distinguished.
First, if the particle loops one time the tracking algorithm will reconstruct two charged particles with opposite charge,
q i q j < 0,
going
back-to-back. Second, if the particle loops a second time the second
loop will be reconstructed as a particle with the same charge,
and same angles
Θ
and
q i q j > 0,
φ.
Looper candidates are selected by applying the following cuts:
where
qi
q i q j > 0,
|∆φ| < 0.1
if
|π − ∆φ| < 0.1
if
|π + ∆φ| < 0.1
if
|∆Θ| < 0.1
if
π
,
2
π
q i q j < 0, ∆φ < − ,
2
i j
q q > 0,
|π − ∆Θ| < 0.1
if
q i q j < 0,
and
qj
q i q j < 0, ∆φ >
are the charges of the rst and second looper can-
didates, respectively. Finally, the candidate with the smallest impact
parameter
dz
is retained while the other looper candidates are rejected.
7. If the tracking algorithm reconstructs two tracks from a single physical
particle we call one of them a ghost track.
The tracking algorithm
splits the DCH hits of the physical particle into two separate tracks
both following the same trajectory. Ghost candidates are selected by
applying the following cuts on pairs of charged particles:
pit < 350 MeV/c,
|∆pt | < 150 MeV/c,
|∆φ| < 0.1,
|∆Θ| < 0.1,
1
2
NDCH
< 45 − NDCH
,
where
1
NDCH
is the number of hits in the DCH. Track
1
is taken to
have the greater number of DCH hits. From selected pairs of charged
ghost track candidates the one with the greater number of hits in the
DCH is retained while the second track is rejected.
Measurement of Hadronic Mass Moments
46
Polar angle
Momentum
Impact parameter
Transverse momentum
Looper rejection
Ghost rejection
0.41 ≤ θ ≤ 2.54
p < 10 GeV/c
|dxy | < 1.5 cm, |dz | < 5 cm
pt > 60 MeV/c
pt < 200 MeV/c for SVT-only tracks
pit < 250 MeV/c
cos θi < 0.2
|∆pt | < 120 MeV/c
|∆φ| < 0.1 if q i q j > 0
|π − ∆φ| < 0.1 if q i q j < 0, ∆φ > π2
|π + ∆φ| < 0.1 if q i q j < 0, ∆φ < − π2
|∆Θ| < 0.1 if q i q j > 0
|π − |∆Θ|| < 0.1 if q i q j < 0
retain candidate with the smallest dz
pit < 350 MeV/c
|∆pt | < 150 MeV/c
|∆φ| < 0.1
|∆Θ| < 0.1
1
2
NDCH
< 45 − NDCH
Track 1 with the greater number
of DCH hist is retained
Table 5.4: Summary of track selection criteria. A detailed discussion can be
found in section 5.3.1.
5.3 Particle Selection for the Signal B Meson
47
5.3.2 Neutral Candidate Selection Criteria
Photons are measured as clusters in the EMC that comply with the following
criteria:
1.
2.
Ncrys > 2,
with
Ncrys
Eclus > 50 MeV,
the number of crystals in the cluster.
Eclus
where
is the deposited energy in the EMC clus-
ter.
3. LAT
< 0.6.
The variable LAT is the lateral moment of EMC cluster,
dened as
PNcrys
LAT
(5.6)
i. Energies are ordered starting from
Ecrys,1 and Ecrys,2 such that E1 > E2 >
. . . > ENcrys . The parameter ri denes the distance of crystal i to
the center of the cluster while r0 is dened as the average distance of
two adjacent crystals, approximately 5 cm for the EMC of the BABAR
with
Ecrys,i
Ecrys,i ri2
,
PNcrys
Ecrys,1 r02 + Ecrys,2 r02 + i=3
Ecrys,i ri2
i=3
=
the energy of crystal
the highest energetic crystals
detector.
The variable LAT describes the lateral energy distribution of the cluster and varies between
0
and
1.
Electromagnetic showers like those
produced by electrons or photons are characterized by small values of
LAT, i.e. energies are mostly concentrated in few central crystals. In
contrast hadronic showers give large values of LAT.
4.
0.32 < θclus < 2.44,
where
θclus
is the polar angle of the cluster's cen-
troid. This selection criteria ensures that the whole cluster is entirely
located inside the EMC.
5.
∆α ≡ arccos [cos θclus cos θtrk + sin θclus sin θtrk cos(φclus − φtrk )] > 0.08
is calculated as angular dierence between the angles θclus and φclus of
a measured cluster and the angles θtrk and φtrk of the impact point
of the nearest track at the surface of the EMC. This selection criteria
aims at the suppression of split-o clusters that are separated from the
points of impact of their associated track and thus remain erroneously
unassigned. If the nearest track is identied as an electron no photon
selection based on
∆α
is performed.
5.3.3 Particle Identication
In order to distinguish between dierent particle types combined information
obtained from the dierent
and IFR is used.
chosen among
BABAR subdetectors SVT, DCH, DIRC, EMC,
Each charged track is assigned a specic particle type
e± , µ± , K ± , (p)
, and
π±.
The mass of the specic particle is
Measurement of Hadronic Mass Moments
48
Number of crystals
Cluster energy
Lateral energy moment
Polar angle
Dierence of angle to nearest track
Ncrys > 2
Eclus > 50 MeV
LAT < 0.6
0.32 < θclus < 2.44
∆α > 0.08
Table 5.5: Summary of neutral candidate selection criteria. A detailed discussion can be found in section 5.3.2.
chosen accordingly. For the particle type assignment dierent selectors are
utilized.
1. A likelihood based selector is used for the identication of electrons
( PidLHElectrons) [30] that combines information from EMC, DCH,
and DIRC. Discriminating variables are the deposited energy in the
EMC
Edep /p,
the lateral shower shape LAT as dened in equation 5.6,
θC measured
in the DIRC, and the energy loss per unit distance dE/dx in the DCH.
Overall a selection eciency above 90% is achieved for electron momenta above p = 0.8 GeV/c in the laboratory. The misidentication
probabilities in this momentum region are below 2% and 5% for π
and K , respectively.
the longitudinal EMC shower shape, the ƒerenkov angle
2. We utilize a neural net based selector for the identication of muons
( muNNTight) [31] mainly based on information obtained from the IFR.
65%
50% for
The achieved muon identication eciency is found to be between
and
80%
for momenta above
p = 1.5 GeV/c
and drops to only
p = 0.9 GeV/c. Misidentifaction probabilities
4% for π and between 0.5% and 1% for K .
momenta at
2%
and
lie between
3. Kaons are identied using a neural net based kaon selector ( KLHTight)
80% and 90% for mo0.5 GeV/c and 3 GeV/c. The probability to misidentify
below 2% in the aforementioned momentum region.
[32] with an overall selection eciencies between
menta between
a
π
as a
K
is
4. Protons are identied utilizing a standard
selector ( pLHTight).
BABAR
likelihood based
85%
achieved with a maximum eciency near 100% for p = 600 MeV/c.
Overall, selection eciencies above
are
Pi-
ons and kaons are misidentied as protons with probabilities between
1%
and
2%
for momenta above
menta below
1.2 GeV/c
and well below
1%
for mo-
1 GeV/c.
Tracks not identied as leptons, kaons, or protons are assumed to be pions.
5.4 Reconstruction of the Hadronic System
49
5.4 Reconstruction of the Hadronic System
The selected charged tracks and photons recoiling against the fully reconstructed
Breco
are associated to the decay of the second
event referred to as
B → Xc `ν
decay
BSL .
The hadronic system
B
meson in the
of the semileptonic signal
is reconstructed from those particles that have not been
e
identied as lepton,
or
µ,
by calculating its four-momentum,
Ntrk,X
X
PX =
Nγ,X
Pi,trk +
i=1
where
Xc
X
Pj,γ ,
(5.7)
j=1
Ntrk and Nγ are the number of tracks and photons not associated
Breco candidate or identied as lepton. The four-momentum of the
with the
unmeasured neutrino is estimated by calculating the missing four-momentum
in the event,
Pmiss = PΥ (4S) − PBreco − PX − P` ,
with
and
(5.8)
PΥ (4S) the four-momentum of the Υ (4S) produced in the e+ e− collision
P` the four-momentum of the lepton originating from the semileptonic
decay.
The four-momentum of the
BSL is calculated from PBSL = PΥ (4S) −PBreco
and will be used in the following to calculate kinematic quantities in the
restframe of the recoiling
BSL .
In the following sections further steps in the reconstruction of the hadronic
Xc
system will be discussed. In section 5.4.1 event based selection criteria
will be discussed followed by a description of the kinematic t in section 5.4.2
used to improve the resolution and reduce the bias of the
Xc
reconstruction.
A summary of all event based selection criteria can be found in table 5.6.
5.4.1 Event Selection Criteria
We apply the following event based selection criteria to improve the selection
of well reconstructed semileptonic signal decays and suppress background.
1.
Pint > 60%, where Pint
is the integrated purity of the
The integrated purity of a given
purity of all individual modes
Breco
i with
P
mode
j
Breco
candidate.
is dened as the overall
greater or equal purity,
Pi ≥ Pj ,
Nsignal,i
P
,
i,Pi ≥Pj Nsignal,i +
i,Pi ≥Pj Nbg,i
Pint,j = P
where
Nsignal,i
events of mode
and
i,
Nsignal,i
i,Pi ≥Pj
(5.9)
are the number of signal and background
respectively. As described in section 5.2.3 all
reconstruction modes are ranked according to their purity.
Breco
Starting
from the decay mode with the highest purity and adding modes one
after another with decreasing purity the purity of the resulting sample
is called integrated purity.
Measurement of Hadronic Mass Moments
50
2.
|∆E| < 90 MeV, where ∆E is
constructed Breco candidate as
the energy dierence of the fully redened in equation 5.2. The
90 MeV
∆E
window corresponds to about three standard deviations of the
distribution.
3.
mES > 5.27 GeV/c2 , with mES
the energy substituted mass of the
Breco
candidate as dened in equation 5.3. The cut denes the signal region
of the
mES
distribution and is ecient in suppressing background com-
Breco
prising of misreconstructed
candidates. Remaining combinato-
rial background present in the signal region will be subtracted using
the sideband of the distribution. A detailed discussion of the applied
procedure can be found in section 5.6.1.
4. We require that exactly one lepton, either an electron or muon, is identied among the tracks not associated with the
Breco
candidate. By
requiring exactly one lepton additional unmeasured neutrinos stemming from secondary semileptonic decays are suppressed which would
otherwise contribute to the missing momentum and energy in the event.
5.
qbreco q` < 0.
The charge of the single reconstructed lepton is required to
be compatible with that of a primary lepton emerging from a semileptonic
B
decay. In this context we exploit the observation that the pri-
mary lepton's charge and the avor of the
The expression
tons, with
q`
qbreco
qbreco q` < 0
Breco
meson are correlated.
is fullled for right-charged primary lep-
the charge of the
the charge of the lepton.
b
or
b
quark in the
Breco
meson and
Apart from that originating from
B 0B 0
osciallations, a large amount of residual background stemming from
subsequent decays of the
wrong-charged leptons,
6.
p∗` ≥ p∗`,cut GeV/c.
BSL is suppressed
qbreco q` > 0.
by rejecting events with
The analysis is performed for several cuts on the
p∗`,cut = 1.9 GeV/c.
The lepton's momentum is calculated in the restframe of the BSL me1
∗
son . The smallest accepted lepton momentum p`,cut = 0.8 GeV/c is
lepton momentum between
p∗`,cut = 0.8 GeV/c
and
chosen to avoid regions of the phase space in which the used particle
type selectors are characterized by higher probabilities to misidentify
a hadron as a lepton.
7.
|Emiss − |~
pmiss || < 0.5 GeV, Emiss > 0.5 GeV,
and
The dierence of missing energy and momentum,
|~
pmiss | > 0.5 GeV/c.
|Emiss − |~
pmiss ||, is
equal to zero within the resolution of the detector if only the neutrino
from the semileptonic decay remains unmeasured. If there are unmeasured tracks or photons in the event positive values of
|Emiss − |~
pmiss ||
are expected while additionally measured particles lead to negative
1
In the following all kinematic quantities accented with ∗ will be calculated in the
restframe of the BSL
5.4 Reconstruction of the Hadronic System
5
5
4
Emiss [GeV]
4
Emiss [GeV]
51
3
2
1
0
3
2
1
0
0 0.5 1 1.5 2 2.5 3 3.5
|p | [GeV/c]
0 0.5 1 1.5 2 2.5 3 3.5
|p | [GeV/c]
miss
Figure 5.4: Distributions of
Emiss
miss
versus
|~
pmiss | for simulated signal
(a) and
background decays (b). The dashed red lines indicate the applied selection
criteria of
Emiss > 0.5 GeV
and
3000 2500
2000
1500
1000
500
0
-1 -0.5 0
400
300
200
100
0
0.5 1 1.5 2 2.5
|E - p | [GeV]
miss
500 entries / 80 MeV
entries / 80 MeV
|~
pmiss | > 0.5 GeV/c.
-1 -0.5 0
miss cut
0.5 1 1.5 2 2.5
Emiss - |p | [GeV]
miss cut
Emiss − |~
pmiss | in data for dierent cuts p∗` ≥
≥ 1.9 GeV/c (b). The dashed lines indicate the applied
|Emiss − |~
pmiss || < 0.5 GeV.
Figure 5.5: Distributions of
0.8 GeV/c
∗
(a) and p`
selection criteria of
|Emiss − |~
pmiss ||. If all particles except the neutrino are measured, Emiss and |~
pmiss | are positive quantities. Figure 5.4 illustrates
the situation showing the distributions of Emiss versus |~
pmiss | for simvalues of
ulated signal and background decays. Figure 5.5 illustrates the applied
selection criteria on
Emiss − |~
pmiss |.
8.
Ntrk,X ≥ 1, with Ntrk,X
9.
|qtot | ≡ |qBreco + q` + qX | ≤ 1.
the number of charged track in the
structed charge in the event.
The quantity
qtot
Xc
system.
is the total recon-
A small charge imbalance is tolerated
to account for unmeasured low momentum tracks that inuence the
Xc reconstruction only slightly. In case only one
Ntrk,X = 1, we require charge conservation, |qtot | = 0.
quality of the
is found,
track
Measurement of Hadronic Mass Moments
52
Pint > 60%
|∆E| < 90 MeV
mES > 5.27 GeV/c2
Integrated purity
Breco
Breco
energy dierence
energy substituted mass
Lepton measurement
Exactly one recoiling lepton
Lepton charge correlation
qbreco q` < 0
p∗` ≥ 0.8 . . . 1.9 GeV/c
Emiss > 0.5 GeV
|~
pmiss | > 0.5 GeV/c
|Emiss − |~
pmiss || < 0.5 GeV
Ntrk,X ≥ 1
|qtot | ≤ 1 if Ntrk,X ≥ 2
|qtot | = 0 if Ntrk,X = 1
Lepton momentum
Missing energy and momentum
Number of tracks
Total charge
Table 5.6: Summary of event selection criteria. A detailed discussion can be
found in section 5.4.1.
5.4.2 Kinematic Fit
Starting from a kinematically well dened initial state
e+ e− → Υ (4S) → BB
additional knowledge about the kinematics of the hypothetical semileptonic
nal state can be exploited to improve the overall resolution and reduce the
bias of the measured hadronic system. The applied procedure called method
of least squares determines corrections to the four-vectors of the measured
particles on the basis of kinematic constraints.
A detailed discussion of the applied mathematical procedure described in
the following sections 5.4.2.1 to 5.4.2.3 can be found in [33]. Section 5.4.2.4
discusses the application to the given problem utilizing a general purpose
kinematic tter package implemented in
5.4.2.1
C++.
It is documented in [34].
Method of Least Squares with Constraints
We start our considerations with a general problem consisting of
quantities
~ȳ
and
~ā
~y
and
p unmeasured parameters ~a.
n measured
The true underlying quantities
comply with the model,
f1 (ā1 , ā2 , . . . , āp , ȳ1 , ȳ2 , . . . , ȳn ) =0,
f2 (ā1 , ā2 , . . . , āp , ȳ1 , ȳ2 , . . . , ȳn ) =0,
.
.
.
(5.10)
fm (ā1 , ā2 , . . . , āp , ȳ1 , ȳ2 , . . . , ȳn ) =0,
dened by
tities
m
equations. Due to uncertainties in the measurement the quan-
~y will in general not fulll these conditions.
The equations 5.10 provide
additional information that allow to improve the knowledge of the measured
quantities by determining corrections
∆~y
so that the corrected quantities
5.4 Reconstruction of the Hadronic System
~y 0 = ~y + ∆~y
will comply with all constraints.
53
Furthermore, the unmea-
sured parameters can be determined with the help of the equations
fi .
The
corrections are determined by minimizing the squared sum,
S(~y ) = ∆~y T V −1 ∆~y ,
with
V
(5.11)
being the covariance matrix of the measured parameters.
A general approach to determine local extrema of functions with multiple variables and constraints is the method of Langrange Multipliers.
introduces
m
additional parameters
each constraint.
λk ,
It
the Lagrange Multipliers, one for
A solution of the problem can be found by searching for
extrema of the function,
L(~y , ~a, ~λ) = S(~y ) + 2
m
X
λk fk (~y , ~a),
(5.12)
k=1
with respect to all parameters
~y , ~a,
~λ.
and
It is equivalent to the condition
S(~y ) under the constraint fk (~y , ~a) = 0.
fk are linear the solution can be found
for a local minimum of
If the constraints
in one step.
Otherwise it has to be found iteratively linearizing the problem in every
iteration.
5.4.2.2
Linearization
A linearized form of the constraints as dened in equation 5.10 can be written
as,
fk (~y 0 , ~a0 ) ≈ f (~y ∗ , ~a∗ ) +
p
X
∂fk
j=1
∂aj
· (∆aj − ∆a∗j )
+
n
X
∂fk
i=1
∂yi
· (∆yi − ∆yi∗ ) ≈ 0,
(5.13)
~y (~a) contains the start values of the measured (unmeasured) parameters, ~
y ∗ (~a∗ ) contains the values of the measured (unmeasured) parameters
after the previous iteration, and ~
y 0 (~a0 ) contains the values of the measured
where
(unmeasured) parameters of the next iteration.
All functional values and
derivatives are calculated at ~
y∗
= ~a + ∆~a∗ . The dierence
= ~y +∆~y and ~a0 = ~a +∆~a.
=
~y + ∆~y ∗ and
~a∗
of start values and the next iteration is given by ~
y0
Rewriting equation 5.13 in matrix notation results in
f~∗ + A(∆~a − ∆~a∗ ) + B(∆~y − ∆~y ∗ ) ≈ 0
(5.14)
or
A∆~a + B∆~y − ~c = 0
with
~c = A∆~a∗ + B∆~y ∗ − f~∗ .
(5.15)
Measurement of Hadronic Mass Moments
54
~c in equation 5.15 for iteration n depends only on quanprevious iteration n − 1. The matrices A and B appearing in
The constant vector
tities of the
both equations 5.14 and 5.15 are dened as,
A=
∂ f~
∂~a
and
B=
∂ f~
.
∂~y
(5.16)
For illustration we give the components of the matrices:

∂f1 /∂a1 ∂f1 /∂a2 . . . ∂f1 /∂ap
 ∂f2 /∂a1 ∂f2 /∂a2 . . . ∂f2 /∂ap

A=
.
.

.
∂fm /∂a1 ∂fm /∂a2 . . . ∂fm /∂ap

∂f1 /∂y1 ∂f1 /∂y2 . . . ∂f1 /∂yn
 ∂f2 /∂y1 ∂f2 /∂y2 . . . ∂f2 /∂yn

B=
.
.

.
∂fm /∂y1 ∂fm /∂y2 . . . ∂fm /∂yn


f1 (~a∗ , ~y ∗ )
 f2 (~a∗ , ~y ∗ ) 


∗
~
f =

.
.


.
∗
∗
fm (~a , ~y )
The function
∆~a,
and
~λ





(5.17)





(5.18)
(5.19)
L that has to be minimized with respect to the parameters ∆~y ,
can now be written as,
L = ∆~y T V −1 ∆~y + 2λT (A∆~a + B∆~y − ~c).
The condition for an extremum of
L
(5.20)
is obtained by dierentiation,
V −1 ∆~y + B T ~λ = 0,
AT ~λ = 0,
(5.21)
B∆~y + A∆~a = ~c,
n + p + m dierential
values of ∆~
y , ∆~a, and ~λ.
resulting in a coupled system of
to be solved for the unknown
5.4.2.3
equations that need
Solution
The system of coupled dierential equations dened in equation 5.21 can be
written in only one equation with partitioned matrices:

  
V −1 0 B T
∆~y
0
T
 0




0 A
∆~a
0 .
=
B A 0
λ
c

(5.22)
5.4 Reconstruction of the Hadronic System
55
In order to nd a solution the inverse of the matrix given in equation 5.22
has to be calculated. Writing its inverse matrix in a partitioned form,
−1 

