148
Copyright by Chris James Schilling 2008
Copyright
by
Chris James Schilling
2008
The Dissertation Committee for Chris James Schillingcertifies that this is the approved version of the following dissertation:
A Study of Angular Asymmetries in the Rare Decay
B → K∗`+`−
Committee:
Jack L. Ritchie, Supervisor
Sacha E. Kopp
Pawan Kumar
Karol Lang
Roy F. Schwitters
A Study of Angular Asymmetries in the Rare Decay
B → K∗`+`−
by
Chris James Schilling, B.S.
DISSERTATION
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
THE UNIVERSITY OF TEXAS AT AUSTIN
August 2008
A Study of Angular Asymmetries in the Rare Decay
B → K∗`+`−
Publication No.
Chris James Schilling, Ph.D.
The University of Texas at Austin, 2008
Supervisor: Jack L. Ritchie
This dissertation describes studies of the rare quark transition process
b → s`+`−, in particular the B meson decay B → K∗`+`− where the `+`−
is either e+e− or µ+µ−. These decays are highly suppressed in the Standard
Model and could be strongly affected by new physics.
The angular observables describing the lepton forward-backward asym-
metry and the longitudinal K∗ polarization are measured in this mode. The
measurements were performed using the BABAR detector at the SLAC PEP-
II storage ring running at the Υ (4S) resonance. The analysis was performed
on a 349 fb−1 sample corresponding to 384 million BB pairs. The data was
collected over a period of six years beginning in 1999.
For low dilepton invariant masses, m`` < 2.5 GeV/c2, we measure a
lepton forward backward asymmetry AFB = 0.24+0.18−0.23 ± 0.05 and a K∗ lon-
gitudinal polarization FL = 0.35 ± 0.16 ± 0.04. For m`` > 3.2 GeV/c2, we
measure AFB = 0.76+0.52−0.32 ± 0.07 and FL = 0.71+0.20
−0.22 ± 0.04.
iv
Table of Contents
Abstract iv
List of Tables viii
List of Figures x
Chapter 1. Introduction 1
1.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 CKM Matrix and Flavor Changing Neutral Currents . . 6
1.2 e+e− Collisions and the B Meson . . . . . . . . . . . . . . . . 8
1.3 The b→ s`+`− Transition . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Effective Hamiltonian and the Operator Product Expansion 11
1.3.2 Theoretical Predictions and Previous Measurements . . 12
1.3.3 Branching Fractions . . . . . . . . . . . . . . . . . . . . 14
1.3.4 K∗ Polarization and Lepton Forward-Backward Asymmetry 17
1.3.5 New Physics and Supersymmetry . . . . . . . . . . . . . 23
1.4 Other B Physics at BABAR . . . . . . . . . . . . . . . . . . . . 26
Chapter 2. PEP-II and the BaBar Detector 31
2.1 PEP-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 BaBar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.1 SVT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.2 DCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.3 DIRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.4 EMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.5 IFR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2.6 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Dataflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
v
Chapter 3. Event Selection 49
3.1 Monte Carlo Samples . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Selection of Neutrals and Tracks . . . . . . . . . . . . . . . . . 51
3.2.1 Electron Identification . . . . . . . . . . . . . . . . . . . 53
3.2.2 Muon Identification . . . . . . . . . . . . . . . . . . . . 57
3.2.3 Kaon Identification . . . . . . . . . . . . . . . . . . . . . 59
3.3 Kinematic Selection . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4 Background Suppression . . . . . . . . . . . . . . . . . . . . . 65
3.4.1 Charmonium Vetoes . . . . . . . . . . . . . . . . . . . . 66
3.4.2 Vetoes Against B → Dπ Backgrounds . . . . . . . . . . 68
3.4.3 Continuum Suppression with Neural Networks . . . . . 69
3.5 Multiple Candidate Selection . . . . . . . . . . . . . . . . . . . 72
3.6 Cut Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6.1 Kinematic Regions . . . . . . . . . . . . . . . . . . . . . 77
3.6.2 Selection Efficiencies . . . . . . . . . . . . . . . . . . . . 78
3.6.3 Expected Signal and Background Yields . . . . . . . . . 79
Chapter 4. Fit Procedure 83
4.1 Fit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.1.1 Signal PDFs . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1.2 Combinatoric Background PDFs . . . . . . . . . . . . . 91
4.1.3 Hadronic Peaking Background PDFs . . . . . . . . . . . 93
4.1.4 Crossfeed PDFs . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Fit Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Tests of Fits in Control Samples . . . . . . . . . . . . . . . . . 99
4.3.1 Fits to Charmonium Control Sample . . . . . . . . . . . 100
4.3.2 Tests of Fits in Simulation . . . . . . . . . . . . . . . . 101
4.3.3 Good Fits and Strategy . . . . . . . . . . . . . . . . . . 106
vi
Chapter 5. Results 109
5.1 Fit Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.1.1 Unblinding Strategy . . . . . . . . . . . . . . . . . . . . 109
5.1.2 B+ → K+`+`− and B → K∗`+`− mES Fit Results . . . 111
5.1.3 B+ → K+`+`− AFB Fits . . . . . . . . . . . . . . . . . 113
5.1.4 B → K∗`+`− Angular Fits . . . . . . . . . . . . . . . . 113
5.2 Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2.1 Signal Yield Systematic . . . . . . . . . . . . . . . . . . 117
5.2.2 FL Fit Systematics . . . . . . . . . . . . . . . . . . . . . 118
5.2.3 Combinatorial Background Systematics . . . . . . . . . 118
5.2.4 Crossfeed and Signal Shape Systematics . . . . . . . . . 119
5.2.5 Signal Model Systematics . . . . . . . . . . . . . . . . . 120
5.2.6 Fit Bias Systematics . . . . . . . . . . . . . . . . . . . . 121
5.2.7 ∆E Fit Window Systematics . . . . . . . . . . . . . . . 122
5.2.8 Peaking Background Systematics . . . . . . . . . . . . . 126
5.2.9 Total Systematic Error . . . . . . . . . . . . . . . . . . 127
5.3 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . 127
Bibliography 131
Vita 136
vii
List of Tables
1.1 Standard Model gauge bosons . . . . . . . . . . . . . . . . . . 3
1.2 Standard Model quarks and leptons . . . . . . . . . . . . . . . 5
1.3 Properties of the B meson. . . . . . . . . . . . . . . . . . . . . 10
1.4 Current SM Wilson coefficient predictions. . . . . . . . . . . . 13
1.5 Current B → K(∗)`+`− Branching Fraction averages and pre-dictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Current Standard Model predictions for FL and AFB in B →K(∗)`+`−. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Cross-sections at BABAR . . . . . . . . . . . . . . . . . . . . . 32
3.1 Simulated generic samples. . . . . . . . . . . . . . . . . . . . . 51
3.2 MC simulated signal samples . . . . . . . . . . . . . . . . . . . 52
3.3 s regions to be measured for B → K (∗)`+`− . . . . . . . . . . 76
3.4 Optimized ∆E and hadronic mass cuts . . . . . . . . . . . . . 77
3.5 Final Reconstruction Efficiency for Signal Events By Mode ands Bin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.6 Expected Signal and Background Yields By s Bin . . . . . . . 81
3.7 Expected Signal Yields and Background Yields By IndividualMode and s Bin . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1 Hadronic Peaking Background by Mode and s Bin . . . . . . . 95
4.2 Mode-wise signal to self-crossfeed fractions in each s bin . . . 97
4.3 J/ψ BF by Mode and s Bin NN Cuts . . . . . . . . . . . . . . 102
4.4 J/ψ mES PDF Shape Parameters by Mode . . . . . . . . . . . 104
4.5 Embedded Toy NS(s) Pull Results . . . . . . . . . . . . . . . 105
4.6 Embedded Toy FL Pull Results . . . . . . . . . . . . . . . . . 105
4.7 Embedded Toy AFB Pull Results . . . . . . . . . . . . . . . . 106
viii
5.1 B+ → K+`+`− Expected and Observed Signal and BackgroundYields in Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2 B+ → K+`+`− AFB in Data . . . . . . . . . . . . . . . . . . . 113
5.3 Signal Yield Systematics . . . . . . . . . . . . . . . . . . . . . 117
5.4 FL Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.5 Combinatorial background systematics. . . . . . . . . . . . . . 119
5.6 Signal model systematics . . . . . . . . . . . . . . . . . . . . . 121
5.7 Varied Wilson Coefficient Toy FL . . . . . . . . . . . . . . . . 123
5.8 Varied Wilson Coefficient Toy AFB . . . . . . . . . . . . . . . 124
5.9 Fit bias systematic error. . . . . . . . . . . . . . . . . . . . . . 124
5.10 ∆E systematics . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.11 Total Systematic Errors . . . . . . . . . . . . . . . . . . . . . 127
ix
List of Figures
1.1 Standard Model diagrams for the decays B → K (∗)`+`−. . . . 8
1.2 Upsilon resonances and possible interactions of the b quarks. . 10
1.3 Current Branching Fraction Results in B → K (∗)`+`−. . . . . 16
1.4 Current Partial BF SM predictions for B → K (∗)`+`−. . . . . 18
1.5 The s dependence of AFB is shown in the top plot and thedependence of FL is shown in the bottom plot. The color codeis given in the text. . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 Previous BABAR and BELLE AFB results. . . . . . . . . . . . 24
1.7 Possible new physics contributions to b→ s`+`− . . . . . . . . 25
1.8 The Unitarity Triangle. . . . . . . . . . . . . . . . . . . . . . . 27
1.9 Current constraints on the Unitarity Triangle. . . . . . . . . . 28
1.10 Current B-factory CP measurements in penguin decays. . . . 30
2.1 Integrated luminosity as a function of time. . . . . . . . . . . 33
2.2 The BABAR detector. . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 An example of a b → s`+`− decay in the BABAR detector. Thecyan lines represent electrons. The pion (red) and kaon (yellow)are back-to-back with the di-lepton system. . . . . . . . . . . . 36
2.4 Longitudinal schematic of the SVT. . . . . . . . . . . . . . . . 37
2.5 dE/dx in the DCH as a function of track momentum for differ-ent charged particles. . . . . . . . . . . . . . . . . . . . . . . . 39
2.6 Geometry of the DIRC. . . . . . . . . . . . . . . . . . . . . . . 41
2.7 DIRC Cherenkov angle versus energy with curves for variousparticle hypotheses. Muon data points are not shown, howeverthe hypothesis curve is. . . . . . . . . . . . . . . . . . . . . . . 42
2.8 Geometry of the EMC. . . . . . . . . . . . . . . . . . . . . . . 43
2.9 EMC resolution as a function of photon energy. . . . . . . . . 44
2.10 Design of the BABAR instrumented flux return. . . . . . . . . . 45
2.11 Schematic drawing of a BABAR RPC. . . . . . . . . . . . . . . 46
x
2.12 Muon efficiency before the LSTs (blue) and after the LST up-grade (red) and pion misID before the LSTs (green) and afterthe LST upgrade (magenta) . . . . . . . . . . . . . . . . . . . 47
3.1 Electron efficiency in the e+e− → e+e−γ data and MC controlsample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Pion mis ID rate for the electron selector. . . . . . . . . . . . 56
3.3 Bremsstrahlung recovery in charmonium . . . . . . . . . . . . 57
3.4 Muon selection efficiency as a function of momentum. . . . . . 60
3.5 Pion mis ID rate of the muon selector. . . . . . . . . . . . . . 61
3.6 Kaon efficiency as a function of momentum. . . . . . . . . . . 63
3.7 Pion mis ID rate of the kaon selector. . . . . . . . . . . . . . . 64
3.8 B+ → K+e+e− Charmonium Veto Region. . . . . . . . . . . . 68
3.9 B0 → K+π−e+e− Low s NN Inputs . . . . . . . . . . . . . . . 73
3.10 B0 → K+π−e+e− Low s BB NN Output . . . . . . . . . . . . 74
3.11 B0 → K+π−e+e− Low s background rejection versus signalefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1 Angular fits to generated B0 → K+π−e+e− signal MC events. 87
4.2 FL and AFB distributions as a function of s as modeled in thesignal MC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3 B+ → K+π0e+e− efficiency as a function of cos θK in the highs bin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4 B0 → K+π−e+e− efficiency as a function of cos θK in the low sbin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5 B+ → K+π0e+e− efficiency as a function of cos θ` in the high sbin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.6 B0 → K+π−e+e− efficiency as a function of cos θ` in the low sbin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.7 cos θ` GSB data distributions for standard and LFV events withrelaxed NN cuts. . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.8 Gaussian + ARGUS fits to the hadronic peaking control samplein low s (left) and high s (right) . . . . . . . . . . . . . . . . . 94
4.9 J/ψ fits for B+ → K0Sπ+µ+µ− mode . . . . . . . . . . . . . . . 103
4.10 Embedded Toy Low mES Pull . . . . . . . . . . . . . . . . . . 105
4.11 Embedded Toy Low FL Pull . . . . . . . . . . . . . . . . . . . 106
xi
4.12 Embedded Toy Low AFB Pull . . . . . . . . . . . . . . . . . . 106
4.13 FL errors from embedded toys: low s bin (left) and high s bin(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.14 AFB errors from embedded toys: low s bin (left) and high s bin(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.1 B+ → K+`+`− low mES fit . . . . . . . . . . . . . . . . . . . . 112
5.2 B → K∗`+`− mES fit . . . . . . . . . . . . . . . . . . . . . . . 112
5.3 B+ → K+`+`− AFB fits . . . . . . . . . . . . . . . . . . . . . 114
5.4 B → K∗`+`− FL and AFB fits . . . . . . . . . . . . . . . . . . 115
5.5 Values of FL for Wilson coefficient variations by s bin . . . . . 122
5.6 Values of AFB for Wilson coefficient variations by s bin . . . . 125
5.7 FL and AFB Results and Theory Expectations . . . . . . . . . 129
xii
Chapter 1
Introduction
The goal of particle physics is to understand the fundamental constituents
of matter and their interactions. The current theory used to describe this is
known as the Standard Model (SM) of particle physics. The SM has been
highly successful in incorporating the known particles and forces (excluding
gravity) into a framework that can be used to predict particle interaction
phenomena. The SM has survived over three decades of experimental tests of
these predictions. It is, however, widely believed that the SM can only be an
approximation of a more fundamental theory.
Studies of the B meson system allow for precision tests of the Standard
Model. The b-quark to s-quark transition is particularly interesting because
it is a flavor changing neutral current and is forbidden at the tree level in the
SM. Loop diagrams known as penguin diagrams are the leading contribution to
the amplitude. Flavor changing neutral currents are sensitive to new physics
particles that can enter the loop and enhance Standard Model predictions.
The decay B → K∗`+`− is one such example.
Angular asymmetries such as the the lepton forward-backward asym-
metry have precise Standard Model predictions. Significant deviations from
1
the SM predictions could be a sign of new physics. The goal of this analysis
is to measure the fraction of longitudinal K∗ polarization (FL) and the lepton
forward-backward asymmetry (AFB) in two bins of the di-lepton mass above
and below the J/ψ mass peak. A 349 fb−1 data sample was collected at the
Υ (4S) resonance. This corresponds to approximately 384 million BB pairs.
A total of six flavor tagged B → K∗`+`− modes are used. The B →
K`+`− modes are used as a control sample. B → K`+`− is the rarest decay
observed at the B-factories. The current world average branching fraction
measurement of B → K`+`− is B = (5.4 ± 0.8) × 10−7. The average for
B → K∗`+`− is B = (1.05 ± 0.2) × 10−6 [1]. The final data sample used to
extract the angular asymmetries is extremely statistics limited. The analysis
requires a comprehensive understanding of peaking and random combinatoric
backgrounds in order to maximize the signal significance. Monte Carlo samples
are used to aid in understanding background sources. A complicated maximum
likelihood fitting technique is used to extract the signal yield, FL and AFB in
the two s bins. Control samples from Monte Carlo and data are used to study
and test all techniques used in the analysis.
This chapter provides a brief introduction to the Standard Model and
the theoretical framework used to make predictions. The next chapter discuss
the BABAR experiment including the Stanford Linear Accelerator Center fa-
cilities and the detector. The following chapters discuss the analysis method
including selection of B meson events and an explanation of the maximum like-
lihood fit technique used to extract FL and AFB. The last chapter concludes
2
with the results and outlook for future improvements of this measurement.
1.1 The Standard Model
The Standard Model is a theory of fundamental particles and the electro-
magnetic, weak, and strong interactions that act upon them. The interac-
tions of the Standard Model are described by SU(3) × SU(2) × U(1) gauge
theory, where the SU(3) subgroup characterizes the strong interaction and
SU(2)×U(1) describes the electromagnetic and weak interactions formulated
by the Glashow-Weinberg-Salam (GWS) model [2–4]. The interactions of the
SM are mediated by spin-1 gauge bosons: the electromagnetic force is medi-
ated by massless photons, the weak force by massive W and Z bosons, and the
strong force by massless gluons (Table 1.1)[1].
Gauge boson m ( GeV/c2) Charge Mediatesγ < 6 × 10−17 0 electromagneticg 0 0 strong forceZ0 91.18746 ± 0.0021 0 weak forceW+ 80.376 ± 0.029 +1 weak force
Table 1.1: Properties of the Standard Model gauge bosons. From left: particleidentity, mass, electric charge, and the force mediated by the particle. Themass of the Z0 and W+ are experimentally measured quantities. The upperlimit on the photon mass is an experimentally derived quantity. The gluonmass is the theoretical value in the Standard Model.
The SU(2) component is composed of a triplet of vector bosons which
couple to quantum numbers called weak isospin in the GWS model:
Wµ =
W µ1
W µ2
W µ3
.