T
T
V −1 0 B T
C11 C21
C31
T
T ,
 0
0 AT  =  C21 C22
C32
B A 0
C31 C32 C33

(5.23)
and using the abbreviations
VB = BVB T
−1
VA = AT VB A
and
(5.24)
gives
C11 = V − VB T VB BV
+ VB T VB AVA −1 AT VB BV
C21 = −VA −1 AT VB BV
C22 = VA −1
(5.25)
C31 = VB BV − VB AVA −1 AT VB BV
C32 = VB AVA −1
C33 = −VB + VB AVA −1 AT VB
The corrections
∆~y
and
∆~a
as well as the Lagrange Multipliers
~λ
are calcu-
lated by multiplication,
T
∆~y = C31
~c = VB T VB − VB T VB AVA −1 AT VB ~c,
T
∆~a = C32
~c = VA −1 AT VB ~c,
~λ = C33~c = −VB + VB AVA −1 AT VB ~c.
(5.26)
The iteration is repeated until certain convergence criteria are met guaranteeing that the squared sum
S
has reached a minimum and all constraints
are fullled. Therefore, we dene convergence criteria,
S(n − 1) − S(n)
< S
ndf
where
n
m
X
and
(n)
fk (~y , ~a) < F ,
(5.27)
k=1
denotes the number of iterations and
ndf
denes the number of
degrees of freedom which is given by dierence of the number of constraints
and the number of unmeasured parameters,
ndf = m − p.
S
from one iteration to the next.
denes the size of the allowed change of
S
The parameter
The precision by which the constraints have to be fullled is dened by the
parameter
F .
Measurement of Hadronic Mass Moments
56
5.4.2.4
Application to Semileptonic Decays
After having discussed the general approach of the kinematic t we will now
describe its application to the signal decay
consider the following
•
Υ (4S) → Breco BSL (Xc `ν).
We
measured parameters:
Eight measured parameters for the eight components of the
Xc
•
11
four vectors in cartesian coordinates,
Breco
EBreco , p~Breco , EXc ,
and
and
p~Xc .
Three measured parameters for the three independent components of
the
`
four vector,
mass [12] with
p~` .qThe
m2`
E` =
energy of the lepton is constrained by its
+ p~2` .
Furthermore, three parameters have to be considered for the unmeasured
neutrino momentum,
p~ν ,
constraining its mass to
mν = 0 GeV/c2 .
The covariance matrix utilized in the kinematic t is comprised of the
dierent covariance matrices of the measured particles. For the lepton the
covariance obtained by the track t is used.
Breco
For the fully reconstructed
candidate the covariance matrix obtained from the nal kinematic t
described in section 5.2.3 is used.
the covariance of the hadronic
Xc
The situation is more complicated for
system.
Due to the limited acceptance
by particles that have not been measured.
Therefore, we identify the av-
BABAR detector the precision of the Xc
of the
erage covariances of the
Xc
EXc (meas.) − EXc (true)
and
measurement is dominated
four vector components with their resolutions
p~Xc (meas.) − p~(true)
calculated as dierence
of measured and true quantities and determined in simulated signal decays.
A diagonal covariance matrix is assumed.
The studied resolutions vary with the
the
Xc
momentum
ΘXc .
ΘXc ,
multiplicity and polar angle of
MultXc , [1, 4, 7, 10, 13, 21],
[0, 0.52, 1.05, 1.57, 2.09, π].
quantities. We choose ve bins in
bins in
Xc
Therefore, they are determined in bins of theses
as well as ve
Figure 5.6 shows examples of the extracted energy resolutions. While for
high-multiplicity events the resolution distributions are similar to those of
Gauss functions, long tails to negative values are observed for low multiplicity
events.
However, the least squares approach used in the kinematic t is the optimal estimator only for Gaussian distributed input parameters.
Several
dierent parameterizations of the input four-vectors like spherical coordinates have been studied but did not result in signicant improvements of
the observed shapes. Nevertheless, even with the non-Gaussian distributed
input for some of the
well.
Xc
covariances the performed t behaves suciently
We estimate the widths of the extracted resolution distributions by
performing ts with Gauss functions. Figure 5.7 summarizes all extracted
widths.
5.4 Reconstruction of the Hadronic System
900
4 ≤ Mult ≤ 6
800 1.571 ≤ ΘX < 2.094
X
700 σ = 0.45 GeV
600
500
400
300
200
100
0
-5 -4 -3 -2 -1 0 1 2
EX (meas.) - EX (true) [GeV]
entries / 100 MeV
entries / 100 MeV
2000
1800 10 ≤ MultX ≤ 12
1600 0.524 ≤ ΘX < 1.047
1400 σ = 0.45 GeV
1200
1000
800
600
400
200
0
-5 -4 -3 -2 -1 0 1 2
E X(meas.) - E X(true) [GeV]
entries / 100 MeV
entries / 100 MeV
3500 7 ≤ MultX ≤ 9
3000 1.047 ≤ ΘX < 1.571
2500 σ = 0.39 GeV
2000
1500
1000
500
0
-5 -4 -3 -2 -1 0 1 2
EX (meas.) - EX (true) [GeV]
600
500
7 ≤ MultX ≤ 9
1.571 ≤ ΘX < 2.094
σ = 0.37 GeV
400
300
200
100
0
-5 -4 -3 -2 -1 0 1 2
EX (meas.) - EX (true) [GeV]
800
10 ≤ MultX ≤ 12
700 1.047 ≤ Θ < 1.571
X
600 σ = 0.37 GeV
500
400
300
200
100
0
-5 -4 -3 -2 -1 0 1 2
E X(meas.) - E X(true) [GeV]
10 ≤ MultX ≤ 12
100 1.571 ≤ ΘX < 2.094
σ = 0.33 GeV
80
60
40
20
0
-5 -4 -3 -2 -1 0 1 2
E X(meas.) - E X(true) [GeV]
Figure 5.6: Examples of energy resolutions of the hadronic
dierent bins of
ΘXc
and
MultXc
120
entries / 100 MeV
entries / 100 MeV
entries / 100 MeV
4000
4
≤
Mult
≤
6
X
3500
1.047 ≤ ΘX < 1.571
3000 σ = 0.44 GeV
2500
2000
1500
1000
500
0
-5 -4 -3 -2 -1 0 1 2
EX (meas.) - EX (true) [GeV]
8000
7 ≤ MultX ≤ 9
7000 0.524 ≤ Θ < 1.047
X
6000 σ = 0.46 GeV
5000
4000
3000
2000
1000
0
-5 -4 -3 -2 -1 0 1 2
EX (meas.) - EX (true) [GeV]
entries / 100 MeV
entries / 100 MeV
8000
7000 4 ≤ MultX ≤ 6
0.524 ≤ ΘX < 1.047
6000
σ = 0.52 GeV
5000
4000
3000
2000
1000
0
-5 -4 -3 -2 -1 0 1 2
EX (meas.) - EX (true) [GeV]
57
Xc
system for
(histograms a-i). The resolutions are esti-
mated by tting the distributions with Gauss functions (red lines).
σp
X,z
/ GeV/c
1
0.8
0.6
0.4
0.2
0
00.5
11.5
Θ 22.5
X
X,y
σp
3 20
15
10
Mult X
5
3 20
15
10
Mult X
0.5
0.4
0.3
0.2
0.1
0
00.5
11.5
Θ 22.5
3 20
15
10
Mult X
1
0.8
0.6
0.4
0.2
0
00.5
11.5
Θ 22.5
3 20
15
10
Mult X
X
X
X
σE / GeV
σp
X,x
/ GeV/c
0.5
0.4
0.3
0.2
0.1
0
00.5
11.5
Θ 22.5
/ GeV/c
Measurement of Hadronic Mass Moments
58
5
X
Figure 5.7: Energy and momentum resolution of the measured
in bins of
ΘXc
and
5
5
Xc four vector
MultXc .
The kinematics of the signal decay is constrained by the following relations:
f1 = EBreco + EXc + E` + Eν − EΥ (4S) = 0,
f2,3,4 = p~Breco + p~Xc + p~` + p~ν − p~Υ (4S) = 0,
f5 = mBreco − mBSL
i
2
h
2
2
2
= EB
−
p
~
−
(E
+
E
+
E
)
−
(~
p
+
p
~
+
p
~
)
= 0.
ν
ν
Xc
Xc
`
`
Breco
reco
(5.28)
The rst four constraints ensure energy and momentum conservation in the
event.
The fths constraint demands mass equality of the reconstrcuted
2
Breco and BSL mesons. As a consequence the missing mass, m2miss = Emiss
−
2
p~miss , of the event is constraint to zero. By requiring equal masses rather
than xing both masses to their nominal values avoids small inconsitencies
introduced by imprecise beam energy measurements.
The performance of the kinematic t is depicted in gure 5.8. Presented
is the resolution of the hadronic mass
mX −mX,true before and after the kine−254 MeV/c2
matic t in left plot. A signicant reduction of the overall bias,
−96 MeV/c2 after the t, as well as improvement of the resolution,
425 MeV/c2 before and 355 MeV/c2 after the t, is observed.
before and
Figure 5.9 illustrates the eect of the kinematic t on resolution and bias
of the
mX
measurement in bins of
Emiss −|~
pmiss | and the hadronic nal state
5.5 Comparison of Data and MC Simulations
entries / 0.01
entries / 10 MeV/c2
2000
1800
1600
1400
1200
1000
800
600
400
200
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
mX - mX (true) [GeV/c2]
2000
1800
1600
1400
1200
1000
800
600
400
200
0
0
0.2
Figure 5.8: (a) Resolution of measured () and tted
χ2
bility of
Xc .
59
0.4
0.6
mX ().
0.8
1
P(χ2)
(b) Proba-
for the nal t performed on data.
In both cases, a signicant reduction of the bias is observed. Improve-
ments of the resolution are observed for all hadronic nal states. Studying
the resolution as function of
Emiss − |~
pmiss |,
improvements are achieved
predominantly for positive values where most of the measured decays are
accumulated. For negative values of
the
mX
Emiss − |~
pmiss |
a slight degradation of
resolution is observed.
The overall
χ2
probability
P(χ2 )
of the nal t performed on data in
shown in the right plot in gure 5.8. While the overall distribution is reasonably at and only slightly raising toward
bin that contains about
perfect modelling of
Xc
10%
P(χ2 ) = 1
a peak in the rst
of the events is visible. It is caused by the im-
energy and momentum covariances. Since rejecting
these events does not result in an improved background rejection rate and
the overall
P(χ2 )
distribution is well modelled in MC simulations we decide
to keep them in the analysis.
5.5 Comparison of Data and MC Simulations
In the following, the agreement of dierent kinematic variables in data and
MC simulations relevant for the presented measurement will be discussed.
Figure 5.10 compares the distributions of missing energy and momentum
Emiss − |~
pmiss |, Emiss ,
and
|~
pmiss |.
We scale the extracted MC distributions
according to the ratio of the number of fully reconstructed
MC. While the positive tails of the
Emiss − |~
pmiss |
Breco
in data and
distributions agree very
well, discrepancies are observed in the negative part of the distributions. In
the mentioned region an excess of MC events is observed indicating an excess
of additionally measured particles with respect to data. The distributions
of
Emiss
and
|~
pmiss |
show a good agreement of data and MC simulated
distributions.
Comparing the distributions of charged track
plicity
Nγ,X
of the
Xc
Ntrk,X
and photon multi-
system (cf. g. 5.11) shows that the simulated
Nγ,X
hadronic mode
(right
hadronic decay
-0.4
-0.2
0
|E
miss
miss
D*π
Dπ
D1’
0.44
0.42
0.4
0.38
0.36
0.34
0.32
0.3
0.28
0.26
0.2 0.4
- p | [GeV]
hadronic mode
< mX − mX,true > (left column) and resolution σ(mX −
column) of mX in bins of Emiss − |~
pmiss | (a,b) and the
mode (c,d). Shown are bias and resolution before (◦) and
Figure 5.9: Bias
mX,true )
0.2
Ds *
D*π
D0 *
Ds *
D1
D*
D
X
0.25
D0 *
miss
0.3
D1
miss
0.2 0.4
- p | [GeV]
0.35
D*
0
|E
0.4
D
X
-0.2
σ( m - m X (true) ) [GeV/c 2]
-0.4
Dπ
< mX - m X (true) > [GeV/c2]
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35
D1’
< m X - m X (true) > [GeV/c2]
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
σ( m - m X (true) ) [GeV/c 2]
Measurement of Hadronic Mass Moments
60
after (•) the kinematic is performed.
5.6 Background Subtraction
61
distribution is shifted to higher values with respect to the corresponding distribution in data. The simulation contains on average
0.3 additional photons
per event. By contrast the charged track multiplicities agree reasonably well
in data and MC simulation.
Figure 5.12 compares the distributions of the energies of all individual
photons,
PNγ,X
j=1
Eγ ,
Ej,γ .
and the total neutral energy of the hadronic system,
Eγ,Xc =
As mentioned before, the MC simulation contains additional
photons not present in data. These photons appear in the simulation as a
signicant excess of low-energetic photons with
Eγ < 100 MeV
or
Eγ,Xc <
300 MeV.
Even though discrepancies between data and MC simulations are observed especially in the modelling of low-energetic photons, their overall
agreement is generally good.
The inuence of the observed disagreement
on the extracted results will be subject to detailed systematic studies described in section 6.
5.6 Background Subtraction
In the following section procedures applied for the subtraction of background
are discussed. Two sources of background are distinguished:
1. Combinatorial background stemming from misreconstructed
Breco can-
didates. The shape and magnitude of this background is determined
directly on data utilizing the
mES
sideband.
2. Residual background in the reconstruction and selection of the recoiling
hadronic system that does not originate from
background peaks in the
mES
B → Xc `ν
decays. This
signal region and is subtracted utilizing
MC simulations.
5.6.1 Combinatorial Breco Background
Combinatorial background originates from
Breco
candidates that are recon-
structed in decay modes dierent from their true underlying modes. Such
misreconstruction results from random combination and exchange of tracks
and photons. The region under the
mES
signal peak is contaminated with
combinatorial background which needs to be subtracted. Since the
mES
side-
band region is signal free and contains pure combinatorial background we
use it to determine the shape of the combinatorial background distribution
assuming that it agrees with the combinatorial background shape under the
signal peak. This procedure holds the advantage that it does not rely on MC
simulations. We dene the
mES
signal and sideband regions as follows:
5.27 ≤mES,signal ≤ 5.289 GeV/c2
5.21 ≤mES,sb ≤ 5.255 GeV/c2
(5.29)
Measurement of Hadronic Mass Moments
62
2500
2000
1500
1000
500
0
500
entries / 80MeV
entries / 80MeV
3000
-1 -0.5 0
400
300
200
100
0
0.5 1 1.5 2 2.5
Emiss - |p | [GeV]
-1 -0.5 0
miss
800
600
400
200
0
entries / 80MeV
entries / 81MeV
1000
1400
1200
1000
800
600
400
200
0
0
1
2
3
4
5
Emiss [GeV]
entries / 80MeV
entries / 81MeV
1200
0.5 1 1.5 2 2.5
Emiss - |p | [GeV]
miss
!" #
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
|p | [GeV/c]
miss
160
140
120
100
80
60
40
20
0
!" #
220
200
180
160
140
120
100
80
60
40
20
0
!" #
0
1
2
3
4
5
Emiss [GeV]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
|p | [GeV/c]
miss
Figure 5.10: Comparison of data (•) and MC simulated distributions (red
histogram) for
Emiss −|~
pmiss | (a,b), Emiss
cuts on the minimum lepton momenta,
p∗` ≥ 1.9 GeV/c
(right column).
(c,d), and |~
pmiss | (e,f ) for dierent
p∗` ≥ 0.8 GeV/c (left column) and
3500
3000
2500
2000
1500
1000
500
500
200
100
1200
!" #
1000
entries
entries
2 4 6 8 10 12 14 16 18 20 22
N trk,X + Nγ ,X
8000
7000
6000
5000
4000
3000
2000
1000
800
600
400
200
0
1
2
3
4
5
6
7
N trk,X
0
500
1
2
3
4
5
6
7
N trk,X
!" #
400
entries
entries
!" #
300
2 4 6 8 10 12 14 16 18 20 22
N trk,X + Nγ ,X
3500
3000
2500
2000
1500
1000
500
0
63
400
entries
entries
5.6 Background Subtraction
300
200
100
0
2
4
6
8
10 12
N γ ,X
0
0
2
4
6
8
10 12
N γ ,X
Figure 5.11: Comparison of data (•) and MC simulated distributions (red
histogram) for
Ntrk,X + Nγ,X
(a,b),
Ntrk,X (c,d), and Nγ,X (e,f ) for dierent
p∗` ≥ 0.8 GeV/c (left column) and
cuts on the minimum lepton momenta,
p∗` ≥ 1.9 GeV/c
(right column).
Measurement of Hadronic Mass Moments
entries / 40MeV
600
2000
1500
1000
500
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Eγ
500
400
300
200
100
0.5
1
1.5
2
!
2500
entries / 40MeV
20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Eγ
100
entries / 40MeV
entries / 40MeV
64
!
80
60
40
20
2.5
EX, γ
0.5
1
1.5
2
2.5
EX, γ
Figure 5.12: Comparison of data (•) and MC simulated distributions (red histogram) of the energies of individual photons
column).
(a,b) and the total neutral
Eγ,Xc (c,d) for dierent cuts on the minimum
p∗` ≥ 0.8 GeV/c (left column) and p∗` ≥ 1.9 GeV/c (right
energy of the hadronic system
lepton momenta,
Eγ
5.6 Background Subtraction
65
15 MeV/c2
Thereby, signal and sideband region are separated by
to avoid
leakage of signal events from the signal region into the sideband region.
To quantify the amount of combinatorial background under the
signal peak we parametrize the
mES
mES
distribution utilizing the sum of an
ARGUS [35] and Crystal Ball [36] function. The ARGUS function describes
the shape of the combinatorial background. It is dened as
r
1−(
fARGUS (mES ; N, χ, mES,max ) = N mES
ES )2 )
mES 2 −χ(1−( mm
ES,max
) e
,
mES,max
(5.30)
where
N
determines its overall normalization and the parameter
curvature. The mass
mES,max
χ denes its
describes the kinematic endpoint constrained
by the beam energies and is xed to
mES,max = 5.289 GeV/c2 .
The signal
shape is described by a Crystal Ball function:
 m −m 2
ES
0
−

2σ

e




if mES > m0 − ασ,
−n
fCB (mES ; N, m0 , σ, α, n) = N σ · 2
2
(mES −m0 )