3
The U(1) component contributes a single Bµ boson coupling to weak hyper-
charge. The Standard Model contains left-handed doublets of quarks (Qi) and
leptons (Li), and right-handed singlets of leptons (eRi) and up and down-type
quarks (uRiand dRi
):
L1 =
(
e−
νe
)
L
eR1 = e−1R Q1 =
(
ud
)
L
uR1 = u, dR1 = d
L2 =
(
µ−
νµ
)
L
eR2 = µ−1R Q2 =
(
cs
)
L
uR2 = c, dR2 = s
L3 =
(
τ−
ντ
)
L
eR3 = τ−1R Q3 =
(
tb
)
L
uR3 = t, dR3 = b
.
Given this structure, the Lagrangian for the SM can be written as
L = −1
4Ga
µνGµνa − 1
4W a
µνWµνa − 1
4Ba
µνBµνa
+ LiiDµγµLi + ¯eRi
iDµγµeRi
+ QiiDµγµQi + uRi
iDµγµuRi
+ dRiiDµγ
µdRi.
Dµ indicates a covariant derivative that can be expressed in terms of the
couplings gi and the hypercharge Y as:
Dµ = δµ − igs − ig21
2σaW
aµ − ig1
Yq
2Bµ.
The covariant derivative acts on the fields G, W, and B, associated with the
strong, weak, and electromagnetic interactions respectively. Properties of the
Standard Model quarks and leptons are given in Table 1.2.
In the GWS theory, the SU(2) × U(1) symmetry is broken since the
third component of the W triplet and the B boson of the U(1) symmetry
4
Leptons m ( GeV/c2) Charge Interactionse 0.000511 −1 weak, EMνe < 3 × 10−9 0 weakµ 0.106 −1 weak, EMνµ < 1.9 × 10−4 0 weakτ 1.7770+0.00029
−0.00026 −1 weak, EMντ < 0.018 0 weak
Quarksu 0.0015 to 0.003 +2/3 strong,weak,EMd 0.003 to 0.007 −1/3 strong,weak,EMc 1.25 ± 0.09 +2/3 strong,weak,EMs 0.095 ± 0.025 −1/3 strong,weak,EMt 174.2 ± 3.3 +2/3 strong,weak,EMb 4.70 ± 0.07 −1/3 strong,weak,EM
Table 1.2: Properties of the Standard Model quarks and leptons. From left:particle identity, mass, electric charge, and the interactions the particle isinvolved in. The charged lepton masses are experimentally determined. Forthe neutrinos, upper limits are experimentally determined. The mass of the u,d and s quarks are extracted from kaon and pion masses using chiral symmetry.The c and b quark masses are the masses in the MS scheme. The t-quark massis from direct observation of top events.
mix through the weak mixing angle θW into two linear combinations that
correspond to the neutral Z0 boson and the photon (the carriers of the neutral
weak current and the electromagnetic interactions, respectively):
Bµ = Aµ cos θW − Zµ sin θW
W 3µ = Aµ sin θW + Zµ cos θW
The origin of the SU(2) × U(1) symmetry breaking is the Higgs mechanism
[5, 6] which is responsible for the mass of the fermions and the W± and the
Z0 bosons in the SM. Such spontaneous symmetry breaking is accomplished
5
by adding to the Lagrangian an additional term of the form:
L = (DµΦ)†(DµΦ) − µ2Φ†Φ − λ(Φ†Φ)2,
where Φ is a doublet of scalar fields:
Φ =
(
φ+
φ0
)
.
With µ2 > 0, the potential term µ2Φ†Φ − λ(Φ†Φ)2 has a minimum at 0, as
expected for a massless gauge boson. With µ2 < 0, the potential has a min-
ima at non-zero values of the vacuum expectation value v, where v2 = −µ2/λ.
After an appropriate gauge transformation, this spontaneous symmetry break-
ing allows the W and Z bosons to acquire masses of MW = 12vg2 and MZ =
12v√
g22 + g2
1 while the photon remains massless.
One physical degree of freedom remains after the symmetry is broken.
This corresponds to a neutral scalar Higgs boson of mass MH =√
−2µ2 which
must be determined experimentally. The Higgs boson remains unobserved.
One of the main goals of the Large Hadron Collider experiments is to find and
determine the mass of the Higgs boson.
1.1.1 CKM Matrix and Flavor Changing Neutral Currents
The charged-current Lagrangian described previously appears to conserve quark
generation, as the W± transformation operates only within a flavor doublet. In
weak neutral-current interactions that occur through Z0 exchange, the quark
flavor is conserved at the vertex. However, charged W±-interactions always
change the quark flavor.
6
Flavor mixing in the quark sector is described in the Standard Model
through the Cabbibo-Kabayashi-Maskawa (CKM) mechanism [7, 8]. Weak fla-
vor eigenstates of the SM do not correspond exactly to the mass eigenstates
of the Hamiltonian, which governs how particles propagate through space. In
other words, the physical up or down type quarks are actually admixtures of
different flavors. The mass and flavor eigenstates are related by the CKM
matrix:
d′
s′
b′
mass
=
Vud Vus Vub
Vcd Vcs Vcb
Vtd Vts Vtb
dsb
. (1.1)
The CKM matrix is a 3 × 3 complex, unitary matrix, and thus can
be parameterized by three angles and six complex phases. Five of the phases
can be removed by redefining unphysical spaces in the quark fields of the
Lagrangian. We are thus left with 4 independent parameters: three rotation
angles and one irreducible complex phase. The irreducible phase allows for
an asymmetry under the combined discrete operations of charge conjugation
(C) and parity reversal (P). In the SM, the weak phase is the only source of
CP violating asymmetries in the quark sector. To date all measurements are
consistent with this hypothesis [9, 10].
The matrix can be written in terms of the Wolfenstein parameterization
[11] in which the matrix elements are give as expansions in the parameter
λ = |Vus| ≈ 0.23:
VCKM =
1 − 12λ2 λ Aλ3(ρ− iη)
−λ 1 − 12λ2 Aλ2
Aλ3(1 − ρ− iη) −Aλ2 1
+O(λ4) (1.2)
7
where the parameters A,ρ and η are of order unity.
Flavor changing neutral currents (FCNCs) do not occur at the tree level
in the SM. FCNC are, however, allowed in higher-order processes such as the
(so-called) penguin and box diagrams involving heavy virtual particles. Ex-
amples of such diagrams are shown in Figure 1.1, which depict the electroweak
decay of a B meson at the quark level. In Figure 1.1, the loop diagrams are
called penguins since the first order contributions come from what are nor-
mally radiative corrections to the tree diagram. The rates are suppressed due
to the absence of the tree diagram and further suppression is caused by the
GIM mechanism [4].
q q
b st
W
γ , Z
l +
l −
q q
b st
W +W − ν
l − l +
Figure 1.1: Standard Model diagrams for the decays B → K (∗)`+`−.
1.2 e+e− Collisions and the B Meson
To facilitate an understanding of SM predictions, particle accelerators are
used to create new particles which can be detected and measured. Particle
accelerators have existed since the 1950s. In these early experiments, a heavy-
8
nucleus material (fixed-target) was bombarded by subatomic particles (usually
protons). Accelerators have evolved into sophisticated high-energy physics
experiments with large linear or circular colliders and storage rings colliding
nucleons or electron-positron pairs.
The PEP-II storage ring at the BABAR experiment is used to collide
electrons and positrons. A two mile long linear accelerator is used to inject
the electrons and positrons into the PEP-II accelerator ring. These facilities
are discussed in greater detail in the next chapter. Electron-positron collisions
are much cleaner than hadronic collisions because the physics is governed by
clean QED processes whereas QCD strong interaction processes dominate in
high-energy hadronic collisions. Further, synchrotron losses are minimal in a
linear accelerator. When an electron and positron collide, they annihilate to
a virtual particle photon or Z boson. The virtual particle almost immediately
decays into other elementary particles which are then detected by massive
detectors.
At BABAR, electrons and positrons are collided at a center-of-mass en-
ergy of 10.58 GeV which corresponds to the mass of the Υ (4S) meson. The Υ
system refers to the family of bound states of a b quark and a b quark. The
system is bound somewhat analogously to the electron and proton of a hydro-
gen atom. Various excited states (resonances) of this system can be created by
tuning the accelerator energy. Resonances below the Υ (4S) can only decay by
the b quark and b quark annihilating. At the Υ (4S) resonance there is enough
energy in the excited state to create a light quark/ anti-quark pair, producing
9
Parameter Valuemass (mB) 5.279 ± 0.0005 GeV/c2
(lifetime) τB± (1.671 ± 0.018) × 10−12 sτB+/τB0 1.086 ± 0.017
Table 1.3: Some properties of the B meson.
a pair of B mesons (Figure 1.2). Some properties of the B meson are given in
Table 1.3.
Figure 1.2: Upsilon resonances and possible interactions of the b quarks. Res-onances below the Υ (4S) can only decay through the b quark and b quarkannihilating. This is illustrated in the diagrams (a-c) on the right. At theΥ (4S) resonance there is enough energy in the excited state to create a pairof B mesons. This is illustrated in diagram d.
1.3 The b→ s`+`− Transition
The first evidence of a b → s penguin process was observed in 1993 by the
CLEO collaboration in a signal of B → K∗(892)γ decays. The Feynman dia-
gram is similar to the b→ s`+`− diagram in Figure 1.1 with the photon being
real (no W box diagram exists in this case). The decay rate for b → s`+`− is
10
suppressed by another vertex coupling constant compared to b → sγ. These
rare decays have three amplitudes contributing differently at different recoil
energies (m2`+`− = q2), and thus they have non-trivial kinematic properties
which can be predicted and measured.
1.3.1 Effective Hamiltonian and the Operator Product Expansion
The physics of heavy quark transitions is often described using the Operator
Product Expansion (OPE) [12], which separates the decay amplitude into
a short distance perturbative portion and a long distance non-perturbative
piece. In this framework, the effective low-energy Hamiltonian relevant to the
b→ s`+`− process can be written as [13]:
H(b → s`+`−) = −4GF√2V ∗
tsVtb
10∑
i=1
Ci(µ)Oi(µ) (1.3)
where GF ≡√
28
(
g2
MW
)2
is the Fermi coupling constant and V ∗tsVtb are the CKM
matrix elements which dominate.
The terms Ci(µ) are the Wilson coefficients [14] which describe the short
distance physics above the energy scale µ; the terms O(µ) are local operators
describing the non-perturbative physics at scales below µ. Both the operators
and the Wilson coefficients depend on the scale at which they are calculated,
while the resulting Hamiltonian is scale independent.
The Wilson coefficients are customarily calculated at the scale of MW
and must be scaled (using a renormalization in the MS scheme [15]) down to
the b mass. In the OPE, physical observables are rewritten in terms of “effec-
11
tive” Wilson coefficients Ceffi which are independent of the renormalization
scheme.
1.3.2 Theoretical Predictions and Previous Measurements
Inclusive decays such as B → Xs`+`−, where Xs refers to any system of
hadrons containing at least one kaon, are the simplest to study theoretically.
In this case heavy quark expansion parameters (HQE) [16, 17] can be used
to make reliable predictions. Three of the ten Wilson coefficients Ceffi are
relevant to the b → s`+`− decay: the electromagnetic operator Ceff7 , and the
vector and axial vector terms Ceff9 and Ceff
10 . The resulting dependence of the
branching fraction as a function of s is given by [18]:
dΓ(B → Xs`+`−)
ds∝ (1 − s)2((1 + 2s)(|Ceff
9 |2 + |Ceff10 |2) (1.4)
+ 4(1 +2
s)|Ceff
7 |2 + 12Re(Ceff7 Ceff∗
9 )),
where s ≡ q2/m2b and q2 ≡ m2
`+`−. For very small values of m``, the rate is
dominated by the second term, proportional to the magnitude of Ceff7 . For
large m``, the rate is dominated by the first term and is proportional to the
magnitude of Ceff9 and Ceff
10 . In the SM, the Wilson coefficients are given in
Table 1.4 [18].
The b-quark mass mb is the largest contribution to the theoretical un-
certainty. To avoid the large uncertainty (≈ 15%), it has become customary
to normalize the branching fraction to the experimentally measured b → ceν
12
Coefficient Value
Ceff7 −0.3094
Ceff9 4.2978
Ceff10 −4.4300
Table 1.4: Wilson coefficients in the Standard Model.
branching fraction:
B(B→Xs`+`−)(s) =B
exp
(B→Xceν)
dΓ(B → Xceν)
dΓ(B → Xs`+`−)
ds(1.5)
The expression for the semileptonic decay width dΓ(B → Xceν) can be found
in Ref. [19].
From an experimental standpoint, exclusive decays such as B → K`+`−
and B → K∗`+`− are easier to measure than the inclusive decays. These
exclusive decays have well defined kinematic properties that can be used to
select events. On the other hand, the use of exclusive decay modes introduce
complications with theory predictions due to strong interaction effects involved
in theB → K(∗) transition. Theoretical calculations rely on form factor models
to describe the hadronic effects. Form factors for the B → K (∗) (where the
() notation can mean K or K∗) transition are calculated in terms of matrix
elements. The matrix element describing the standard weak B → K (∗) current
is [20]:
〈K∗(pK∗)|uγµb|B(pB)〉 =
{
(pK∗ + pB)µ − m2B −m2
K∗
q2qµ
}
fK∗
+ (q2)
+
{
m2B −m2
K∗
q2qµ
}
fK∗
0 (q2). (1.6)
13
The matrix element describing the B → K∗ penguin current is given by:
〈K∗(pK∗)|dσµνqν(1 − γ5)b|B(pB)〉 =
i
mB +mK∗
{q2(pK∗ + pB)µ − (m2B −m2
K∗)qµ}fK∗
T (q2, µ). (1.7)
Here, pB and pK∗ are the B and K∗ meson momenta, q = pB − pK∗, mB and
mK∗ are the meson masses.
In semileptonic decays, the physical range of q2 is 0 GeV/c2 ≤ q2 ≤
(mB − mK∗)2. The form factors f+ and f0 are also relevant to B → π`ν
decays, while fT is relevant only for penguin decays. The signal model used
in this analysis uses form-factor predictions of Ball and Zwicky. This model
includes radiative corrections and the most recent input parameters. Light-
cone QCD sum rules (LQSR) are used to calculate the form factors [20, 21]. In
this framework, the final-state meson is required to have E >> ΛQCD; thus the
calculations only cover 0 ≤ q2 ≤ 14 GeV/c2. Other techniques which have been
used to calculate these form factors include the lattice-constrained constituent
quark model [22] and three-point QCD sum rules [23, 24].
1.3.3 Branching Fractions
Evidence of a radiative penguin decay was first established by the CLEO-II ex-
periment in 1993 in B → K∗(892)γ. CLEO observed 10 events corresponding
to a branching fraction of (4.1± 1.5± 0.9) × 10−5 [25]. Prior to the discovery
of the B → K`+`− and the B → K∗`+`− transitions, searches were conducted
by a number of experiments. Most notably CLEO [26, 27], CDF [28, 29], and
14
Mode World Average (×10−6) Ali et al. (×10−6)B → K`+`− 0.54 ± 0.08 0.35 ± 0.12B → K∗`+`− 1.05 ± 0.20 1.19 ± 0.39
Table 1.5: World average B → K (∗)`+`− branching fractions, compared to arecent SM based prediction.
BABAR [30]. The B → K`+`− decay was first observed by the Belle collabora-
tion in 2002 in a sample of 31 million BB decays [31]. The first evidence for
the B → K∗`+`− decay was reported by in 2003 BABAR using a sample of 123
million BB pairs [32] and by Belle using 152 million BB pairs [33]. Addition-
ally, both Belle and BABAR have reported measurements of the semi-inclusive
B → Xs`+`− rate, where Xs represents a final state with a kaon plus up to
three pions [34, 35].
Table 1.5 and Figure 1.3 show the current measurements of the B →
K(∗)`+`− branching fraction results [1] along with the theoretical predictions
[18]. Existing measurements are consistent with the range of SM predictions.
Currently, the experimental errors in these measurements are comparable to
or smaller than the theoretical uncertainties due to the hadronic form fac-
tors. In the absence of improvements to the form factor calculations, the total
branching fractions in the exclusive modes will not provide a precision test of
the SM.
In the previous section, it was shown that the relationship between
the Wilson coefficients and the branching fraction has a strong dependence
on m``. A measurement of the partial branching fraction as a function of
15
0 0.5 1 1.5 2 2.5Branching Fraction
-l+Kl
-l+l*
K
-1BaBar, 349 fb2008 preliminary
-1BaBar, 208 fbPRD 73, 092001 (2006)
-1CDF 1 fb2006 preliminary
-1Belle, 253 fb2004 preliminaryAli ’02Zhong ’02
-6 10×
Figure 1.3: Current Branching Fraction Results in B → K (∗)`+`− overlayedwith the theoretical predictions.
m`` is sensitive to the relative contribution of the Wilson coefficients. The
b → sγ branching fraction is proportional to the amplitude of the photon
penguin corresponding to the Ceff7 Wilson coefficient. The fact that the mea-
sured branching fraction [36–39] is in excellent agreement with the Standard
Model prediction places strong constraints on how new physics can affect the
magnitude of Ceff7 . However, the sign of Ceff
7 cannot be determined from
measurements of b→ sγ branching fractions. There are new physics scenarios
which allow opposite sign solutions and cannot be ruled out by the b → sγ
branching fraction alone. In Figure 1.4, the partial branching fractions are
illustrated for the Standard Model and several new physics models in which
the value of Ceff7 is modified within the bounds allowed by the b → sγ mea-
surement.