− α2
α
α

1
−
−
·
e

n
n
σ



if mES ≤ m0 − ασ.
(5.31)
fCB
is constituted of a central Gaussian with mean
power law tail to low energies of order
at
m0 − ασ , with α > 0.
n
m0
and width
σ
and a
continously joined to the Gaussian
The power law tail is needed to model energy losses
in the EMC on the reconstruction of
π0
mesons.
The relative size of the combinatorial background in the signal region
with respect to the sideband region can be determined by tting over the
full
mES
range,
5.21 ≤ mES ≤ 5.289 GeV/c2 .
It is given by the ratio of the
two integrals,
R
ζsb =
signal fARGUS (mES )dmES
R
sb fARGUS (mES )dmES
The number of well reconstructed
Breco
.
(5.32)
candidates is given by
NBreco = Nsignal − ζsb Nsb ,
with
Nsignal
(5.33)
the total number of events in the signal region and
Nsb
the
number of events in the sideband region.
Studies reveal a dependence of the scaling factor
hadronic mass
mX
gurations with regard to dierent regions in the
fore, the ts are performed in bins of
the
mX
ζsb
on the reconstructed
resulting from dierent momentum and multiplicity con-
mX
mX
phase space. There-
(see Fig. 5.13 and 5.14).
For
dependent rescaling of the combinatorial background shape, mea-
a cubic spline [37].
ranging between
p∗`,cut ,
ζsb are interpolated with
p∗` ≥ 0.8 GeV/c we nd scaling factors
0.61 ± 0.12, 0.40 ≤ mX < 1.44 GeV/c2 , and 0.24 ± 0.03,
sured separately for every
For the cut
the extracted
Measurement of Hadronic Mass Moments
66
2.40 ≤ mX < 2.72 GeV/c2 . Only a small variation of the extracted ζsb is
∗
found for the cut p` ≥ 1.9 GeV/c. The extracted ζsb are 0.57 ± 0.14m for
0.24 ≤ mX < 1.60 GeV/c2 . and 0.71 ± 0.16m for 0.24 ≤ mX < 2.52 GeV/c2 .
Fig. 5.15 shows the extracted scaling factors (left) together with the corresponding combinatorial background distributions (right). For comparison
the background distributions scaled with a single factor determined from a
mX
t over the full
range are superimposed (red histogram).
5.6.2 Residual Background
We call background that originates from the selection of decays other that
B → Xc `ν
residual background. While combinatorial background stemming
from misreconstructed
Breco
can be studied directly in data the subtraction
of residual background requires studies with MC simulations.
Residual background shapes are extracted from MC. Combinatorial background is subtracted like in data following the procedure described in the
previous section 5.6.1. The extracted spectra are normalized to the number
of signal
Breco
candidates in data,
NBreco ,data = NBreco ,M C .
To improve the quality of the MC simulation we rescale the simulated
spectra to match branching fraction measurements of the corresponding processes. Table 5.7 summarizes all applied MC scaling factors.
Fig. 5.16 shows a breakdown of all residual background contributions for
dierent cuts on
p∗` .
In the following, dominant sources of residual back-
ground will be described and current branching fraction measurements will
be summarized.
5.6.2.1
Flavor Anticorrelated
D and Ds+ Decays
While the
B
b → cW ∗+ into a avor correlated
anticorrelated c quarks is also possible
c
mesons decay dominantly via
quark, the production of avor
through the decay of the virtual
W ∗+ .
Fig. 5.17 shows the Feynman graphs of
the involved dominant Cabibbo-allowed processes where the
into a
cs
(∗)+
quark pair producing Ds
,
nal state.
D(∗)0 K + , and
W ∗+
fragments
D(∗)+ K 0 mesons in the
The charmed mesons can subsequently decay semileptonically
into right-charged leptons which cannot be distiguished experimentally from
semileptonic
•
B
decays.
The branching fractions of avor anticorrelated
B → D+ X
D0 X decays have been measured recently by the
and
B→
BABAR collaboration
[38]:
B(B 0 → D0 X) = (8.1 ± 1.5)%,
B(B + → D0 X) = (8.6 ± 0.7)%,
B(B 0 → D+ X) = (2.3 ± 1.1)%,
B(B + → D+ X) = (2.5 ± 0.5)%.
(5.34)
5.6 Background Subtraction
entries / 2.5MeV/c 2
entries / 2.5MeV/c 2
2
N B = 16391 ± 136
reco
pur = 0.88 ± 0.01
ζ = 0.34 ± 0.01
5000
4000
sb
3000
2000
1000
0
5.2
NB
300
1500
100
1000
500
0
5.2
1200
1000
sb
200
sb
150
100
50
0
5.2
5.25
5.3
mES [GeV/c2 ]
Figure 5.13: Fits of
the full
mX
5.2
600
N B = 3284 ± 62
reco
pur = 0.88 ± 0.01
ζ = 0.38 ± 0.04
sb
600
400
200
2
N B = 2443 ± 52
reco
pur = 0.90 ± 0.02
ζ = 0.46 ± 0.06
sb
400
200
0
5.25
5.3
mES [GeV/c 2]
600
2.08 ≤ m X < 2.40 GeV/c
5.2
800
1.44 ≤ m X < 1.76 GeV/c
500
400
300
5.2
5.25
5.3
mES [GeV/c2 ]
2.40 ≤ m X < 2.72 GeV/c
2
N B = 1421 ± 36
reco
pur = 0.87 ± 0.02
ζ = 0.24 ± 0.03
sb
200
100
5.25
5.3
mES [GeV/c 2]
0
5.2
5.25
5.3
mES [GeV/c2 ]
2
300 3.04 ≤ m X < 4.08 GeV/c
entries / 2.5MeV/c 2
entries / 2.5MeV/c 2
300 2.72 ≤ m X < 3.04 GeV/c2
pur = 0.70 ± 0.03
ζ = 0.34 ± 0.04
800
0
5.25
5.3
mES [GeV/c2 ]
250 N Breco = 631 ± 29
sb
1000
2
entries / 2.5MeV/c 2
entries / 2.5MeV/c 2
N B = 6556 ± 83
reco
pur = 0.94 ± 0.01
ζ = 0.40 ± 0.04
2000
ζ = 0.61 ± 0.12
200
2
2500
= 1292 ± 34
400 purreco= 0.87 ± 0.03
0
5.25
5.3
mES [GeV/c2 ]
1.76 ≤ m X < 2.08 GeV/c
2
500 0.40 ≤ m X < 1.44 GeV/c
entries / 2.5MeV/c 2
0.40 ≤ m X < 4.08 GeV/c
6000
entries / 2.5MeV/c 2
7000
67
mES
250 N Breco = 667 ± 31
pur = 0.60 ± 0.03
200 ζ = 0.27 ± 0.02
sb
150
100
50
0
5.2
5.25
5.3
mES [GeV/c 2]
distributions in data (•) for
range (upper left plot) and in bins of
mX
p∗` ≥ 0.8 GeV/c
over
(other plots). The t
is performed using a function constituted of an ARGUS and a Crystal Ball
function.
The t results are shown (black line) together with their signal
and and background shapes (red dashed lines).
Measurement of Hadronic Mass Moments
300
0.24 ≤ m X < 2.52 GeV/c
2
N B = 2207 ± 49
reco
pur = 0.90 ± 0.02
ζ = 0.64 ± 0.11
sb
5.2
5.25
5.3
mES [GeV/c2 ]
Figure 5.14: Fits of
the full
mX
mES
250
200
150
0.24 ≤ m X < 1.60 GeV/c
2
600 1.60 ≤ m X < 2.52 GeV/c
2
entries / 2.5MeV/c 2
900
800
700
600
500
400
300
200
100
0
entries / 2.5MeV/c 2
entries / 2.5MeV/c 2
68
N B = 697 ± 28
reco
pur = 0.87 ± 0.04
ζ = 0.57 ± 0.14
sb
100
50
0
5.2
500 N Breco = 1505 ± 40
pur = 0.90 ± 0.03
400 ζ = 0.71 ± 0.16
sb
300
200
100
0
5.25
5.3
mES [GeV/c 2]
distributions in data (•) for
range (upper left plot) and in bins of
mX
5.2
5.25
5.3
mES [GeV/c2 ]
p∗` ≥ 1.9 GeV/c
over
(other plots). The t
is performed using a function constituted of an ARGUS and a Crystal Ball
function.
The t results are shown (black line) together with their signal
and and background shapes (red dashed lines).
entries / 40 MeV
0.7
0.6
ζ
sb
0.5
0.4
0.3
0.2
0.5 1 1.5 2
#
$! "%&
entries / 40 MeV
ζ
sb
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
2.5 3 3.5 4
mX [GeV/c2]
0.5
1
1.5
2
2.5
mX [GeV/c2]
90
80
70
60
50
40
30
20
10
0
24
22
20
18
16
14
12
10
8
6
4
2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
! "
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
ζsb extracted from mES ts in bins of
p∗` ≥ 0.8 GeV/c (a) and p∗` ≥ 1.9 GeV/c (c). The ζsb are
Figure 5.15: Sideband scaling factors
mX
for dierent cuts
interpolated utilizing a cubic spline. The corresponding rescaled combinatorial background distributions are shown in (b,d) (◦). Distributions scaled
with a single scaling factor extracted over the full
line).
mX
range are overlaid (red
5.6 Background Subtraction
30
entries / 40 MeV/c 2
69
25
20
15
10
5
0
0
12
0.5
1
1.5
2
2.5
3
3.5
4
m X [GeV/c2]
entries / 40 MeV/c 2
10
8
6
4
2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
m X [GeV/c2]
p∗` ≥ 0.8 GeV/c (a) and
≥ 1.9 GeV/c (b). The background contributions are B → Xu `ν (),
+,0 → D +,0 X → Y `+ ν (), B +,0 → D + X →
misidentied hadrons (), B
s
Y `+ ν (), B +,0 (→ Ds+ ) → τ + → `+ (), photon conversions γ → `+ `−
0
+ − (), B → J/ψ , ψ(2S) → `+ `− (), other
(), Dalitz decays π → γ` `
0 0
(), B B Mixing (), and mis tagged events ().
Figure 5.16: Residual background distributions for
p∗`
Measurement of Hadronic Mass Moments
70
Decay
BMC [%]
Bdata [%]
S
B →D X →Y` ν
B 0 → D + X → Y `+ ν
B + → D 0 X → Y `+ ν
B 0 → D 0 X → Y `+ ν
B + → D 0 , D − X → Y `− ν
B 0 → D 0 , D − X → Y `− ν
0.56
0.53
0.51
0.60
6.60
8.90
0.40 ± 0.08 [38, 39]
0.37 ± 0.18 [38, 39]
0.56 ± 0.05 [38, 39]
0.52 ± 0.10 [38, 39]
7.60 ± 0.47 [38, 39, 40]
7.13 ± 0.52 [38, 39, 40]
0.719 ± 0.145
0.706 ± 0.338
1.086 ± 0.095
0.871 ± 0.164
1.151 ± 0.071
0.801+0.058
−0.055
B + → Ds+ X → Y `+ ν
B 0 → Ds+ X → Y `+ ν
0.81
0.77
0.61+0.12
−0.10 [38, 41, 12]
0.79+0.16
−0.14 [38, 41, 12]
0.753+0.146
−0.118
1.035+0.215
−0.185
B + → τ + → `+
B 0 → τ + → `+
B + → Ds+ → τ + → `+
B 0 → Ds+ → τ + → `+
0.52
0.52
0.17
0.16
[12, 26, 42]
[12, 26, 42]
[12, 26, 42]
[12, 26, 42]
0.905 ± 0.098
0.834 ± 0.092
0.524+0.158
−0.146
0.721+0.224
−0.211
6.06 × 10−2
0.23 × 10−2
(6.46 ± 0.20) × 10−2 [12]
(0.226 ± 0.017) × 10−2 [12]
1.066 ± 0.034
0.982 ± 0.088
0.210
0.222 ± 0.033 [42]
1.057 ± 0.157
+
+
+
B → J/ψ → `+ `−
B → ψ(2S) → `+ `−
B → Xu `ν
0.47 ± 0.05
0.43 ± 0.05
0.09+0.03
−0.02
0.12+0.04
−0.03
Table 5.7: Summary of scaling factors
of background processes in MC
BMC
S
used to correct braching fractions
to measured branching fractions
Bdata
as discussed on pages 66 - 74.
Ds
s
q D
(a)
B
W
b
q
+
( )
D
u
(b)
+
b
q
u K
s
q D
D
d
()
( )0
W
B
( )+
( )+
W
+
B
( )
b
q
B→
→Y
`+ ν (a),
B→
D+ X
→Y
`+ ν (b),
d K
s
q D
0
( )
D and Ds+ Decays,
B → D0 X → Y `+ ν (c).
Figure 5.17: Dominant Feynman graphs of right-charged
Ds+ X
+
5.6 Background Subtraction
71
The CLEO collaboration measured the absolute semileptonic
D-meson
branching fractions [39],
B(D0 → X`+ ν) = (6.46 ± 0.21)%,
(5.35)
B(D+ → X`+ ν) = (16.13 ± 0.39)%,
resulting in
B(B + → D+ X → Y `+ ν) = (0.40 ± 0.08)%,
B(B 0 → D+ X → Y `+ ν) = (0.37 ± 0.18)%,
(5.36)
B(B + → D0 X → Y `+ ν) = (0.56 ± 0.05)%,
B(B 0 → D0 X → Y `+ ν) = (0.52 ± 0.10)%.
•
The branching fractions of avor anticorrelated
Ds+ X
B 0 → Ds+ X
and
B+ →
decays have been measured in [38] resulting in
B(B + → Ds+ X) = 7.9+1.5
−1.2 %,
(5.37)
B(B 0 → Ds+ X) = 10.3+2.1
−1.8 %.
Since the actual size of the total semileptonic
Ds+ -meson
branching
fraction is unmeasured we estimate it by using the lifetime ratio τD /τ + .
Ds
Assuming equal semileptonic decay rates of
D
and
Ds+
mesons [41] one
obtains the following expression
B(Ds+ → X`+ ν) = B(D → X`+ ν)
Using the measured lifetimes
9) fs
τD
.
τDs+
τD0 = (410.3 ± 1.5) fs
and
(5.38)
τDs+ = (490 ±
[12] we obtain
B(Ds+ → X`+ ν) = (7.71 ± 0.29)%.
(5.39)
Combining both measurements the total branching fractions are quantied as
B(B + → Ds+ X → Y `+ ν) = (0.61+0.12
−0.10 )%,
B(B 0 → Ds+ X → Y `+ ν) = (0.79+0.16
−0.14 )%.
5.6.2.2
(5.40)
τ Decays
There are two relevant sources of background from
τ
decays which have to
be accounted for:
•
Semileptonic B decays into
τ
leptons that subsequently decays into
right-charged leptons produce irreducible background that has to be
subtracted using MC simulated decays. Since there are no measurements of the semileptonic branching fractions of
B
mesons
B(B 0 →
Measurement of Hadronic Mass Moments
72
Xτ + ντ )
and
B(B + → Xτ + ντ )
available we utilize the measurement
of inclusive semileptonic branching fraction in the admixture of
Bs ,
Λb
and
B,
to calculate both branching fractions following the dis-
cussion in [26, p. 48]. This branching fraction was measured by the
B(Xb → Xc τ
Xc τ + ντ ) is composed of
LEP experiments to be
B(Xb →
+ν )
τ
= (2.48 ± 0.26)%
[12] where
B(Xb → Xc τ + ντ ) =fu,d B(B 0 → Xτ + ντ ) + fu,d B(B + → Xτ + ντ )+
fs B(Bs → Xτ + ντ ) + fΛb B(Λb → Xτ + ντ ).
(5.41)
fu,d = (39.8 ± 1)%, fs = (10.4 ± 1.4)%,
the production fractions.
assumed. Implying equal semileptonic
contributing to
Xb
eq. 5.41 can be rewritten as
B(B 0 → Xτ + ντ ) = +
fΛb = (9.9 ± 1.7)% [42] are
B 0 and B + decays are
decay rates ΓSL for all hadrons
and
Equal fractions for
B(Xb → Xc τ + ντ )
fu,d (1 +
τ
τΛb
+ fs τBs0 + fΛb τ
B
,
B0
B(Xb → Xc τ + ντ )
+
B(B → Xτ ντ ) = τB +
τB 0 )
fu,d (1 +
τB 0
τB + )
τ
τΛb
+ fs τ B+s + fΛb τ
B
(5.42)
.
B+
τB + = (1.643±0.010) ps, τB 0 = (1.527±0.008) ps, τBs = 1.454±
0.040) ps, and τΛb = (1.288 ± 0.065) ps [42] we get
Using
B(B 0 → Xτ + ντ ) = (2.46 ± 0.27)%,
(5.43)
B(B + → Xτ + ντ ) = (2.67 ± 0.29)%.
Combining with the average branching fraction [12]
B(τ + → `+ ν τ ν) = (17.60 ± 0.04)%
(5.44)
results in
B(B 0 → Xτ + ντ → `+ ν τ ν) = (0.43 ± 0.05)%,
B(B + → Xτ + ντ → `+ ν τ ν) = (0.47 ± 0.05)%.
• Ds+
mesons decaying leptonically into a right-charged
an additional source of background.
of this decay is known to be
τ+
(5.45)
contribute
The average branching fraction
B(Ds+ → τ + ντ ) = (6.4 ± 1.5)%
[12].
Combining with eq. 5.37 and 5.44 results in
B(B + → Ds+ → τ + → `+ ) = (0.09+0.03
−0.02 )%
B(B 0 → Ds+ → τ + → `+ ) = (0.12+0.04
−0.03 )%.
(5.46)
5.6 Background Subtraction
5.6.2.3
J/ψ
J/ψ and ψ(2S) Decays
and
J/ψ
73
ψ(2S) Decays
ψ(2S) decaying into two leptons of which only the right-charged
and
lepton is measured contribute to the background.
The average branching
`+ `− )
B(J/ψ →
= (5.91 ± 0.07)% and B(ψ(2S) → `+ `− ) =
(0.736 ± 0.042)% [12]. Combining with B(B → J/ψ X) = (1.094 ± 0.032)%
and B(B → ψ(2S)X) = (0.307 ± 0.021)% [12] results in
fractions are
B(B → J/ψ → `+ `− ) = (6.46 ± 0.20) × 10−4
and
(5.47)
B(B → ψ(2S) → `+ `− ) = (0.226 ± 0.017) × 10−4 ,
respectively.
5.6.2.4
B 0B 0 Mixing
After subtraction of all background sources described so far there is still
B 0B 0 oscillations. These oscillations result in
events with two equally avored B mesons.
B 0 mesons stemming from B 0B 0 events are dominantly selected via
0
0
−
−
wrong-charged leptons from B → (D , D )X → Y ` ν cascades. The size
background remaining from
of the total avor-correlated charm production has been measured in [38] to
B(B 0 → (D0 , D− )X) = (84.30+4.41
−4.18 )%. Subtracting
semileptonic decays, B(B → Xc `ν) = (10.611 ± 0.175)%
be
contributions from
[40], leads to
B(B 0 → (D0 , D− )X) = (63.08+4.42
−4.19 )%.
In combination with the average semileptonic
D
(5.48)
meson branching fraction
[39],
B(D+,0 → X`+ ν) = (11.30 ± 0.22)%.
(5.49)
B(B 0 → (D0 , D− )X → Y `− ν) = (7.13+0.52
−0.49 )%.
(5.50)
we estimate
The time-integrated mixing probability
χd
of
B0
mesons oscillating into
B 0 mesons and vice versa has been measured to be
χd,data = 0.188 ± 0.002
χd,M C =
χd,data /χd,M C = 1.038 ± 0.017.
[42]. The MC simulation implements an oscillation probability of
0.181
resulting in a small scaling factor of
Since this scaling factor and its uncertainty are negligible small compared
to the eect observed in the rescaling of the branching fraction
(D0 , D− )X → Y `− ν)
a rescaling of χd .
5.6.2.5
B(B 0 →
we stick to the original MC value and do not perform
Mis Tagged Events
We call events with
Breco
candidates that are reconstructed in decay modes
dierent from the underlying true decay modes mis tagged events.
Even
Measurement of Hadronic Mass Moments
74
though most of the mistagging background is subtracted as combinatorial
background there is still a small contribution of quasi well reconstructed
Breco
candidates remaining. Those events occure mainly by exchange of slow
Breco candi17.8% as a charged Breco
charged or neutral pions. MC studies show that true neutral
dates are reconstructed with a probability of about
candidate while only
4.84% of all charged Breco
candidates are reconstructed
as neutral candidates.
Only a small fraction of those events results in a reversal of the charge
correlation between lepton and
0
structed as a B or
B−.
Breco
avor, i.e. when a true
B0
is recon-
The probabilities to reconstruct true neutral or
B mesons as Breco candidates with dierent b quark avor are 1.9%
and 1.8%, respectively. This type of mis tagged events is dominantly selected
+,0 → (D 0 , D − )X → Y `− ν cascades. The
via wrong-charged leptons from B
0
0
−
−
branching fraction B(B → (D , D )X → Y ` ν) is calculated in sec. 5.6.2.4.
charged
Accordingly, we get
B(B + → (D0 , D− )X → Y `− ν) = (7.60 ± 0.47)%,
using
B(B + → (D0 , D− )X) = (67.28±3.91)% [38].
(5.51)
After subtraction of com-
bionatorial background only a small but signicant fraction of those events
is retained.
5.6.2.6
Misidentied Hadrons
Background stemming from hadrons like
π , K , and p that have been misiden-
tied as electrons or muons is estimated from MC simulations corrected for
eciency and misidentication rate dierences in data and MC. The correction factors are estimated from pure data control samples [30,31] and applied
using the so called tweaking method [43].
5.6.2.7
Semileptonic
Semileptonic
B
B → Xu `ν Decays
meson decays
B → Xu `ν
into charmless hadronic nal states
contribute to the background. The current world average [42],
B(B → Xu `ν) = (2.22 ± 0.33) · 10−3 ,
(5.52)
has been calculated within the BLNP framework [44]. It uses Heavy Quark
parameter input from a global t to
B → Xc `ν
and
B → Xs γ
moments [11]
in the Kinetic Scheme [14].
5.6.3 Summary of Background Subtraction
5.6.3.1
Normalization of Background Distributions
As described above, the overall normalization for combinatorial
ground is extracted from a t to the
mES
Breco
back-
distribution while the residual
45
40
35
30
25
20
15
10
5
0
entries / 320 MeV/c2
450
400
350
300
250
200
150
100
50
0
χ2 (NDF) = 16.6 (10)
Scale BG = 0.99 ± 0.03
(" )
χ (NDF) = 3.1 (5)
Scale BG = 0.98 ± 0.12
2
120
100
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
75
!"$#%&'
χ2 (NDF) = 7.5 (8)
Scale BG = 0.96 ± 0.06
80
60
40
20
0
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
entries / 320 MeV/c2
entries / 320 MeV/c2
entries / 320 MeV/c2
5.6 Background Subtraction
22
20
18
16
14
12
10
8
6
4
2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
!" *&'
χ (NDF) = 1.9 (4)
Scale BG = 0.88 ± 0.34
2
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
Figure 5.18: Spectra of wrong-charged events in a sample of charged
Breco
candidates. The overall normalization of the background histograms is determined in a t to the observed spectrum in data.
The histograms are:
observed spectrum in data (•), combinatorial background (), and residual
background ().
background extracted from MC is normalized to the number of signal
candidates,
NBreco ,
Breco
in data.
We check the normalization utilizing a control sample with inverted lepton charge correlation. Since we aim for a sample containing only background
±
Breco
candidates in the selection. Thereby,
0
0
we reject events which undergo B B oscillations and hence contain wrongcharged signal leptons. Fig. 5.18 shows a comparison of the extracted mX
events we consider only charged
spectra with combinatorial background and simulated background distributions for dierent cuts on
p∗` .
The background shapes are tted with a single
free scaling factor to the full spectrum. All tted scaling factors, as given
in the gure, are compatible with unity within errors.
Together with the
reasonable agreement of the simulated and measured spectra we are condent that these results are transferable to the corresponding right-charged
samples.
As an additional cross-check we repeat the same procedure with the corresponding control sample of neutral
Breco
events. The results are shown in
Fig. 5.19. Even though this sample contains a signicant contribution of signal events the overall agreement of data and MC distributions is reasonable.
Measurement of Hadronic Mass Moments
1000
χ2 (NDF) = 18.5 (10)
Scale BG = 1.00 ± 0.02
800
600
400
200
0
entries / 320 MeV/c 2
entries / 320 MeV/c 2
1200
500
450
400
350
300
250
200
150
100
50
0
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
! $
χ (NDF) = 9.6 (6)
Scale BG = 1.00 ± 0.04
2
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
entries / 320 MeV/c 2
entries / 320 MeV/c 2
76
1000
900
800
700
600
500
400
300
200
100
0
160
140
120
100
80
60
40
20
0
! "#
χ2 (NDF) = 7.5 (8)
Scale BG = 1.00 ± 0.03
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
! %
χ (NDF) = 3.8 (5)
Scale BG = 1.01 ± 0.07
2
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
Figure 5.19: Spectra of wrong-charged events in a sample of neutral
Breco
candidates. The overall normalization of the background histograms is determined in a t to the observed spectrum in data.
The histograms are:
observed spectrum in data (•), combinatorial background (), residual background (), and signal decays in data () using
χd = 0.181.
5.7 Extraction of Moments
pl,cut
[ GeV/c]
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
77
Ntotal
Nsb
Nres
Nbg
Nsignal
19212 ± 138
18084 ± 134
16922 ± 130
15637 ± 125
14262 ± 119
12764 ± 113
11152 ± 105
9473 ± 97
7741 ± 88
5902 ± 76
4118 ± 64
2527 ± 50
2429 ± 43
2067 ± 39
1835 ± 38
1622 ± 35
1410 ± 33
1266 ± 33
1081 ± 31
918 ± 27
766 ± 23
604 ± 22
427 ± 19
271 ± 17
1696 ± 19
1443 ± 17
1212 ± 16
1033 ± 15
884 ± 14
763 ± 12
642 ± 11
553 ± 10
474 ± 10
389 ± 9
310 ± 8
248 ± 7
4126 ± 47
3510 ± 43
3047 ± 41
2655 ± 38
2295 ± 36
2029 ± 35
1724 ± 33
1471 ± 29
1240 ± 25
993 ± 24
738 ± 21
520 ± 18
15085 ± 146
14573 ± 141
13874 ± 136
12981 ± 130
11966 ± 124
10734 ± 118
9427 ± 110
8001 ± 101
6500 ± 91
4908 ± 80
3379 ± 67
2006 ± 53
Table 5.8: Number of signal and background events for dierent cuts on
p∗` .
Number of total (Ntotal ), sideband background (Nsb ), residual background
(Nres ), total background (Nbg ), and signal events (Nsignal ).
5.6.3.2
Calculation of Background Subtraction Factors
The left column of Fig. 5.20 shows the measured
mX
wi
spectra together with
the extracted combinatorial and residual background distributions for dierent cuts on
p∗` .
The corresponding total number of signal and background
events are summarized in table 5.8. For each bin in
traction factors
wi
wi (mX ) =
The factors
mass
mX
wi
mX
background sub-
are determined,
Ntotal (mX ) − Nbg (mX )
.
Ntotal (mX )
(5.53)
dene the probability for an event with the invariant hadronic
to be a signal event. For the nal background subtraction we t
the calculated
wi
with a polynomial of order 4 (see Fig. 5.20, right column).
5.7 Extraction of Moments
In the following chapter will be described the method utilized to measure
the moments of the invariant hadronic mass distribution
< mnX >
with
n = 1 . . . 6 as functions of p∗`,cut . Section 5.7.1 will give an introduction to the
applied calibration method. Details of the mX calibration will be discussed
in section 5.7.2. Section 5.7.3 summarizes additional bias corrections that
have to be applied. Statistical uncertainties and correlations will be discussed
in section 5.7.4. The verication of the overall procedure will be presented
in section 5.7.5.
Measurement of Hadronic Mass Moments
i
600
400
200
i
)(*,+!#"%$'&
signal probability w
1000
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
800
600
400
200
0
200
)(* -!#"%$'&
i
600
500
400
300
200
100
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
signal probability w
entries / 40 MeV/c2
signal probability w
800
0
entries / 40 MeV/c2
1000
0
entries / 40 MeV/c2
!#"%$'&
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
)(* .!#"%$'&
150
100
50
0
0 0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
i
1200
signal probability w
entries / 40 MeV/c2
78
1.4
!#"%$'&
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
)(*,+!#"%$'&
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
)(* -!#"%$'&
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
1.4
)(* .!#"%$'&
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
0.5 1 1.5 2 2.5 3 3.5 4
mX [GeV/c2]
Figure 5.20: The left column shows the measured
mX
spectra in data (•)
together with the extracted combinatorial () and residual () background
p∗` . The corresponding background subNtotal −Nbg
are shown in the right column together with
Ntotal
distributions for dierent cuts on
traction factors
wi =
the t of a polynomial of 4th order.
5.7 Extraction of Moments
79
5.7.1 Overview
As discussed in section 5.4.2 the kinematic t already provides an instrument
that allows a reliable measurement of the hadronic system in semileptonic
decays
B → Xc `ν .
mX,reco
It signicantly reduces the average bias of the measured
distribution with respect to the true underlying distribution, i.e. the
mean of the distribution
mX,reco − mX,true
is close to zero for all nal states.
Nevertheless, additional corrections have to be applied to correct remaining
eects resulting from the limited acceptance and resolution of the detector.
For the extraction of moments we utilize a method that calibrates the
measured and kinematically tted
mX,reco
by applying correction factors on
an event-by-event basis. Thereby, the moments
directly from the measured
mX,reco .
< mnX >
can be calculated
These factors are chosen in a way that
allows the correction of the moments of the measured distribution rather
than the deconvolution of the distribution itself.
The background is subtracted by weighting events according to the fraction of signal events expected in the corresponding part of the
trum.
torial
mX,reco
spec-
The extraction of the background spectrum composed of combina-
Breco
and residual contributions has already been described in the
previous chapter 5.6. Fig. 5.20 summarizes the extracted weighting factors
wi (mX ).
We determine the background subtracted moments of order
n
by calcu-
lating the weighted mean,
PNevt
<
mnX
>=
wi (mX )mnX,calib,i
× Ccalib × Ctrue ,
PNevt
wi (mX )
i
i=1
(5.54)
wi (mX ) the mX,reco -dependent background subtraction factors and
mX,calib,i the calibrated mass. Nevertheless, additional corrections have to
with
be applied to correct for a remaining small bias after the calibration has been
performed. It is distinguished between two categories:
•
•
The factor
The factor
Ccalib corrects for an observed bias of the calibration method.
Ctrue
corrects for two small principal bias sources: First,
for varying selection eciencies with respect to dierent exclusive decay channels that can be introduced by the selection cuts. Second, it
corrects eects introduced by photons emitted in nal state radiation.
5.7.2 Calibration
5.7.2.1
General Construction Principle
mX
n
distribution < mX,reco > with the moments of the true underlying mass
n
distribution < mX,true,cut > after application of all selection criteria. For
the construction of the calibration, decays B → Xc `ν are selected from the
The calibration is performed by relating the moments of the measured
Measurement of Hadronic Mass Moments
80
5
entries / 40 MeV/c2
entries / 40 MeV/c2
10
104
3
10
102
1.5
2
2.5
3
3.5 4 4.5
mX,true [GeV/c2]
mX,true
Figure 5.21: Spectra of
(a) and
8000
7000
6000
5000
4000
3000
2000
1000
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
m X,reco [GeV/c2]
mX,reco
(b) with
p∗` ≥ 0.8 GeV/c
for dierent signal decay channels. All selection criteria have been applied.
B → D`ν (), B → D∗ `ν (), B → D∗∗ `ν
B→
B → Xc `ν
The dotted lines in (a) indicate the default mX,true binning used in the
Shown are spectra for the decays
(),
().
D(∗) π`ν (), and the sum of all signal decays spectra
construction of calibration curves.
sample of cocktail MC simulations. The same subset of events is used for
the calculation of
< mnX,true,cut >
< mnX,reco >
and
n
bution mX,true into nine bins.
dividing the distri-
For the rst moment < mX > we choose
[1.8, 1.92, 2.04, 2.36, 2.52, 2.72, 2.92, 3.24, 3.52, 4.0] GeV/c2 . Fig. 5.21 shows
the spectra of true and measured mX for dierent signal decay channels
together with the chosen binning. The rst and the second bin correspond
D∗ mesons, respectively. The remaining bins
∗∗
(∗) π`ν . For moments of
comprise a mixture of decays B → D `ν and B → D
to the decays into true
D
and
higher order a corresponding binning is chosen. Bins containing few events,
e.g. bins with high
mnX
for high cuts on