16
1.3.4 K∗ Polarization and Lepton Forward-Backward Asymmetry
In the expression for the total and partial rates (given in Equation 1.5), the
Wilson coefficients enter quadratically, meaning that their magnitude can be
constrained from branching fraction measurements while their relative sign
cannot. Measurements of angular distributions, such as the di-lepton forward-
backward asymmetry AFB and polarization of the K∗ help to resolve this
ambiguity. The forward backward asymmetry as a function of s is defined to
be [40]:
AFB(s) =
∫ 1
0d cos θ`
d2Γ(B→K(∗)`+`−)d cos θ`ds
−∫ 0
−1d cos θ`
d2Γ(B→K(∗)`+`−)d cos θ`ds
dΓ(B → K(∗)`+`−)/ds(1.8)
where θ` is the angle of the lepton with respect to the flight direction of the B
meson. The angle θ` is defined in the dilepton rest frame with a sign determined
by the flavor of the B meson. This analysis will follow the sign convention in
[41]. For a B+ or B0 meson, θ` is the angle between the negatively charged
lepton and the B. For a B− or B0, θ` is the angle between the positively
charged lepton and the B. Decays with cos(θ`) > 0 are defined as “forward”,
while decays with cos(θ`) < 0 are defined as “backward”. Note that only
modes for which the B flavor can be determined are used for this analysis.
For instance, the B0 → K0Sπ0µ+µ− decay cannot be used. The forward-
backward asymmetry is non zero only for B → K∗`+`− decays in which the
K∗ is polarized. The K∗ polarization FL is also dependent on m``. In the
B → K∗`+`− mode, the longitudinal polarization is defined in terms of the
angle θK : the angle between the kaon and the B calculated in the K∗ rest
17
Figure 1.4: Partial branching fractions in B → K`+`− (top) and B → K∗`+`−
(bottom) as a function of s ≡ (m2``). The solid line corresponds to the Standard
Model, with the shaded area representing the uncertainty due to the formfactors. The dotted and long-short dashed lines represent allowed points intwo supersymmetric models [18].
18
frame.
The distribution of AFB as a function of s in B → K∗`+`− depends on
the Wilson coefficients as [40]:
dAFB(B → K∗`+`−)
ds∝ Ceff
10
[
Re(Ceff9 ) +
Ceff7
s
]
. (1.9)
There are then four categories defined by the relative sign of the Wilson coeffi-
cients. If Ceff9 has the sign expected in the SM, the four cases are (illustrated
in Figure 1.5):
• Ceff7 > 0, Ceff
9 Ceff10 > 0. The forward backward asymmetry is positive
at very low s and negative at high s (red squares).
• Ceff7 > 0, Ceff
9 Ceff10 < 0. The forward backward asymmetry is positive
for all s (green circles).
• Ceff7 < 0, Ceff
9 Ceff10 > 0. The forward backward asymmetry is negative
for all s (magenta squares).
• Ceff7 < 0, Ceff
9 Ceff10 < 0. This is the Standard Model case. The forward
backward asymmetry is negative at very low s and positive at high s
(blue dots).
19
The most dramatic deviations from the Standard Model can occur in
cases 1 and 3 when the product Ceff9 Ceff
10 has the same magnitude but opposite
relative sign as the Standard model. A similar effect occurs at very low s where
Ceff7 is the dominant term. However, since the magnitude of the asymmetry
is smaller at low s, it is more difficult to distinguish in this case. In contrast
to AFB, FL is most sensitive to the sign of Ceff7 .
These cases have been considered in a number of specific new physics
scenarios, particularly the case in which Ceff7 has a similar magnitude but
opposite sign as expected in the SM is a common feature of supersymmetric
theories with a large tan(β) [40, 42]. Scenarios resulting in a large negative
asymmetry at high s are investigated in Refs. [40, 41, 43–46].
In the B → K`+`− decay mode, the forward-backward asymmetry is
predicted to be zero for all regions of s in all these cases. The only exception
to this comes if new scalar amplitudes are introduced [46], however any asym-
metry is expected to be of order 0.01 or less [47]. In this analysis, B → K`+`−
serves as a control sample and a cross check for the fit method.
A precise measurement of the shape of the AFB and FL distributions
would require extremely large data samples. For this analysis, the current
BABAR dataset allows for a measurement of AFB and FL in two bins of the
dilepton mass: a “low” region above the photon pole but below the J/ψ res-
onance 0.1 < s < 6.5 GeV/c22
and a high region above the J/ψ resonance
excluding the ψ(2S) resonance s > 10.24 GeV/c22. The theoretical predictions
for AFB and FL are listed in Table 1.6.
20
)4/c2
(GeV2q0 2 4 6 8 10 12 14 16 18 20
FB
A
-1
-0.8
-0.6
-0.4
-0.2
-0
0.2
0.4
0.6
0.8
1
Low High
)4/c2
(GeV2q0 2 4 6 8 10 12 14 16 18 20
0F
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Low High
Figure 1.5: The s dependence of AFB is shown in the top plot and the de-pendence of FL is shown in the bottom plot. The color code is given in thetext.
Mode FL (low) FL (high) AFB (low) AFB (high)B → K`+`− N/A N/A 0 0B → K∗`+`− 0.67 0.48 0.03 0.36
Table 1.6: Current B → K(∗)`+`− predictions for FL and AFB.
21
Previous measurements of the lepton AFB have been performed by the
BABAR and BELLE experiments. The previous BABAR analysis [48] was based
on half the data used in this analysis and was only able to set a lower-limit on
the value of AFB in the lowest s bin. In BELLE’s analysis [49], the SM value
of FL (along with the SM value of the Ceff7 Wilson coefficient) were assumed.
The results are illustrated in Figure 1.6.
22
1.3.5 New Physics and Supersymmetry
Since the b → s`+`− transitions proceed through weakly-interacting particles
with virtual energies near the electro-weak scale, they provide a promising
means to search for effects from new flavor-changing interactions. In many
theories beyond the SM, new heavy particles can replace the t orW in the loop.
The effects have been studied in detail in various supersymmetric theories.
Another source of new physics effects is the potential existence of other penguin
diagrams. In particular, the γ or Z boson can be replaced with a neutral
Higgs boson coupling to the lepton pair. Figure 1.7 illustrates new physics
contributions to the b→ s`+`− decay.
Despite the impressive agreement between experiment and the SM,
there are reasons that there must be sources for new physics. One such reason
is the so-called hierarchy problem (or Higgs divergence): experimental con-
straints indicate that the SM Higgs boson should have a mass of ≈ 250 GeV/c2
or less, however energy corrections to the Higgs mass are quadratically diver-
gent and can be many orders of magnitude larger than this. Another reason
is the unification of forces. The strength of the strong, weak, and electromag-
netic gauge couplings evolve as a function of energy scale. In the SM, these
couplings never unify.
Supersymmetric models are among the most well-motivated extensions
to the SM; they are also the most thoroughly studied. Supersymmetry intro-
duces a new set of ’superpartner’ bosons for each of the SM fermions (and
vice-versa). The SM quarks and leptons are paired with squarks (q) and slep-
23
Figure 1.6: Previous BABAR (top) and BELLE (bottom) AFB results.
24
b t,c,u ss
-H(a)
b u~, c~, t~ ss
-χ(b)
b d~
, s~, b~ ss
0χ, g~(c)
Figure 1.7: Possible new physics contributions to b→ s`+`−.
tons (l); the gluons are paired with gluinos (g). In the minimal extension to
the SM (MSSM), the Higgs sector is expanded to include two Higgs doublets
whose ratio of vacuum expectation values is a free parameter tan(β). The
extra degrees of freedom are expressed in additional Higgs bosons (charged
and neutral).
Supersymmetry addresses the outstanding problems with the SM dis-
cussed above. First, the superpartners partially cancel the quadratic diver-
gences in the Higgs mass. Second, the gauge couplings are unified at an energy
scale of order (≈ 1016 GeV).
Superpartners to the SM have yet to be observed. This implies that
supersymmetry must be a broken symmetry (between the masses of SM par-
ticles and their superpartners). It is generally argued that supersymmetry
should be visible at the TeV energy scale in order to have a stabilizing effect
on the Higgs mass. The other major goal of the LHC is to find and identify
superpartners to the SM particles.
25
1.4 Other B Physics at BABAR
The initial goal of the BABAR experiment is to study CP violation in the B-
meson system. As previously stated, the only source of CP violation in the
SM is due to the irremovable phase in the CKM matrix. The unitarity of the
CKM matrix implies various relations among its elements. One relation is very
useful for understanding the SM predictions for CP violation in the B system:
VudV∗ub + VcdV
∗cb + VtdV
∗tb = 0. (1.10)
From this relation, it is possible to define the Unitarity Triangle (Figure 1.8).
The triangle is derived from Eq. 1.10 by choosing a phase convention such that
VcdV∗cb is real and dividing the lengths of all sides by |VcdV
∗cb|. The angles α
and β can be defined in terms of the Wolfenstein parametrization:
sin 2α =2η[η2 + ρ(ρ− 1)]
[η2 + ρ(ρ− 1)][η2 + ρ2](1.11)
sin 2β =2η(1 − ρ)
η2 + (1 − ρ)2(1.12)
where ρ = ρ(1 − λ2/2) and η = η(1 − λ2/2).
In the B-meson system, there are three possible manifestations of CP
violation: the so called CP violation in decay, in mixing, and in the interference
between mixing and decay. CP violation in decay, or “direct” CP violation, is
observed as the difference between the decay rate of a particle to a final state
and the decay rate of its antiparticle to the corresponding charge-conjugate fi-
nal state. CP violation in mixing, otherwise known as “indirect” CP violation,
occurs when the neutral B-meson flavor eigenstates and the mass eigenstates
26
1.4 Violation in the Standard Model 21
!
" #
$
A%
(b) 7204A57Ð92
1
VtdVtb&
|VcdVcb|&
VudVub&
|VcdVcb|&
VudVub&
VtdVtb&
VcdVcb&
$
#
"
0
0
(a)
Figure 1-2. The rescaled Unitarity Triangle, all sides divided by .
The rescaled Unitarity Triangle (Fig. 1-2) is derived from (1.82) by (a) choosing a phase convention
such that is real, and (b) dividing the lengths of all sides by ; (a) aligns one side
of the triangle with the real axis, and (b) makes the length of this side 1. The form of the triangle
is unchanged. Two vertices of the rescaled Unitarity Triangle are thus Þxed at (0,0) and (1,0). The
coordinates of the remaining vertex are denoted by . It is customary these days to express the
CKM-matrix in terms of four Wolfenstein parameters with playing
the role of an expansion parameter and representing the -violating phase [27]:
(1.83)
is small, and for each element in , the expansion parameter is actually . Hence it is sufÞcient
to keep only the Þrst few terms in this expansion. The relation between the parameters of (1.78)
and (1.83) is given by
(1.84)
This speciÞes the higher order terms in (1.83).
REPORT OF THE BABAR PHYSICS WORKSHOP
Figure 1.8: The Unitarity Triangle.
cannot be chosen to be the same. The final form of CP violation can occur
in B0 and B0 decays to the same final state. In this situation there can be
interference between the direct decay of the meson into the final state and the
alternate path of first mixing into the anti-meson and then decaying into the
final state. All three types of CP violation involve interference between several
amplitudes that lead to the same final state with different phases.
Measurements of the angles and sides of the Unitarity triangle through
processes dominated by tree-level amplitudes have so far shown that the CKM
picture is, to first order, the correct description of CP -violating phenomena
in the SM. In particular, decays of B0 mesons to ccs CP eigenstates (e.g.
B → J/ψK0S) provide the cleanest channel to constrain the parameter sin 2β
[50]. Figure 1.9 shows current agreement of these measurements in the (ρ, η)
plane.
The amount of CP violation originating from the CKM mechanism is
not enough to account for the observed matter-antimatter asymmetry in the
27
Figure 1.9: Current tree-level constraints on the Unitarity Triangle [10].
universe. Having established the CKM mechanism, the aim of the B factories
has extended to the search for signatures of new physics in the form of small
deviations from CKM predictions in highly precise measurements of B decays.
A particular focus is on finding additional sources of CP violation phases from
new physics.
It is not surprising that there is an absence of sizable non-SM effects in
tree-level B decays. The new amplitudes would involve highly off-shell massive
bosons that are suppressed relative to the SM weak amplitudes. A promising
28
area to search for new physics signatures is in the penguin decays described in
§ 1.1.1. As stated, these amplitudes are suppressed in the SM but NP particles
can enter the loop at comparable strength to SM particles because the loop
is virtual. Any CP violating structure in these processes that differs from
SM predictions would indicate the presence of NP contributions. Overall, no
significant deviations from SM predictions have been found in penguin modes
(Figure 1.10). A detailed discussion of the CP analyses performed at BABAR
is beyond the scope of this thesis.
29
sin(2βeff) ≡ sin(2φe1ff)
HF
AG
LP 2
007
HF
AG
LP 2
007
HF
AG
LP 2
007
HF
AG
LP 2
007
HF
AG
LP 2
007
HF
AG
LP 2
007
HF
AG
LP 2
007
HF
AG
LP 2
007
HF
AG
LP 2
007
HF
AG
LP 2
007
b→ccs
φ K
0
η′ K
0
KS K
S K
S
π0 KS
ρ0 KS
ω K
S
f 0 K
0
π0 π0 K
S
K+ K
- K0
-2 -1 0 1 2
World Average 0.68 ± 0.03BaBar 0.21 ± 0.26 ± 0.11Belle 0.50 ± 0.21 ± 0.06Average 0.39 ± 0.17BaBar 0.58 ± 0.10 ± 0.03Belle 0.64 ± 0.10 ± 0.04Average 0.61 ± 0.07BaBar 0.71 ± 0.24 ± 0.04Belle 0.30 ± 0.32 ± 0.08Average 0.58 ± 0.20BaBar 0.40 ± 0.23 ± 0.03Belle 0.33 ± 0.35 ± 0.08Average 0.38 ± 0.19BaBar 0.61 +-
00..2224 ± 0.09 ± 0.08
Average 0.61 +-00..2257
BaBar 0.62 +-00..2350 ± 0.02
Belle 0.11 ± 0.46 ± 0.07Average 0.48 ± 0.24BaBar 0.90 ± 0.07Belle 0.18 ± 0.23 ± 0.11Average 0.85 ± 0.07BaBar -0.72 ± 0.71 ± 0.08Belle -0.43 ± 0.49 ± 0.09Average -0.52 ± 0.41BaBar 0.76 ± 0.11 +-
00..0074
Belle 0.68 ± 0.15 ± 0.03 +-00..2113
Average 0.73 ± 0.10
H F A GH F A GLP 2007
PRELIMINARY
Figure 1.10: Current B-factory CP measurements in penguin decays.
30
Chapter 2
PEP-II and the BaBar Detector
The BABAR experiment is located at the Stanford Linear Accelerator Center
(SLAC) at Stanford University. The primary goal of BABAR is the precision
study of CP violation and rare decay processes in the B meson system. Com-
plementary programs in charm and τ physics are also conducted. This section
describes the BABAR detector and the environment in which it operates.
2.1 PEP-II
The PEP-II facility is an asymmetric e+e− collider, in which the SLAC LINAC
is used to inject 9.0 GeV electrons and 3.1 GeV positrons into separate high-
energy (HER) and low-energy (LER) storage rings. The beams collide at a
center-of-mass energy equal to the mass of the Υ (4S) particle (10.58 GeV/c2)
which has a branching fraction to B-meson (mB = 5.279 GeV/c2) pairs of
nearly 100% [1]. The Υ (4S) system is Lorentz boosted by a factor βγ = 0.56.
The boost allows for the measurement of the B and B decay times critical
for studying time-dependent CP violation. This is much less important for
analyses of rare decays such as B → K (∗)`+`−.
Other processes occur at a high rate in e+e− collisions. These processes
31
Decay Cross-section ( pb)Υ (4S) 1.05cc 1.300uds 2.090τ+τ− 0.900
Table 2.1: Cross-sections for 10.58 GeV center-of-mass energy.
include Bhabhas where e+e− → e+e− or e+e− → e+e−γ, e+e− → µ+µ−γ,
e+e− → τ+τ− and e+e− → qq continuum QED processes where q = udsc type
quarks. Bhabha and µ+µ−γ events are easy to identify and are not important
background sources in this analysis. However, the continuum quark processes
do enter as an important background sources. The cross section for these and
the Υ (4S) process are given in Table 2.1.
The machine has operated efficiently since 1999, delivering a total in-
tegrated luminosity of 553.84 fb−1 in seven different run periods (Figure 2.1).
432.89 fb−1 was dedicated to running at the Υ (4S) resonance. PEP-II also
ran at the Υ (3S) resonance, delivering 30.23 fb−1, and the Υ (2S) resonance,
delivering 14.45 fb−1. Approximately 10% of the data is collected at energies
40 MeV below the Υ (4S) resonance. This data is called OffPeak and is used
to study continuum backgrounds.
2.2 BaBar
BABAR is a general purpose detector designed to support a wide variety of
analyses in flavor physics. To accommodate the large number of analyses to be
performed by BABAR physics program, the detector must satisfy the following
32
]-1
Inte
gra
ted
Lu
min
osi
ty [
fb
0
100
200
300
400
500
Delivered LuminosityRecorded LuminosityRecorded Luminosity Y(4s)Recorded Luminosity Y(3s)Recorded Luminosity Y(2s)Off Peak
BaBarRun 1-7
PEP II Delivered Luminosity: 553.48/fbBaBar Recorded Luminosity: 531.43/fb
BaBar Recorded Y(4s): 432.89/fbBaBar Recorded Y(3s): 30.23/fbBaBar Recorded Y(2s): 14.45/fbOff Peak Luminosity: 53.85/fb
BaBarRun 1-7
PEP II Delivered Luminosity: 553.48/fbBaBar Recorded Luminosity: 531.43/fb
BaBar Recorded Y(4s): 432.89/fbBaBar Recorded Y(3s): 30.23/fbBaBar Recorded Y(2s): 14.45/fbOff Peak Luminosity: 53.85/fb
As of 2008/04/11 00:00
2000
2001
2002
2003
2004
2005
2006
2007
2008
Figure 2.1: Integrated luminosity as a function of time.
33
requirements:
• Excellent vertex reconstruction in the tracker
• Large acceptance, including at small polar angles relative to the boost
direction
• Excellent reconstruction efficiency and good momentum resolution for
charged particles and photons.
• Particle identification to separate lepton, pion, and kaon candidates
• The detector was also built to withstand long-term damage from radia-
tion.
The BABAR detector was built with several detector systems that to-
gether satisfy these requirements. The inner detector includes a silicon ver-
tex tracker (SVT), drift chamber (DCH), ring-imaging Cherenkov detector
(DIRC), and an electromagnetic calorimeter (EMC). Surrounding the inner
detector is a superconducting solenoid producing a 1.5T magnetic field. The
steel flux return is instrumented for muon and neutral hadron identification.
Figure 2.2 gives an overview of the major components of the BABAR detector.
An example of a B meson decaying as B0 → K+π−e+e− in the BABAR
detector can be seen in the event display (Figure 2.3). The cyan lines are
electrons. The pion (red) and the oppositely charged kaon (yellow) are back-to-
back with the di-lepton system. The areas in green represent energy deposits
in the electromagnetic calorimeter.
34
��
� �
���� ����
����
����
����
���
����
��������
�
���
����
�
�
�
��
Scale
BABAR Coordinate System
0 4m
Cryogenic Chimney
Magnetic Shield for DIRC
Bucking Coil
Cherenkov Detector (DIRC)
Support Tube
e– e+
Q4Q2
Q1
B1
Floor
yx
z1149 1149
Instrumented Flux Return (IFR))
BarrelSuperconducting
Coil
Electromagnetic Calorimeter (EMC)
Drift Chamber (DCH)
Silicon Vertex Tracker (SVT)
IFR Endcap
Forward End Plug
1225
810
1375
3045
3500
3-2001 8583A50
1015 1749
4050
370
I.P.
Detector CL
�
�
�� ��
IFR Barrel
Cutaway Section
ScaleBABAR Coordinate System
y
xz
DIRC
DCH
SVT
3500
Corner Plates
Gap Filler Plates
0 4m
Superconducting Coil
EMC
IFR Cylindrical RPCs
Earthquake Tie-down
Earthquake Isolator
Floor
3-2001 8583A51
Figure 2.2: The BABAR detector.
35
Figure 2.3: An example of a b→ s`+`− decay in the BABAR detector. The cyanlines represent electrons. The pion (red) and kaon (yellow) are back-to-backwith the di-lepton system.
2.2.1 SVT
The SVT consists of five layers of double-sided silicon sensors. The SVT
was designed to accurately measure the positions and decay vertices of B
mesons and other particles. The SVT also contributes to hadron identification
through measuring track ionization losses (dE/dx) as a function of position.
36
See Figure 2.4.
580 mm
350 mrad520 mrad
ee +-
Beam Pipe
Space Frame
Fwd. support cone
Bkwd. support cone
Front end electronics
Figure 2.4: Longitudinal schematic of the SVT.
The first three layers sensors are located as close to the beampipe as
possible in order to maximize vertex resolution. The outer two layers are closer
to the DCH to facilitate matching of SVT tracks with DCH tracks. The SVT
polar angle coverage is 20 < θ < 150 degrees, corresponding to 90% coverage
of the solid angle in the CM frame.
The SVT measures the track vertices in z with a resolution of 20µm to
40µm compared to the 250µm mean ∆z separation between the decay vertices
of the two B mesons produced in an event. For particles with transverse
momentum below 120 MeV/c the SVT provides the only tracking information,
as these particles may not reach the drift chamber. The hit reconstruction
efficiency of the SVT is 97% as measured from clean sources such as Bhabha
and dimuon events.
37
2.2.2 DCH
The drift chamber is the primary tracker used for the measurement of the
momenta of charged particles. The DCH is also used to identify low momenta
particles by measuring the ionization loss dE/dx.
The chamber is 2.8 m long and consists of 40 cylindrical layers of 12 mm
by 19 mm hexagonal cells, each consisting of six field wires at the corners and
one sense wire in the center. Approximately half of the layers are oriented
at angles with respect to the z-axis in order to give longitudinal position
information. The space around the wires is filled with a gas mixture containing
80% helium and 20% isobutane. The DCH wire layers operate at a voltage of
approximately 1930V.
The track reconstruction uses a Kalman filter algorithm to find helical
tracks in the DCH. Secondary algorithms then attempt to associate unassigned
hits to tracks and to find tracks that do not originate from the interaction
point. The tracks are then extrapolated and matched to any associated hits
in the SVT.
The DCH has demonstrated excellent performance throughout the life
of BABAR. Track reconstruction efficiencies are at the 96% level, relative to
the number of tracks found in the SVT. For momenta below 0.7 GeV/c the
dE/dx resolution is approximately 7%; this allows greater than 2σ separation
of charged kaons and pions (Figure 2.5).
38
104��
103��
10–1 101
eµ
π
K
pd
dE/d
x
Momentum (GeV/c)1-2001 8583A20
Figure 2.5: dE/dx in the DCH as a function of track momentum for differentcharged particles.
2.2.3 DIRC
The detector of internally reflected Cherenkov light is a ring imaging detector
for charged-particle identification. In particular, the DIRC provides greater
than 2σ separation of kaons and pions from 700 MeV/c to about 4.0 GeV/c.
Cherenkov devices detect photons radiated by particles that move faster than
the speed of light in a given medium. The Cherenkov angle, θC , of the radiated
39
photons is given by
cos(θC) =1
nβ=
c
nv(2.1)
where n is the index of refraction of the medium and v is the particle’s velocity.
For a given momentum, particles of different mass will have different velocities.
This information can be used to differentiate particle mass hypotheses for a
track.
Cherenkov light from a charged particle is transmitted via total inter-
nal reflection within a set of 144 quartz bars. The light is delivered to a water
filled standoff box at the backward end of the detector. The forward end of
the detector is uninstrumented and so a mirror is used to reflect light toward
the instrumented backward end. The rear surface of the standoff box is in-
strumented with 12 sectors of 896 photomultiplier tubes (PMTs) each. The
opening angle of the ring of Cherenkov light emitted by the particle can then
be reconstructed, correcting for the small difference between the index of re-
fraction of quartz (n = 1.473) and the water (n ≈ 1.346). Figure 2.6 shows
the geometry of the DIRC.
The DIRC Cherenkov angle resolution is 2.5 mrad. The DIRC is effec-
tive at separating charged kaons and pions with lab momenta above 0.7 GeV/c,
as illustrated in Figure 2.7. The separation power of the DIRC is quantified by
the difference in the mean measured angle divided by the angular resolution.
At momentum of 3 GeV/c the separation is 4σ; at 4.1 GeV/c the separation
is 2.5σ.
40
Mirror
4.9 m
4 x 1.225m Bars glued end-to-end
Purified Water
Wedge
Track Trajectory
17.25 mm Thickness (35.00 mm Width)
Bar Box
PMT + Base 10,752 PMT's
Light Catcher
PMT Surface
Window
Standoff Box
Bar
{ {1.17 m
8-2000 8524A6
Figure 2.6: Geometry of the DIRC.
2.2.4 EMC
Measuring the properties of photons and neutral hadrons is accomplished using
an electromagnetic calorimeter. The calorimeter allows the identification of
photons and the reconstruction of neutral π0 and η mesons which decay to
two photons. Further, the ratio E/p (energy deposited in the EMC over the
momentum measured in the DCH) and the shower shape measured in the
EMC provide the primary means for identifying electrons.
The EMC consists of 6580 thalium-doped cesium-iodide crystals split
into a barrel and a forward endcap detector. The crystals in the barrel are
arranged in 48 axially symmetric rings while the endcap is a conic section
41
LAB momentum at DIRC, GeV/c0 1 2 3 4 5 6
DIR
C C
her
enko
v an
gle
0.7
0.72
0.74
0.76
0.78
electrons
muons
(for clarity, muons are not shown)
pions
kaons
protons
0.8
0.82
0.84
Figure 2.7: DIRC Cherenkov angle versus energy with curves for various par-ticle hypotheses. Muon data points are not shown, however the hypothesiscurve is.
consisting of 8 rings in which the front and back surfaces are tilted 22.7◦ to the
vertical. The polar angle coverage of the EMC ranges from 15.8 < θ < 141.8
degrees which corresponds to about 90% coverage of the total solid angle.
Figure 2.8 illustrates the geometry of the EMC.
The energy response of the EMC is calibrated using low-energy pho-
tons from a radioactive source and π0 decays. High-energy photons from e+e−
Bhabha events and µ+µ−γ events are used to calibrate the higher energy
ranges. The energy resolution of the EMC is expressed as a term propor-
tional to the inverse fourth root of the energy added in quadrature with a
42
11271375920
1555 2295
2359
1801
558
1979
22.7˚
26.8˚
15.8˚
Interaction Point 1-2001 8572A03
38.2˚
External Support
Figure 2.8: Geometry of the EMC.
constant term. The energy resolution is determined to be (Figure 2.9)
σE
E=
(2.32 ± 0.30)%4√
E( GeV)⊕ (1.85 ± 0.12)% (2.2)
2.2.5 IFR
Outside the EMC, a superconducting solenoid provides a magnetic field of 1.5T
needed to measure charged-particle momenta. The solenoid is surrounded by
steel plates which function as a magnetic flux return. The IFR is instrumented
with limited streamer tubes (LSTs) or resistive plate chambers (RPCs) which
function as the muon detector. The instrumented flux return consists of ap-
proximately 200 m2 of muon chambers. Figure 2.10 gives an overview of the
IFR design.
The IFR is divided into a barrel and two endcaps. The barrel region
was designed with 19 layers of RPCs and has a total iron thickness of 65 cm.
43
γγ→ 0πBhabhas
c
MonteCarloγψ J/→ χ
3-2001 8583A41 Photon Energy (GeV)
10-1 1.0 10.0
σE /
E
0.02
0.02
0.04
0.06
Figure 2.9: EMC resolution as a function of photon energy.
Each endcap has 18 layers of RPCs and a total iron thickness of 60 cm. The
RPCs are designed to detect streamers (electrical discharges in the gas) from
ionizing particles passing through the IFR. Each RPC consists of a 2 mm
gap filled with a gas mixture of isobutane, argon, and freon. Surrounding
the gap are two layers of bakelite covered with linseed oil. Streamers passing
through the chamber induce signals in rows of aluminum strips which cover the
bakelite. The signals from the strips provide a two-dimensional measurement
of the streamer position. The penetration depth of a track in the IFR is used
to distinguish muons. Figure 2.11 illustrates the RPC design.
44
Barrel 342 RPC Modules
432 RPC Modules End Doors
19 Layers
18 LayersBW
FW
3200
3200
920
12501940
4-2001 8583A3
Figure 2.10: Design of the BABAR instrumented flux return.
Shortly after the start of data-taking with BABAR in 1999, the perfor-
mance of the RPCs started to deteriorate rapidly. Many RPCs began drawing
currents and developing large areas of low efficiency. The overall efficiency of
the RPCs dropped and the number of non-functional chambers rose dramati-
cally. This had a deteriorating effect on the muon identification.
In the lifetime of the BABAR detector, the IFR has undergone the most
major upgrades of any other subsystem. The forward endcap with retrofitted
with improved RPCs in 2002. During the summer of 2004 and the summer
of 2006, the IFR was reinstrumented with limited streamer tube technology.
The LSTs consist of a PVC comb of eight 15 mm by 17 mm cells. The comb
is about 3.5 m in length encased in a PVC sleeve with a 100 µm thick gold-
plated beryllium-copper wire running down the center of each cell. The cells
in the comb are covered with graphite, which is grounded, while the wires
45
AluminumX StripsInsulator
2 mm
GraphiteInsulator
SpacersY Strips
Aluminum
H.V
.
Foam
Bakelite
BakeliteGas
Foam
Graphite
2 mm2 mm
8-2000 8564A4
Figure 2.11: Schematic drawing of a BABAR RPC.
are held at 5500 V. The gas mixture is 3.5% argon, 8% isobutane and 88.5%
carbon dioxide. The LSTs also detect streamers in the gas mixture. The
LSTs boost the muon efficiency by 20% at low momenta while also providing
a considerable decrease in the pion mis-identification rate. (See Figure 2.12.)
2.2.6 Trigger
Data relevant for the BABAR physics program is selected for storage from the
flow of collision information collected by the detector using a two-level trigger
system. The BABAR trigger is designed to maintain near 100% efficiency for
BB events. Final event rates are approximately 300 Hz for the data used
in this analysis. Th trigger is implemented in two levels. A level 1 (L1)
hardware trigger selects physics events based on simple detector signals and is
46
Figure 2.12: Muon efficiency before the LSTs (blue) and after the LST upgrade(red) and pion misID before the LSTs (green) and after the LST upgrade(magenta)
used to reduce beam-background to a level acceptable for the software trigger
(≈ 2kHz). The software trigger (L3) decides which events will be stored for
offline processing.
The L1 trigger decision is based on track segments from hits in the
DCH, showers in the EMC and hits in the IFR. IFR information is used
mainly for triggering di-muon events and cosmics, while the DCH and EMC
provide the main trigger inputs for B-physics processes. The L3 software
trigger further refines the event selection in order to reduce the rate Bhabhas
and beam background events. About 75% of the total trigger rate is allocated
to physics events, and the remainder to calibration samples used to study
47
performance.
2.3 Dataflow
After passing through the trigger system, the data is sent through a set of
offline filters before being fully reconstructed. These filters primarily remove
calibration events needed by the detector systems. The first level of recon-
struction after the filters involves track finding in the DCH and cluster finding
in the EMC. Another filter classifies events based on the tracks and clusters.
Events are divided into multi-hadron, e+e− → leptons, photon events, Bhabha
events, etc. This information is then used to choose events that will be fully
reconstructed. Approximately 35% of the events passing the triggers are fully
reconstructed.
The raw data is sent to a computing “farm” in Padova where it under-
goes the full reconstruction and is then sent back to SLAC. The final datasets
containing the fully reconstructed events are in ROOT file format. These
datasets can then be “skimmed” into different types of physics events. The
skim used in this analysis has a rate of approximately 2% (only 2% of the
fully reconstructed events are selected from the final datasets). This greatly
reduces the processing time necessary to create ROOT ntuples which are used
to perform the final analysis described in the next chapters.
48
Chapter 3
Event Selection
The following decay modes of the B meson are studied in this analysis:
• B+ → K∗+`+`− where K∗+ → K+π0 and π0 → γγ
• B+ → K∗+`+`− where K∗+ → K0Sπ+ and K0
S→ π+π−
• B+ → K∗0`+`− where K∗0 → K+π−
where the `+`− pair can be either e+e− or µ+µ−. Charge conjugates are alway
considered. This gives a total of six signal modes used to determine FL and
AFB for the final result. Several control samples are also used to study signal
and background distributions. Most notably, the B → K`+`− modes are
reconstructed. A wrong lepton-flavor control sample is also reconstructed. In
this sample, the `+`− pair is replaced by an e±µ∓ pair. These control samples
will be discussed in more detail in later sections.
Mesons are reconstructed from their decay products, which are detected
as charged tracks or as clusters of energy deposits in the detector. This analysis
relies heavily on the ability of the BABAR detector to identify these tracks as
pions, kaons, or leptons. This section details the algorithms used to identify
charged tracks as well as the final event selection criteria.
49
3.1 Monte Carlo Samples
Various Monte Carlo samples are used to study signal and background behavior
in this analysis. A full model of the BABAR detector is realized using the
GEANT4 toolkit [51]. GEANT4 is used to simulate the passage of particles
through matter. The PHOTOS [52] Monte Carlo algorithm is also used to
implement QED interference and multiple-photon radiation. The Monte Carlo
samples are in the same ROOT form as the data event collections.
• A sample of “generic” BB backgrounds is used to study random combi-
natoric backgrounds from B meson decays. This sample is generated at
approximately three times the luminosity of the data and contains most
known B meson decays.
• A sample of “generic” backgrounds from uds and cc continuum events are
simulated to study random combinatoric backgrounds. The quark level
QED uds processes are generated at approximately the same luminosity
as the data. The cc events are generated at two times the luminosity
expected in the data.
• “Exclusive signal” Monte Carlo is generated for each of the decay modes
using the theoretical framework described in the introduction, including
J/ψ and ψ(2S) final states. These samples are used to study reconstruc-
tion efficiencies and expected signal shapes. They are also used to study
backgrounds due to mis-reconstructing a true signal decay. These back-
50
grounds are called “crossfeed” and are discussed in greater detail in the
next chapter.
The number of simulated events for each MC sample used, and the ratio of
Data to Monte Carlo is given in Tables 3.1-3.2.
Mode Cross-section Events Data/MC( nb) (lumi.)
Generic B+B− 0.525 555572000 0.33Generic B0B0 0.525 552414000 0.33Continuum cc 1.30 591198000 0.77Continuum uds 2.05 695820000 1.03
Table 3.1: Number of MC simulated generic events and the ratio of the numberof BB decays (or, for continuum events, scaled cross-section) in data to thenumber simulated.