events are accumulated in each
mX,true
p∗` ,
are merged so that at least
20
bin.
Although the calibration curves are calculated from the cocktail MC
simulations that already provide a clean sample of
Breco
candidates there
is still a small amount of combinatorial background existing.
The general
approach to subtract combinatorial background utilizing a tting technique
See section 5.6.1 for details.
has already been discussed.
Unfortunately,
the small amount of combinatorial background in the cocktail MC sample
does not allow to t the
mES
distribution.
Therefore, we simply estimate
the shape and size of the combinatorial background in the
mES
signal region
mES
mES signal and background regions,
by scaling the combinatorial background spectrum as determined in the
sideband region by the relative size of the
R
ζsb =
signal dmES
R
sb dmES
and assume a relative uncertainty of
100%
,
on the scaling factor
(5.55)
ζsb .
This
sideband subtraction has only a very small eect on the calculated moments.
5.7 Extraction of Moments
5.7.2.2
81
Binning of Calibration
< mnX,reco > is not constant over
the whole phase space but depends on the resolution and multiplicity MultXc
of the measured hadronic system in the event. The observable Emiss −|~
pmiss |
is closely connected to the resolution of mX,reco since it is a measure for
Studies show that the bias of the measured
missing (positive values) or additionally measured (negative values) photons
and charged particles in the event. Furthermore, we nd that the bias of the
< mnX,reco > measurement does also depend on the momentum of the lepton
∗
in the semileptonic decay. The p` dependence is clearly evident especially for
∗
∗
∗
high p` while its p` dependence is small for low p` . It is indirectly introduced
by the observation that the momentum spectrum of the hadronic system
varies with dierent
p∗`
and the observed bias and resolution of the
mX,reco
measurement are functions of the momentum of the hadronic system.
Therefore, the calibration is performed in bins of the mentioned quantities. The following binning is chosen:
p∗` : eleven bins
1.9 GeV/c and one
1. Twelve bins in
0.8 GeV/c
and
100 MeV/c between
∗
with p` ≥ 1.9 GeV/c.
with widths of
additional bin
Emiss − |~
pmiss |: −0.5 GeV to 0.05 GeV, 0.05 GeV to
0.2 GeV, and 0.2 GeV to 0.5 GeV. Due to limited statistics for high
p∗` we do not adopt this binning for p∗` ≥ 1.7 GeV/c.
2. Three bins in
MultXc : 1 to 5, 6 to 7, and ≥ 8. Like for the Emiss −
|~
pmiss | binning, MultXc is not binned for lepton momenta p∗` ≥ 1.7 GeV/c
3. Three bins in
due to limited MC statistics in that region of the phase space.
Overall the calibration is performed in 75 bins for each moment
∗
9 for each p` bin below
∗
remaining p` bins.
5.7.2.3
p∗`
= 1.7 GeV/c
< mnX >:
and three additional ones for the
Linear Dependence
< mnX,reco > vs. < mnX,true,cut >
Emiss − |~
pmiss |, MultXc , and p∗` . A clear
Figures 5.22 - 5.25 show the extracted moments
for
n = 1, 2, 4, 6
in dierent bins of
linear dependence between the measured and true quantities is visible. We
take advantage of this feature by performing a linear t that allows to obtain
the true moments
<
mnX,reco
< mnX,true,cut >
in terms of the reconstructed moments
>.
As shown in gures 5.22 - 5.25 the tted calibration curves are characterized by comparable slopes but dierent osets and follow the linear
shape of the calculated moments.
A signicant deviation from the ideal
case of a perfect reconstruction is observed which would imply the equality
< mX,reco >=< mX,true,cut >
over the full
mX,true
range.
The observed linear dependence allows to draw some interesting conclusions. The applied method is as far as possible independent of the underlying
Measurement of Hadronic Mass Moments
82
physics model as the dierent decays correspond to dierent physical processes. By the observed linearity decays to higher masses like
B → D∗∗ `ν
(∗) π`ν can be described solely from the rst two points in the
and B → D
∗
calibration curves which correspond to the decays B → D`ν and B → D `ν .
This observation supports the assumption that unknown contributions from
decays not included in the MC simulation, e.g.
B → D(∗) ππ`ν ,
are also
properly described by the calibration technique.
In addition the calibration method is independent of the branching fractions of the various contributing decay channels implemented in MC since
only the mean values of their
5.7.2.4
mnX
distributions enter.
From the Measured to the Corrected (Calibrated) Masses
The set of linear calibration curves provide a tool for the correction of the
measured
mnX,reco .
For each event
i
the calibrated mass
mX,calib
can be
calculated directly from the measured mass by inverting the linear function,
mX,calib,i =
with
A
mX,reco − A(Emiss − |~
pmiss |, MultXc , p∗` )
,
B(Emiss − |~
pmiss |, MultXc , p∗` )
the oset and
B
(5.56)
the slope of the corresponding linear calibration
curve. Thereby, the measured quantities are on average corrected back to
the true masses. The performance of this method is illustrated in gures 5.26
and 5.27 which compare measured, calibrated, and the corresponding true
underlying moments with lower cuts on
p∗`
and binned in
p∗` ,
respectively,
extracted from the sample of generic MC simulations. Comparing calibrated
and measured moments we nd an overall correction of the rst moment
< mX > between 5% and 16% for moments between p∗` ≥ 0.8 GeV/c and
p∗` ≥ 1.9 GeV/c, respectively. Accordingly, we nd relative correction of
8 − 30% for < m2X >, 10 − 43% for < m3X >, 10 − 54% for < m4X >,
10 − 63% for < m5X >, and 21 − 72% for < m6X >. Comparing calibrated
and true moments we nd a small bias of the size of a few per mille up to
1.5% for all moments. Only for < m6X > the calibration is resulting in larger
∗
∗
biases between 3% and 11% for the lowest p`,cut between p` ≥ 0.8 GeV/c and
∗
p` ≥ 1.2 GeV/c. The correction of the remaining bias stemming from the
calibration will be discussed in section 5.7.3.1.
5.7.3 Bias Corrections
In the following section we discuss sources of small biases that remain after
the calibration and need to be corrected. Table 5.9 summarizes all correction
factors used in the nal moment extraction and their statistical uncertainties.
We divide the bias corrections in
Ccalib , as described in the previous section
Ctrue , the ratio between true moments
and shown in gures 5.26 and 5.27, and
before and after analysis cuts.
3.8 -0.5 ≤ E -|p | < 0.05 GeV
miss
miss
3.6
3.4 8 ≤ Mult ≤ 20
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.5 2 2.5 3 3.5
<mX,true,cut > [GeV/c 2]
3.8 0.05 ≤ E -|p | < 0.2 GeV
miss
miss
3.6
3.4 6 ≤ Mult ≤ 7
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.5 2 2.5 3 3.5
<mX,true,cut > [GeV/c 2]
3.8 0.05 ≤ E -|p | < 0.2 GeV
miss
miss
3.6
3.4 8 ≤ Mult ≤ 20
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.5 2 2.5 3 3.5
<mX,true,cut > [GeV/c 2]
<mX,reco> [GeV/c 2]
<mX,reco> [GeV/c 2 ]
<mX,reco> [GeV/c 2]
3.8 0.05 ≤ E -|p | < 0.2 GeV
miss
miss
3.6
3.4 1 ≤ Mult ≤ 5
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.5 2 2.5 3 3.5
<mX,true,cut > [GeV/c 2]
83
<mX,reco> [GeV/c 2 ]
3.8 -0.5 ≤ E -|p | < 0.05 GeV
miss
miss
3.6
3.4 6 ≤ Mult ≤ 7
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.5 2 2.5 3 3.5
<mX,true,cut > [GeV/c 2]
<mX,reco> [GeV/c 2 ]
3.8 -0.5 ≤ E -|p | < 0.05 GeV
miss
miss
3.6
3.4 1 ≤ Mult ≤ 5
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.5 2 2.5 3 3.5
<mX,true,cut > [GeV/c 2]
<mX,reco> [GeV/c 2 ]
<mX,reco> [GeV/c 2 ]
<mX,reco> [GeV/c 2 ]
<mX,reco> [GeV/c 2]
5.7 Extraction of Moments
3.8 0.2 ≤ E -|p | < 0.5 GeV
miss
miss
3.6
3.4 1 ≤ Mult ≤ 5
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.5 2 2.5 3 3.5
<mX,true,cut > [GeV/c 2 ]
3.8 0.2 ≤ E -|p | < 0.5 GeV
miss
miss
3.6
3.4 6 ≤ Mult ≤ 7
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.5 2 2.5 3 3.5
<mX,true,cut > [GeV/c 2 ]
3.8 0.2 ≤ E -|p | < 0.5 GeV
miss
miss
3.6
3.4 8 ≤ Mult ≤ 20
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.5 2 2.5 3 3.5
<mX,true,cut > [GeV/c 2 ]
< mX > in bins of MultXc ,
Emiss − |~
pmiss |
Shown are the extracted < mX,reco > versus <
mX,true,cut > in bins of mX,true for 0.8 ≤ p∗` < 0.9 GeV/c (•), 1.4 ≤ p∗` <
1.5 GeV/c (◦), and p∗` ≥ 1.9 GeV/c (). The results of ts of linear functions
are overlaid as solid lines. A reference line with < mX,reco >=< mX,true,cut >
Figure 5.22: Examples of calibration curves for
∗
and p` .
is superimposed as dashed line.
8
6
4
2
12
10
8
6
4
2
12
-0.5 ≤ E miss-|p | < 0.05 GeV
miss
6 ≤ Mult ≤ 7
10
8
6
4
<m2X,reco> [(GeV/c2) 2]
10
15
<m2X,true,cut > [(GeV/c2) 2]
<m2X,reco> [(GeV/c2) 2]
5
14
2
14
12
10
8
6
4
<m2X,reco> [(GeV/c2) 2]
<m2X,reco> [(GeV/c2) 2]
0.05 ≤ E miss-|p | < 0.2 GeV
miss
6 ≤ Mult ≤ 7
8
6
4
2
10
8
6
4
5
10
15
<m2X,true,cut > [(GeV/c2 )2 ]
14
12
8
6
4
2
10
15
<m2X,true,cut > [(GeV/c2) 2]
12
0.05 ≤ E miss-|p | < 0.2 GeV
miss
8 ≤ Mult ≤ 20
10
8
6
4
2
5
10
15
<m2X,true,cut > [(GeV/c2) 2]
0.2 ≤ E miss-|p | < 0.5 GeV
miss
6 ≤ Mult ≤ 7
10
5
14
0.2 ≤ E miss-|p | < 0.5 GeV
miss
1 ≤ Mult ≤ 5
2
10
10
15
<m2X,true,cut > [(GeV/c2) 2]
12
12
10
15
<m2X,true,cut > [(GeV/c2) 2]
2
-0.5 ≤ E miss-|p | < 0.05 GeV
miss
8 ≤ Mult ≤ 20
14
5
5
14
0.05 ≤ E miss-|p | < 0.2 GeV
miss
1 ≤ Mult ≤ 5
<m2X,reco> [(GeV/c2 )2 ]
10
14
<m2X,reco> [(GeV/c2) 2]
12
-0.5 ≤ E miss-|p | < 0.05 GeV
miss
1 ≤ Mult ≤ 5
<m2X,reco> [(GeV/c2) 2]
<m2X,reco> [(GeV/c2 )2 ]
14
<m2X,reco> [(GeV/c2 )2 ]
Measurement of Hadronic Mass Moments
84
5
10
15
<m2X,true,cut > [(GeV/c2 )2 ]
14
12
0.2 ≤ E miss-|p | < 0.5 GeV
miss
8 ≤ Mult ≤ 20
10
8
6
4
2
5
10
15
<m2X,true,cut > [(GeV/c2) 2]
5
10
15
<m2X,true,cut > [(GeV/c2 )2 ]
< m2X > in bins of MultXc ,
Emiss − |~
pmiss | and p∗` . Shown are the extracted < mX,reco > versus <
mX,true,cut > in bins of mX,true for 0.8 ≤ p∗` < 0.9 GeV/c (•), 1.4 ≤ p∗` <
1.5 GeV/c (◦), and p∗` ≥ 1.9 GeV/c (). The results of ts of linear functions
are overlaid as solid lines. A reference line with < mX,reco >=< mX,true,cut >
Figure 5.23: Examples of calibration curves for
is superimposed as dashed line.
102
102
10
102
10
102
10 4
<mX,true,cut > [(GeV/c2 )4 ]
-0.5 ≤ E miss-|p | < 0.05 GeV
miss
6 ≤ Mult ≤ 7
0.05 ≤ E miss-|p | < 0.2 GeV
miss
6 ≤ Mult ≤ 7
0.2 ≤ E miss-|p | < 0.5 GeV
miss
6 ≤ Mult ≤ 7
102
102
<m4X,reco> [(GeV/c2) 4]
102
10 4
<mX,true,cut > [(GeV/c2) 4]
10
10
102
10
10
10 4
<mX,true,cut > [(GeV/c2) 4]
102
10 4
<mX,true,cut > [(GeV/c2 )4 ]
-0.5 ≤ E miss-|p | < 0.05 GeV
miss
8 ≤ Mult ≤ 20
0.05 ≤ E miss-|p | < 0.2 GeV
miss
8 ≤ Mult ≤ 20
0.2 ≤ E miss-|p | < 0.5 GeV
miss
8 ≤ Mult ≤ 20
10
102
10
102
10 4
<mX,true,cut > [(GeV/c2) 4]
<m4X,reco> [(GeV/c2) 4]
10
10 4
<mX,true,cut > [(GeV/c2) 4]
102
2
<m4X,reco> [(GeV/c2) 4]
2
<m4X,reco> [(GeV/c2) 4]
0.2 ≤ E miss-|p | < 0.5 GeV
miss
1 ≤ Mult ≤ 5
102
10 4
<mX,true,cut > [(GeV/c2) 4]
<m4X,reco> [(GeV/c2) 4]
<m4X,reco> [(GeV/c2) 4]
10
0.05 ≤ E miss-|p | < 0.2 GeV
miss
1 ≤ Mult ≤ 5
85
<m4X,reco> [(GeV/c2 )4 ]
-0.5 ≤ E miss-|p | < 0.05 GeV
miss
1 ≤ Mult ≤ 5
<m4X,reco> [(GeV/c2 )4 ]
<m4X,reco> [(GeV/c2 )4 ]
5.7 Extraction of Moments
102
10
102
10 4
<mX,true,cut > [(GeV/c2) 4]
102
10 4
<mX,true,cut > [(GeV/c2 )4 ]
< m4X > in bins of MultXc ,
Emiss − |~
pmiss |
Shown are the extracted < mX,reco > versus <
mX,true,cut > in bins of mX,true for 0.8 ≤ p∗` < 0.9 GeV/c (•), 1.4 ≤ p∗` <
1.5 GeV/c (◦), and p∗` ≥ 1.9 GeV/c () on a double logarithmic scale. The
Figure 5.24: Examples of calibration curves for
∗
and p` .
results of ts of linear functions are overlaid as solid lines. A reference line
with
< mX,reco >=< mX,true,cut >
is superimposed as dashed line.
3
3
3
6
102
3
10
102
6
0.05 ≤ E miss-|p | < 0.2 GeV
miss
6 ≤ Mult ≤ 7
0.2 ≤ E miss-|p | < 0.5 GeV
miss
1 ≤ Mult ≤ 5
102
10
6
6
<mX,true,cut > [(GeV/c2 ) ]
6
102
<m6X,reco> [(GeV/c2) 6]
10
10
102
10
6
6
<mX,true,cut > [(GeV/c2) ]
3
0.2 ≤ E miss-|p | < 0.5 GeV
miss
6 ≤ Mult ≤ 7
10
102
102
103
<m6X,true,cut > [(GeV/c2) 6]
102
103
<m6X,true,cut > [(GeV/c2) 6]
102
103
<m6X,true,cut > [(GeV/c2 )6 ]
-0.5 ≤ E miss-|p | < 0.05 GeV
miss
8 ≤ Mult ≤ 20
0.05 ≤ E miss-|p | < 0.2 GeV
miss
8 ≤ Mult ≤ 20
0.2 ≤ E miss-|p | < 0.5 GeV
miss
8 ≤ Mult ≤ 20
10
102
102
103
<m6X,true,cut > [(GeV/c2) 6]
3
10
102
6
<m6X,reco> [(GeV/c2) 6]
<m6X,reco> [(GeV/c2) 6]
<m6X,reco> [(GeV/c2) 6]
-0.5 ≤ E miss-|p | < 0.05 GeV
miss
6 ≤ Mult ≤ 7
3
3
102
10
6
6
<mX,true,cut > [(GeV/c2) ]
3
6
102
<mX,reco> [(GeV/c2 ) ]
10
0.05 ≤ E miss-|p | < 0.2 GeV
miss
1 ≤ Mult ≤ 5
<mX,reco> [(GeV/c2) ]
102
3
6
10
-0.5 ≤ E miss-|p | < 0.05 GeV
miss
1 ≤ Mult ≤ 5
<mX,reco> [(GeV/c2) ]
<m6X,reco> [(GeV/c2 )6 ]
3
<m6X,reco> [(GeV/c2 )6 ]
Measurement of Hadronic Mass Moments
86
102
103
<m6X,true,cut > [(GeV/c2) 6]
3
10
102
102
103
<m6X,true,cut > [(GeV/c2 )6 ]
< m6X > in bins of MultXc ,
Emiss − |~
pmiss |
Shown are the extracted < mX,reco > versus <
mX,true,cut > in bins of mX,true for 0.8 ≤ p∗` < 0.9 GeV/c (•), 1.4 ≤ p∗` <
1.5 GeV/c (◦), and p∗` ≥ 1.9 GeV/c () on a double logarithmic scale. The
Figure 5.25: Examples of calibration curves for
∗
and p` .
results of ts of linear functions are overlaid as solid lines. A reference line
with
< mX,reco >=< mX,true,cut >
is superimposed as dashed line.
2.1
2.05
2
1.95
1.9
1.85
1.8
1.75
<m 2X > [(GeV/c2)2]
<m X > [GeV/c 2]
5.7 Extraction of Moments
0.8
1
87
4.6
4.4
4.2
4
3.8
3.6
3.4
3.2
1.2 1.4 1.6 1.8
p * [GeV/c]
0.8
1
10
9.5
9
8.5
8
7.5
7
6.5
6
l,cut
<m 4X > [(GeV/c2)4]
<m 3X > [(GeV/c2)3]
l,cut
0.8
1
24
22
20
18
16
14
12
1.2 1.4 1.6 1.8
p * [GeV/c]
0.8
1
1.2 1.4 1.6 1.8
p * [GeV/c]
l,cut
<m 6X > [(GeV/c2)6]
<m 5X > [(GeV/c2)5]
l,cut
55
50
45
40
35
30
25
20
1.2 1.4 1.6 1.8
p * [GeV/c]
140
120
100
80
60
40
0.8
1
1.2 1.4 1.6 1.8
p * [GeV/c]
l,cut
0.8
1
1.2 1.4 1.6 1.8
p * [GeV/c]
l,cut
Figure 5.26: Moments calculated in MC simulations for dierent cuts on
p∗` .
The plotted moments are: measured uncalibrated moments (), calibrated
moments (•), and true moments after the application of all analysis cuts (◦).
Measurement of Hadronic Mass Moments
88
<m 2X > [(GeV/c2)2]
<m X > [GeV/c 2]
2.3
2.2
2.1
2
1.9
1.8
1
1.5
2
p*
5.5
5
4.5
4
3.5
2.5
3
[GeV/c]
1
1.5
2
1
p*
2.5
3
[GeV/c]
p*
2.5
3
[GeV/c]
l,cut
1.5
2
35
30
25
20
15
10
2.5
3
p * [GeV/c]
1
1.5
2
l,cut
110
100
90
80
70
60
50
40
30
20
l,cut
<m 6X > [(GeV/c2)6]
<m 5X > [(GeV/c2)5]
2.5
3
[GeV/c]
40
<m 4X > [(GeV/c2)4]
<m 3X > [(GeV/c2)3]
l,cut
15
14
13
12
11
10
9
8
7
6
p*
1
1.5
2
p*
2.5
3
[GeV/c]
500
450
400
350
300
250
200
150
100
50
1
1.5
2
l,cut
Figure 5.27: Moments calculated in MC simulations in bins of
l,cut
p∗` .
The plot-
ted moments are: measured uncalibrated moments (), calibrated moments
(•), and true moments after the application of all analysis cuts (◦).
Ctrue
0.985 ± 0.003
0.991 ± 0.003
0.993 ± 0.003
0.996 ± 0.003
0.997 ± 0.003
1.001 ± 0.002
1.004 ± 0.002
1.006 ± 0.002
1.005 ± 0.002
1.004 ± 0.002
1.004 ± 0.002
1.002 ± 0.002
< m4X >
Ccalib
1.011 ± 0.005
1.009 ± 0.005
1.010 ± 0.005
1.006 ± 0.005
1.010 ± 0.005
1.010 ± 0.005
1.011 ± 0.005
1.012 ± 0.006
1.015 ± 0.006
1.015 ± 0.007
1.011 ± 0.008
1.026 ± 0.010
and
Ctrue
0.990 ± 0.006
0.993 ± 0.006
0.998 ± 0.006
0.997 ± 0.006
1.002 ± 0.006
1.008 ± 0.006
1.014 ± 0.006
1.018 ± 0.006
1.020 ± 0.007
1.018 ± 0.008
1.014 ± 0.010
1.032 ± 0.012
Ccalib × Ctrue
0.997 ± 0.002
0.995 ± 0.002
0.994 ± 0.002
0.995 ± 0.002
0.992 ± 0.002
0.990 ± 0.002
0.988 ± 0.002
0.987 ± 0.002
0.985 ± 0.003
0.986 ± 0.003
0.988 ± 0.004
0.983 ± 0.005
Ccalib × Ctrue
Ccalib
0.975 ± 0.004
0.984 ± 0.004
0.988 ± 0.004
0.992 ± 0.004
0.993 ± 0.004
0.999 ± 0.004
1.004 ± 0.003
1.006 ± 0.003
1.005 ± 0.003
1.004 ± 0.003
1.004 ± 0.003
1.002 ± 0.003
Ctrue
1.004 ± 0.001
1.002 ± 0.001
1.001 ± 0.001
1.000 ± 0.001
0.999 ± 0.001
0.998 ± 0.001
0.997 ± 0.001
0.997 ± 0.001
0.997 ± 0.001
0.997 ± 0.001
0.997 ± 0.001
0.998 ± 0.001
Ctrue
The bias correction factor
1.015 ± 0.008
1.010 ± 0.008
1.010 ± 0.007
1.005 ± 0.007
1.009 ± 0.007
1.009 ± 0.007
1.010 ± 0.007
1.012 ± 0.007
1.015 ± 0.008
1.014 ± 0.008
1.010 ± 0.010
1.031 ± 0.013
< m5X >
Ccalib
0.993 ± 0.002
0.994 ± 0.002
0.993 ± 0.002
0.995 ± 0.002
0.993 ± 0.002
0.992 ± 0.002
0.991 ± 0.002
0.990 ± 0.003
0.988 ± 0.003
0.988 ± 0.003
0.990 ± 0.004
0.984 ± 0.005
< m2X >
Ccalib
0.963 ± 0.006
0.974 ± 0.006
0.980 ± 0.006
0.986 ± 0.005
0.987 ± 0.005
0.995 ± 0.005
1.002 ± 0.005
1.006 ± 0.004
1.005 ± 0.004
1.004 ± 0.004
1.004 ± 0.004
1.001 ± 0.004
Ctrue
1.008 ± 0.002
1.005 ± 0.002
1.003 ± 0.002
1.001 ± 0.002
1.000 ± 0.002
0.998 ± 0.002
0.996 ± 0.002
0.995 ± 0.002
0.996 ± 0.002
0.996 ± 0.002
0.997 ± 0.002
0.998 ± 0.002
Ctrue
1.070 ± 0.009
1.053 ± 0.008
1.037 ± 0.008
1.022 ± 0.008
1.018 ± 0.008
1.018 ± 0.008
1.015 ± 0.008
1.019 ± 0.008
1.019 ± 0.008
1.016 ± 0.009
1.012 ± 0.011
1.036 ± 0.015
Ccalib × Ctrue
0.999 ± 0.003
0.997 ± 0.003
0.994 ± 0.003
0.995 ± 0.003
0.991 ± 0.003
0.988 ± 0.003
0.986 ± 0.003
0.984 ± 0.004
0.982 ± 0.004
0.983 ± 0.005
0.986 ± 0.006
0.977 ± 0.008
Ccalib × Ctrue
account for the remaining bias of the
1.111 ± 0.011
1.081 ± 0.011
1.058 ± 0.010
1.037 ± 0.010
1.031 ± 0.009
1.023 ± 0.009
1.013 ± 0.009
1.013 ± 0.009
1.013 ± 0.009
1.012 ± 0.010
1.008 ± 0.012
1.035 ± 0.015
< m6X >
Ccalib
0.991 ± 0.004
0.992 ± 0.004
0.991 ± 0.004
0.994 ± 0.004
0.991 ± 0.004
0.990 ± 0.004
0.989 ± 0.004
0.989 ± 0.004
0.986 ± 0.004
0.986 ± 0.005
0.989 ± 0.006
0.979 ± 0.008
< m3X >
Ccalib
of
Ccalib
and
Ctrue
see sections 5.7.3.1 and 5.7.3.2.
calibration method and the residual bias introduced by analysis cuts and nal state radiation, respectively. For the denitions
Table 5.9:
0.996 ± 0.005
1.000 ± 0.004
1.003 ± 0.004
1.002 ± 0.004
1.007 ± 0.004
1.011 ± 0.005
1.015 ± 0.005
1.018 ± 0.005
1.020 ± 0.005
1.019 ± 0.006
1.015 ± 0.008
1.029 ± 0.010
Ccalib × Ctrue
0.997 ± 0.001
0.997 ± 0.001
0.996 ± 0.001
0.996 ± 0.001
0.995 ± 0.001
0.994 ± 0.001
0.993 ± 0.001
0.992 ± 0.001
0.991 ± 0.001
0.991 ± 0.002
0.992 ± 0.002
0.990 ± 0.003
Ccalib × Ctrue
Summary of bias corrections.
1.001 ± 0.000
1.000 ± 0.000
1.000 ± 0.000
1.000 ± 0.000
0.999 ± 0.000
0.999 ± 0.000
0.998 ± 0.000
0.998 ± 0.000
0.998 ± 0.000
0.999 ± 0.000
0.999 ± 0.000
0.999 ± 0.000
0.996 ± 0.001
0.996 ± 0.001
0.996 ± 0.001
0.997 ± 0.001
0.995 ± 0.001
0.995 ± 0.001
0.994 ± 0.001
0.994 ± 0.001
0.993 ± 0.002
0.993 ± 0.002
0.993 ± 0.002
0.991 ± 0.003
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Ctrue
< mX >
Ccalib
p∗l,cut
[GeV /c]
5.7 Extraction of Moments
89
Measurement of Hadronic Mass Moments
90
5.7.3.1
Bias Correction of Calibration
As pointed out in section 5.7.2.4 the nal extraction of moments suers from
a bias that is inherent in the applied method itself. Possible sources are the
subtraction of combinatorial background and the calibration of moments,
i.e. small non-linearities of the calibration curves that are not accounted for
in its construction where a linear dependence of measured
true
<
mnX,true,cut
>
< mnX,reco >
and
quantities is assumed. Although, the observed bias is
small compared to the overall size of the calibration it needs to be corrected.
Figure 5.28 shows the dierence between the calibrated and true moments
together with its statistical uncertainty. We observe a signicant deviation
from zero for all moments and an uncommonly larger bias for the moments
< m6X >
with lepton momenta cuts below
p∗` ≥ 1.2 GeV/c.
The extracted moments are corrected by applying a factor that is calculated from the rate of true and calibrated moments,
Ccalib =
The extracted correction factors
< mnX,true,cut >
< mnX,calib >
Ccalib
.
(5.57)
are summarized in table 5.9 and il-
lustrated in gure 5.29. They are typically ranging between
< m6X > we observe biases between 3%
∗
∗
between p` ≥ 0.8 GeV/c and p` ≥ 1.2 GeV/c.
For
5.7.3.2
and
11%
0.4%
and
for the lowest
1.5%.
p∗`,cut
Residual Bias Correction
The calibration curves are constructed utilizing the linear dependence of
measured
< mnX,reco >
and true
< mnX,true,cut >
moments where both
are calculated for the same events after application of the selection criteria. Thus, applying the calibration procedure, measured moments can only
< mnX,true,cut >.
< mnX,true,cut > are aected
be corrected towards
Therefore, we have to check, if the
moments
by an intrinsic bias that might be
introduced by the applied selection criteria. If the applied selection criteria
select dierent exclusive decay channels with dierent relative eciencies the
calculated true moments are biased with respect to moments calculated for
events that are selected only via their true lepton momenta.
The second eect that needs to be accounted for is caused by photons
emitted in nal state radiation. Since the theoretical interpretation will be
based on calculations that do not take into account nal state radiation but
it is present in data as well as in the MC simulations, we need to correct this
eect.
The residual bias correction factor
Ctrue
is calculated from MC simula-
tions alone. It is dened by
Ctrue =
< mnX,true >
< mnX,true,cut >
,
(5.58)
1
l,cut
0.8
1
1.2 1.4 1.6 1.8 2
p * [GeV/c]
l,cut
1.5
1
0.5
0
-0.5
0.8
1
1.2 1.4 1.6 1.8 2
p * [GeV/c]
l,cut
Figure 5.28:
decays.
91
0.1
0.08
0.06
0.04
0.02
0
0.8
1
1.2 1.4 1.6 1.8 2
p * [GeV/c]
l,cut
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
0.8
1
1.2 1.4 1.6 1.8 2
p * [GeV/c]
l,cut
6
3
3
<m X,calib
> - <mX,true,cut
> / (GeV/c 2)
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05
5
5
5
<m X,calib
> - <mX,true,cut
> / (GeV/c 2)
1.2 1.4 1.6 1.8 2
p * [GeV/c]
2
2
<m X,calib
> - <mX,true,cut
> / (GeV/c 2) 2
0.8
4
4
<m X,calib
> - <mX,true,cut
> / (GeV/c 2) 4
6
6
<m X,calib
> - <mX,true,cut
> / (GeV/c 2)
0.03
0.025
0.02
0.015
0.01
0.005
0
-0.005
3
<m X,calib > - <mX,true,cut> / GeV/c 2
5.7 Extraction of Moments
16
14
12
10
8
6
4
2
0
-2
0.8
1
1.2 1.4 1.6 1.8 2
p * [GeV/c]
l,cut
Bias after application of the calibration on simulated signal
Measurement of Hadronic Mass Moments
1.002
1
0.998
0.996
0.994
0.992
0.99
0.988
<m 2X,true,cut> / <m 2X,calib >
<m X,true,cut> / <m X,calib >
92
0.8
1
1
0.995
0.99
0.985
0.98
1.2 1.4 1.6 1.8 2
p * [GeV/c]
0.8
1
1.005
1
0.995
0.99
0.985
0.98
0.975
0.97
0.965
0.96
l,cut
<m 4X,true,cut> / <m 4X,calib >
3
<m 3X,true,cut> / <m X,calib >
l,cut
0.8
1
1.2 1.4 1.6 1.8 2
p * [GeV/c]
1.005
1
0.995
0.99
0.985
0.98
0.975
0.97
0.965
0.96
0.8
1
l,cut
<m 6X,true,cut> / <m X,calib >
0.99
6
5
<m 5X,true,cut> / <m X,calib >
1
0.98
0.97
0.96
0.95
0.8
1
1.2 1.4 1.6 1.8 2
p * [GeV/c]
1.02
1
0.98
0.96
0.94
0.92
0.9
0.88
l,cut
Figure 5.29:
mnX,calib
>.
Rate of true
1.2 1.4 1.6 1.8 2
p * [GeV/c]
l,cut
1.01
1.2 1.4 1.6 1.8 2
p * [GeV/c]
< mnX,true,cut >
0.8
1
1.2 1.4 1.6 1.8 2
p * [GeV/c]
l,cut
and calibrated moments
<
5.7 Extraction of Moments
with
< mnX,true,cut >
93
the true moment calculated after all selection criteria
< mnX,true >
have been applied whereas
∗
only the cut on p` applied.
simulated nal state radiation.
summarized in table 5.9.
is the true moment calculated with
< mnX,true >
The moments
do not include
The extracted correction factors
Ctrue
are
Their deviations from unity are typically of the
size of a few per mille for all moments.
5.7.4 Statistical Uncertainties and Correlations
Statistical uncertainties stemming from the following sources related to the
overall number of signal and background events in data and MC are considered.
1. Moments are extracted from data by calculating the weighted mean,
PNevt
<
where the weights
wi
mnX
wi mnX,calib,i
,
PNevt
i=1 wi
i=1
>=
correspond to the relative amount of signal at
the specic location of the
mX
spectrum. The sums in the nominator
and denominator run over all events. To calculate the variance
V
of
the extracted moments related to the number of observed events, the
following expression is used:
PNevt
V (<
mnX
>) = V
wi mnX,calib,i
PNevt
i=1 wi
!
i=1
1
= P
Nevt
i=1
wi
2 V
N
evt
X
1
= P
Nevt
i=1
N
evt
X
wi
2
!
wi mnX,calib,i
i=1
(5.59)
wi2 V
mnX,calib,i
i=1
PNevt 2
w
= P i=1 i2 V mnX,calib .
Nevt
i=1 wi
Assuming here that the weights
wi
and the calibrated masses
do not have uncertainties themselves.
mnX,calib,i
It should be noted that the
overall statistical variance is given by the variance of the unweighted
mX,calib
distribution multiplied with the ratio of the sum of squared
weights and the squared sum of weights.
2. The statistical uncertainty related to the subtraction of combinatorial and residual background, i.e. the background subtraction factors
wi ,
is estimated by varying the factors
ζsb
used for the rescaling of
Measurement of Hadronic Mass Moments
94
background extracted from the
mES
sideband randomly within their
statistical uncertainties (see sections 5.6.1 and 5.6.2 for details). The
ζsb
are varied in data and MC and all moments are extracted based on
background distribution changed within their statistical uncertainties.
We repeat this procedure 250 times and take the RMS of observed
moment variation as measure for the statistical uncertainty related to
the background subtraction.
3. The statistical uncertainty related to the extraction of the calibration
curves, i.e. the statistical uncertainty in the determination of
mX,calib ,
is estimated by varying all calibration curves randomly within their
statistical uncertainties resulting in a changed set of curves used for
the calibration. We repeat this procedure 250 times and take the RMS
of observed moment variation as measure for the statistical uncertainty.
4. Further on we extract additional contributions related to bias corrections factors
Ccalib
and
Ctrue . Ccalib
and
Ctrue
are given by fractions
of moments (see Eq. 5.57 and 5.58) whose statistical uncertainties are
calculated using Eq. 5.59.
All contributions are added in quadrature to form the total statistical uncertainty.
The obtained statistical uncertainties are summarized in table
5.10. In total the statistical uncertainties rst decrease going to higher cuts
on
p∗`
and start increasing again. This behaviour can be reasoned directly
from equation 5.59.
For higher cuts on
p∗`
and therefore decreasing num-
ber of events the statistical uncertainty is expected to drop since the factor
PNevt
i=1
wi2 /
P
Nevt
i=1
wi
2
decreases, i.e. the statistical uncertainty is expected
to increase for higher cuts on
p∗` .
The second eect contributing arises from
mX spetrum. Since the
∗
spectrum becomes wider for lower cuts on p` the statistical uncertainty is
second term in 5.59, the variance of the measured
expected to increase again.
Since all extracted moments share subsets of events among each other
they are known to be strong correlated.
The overall correlation will be
important for the interpretations of the obtained results. We calculate the
correlation matrix from rst principles. Two arbitrary moments of dierent
orders
k
and
l
from dierent event samples
a
and
c
with
a
being a subset of
5.7 Extraction of Moments
c
95
are dened by the weighted means,
PNa
<
mkX
>a =
< mlX >c =
=
=
k
i=1 wi mX,calib,i
,
PNa
w
i
i=1
PNc
l
w
i=1 i mX,calib,i
PNc
i=1 wi
PNa
PNc
l
l
i=1 wi mX,calib,i +
j=Na +1 wi mX,calib,i
PNc
i=1 wi
PNa
P c
l
< mX >a i=1 wi + < mlX >c N
i=Na +1 wi
.
PNc
i=1
c
The event sample
(5.60)
is splitted here into two independent subsamples. The
covariance of both moments is then given by
C < mkX >a , < mlX >c
=C
< mkX >a ,
PNa
wi C < mkX
w
i=1 i
i=1
= PN
c
PNa
l
i=1 wi + < mX >b
PNc
i=1 wi
>a , < mlX >a
< mlX >a
!
PNc
where we make use of the fact that the subsamples
i=Na +1 wi
a
and
b
wi
(5.61)
are statistically
independent by denition. It follows
C < mkX >a , < mlX >c
!
PNa
PNa
PNa
k
l
w
m
w
m
i
j
w
i=1
j=1
i
X,calib,i
X,calib,j
i=1
= PN
C
,
PNa
PNa
c
w
w
w
i=1 i
i=1 i
j=1 j