3.2 Selection of Neutrals and Tracks
Photons are reconstructed as clusters in the calorimeter that cannot be as-
sociated to any tracks in the DCH. Lateral and longitudinal shape variables
provide discrimination between photons or electrons and hadrons. The lateral
shape is given by
LAT =
∑N
i=1Eir2i
∑N
i=1EiR2i + E1r
20 + E2r
20
(3.1)
where N is the number of crystals associated with a shower, Ei is the energy
deposited in the i-th crystal, ri is the lateral distance from the center of the
shower and the i-th crystal, and r0 is the average distance between two crystals.
51
Mode B(/10−6) Events Data/MC (/10−3)K+e+e− 0.31 530000 0.22K+µ+µ− 0.31 530000 0.23K+π−e+e− 0.69 530000 0.50K+π−µ+µ− 0.52 530000 0.38K0
Sπ+e+e− 0.23 530000 0.17
K0Sπ+µ+µ− 0.17 530000 0.12
K+π0e+e− 0.35 530000 0.25K+π0µ+µ− 0.26 530000 0.19Charmonium B(/10−3)J/ψK+ 1.008 18504000 20.9J/ψK+π− 0.8 8440000 36.4J/ψK+π0 0.47 8638000 20.7J/ψK0
Sπ+ 0.32 8638000 14.3
ψ2sK+ 0.67 985000 261ψ2sK+π− 0.48 898000 205ψ2sK+π0 0.21 862000 93ψ2sK0
Sπ+ 0.14 862000 64
Table 3.2: Number of MC simulated signal events and the ratio of the numberof BB decays in data to the number simulated. In each mode with multiplefinal state hadrons, the generated events decay through the K∗(892) resonance.
The longitudinal shape information is described by ∆Φ which is the difference
between the polar angle where the track intersects the crystal face and the
shower center. Photons are required to deposit a minimum energy of 30 MeV
in the EMC. The lateral moment (LAT) is required to be less than 0.8.
Charged tracks are required to have a distance-of-closest approach
(doca) to the e+e− interaction point of less than 1.5 cm in the x-y plane and
less than 10 cm in the z-direction. Leptons are required to have a minimum of
12 hits in the DCH. Strict particle identification is required for both leptons
52
and kaons. The charged pion candidate from the K∗ decay must fail the kaon
identification algorithm.
Particle identification (PID) is performed using multivariate techniques
such as neural networks or likelihood ratios that combine information from the
various sub-detector systems.
3.2.1 Electron Identification
Tracks identified as electrons must be within the acceptance of the tracking
and EMC detectors. A minimum of four EMC crystals must be associated
to the track in the DCH and the measured energy deposit should be close
to the track momentum. This pre-selection separates electrons from muons.
To separate hadrons from electrons, the following information from the EMC,
DIRC, and DCH is combined to form a likelihood ratio:
• The ratio E/p of the shower energy deposited in the EMC to the track
momentum measured in the DCH.
• The shower shape (LAT, ∆Φ) of the cluster in the EMC.
• The difference between the dE/dx measured in the DCH and the ex-
pected dE/dx under the electron hypothesis.
• The Cerenkov angle θc measured in the DIRC.
The efficiency of the electron selection is evaluated using samples of
electrons selected from e+e− → e+e−γ events. Pion misidentification rates
53
are evaluated from τ and K0S
decays. Overall, the selection efficiency from
this algorithm is above 92% corresponding to a pion misidentification rate of
less than 2%. Figure 3.1 shows the efficiency for e+ and e− as a function of
momentum in the e+e− → e+e−γ data and Monte Carlo control samples. The
pion misidentification rate is shown in Figure 3.2. For this analysis, electrons
are chosen to have a momentum greater than 300 MeV/c.
Bremsstrahlung radiation is also taken into consideration for electrons.
When ultra-relativistic particles are deflected by the field surrounding an
atomic nuclei, they emit photons to conserve four-momentum. For energies
relevant to BABAR, only electrons (due to their small mass) display measurable
losses due to Bremsstrahlung radiation.
Bremsstrahlung is recovered by combining the electron candidates with
nearby photons. The photon must lie within an angular region in the polar
angle θ: |θe − θγ | < 35 mrad and within the following region in the azimuth
angle φ:
φe−0 − 50 mrad < φγ < φe−
cent
φe+cent mrad < φγ < φe+
0 + 50 mrad.
Here, (θ0, φ0) is the initial direction of the electron track evaluated at the
interaction point and (θcent, φcent) is the centroid position of the shower in
the calorimeter. Figure 3.3 shows the effect of the bremsstrahlung recovery
on the e+e− invariant mass distribution (from Monte Carlo simulations of
B → J/ψK(∗) events).
54
p [GeV/c]1 2 3 4 5
∈ef
fici
ency
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0.8
0.9
1 < 141.72θ ≤22.18
, Data+e, MC+e
p [GeV/c]1 2 3 4 5
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ency
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0.9
1 < 141.72θ ≤22.18
, Data-e, MC-e
p [GeV/c]1 2 3 4 5
MC
∈ /
data
∈
0.9
0.95
1
1.05
1.1 < 141.72θ ≤22.18
+e-e
Selector : PidLHElectronSelector Dataset : all-r18b Tables created on 18/1/2007 (Data) , 17/1/2007 (MC)
Figu
re3.1:
Electron
efficien
cyin
thee+e −
→e+e −γ
data
and
MC
control
sample.
55
p [GeV/c]1 2 3
∈ef
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ency
0.001
0.002
0.003
< 137.38θ ≤22.18 , Data+π, MC+π
p [GeV/c]1 2 3
∈ef
fici
ency
0.001
0.002
0.003
< 137.38θ ≤22.18 , Data-π, MC-π
p [GeV/c]1 2 3
MC
∈ /
data
∈
1
1.5
2
2.5 < 137.38θ ≤22.18
+π-π
Selector : PidLHElectronSelector Dataset : all-r18b Tables created on 18/1/2007 (Data) , 17/1/2007 (MC)
Figu
re3.2:
Pion
mis
IDrate
forth
eelectron
selector.
56
2GeV/c2.2 2.4 2.6 2.8 3 3.2
0
50
100
150
200
250
300
350
400
450
) massγ (- e+ -> eψJ/
Bremsstrahlung recovered
invariant mass- e+pure track e
Figure 3.3: Bremsstrahlung recovery in B → K±J/ψ (→ e+e−) charmoniumevents. Much of the tail in the invariant mass distribution can be recoveredusing the recovery method described above.
3.2.2 Muon Identification
Muons are identified by their penetration depth in the IFR. Energy deposited
in the EMC is used to distinguish between muons and electrons, while IFR
information is used to remove hadrons. A neural network algorithm is imple-
mented with the following detector quantities as inputs:
• The number of measured interaction lengths of the muon candidate in
the IFR.
• The difference between the number of measured interaction lengths and
the expected number of interaction lengths under the muon hypothesis.
57
• The continuity of the track in the IFR, defined as:
continuity =Nlayers
Llast − Lfirst + 1,
where Lfirst is the innermost layer hit, Llast is the outermost layer hist,
andNlayers is the total number of layers hit in a three-dimensional cluster.
• The average multiplicity of strips hit per layer.
• The standard deviation of the average strip multiplicity.
• The goodness of fit (χ2/dof) of a third order polynomial fit to the hits
in the three-dimensional cluster.
• The goodness of fit with respect to the track extrapolation from the
DCH.
• The energy deposited in the EMC.
The efficiency of the muon selection is evaluated using a control sample
of e+e− → µ+µ−γ. Pions are most often mis-identified as muons, and the rate
is evaluated using control samples derived from clean D∗ decays. Different
levels of muon efficiency and hadron rejection can be achieved by changing the
cut on the neural network output values. The selection used to reconstruct
signal candidates in this analysis is rather tight and corresponds to an efficiency
of about 70% for momenta higher than 1 GeV with a pion misidentification rate
of 2 − 3%. Muons are selected with a momentum of greater than 700 MeV/c
to reduce the pion fake rate. Figure 3.4 shows the efficiency as a function of
58
momentum. The misidentification rate as a function of momentum is shown
in Figure 3.5.
A looser selection (i.e. looser cut on the NN output) of muons is also
studied as a control sample for studying hadronic peaking backgrounds. This
loose selection gives a muon identification efficiency of 90% corresponding to a
pion mis-identifiaction rate of 8%. This control sample will be discussed later.
3.2.3 Kaon Identification
Kaon identification algorithms combine information from the SVT, DCH, and
DIRC into a likelihood function:
LK = LSV TK × L
DCHK × L
SV TK
The detector quantities considered in the likelihood are:
• The difference between the dE/dx measured in the DCH and the ex-
pected dE/dx under the kaon hypothesis.
• The difference between the dE/dxmeasured in the SVT and the expected
dE/dx under the kaon hypothesis.
• The Cerenkov angle θc measured in the DIRC.
• The number of observed photons in the DIRC.
• The quality of the track prior to reaching the DIRC.
59
p [GeV/c]1 2 3 4 5
∈ef
fici
ency
0.4
0.6
0.8
1 < 147.00θ ≤17.00
, Data+µ, MC+µ
p [GeV/c]1 2 3 4 5
∈ef
fici
ency
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0.6
0.8
1 < 147.00θ ≤17.00
, Data-µ, MC-µ
p [GeV/c]1 2 3 4 5
MC
∈ /
data
∈
0.7
0.8
0.9
1
1.1 < 147.00θ ≤17.00
+µ-µ
Selector : NNTightMuonSelection Dataset : all-r18b Tables created on 18/1/2007 (Data) , 17/1/2007 (MC)
Figu
re3.4:
Muon
selectioneffi
ciency
asa
function
ofm
omen
tum
.
60
p [GeV/c]1 2 3 4 5
∈ef
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ency
0
0.02
0.04
0.06
0.08
0.1 < 147.00θ ≤17.00
, Data+π, MC+π
p [GeV/c]1 2 3 4 5
∈ef
fici
ency
0
0.02
0.04
0.06
0.08
0.1 < 147.00θ ≤17.00
, Data-π, MC-π
p [GeV/c]1 2 3 4 5
MC
∈ /
data
∈
0.6
0.8
1
1.2
1.4
< 147.00θ ≤17.00
+π-π
Selector : NNTightMuonSelection Dataset : all-r18b Tables created on 18/1/2007 (Data) , 17/1/2007 (MC)
Figu
re3.5:
Pion
mis
IDrate
ofth
em
uon
selector.
61
Kaon candidates are also required to fail the electron identification algorithm.
Kaon efficiency is evaluated using a sample of kaons from the decay
D → Kπ where the D is selected from the decay of a D∗. Kaons are most
often misidentified as pions. The pion mis-identification rate is evaluated using
pions from the same source. The efficiency of the kaon selection is more than
80% for most of the momentum spectrum with a misidentification from pions
of 2−3%. Figure 3.6 shows the efficiency of the kaon identification algorithm as
a function of momentum. The pion mis-identification rate of the kaon selector
is shown in Figure 3.7. As previously stated, pions are required to fail the
kaon identification algorithm.
3.3 Kinematic Selection
B meson candidates are formed by combining their decay products (i.e. adding
four-momenta). A constraint forces the decay products to originate from the
same decay vertex. A B candidate is selected if the decay products of the B
satisfy the following requirements:
• Electron momentum: pLAB ≥ 0.3 GeV/c.
• Muon momentum: pLAB ≥ 0.7 GeV/c.
• K0S
candidates must satisfy 0.4887 < MK0S< 0.5073 GeV/c2.
• π0 candidates are mass constrained after the following cuts are applied
to the photons:
62
p [GeV/c]1 2 3 4 5
∈ef
fici
ency
0.7
0.8
0.9
1 < 146.10θ ≤25.78
, Data+K, MC+K
p [GeV/c]1 2 3 4 5
∈ef
fici
ency
0.6
0.8
1 < 146.10θ ≤25.78
, Data-K, MC-K
p [GeV/c]1 2 3 4 5
MC
∈ /
data
∈
0.9
0.95
1
1.05
1.1 < 146.10θ ≤25.78
+K-K
Selector : TightLHKaonMicroSelection Dataset : all-r18b Tables created on 18/1/2007 (Data) , 17/1/2007 (MC)
Figu
re3.6:
Kaon
efficien
cyas
afu
nction
ofm
omen
tum
.
63
p [GeV/c]1 2 3 4 5
∈ef
fici
ency
0
0.05
0.1
0.15
0.2 < 146.10θ ≤25.78
, Data+π, MC+π
p [GeV/c]1 2 3 4 5
∈ef
fici
ency
0
0.05
0.1
0.15
0.2 < 146.10θ ≤25.78
, Data-π, MC-π
p [GeV/c]1 2 3 4 5
MC
∈ /
data
∈
0
1
2
3 < 146.10θ ≤25.78
+π-π
Selector : TightLHKaonMicroSelection Dataset : all-r18b Tables created on 18/1/2007 (Data) , 17/1/2007 (MC)
Figu
re3.7:
Pion
mis
IDrate
ofth
ekaon
selector.
64
– 0.115 < Mγγ < 0.150 GeV/c2
– Eγ > 0.05 GeV
– LATγ < 0.8
The fully reconstructed exclusive final states B → K (∗)`+`− can be
discriminated from continuum events, and from other B decay backgrounds,
by using the kinematic quantities:
mES =
√
s
2+
(p0.pB)2
E20
− p2B ∆E = E∗
B −√s
2
where pB is the B momentum in the lab frame, E∗B is the B energy in the
center-of-mass (CM) frame, E0 and p0 are the energy and momentum of the
Υ (4S) in the lab frame, and√s is the total CM energy. For the decay modes
with a K∗ in the final state, the reconstructed Kπ mass of K∗ candidates,
mKπ, is also useful. Correctly reconstructed signal events will peak at the B
mass in mES and near zero for ∆E.
3.4 Background Suppression
Backgrounds relevant for this analysis are divided into two categories: com-
binatoric backgrounds and backgrounds that peak in mES and ∆E. Com-
binatoric background is reduced by implementing neural network (NN) al-
gorithms that combine several discriminating event shape variables. Peaking
backgrounds are vetoed where possible, or measured using data or Monte Carlo
control samples.
65
The largest source of peaking backgrounds are decays of the type B →
J/ψK(∗) and B → ψ(2S)K(∗) where the J/ψ or ψ(2S) decays to a `+`−
pair. These events enter the B → K (∗)`+`− sample at a rate of 1000 times the
expected branching fraction and also serve as a control sample.
The other significant source of peaking backgrounds are hadronic B
decays to final states of the type K (∗)h+h−, where the hadron, h, can be
either a charged kaon or pion. These will fake a signal candidate when both
hadrons are misidentified as muons. The majority of these events originate
from the decay B → Dπ where D → K∗π. These can be vetoed by assuming
the µ is a π and removing events where the K∗µ invariant mass is consistent
with the D mass.
3.4.1 Charmonium Vetoes
For the electron modes, the J/ψ veto region is the union of the following three
regions in the ∆E −m`` plane, where ∆E is in GeV and m`` is in GeV/c2:
• A di-lepton mass cut 2.90 < m`` < 3.20
• For m`` > 3.20, a band in the ∆E −m`` plane, 1.11m`` − 3.58 < ∆E <
1.11m`` − 3.25
• For m`` < 2.90, a triangle in the ∆E −m`` plane, ∆E < 1.11m`` − 3.25
For the muon modes, the J/ψ veto region is the union of the following three
regions in the ∆E −m`` plane:
66
• A di-lepton mass cut 3.00 < m`` < 3.20
• For m`` > 3.20, a band in the ∆E −m`` plane, 1.11m`` − 3.53 < ∆E <
1.11m`` − 3.31
• Form`` < 3.00 GeV/c2, a triangle in the ∆E−m`` plane, ∆E < 1.11m``−
3.31
For both the electron and muon modes, the ψ(2S) veto region is the union of
the following three regions in the ∆E −m`` plane:
• A di-lepton mass cut 3.60 < m`` < 3.75
• For m`` > 3.75, a band in the ∆E −m`` plane, 1.11m`` − 4.14 < ∆E <
1.11m`` − 3.97
• For m`` < 3.60, a triangle in the ∆E −m`` plane, ∆E < 1.11m`` − 3.97
Figure 3.8 shows the vetoed region for the B+ → K+e+e− final state. There
is an additional charmonium veto imposed on the electron modes for those
events which escape the vetoes described above. If a photon which does not
arise from electron bremsstrahlung is incorrectly associated with an electron,
the event could escape the veto on the m`` mass. This possibility is reduced by
requiring that the original dielectron mass, without bremsstrahlung recovery,
does not lie in the regions 2.90 GeV/c2 < m`` < 3.20 GeV/c2 or 3.60 GeV/c2 <
m`` < 3.75 GeV/c2.
67
Figure 3.8: B+ → K+e+e− Charmonium Veto Region.
3.4.2 Vetoes Against B → Dπ Backgrounds
Because of the much higher fake rates for muons compared to electrons, ve-
toes against B → Dπ backgrounds are applied only in the di-muon modes.
Approximately 100, 000 decays of the type B → D(→ Kπ)π are expected in
the data sample. If both pions are misidentified as muons, decays of this type
will satisfy the selection requirements and peak in the signal region for the
K`` modes. Similarly, a decay of the form B → D(→ K∗π)π could peak
in the signal region for the K∗`` modes. For the K`` modes there can also
be a triple-fake background if a charged kaon and an opposite-sign pion are
misidentified as muons and the remaining pion is misidentified as a kaon.
The K±, K∗0, or K∗± candidate 4-vector is combined with a muon
candidate 4-vector whose charge is consistent with the appropriate D decay.
The invariant mass of the K(∗)µ system is calculated assuming the muon is a
68
pion, and the event is vetoed if the mass lies between 1.84 and 1.90 GeV/c2.
In the B± → K±µ+µ− mode the event is also vetoed if the invariant mass of
the µ+µ− pair, with one muon assumed to be a kaon and the other a pion, is
consistent with a D decay. This vetoes triple fakes.
3.4.3 Continuum Suppression with Neural Networks
Neural networks (NN) are used to suppress combinatorial backgrounds from
continuum and BB events. These are trained with signal MC and generic
continuum and BB simulated events. The events selected for the NN training
and testing samples are required to pass both the skim cuts and the selection
criteria discussed above. The NN training and selection is separated into two
bins of low and high di-lepton mass, which are separated by the J/ψ resonance.
NNs are trained separately for udsc continuum and BB backgrounds. In the
low-mass region a further cut of m`` > 0.1 is made, in order to suppress the
large contribution to the di-electron K∗ modes from the pole region. There
are four separate NNs trained for each signal mode: continuum suppression
at low di-lepton mass; continuum suppression at high di-lepton mass; BB
suppression at low di-lepton mass; BB suppression at high di-lepton mass.
With eight signal modes (6 K∗`` and 2 K``), this gives a total of 32 NNs to
be trained.
In general event shape variables are used to characterize an event.
Υ (4S) events tend to have a spherical shape in the CMS (the Bs decay isotrop-
ically) whereas continuum quark events tend to be jet-like. The jet-like shape
69
of the event is given in terms of the thrust:
Thrust = max~n
∑
i |~pi · ~n|∑
i |~pi|. (3.2)
Here, the sum goes over final state hadrons with momenta pi in the CMS,∑
i pi = 0, and the unit vector ~n is chosen in such a way that the r.h.s takes
the maximally possible value. Preferable direction defined by the vector ~n is
called the thrust axis.
Thirteen variables were used as inputs for the NNs:
• The ratio of Fox-Wolfram moments R2 = H2/H0, computed in the CMS
using all tracks and neutral clusters in the event [53].
• The ratio of Legendre moments L2/L0, computed in the CMS using all
tracks and neutral clusters in the event [54].
• The mES of the rest of the event (ROE), mROEES , computed in the lab
frame by summing all tracks and neutral clusters which are not used to
reconstruct the signal candidate.
• The ∆E of the rest of the event, ∆EROE , computed in the CMS from
the same recoiling B candidate used in the calculation of mROEES .
• The magnitude of the total transverse vector momentum of an event,
computed in the lab frame using all tracks and neutral clusters
• The distance of closest approach along the z-axis to the primary inter-
action point by the di-lepton system.
70
• The distance of closest approach in the xy-plane to the primary interac-
tion point by the di-lepton system.
• The following function of the vertex probability of the B candidate:
acos([log10(vtxBprob) + 10]/10)
2π(3.3)
• The same functional form as directly above except substituting the vertex
probability of the di-lepton system.
• The value cos θB, where θB is the angle between the B candidate’s mo-
mentum and the z axis in the CM frame.
• The value cos θthrust, where θthrust is the angle between the event’s thrust
axis and the z axis in the CM frame.
• The value cos θROEthr , where θROE
thrust is the angle between the ROE thrust
axis (i.e., calculated with respect to the tracks and clusters comprising
the B candidate used for mROEES and ∆EROE) and the z axis in the CM
frame.
• The value ∆ cos θthrust, which is the cosine of the opening angle in the
CM frame between the angles which are the arguments of cos θROEthr and
cos θthrust.
A slightly different selection of the above inputs is used for each of
the four neural networks belonging to a particular mode. The assignment
71
of the input parameters to a particular NN is based on their discriminating
power against background events. Figure 3.9 shows the 13 input variables
for signal, BB background, and udsc continuum events for B0 → K+π−e+e−
events in the low s region. Figure 3.10 shows the output distributions for the
BB and continuum NNs. Figure 3.11 shows the background rejection versus
signal efficiency curve for B0 → K+π−e+e− NNs in the low s region for the
continuum NN and the BB NNs.
3.5 Multiple Candidate Selection
After all the selection criteria have been applied, 1-2% of the events still contain
more than one reconstructed B candidate in a particular mode. A multiple
candidate selection is performed after all other analysis cuts to select a single
candidate. This includes the final cuts on ∆E, the hadronic mass and the
event selection NNs, which are described below. Typically such events will
have two or three different candidates, where it is usually one (or more) of
the hadron tracks that differs among them. The multiplicity of candidates in
the K∗ modes is somewhat higher than in the K modes, reaching 20% for the
final states containing a π0. A single candidate in a particular mode is selected
according to these rules:
• For B+ → K+`+`− events, the candidate with the largest number of
DCH hits on the K± track.
• For B0 → K∗0`+`− and B+ → K∗+`+`− events with a K0S, the candidate
72
Figure 3.9: B0 → K+π−e+e− Low s NN inputs. The sig-nal events ( green), generic BB MC (blue), and continuum MCdistributions (red) are each normalized to unit area. The plotlabels correspond to the following scaled NN inputs described inthis section: R2= R2; L20=L2/L0; mESr=mROE
ES ; dEr=∆EROE ;pT=pt; dz=DOCA(z); dxy=DOCA(xy); vtxB=vtx(B); vtxlep=vtx(``);cosPcm=cos θB; costhz=cos θthrust; costhzr=cos θROE
thr ; cosththr=∆ cos θthrust.
73
Figure 3.10: Top: B0 → K+π−e+e− low s BB neural network output. Bot-tom: B0 → K+π−e+e− low s udsc neural network. Signal is green, backgroundis blue.
74
Signal Efficiency0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Bkg
Rej
ecti
on
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Signal Efficiency0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Bkg
Rej
ecti
on
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3.11: Top: B0 → K+π−e+e− low s BB background rejection curve.Bottom: udsc NN background rejection curve.
75
with the largest number of SVT hits on the π± track.
• For B+ → K∗+`+`− events with a π0, the candidate with the π0 mass
closest to 135 MeV/c2.
• In the very small number of cases in which the above rules results in a
tie, the first candidate that appears in our ntuple for that event.
3.6 Cut Optimization
Initially, the goal of the analysis was to make measurements in four disjoint
bins of s. It turns out this goal was too optimistic. However, cut optimization
was performed in four bins: two bins below the J/ψ , one bin between the J/ψ
and ψ(2S), and a bin above the ψ(2S). These bins are defined in Table 3.3.
Finally, events in the LOW− and LOW+ bin are combined into a single “low”
Region s min ( GeV2) s max ( GeV2)LOW− 0.10 4.20
LOW+ 4.20 6.25
J/ψ 8.41 10.24MID 10.24 12.96
ψ(2S) 12.96 14.06HIGH 14.06 (mB −mK(∗))2
Table 3.3: s regions (in bold) to be measured for B → K (∗)`+`−. The vetoedcharmonium regions are listed for reference.
bin and events in the MID and HIGH bins are combined into a single “high”
bin.
76
Mode Mass bin low ∆E high ∆E low M(K∗) high M(K∗)µ+µ− low s −0.04 0.04 0.82 0.97e+e− low s −0.07 0.04 0.82 0.97µ+µ− high s −0.05 0.05 0.82 0.97e+e− high s −0.08 0.05 0.82 0.97
Table 3.4: Optimized ∆E and hadronic mass cuts in GeV and GeV/c2 respec-tively, for muon and electron modes in low and high di-lepton mass bins.
The final selection is optimized for signal significance. The ∆E range
for each mode, and the hadronic mass for the K∗ modes, are varied simulta-
neously with the NN selections. This is done separately for each signal mode
in the low and high di-lepton mass bins. The value of S/√S +B is calculated
for mES > 5.27, where S and B are the number of signal and random com-
binatoric background events, respectively. The result of this exercise is four
separate sets of cuts on ∆E and (where appropriate) the hadronic mass for
muon and electron modes, both above and below the J/ψ mass. These are
listed in Table 3.4.
The second stage of the event selection optimization seeks to simultane-
ously optimize across all contributing signal modes the NN selections in each
di-lepton mass bin after fixing the ∆E and M(K∗) cuts given in Table 3.4.
Again the value of S/√S +B is calculated for mES > 5.27.
3.6.1 Kinematic Regions
Once the final selection cuts are finalized, two regions in the space spanned by
mES,∆E, and the mKπ become relevant for the rest of the analysis.
77
• Signal Region: This region contains essentially all of the signal. It is
used to do the angular fits. It is defined by
5.27 GeV/c2 < mES < 5.29 GeV/c2
−0.08 < ∆E < 0.05 GeV (or slightly narrower depending on the
signal mode)
0.82 GeV/c2 < mKπ < 0.97 GeV/c2 for K∗ `+`− modes
• mES Sideband: This region is defined by:
5.2 < mES < 5.27 GeV/c2
−0.08 < ∆E < 0.05 GeV (or slightly narrower depending on the
signal mode)
0.82 GeV/c2 < mKπ < 0.97 GeV/c2 for K∗ `+`− modes
3.6.2 Selection Efficiencies
Once all selection criteria have been established, the selection efficiency is eval-
uated using simulated signal events. The candidate reconstruction and event
selection described in this chapter is implemented in this so called signal MC
sample. The efficiency is then just the number of selected signal events divided
by the total number of events generated. The efficiencies would normally be
used to make branching fraction calculations. In this analysis, they are used
to compare to efficiencies in the J/ψ and ψ(2S) control sample (for which the
branching fractions are calculated as a crosscheck of the fit described in the
next chapter).
78
Table 3.5 shows the final efficiency for reconstructing signal events in
each mode and s bin after all event selection cuts have been imposed.
3.6.3 Expected Signal and Background Yields
Table 3.6 shows the expected yields of signal and combinatorial background
in the region 5.274 < mES < 5.286 for the ∼ 349 fb−1 dataset. The expected
yields are computed by summing over, respectively, all six K∗ and both K
modes using the sets of cuts resulting from the event selection optimization
exercise described above, along with all other event selection cuts described
in this section, for each of the four s bins. The expected signal yields are
calculated assuming the previous BABAR measurement of the BFs. Table 3.7
lists the expected signal and background yields for each individual mode over
the whole fit range mES > 5.2.
79
Mode s bin Reconstruction Efficiency (%)B+ → K+µ+µ− LOW− 6.2
LOW+ 14.0MID 15.3HIGH 15.4
B+ → K+e+e− LOW− 21.6LOW+ 21.8MID 21.3HIGH 19.5
B+ → K+π0µ+µ− LOW− 1.5LOW+ 3.2MID 4.3HIGH 5.7
B+ → K0Sπ+µ+µ− LOW− 3.6
LOW+ 6.1MID 6.0HIGH 8.4
B0 → K+π−µ+µ− LOW− 4.5LOW+ 6.6MID 9.4HIGH 9.5
B+ → K+π0e+e− LOW− 5.3LOW+ 7.0MID 6.5HIGH 12.4
B+ → K0Sπ+e+e− LOW− 11.2
LOW+ 8.5MID 7.7HIGH 10.6
B0 → K+π−e+e− LOW− 8.7LOW+ 10.0MID 10.9HIGH 12.6
Table 3.5: Final reconstruction efficiency for signal events by mode and s bin.
80
Mode s bin Signal Background Background Significance(Sig. Reg.) (Sig. Reg.) (Fit Reg.) (σ)
B+ → K+`+`− LOW− 8.1 2.5 48.1 2.5LOW+ 10.3 1.0 77.2 2.6MID 6.1 1.0 67.2 1.9HIGH 9.4 2.1 47.8 2.7
B → K∗`+`− LOW− 8.9 6.5 82.3 2.3LOW+ 9.4 5.6 80.9 2.4MID 8.4 6.2 122.7 2.2HIGH 13.0 12.4 162.6 2.6
Table 3.6: Expected signal and combinatorial background yields in the signalregion 5.274 < mES < 5.286, and the fit region 5.2 < mES assuming theprevious BABAR branching fraction result. The last column gives the expectedstatistical significance of the signal yield.
81
Mode s bin Signal Events Background Background(Signal Region) (Signal Region) (Fit Region)
B+ → K+µ+µ− LOW− 1.7 0.3 6.0LOW+ 4.0 0.7 33.0MID 2.6 0.7 19.2HIGH 4.3 2.1 32.3
B+ → K+e+e− LOW− 6.4 2.2 42.1LOW+ 6.3 0.3 44.2MID 3.5 0.3 48.0HIGH 5.0 0.0 15.5
B+ → K+π0µ+µ− LOW− 0.4 1.4 9.8LOW+ 0.6 1.0 9.2MID 0.7 0.3 17.2HIGH 1.2 5.0 28.9
B+ → K0Sπ+µ+µ− LOW− 0.6 0.0 7.2
LOW+ 0.8 0.3 9.0MID 0.7 0.3 5.6HIGH 1.2 1.0 26.6
B0 → K+π−µ+µ− LOW− 2.1 0.8 17.7LOW+ 2.7 1.0 16.8MID 3.0 2.8 52.5HIGH 3.9 2.2 33.4
B+ → K+π0e+e− LOW− 1.3 1.4 9.2LOW+ 1.4 1.8 26.6MID 1.0 1.0 16.1HIGH 2.3 2.2 50.5
B+ → K0Sπ+e+e− LOW− 1.8 0.3 21.2
LOW+ 1.1 0.7 5.5MID 0.8 0.3 8.1HIGH 1.3 1.0 5.6
B0 → K+π−e+e− LOW− 2.8 2.6 17.2LOW+ 2.7 0.7 13.7MID 2.2 1.4 23.1HIGH 3.1 1.0 17.6
Table 3.7: Expected signal and combinatorial background yields in signal re-gion 5.274 < mES < 5.286 and in the fit region mES > 5.2 in each mode andeach s bin.
82
Chapter 4
Fit Procedure
To extract the B → K∗`+`− signal yield, longitudinal K∗ polarization, FL, and
the lepton forward-backward asymmetry AFB in each s bin, an unbinned ex-
tended maximum likelihood fit is implemented. For the B → K∗`+`− a three-
dimensional probability density function (PDF) in the variables is mES, cos θK ,
and cos θ` is used. B+ → K+`+`− modes are fit using a two-dimensional PDF
in the variables mES and cos θ` to extract the signal yield and AFB.
The PDF’s are built using the RooFit package in the ROOT framework.
The MINUIT algorithm [55] is used to optimize the floating parameters in the
fit by minimizing the log of the likelihood function − log(L). The likelihood
function used for the fit to the B → K∗`+`− data is:
L = exp
−Nhyp∑
i=1
ni
Nk∏
j=1
Nhyp∑
i=1
niP(xj; αi)
where Nhyp is the number of event hypotheses, ni is the yield of each hy-
pothesis, and Nk is the number of candidate events observed in data. Since
correlations among observables (mES, cos θK and cos θ`) )are found to be small,
the PDF P(xj; αi) for the ith event hypothesis is defined as the product of in-
dividual PDFs for each fit observable xj given the set of parameters αi. This
83
analysis considers several hypotheses for the origin of the events: signal, com-
binatoric backgrounds from continuum and BB events, peaking backgrounds
and mis-reconstructed signal decays. These backgrounds and the sources for
the PDFs are described in this chapter.
Various control samples are used to study the fit method and also pro-
vide a source for some of the components of the fit. This chapter describes
these control samples and the PDFs used to model the signal and background
shapes used in the fit.
4.1 Fit Model
Analytic functions are used to model the line shape of the peaking signal
distributions in each dimension of the fit. When possible, peaking backgrounds
are modeled using data control samples. There are cases when the Monte Carlo
must be used to model line shapes.
Backgrounds modeled in the fit include:
• combinatoric background from B decays and continuum events
• peaking hadronic background from B → K (∗)µh events
• crossfeed from mis-reconstructed signal events.
The sources for the PDF shapes are discussed in this section.
84
4.1.1 Signal PDFs
The signal yield is extracted from a fit to the mES distribution. The probability
distribution function (PDF) used to model the signal component is parame-
terized as a Gaussian in which the mean and the width of the Gaussian are
fixed to the values from fits to the B → J/ψK (∗) control sample. These fits
will be discussed in the next section.
To extract FL and AFB, the angular distributions cos θK and cos θ` are
added as extra fit dimensions. As described in section 1.3.4, the kaon decay
angle θK is defined as the angle between the kaon and the B measured in the
K∗ rest frame. θ` is defined as the angle between the `−(`+) and the B (B)
measured in the dilepton rest frame. The signal shape in cos θK is described
by an underlying differential distribution which depends on the longitudinal
polarization FL as [56]:
1
Γ
dΓB → K∗`+`−
d cos θK
=3
2FL cos2 θK +
3
4(1 − FL)(1 − cos2 θK) (4.1)
The differential decay rate for signal in cos θ` is then described in terms of
FL and the forward-backward asymmetry term AFB which enters linearly in
cos θ` [56]:
1
Γ
dΓ(B → K∗`+`−)
d cos θ`
=3
4FL(1 − cos2 θ`)
+3
8(1 − FL)(1 + cos2 θ`) + AFB cos θ`. (4.2)
In the B → K`+`− mode (where there is no K∗ polarization distribu-
85
tion), the most general form for the angular distribution is:
1
Γ
dΓ(B → K`+`−)
d cos θ`
=3
4(1 − FS)(1 − cos2 θ`)
+1
2FS + AFB cos θ` (4.3)
where FS is the scalar contribution. As discussed in § 1.3.4, the scalar compo-
nent is expected to be small even in the presence of new physics. In the limit
of zero scalar contribution, the distribution reduces to:
1
Γ
dΓ(B → K`+`−)
d cos θ`
=3
4(1 − cos2 θ`) + AFB cos θ` (4.4)
To extract the values of AFB and FL modeled in the signal MC, these PDFs
can be fit to the generated events in various s bins. An example of such a fit
is given in Figure 4.1. Fitting in a range of s bins will yield the distribution
of FL and AFB modeled in the MC (Figure 4.2).