Na
Na
X
X
1
= PNc
C
wi mkX,calib,i
wj mlX,calib,j 
PNa
w
w
i=1 i
i=1 i
i=1
j=1
= PNc
1
PNa
= PNc
1
PNa
i=1 wi
i=1 wi
Na X
Na
X
i=1 wi i=1 j=1
Na X
Na
X
i=1 wi i=1 j=1
(5.62)
wi wj C mkX,calib,i , mlX,calib,j
wi wj δij C mkX,calib , mlX,calib .
The last step follows from the fact that two dierent
mkX,calib,i
and
mlX,calib,j
are statistical independent but random samples of the same pdf. Finally, we
Measurement of Hadronic Mass Moments
96
arrive at
C < mkX >a , < mlX >c
PNa 2
w
= PNc i=1PNi a
C mkX,calib , mlX,calib
a
i=1 wi
i=1 wi
PNa 2
D
E
w
= PNc i=1PNi a
mkX,calib − < mkX,calib > mlX,calib − < mlX,calib >
a
i=1 wi
i=1 wi
PNa 2
w
k
l
= PNc i=1PNi a
< mk+l
X >a − < mX >a < mX >a .
i=1 wi
i=1 wi
(5.63)
The corresponding correlations
ρ
are calculated from the covariance by
ρ < mkX >a , < mlX >c
C < mkX >a , < mlX >c
=q
k
l
V < mX >a V < mX >c
sP
Na
k > < ml >
2
< mk+l
a
a
X
i=1 wi
X >a − < m
q
qX
.
= PN
·
c
2
k >2 < m2l > − < ml >2
< m2k
>
−
<
m
i=1 wi
c
a
a
c
X
X
X
X
(5.64)
k and l
k , l, k + l, 2k ,
That is, the calculation of the correlation of two moments of orders
requires the calculation of the unweighted moments of order
and
2l.
The tables A.1 - A.6 summarize all extracted correlation coecients.
They vary between
4%
and
99%.
5.7.5 Verication of the Calibration Procedure
In order to verify the method used for the moment extraction, i.e.
the
calibration and nal bias correction, two cross checks are performed.
1. The calibration is checked on simulated exclusive signal nal states.
2. A complete check of the entire analysis chain is performed with MC
simulation treating a subset of simulated events like data and the remaining events like MC.
The obtained results will be discussed in the following.
5.7.5.1
Calibration of Exclusive Signal Decay Modes
The calibration curves are constructed using the sum of dierent simulated
exclusive nal states. As was already shown in section 5.7.2.4 the calibration
2
<m 2X, calib> [(GeV/c2)2]
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
2.5
3
<m X, true, cut> [GeV/c2]
102
<m 6X, calib > [(GeV/c2)6]
<m 4X, calib > [(GeV/c2)4]
<m X, calib> [GeV/c 2]
5.7 Extraction of Moments
10
9
8
7
6
5
4
3
97
4
6
8
10
<m 2X, true, cut> [(GeV/c2)2]
10
3
102
10
10
<m 4X, true, cut>
3
102
[(GeV/c2)4]
102
10
<m 6X, true, cut> [(GeV/c2)6]
p∗` ≥ 0.8 GeV/c for
6
and < mX > (d). A
Figure 5.30: Calibration of exclusive decays modes with
the moments
< mX >
m4X
<
> (b), <
> (c),
< mX,reco >=< mX,true,cut > is
(a),
reference line complying
m2X
superimposed.
method works reliably, apart from a small remaining bias that needs to be
accounted for.
We check whether the calibration is also able to correct exclusive nal
states back to their true masses by applying the calibration on single signal
decay channels.
The test is performed for the decays
B → D`ν , B →
D∗ `ν , four resonant decays B → D∗∗ `ν , and two non-resonant decays B →
D(∗) π`ν . We use the sample of cocktail MC simulations that is also used for
the construction of the calibration curves.
The gures 5.30 and 5.31 show the obtained results for dierent cuts
p∗` ≥ 0.8 GeV/c and p∗` ≥ 1.9 GeV/c.
mX,reco >=< mX,true,cut > implying a
The dashed line corresponds to
<
perfect calibration. As can be seen
the calibrated moments line up nicely along this line.
5.7.5.2
Calibration of Generic MC Simulations
We check the procedure applied for the extraction of hadronic mass moments
on the sample of generic MC simulations. Thereby, it is split into two even
sized subsets which are used for dierent purposes:
1. The rst half is used like data in the nal analysis, i.e.
only mea-
sured quantities and no truth information is used. Moments are calcu-
2
2.5
3
<m X, true, cut> [GeV/c2]
102
<m 6X, calib > [(GeV/c2)6]
<m X, calib> [GeV/c 2]
<m 4X, calib > [(GeV/c2)4]
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
<m 2X, calib> [(GeV/c2)2]
Measurement of Hadronic Mass Moments
98
10
9
8
7
6
5
4
3
4
6
8
10
<m 2X, true, cut> [(GeV/c2)2]
10
3
102
10
10
<m 4X, true, cut>
3
102
[(GeV/c2)4]
102
10
<m 6X, true, cut> [(GeV/c2)6]
p∗` ≥ 1.9 GeV/c for
6
and < mX > (d). A
Figure 5.31: Calibration of exclusive decays modes with
the moments
< mX >
m4X
<
> (b), <
> (c),
< mX,reco >=< mX,true,cut > is
(a),
reference line complying
m2X
superimposed.
5.8 Results
99
lated by applying the default set of calibration curves constructed on
the independent cocktail MC sample and subtraction of combinatorial
background.
2. The second half is treated like MC in the nal analysis. It is used for
the subtraction of residual background and for the determination of
bias correction factors.
This test allows to check the performance of the analysis procedure with
respect to some important issues:
•
It allows to perform a robust test of the extraction formalism. If the
true moments are reobtained we can deduce that the overall procedure
works reliably stable and delivers self-consistent results.
•
Since we utilize a sample of cocktail MC simulations that provide a
large quantity of clean
Breco
candidates for the construction of the cal-
ibration curves it has to be checked if systematic eects are introduced
in the extraction of moments. If the calibration applied on generic MC
is able to extract the true underlying moments it can be concluded
that systematic eects stemming from the dierence of cocktail and
generic MC are small.
Figure 5.32 shows the obtained results for all moments and dierent lower
cuts on
p∗` .
Since the dierent moments are highly correlated we extracted
the same quantities also in statistical independent bins of
p∗`
as shown in g-
ure 5.33. Applying the analysis procedure the true moments are reobtained
within
2σ
p∗` ≥ 1.2 GeV/c are
∗
of p` match the true
whereas the moments with higher cuts above
reobtained within
1σ .
All moments extracted in bins
underlying moments within
1.5σ .
Overall, we conclude that the performed
test is able to verify the applied analysis procedure within its statistical
uncertainties.
5.8 Results
< mnX > for n = 1 . . . 6 as functions
∗
of lower cuts on the lepton momentum p` . The moments are shown together
Figure 5.34 shows the extracted moments
with their statistical and systematic uncertainties. In addition all numerical
results are summarized in the table 5.10.
As expected a signicant
p∗`
dependence is observed manifesting in de-
creasing moments for higher cuts on
p∗` .
It is resulting from a reduced con-
tribution of higher mass nal states to the overall
mX
spectrum as depicted
in Fig. 5.20. For the higher lepton momenta cuts above
p∗` ≥ 1.6 GeV/c
a
attening of the measured slope is observed occuring in the region of the
phase space where only few higher mass nal states with masses above those
of the
D∗
meson contribute.
<m X,calib > - <mX,true> / GeV/c 2
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
0.8
1
1.2 1.4 1.6 1.8
p * [GeV/c]
2
2
<m X,calib
> - <mX,true
> / (GeV/c 2) 2
Measurement of Hadronic Mass Moments
100
0.1
0.05
0
-0.05
-0.1
-0.15
0.8
1
3
3
<m X,calib
> - <mX,true
> / (GeV/c 2)
3
0.3
0.2
0.1
-0
-0.1
-0.2
-0.3
-0.4
0.8
1
1.2 1.4 1.6 1.8
p * [GeV/c]
l,cut
4
4
<m X,calib
> - <mX,true
> / (GeV/c 2) 4
l,cut
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0.8
1
l,cut
2
1
0
-1
-2
-3
0.8
1
1.2 1.4 1.6 1.8
p * [GeV/c]
1.2 1.4 1.6 1.8
p * [GeV/c]
l,cut
6
6
6
<m X,calib
> - <mX,true
> / (GeV/c 2)
5
5
<m X,calib
> - <mX,true
> / (GeV/c 2)
5
1.2 1.4 1.6 1.8
p * [GeV/c]
l,cut
8
6
4
2
0
-2
-4
-6
-8
-10
0.8
1
1.2 1.4 1.6 1.8
p * [GeV/c]
l,cut
Figure 5.32: Verication of the analysis procedure on MC simulations for all
moments and dierent lower cuts on
p∗` .
The moments are calculated on a
subset of events while the while the events are used for the subtraction of
residual background and calculation of bias correction factors. Two checks
are performed with the default subdivision () and interchanged samples
(•).
0.06
0.04
0.02
0
-0.02
-0.04
1
1.5
2
2.5
3
p * [GeV/c]
2
2
<m X,calib
> - <mX,true
> / (GeV/c 2) 2
<m X,calib > - <mX,true> / GeV/c 2
5.8 Results
101
0.3
0.2
0.1
0
-0.1
-0.2
1
1.5
2
p*
1.5
1
0.5
0
-0.5
-1
1
1.5
2
p*
2.5
3
[GeV/c]
l,cut
4
4
<m X,calib
> - <mX,true
> / (GeV/c 2) 4
3
3
<m X,calib
> - <mX,true
> / (GeV/c 2)
3
l,cut
2
8
6
4
2
0
-2
-4
1
1.5
2
p*
20
10
0
-10
-20
1
1.5
2
p*
2.5
3
[GeV/c]
6
6
<m X,calib
> - <mX,true
> / (GeV/c 2)
5
5
5
<m X,calib
> - <mX,true
> / (GeV/c 2)
30
2.5
3
[GeV/c]
l,cut
6
l,cut
2.5
3
[GeV/c]
150
100
50
0
-50
1
1.5
2
p*
l,cut
2.5
3
[GeV/c]
l,cut
Figure 5.33: Verication of the analysis procedure on MC simulations for all
moments in bins of
p∗` .
The moments are calculated on a subset of events
while the while the events are used for the subtraction of residual background
and calculation of bias correction factors.
Two checks are performed with
the default subdivision () and interchanged samples (•).
Measurement of Hadronic Mass Moments
102
Correlation coecients for all moments measured are summarized in tables A.1 - A.6 in the appendix. We obtain correlations ranging between
and
99%.
4%
5.8 Results
<m 2X > [(GeV/c2)2]
<m X > [GeV/c 2]
2.1
2.08
2.06
2.04
2.02
2
0.8
1
103
4.5
4.4
4.3
4.2
4.1
4
1.2 1.4 1.6 1.8
p * [GeV/c]
0.8
1
l,cut
1.2 1.4 1.6 1.8
p * [GeV/c]
l,cut
<m 4X > [(GeV/c2)4]
<m 3X > [(GeV/c2)3]
10
9.5
9
8.5
8
0.8
1
1.2 1.4 1.6 1.8
p * [GeV/c]
22
21
20
19
18
17
16
0.8
1
l,cut
<m 6X > [(GeV/c2)6]
<m 5X > [(GeV/c2)5]
l,cut
54
52
50
48
46
44
42
40
38
36
34
32
0.8
1
1.2 1.4 1.6 1.8
p * [GeV/c]
1.2 1.4 1.6 1.8
p * [GeV/c]
130
120
110
100
90
80
70
0.8
1
l,cut
Figure 5.34: Moments calculated for dierent cuts on
1.2 1.4 1.6 1.8
p * [GeV/c]
l,cut
p∗` .
The inner error bars
correspond to the statistical uncertainties while the full error bars correspond
to the square sum of statistical and systematic uncertainties.
Measurement of Hadronic Mass Moments
104
pl,cut
[GeV /c]
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
< mX > [GeV /c2 ]
< m2X > [(GeV /c2 )2 ]
2.0951 ± 0.0086 ± 0.0119
2.0912 ± 0.0074 ± 0.0097
2.0860 ± 0.0072 ± 0.0093
2.0785 ± 0.0073 ± 0.0081
2.0708 ± 0.0074 ± 0.0095
2.0653 ± 0.0077 ± 0.0121
2.0571 ± 0.0080 ± 0.0118
2.0500 ± 0.0080 ± 0.0117
2.0410 ± 0.0089 ± 0.0115
2.0349 ± 0.0106 ± 0.0161
2.0357 ± 0.0138 ± 0.0170
2.0325 ± 0.0176 ± 0.0230
4.455 ± 0.041 ± 0.058
4.429 ± 0.035 ± 0.043
4.404 ± 0.032 ± 0.041
4.365 ± 0.031 ± 0.034
4.327 ± 0.031 ± 0.039
4.297 ± 0.031 ± 0.049
4.260 ± 0.032 ± 0.046
4.224 ± 0.031 ± 0.043
4.178 ± 0.034 ± 0.039
4.145 ± 0.039 ± 0.055
4.142 ± 0.050 ± 0.058
4.120 ± 0.064 ± 0.079
< m3X > [(GeV /c2 )3 ]
< m4X > [(GeV /c2 )4 ]
9.67 ± 0.15 ± 0.23
9.54 ± 0.12 ± 0.16
9.45 ± 0.11 ± 0.15
9.29 ± 0.10 ± 0.12
9.16 ± 0.10 ± 0.13
9.04 ± 0.10 ± 0.16
8.91 ± 0.10 ± 0.14
8.77 ± 0.09 ± 0.12
8.60 ± 0.10 ± 0.10
8.48 ± 0.11 ± 0.14
8.44 ± 0.15 ± 0.15
8.34 ± 0.18 ± 0.21
21.60 ± 0.55 ± 0.81
21.00 ± 0.41 ± 0.56
20.68 ± 0.37 ± 0.51
20.15 ± 0.32 ± 0.40
19.70 ± 0.31 ± 0.40
19.29 ± 0.29 ± 0.47
18.89 ± 0.29 ± 0.40
18.41 ± 0.27 ± 0.33
17.83 ± 0.27 ± 0.25
17.43 ± 0.30 ± 0.35
17.27 ± 0.38 ± 0.36
16.93 ± 0.46 ± 0.50
< m5X > [(GeV /c2 )5 ]
< m6X > [(GeV /c2 )6 ]
49.93 ± 1.95 ± 2.87
47.44 ± 1.40 ± 1.97
46.40 ± 1.18 ± 1.75
44.64 ± 1.00 ± 1.34
43.26 ± 0.92 ± 1.22
41.91 ± 0.84 ± 1.37
40.72 ± 0.83 ± 1.14
39.12 ± 0.73 ± 0.89
37.33 ± 0.72 ± 0.59
36.12 ± 0.79 ± 0.83
35.52 ± 0.97 ± 0.83
34.44 ± 1.13 ± 1.14
120.04 ± 6.78 ± 11.89
109.85 ± 4.58 ± 8.37
106.58 ± 3.73 ± 7.07
100.73 ± 3.13 ± 5.29
96.95 ± 2.65 ± 3.98
92.83 ± 2.41 ± 3.88
89.38 ± 2.31 ± 3.05
84.32 ± 1.98 ± 2.48
79.06 ± 1.88 ± 1.56
75.53 ± 2.01 ± 1.98
73.58 ± 2.39 ± 1.91
70.37 ± 2.71 ± 2.59
Table 5.10: Summary of extracted moments
< mnX >
The uncertainties given are statistical and systematic.
.
as function of
p∗`,cut .
Chapter 6
Systematic Studies
In the following section several potential sources of systematic uncertainties
will be discussed. We will focus on studies associated with the calibration
procedure, like the bias corrections (6.2) and the chosen calibration curve
binning (6.3), the model used for the simulation of signal decays (6.4), residual and combinatorial background subtraction (6.5), eects stemming from
mis modeling in MC like track and photon selection eciencies (6.6, 6.7),
hard and soft photon emission (6.8, 6.9), as well as checks testing the stability of the extracted results (6.10). All determined systematic uncertainties
are summarized in tables 6.1 - 6.6 at the end of this chapter.
6.1 General Procedure
If not stated otherwise the following procedure is followed in the determination of systematic uncertainties:
1. Starting from the nominal analysis procedure, corrections to MC are
applied to the quantities in question.
2. Reapplying the extraction procedure to the changed setup, new moments are calculated.
3. The observed dierence with respect to the nominal value is taken as
systematic uncertainty.
The described procedure is repeated for every source of systematic uncertainty. The total systematic uncertainty is calculated as the square sum of
all of the individual uncertainties
6.2 Bias of Calibration
As discussed in section 5.7.3.1 the calibration method is aected by a small
bias.
The extracted moments are corrected for this bias.
The statistical
105
Systematic Studies
106
uncertainty of the determined bias correction is added to the statistical error.
In addition half of the overall bias correction is taken as systematic
uncertainty.
6.3 Binning of Calibration Curves
Resolution and bias of the measured hadronic mass
mX,reco
varies with the
multiplicity of the hadronic system and the missing energy and momentum
Emiss − |~
pmiss |.
This eect is regarded to by the construction of the cal-
ibration curves binned in these quantities.
and
Emiss − |~
pmiss |
Dierences in the multiplicity
distributions between data and MC might result in the
migration of events between bins. To estimate such eects we perform a rebinning of the calibration curves. The nominal calibration curves are binned
Emiss − |~
pmiss | ([−0.5, 0.05, 0.2, 0.5] GeV).
This default 3 × 3 binning is changed into a 2 × 2 binning. We choose
[1, 6, 50] for the multiplicity binning and [−0.5, 0.125, 0.5] GeV for the binning
in Emiss − |~
pmiss |. We repeat the measurement with the modied calibration
in multiplicity ([1, 5, 7, 50]) and
curves and take the observed dierence as systematic uncertainty.
6.4 Simulation Model of Signal Decays
The analysis technique applied to measure the moments of the invariant
hadronic mass distribution relies in important parts on the usage of MC
simulations. For the extraction of moments the dependence exists especially
with respect to the model utilized to simulate signal decays. Thereby, especially high mass contributions to the full semileptonic decay width apart
from
B → D`ν
and
B → D∗ `ν
decays are scarcely known. The simulation
B → D∗∗ `ν with four exclusive channels
B → D(∗) π`ν nal states with also four exclu-
contains decays to excited states
and decays into non-resonant
sive channels. The systematic uncertainty connected with the chosen signal
model is evaluated by changing its composition, i.e. dropping one or more
exclusive channels. We study the simulation model dependence of the construction of calibration curves and of the residual bias correction.
6.4.1 Simulation Model Dependence of Calibration
The systematic uncertainty of the calibration method with respect to the
chosen signal model is checked by applying the following two procedures
described below:
1. The compostion of the signal decay distribution is changed by dropping
single exclusive modes in the construction of the calibration curves.
Thereby, we get eight dierent sets of calibration curves to be used in
the moment extraction.
6.5 Background Subtraction
107
2. We check the dependence of the calibration method on the simulation
of the high mass tail by cutting out events above certain thresholds
mX,true ≥ 2.5, 2.7, 2.9, 3.2, 3.5, 4.0 GeV/c2
corresponding to the highest
bins in the calibration curve construction. By doing so the high mass
dependence of the calibration curve is changed.
Overall we construct ten dierent sets of calibration curves with signal decay
models deviating from the nominal model. Figure 6.1 shows distributions of
moments under variation of the signal model for dierent cuts
p∗`,cut .
The
RMS of the observed variations is taken as measure for the systematic uncertainty.
6.4.2 Model Dependence of Residual Bias Correction
The simulation model used in the residual bias correction is changed by
simultaneously dropping one or more exclusive modes,
D(∗) π`ν , in enumerator and denominator of
Ctrue =
B → D∗∗ `ν
and
B→
<mn
X,true >
. To keep
<mn
X,true,cut >
the total semileptonic branching fraction constant the branching fractions of
the remaining high mass channels are scaled up to compensate the dropped
modes.
The observed variations of the residual bias correction factors are
shown in Fig. 6.2. We take the RMS of the observed variations as systematic
uncertainties.
6.5 Background Subtraction
6.5.1 Combinatorial Background Subtraction
The shape of the combinatorial background distribution is determined in the
5.21 ≤ mES,sb ≤ 5.255 GeV/c2 .
We vary the
upper and lower bounds of the sideband window denition by
±2.5 MeV/c2
mES
sideband region dened as
separately and concurrently to study the eect of signal events leaking from
the signal region into the sideband region. The size of the performed variation corresponds to the usual
σ of the signal peak in the mES distribution and
is chosen to be relatively small to avoid additional statistical uctuations.
The RMS of the observed variations with respect to the nominal value
is taken as measure for possible systematic deviations. Since the observed
variation is small and fully covered by the statical uncertainty on the background subtraction we interpret it as a statistical uctuation and do not
consider it as an additional systematic uncertainty.
6.5.2 Residual Background Subtraction
The subtraction of residual background requires the simulation of several decays channels. The branching fractions of those simulated decays are rescaled
to meet experimental measurements. The applied procedure is described in
<(<m X >)> = 2.0962 GeV/c
RMS(<m X >) = 0.0007 GeV/c2
2.095
2.1
<m X > [GeV/c2]
5
2
<(<m X >)> = 2.0716 GeV/c
RMS(<m X >) = 0.0009 GeV/c2
4
3
2
1
2.065
7
6
5
2.075
<m X > [GeV/c2]
2
<(<m X >)> = 2.0428 GeV/c
RMS(<m X >) = 0.0010 GeV/c2
4
3
2
1
0
2.035
8
7
6
5
4
3
2
1
0
2.07
2.04
2.045
2.05
<m X > [GeV/c2]
2
<(<m X >)> = 2.0356 GeV/c
RMS(<m X >) = 0.0020 GeV/c2
2.025
2.03
entries / 0.011 (GeV/c 2) 4
2.09
entries / 0.011 (GeV/c 2) 4
entries / 0.233 MeV/c 2
2
4
3
2
1
0
0
entries / 0.233 MeV/c 2
2.035 2.04
<m X > [GeV/c2]
entries / 0.011 (GeV/c 2) 4
entries / 0.233 MeV/c 2
entries / 0.233 MeV/c 2
7
6
5
entries / 0.011 (GeV/c2) 4
Systematic Studies
108
3.5
3
2.5
2
1.5
1
0.5
0
3.5
3
2.5
2
1.5
1
0.5
0
7
6
5
<(<m 4X>)> = 21.5683 (GeV/c2 )4
RMS(<m 4X>) = 0.0871 (GeV/c2 )4
21.2
21.6 21.8 22
<m 4X > [(GeV/c2)4]
<(<m 4X>)> = 19.6519 (GeV/c2 )4
RMS(<m 4X>) = 0.1062 (GeV/c2 )4
19.5
20
4
<m X > [(GeV/c2)4]
<(<m 4X>)> = 17.8486 (GeV/c2 )4
RMS(<m 4X>) = 0.0498 (GeV/c2 )4
4
3
2
1
0
17.4
8
7
6
5
4
3
2
1
0
21.4
17.6
17.8
18
18.2
<m 4X > [(GeV/c2)4]
<(<m 4X>)> = 16.9775 (GeV/c2 )4
RMS(<m 4X>) = 0.0509 (GeV/c2 )4
16.6
16.8
17
17.2
<m 4X > [(GeV/c2)4]
Variation of measured moments < mX > (left column) and
< m4X > (right column) under variation of the underlying signal model used
Figure 6.1:
to construct the calibration curves. The arrow indicates the nominal result.
The histograms are: model changed by dropping single exclusive channels
(), model changed by cutting out the high mass tail ().
20
18
16
14
12
10
8
6
4
2
0
25
20
p∗
ℓ ≥ 0.8 GeV/c
0.99 0.995
entries / 0.0004
<Ctrue> = 1.0008
RMS(C true) = 0.0021
1
1.005
Ctrue
p∗
ℓ ≥ 1.2 GeV/c
<Ctrue> = 0.9991
RMS(C true) = 0.0016
0.99
entries / 0.0004
20
18
16
14
12
10
8
6
4
2
0
0.995
1
1.005
Ctrue
p∗
ℓ ≥ 1.6 GeV/c
<Ctrue> = 0.9983
RMS(C true) = 0.0010
15
10
5
0
0.99
0.995
8
7
6
5
4
3
2
1
0
14
12
10
8
6
4
2
0
109
p∗
ℓ ≥ 0.8 GeV/c
<Ctrue> = 1.0127
RMS(C true) = 0.0137
0.95
1
1.05
Ctrue
p∗
ℓ ≥ 1.2 GeV/c
<Ctrue> = 1.0013
RMS(C true) = 0.0103
0.95
1
Ctrue
entries / 0.0004
entries / 0.0004
entries / 0.0004
1
8
7
6
5
4
3
2
1
0
p∗
ℓ ≥ 1.6 GeV/c
<Ctrue> = 0.9945
RMS(C true) = 0.0061
0.96
0.98
1
Ctrue
50
40
30
10
p∗
ℓ ≥ 1.9 GeV/c
<Ctrue> = 0.9991
RMS(C true) = 0.0005
entries / 0.0004
(a)
(b)
()
(d)
(e)
(f )
(g)
(h)
(i)
(j)
(k)
entries / 0.0004
PSfrag replaements
entries / 0.0004
6.5 Background Subtraction
20
10
0
0.995
8
6
p∗
ℓ ≥ 1.9 GeV/c
<Ctrue> = 0.9978
RMS(C true) = 0.0031
4
2
0
1
0.98
1
Ctrue
Ctrue
Figure 6.2: Variation of the residual bias correction factor
sured moments
< mX >
m4X
<
B → D∗∗ `ν
(left column) and
variation of the underlying model of
1.02
Ctrue
The arrow indicates the nominal result.
>
and
Ctrue
for the mea-
(right column) under
B → D(∗) π`ν
decays.
Systematic Studies
110
detail in section 5.6.2. Table 5.7 summarizes the applied correction factors
along with their experimental uncertainties aecting the background estimate. In order to evaluate the resulting systematic uncertainty the MC is
adjusted in a way that the branching fraction of the decay type studied
matches the measured scaling factor plus or minus its experimental uncertainty. The average deviation from the nominal value is taken as systematic
uncertainty.
The errors determined for the several background types are added in
quadrature to obtain the full uncertainty on the residual background subtraction. A clear
p∗`,cut
dependence is observed. While for low
p∗`,cut
most of
the studied background channels contribute to the systematic uncertainty it
is purely dominated by background stemming from
high
p∗`,cut .
Contributions from
J/ψ
and
ψ(2S)
B → Xu `ν
decays for
decays are found to be neg-
ligible. The full uncertainty stays constant over a large range of
∗
raises slightly for highest cuts on p` .
p∗`,cut
and
6.6 Track Selection Eciency
MC simulations are corrected for dierences in the selection eciency of
charged tracks between data and MC. We perform a at correction which
wevt,trk =
N
trk
(1 − 0.005)
. The correction of (−0.5 ± 1.2)% is determined on a sam+ − → τ + τ − decays where one τ decays semileptonically and the
ple of e e
weights MC events according to their charged track multiplicity
other into three charged tracks
τ ± → h± h± h∓ ντ
[45]. The correction fac-
tors are determined for dierent standard track denition. Since the track
selection applied in this analysis does not match one of these standard selections but rather lies in between the denitions of GoodTracksLoose
and GoodTracksVeryLoose we add the dierence of the correction factors
0.3%, quadratically as additional uncertainty. The total un1.2%. The systematic uncertainty on the tracking eciency is
for both lists,
certainty is
determined by applying the default tracking eciency correction plus or minus its uncertainty,
wevt,trk = (1 − 0.005 ± 0.012)Ntrk , and taking the average
deviation from the nominal result as systematic uncertainty.
6.7 Photon Selection Eciency
π 0 selection eciencies have been studied on samples of e+ e− → τ + τ − decays
± → e± ν ν and the other into τ ± →
where one of the τ decays leptonically, τ
e τ
ρ± ντ [46]. In the aforementioned study also selection eciencies and their
uncertainties for single photons have been determined. No correction has to
be applied to MC with an uncertainty of
1.8%.
We determine the systematic
uncertainty on the photon selection eciency by reweighting events in MC
according to their photon multiplicity,
wγ = (1 ± 0.018)Nγ ,
and taking the
average deviation from the nominal result as systematic uncertainty.
6.8 Hard Photon Emission
111
6.8 Hard Photon Emission