The true angular distributions will be modified by detector acceptance
and efficiency effects. To account for this, signal shape PDFs are defined as
the product of the true angular distribution with non-parametric histogram
PDFs describing the efficiency as a function of cos θK or cos θ`. These are
derived separately for each decay channel using signal Monte Carlo simulation.
Example efficiency PDFs for B+ → K+π0e+e− modes in the high s and B0 →
K+π−e+e− modes in the low s as a function of cos θK and cos θ` in the low s
region are given in Figures 4.3-4.6.
86
kHel-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.02
)
0
100
200
300
400
500
600
700
800
A RooPlot of "kHel"
kHel-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.02
)
0
100
200
300
400
500
600
700
800
A RooPlot of "kHel"
lepHel-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.02
)
0
100
200
300
400
500
A RooPlot of "lepHel"
lepHel-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.02
)
0
100
200
300
400
500
A RooPlot of "lepHel"
Figure 4.1: Fits to generated B0 → K+π−e+e− signal MC events. Top: Fitto cos θK , Bottom: fit to cos θ`
87
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
FL vs q2: Kpi0ee Generator Truth
q20 2 4 6 8 10 12 14 16 18 20
AF
B
-0.2
-0.1
0
0.1
0.2
0.3
0.4
AFB vs q2: Kpi0ee Generator Truth
Figure 4.2: FL (top) and AFB (bottom) distributions as a function of s asmodeled in the signal MC. Each point is the result of a fit of cos θK and cos θ`
for FL and AFB in a narrow s range surrounding each point. The s value atwhich the result of each fit is reported is the center of the narrow s region.
88
truKHist103__kHelEntries 4470Mean 0.1002RMS 0.5321Underflow 0Overflow 0Integral 2.995
kHel-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.04
)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
truKHist103__kHelEntries 4470Mean 0.1002RMS 0.5321Underflow 0Overflow 0Integral 2.995
Histogram of truKHist103__kHel
Figure 4.3: B+ → K+π0e+e− efficiency as a function of cos θK in the high sbin.
truKHist120__kHelEntries 7926Mean 0.04265RMS 0.568Underflow 0Overflow 0Integral 4.154
kHel-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.04
)
0
0.02
0.04
0.06
0.08
0.1
0.12
truKHist120__kHelEntries 7926Mean 0.04265RMS 0.568Underflow 0Overflow 0Integral 4.154
Histogram of truKHist120__kHel
Figure 4.4: B0 → K+π−e+e− efficiency as a function of cos θK in the low sbin.
89
truLepHist103__lepHel
Entries 4470Mean 0.03136RMS 0.5811Underflow 0Overflow 0Integral 2.906
lepHel-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.04
)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
truLepHist103__lepHel
Entries 4470Mean 0.03136RMS 0.5811Underflow 0Overflow 0Integral 2.906
Histogram of truLepHist103__lepHel
Figure 4.5: B+ → K+π0e+e− efficiency as a function of cos θ` in the high sbin.
truLepHist120__lepHel
Entries 7926Mean -0.007748RMS 0.4668Underflow 0Overflow 0Integral 3.974
lepHel-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.04
)
0
0.02
0.04
0.06
0.08
0.1
0.12
truLepHist120__lepHel
Entries 7926Mean -0.007748RMS 0.4668Underflow 0Overflow 0Integral 3.974
Histogram of truLepHist120__lepHel
Figure 4.6: B0 → K+π−e+e− efficiency as a function of cos θ` in the low s bin.
90
4.1.2 Combinatoric Background PDFs
The combinatoric backgrounds are dominated by events with two semileptonic
B decays. Additional combinatoric backgrounds can enter from continuum
(qq) events, and cascade semi-leptonic decays of the form B → D`ν followed
by D → K`ν, and events where one of the leptons is misidentified as a hadron.
In mES, such combinatoric backgrounds can be modeled using the AR-
GUS threshold function [57]:
f(x) ∝ x√
(1 − x2) exp[−ζ(1 − x2)], (4.5)
where ζ is a fit parameter and x = mES/E∗B.
Each of the combinatoric backgrounds has a non-trivial angular dis-
tribution in cos θ`, resulting in background shapes which are highly forward-
backward asymmetric. Rather than using an arbitrary high-order polynomial
to describe these shapes, they are modeled using data which is drawn from the
cos θ` distribution in the mES sideband region defined in the previous chapter.
The wrong lepton-flavor control sample (in which the hadronic system does
not change but the dilepton system is now either e+µ− or µ+e−) provides an
excellent control sample to study semileptonic B backgrounds. Widening the
window on ∆E and mES provides a larger sideband for studying these events.
In this grand side band (5.0 < mES < 5.29 and |∆E| < 0.25), the peaking
nature of the semi-leptonic decays becomes clear (Figure 4.7). A strong peak
can be seen for the e+e− and the µ+e− events. In these two cases, the electron
originates from the semi-leptonically decaying D-meson (which originates from
91
Figure 4.7: cos θ` GSB data distributions for standard and LFV events withrelaxed NN cuts.
the semi-leptonic decay of the B). This peak is inhabited by wrongly recon-
structed B mesons in which the electron from the D decay has momentum less
than 700 MeV/c. Note that the muons have a cut of greater than 700 MeV so
the peak at high cos θ` is not as apparent.
These events in the mES sideband region are added to the datasets
to give a better model and to increase the statistics in the PDF. A binned
PDF, in which the height of each bin is allowed to float, is used to model the
combinatoric backgrounds in the angular distributions. This is different than
using a histogram shape as the histogram shape would be a fixed shape. As
stated the model for this background comes from data. It is the combination
of the wrong-lepton flavor control sample and the lepton flavor conserving data
92
in the mES sideband region. This sample is fit simultaneously with the data in
the mES signal region thus allowing the background dataset to constrain the
shape of the binned PDF. Fitting in this way allows a calculation of the errors
due to bin-by-bin fluctuations in the case where the sideband is not a perfect
representation of the signal region.
4.1.3 Hadronic Peaking Background PDFs
Non-crossfeed backgrounds that peak in mES are small in this analysis. Care
is taken to model these backgrounds in the fit. As discussed, vetoes are im-
plemented which remove most of the peaking backgrounds, however some still
escape our vetoes.
The hadronic peaking backgrounds that remain after the vetoes come
from three-body B decays such as B → K (∗)π+π−, B → K(∗)K∗+π−, and B →
K(∗)K+K−. Since the branching fractions for these decays are not precisely
measured, this background is estimated by constructing a data control sample.
The control sample consists of events reconstructed as B → K (∗)µh, where the
h is either a K or a π. The µ is required to pass the looser of the two muon
selections (discussed in § 3.2.2) in order to reduce the rate of reconstruction of
this final state. The h is required to fail electron or muon identification, thus
resulting in a sample which is composed of hadronic B decays. The events
are required to pass all analysis cuts discussed in the previous chapter. This
inclusive sample is then weighted by the probability to misidentify hadrons
as muons. The peaking component of this weighted sample is extracted from
93
)2 (GeV/cESm5.23 5.24 5.25 5.26 5.27 5.28 5.29
)2E
ven
ts /
( 0.
002
GeV
/c
0
0.1
0.2
0.3
0.4
0.5m0 = 5.2890
mB = 5.2778
mESWidth = 0.0025769
mESslope = -27.4634
nbkg = 8.0440
nsig = 0.43216
)2 (GeV/cESm5.23 5.24 5.25 5.26 5.27 5.28 5.29
)2E
ven
ts /
( 0.
002
GeV
/c
0
0.1
0.2
0.3
0.4
0.5
, LQ+0.161
-0.141nsig: 0.432
)2 (GeV/cESm5.23 5.24 5.25 5.26 5.27 5.28 5.29
)2E
ven
ts /
( 0.
002
GeV
/c
0
0.2
0.4
0.6
0.8
1
1.2
1.4
m0 = 5.2892
mB = 5.2782
mESWidth = 0.0034749
mESslope = -43.2317
nbkg = 29.267
nsig = 1.6157
)2 (GeV/cESm5.23 5.24 5.25 5.26 5.27 5.28 5.29
)2E
ven
ts /
( 0.
002
GeV
/c
0
0.2
0.4
0.6
0.8
1
1.2
1.4
, HQ+0.724
-0.570nsig: 1.616
Figure 4.8: Gaussian + ARGUS fits to the hadronic peaking control samplein low s (left) and high s (right)
a ARGUS (background) plus Gaussian (peaking) fit to the mES distribution.
This peaking Gaussian is added to the final mES fit. Figure 4.8 shows the fits
to the hadronic peaking background control sample. The angular cos θK and
cos θ` distributions are taken from this sample as histogram PDFs in the mES
signal region of the reweighted distributions in each s bin.
4.1.4 Crossfeed PDFs
Crossfeed is defined to be true signal events that are mis-reconstructed as
the wrong signal final state. For this analysis, two types of crossfeed are
defined. Self-crossfeed occurs when the true signal mode is reconstructed in
the correct final state, however one of the final state particles was incorrect.
This occurs mostly when swapping the correct π± with another (the wrong)
π± in the event. Feed-across backgrounds occur when a true signal decay is
reconstructed as a different signal decay used in the analysis. This occurs,
for instance, when swapping the correct π0 with a random π± in the event.
Crossfeed backgrounds are modeled for each mode using signal Monte Carlo
94
Mode s bin N Peaking (+) error (−) errorB+ → K+µ+µ− LOW− 0.174 +0.054 −0.052
LOW+ 0.085 +0.066 −0.074MID 0.133 +0.122 −0.096HIGH 0.890 +0.327 −0.298
B+ → K+π0µ+µ− LOW− 0.125 +0.042 −0.040LOW+ 0.055 +0.070 −0.047MID 0.114 +0.055 −0.103HIGH 0.336 +0.142 −0.125
B+ → K0Sπ+µ+µ− LOW− 0.084 +0.038 −0.036
LOW+ 0.087 +0.059 −0.048MID 0.079 +0.034 −0.036HIGH 0.088 +0.185 −0.081
B0 → K+π−µ+µ− LOW− 0.185 +0.112 −0.089LOW+ 0.183 +0.117 −0.095MID 0.354 +0.157 −0.138HIGH 0.321 +0.173 −0.113
Table 4.1: Hadronic peaking background by mode and s bin.
95
generated for that mode.
In mES, both types of crossfeed have a long tail and cannot be modeled
using a Gaussian. Instead, a Crystal Ball function is used. The Crystal Ball
function is an empirically defined function which has a Gaussian core with a
power law tail [58]:
f(x) ∝{
exp(
− (x−x)2
2σ2
)
: (x− x)/σ > α
A×(
B − x−xσ
)−n: (x− x)/σ < α
(4.6)
where A ≡ (n/|α|)n × exp(−|α|2/2) and B ≡ n/|α| − |α|. x and σ are the
Gaussian peak and width. α is the point at which the function transitions
to the power function and n is the exponent of the power function. The four
parameters of the PDF are fixed from fits to the mES crossfeed distributions
in signal MC (see Figure ??). In mES self crossfeed is treated as signal by
summing the Gaussian signal shape and the crystal ball self-crossfeed shape.
The fraction of signal to self crossfeed is fixed from the signal MC. For the
combined fit, an efficiency weighted sum of the self crossfeed PDFs for each
mode is used. The weights and signal-to-crossfeed fractions are given in Table
4.2. The feed-across background is not treated in this manner. In this case, a
separate PDF is used and the normalization of the feed-across backgrounds is
fixed from the signal MC. Feed-across backgrounds are much smaller than the
self-crossfeed: 0.2 events are expected in low s and 0.8 events are expected in
the high s region.
96
mode low s high s efficiency-corrected weightB+ → K+π0µ+µ− 0.83 0.60 0.13B+ → K0
Sπ+µ+µ− 0.65 0.70 0.09
B0 → K+π−µ+µ− 0.93 0.87 0.26B+ → K+π0e+e− 0.77 0.56 0.17B+ → K0
Sπ+e+e− 0.78 0.68 0.12
B0 → K+π−e+e− 0.93 0.85 0.23
Table 4.2: Mode-wise signal to self-crossfeed fractions in each s bin
4.2 Fit Strategy
In order to extract unbiased values of FL and AFB a complicated strategy was
established. Because the final B → K∗`+`− dataset is extremely statistics
limited, it was not possible to perform a full 3-dimensional fit in mES, cos θK ,
and cos θ`. Thus, the fit is performed in three stages as follows:
Fit 1 The B → K∗`+`− candidate events are combined to make a mES distri-
bution in the mES sideband plus signal region for each s bin.
Float parameters:
– The signal yield in each s bin, NS(s).
– The combinatorial background yield, NB(s).
– The combinatorial background ARGUS shape parameter, ξ(s).
Fixed parameters:
– The endpoint of the combinatorial background at mES =5.29 GeV.
– The ratio of the crossfeed to the signal yield from signal MC.
97
– The shape of the signal Gaussian from charmonium fits.
– The shape of the crossfeed contributions from signal MC.
Fit 2 cos θK as an additional fit variable in each s bin.
Float parameters:
– The K∗ polarization of the signal in each s bin, FL(s).
Fixed parameters:
– All fixed and floating parameters from Fit 1.
– The cos θK shape of the combinatorial background.
– The cos θK shape of the signal from theory and MC efficiency cor-
rections.
– The cos θK shape of the crossfeed contributions from signal MC.
A combined fit to mES and cos θK is performed in the signal region
in order to minimize any bias arising from correlations between these
variables for any of the PDFs. This improves the discrimination between
signal and background events in the likelihood fit.
Fit 3 cos θ` is added as the final fit variable in each s bin.
Float parameters:
– The forward-backward lepton asymmetry of the signal in each s
bin, AFB(s).
98
Fixed parameters:
– All fixed and floating parameters from Fit 2.
– The cos θ` shape of the combinatorial background.
– The cos θ` shape of the signal from theory and MC efficiency cor-
rections.
– The cos θ` shape of the crossfeed contributions from signal MC.
A combined fit to mES, cos θK and cos θ` is performed in the signal region
in order to minimize any bias arising from correlations between these
variables for any of the PDFs. This improves the performance of the
likelihood fit.
4.3 Tests of Fits in Control Samples
Various control samples are used to test the fit procedure. The K∗ polariza-
tion and dilepton forward-backward asymmetry are known for charmonium
events, thus the fit is performed on events in the charmonium veto region.
B+ → K+`+`− events are expected to have null AFB in the limit that FS is
zero. These events are fit using the same strategy as above, however there is
no K∗ polarization, so the second fit is removed (this fit is discussed in the
next chapter). Finally, ensembles of Monte Carlo samples are generated to
resemble the expected B → K∗`+`− samples. Each of these “toy” datasets
is fit, allowing for an estimate of the bias inherent in the fit along with an
estimate of the expected errors. These tests are discussed in this section.
99
4.3.1 Fits to Charmonium Control Sample
Charmonium events enter our sample at a rate greater than 1000 times the
B → K(∗)`+`− rate, and is thus a much larger sample of events that have the
same topology as the B → K(∗)`+`− events. The charmonium control samples
are used to calibrate the signal mES Gaussian and as a crosscheck of the fit
method.
B → J/ψK(∗) and B → ψ(2S)K(∗) events are selected by reversing
the charmonium vetoes described is the previous chapter. All other selection
requirements are identical to those used for B → K (∗)`+`− signal candidates.
Recall that NNs were trained in a low and a high s region which excluded the
charmonium region and that cuts were optimized in four s bins. To study the
performance of the NNs, all four sets of cuts for the two NNs are implemented
in the J/ψ sample. For instance, the udsc and BB NN output for low s are
calculated for each B → J/ψK(∗) event. Then, the LOW− and LOW+ NN
cuts are applied. For each of the two sets of cuts, the efficiencies, branching
fractions, PDF shapes are recalculated. This is repeated for the MID and
HIGH bins using the high s NN output. For the ψ(2S) events, only the HIGH
cuts and high s NN are used.
The fit in this case does not contain a hadronic peaking background
component. The crossfeed PDF shapes are rederived from J/ψ and ψ(2S)
exclusive signal Monte Carlo. The fit method is implemented on each channel
separately (the large statistics of these samples permit this strategy). The
fit in which the 6 signal modes (and two K`+`−) modes are combined is also
100
performed in this control sample. This combined fit is used to fix the mES
signal Gaussian shape. The exclusive fits are crosschecks of the fit method
when determining FL and AFB. The branching fractions are also calculated
to check the mES fit method:
B =NBfit
ε× B(J/ψ ) × B(K∗) ×NBtot
where NBfitis the number of signal events from the fit, ε is the efficiency from
the signal MC, B(J/ψ ) is the branching fraction of J/ψ decaying to e+e− or
µ+µ−, B(K∗) is the branching fraction of the K∗ decaying to Kπ, and NBtot
is the total number of B mesons in the dataset.
Figure 4.9 shows the combined fit in each dimension for the B+ →
K0Sπ+µ+µ− modes. Table 4.3 summarizes the branching fractions, FL, and
AFB results from each fit in this sample. In general, the results agree very
well with the previous measurements [1] for both B → J/ψK and B → J/ψK∗.
Similar studies were performed for the ψ(2S) modes and the results were also
found to agree very well with expected values.
As stated, the charmonium fits in mES are used to constrain the B →
K(∗)`+`− signal Gaussian shape mean and width. The parameters from the
fit are given in table 4.4.