In the MC simulation nal state radiation is modeled utilizing the
PHOTOS
O(α2 )
package [22]. The simulation takes double photon emission of order
into account.
photons by the
Thereby,
W
PHOTOS
does not simulate the emission of hard
boson or quarks in the semileptonic decay. The eect is
assumed to be small.
The dependence of the moment measurement on hard photons in the
event is checked by rejecting events that contain photons above certain
∗
Eγ,max
, the energy of the highest energetic photon candidate in
∗
Eγ,max
is measured in the BSL restframe. If a sizable amount of
the event.
hard pho-
tons are present in data the measured moments are expected to vary with
dierent cuts on
∗
Eγ,max
.
On the other hand, cutting on
∗
Eγ,max
might in-
troduce a bias into the measurement by varying selection eciencies even
without hard photon emission in the event. Nevertheless, such a bias will
be corrected following the procedure described in sections 5.7.3 resulting in
constant moments with varying
constant moments with varying
∗
Eγ,max
cuts in MC by construction. Thus,
∗
Eγ,max cut are also expected in data if hard
photons are simulated properly.
The study is implemented by performing measurements with several cuts
∗
Eγ,max
= {0.55, 0.60, 0.70, 0.80, 0.90, 1.00, 1.25, 1.50, 2.00} GeV
as illustrated
in Fig. 6.3. Fig. 6.4 shows the observed variation of the extracted moments.
The errors plotted correspond to the statistical errors of the uncorrelated
events with respect to the default event sample. As expected the MC shows
no variation within statistical uncertainties.
from the nominal moment are observed.
In data signicant deviations
The size and direction of those
p∗` cut applied. While variations to higher
∗
moments are observed for low p`,cut , the extracted moments tend to lower
∗
moments for high cuts on p` . This behavior is a hint that the observed eect
deviations depend strongly on the
does not stem from hard photon radiation alone but might also be caused by
other sources of hard photons, like
π0
decays, and related dierences in data
and MC. Nevertheless, the observed eect has to be treated as systematic
uncertainty. We take the RMS of the observed variation in data as systematic
uncertainty. They are depicted as dashed lines in Fig. 6.4.
6.9 Soft Photons
As discussed in section 5.5 even after applying cuts to improve the agreement
of data and MC there remains an excess of low energetic photons in the
simulation. We study its inuence on the moment measurement by applying
tighter cuts in the selection of photons, i.e. the cut on the minimal photon
Eγ∗ > 50 MeV to Eγ∗ > 100 MeV. Since changing the
∗
Eγ cut does also change the measured Emiss and |~
pmiss | spectra the selected
energy is raised from
sample of events is changed and statistical uctuations are expected. In all
Systematic Studies
112
104
entries / 80MeV
entries / 80MeV
104
3
10
102
10
1
10-1 0
0.5
1
Figure 6.3: Spectra of
for cuts
3
102
10
1
10-1 0
1.5
2
2.5
*
Eγ ,max [GeV]
∗
Eγ,max
10
0.5
1
1.5
2
2.5
*
Eγ ,max [GeV]
p∗` ≥ 0.8 GeV/c (a) and p∗` ≥ 1.9 GeV/c
(b). The dashed lines correspond to the cuts applied in the study of hard
phton emission (see sec. 6.8).
cases the observed shifts are well below the statistical errors. Thus, we do
not consider them as additional systematic uncertainty.
6.10 Stability Checks
In order to test the stability of the extracted moments we perform two tests,
variations of the applied cuts on
Emiss − |~
pmiss |
and moment measurements
on disjoined statistical independent data samples.
6.10.1
Emiss − |~
pmiss |
Cut Variation
The stability of the extracted moments as functions of the cut on
|~
pmiss |
is shown in gure 6.5.
measured with changed cuts on
Emiss −
The plots show the dierence of moments
Emiss − |~
pmiss | and the nominal results.
The
errors correspond to the statistical uncertainties of the uncorrelated events
in the changed and default event samples.
Two dierent types of cuts are studied, symmetrical cuts of
|Emiss − |~
pmiss ||
Emiss −
around zero and cuts selecting the negative or positive parts of the
|~
pmiss |
distribution, respectively. The later cuts are motivated by the fact
that the largest dierences in the distributions of data and MC are observed
for negative values of
Emiss − |~
pmiss |.
As shown, signicant deviations from the nominal moment values are observed. The scans versus
Emiss − |~
pmiss | cuts are sensitive to mis modeling of
the simulation. The following systematic eects stemming from sources that
have been discussed so far are expected to contribute to observed variation:
1. Dierent cuts on
Emiss − |~
pmiss |
result in changed calibration curves.
The bias associated with the calibration and its systematic uncertainty
has been discussed in section 5.7.3.1 and 6.2.
does also contribute to the observed variation.
This systematic eect
6.10 Stability Checks
2.088 %$&(')!#"
2.086
2
2.084 RMS = 0.0023 GeV/c
2.082
2.08
2.078
2.076
2.074
2.072
2.07
2.068 0.6 0.8 1 1.2 1.4 1.6 1.8 2
*
Eγ ,max [GeV]
2.04
2.03
2.02
2.01
0.6 0.8 1 1.2 1.4 1.6 1.8 2
*
Eγ ,max [GeV]
*
<m 4X > / (GeV/c2)4
RMS = 0.671 (GeV/c )
23
22.5
22
21.5
<m 4X > / (GeV/c2)4
2 4
18.2
18.1
18
17.9
17.8
17.7
17.6
0.6 0.8 1 1.2 1.4 1.6 1.8 2
*
Eγ ,max [GeV]
%$& *!#"
2 4
RMS = 0.057 (GeV/c )
<m 4X > / (GeV/c2)4
<m 4X > / (GeV/c2)4
23.5
!#"
RMS = 0.0063 GeV/c
2
2
Eγ ,max [GeV]
24
%$& +!#"
<m X > / GeV/c2
<m X > / GeV/c2
<m X > / GeV/c2
<m X > / GeV/c2
!#"
2.125
RMS = 0.0073 GeV/c2
2.12
2.115
2.11
2.105
2.1
2.095
0.6 0.8 1 1.2 1.4 1.6 1.8 2
*
Eγ ,max [GeV]
2.046 %$& *!#"
2.044 RMS = 0.0024 GeV/c2
2.042
2.04
2.038
2.036
2.034
2.032 0.6 0.8 1 1.2 1.4 1.6 1.8 2
113
21.2 %$&(')!#"
2
4
21 RMS = 0.215 (GeV/c )
20.8
20.6
20.4
20.2
20
19.8
19.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2
*
Eγ ,max [GeV]
17.2
17
%$& +!#"
2 4
RMS = 0.119 (GeV/c )
16.8
16.6
16.4
16.2
0.6 0.8 1 1.2 1.4 1.6 1.8 2
*
Eγ ,max [GeV]
0.6 0.8 1 1.2 1.4 1.6 1.8 2
*
Eγ ,max [GeV]
< mX > (left column) and < m4X >
∗
(right column) under variation of Eγ,max for data (•) and MC (◦) together
Figure 6.4: Variation of the moments
with the extracted nominal results (solid lines). The observed RMS in data
is indicated by the dashed line.
Systematic Studies
114
2. Systematic eects stemming from mis modeling in the track and photon
selection eciencies are expected to contribute.
3. To measure a moment with changed
Emiss − |~
pmiss |
cut a new set of
calibration curves with changed binning adapted to the new cut has
to be constructed.
Associated to the chosen binning is a systematic
uncertainty as discussed in 6.3.
The square sum of those systematic uncertainties is overlaid as dotted line
in Fig. 6.5.
In cases where the observed variation exceeds the expected
variation additional systematic eects contribute that need to be addressed
by an extra contribution to the systematic uncertainty. Its size is estimated
from the square dierence of the observed RMS, indicated by a dashed line,
and the total systematic uncertainty stemming from the source discussed
above.
6.10.2 Measurements on Disjoined Data Samples
The stability of the extracted invariant hadronic mass moments is checked
by performing the measurement on statistical independent subsamples. The
following subsamples are considered:
1. Measurements on datasets of dierent
BABAR
run periods (Run 1-3
and Run4) are performed.
Emiss − |~
pmiss | that select
≤ Emiss − |~
pmiss | < 0 GeV) and positive part
(0 ≤ Emiss − |~
pmiss | ≤ 0.5 GeV), respectively.
2. The measurement is repeated with cuts on
the negative tail (−0.5
of the distribution
Breco can0
0
didate, i.e. dierent sources of background and B B oscillations, two
3. In order to study eects associated with the charge of the
samples with charged and neutral candidates are selected.
4. We split the total dataset into two statistical independent samples
containing events with selected electrons and muons.
Thereby, ef-
fects stemming from dierent selection eciencies and misidentication rates are studied.
Fig. 6.6 shows moments measured on these subsamples for dierent cuts on
p∗` .
The extracted moments are compatible with each other within statistical
precision.
Breco
Only for the measurements on samples of charged and neutral
2.5σ is observed. It is resulting from a uctuation
∗
∗
the p` spectrum, 1.5 ≤ p` < 1.6 GeV/c, as illustrated
∗
other moments measured in dierent bins of p` show
tags a deviation of
in a distinct region of
in Fig.
6.7. Since all
no signicant deviation, no additional systematic uncertainty is considered.
miss cut
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
0.2 0.4 0.6 0.8 1
|E - p |
diff. <m4X > [(GeV/c2)4]
miss
0.5
0
-0.5
-1
-1.5
0.2 0.4 0.6 0.8 1
|E - p |
miss
diff. <m6X > [(GeV/c2)6]
miss cut
miss cut
5
0
-5
-10
-15
0.2 0.4 0.6 0.8 1
|E - p |
miss
Figure 6.5:
miss cut
0
-0.02
-0.04
-0.06
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
0.5
1.2
[GeV]
miss cut
1.2
[GeV]
0.2 0.4 0.6 0.8 1
|E - p |
miss
miss cut
1.2
[GeV]
0
-0.5
-1
-1.5
1.2
[GeV]
miss
1.2
[GeV]
115
0.2 0.4 0.6 0.8 1
|E - p |
diff. <m2X > [(GeV/c2)2]
diff. <m2X > [(GeV/c2)2]
miss
0.02
1.2
[GeV]
diff. <m4X > [(GeV/c2)4]
0.2 0.4 0.6 0.8 1
|E - p |
diff. <mX > [GeV/c 2]
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
-0.02
-0.025
0.2 0.4 0.6 0.8 1
|E - p |
miss
diff. <m6X > [(GeV/c2)6]
diff. <mX > [GeV/c 2]
6.10 Stability Checks
4
2
0
-2
-4
-6
-8
miss cut
1.2
[GeV]
0.2 0.4 0.6 0.8 1
|E - p |
miss
miss cut
1.2
[GeV]
Variation of the measured moments as a function of the cut
Emiss − |~
pmiss | for dierent lepton momenta cuts p∗` ≥ 0.9 GeV/c (left
∗
column) and p` ≥ 1.5 GeV/c (right column). Shown is the dierence between measurements with changed and nominal cuts. |Emiss − |~
pmiss || ≤
|Emiss − |~
pmiss ||cut (•) , −0.5 ≤ Emiss − |~
pmiss | < 0 GeV (◦), 0 ≤ Emiss −
|~
pmiss | ≤ 0.5 GeV (). Contributing systematic eects as discussed in section
on
6.10.1 are indicated by the dottedlines. The dashed line shows the oberved
RMS wrt. the nominal result.
Systematic Studies
116
&"' % !
KYNQP^]T
!
KYNQP[Z\T
KV'W KVXNQPSRUT
KMLONQPSRUT
H67 9C;D=I= 4@0 B 9C;>=?= @ 6 J5
4 5687:9<;>=?= 4A@0 B 9C;D=?= F
@ E G ()*,+.-0/
(1)*32
2
2.05
2.1
[GeV/c2]
<mX>
KYNQP^]T
#"$ %
KYNQP[Z\T
KV'W KVXNQPSRUT
KMLONQPSRUT
H67 9C;D=I= 4@0 B 9C;>=?= @ 6 J5
4 5687:9<;>=?= 4A@0 B 9C;D=?= F
@ E G ()*,+.-0/
(1)*32
16
18
20
<m4X>
Figure 6.6:
Comparison of moments
< m1X >
(a) and
22
[(GeV/c2 )4]
< m4X >
(b)
measured on statistical independent subsamples of data. Shown are results
for dierent cuts on the lepton momentum
1.9 GeV/c ()
p∗` ≥ 0.8 GeV/c (◦)
and
p∗` ≥
together with the default measurement (black and red lines).
2
1
0
X
-1
-2
1
Figure 6.7:
(<m 4 > B± - <m 4X >B 0 ) / σ
X
(<m > B± - <m X >B 0 ) / σ
6.11 Summary of Systematic Uncertainties
1.5
2
1
0
-1
-2
-3
2.5
3
*
p [GeV/c]
Comparison of moments
1
< m1X >
p∗` .
1.5
(a) and
measured on disjoined samples of charged and neutral
bins of
117
2
2.5
3
*
p [GeV/c]
< m4X >
Breco
(b)
candidates in
Plotted is the relative dierence between moments extracted on
both samples.
6.11 Summary of Systematic Uncertainties
Tables 6.1 - 6.6 summarize the results of all systematic studies discussed.
Dominating uncertainties are connected to the bias correction of the calibrated moments, the residual background subtraction, and the stability of
the extracted results under variation of the
Emiss − |~
pmiss | cut as well as the
emission of hard photons. For the residual background subtraction especially
B → Xu `ν decays give large contributions also
∗
for high p`,cut . In most cases we nd systematic uncertainties that exceed
∗
the statistical uncertainty by a factor of 1.5. For higher moments and p`,cut
uncertainties stemming from
systematical and statistical uncertainties are of comparable size.
Systematic Studies
118
Sys. error total
Emiss − |~
pmiss |cut stability
B → Xu eν
B (+,0) → D0 → e+
B (+,0) → D̄0 , D− → e−
B (+,0) → D+ → e+
B (+,0) → Ds+ → e+
B (+,0) → τ + → e+
B (+,0) → Ds+ → τ + → e+
B(→ X) → J/Ψ → e+ e−
B(→ X) → Ψ(2S) → e+ e−
MC model calibration
MC model res. bias corr.
Bias correction calibr.
Track selection e.
Photon selection e.
Stat. error (total)
Stat. error (data)
Stat. error (calibration)
Stat. error (bg. subtr)
Moment < mX >
pl,cut [GeV /c]
0.0147
0.0119
0.0044
0.0014
0.0072
0.0000
0.0038
0.0003
0.0005
0.0004
0.0021
0.0002
0.0004
0.0000
0.0000
0.0007
0.0020
0.0042
0.0029
0.0063
0.0086
0.0059
0.0013
0.0061
2.0951
0.8
0.0122
0.0097
0.0041
0.0009
0.0041
0.0024
0.0038
0.0002
0.0003
0.0003
0.0014
0.0002
0.0003
0.0000
0.0000
0.0006
0.0015
0.0040
0.0027
0.0054
0.0074
0.0057
0.0013
0.0045
2.0912
0.9
0.0118
0.0093
0.0040
0.0008
0.0036
0.0025
0.0039
0.0001
0.0002
0.0002
0.0010
0.0002
0.0003
0.0000
0.0000
0.0007
0.0013
0.0043
0.0026
0.0048
0.0072
0.0057
0.0013
0.0042
2.0860
1.0
0.0014
0.0008
0.0109
0.0081
0.0041
0.0010
0.0024
0.0021
0.0040
0.0001
0.0001
0.0001
0.0007
0.0002
0.0002
0.0000
0.0000
0.0008
0.0012
0.0035
0.0023
0.0044
0.0073
0.0057
0.0014
0.0043
2.0785
1.1
0.0012
0.0011
0.0121
0.0095
0.0042
0.0013
0.0022
0.0043
0.0042
0.0000
0.0000
0.0000
0.0005
0.0001
0.0002
0.0000
0.0000
0.0009
0.0012
0.0047
0.0024
0.0042
0.0074
0.0059
0.0015
0.0043
2.0708
1.2
0.0011
0.0044
0.0144
0.0121
0.0045
0.0009
0.0022
0.0082
0.0044
0.0000
0.0000
0.0000
0.0004
0.0002
0.0001
0.0000
0.0000
0.0009
0.0009
0.0053
0.0024
0.0043
0.0077
0.0061
0.0015
0.0044
2.0653
1.3
0.0021
0.0073
0.0143
0.0118
0.0048
0.0003
0.0008
0.0076
0.0048
0.0000
0.0000
0.0000
0.0002
0.0001
0.0001
0.0000
0.0000
0.0009
0.0007
0.0058
0.0024
0.0043
0.0080
0.0065
0.0016
0.0043
2.0571
1.4
0.0022
0.0084
0.0142
0.0117
0.0053
0.0006
0.0025
0.0059
0.0053
0.0000
0.0000
0.0000
0.0001
0.0001
0.0000
0.0000
0.0000
0.0010
0.0006
0.0062
0.0025
0.0047
0.0080
0.0071
0.0017
0.0032
2.0500
1.5
0.0031
0.0138
0.0146
0.0115
0.0060
0.0009
0.0024
0.0000
0.0060
0.0000
0.0000
0.0000
0.0001
0.0001
0.0000
0.0000
0.0000
0.0010
0.0006
0.0075
0.0026
0.0051
0.0089
0.0080
0.0018
0.0035
2.0410
1.6
0.0037
0.0147
0.0193
0.0161
0.0035
0.0092
0.0072
0.0072
0.0000
0.0000
0.0000
0.0001
0.0001
0.0000
0.0000
0.0000
0.0012
0.0007
0.0075
0.0028
0.0066
0.0106
0.0094
0.0019
0.0045
2.0349
1.7
0.0034
0.0235
0.0219
0.0170
0.0035
0.0094
0.0089
0.0089
0.0000
0.0000
0.0000
0.0001
0.0000
0.0000
0.0000
0.0000
0.0015
0.0005
0.0068
0.0030
0.0072
0.0138
0.0119
0.0026
0.0064
2.0357
1.8
0.0020
0.0164
0.0290
0.0230
0.0061
0.0137
0.0118
0.0118
0.0000
0.0000
0.0000
0.0001
0.0000
0.0000
0.0000
0.0000
0.0020
0.0006
0.0092
0.0036
0.0080
0.0176
0.0158
0.0036
0.0068
2.0325
1.9
Res. background total
Calib. curve binning
Hard photon emission
Error total
0.0015
0.0000
GeV/c2 .
0.0014
0.0013
All quantitites are given in units of
0.0020
0.0023
< mX >.
mES sideband denition
Soft photons
Table 6.1: Summary of systematic uncertainties for the measurement of
For the sake of completeness the two systematic studies mES sideband denition and Soft photons are listed but not included
in the total systematic uncertainty. Details can be found in the sections 6.5.1 and 6.9, respectively.
0.013
0.001
0.002
0.001
0.007
0.001
0.002
0.000
0.000
0.013
0.002
0.003
0.002
0.011
0.001
0.002
0.000
0.000
0.018
0.005
0.041
0.000
0.058
0.071
0.011
0.003
B → Xu eν
B (+,0) → D0 → e+
B (+,0) → D̄0 , D− → e−
B (+,0) → D+ → e+
B (+,0) → Ds+ → e+
B (+,0) → τ + → e+
B (+,0) → Ds+ → τ + → e+
B(→ X) → J/Ψ → e+ e−
B(→ X) → Ψ(2S) → e+ e−
Res. background total
Calib. curve binning
Hard photon emission
Emiss − |~
pmiss |cut stability
Sys. error total
Error total
mES sideband denition
0.006
0.001
0.055
0.043
0.015
0.003
0.025
0.007
0.007
0.003
0.052
0.041
0.014
0.003
0.023
0.012
0.013
0.000
0.001
0.001
0.005
0.001
0.001
0.000
0.000
0.005
0.007
0.015
0.010
0.020
0.032
0.025
0.006
0.020
4.404
1.0
0.005
0.006
0.046
0.034
0.014
0.004
0.016
0.009
0.014
0.000
0.000
0.001
0.004
0.001
0.001
0.000
0.000
0.006
0.006
0.011
0.009
0.018
0.031
0.024
0.006
0.019
4.365
1.1
0.004
0.006
0.050
0.039
0.014
0.005
0.013
0.020
0.014
0.000
0.000
0.000
0.002
0.001
0.001
0.000
0.000
0.006
0.007
0.016
0.009
0.017
0.031
0.024
0.006
0.018
4.327
1.2
0.008
0.037
0.053
0.043
0.017
0.001
0.008
0.025
0.017
0.000
0.000
0.000
0.001
0.000
0.000
0.000
0.000
0.005
0.002
0.020
0.009
0.017
0.031
0.028
0.007
0.012
4.224
1.5
0.011
0.056
0.052
0.039
0.020
0.003
0.008
0.000
0.020
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.004
0.003
0.025
0.009
0.019
0.034
0.030
0.007
0.013
4.178
1.6
0.013
0.060
0.067
0.012
0.092
0.076
0.058
0.012
0.033
0.012
0.032
0.055
0.029
0.029
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.006
0.002
0.020
0.011
0.025
0.050
0.043
0.009
0.023
4.142
1.8
0.024
0.024
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.004
0.003
0.024
0.010
0.023
0.039
0.035
0.007
0.017
4.145
1.7
in the total systematic uncertainty. Details can be found in the sections 6.5.1 and 6.9, respectively.
For the sake of completeness the two systematic studies mES sideband denition and Soft photons are listed but not included
( GeV/c2 )2 .
0.007
0.065
0.102
0.079
0.020
0.047
0.039
0.039
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.007
0.003
0.033
0.013
0.028
0.064
0.057
0.013
0.027
4.120
1.9
All quantitites are given in units of
0.008
0.032
0.056
0.046
0.016
0.002
0.004
0.033
0.016
0.000
0.000
0.000
0.001
0.001
0.000
0.000
0.000
0.005
0.003
0.019
0.009
0.016
0.032
0.026
0.007
0.018
4.260
1.4
< m2X >.
0.004
0.020
0.058
0.049
0.015
0.004
0.013
0.036
0.015
0.000
0.000
0.000
0.002
0.001
0.001
0.000
0.000
0.005
0.004
0.018
0.009
0.016
0.031
0.025
0.006
0.018
4.297
1.3
Table 6.2: Summary of systematic uncertainties for the measurement of
Soft photons
0.004
0.008
0.014
0.011
0.023
0.003
0.011
0.016
0.013
0.029
MC model calibration
MC model res. bias corr.
Bias correction calibr.
Track selection e.
Photon selection e.
0.035
0.026
0.006
0.022
4.429
0.041
4.455
0.9
Stat. error (total)
>
0.8
0.027
0.006
0.030
m2X
Stat. error (data)
Stat. error (calibration)
Stat. error (bg. subtr)
Moment <
pl,cut [GeV /c]
6.11 Summary of Systematic Uncertainties
119
Systematic Studies
120
Res. background total
Calib. curve binning
Hard photon emission
−
stability
B → Xu eν
B (+,0) → D0 → e+
B (+,0) → D̄0 , D− → e−
B (+,0) → D+ → e+
B (+,0) → Ds+ → e+
B (+,0) → τ + → e+
B (+,0) → Ds+ → τ + → e+
B(→ X) → J/Ψ → e+ e−
B(→ X) → Ψ(2S) → e+ e−
MC model calibration
MC model res. bias corr.
Bias correction calibr.
Track selection e.
Photon selection e.
Stat. error (total)
Stat. error (data)
Stat. error (calibration)
Stat. error (bg. subtr)
3
Moment < mX
>
pl,cut [GeV /c]
0.23
0.06
0.01
0.17
0.00
0.04
0.01
0.01
0.01
0.04
0.00
0.01
0.00
0.00
0.01
0.05
0.04
0.05
0.11
0.15
0.10
0.02
0.10
9.67
0.8
0.20
0.16
0.05
0.01
0.11
0.00
0.04
0.00
0.01
0.01
0.03
0.00
0.01
0.00
0.00
0.02
0.03
0.04
0.04
0.08
0.12
0.09
0.02
0.07
9.54
0.9
0.18
0.15
0.04
0.01
0.10
0.04
0.04
0.00
0.00
0.00
0.02
0.00
0.00
0.00
0.00
0.03
0.03
0.04
0.03
0.07
0.11
0.09
0.02
0.07
9.45
1.0
0.02
0.03
0.16
0.12
0.04
0.01
0.07
0.03
0.04
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.03
0.02
0.03
0.03
0.06
0.10
0.08
0.02
0.06
9.29
1.1
0.01
0.02
0.16
0.13
0.04
0.02
0.06
0.07
0.04
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.03
0.03
0.04
0.03
0.05
0.10
0.08
0.02
0.06
9.16
1.2
0.01
0.07
0.18
0.16
0.04
0.01
0.06
0.12
0.04
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.02
0.02
0.04
0.03
0.05
0.10
0.08
0.02
0.05
9.04
1.3
0.02
0.10
0.17
0.14
0.04
0.01
0.02
0.11
0.04
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.01
0.05
0.03
0.05
0.10
0.08
0.02
0.05
8.91
1.4
0.02
0.12
0.15
0.12
0.04
0.00
0.02
0.08
0.04
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.01
0.05
0.03
0.05
0.09
0.08
0.02
0.04
8.77
1.5
0.03
0.17
0.14
0.10
0.05
0.01
0.02
0.00
0.05
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.06
0.03
0.05
0.10
0.09
0.02
0.04
8.60
1.6
0.04
0.18
0.18
0.14
0.03
0.09
0.06
0.06
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.06
0.03
0.06
0.11
0.10
0.02
0.05
8.48
1.7
0.03
0.27
0.21
0.15
0.03
0.09
0.07
0.07
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.01
0.05
0.03
0.07
0.15
0.12
0.02
0.08
8.44
1.8
0.02
0.19
0.27
0.21
0.05
0.12
0.10
0.10
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.01
0.09
0.03
0.07
0.18
0.16
0.03
0.08
8.34
1.9
|~
pmiss |cut
Sys. error total
0.27
0.02
0.02
Emiss
Error total
0.02
0.01
All quantitites are given in units of
0.04
0.01
3 >.
< mX
mES sideband denition
Soft photons
Table 6.3: Summary of systematic uncertainties for the measurement of
( GeV/c2 )3 .
For the sake of completeness the two systematic studies mES sideband denition and Soft photons are listed but not included
in the total systematic uncertainty. Details can be found in the sections 6.5.1 and 6.9, respectively.
0.09
0.02
0.02
0.02
0.11
0.01
0.02
0.00
0.00
0.10
0.02
0.04
0.04
0.17
0.01
0.02
0.00
0.00
0.21
0.03
0.64
0.00
0.81
0.98
0.17
0.14
B → Xu eν
B (+,0) → D0 → e+
B (+,0) → D̄0 , D− → e−
B (+,0) → D+ → e+
B (+,0) → Ds+ → e+
B (+,0) → τ + → e+
B (+,0) → Ds+ → τ + → e+
B(→ X) → J/Ψ → e+ e−
B(→ X) → Ψ(2S) → e+ e−
Res. background total
Calib. curve binning
Hard photon emission
Emiss − |~
pmiss |cut stability
Sys. error total
Error total
mES sideband denition
0.08
0.08
0.70
0.56
0.15
0.01
0.42
0.00
0.07
0.07
0.63
0.51
0.12
0.01
0.39
0.09
0.09
0.01
0.01
0.02
0.07
0.01
0.02
0.00
0.00
0.13
0.10
0.10
0.10
0.21
0.37
0.29
0.05
0.22
20.68
1.0
0.05
0.10
0.51
0.40
0.11
0.03
0.28
0.08
0.09
0.00
0.00
0.01
0.05
0.01
0.01
0.00
0.00
0.12
0.08
0.06
0.08
0.17
0.32
0.26
0.05
0.19
20.15
1.1
0.04
0.07
0.50
0.40
0.10
0.04
0.21
0.21
0.09
0.00
0.00
0.00
0.03
0.01
0.01
0.00
0.00
0.11
0.09
0.09
0.08
0.15
0.31
0.25
0.05
0.18
19.70
1.2
0.06
0.31
0.49
0.40
0.10
0.03
0.09
0.32
0.10
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.08
0.03
0.11
0.07
0.13
0.29
0.23
0.05
0.16
0.06
0.33
0.43
0.33
0.11
0.01
0.05
0.24
0.11
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.07
0.03
0.11
0.07
0.13
0.27
0.23
0.05
0.12
18.41
1.5
0.09
0.46
0.37
0.25
0.12
0.01
0.05
0.00
0.12
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.05
0.04
0.14
0.07
0.14
0.27
0.24
0.05
0.11
17.83
1.6
0.09
0.50
0.46
0.07
0.73
0.52
0.36
0.08
0.21
0.07
0.21
0.35
0.17
0.17
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.04
0.02
0.10
0.07
0.17
0.38
0.33
0.06
0.18
17.27
1.8
0.14
0.14
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.04
0.04
0.13
0.07
0.17
0.30
0.27
0.04
0.13
17.43
1.7
in the total systematic uncertainty. Details can be found in the sections 6.5.1 and 6.9, respectively.
For the sake of completeness the two systematic studies mES sideband denition and Soft photons are listed but not included
( GeV/c2 )4 .
0.05
0.52
0.68
0.50
0.11
0.30
0.23
0.23
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.05
0.02
0.22
0.08
0.18
0.46
0.41
0.08
0.20
16.93
1.9
All quantitites are given in units of
18.89
1.4
< m4X >.
0.04
0.22
0.55
0.47
0.10
0.02
0.20
0.35
0.09
0.00
0.00
0.00
0.02
0.01
0.01
0.00
0.00
0.09
0.05
0.10
0.08
0.14
0.29
0.23
0.06
0.16
19.29
1.3
Table 6.4: Summary of systematic uncertainties for the measurement of
Soft photons
0.11
0.12
0.10
0.12
0.27
0.09
0.16
0.12
0.15
0.36
MC model calibration
MC model res. bias corr.
Bias correction calibr.
Track selection e.
Photon selection e.
0.41
0.32
0.05
0.25
21.00
0.55
21.60
0.9
Stat. error (total)
>
0.8
0.38
0.06
0.40
m4X
Stat. error (data)
Stat. error (calibration)
Stat. error (bg. subtr)
Moment <
pl,cut [GeV /c]
6.11 Summary of Systematic Uncertainties
121
Systematic Studies
122
Res. background total
Calib. curve binning
Hard photon emission
−
stability
B → Xu eν
B (+,0) → D0 → e+
B (+,0) → D̄0 , D− → e−
B (+,0) → D+ → e+
B (+,0) → Ds+ → e+
B (+,0) → τ + → e+
B (+,0) → Ds+ → τ + → e+
B(→ X) → J/Ψ → e+ e−
B(→ X) → Ψ(2S) → e+ e−
MC model calibration
MC model res. bias corr.
Bias correction calibr.
Track selection e.
Photon selection e.
Stat. error (total)
Stat. error (data)
Stat. error (calibration)
Stat. error (bg. subtr)
5
Moment < mX
>
pl,cut [GeV /c]
2.87
0.71
0.02
2.31
0.00
0.29
0.09
0.16
0.14
0.61
0.02
0.08
0.00
0.00
0.42
0.56
0.37
0.50
1.24
1.95
1.37
0.23
1.37
49.93
0.8
2.42
1.97
0.46
0.00
1.54
0.00
0.24
0.06
0.09
0.07
0.37
0.03
0.07
0.01
0.00
0.49
0.39
0.23
0.36
0.85
1.40
1.11
0.19
0.84
47.44
0.9
2.11
1.75
0.34
0.03
1.41
0.23
0.23
0.02
0.04
0.05
0.23
0.03
0.05
0.00
0.00
0.53
0.33
0.24
0.28
0.63
1.18
0.94
0.18
0.68
46.40
1.0
1.67
1.34
0.29
0.05
1.02
0.22
0.23
0.01
0.02
0.03
0.17
0.03
0.04
0.00
0.00
0.48
0.26
0.10
0.24
0.50
1.00
0.82
0.17
0.54
44.64
1.1
0.11
0.22
1.53
1.22
0.24
0.08
0.72
0.63
0.22
0.01
0.01
0.01
0.09
0.02
0.03
0.00
0.00
0.40
0.30
0.20
0.22
0.42
0.92
0.75
0.15
0.51
43.26
1.2
0.12
0.67
1.61
1.37
0.23
0.05
0.65
1.02
0.22
0.00
0.01
0.00
0.07
0.02
0.02
0.00
0.00
0.32
0.17
0.19
0.20
0.37
0.84
0.69
0.15
0.44
41.91
1.3
0.14
0.89
1.41
1.14
0.23
0.08
0.31
0.92
0.23
0.00
0.00
0.00
0.03
0.01
0.01
0.00
0.00
0.30
0.09
0.21
0.18
0.33
0.83
0.67
0.14
0.47
40.72
1.4
0.17
0.92
1.15
0.89
0.24
0.04
0.12
0.68
0.24
0.00
0.00
0.00
0.02
0.01
0.00
0.00
0.00
0.23
0.09
0.23
0.18
0.34
0.73
0.65
0.14
0.31
39.12
1.5
0.22
1.21
0.93
0.59
0.27
0.02
0.13
0.00
0.27
0.00
0.00
0.00
0.01
0.01
0.00
0.00
0.00
0.15
0.12
0.28
0.17
0.34
0.72
0.64
0.13
0.30
37.33
1.6
0.21
1.30
1.15
0.83
0.17
0.52
0.31
0.31
0.00
0.00
0.00
0.01
0.01
0.00
0.00
0.00
0.11
0.13
0.25
0.17
0.41
0.79
0.70
0.11
0.36
36.12
1.7
0.17
1.88
1.28
0.83
0.21
0.49
0.39
0.39
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.11
0.07
0.18
0.18
0.42
0.97
0.83
0.15
0.48
35.52
1.8
0.11
1.31
1.61
1.14
0.24
0.68
0.52
0.52
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.12
0.05
0.51
0.19
0.43
1.13
1.00
0.21
0.49
34.44
1.9
|~
pmiss |cut
Sys. error total
3.47
0.15
0.36
Emiss
Error total
0.25
0.26
( GeV/c2 )5 .
0.30
0.37
All quantitites are given in units of
0.61
0.76
5 >.
< mX
mES sideband denition
Soft photons
Table 6.5: Summary of systematic uncertainties for the measurement of
For the sake of completeness the two systematic studies mES sideband denition and Soft photons are listed but not included
in the total systematic uncertainty. Details can be found in the sections 6.5.1 and 6.9, respectively.
0.56
0.20
0.29
0.24
1.21
0.10
0.20
0.02
0.00
0.76
0.31
0.54
0.49
2.04
0.04
0.24
0.01
0.00
2.33
0.13
8.22
0.00
11.89
13.69
2.04
4.97
B → Xu eν
B (+,0) → D0 → e+
B (+,0) → D̄0 , D− → e−
B (+,0) → D+ → e+
B (+,0) → Ds+ → e+
B (+,0) → τ + → e+
B (+,0) → Ds+ → τ + → e+
B(→ X) → J/Ψ → e+ e−
B(→ X) → Ψ(2S) → e+ e−
Res. background total
Calib. curve binning
Hard photon emission
Emiss − |~
pmiss |cut stability
Sys. error total
Error total
mES sideband denition
1.01
2.46
9.55
8.37
1.42
0.10
5.64
0.00
0.77
1.29
7.99
7.07
0.96
0.10
5.08
0.00
0.53
0.07
0.13
0.18
0.75
0.08
0.16
0.00
0.00
3.10
0.96
2.92
0.76
1.89
3.73
3.04
0.62
2.07
106.58
1.0
0.46
1.30
6.15
5.29
0.77
0.11
3.79
0.00
0.52
0.04
0.05
0.09
0.53
0.07
0.13
0.01
0.00
2.59
0.77
1.79
0.65
1.45
3.13
2.56
0.58
1.69
100.73
1.1
0.34
0.63
4.78
3.98
0.58
0.08
2.43
0.75
0.48
0.02
0.05
0.04
0.30
0.04
0.08
0.01
0.00
2.07
0.96
1.46
0.58
1.15
2.65
2.26
0.58
1.27
96.95
1.2
0.43
2.42
3.17
2.48
0.55
0.05
0.36
1.91
0.54
0.00
0.01
0.01
0.05
0.03
0.01
0.00
0.00
0.87
0.29
0.54
0.43
0.86
1.98
1.77
0.48
0.73
84.32
1.5
0.54
3.10
2.44
1.56
0.60
0.03
0.35
0.66
0.60
0.00
0.00
0.00
0.04
0.03
0.01
0.00
0.00
0.50
0.37
0.52
0.42
0.83
1.88
1.69
0.37
0.72
79.06
1.6
0.49
3.30
2.82
0.36
4.71
3.06
1.91
0.51
1.13
0.40
1.29
1.98
0.87
0.87
0.00
0.00
0.00
0.03
0.01
0.00
0.00
0.00
0.30
0.21
0.30
0.41
0.99
2.39
2.07
0.34
1.14
73.58
1.8
0.70
0.70
0.00
0.00
0.00
0.02
0.02
0.00
0.00
0.00
0.32
0.40
0.46
0.41
0.98
2.01
1.78
0.28
0.88
75.53
1.7
All quantitites are given in units of
0.39
2.40
3.83
3.05
0.53
0.07
1.03
2.29
0.52
0.00
0.01
0.00
0.10
0.03
0.02
0.00
0.00
1.18
0.29
0.58
0.47
0.83
2.31
1.90
0.52
1.21
89.38
1.4
< m6X >.
0.42
1.83
4.57
3.88
0.54
0.01
2.09
2.29
0.49
0.00
0.02
0.01
0.21
0.06
0.07
0.01
0.00
1.60
0.55
1.06
0.52
0.95
2.41
2.02
0.56
1.18
92.83
1.3
( GeV/c2 )6 .
0.27
3.25
3.75
2.59
0.52
1.56
1.15
1.15
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.33
0.14
1.18
0.44
0.99
2.71
2.41
0.44
1.17
70.37
1.9
in the total systematic uncertainty. Details can be found in the sections 6.5.1 and 6.9, respectively.
For the sake of completeness the two systematic studies mES sideband denition and Soft photons are listed but not included
Table 6.6: Summary of systematic uncertainties for the measurement of
Soft photons
3.09
1.22
4.10
1.03
2.70
3.09
1.83
6.02
1.51
4.13
MC model calibration
MC model res. bias corr.
Bias correction calibr.
Track selection e.
Photon selection e.
4.58
3.67
0.69
2.66
109.85
6.78
120.04
0.9
Stat. error (total)
>
0.8
4.71
0.89
4.79
m6X
Stat. error (data)
Stat. error (calibration)
Stat. error (bg. subtr)
Moment <
pl,cut [GeV /c]
6.11 Summary of Systematic Uncertainties
123
Chapter 7
Interpretation of Results and
Extraction of
|Vcb|
In the following section the obtained results will be interpreted in the context
of the
Heavy Quark Expansion
(HQE) in the kinetic scheme. An introduc-
tion to the HQE in the kinetic scheme can be found in section 2.5.2.
We
|Vcb |, the quark masses mb and
mc , the total semileptonic branching fraction B(B → Xc `ν), as well as the
2
2
3
3
four non-perturbative HQE parameters µπ , µG , ρLS , and ρD in a global t
extract results for the CKM-Matrix element
combining the presented results with additional measurements of moments of
< E`n >, in decays B → Xc `ν and moments of
n
spectrum, < Eγ >, in decays B → Xs γ . The performed
the lepton energy spectrum,
the photon energy
t is closely related to the work presented in [11]. It is repeated by replacing
the previous
BABAR measurement of invariant hadronic mass moments [47]
with the moments extracted in this analysis.
The implemented tting procedure will be described in section 7.1. Section 7.2 will summarize the experimental input used in the combined t
followed by a discussion of the obtained results as well as a comparison with
other measurements in section 7.3.
7.1 Extraction Formalism
The utilized extraction formalism is based on a
χ2
minimization technique,
T
~ exp − M
~ HQE
~ exp − M
~ HQE ,
χ2 = M
(Cexp + CHQE )−1 M
(7.1)
to determine the best t of the HQE predictions to the measurements. The
vectors
~ exp
M
and
~ HQE
M
contain the measured moments included in the t
and the corresponding predicted moments, respectively.
Furthermore, the
expression in equation 7.1 contains the sum of the experimental,
theoretical,
CHQE ,
Cexp ,
and
covariance matrices.
125
Interpretation of Results and Extraction of |Vcb |
126
The total semileptonic branching fraction,
B(B → Xc `ν),
is extracted in
Bp∗`,cut (B →
∗
p`,cut GeV/c to the full lepton energy spectrum. Using
the t by extrapolating measured partial branching-fractions,
Xc `ν), with p∗`
≥
HQE predictions of the relative decay fraction
R
Rp∗`,cut =
dΓSL
∗
p∗`,cut dE`∗ dE`
,
R dΓSL
∗
dE
0 dE`∗
(7.2)
`
the total branching fraction can be introduced as a free parameter in the t.
It is given by
B(B → Xc `ν) =
Bp∗`,cut (B → Xc `ν)
Rp∗`,cut
.
(7.3)
The total branching fraction can be used together with the average
lifetime
τB
B
meson
to calculate the total semileptonic rate,
ΓSL =
propotional to
|Vcb |2 .
B(B → Xc `ν)
∝ |Vcb |2 ,
τB
Finally,
|Vcb |
(7.4)
can be extracted by means of a HQE
prediction for the total semileptonic rate and introducing an additional free
parameter to the t.
ρ3LS are estimated from B -B ∗
2
2
mass splitting and heavy-quark sum rules to be µG = (0.35 ± 0.07) GeV and
3
3
ρLS = (−0.15 ± 0.10) GeV [11], respectively. Both parameters are restricted
The non-perturbative parameters
µ2G
and
in the t by imposing Gaussian error constraints.
As discussed in [11] and specied in [14] the following theoretical uncertainties are taken into account:
1. The uncertainty related to the uncalculated perturbative corrections
to the Wilson coecients of nonperturbative operators is assumed to
be
20%
due to
µ2π
and
µ2G
and
30%
due to
ρ3D
and
ρ3LS .
It is esti-
mated by varying the corresponding parameters accordingly around
µ2π = 0.4 GeV2 , µ2G = 0.35 GeV2 ,
= −0.15 GeV3 .
theor theoretically expected values
ρ3D
= 0.2 GeV
3
3
, and ρLS
2. Theoretical uncertainties for the perturbative corrections are estimated
by varying
αs
by
0.1
0.04
αs = 0.22.
for the hadronic mass moments and by
the lepton energy moments around its nominal value
for
3. Theoretical uncertainties in the perturbative corrections of the quark
mb
and
mc
are addressed by varying both by
2
nominal values of 4.6 GeV/c and
4. For the extracted value of
20 MeV/c2
around their
1.18 GeV/c2 , respectively.
|Vcb | and additional theoretical error of 1.4%
is added for the uncertainty in the expansion of the semileptonic rate
7.2 Experimental Input
ΓSL
127
[15, 48]. It accounts for remaining uncertainties in the perturba-
tive corrections to the leading operator
bb,
uncalculated perturbative
corrections to the chromomagnetic and Darwin operator, higher order
power corrections, and possible nonperturbative eects in the operators with charm elds. This uncertainty will not be included into the
theoretical covariance matrix
nal t result of
CHQE
but will be added separately to the
|Vcb |.
5. For the predicted photon energy moments
< Eγn >, additional theoret-
ical uncertainties are taken into account. As outlined in [16] additional
uncertainties of
30% of the applied bias correction as well as half the dif-
ference in the moments derived from two dierent distribution-function
ansätze have to be considered.
The theoretical covariance matrix
CHQE
is constructed by assuming fully
correlated theoretical uncertainties for a given moment with dierent lepton
momentum or photon energy cuto and assuming uncorrelated theoretical
uncertainties for moments of dierent orders and types.
Theoretical expressions for predictions of hadronic mass, lepton energy,
and photon energy moments in the kinetic scheme are calculated in [14, 16].
Calculations for the total semileptonic rate can be found in [15]. A discussion
of perturbative corrections to the hadronic mass moments can be found in
[49, 13, 50].
7.2 Experimental Input
The performed global t combines measurements of moments of the invariant
hadronic mass,
< mnX >,
< E`n >, in decays
n
spectrum, < Eγ >, in
and lepton energy distribution,
B → X`ν as well as moments of the photon energy
decays B → Xs γ of dierent order n and for dierent
∗
lepton momentum, p`,cut , and photon energy Eγ,cut .
cuts on the minimum
Beside the measure-
ment presented in this analysis, currently available experimental moment
measurements that are used in the combined t are in detail:
1. The
BABAR collaboration measures zeroth to third order moments of
B → Xc `ν [51] for
∗
from p` ≥ 0.6 GeV/c.
lepton energy distribution in decays
dierent cuts
on the lepton momentum starting
The rst and
second moments of the photon energy distribution in decays
B → Xs γ
are measured utilizing two dierent experimental methods. First, by
measuring the photon energy spectrum from a sum of exclusive nal
states [52] and second with a fully inclusive approach [53].
2. The BELLE collaboration performed inclusive measurements of the
rst and second photon energy moments [54, 55].
3. The CDF collaboration measures invariant hadronic mass moments
with a minimum lepton momentum
p∗` ≥ 0.7 GeV/c
[56].
Interpretation of Results and Extraction of |Vcb |
128
4. The CLEO collaboration provides measurements of the second and
fourth hadronic mass moments [57] as well as the rst photon energy
moments [58] with dierent cuts on the lepton momentum and the
photon energy, respectively.
5. Moments of the invariant hadronic mass spectrum and the lepton energy spectrum [59] without restrictions on the lepton momentum are
provided by the DELPHI collaboration.
6. In addition we use the world average of partial branching fraction
Bp∗` ≥0.6 GeV/c (B → Xc `ν)
measurements provided by the Heavy Fla-
vor Averaging Group (HFAG) [42, 11]. The average does not include
measurements already used in the t.
Only a subset of all measurements available with correlations below
is considered.
95%
The inclusion of highly correlated measurements does not
contribute additional information to the t but results in an uninvertible covariance matrix. The hadronic mass moments
< mX > and < m3X > are not
considered in the t since the accuracy of the theoretical expansion of those
moments is believed to be reduced. Table 7.1 summarizes all measurements
used as experimental input for the combined t.
In addition to the experimental input of measured moments the average
B
meson lifetime,
τB = (1.585 ± 0.007) ps
ΓSL .
[42], is used for the calculation of
the total semileptonic rate
7.3 Fit Results and Comparison With Other Measurements
A comparison of the HQE predictions obtained from the best t and the
hadronic mass moment measurement is shown in gure 7.1. The green band
corresponds to the experimental uncertainties obtained by translating the
uncertainties of the extracted t parameters into uncertainties for the individual moments.
The total error calculated as squared sum of theoretical
and experimental uncertainties is depicted as yellow band. Moments with
lled markers are included in the t while moments with open markers are
not included in the t.
The theoretical prediction is in good agreement with all measured moments.
χ2
This is also reected by the excellent minimal
= 24 (ndf = 38)
χ2 found to be
P(χ2 ) = 96%.
2
and corresponding to a χ probability of
In particular, the prediction that is obtained from a t to the measured moments
< m2X >
the measured
< (m2X − < m2X >)2 > is also in good agreement with
3
moments < mX > and < mX > that are not included in the
and
t.
The HQE approach is expected to break down for moments calculated
with high cuts on the lepton momentum.
This behaviour is indicated by
< mnX >
n = 2,
n = 4,
n = 2,
n = 4,
n = 2,
n = 4,
n = 6,
p∗`,cut
p∗`,cut
p∗`,cut
p∗`,cut
p∗`,cut
p∗`,cut
p∗`,cut
= 0.7 GeV/c
= 0.7 GeV/c
= 1.0, 1.5 GeV/c
= 1.0, 1.5 GeV/c
= 0 GeV/c
= 0 GeV/c
= 0 GeV/c
n = 2, p∗`,cut = 0.9, 1.1, 1.3, 1.5 GeV/c
n = 4, p∗`,cut = 0.8, 1.0, 1.2, 1.4 GeV/c
Hadronic mass moments
n = 1,
n = 2,
n = 3,
n = 0,
n = 0,
n = 1,
n = 2,
n = 3,
p∗`,cut
p∗`,cut
p∗`,cut
p∗`,cut
p∗`,cut
p∗`,cut
p∗`,cut
p∗`,cut
< E`n >
= 0 GeV
= 0 GeV
= 0 GeV
= 0.6 GeV
= 0.6, 1.2, 1.5 GeV
= 0.6, 0.8, 1.0, 1.2, 1.5 GeV
= 0.6, 1.0, 1.5 GeV
= 0.8, 1.2 GeV
Lepton energy moments
< Eγn >
n = 1, Eγ,cut = 2.0 GeV
n = 1, Eγ,cut = 1.8, 1.9 GeV
n = 2, Eγ,cut = 1.8, 2.0 GeV
n = 1, Eγ,cut = 1.9, 2.0 GeV
n = 2, Eγ,cut = 1.9 GeV
Photon energy moments
< mnX >,
lepton energy moments
< E`n >,
together with their order
n
and
Group (HFAG) averages several measurements of partial branching fraction not including measurements already used in the t.
cuts. The Heavy Flavor Averaging
< Eγn >
Eγ,cut ,
and photon energy moments
∗
the applied minimum lepton momentum, p`,cut , as well as minimum photon energy,
mass moments
Table 7.1: Summary of measurements used as experimental input for the combined t. The table lists the dierent hadronic
HFAG average [42, 11]
[59]
DELPHI
[57, 58]
CLEO
[56]
CDF
[54, 55]
BELLE
[51, 52, 53]
BABAR
This analysis
Experiment
7.3 Fit Results and Comparison With Other Measurements
129
Interpretation of Results and Extraction of |Vcb |
130
the measurement for moments above
p∗` ≥ 1.6 GeV/c
corresponding to the
∗
region in which the measured p`,cut dependence changes and attens out
∗
with respect to that measured for moments below the mentioned p`,cut . In
contrast the theoretical prediction is not expected to show this behaviour in
the mentioned region of the phase space. However, the measurement cannot
conrm a break down of the HQE approach with the experimental accuracy
available.
As can be seen from gure 7.1, the measured moments presented in this
work are in good agreement with other measurements. Comparing this measurement to a previous measurement performed by the
BABAR
collabora-
tion [47], we nd an agreement within one experimental standard deviation
but with reduced experimental uncertainty in this measurement. While both
analyses obtain the same
p∗`,cut
dependency for the measured moments, the
measurement presented here is oset to higher values. In general the agreement of both measurements improves for higher moments reaching perfect
agreement for the fourth central moments
< (m2X − < m2X >)2 >
(cf. gure
7.1d).
Figure 7.2 presents a comparison of the results obtained in the combined
t (black line) with those obtained from a t to all hadronic mass and lepton
energy moments in decays
B → X`ν
(red green line) and a t combining
information of this measurement and the other
BABAR measurements of lep-
ton energy and photon energy moments (red line) in the
µ2π -mb
plane. The
obtained results agree for all ts. As can bee seen the inclusion of the photon
b-quark
= 1 con-
energy moments signicantly improves the sensitivity to extract the
2
mass mb . The same conclusion can be drawn comparing the ∆χ
tours of the mentioned ts in the
|Vcb |-mb
plane. While the results of the ts
are in good agreement it can be seen that the inclusion of the photon energy
moments adds additional sensitivity to the extracted
b-quark
mass.
Table 7.2 summarizes the t results together with the experimental (∆exp ),
theoretical (∆HQE ), and combined (∆total ) uncertainties. For the extracted
value of
|Vcb |
an additional theoretical error,
∆ΓSL ,
of
uncertainty in the expansion of the semileptonic rate
1.4% is added
ΓSL . We nd,
|Vcb | × 10−3 = (42.06 ± 0.21exp ± 0.35HQE ± 0.59ΓSL ) × 10−3 .
obtaining an overall precision of
for the
(7.5)
1.7% dominated by theoretical uncertainties.
The obtained result is in good agreement with previous measurements:
•
A global analysis of inclusive measurements is performed by Buch-
|Vcb | = (41.96 ± 0.23exp ± 0.35HQE ±
−3
0.6ΓSL )×10 [11]. The presented t is very similar to the one presented
müller and Flächer. They nd:
here but instead of using this measurement of hadronic mass moments
a previous measurement by the
BABAR collaboration is included [40].
While theoretical uncertainties are found to be identical in both ts,
we observe a slight improvement in the experimental uncertainty of the
extracted value for
|Vcb |.
PSfrag replaements
2.15
(a)
2.1
2.05
2
0
0.5
1
1.5
4.6
(b)
4.4
4.2
4
2
0
0.5
1
2
10
2
(d)
2
2
()
9.5
1
2
9
8.5
8
0
0.5
1
1.5
2
p∗ℓ [GeV/℄
3 > [(
2 6
2 GeV/c
< (m2X <(Mx
− < m22X ->)
<Mx
>)3> ) ]
1.5
p∗ℓ [GeV/℄
<Mx
> 2 )4 ]
2 > [(>)
< (m2X<(Mx
− < m2X ->)
GeV/c
2 3
< m3X > [( GeV/c
<Mx3)>]
p∗ℓ [GeV/℄
(f )
(g)
(h)
(i)
(j)
(k)
131
2
<Mx2 )>2 ]
< m2X > [( GeV/c
2
< mX > [ GeV/c
<Mx>]
7.3 Fit Results and Comparison With Other Measurements
10
0
0
0.5
1
1.5
2
p∗ℓ [GeV/℄
BABAR OLD (NOT FIT)
(e)
8
BABAR NEW (NOT FIT)
6
CDF
4
2
0
0
0.5
1
1.5
2
p∗ℓ [GeV/℄
CLEO
(NOT FIT)
DELPHI
(NOT FIT)
HFAG
Figure 7.1: Comparison of measurements (markers) and HQE predictions
(red line) for the best t. The yellow band corresponds to the squared sum
of theoretical and experimental uncertainties. The green band corresponds
to the experimental uncertainties only. Moments with lled markers are included in the t while moments with open markers are not included. The
measurements are in detail: this analysis (/),
BABAR collaboration [47]
(◦), CDF collaboration [56] (F), CLEO collaboration [57] (N/4), and DELPHI collaboration [59] (F).
Interpretation of Results and Extraction of |Vcb |
132
mb(GeV)
4.8
BABAR
!#"%$ only
4.7
5 6 798;:<("%$
clv only
4.6
BABAR only
4.5
& ('*),+-#.0/213+%4
World Average (no BELLE CLV)
4.4
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6 0.65 0.7
2
)
µπ2(GeV
Figure 7.2: Comparison of ts to dierent subsets of moments in the
mb
plane.
2
Shown are the ∆χ
= 1
µ2π -
contours for the combined t to all
moments (black line), a t to all hadronic mass and lepton energy moments
in decays
B → X`ν
only (green line), and a t to moments measured in
this analysis combined with measurements of the lepton energy and photon
energy spectrum performed by the
BABAR collaboration.
7.3 Fit Results and Comparison With Other Measurements
133
4.8
5
6
7*8:9;"!<#,$
4.7
&
('*)+,!-/.102+43
4.6
4.5
4.4
0.41
0.413 0.415 0.418
"!#%$
0.42
0.423 0.425 0.428
Figure 7.3: Comparison of ts to dierent subsets of moments in the
mb
plane.
2
Shown are the ∆χ
= 1
0.43
-1
x 10
|Vcb |-
contours for the combined t to all
moments (black line), a t to all hadronic mass and lepton energy moments
in decays
B → X`ν
only (green line), and a t to moments measured in
this analysis combined with measurements of the lepton energy and photon
energy spectrum performed by the
BABAR collaboration.
Interpretation of Results and Extraction of |Vcb |
134
Parameter
|Vcb | × 10−3
mb [ GeV/c2 ]
mc [ GeV/c2 ]
B(B → Xc `ν) [%]
µ2π [ GeV2 ]
µ2G [ GeV2 ]
ρ3D [ GeV3 ]
ρ3LS [ GeV3 ]
Table 7.2:
Value
∆exp
∆HQE
∆ΓSL
∆total
42.06
4.580
1.131
10.71
0.414
0.293
0.077
−0.165
0.21
0.025
0.037
0.10
0.019
0.024
0.009
0.054
0.35
0.030
0.045
0.08
0.035
0.046
0.022
0.071
0.59
0.72
0.039
0.058
0.13
0.040
0.052
0.024
0.089
Summary of t results obtained from a combined t of HQE
predictions to measurements of moments of the invariant hadronic mass dis-
B → Xc `ν as
well as moments of the photon energy spectrum in decays B → Xs γ (cf. table 7.1). All quantities are given in the kinematic scheme with µ = 1 GeV.
The given uncertainties are experimental (∆exp ), theoretical (∆HQE ), and
the combined experimental and theoretical (∆total ). For |Vcb | an additional
theoretical error, ∆ΓSL , of 1.4% is added for the uncertainty in the expansion
of ΓSL .
tribution and moments of the lepton energy spectrum in decays
•
The Heavy Flavor Averaging Group (HFAG) averages several measurements of
|Vcb |
obtained from exclusive analyses of decays
and performed by the collaborations ALEPH,
DELPHI, and OPAL. They nd:
B → D∗ `ν
BABAR, BELLE, CLEO,
|Vcb | = (41.3 ± 1.0exp ± 1.8theo ) × 10−3
[42].
•
The DELPHI collaboration performed an inclusive measurement of
B→
= 0.
|Vcb | =
hadronic mass and lepton energy moments in semileptonic decays
X`ν
∗
without applying restrictions on the lepton momentum, p`,cut
In a combined t in the kinetic HQE scheme they extract:
(42.1 ± 0.6exp ± 0.6fit ± 0.6theo ) × 10−3
•
Bauer et al. perform a combined t in a dierent HQE scheme, namely
the 1S scheme, to measurements by
DELPHI. They obtain:
•
[59].
BABAR, BELLE, CDF, CLEO, and
|Vcb | = (41.4 ± 0.6 ± 0.1τB ) × 10−3
[60],
A combined t in the kinetic HQE scheme to previous inclusive BABAR
measurements of hadronic mass and lepton energy moments obtains
the result:
|Vcb | = (41.4 ± 0.4exp ± 0.4HQE ± 0.6ΓSL ) × 10−3
Figure 7.4 compares this measurement of
|Vcb |
[40],
with those described above.
PSfrag replaements
(a)
(b) 7.3
()
(d)
(e)
(f )
(g)
(h)
(i)
(j)
(k)
Ge
Fit Results and Comparison With Other Measurements
135
This analysis
(a) HFAG exlusive average
(b) DELPHI ollaboration
() Bauer et al. (1S Sheme)
(d) BABAR ollaboration
(e) Buhmüller et al. (inlusive average)
39
Figure 7.4:
Comparison of extracted
40
|Vcb |
41
42
43
-3
|Vcb| × 10
with other measurements:
(a)
D∗ `ν
B →
= (41.3±1.0exp ±1.8theo )×10−3 ) [42], (b) DELPHI collaboration
inclusive analysis of B → X`ν decays (|Vcb | = (42.1±0.6exp ±0.6fit ±0.6theo )×
10−3 ) [59], (c) combined t in the 1S scheme by Bauer at al. (|Vcb | =
(41.4 ± 0.6 ± 0.1τB ) × 10−3 ) [60], (d) BABAR collaboration inclusive analysis
−3 ) [40],
of B → X`ν decays (|Vcb | = (41.4 ± 0.4exp ± 0.4HQE ± 0.6theo ) × 10
Heavy Flavor Averaging Group (HFAG) average of exclusive
decays (|Vcb |
(e) Global analysis of inclusive measurements by Buchmüller and Flächer
(|Vcb |
= (41.96 ± 0.23exp ± 0.35HQE ± 0.6ΓSL ) × 10−3 )
[11].
Appendix A
Correlation Matrix
137
Correlation Matrix
138
> 0.8 1.00 0.94 0.88
0.9
1.00 0.94
1.0
1.00
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.83
0.88
0.94
1.00
< mX > [GeV /c2 ]
pl,cut [GeV /c] 0.8 0.9 1.0 1.1
< mX
2
< mX
> 0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.79
0.84
0.90
0.95
1.00
1.2
0.74
0.79
0.84
0.88
0.93
1.00
1.3
0.69
0.74
0.79
0.83
0.88
0.94
1.00
1.4
0.64
0.68
0.73
0.77
0.81
0.87
0.93
1.00
1.5
0.58
0.62
0.66
0.70
0.74
0.79
0.84
0.91
1.00
1.6
0.52
0.55
0.59
0.62
0.66
0.70
0.75
0.81
0.89
1.00
1.7
0.45
0.48
0.51
0.54
0.57
0.61
0.65
0.70
0.78
0.87
1.00
1.8
1.3
1.4
1.5
1.6
1.7
0.31
0.34
0.38
0.42
0.46
0.50
0.55
0.61
0.69
0.80
0.95
0.73
1.8
0.23
0.26
0.29
0.32
0.35
0.38
0.42
0.47
0.53
0.62
0.73
0.95
1.9
2
< mX
> [(GeV /c2 )2 ]
0.8 0.9 1.0 1.1 1.2
0.36
0.41
0.45
0.50
0.54
0.59
0.65
0.72
0.82
0.95
0.81
0.62
1.9
0.42
0.47
0.53
0.58
0.63
0.69
0.75
0.84
0.95
0.82
0.70
0.53
0.70
0.78
0.87
0.96
0.88
0.80
0.73
0.65
0.58
0.49
0.42
0.32
0.48
0.54
0.60
0.66
0.71
0.78
0.86
0.96
0.84
0.72
0.61
0.47
0.77
0.86
0.96
0.87
0.80
0.73
0.67
0.60
0.53
0.45
0.38
0.29
0.54
0.60
0.67
0.73
0.80
0.87
0.96
0.86
0.75
0.65
0.55
0.42
0.86
0.96
0.86
0.78
0.72
0.65
0.60
0.54
0.47
0.41
0.34
0.26
0.58
0.65
0.73
0.79
0.87
0.96
0.87
0.78
0.69
0.59
0.50
0.38
0.96
0.86
0.77
0.70
0.64
0.58
0.54
0.48
0.42
0.36
0.31
0.23
0.64
0.72
0.80
0.88
0.96
0.87
0.80
0.71
0.63
0.54
0.46
0.35
0.36
0.38
0.41
0.43
0.45
0.48
0.52
0.56
0.61
0.69
0.79
1.00
0.38
0.43
0.47
0.52
0.57
0.62
0.68
0.76
0.86
1.00
1.00 0.89 0.80 0.73
1.00 0.90 0.82
1.00 0.91
1.00
0.50
0.56
0.63
0.69
0.75
0.82
0.90
1.00
0.44
0.49
0.55
0.61
0.66
0.72
0.79
0.88
1.00
0.25
0.28
0.31
0.34
0.37
0.40
0.44
0.49
0.56
0.65
0.76
1.00
0.61
0.68
0.76
0.83
0.91
1.00
and
0.32
0.36
0.40
0.44
0.48
0.52
0.57
0.64
0.73
0.85
1.00
0.67
0.75
0.84
0.92
1.00
< mX >
2 >.
< mX
0.56
0.63
0.70
0.77
0.83
0.91
1.00
Table A.1: Correlation matrix of the moments
1.3
1.4
1.5
1.6
1.7
1.8
1.9
< m3X >
and
0.43
0.53
0.65
0.75
0.86
1.00
0.43
0.52
0.64
0.74
0.85
0.98
0.86
0.72
0.58
0.48
0.39
0.28
1.3
< m4X >.
0.50
0.61
0.75
0.87
1.00
1.00 0.81 0.67 0.57
1.00 0.82 0.70
1.00 0.86
1.00
0.66
0.81
0.99
0.84
0.74
0.64
0.56
0.47
0.38
0.31
0.26
0.18
0.49
0.60
0.74
0.86
0.98
0.85
0.75
0.63
0.51
0.42
0.34
0.25
0.80
0.99
0.81
0.69
0.60
0.52
0.46
0.38
0.31
0.26
0.21
0.15
0.56
0.69
0.84
0.98
0.86
0.74
0.66
0.55
0.45
0.37
0.30
0.22
0.99
0.80
0.66
0.56
0.49
0.43
0.38
0.31
0.26
0.21
0.17
0.12
< m4X > [(GeV /c2 )4 ]
0.8 0.9 1.0 1.1 1.2
Table A.2: Correlation matrix of the moments
> 0.8 1.00 0.85 0.73 0.64 0.58 0.51 0.46 0.40 0.34 0.29 0.23 0.18
0.9
1.00 0.86 0.76 0.68 0.60 0.54 0.47 0.40 0.34 0.28 0.21
1.0
1.00 0.88 0.79 0.70 0.63 0.54 0.46 0.39 0.32 0.24
1.1
1.00 0.89 0.79 0.71 0.62 0.53 0.44 0.37 0.27
1.2
1.00 0.89 0.80 0.69 0.59 0.49 0.41 0.31
1.3
1.00 0.89 0.77 0.65 0.55 0.45 0.34
1.4
1.00 0.87 0.74 0.62 0.51 0.38
1.5
1.00 0.85 0.72 0.60 0.44
1.6
1.00 0.83 0.70 0.52
1.7
1.00 0.83 0.62
1.8
1.00 0.74
1.9
1.00
< m4X > 0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
<
m3X
< m3X > [(GeV /c2 )3 ]
pl,cut [GeV /c] 0.8 0.9 1.0 1.1 1.2
0.38
0.47
0.57
0.67
0.77
0.88
1.00
0.38
0.46
0.56
0.66
0.75
0.86
0.98
0.82
0.67
0.55
0.45
0.32
1.4
0.32
0.39
0.47
0.55
0.63
0.72
0.83
1.00
0.31
0.38
0.47
0.55
0.63
0.72
0.82
0.99
0.81
0.66
0.55
0.39
1.5
0.26
0.32
0.38
0.45
0.52
0.58
0.67
0.82
1.00
0.25
0.31
0.38
0.45
0.51
0.58
0.66
0.81
0.99
0.79
0.66
0.47
1.6
0.21
0.26
0.32
0.37
0.42
0.48
0.55
0.67
0.80
1.00
0.21
0.26
0.31
0.37
0.42
0.48
0.55
0.66
0.80
0.99
0.82
0.59
1.7
0.17
0.21
0.26
0.30
0.35
0.39
0.45
0.55
0.67
0.82
1.00
0.17
0.21
0.26
0.30
0.34
0.39
0.45
0.54
0.66
0.81
0.99
0.71
1.8
0.12
0.15
0.19
0.22
0.25
0.28
0.33
0.40
0.48
0.59
0.71
1.00
0.12
0.15
0.18
0.22
0.25
0.28
0.32
0.39
0.48
0.58
0.71
0.99
1.9
139
Correlation Matrix
140
> 0.8 1.00 0.78 0.61
0.9
1.00 0.78
1.0
1.00
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.51
0.65
0.84
1.00
0.43
0.56
0.71
0.85
1.00
5
< mX
> [(GeV /c2 )3 ]
pl,cut [GeV /c] 0.8 0.9 1.0 1.1 1.2
5
< mX
6
< mX
> 0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.37
0.48
0.61
0.73
0.85
1.00
1.3
0.32
0.42
0.53
0.63
0.74
0.87
1.00
1.4
0.25
0.32
0.41
0.50
0.58
0.68
0.78
1.00
1.5
0.19
0.25
0.32
0.38
0.45
0.51
0.61
0.78
1.00
1.6
0.15
0.20
0.25
0.31
0.36
0.41
0.48
0.62
0.77
1.00
1.7
0.12
0.16
0.20
0.25
0.29
0.33
0.39
0.50
0.64
0.82
1.00
1.8
1.3
1.4
1.5
1.6
1.7
0.09
0.12
0.16
0.20
0.23
0.27
0.32
0.45
0.60
0.80
0.99
0.65
1.8
0.06
0.08
0.11
0.13
0.16
0.18
0.21
0.30
0.40
0.53
0.66
0.99
1.9
6
< mX
> [(GeV /c2 )4 ]
0.8 0.9 1.0 1.1 1.2
0.11
0.15
0.20
0.25
0.29
0.34
0.40
0.55
0.73
0.99
0.81
0.54
1.9
0.15
0.20
0.26
0.33
0.39
0.45
0.53
0.74
0.99
0.73
0.60
0.40
0.45
0.60
0.81
0.99
0.83
0.70
0.60
0.44
0.32
0.24
0.19
0.13
0.20
0.27
0.36
0.44
0.53
0.62
0.72
0.99
0.73
0.55
0.45
0.30
0.54
0.73
0.99
0.80
0.67
0.57
0.49
0.35
0.25
0.20
0.15
0.10
0.27
0.37
0.50
0.60
0.72
0.85
0.98
0.72
0.53
0.40
0.32
0.22
0.73
0.98
0.74
0.60
0.50
0.43
0.36
0.26
0.19
0.15
0.12
0.08
0.32
0.43
0.57
0.70
0.83
0.99
0.85
0.62
0.45
0.34
0.27
0.18
0.99
0.73
0.55
0.45
0.37
0.32
0.27
0.20
0.14
0.11
0.09
0.06
0.38
0.51
0.68
0.83
0.99
0.83
0.72
0.52
0.38
0.29
0.23
0.16
0.09
0.11
0.14
0.17
0.20
0.23
0.27
0.35
0.44
0.56
0.69
1.00
0.11
0.15
0.20
0.24
0.29
0.34
0.40
0.56
0.74
1.00
1.00 0.74 0.55 0.45
1.00 0.75 0.61
1.00 0.82
1.00
0.20
0.27
0.36
0.44
0.53
0.63
0.73
1.00
0.14
0.19
0.26
0.32
0.39
0.45
0.54
0.74
1.00
0.06
0.08
0.10
0.13
0.16
0.18
0.22
0.30
0.40
0.54
0.66
1.00
0.32
0.43
0.58
0.71
0.84
1.00
and
0.09
0.11
0.15
0.19
0.23
0.27
0.32
0.45
0.61
0.81
1.00
0.38
0.51
0.69
0.84
1.00
5 >
< mX
6 >.
< mX
0.28
0.37
0.50
0.61
0.73
0.86
1.00
Table A.3: Correlation matrix of the moments
0.84
0.98
0.84
0.74
0.66
0.59
0.53
0.46
0.39
0.33
0.27
0.20
0.98
0.83
0.72
0.63
0.57
0.50
0.45
0.39
0.33
0.28
0.23
0.17
< m2X > 0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.72
0.84
0.98
0.86
0.77
0.69
0.62
0.53
0.46
0.38
0.32
0.24
0.65
0.77
0.89
0.78
0.70
0.62
0.56
0.49
0.42
0.35
0.29
0.22
0.63
0.74
0.86
0.98
0.88
0.77
0.70
0.61
0.52
0.43
0.36
0.27
0.58
0.68
0.79
0.89
0.80
0.70
0.63
0.55
0.47
0.40
0.33
0.25
0.57
0.67
0.77
0.88
0.98
0.87
0.78
0.68
0.58
0.49
0.41
0.30
0.51
0.60
0.70
0.79
0.89
0.79
0.71
0.62
0.53
0.44
0.37
0.28
0.50
0.59
0.68
0.77
0.87
0.98
0.88
0.76
0.65
0.55
0.45
0.34
0.46
0.53
0.62
0.70
0.79
0.89
0.80
0.69
0.59
0.50
0.41
0.31
1.3
0.45
0.53
0.62
0.70
0.78
0.88
0.98
0.85
0.73
0.61
0.51
0.38
0.41
0.48
0.56
0.63
0.71
0.80
0.89
0.77
0.66
0.56
0.46
0.35
1.4
0.39
0.46
0.53
0.61
0.68
0.76
0.85
0.98
0.84
0.70
0.59
0.44
0.36
0.42
0.49
0.55
0.62
0.69
0.77
0.89
0.77
0.64
0.54
0.40
1.5
0.33
0.39
0.45
0.52
0.58
0.64
0.72
0.83
0.98
0.82
0.69
0.51
0.30
0.36
0.41
0.47
0.53
0.59
0.66
0.76
0.89
0.74
0.63
0.47
1.6
0.28
0.33
0.38
0.43
0.49
0.55
0.61
0.70
0.82
0.98
0.82
0.61
0.26
0.30
0.35
0.40
0.44
0.50
0.56
0.64
0.75
0.90
0.75
0.56
1.7
0.23
0.27
0.32
0.36
0.40
0.45
0.51
0.58
0.69
0.81
0.98
0.73
0.21
0.25
0.29
0.33
0.37
0.41
0.46
0.53
0.62
0.74
0.89
0.67
1.8
Table A.4: Correlation matrix of the moments
0.76
0.89
0.77
0.67
0.60
0.54
0.48
0.42
0.36
0.30
0.25
0.19
0.89
0.76
0.65
0.57
0.51
0.46
0.41
0.36
0.31
0.26
0.21
0.16
< mX > 0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
< m3X > [(GeV /c2 )3 ]
pl,cut [GeV /c] 0.8 0.9 1.0 1.1 1.2
0.94
0.76
0.62
0.53
0.46
0.40
0.35
0.30
0.25
0.20
0.17
0.12
0.82
0.66
0.55
0.46
0.40
0.35
0.31
0.26
0.22
0.18
0.15
0.11
0.76
0.94
0.77
0.65
0.57
0.50
0.44
0.37
0.30
0.25
0.20
0.15
0.67
0.82
0.67
0.57
0.50
0.43
0.38
0.32
0.27
0.22
0.18
0.13
0.63
0.77
0.94
0.80
0.70
0.60
0.53
0.44
0.37
0.30
0.25
0.18
0.55
0.67
0.82
0.69
0.60
0.52
0.46
0.39
0.32
0.26
0.22
0.16
0.53
0.66
0.80
0.93
0.81
0.70
0.62
0.52
0.43
0.35
0.29
0.21
0.46
0.57
0.70
0.81
0.71
0.61
0.54
0.45
0.38
0.31
0.25
0.19
0.46
0.57
0.70
0.81
0.93
0.80
0.71
0.59
0.49
0.40
0.33
0.24
0.40
0.50
0.61
0.70
0.81
0.70
0.61
0.52
0.43
0.35
0.29
0.21
< m4X > [(GeV /c2 )4 ]
0.8 0.9 1.0 1.1 1.2
and
0.40
0.49
0.60
0.70
0.80
0.93
0.81
0.68
0.56
0.46
0.38
0.27
0.35
0.43
0.52
0.60
0.70
0.81
0.71
0.60
0.50
0.41
0.33
0.24
1.3
< mX >, < m2X >, < m3X >,
0.17
0.20
0.24
0.27
0.30
0.34
0.38
0.44
0.51
0.61
0.73
0.99
0.16
0.19
0.22
0.25
0.28
0.31
0.35
0.40
0.47
0.55
0.67
0.90
1.9
0.30
0.36
0.44
0.52
0.59
0.68
0.78
0.94
0.78
0.63
0.52
0.38
0.26
0.32
0.39
0.45
0.52
0.59
0.68
0.81
0.68
0.55
0.46
0.33
1.5
0.24
0.30
0.36
0.43
0.49
0.55
0.64
0.77
0.94
0.75
0.63
0.45
0.21
0.26
0.32
0.38
0.43
0.48
0.55
0.67
0.81
0.66
0.55
0.40
1.6
< m4X >.
0.35
0.44
0.53
0.62
0.71
0.82
0.93
0.78
0.64
0.53
0.43
0.31
0.31
0.38
0.46
0.54
0.61
0.71
0.80
0.68
0.57
0.46
0.38
0.28
1.4
0.20
0.25
0.30
0.35
0.40
0.46
0.53
0.63
0.77
0.94
0.78
0.57
0.18
0.22
0.26
0.31
0.35
0.41
0.46
0.56
0.67
0.83
0.69
0.50
1.7
0.16
0.20
0.25
0.29
0.33
0.37
0.43
0.52
0.63
0.77
0.94
0.68
0.14
0.18
0.22
0.25
0.29
0.33
0.38
0.45
0.55
0.67
0.83
0.60
1.8
0.12
0.15
0.18
0.21
0.24
0.27
0.31
0.38
0.46
0.56
0.68
0.95
0.10
0.13
0.16
0.19
0.21
0.24
0.27
0.33
0.40
0.49
0.60
0.84
1.9
141
Correlation Matrix
142
< mX
> 0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.75
0.58
0.45
0.37
0.31
0.27
0.23
0.19
0.15
0.12
0.10
0.07
0.58
0.74
0.58
0.48
0.41
0.34
0.29
0.24
0.19
0.15
0.13
0.09
0.45
0.58
0.74
0.61
0.52
0.44
0.38
0.31
0.25
0.20
0.16
0.11
0.44
0.57
0.73
0.87
0.74
0.63
0.54
0.43
0.35
0.28
0.23
0.16
0.37
0.48
0.61
0.73
0.62
0.52
0.45
0.37
0.30
0.24
0.19
0.14
0.38
0.48
0.62
0.74
0.87
0.73
0.63
0.51
0.41
0.32
0.26
0.18
0.31
0.41
0.52
0.62
0.72
0.61
0.53
0.43
0.35
0.28
0.23
0.16
0.32
0.41
0.52
0.62
0.73
0.86
0.74
0.59
0.47
0.38
0.31
0.21
0.26
0.34
0.43
0.52
0.61
0.72
0.62
0.50
0.40
0.32
0.26
0.19
1.3
0.27
0.35
0.45
0.54
0.63
0.74
0.85
0.69
0.55
0.44
0.36
0.25
0.23
0.29
0.38
0.45
0.53
0.62
0.71
0.58
0.47
0.37
0.30
0.21
1.4
0.22
0.28
0.36
0.43
0.51
0.59
0.68
0.87
0.70
0.55
0.46
0.32
0.19
0.24
0.31
0.37
0.43
0.50
0.58
0.73
0.60
0.47
0.39
0.27
1.5
0.17
0.22
0.29
0.35
0.40
0.46
0.54
0.69
0.88
0.68
0.57
0.40
0.15
0.19
0.24
0.29
0.34
0.39
0.46
0.58
0.74
0.58
0.48
0.34
1.6
0.14
0.18
0.23
0.28
0.32
0.37
0.43
0.55
0.70
0.89
0.74
0.51
0.12
0.15
0.20
0.24
0.28
0.32
0.37
0.47
0.60
0.76
0.63
0.44
1.7
0.11
0.15
0.19
0.22
0.26
0.30
0.35
0.45
0.57
0.72
0.89
0.62
0.10
0.12
0.16
0.19
0.22
0.26
0.30
0.38
0.48
0.61
0.76
0.53
1.8
0.08
0.10
0.13
0.16
0.18
0.21
0.25
0.31
0.40
0.51
0.63
0.91
0.07
0.09
0.11
0.14
0.16
0.18
0.21
0.27
0.34
0.43
0.54
0.78
1.9
0.82
0.60
0.45
0.36
0.30
0.25
0.21
0.16
0.12
0.10
0.08
0.05
0.68
0.50
0.37
0.29
0.24
0.20
0.17
0.13
0.10
0.08
0.06
0.04
0.60
0.81
0.60
0.48
0.40
0.33
0.28
0.22
0.17
0.13
0.10
0.07
0.49
0.67
0.49
0.39
0.32
0.27
0.23
0.18
0.14
0.11
0.09
0.06
0.45
0.60
0.81
0.64
0.54
0.45
0.38
0.29
0.22
0.17
0.14
0.09
0.37
0.50
0.66
0.52
0.43
0.36
0.30
0.24
0.18
0.14
0.11
0.08
0.36
0.49
0.66
0.79
0.66
0.55
0.46
0.36
0.27
0.21
0.17
0.11
0.30
0.40
0.53
0.64
0.53
0.44
0.37
0.29
0.23
0.18
0.14
0.10
0.30
0.41
0.54
0.66
0.79
0.65
0.55
0.42
0.33
0.25
0.20
0.14
0.24
0.33
0.44
0.53
0.64
0.53
0.45
0.35
0.27
0.21
0.17
0.12
6
< mX
> [(GeV /c2 )6 ]
0.8 0.9 1.0 1.1 1.2
0.25
0.33
0.45
0.54
0.65
0.78
0.66
0.50
0.39
0.30
0.24
0.16
0.20
0.27
0.36
0.44
0.52
0.63
0.53
0.41
0.32
0.25
0.20
0.14
1.3
0.21
0.29
0.38
0.47
0.56
0.66
0.76
0.58
0.45
0.35
0.28
0.19
0.17
0.23
0.31
0.37
0.45
0.53
0.61
0.48
0.37
0.29
0.23
0.16
1.4
0.16
0.22
0.29
0.36
0.43
0.50
0.58
0.80
0.62
0.47
0.38
0.26
0.13
0.18
0.24
0.29
0.35
0.41
0.48
0.65
0.51
0.39
0.32
0.22
1.5
0.12
0.17
0.22
0.28
0.32
0.38
0.44
0.61
0.82
0.62
0.51
0.34
0.10
0.14
0.18
0.23
0.27
0.31
0.36
0.50
0.67
0.51
0.42
0.29
1.6
0.10
0.13
0.17
0.21
0.25
0.29
0.34
0.47
0.63
0.84
0.69
0.46
0.08
0.11
0.14
0.18
0.21
0.25
0.29
0.39
0.52
0.70
0.57
0.39
1.7
0.08
0.10
0.14
0.17
0.20
0.23
0.27
0.38
0.51
0.67
0.83
0.56
0.06
0.08
0.11
0.14
0.17
0.19
0.23
0.31
0.41
0.55
0.69
0.47
1.8
0.05
0.07
0.09
0.12
0.14
0.16
0.19
0.26
0.34
0.46
0.57
0.86
0.04
0.06
0.08
0.10
0.11
0.13
0.16
0.21
0.29
0.38
0.48
0.72
1.9
5
< mX
> [(GeV /c2 )5 ]
pl,cut [GeV /c] 0.8 0.9 1.0 1.1 1.2
2
< mX
0.53
0.69
0.88
0.72
0.62
0.52
0.45
0.36
0.29
0.23
0.19
0.13
6 >.
< mX
0.68
0.88
0.69
0.57
0.48
0.41
0.35
0.28
0.23
0.18
0.15
0.10
and
0.88
0.68
0.53
0.44
0.38
0.32
0.27
0.22
0.18
0.14
0.11
0.08
2 >, < m5 >,
< mX >, < mX
X
> 0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Table A.5: Correlation matrix of the moments
0.77
0.99
0.77
0.65
0.55
0.47
0.41
0.32
0.25
0.20
0.16
0.11
0.99
0.77
0.60
0.50
0.43
0.36
0.32
0.25
0.19
0.15
0.12
0.09
< m4X > 0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.60
0.77
0.99
0.82
0.70
0.60
0.52
0.41
0.32
0.25
0.20
0.14
0.58
0.75
0.95
0.79
0.67
0.57
0.50
0.39
0.31
0.25
0.20
0.14
0.50
0.65
0.83
0.99
0.84
0.72
0.62
0.49
0.38
0.30
0.25
0.17
0.48
0.62
0.79
0.95
0.81
0.68
0.59
0.47
0.37
0.29
0.24
0.17
0.43
0.55
0.70
0.84
0.99
0.84
0.73
0.57
0.45
0.36
0.29
0.20
0.41
0.53
0.67
0.81
0.94
0.80
0.69
0.55
0.44
0.35
0.28
0.19
0.36
0.47
0.60
0.71
0.84
0.99
0.85
0.67
0.51
0.41
0.33
0.23
0.35
0.45
0.57
0.68
0.80
0.94
0.81
0.64
0.50
0.40
0.32
0.23
1.3
0.32
0.41
0.52
0.62
0.73
0.85
0.98
0.77
0.60
0.48
0.39
0.27
0.30
0.39
0.50
0.59
0.69
0.81
0.94
0.74
0.59
0.46
0.38
0.26
1.4
0.25
0.32
0.41
0.49
0.57
0.67
0.77
0.99
0.78
0.61
0.50
0.35
0.24
0.31
0.39
0.47
0.55
0.64
0.74
0.95
0.75
0.59
0.48
0.34
1.5
0.19
0.25
0.32
0.38
0.45
0.51
0.60
0.77
0.99
0.76
0.63
0.44
0.18
0.24
0.31
0.37
0.43
0.49
0.58
0.74
0.95
0.73
0.61
0.42
1.6
0.15
0.20
0.25
0.30
0.35
0.41
0.48
0.61
0.77
0.99
0.81
0.56
0.15
0.19
0.24
0.29
0.34
0.40
0.46
0.59
0.75
0.96
0.79
0.55
1.7
0.12
0.16
0.20
0.25
0.29
0.33
0.39
0.50
0.63
0.80
0.99
0.68
0.12
0.15
0.20
0.24
0.28
0.32
0.37
0.48
0.61
0.77
0.96
0.66
1.8
Table A.6: Correlation matrix of the moments
0.74
0.95
0.75
0.62
0.53
0.45
0.39
0.31
0.24
0.19
0.16
0.11
0.95
0.74
0.58
0.48
0.41
0.35
0.30
0.24
0.19
0.15
0.12
0.08
<
m3X
> 0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
< m5X > [(GeV /c2 )5 ]
pl,cut [GeV /c] 0.8 0.9 1.0 1.1 1.2
0.96
0.71
0.53
0.43
0.36
0.30
0.26
0.19
0.14
0.11
0.09
0.06
0.91
0.67
0.50
0.40
0.33
0.28
0.24
0.18
0.13
0.10
0.08
0.06
0.71
0.95
0.71
0.58
0.48
0.41
0.35
0.25
0.19
0.14
0.11
0.08
0.67
0.90
0.67
0.54
0.45
0.38
0.32
0.24
0.18
0.14
0.11
0.07
0.53
0.71
0.96
0.77
0.65
0.55
0.47
0.34
0.25
0.19
0.15
0.10
0.50
0.67
0.90
0.72
0.60
0.51
0.43
0.32
0.24
0.19
0.15
0.10
0.43
0.58
0.78
0.95
0.80
0.67
0.57
0.42
0.31
0.24
0.19
0.13
0.41
0.55
0.73
0.89
0.74
0.62
0.53
0.39
0.30
0.23
0.18
0.12
0.36
0.49
0.66
0.80
0.95
0.80
0.68
0.50
0.37
0.29
0.23
0.15
0.34
0.46
0.61
0.74
0.89
0.74
0.63
0.47
0.36
0.27
0.22
0.15
< m6X > [(GeV /c2 )6 ]
0.8 0.9 1.0 1.1 1.2
and
0.30
0.41
0.55
0.67
0.79
0.95
0.81
0.60
0.44
0.34
0.27
0.18
0.28
0.38
0.51
0.62
0.73
0.88
0.75
0.56
0.42
0.32
0.26
0.17
1.3
< m3X >, < m4X >, < m5X >,
0.09
0.11
0.14
0.17
0.20
0.23
0.27
0.34
0.44
0.56
0.68
0.99
0.08
0.11
0.14
0.17
0.19
0.22
0.26
0.33
0.43
0.54
0.66
0.96
1.9
0.19
0.26
0.35
0.43
0.51
0.60
0.69
0.95
0.71
0.54
0.44
0.29
0.18
0.24
0.33
0.40
0.47
0.56
0.65
0.89
0.68
0.52
0.42
0.28
1.5
0.14
0.19
0.26
0.32
0.38
0.44
0.52
0.72
0.96
0.71
0.59
0.39
0.13
0.18
0.24
0.30
0.36
0.42
0.49
0.67
0.91
0.68
0.56
0.37
1.6
< m6X >.
0.26
0.35
0.47
0.58
0.69
0.81
0.94
0.69
0.52
0.39
0.32
0.21
0.24
0.32
0.44
0.53
0.63
0.75
0.86
0.65
0.49
0.38
0.30
0.20
1.4
0.11
0.15
0.20
0.24
0.29
0.34
0.39
0.54
0.72
0.97
0.79
0.52
0.10
0.14
0.19
0.23
0.27
0.32
0.37
0.51
0.69
0.92
0.75
0.50
1.7
0.09
0.12
0.16
0.19
0.23
0.26
0.31
0.43
0.59
0.78
0.97
0.64
0.08
0.11
0.15
0.18
0.22
0.25
0.30
0.41
0.56
0.73
0.92
0.61
1.8
0.06
0.08
0.10
0.13
0.15
0.18
0.21
0.29
0.39
0.52
0.64
0.97
0.06
0.08
0.10
0.12
0.15
0.17
0.20
0.28
0.37
0.49
0.61
0.93
1.9
143
144
Correlation Matrix
List of Tables
2.1
Properties of quark, lepton, and Higgs elds . . . . . . . . . .
2.2
Properties of heavy charmed mesons
. . . . . . . . . . . . . .
14
5.1
Summary of datasets . . . . . . . . . . . . . . . . . . . . . . .
38
5.2
Summary of reconstructed compositions of the hadronic system
5.3
Y+
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of modes considered in the semi-exclusive
Breco
7
39
re-
construction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
5.4
Summary of charged track selection criteria
. . . . . . . . . .
46
5.5
Summary of neutral candidate selection criteria . . . . . . . .
48
5.6
Summary of event selection criteria . . . . . . . . . . . . . . .
52
5.7
Summary of scaling factors used to correct background pro-
5.8
Number of signal and background events for dierent cuts on
cesses in MC
p∗`
5.9
. . . . . . . . . . . . . . . . . . . . . . . . . . .
70
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
Summary of bias corrections . . . . . . . . . . . . . . . . . . .
89
5.10 Summary of extracted moments
< mnX >
6.1
Summary of systematic uncertainties for
6.2
Summary of systematic uncertainties for
6.3
Summary of systematic uncertainties for
6.4
Summary of systematic uncertainties for
6.5
Summary of systematic uncertainties for
6.6
Summary of systematic uncertainties for
as function of
< mX
< m2X
< m3X
< m4X
< m5X
< m6X
>
>
>
>
>
>
p∗`,cut
104
. . . . . . . 118
. . . . . . . 119
. . . . . . . 120
. . . . . . . 121
. . . . . . . 122
. . . . . . . 123
7.1
Summary of measurements used in the combined t . . . . . . 129
7.2
Summary of t results obtained from a combined t of HQE
predictions to measurement moments . . . . . . . . . . . . . . 134
A.1
A.2
A.3
A.4
< mX > and < m2X > . .
3
4
Correlation matrix of the moments < mX > and < mX > . .
5
6
Correlation matrix of the moments < mX > and < mX > . .
2
Correlation matrix of the moments < mX >, < mX >, <
m3X >, and < m4X > . . . . . . . . . . . . . . . . . . . . . . .
Correlation matrix of the moments
138
139
140
141
145
LIST OF TABLES
146
A.5
Correlation matrix of the moments
m5X >,
A.6
and
< m6X >
. . . . . . . . . . . . . . . . . . . . . . . 142
Correlation matrix of the moments
m5X >,
and
< m6X >
< mX >, < m2X >, <
< m3X >, < m4X >, <
. . . . . . . . . . . . . . . . . . . . . . . 143
List of Figures
2.1
Global t of the unitary triangle
. . . . . . . . . . . . . . . .
11
2.2
Spectroscopy of heavy mesons . . . . . . . . . . . . . . . . . .
13
2.3
Weak decay diagrams for semileptonic
b
quark and
B
meson
decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
The Stanford Linear Collider and
BABAR detector
P EP − II e+ e−
18
storage rings 24
3.2
The
. . . . . . . . . . . . . . . . . . . . . . .
25
3.3
The multi-wire Drift Chamber . . . . . . . . . . . . . . . . . .
27
3.4
The detector of Internally Reected ƒerenkov light (DIRC)
.
28
3.5
The Electromagnetic Calorimeter . . . . . . . . . . . . . . . .
29
3.6
The Instrumented Flux Return
31
4.1
Illustration of the studied event structure of decays
Breco BSL (Xc `ν)
Breco
. . . . . . . . . . . . . . . . .
Υ (4S) →
. . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
Number of
5.2
Distributions of
5.3
Distributions of
5.4
Distributions of
5.5
Distributions of
5.6
Examples of energy resolutions of the hadronic
candidates per event after the application of
preselection criteria . . . . . . . . . . . . . . . . . . . . . . . .
mES for the best Breco candidate
∆E for the best Breco candidate
Emiss versus |~
pmiss | for simulated
background decays
dierent bins of
5.7
43
. . . . . . .
44
signal and
. . . . . . . . . . . . . . . . . . . . . . . .
51
p∗`
51
Emiss − |~
pmiss |
ΘXc
ΘXc
mX
and
and
5.8
Resolution of
5.9
Resolution and bias of
in data for dierent cuts on
MultXc
Xc
system for
. . . . . . . . . . . . . . . .
MultXc
Xc
before and after the kinematic t and
mX
in bins of
Emiss − |~
pmiss |
hadronic decay mode . . . . . . . . . . . . . . . . . . . . . . .
|~
pmiss |
and
Nγ,X
59
60
Emiss −
. . . . . . . . . . . . . . . . . . . .
5.11 Comparison of data and MC simulated distributions for
Nγ,X , Ntrk,X ,
58
χ2
and the
5.10 Comparison of data and MC simulated distributions for
and
57
four
. . . . . . . . . . . . . . . .
probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|~
pmiss |, Emiss ,
42
. . . . . . .
Energy and momentum resolution of the measured
vector in bins of
34
62
Ntrk,X +
. . . . . . . . . . . . . . . . . . . . .
63
147
LIST OF FIGURES
148
5.12 Comparison of data and MC simulated distributions for
and
5.13
5.14
mES
mES
Eγ,Xc
Eγ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ts for
ts for
p∗` ≥ 0.8 GeV/c
p∗` ≥ 1.9 GeV/c
67
. . . . . . . . . . . . . . . . . . .
68
5.15 Sideband scaling factors extracted from
mES ts together with
the corresponding background shapes . . . . . . . . . . . . . .
5.16 Residual background distributions for
p∗` ≥ 1.9 GeV/c
64
. . . . . . . . . . . . . . . . . . .
p∗` ≥ 0.8 GeV/c
. . . . . . . . . . . . . . . . . . . . . . . . . .
5.17 Dominant Feynman graphs of right-charged
D
and
Ds+
5.18 Spectra of wrong-charged events in a sample of charged
69
Decays. 70
Breco
candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.19 Spectra of wrong-charged events in a sample of neutral
68
and
75
Breco
candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
5.20 Measured
78
5.21
mX spectra and background subtraction factors wi
Spectra of mX,true and mX,reco for dierent signal decay channels
Examples of calibration curves for < mX > in bins of MultXc ,
Emiss − |~
pmiss | and p∗` . . . . . . . . . . . . . . . . . . . . . .
2
Examples of calibration curves for < mX > in bins of MultXc ,
Emiss − |~
pmiss | and p∗` . . . . . . . . . . . . . . . . . . . . . .
4
Examples of calibration curves for < mX > in bins of MultXc ,
Emiss − |~
pmiss | and p∗` . . . . . . . . . . . . . . . . . . . . . .
6
Examples of calibration curves for < mX > in bins of MultXc ,
Emiss − |~
pmiss | and p∗` . . . . . . . . . . . . . . . . . . . . . .
∗
Moments calculated in MC simulations for dierent cuts on p`
∗ . . . . .
Moments calculated in MC simulations in bins of p`
80
5.22
5.23
5.24
5.25
5.26
5.27
83
84
85
86
87
88
5.28 Bias after application of the calibration on simulated signal
decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
5.29 Rate of true and calibrated moments . . . . . . . . . . . . . .
92
∗
5.30 Calibration of exclusive decays modes with p`
≥ 0.8 GeV/c for
dierent moments . . . . . . . . . . . . . . . . . . . . . . . . .
5.31 Calibration of exclusive decays modes with
97
p∗` ≥ 1.9 GeV/c for
dierent moments . . . . . . . . . . . . . . . . . . . . . . . . .
98
5.32 Verication of the analysis procedure on MC simulations
. . 100
5.33 Verication of the analysis procedure on MC simulations
. . 101
5.34 Moments calculated for dierent cuts on
p∗`
. . . . . . . . . . 103
6.1
Variation of measured moments under variation of the under-
6.2
Variation of the residual bias correction factor under variation
6.3
Spectra of
6.4
Variation of moments under variation
6.5
Variation of the measured moments as a function of the cut
lying model used to cconstructcalibration curves
of the underlying model
on
∗
Eγ,max
. . . . . . . . . . . . . . . . . . . . . 109
for dierent cuts on
Emiss − |~
pmiss | .
. . . . . . . 108
p∗` . . . .
∗
of Eγ,max
. . . . . . . . 112
. . . . . . . . 113
. . . . . . . . . . . . . . . . . . . . . . . . 115
LIST OF FIGURES
149
6.6
Comparison of moments measured on statistical independent
6.7
Comparison of moments measured on disjoined samples of
subsamples of data . . . . . . . . . . . . . . . . . . . . . . . . 116
charged and neutral
Breco
candidates in bins of
p∗`
. . . . . . 117
7.1
Comparison of measurements and HQE predictions for the
7.2
Comparison of ts to dierent subsets of moments in the
7.3
Comparison of ts to dierent subsets of moments in the
best t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
mb
mb
7.4
µ2π -
plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
|Vcb |-
plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Comparison of extracted
|Vcb |
with other measurements
. . . 135
150
LIST OF FIGURES
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