4.3.2 Tests of Fits in Simulation
To estimate the expected precision for FL and AFB and to test for fit biases,
a series of so-called embedded toy experiments is performed. For each toy
101
Mode s cuts J/ψ BF Err J/ψ FL Err. J/ψ AFB ErrB+ → K+µ+µ− LOW− 1.00 0.02 N/A N/A 0.011 0.011
LOW+ 1.03 0.02 N/A N/A 0.006 0.007MID 1.01 0.02 N/A N/A 0.011 0.008HIGH 1.01 0.02 N/A N/A 0.011 0.008
B+ → K+e+e− LOW− 1.03 0.01 N/A N/A 0.006 0.006LOW+ 1.03 0.01 N/A N/A −0.002 0.006MID 1.04 0.01 N/A N/A 0.006 0.005HIGH 1.04 0.01 N/A N/A 0.000 0.007
B → K`+`− LOW− N/A N/A N/A N/A 0.007 0.005LOW+ N/A N/A N/A N/A 0.001 0.005MID N/A N/A N/A N/A 0.005 0.005HIGH N/A N/A N/A N/A 0.001 0.006
B+ → K+π0µ+µ− LOW− 1.45 0.08 0.50 0.05 −0.060 0.06LOW+ 1.45 0.08 0.50 0.05 −0.060 0.06MID 1.50 0.09 0.55 0.05 −0.040 0.05HIGH 1.50 0.08 0.55 0.05 −0.040 0.05
B+ → K0Sπ+µ+µ− LOW− 1.49 0.08 0.43 0.05 0.006 0.053
LOW+ 1.49 0.07 0.44 0.05 0.004 0.051MID 1.43 0.08 0.49 0.05 0.001 0.055HIGH 1.48 0.07 0.47 0.05 0.028 0.047
B0 → K+π−µ+µ− LOW− 1.22 0.03 0.58 0.02 −0.001 0.023LOW+ 1.24 0.03 0.59 0.02 −0.003 0.025MID 1.30 0.03 0.57 0.02 −0.018 0.020HIGH 1.28 0.03 0.58 0.02 −0.010 0.022
B+ → K+π0e+e− LOW− 1.55 0.07 0.60 0.04 0.005 0.037LOW+ 1.54 0.06 0.61 0.04 −0.003 0.034MID 1.57 0.06 0.60 0.03 0.016 0.029HIGH 1.61 0.06 0.60 0.03 0.022 0.030
B+ → K0Sπ+e+e− LOW− 1.48 0.06 0.52 0.04 −0.048 0.039
LOW+ 1.43 0.06 0.52 0.04 −0.017 0.050MID 1.48 0.06 0.53 0.04 −0.019 0.042HIGH 1.49 0.06 0.54 0.04 −0.046 0.039
B0 → K+π−e+e− LOW− 1.38 0.03 0.58 0.02 −0.001 0.020LOW+ 1.38 0.03 0.58 0.02 −0.001 0.020MID 1.35 0.03 0.57 0.02 0.010 0.017HIGH 1.35 0.03 0.58 0.02 −0.004 0.018
B → K∗`+`− LOW− N/A N/A 0.56 0.01 −0.006 0.013LOW+ N/A N/A 0.57 0.01 −0.004 0.014MID N/A N/A 0.56 0.01 −0.001 0.011HIGH N/A N/A 0.57 0.01 −0.004 0.012
Table 4.3: J/ψ BF by Mode and s Bin NN Cuts. The PDG 2006 values forB+ → K+J/ψ(→ `+`−), B0 → K∗0J/ψ(→ `+`−) and B0 → K∗+J/ψ(→`+`−) are ( 1.008 ± 0.0035, 1.33± 0.06, 1.41 ± 0.08) × 10−3, respectively. Thevalue of FL is expected to be 0.56 and the AFB is expected to be zero.
102
mES5.2 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29
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140nsig = 431.8 +/- 23.26
mES5.2 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29
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140
A RooPlot of "mES"
kHel-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.05
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0
5
10
15
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kHel-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
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A RooPlot of "kHel"
lepHel-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.05
)
0
5
10
15
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lepHel-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
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A RooPlot of "lepHel"
Figure 4.9: J/ψ fits for B+ → K0Sπ+ J/ψ (J/ψ → µ+µ−). The colors are: total
(solid blue), signal (dashed blue), combinatoric background (dashed green),self crossfeed (dashed magenta), crossfeed (dashed red).
experiment, the combinatoric background shapes are generated from the PDF
resulting from a fit to the fully simulated generic background sample used
to model the distribution. For the hadronic peaking background, the events
are generated from the PDFs used in the fit. Signal and crossfeed events
are embedded into this background sample directly from the signal Monte
Carlo simulated for each mode. The number of expected events in 350 fb−1 is
generated for each study. Seven hundred experiments are generated for each
s bin for both B → K`+`− and B → K∗`+`−.
These toy experiments allow for an estimate of any inherent bias in
103
Mode Gaussian Mean ( GeV) Gaussian Sigma ( MeV)B+ → K+µ+µ− 5.27852 ± 0.00004 2.571 ± 0.030B+ → K+e+e− 5.27817 ± 0.00004 2.739 ± 0.026Combined K`` 5.27830 ± 0.00003 2.678 ± 0.020
B0 → K+π−µ+µ− 5.27924 ± 0.00007 2.449 ± 0.052B+ → K+π0e+e− 5.27871 ± 0.00016 2.737 ± 0.141B+ → K0
Sπ+e+e− 5.27879 ± 0.00013 2.531 ± 0.108
B0 → K+π−e+e− 5.27927 ± 0.00006 2.617 ± 0.050Combined K∗`` 5.27913 ± 0.00004 2.596 ± 0.030
Table 4.4: J/ψ mES PDF Shape Parameters by Mode. The listed central valuesand errors are the average of the values obtained in the four individual fits,using different sets of NN cuts, in each mode.
the fit procedure as the full-fit is performed on each experiment. For a given
sample the pull of a parameter in the fit can be defined by:
Pull =parexp − parfit
parσ
(4.7)
where parexp is the value of the parameter generated, parfit is the value of the
parameter returned from a fit, and parσ is the error on the parameter returned
from the fit. The distribution of the pulls from an ensemble of experiments
should be a Gaussian shape with a mean of zero and RMS of one. If the mean
is shifted from zero, then this is a sign of bias in the central value of par. If
the RMS is different from one, then it is a sign of bias on the error returned
from the fit. The toy studies also allow for an estimate of the expected error
on FL and AFB. Further, the toys are used to define a good fit. This will be
discussed at the end of this section.
Figure 4.10 shows the signal yield pull distributions in each s bin, and
Table 4.5 tabulates statistical means and rms widths, and the results of single
104
s bin Stat Mean Stat RMS Fit Mean Fit Sigma χ2/ndflow 0.133 1.029 0.143 ± 0.042 0.946 ± 0.035 1.91high 0.211 1.022 0.181 ± 0.043 1.038 ± 0.037 0.86
Table 4.5: Embedded Toy NS(s) Pull Results.
-6 -4 -2 0 2 4 60
10
20
30
40
50
60
70
80
90
nσ)/fit-ngen(n nσ)/fit-ngen(n
-6 -4 -2 0 2 4 60
10
20
30
40
50
60
70
nσ)/fit-ngen(n nσ)/fit-ngen(n
Figure 4.10: Embedded toy low mES pull (left) and high mES pull (right).
Gaussian fits to the pull distributions. Since the goal of this analysis is not
to make branching fraction measurements, there is no reason to perform any
corrections to the signal yields based on the small biases seen.
Figure 4.11 shows the FL pull distributions in each s bin, and Table 4.6
tabulates statistical means and rms widths, and the results of single Gaussian
fits to the pull distributions. There are minimal biases present.
Figure 4.12 shows the AFB pull distributions in each s bin and Table 4.7
gives statistical means and rms widths, and the results of single Gaussian fits
s bin Stat Mean Stat RMS Fit Mean Fit Sigma χ2/ndflow −0.062 1.086 −0.069 ± 0.041 0.962 ± 0.041 0.92high −0.089 1.065 −0.112 ± 0.040 0.993 ± 0.035 0.77
Table 4.6: Embedded toy FL pull results.
105
-6 -4 -2 0 2 4 60
10
20
30
40
50
60
70
80
FLσ)/fit-FLgen
(FL FLσ)/fit-FLgen
(FL
-6 -4 -2 0 2 4 60
10
20
30
40
50
60
70
80
FLσ)/fit-FLgen
(FL FLσ)/fit-FLgen
(FL
Figure 4.11: Embedded toy FL pulls: low s left and high s right.
s bin Stat Mean Stat RMS Fit Mean Fit Sigma χ2/ndflow −0.072 1.11 −0.076 ± 0.047 1.049 ± 0.042 1.11high −0.090 1.13 −0.118 ± 0.052 1.121 ± 0.042 1.04
Table 4.7: Embedded toy AFB pull results.
to the pull distributions. There are minimal biases present.
4.3.3 Good Fits and Strategy
To this point in the analysis, the B → K∗`+`− data in the mES sideband
and signal regions were blind. Prior to looking at the central value of FL or
-6 -4 -2 0 2 4 60
10
20
30
40
50
60
70
AFBσ)/fit-AFBgen
(AFB AFBσ)/fit-AFBgen
(AFB
-6 -4 -2 0 2 4 60
10
20
30
40
50
60
AFBσ)/fit-AFBgen
(AFB AFBσ)/fit-AFBgen
(AFB
Figure 4.12: Embedded toy AFB pulls: low s left and high s right.
106
AFB from the fit to this data, the errors from the fit are studied to determine
whether or not the fit was “good”. The criteria for a good fit is established
using these toy studies. The unblinding strategy is discussed in the next
chapter in more detail.
The error distributions of FL and AFB from the embedded toy fits,
Figures 4.13-4.14, show two features: a small fraction of the fits have very
small errors close to zero from fits that did not converge properly; and a long,
small tail extending to relatively large error values arises from fits in which
the signal yield from the mES fit was very low. In order to reject failed fits,
the values of the error (prior to unblinding) are allowed to be no larger than
the full range of FL and AFB and require:
• the error on FL be 0.05 < σ(FL) < 1.00;
• the error on AFB be 0.05 < σ(AFB) < 2.00
The fraction of good FL fits is >∼ 98% in both s bins. The fraction of good
AFB fits is >∼ 99% in the low s bin and >∼ 92% in the high s bin.
At this stage in the analysis, the techniques undergo an intensive peer
review by BABAR collaborators. The reviewers decide, based on the results
of the validations, whether it is appropriate to proceed with unblinding the
central values from the final fit to the B → K∗`+`− data. This final procedure
is discussed in the next chapter along with the results of the fit and a discussion
of the systematic errors.
107
Entries 700
Mean 0.2184
RMS 0.09934
Underflow 0
Overflow 5
Integral 695
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
Entries 700
Mean 0.2184
RMS 0.09934
Underflow 0
Overflow 5
Integral 695
FL_err Entries 700
Mean 0.2054
RMS 0.07832
Underflow 0
Overflow 5
Integral 695
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
Entries 700
Mean 0.2054
RMS 0.07832
Underflow 0
Overflow 5
Integral 695
FL_err
Figure 4.13: FL errors from embedded toys: low s bin (left) and high s bin(right).
Entries 700
Mean 0.3441
RMS 0.1741
Underflow 0
Overflow 6
Integral 694
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
10
20
30
40
50
60
70
80
Entries 700
Mean 0.3441
RMS 0.1741
Underflow 0
Overflow 6
Integral 694
AFB_err Entries 700
Mean 0.2396
RMS 0.171
Underflow 0
Overflow 6
Integral 694
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
10
20
30
40
50
60
70
80
Entries 700
Mean 0.2396
RMS 0.171
Underflow 0
Overflow 6
Integral 694
AFB_err
Figure 4.14: AFB errors from embedded toys: low s bin (left) and high s bin(right).
108
Chapter 5
Results
With the selection criteria having been established and the fit procedure val-
idated, the final fit is performed on the B → K∗`+`− dataset. The analysis
was “blind” until this point, meaning that the B → K∗`+`− and B → K`+`−
data in the signal region and the mES sideband region were never studied or
examined. This prevents any human introduced biases from entering the fi-
nal result. The final unblinding strategy (described below) was agreed upon
through a peer review process by BABAR collaborators.
This chapter describes the unblinding procedure, systematic errors that
enter into the final result, and the final fit. This thesis concludes with a
discussion of the results and future outlook for this and other b → s`+`−
measurements.
5.1 Fit Results
5.1.1 Unblinding Strategy
The unblinding of the B → K∗`+`− analysis occurs in several phases. Af-
ter each phase, the results of each phase are shared with the group of peer
reviewers. The phases are as follows:
109
Step 1 Before unblinding any of the results in B → K∗`+`−, the fit signal
yield and AFB in B+ → K+`+`− is examined in each s bin. Given
the mode-wise signal efficiencies shown in Table 3.5, signal yields in
each s bin can be validated against Standard Model expectations and
previous measurements. Also, as there is a theoretical expectation of
an essentially null result (at least at the level of our current sensitivity)
for AFB in B+ → K+`+`− regardless of s region, obtaining fit central
values of AFB consistent with a null result will provide an indication of
the robustness of the angular fit model.
Step 2 Given that the results of step 1 reasonably agree with expectations, the
B → K∗`+`− signal yields resulting from the mES fits in each s bin
are unblinded. As with the B+ → K+`+`− mES fit results above, the
B → K∗`+`− signal yields can be compared to expected results.
Step 3 The cos θK fits are then performed. There are no a priori expectations
of any preferred values for FL and the criterion for a successful FL fit in
any particular s bin is (a) that the fit converge and (b) that the error
on FL meet the requirements given in the previous chapter. If the fit in
a particular bin does not meet criteria (a) and (b), a result for FL will
not be reported for that bin.
Step 4 The fits to cos θ` are performed to extract AFB in each s bin. In the case
that the fit to cos θK did not converge for a particular s bin, the value of
FL would be fixed to the SM value appropriate to that bin as determined
110
from the signal Monte Carlo. This would introduce a systematic on AFB.
This systematic is discussed later in this chapter. Analogous to Step 2,
there are no a priori expectations of any preferred values for AFB and
the criterion for a successful fit in any particular s bin is (a) that the
fit converge and (b) that the error meet the requirements given in the
previous chapter. If the fit in a particular bin does not meet criteria (a)
and (b), a frequentist method based on toy MC experiments would be
used to determine 68% and 95% confidence level limits on the value of
AFB in that bin.
5.1.2 B+ → K+`+`− and B → K∗`+`− mES Fit Results
The B+ → K+`+`− and B → K∗`+`− mES distributions are fit in each s
bin. Table 5.1 shows the expected (based on an early branching fraction
measurement by BABAR) and observed yields for B+ → K+`+`− and B →
K∗`+`−. Figure 5.1 shows the B+ → K+`+`− fits in each s bin. Figure 5.2
shows the B → K∗`+`− fits in each s bin. The expected yields are taken from
Table 3.6, above. Although approximately twice as many signal events are
observed than expected, there is good agreement between these results and
Belle and CDF results, while the earlier BABAR analysis tends to disagree with
these results and Belle and CDF results. The disagreement motivated further
study and a new branching fraction result by BABAR is in the process of being
published.
111
s bin Exp. NS(s) Fit NS(s) Exp. NB(s) Fit NB(s)B → K`+`−
low 18.4 26.92 ± 5.79 125.3 81.2 ± 9.3high 15.5 24.92 ± 6.73 165.0 168.1 ± 13.7
B → K∗`+`−
low 8.9 27.0 ± 10.2 87.3 92.8 ± 10.3high 13.0 15.4 ± 7.3 285.3 285.8 ± 19.7
Table 5.1: B+ → K+`+`− Expected (based on early BABAR BF measurement)and observed signal and background yields after the first stage of unblinding.
mES5.2 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29
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nComb13 = 81.20 +/- 9.372
nSig13 = 26.92 +/- 5.791
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A RooPlot of "mES"
mES5.2 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29
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nComb13 = 168.1 +/- 13.73
nSig13 = 24.92 +/- 6.727
mES5.2 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29
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A RooPlot of "mES"
Figure 5.1: B+ → K+`+`− low s mES distribution (left) and high s (right):total (solid blue), signal (dashed blue), random combinatoric (dashed blue)
mES5.2 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29
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nComb0 = 92.06 +/- 10.25
nSig0 = 27.01 +/- 6.337
mES5.2 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29
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A RooPlot of "mES"
mES5.2 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29
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nComb0 = 332.8 +/- 19.71
nSig0 = 37.23 +/- 9.655
mES5.2 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29
Eve
nts
/ (
0.00
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0
5
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A RooPlot of "mES"
Figure 5.2: B → K∗`+`− low s mES distribution (left) and high (right): total(solid blue), signal (dashed blue), random combinatoric (dashed blue), cross-feed (dashed red).
112
s bin AFB
low 0.13+0.10−0.12
high 0.20+0.18−0.18
Table 5.2: B+ → K+`+`− AFB results in data.
5.1.3 B+ → K+`+`− AFB Fits
B → K`+`− is expected to have a null AFB across s. New physics is not
expected to change this prediction in a significant way. The B → K`+`− data
serves to test this hypothesis. cos θK does not exist, so the fit for FL is not
performed. In this case, the scalar component FS is assumed to be 0.
Figure 5.3 shows the B+ → K+`+`− AFB fits in each s bin. The plots
on the left show the cos θ` data in the mES signal region for the low and high
s bins with the fit PDFs overlaid. The solid line is the total fit, the 10-bin
histogram shape is the parametric step function combinatoric PDF taken from
the mES sideband and the other dotted-line curve is the signal PDF. The plots
on the right show the binned combinatoric PDFs including events from the
mES sideband region. Table 5.2 lists the AFB central values (plus errors) from
the fits. Ignoring any systematic error, the central values in each s bin are
consistent with a null result.
5.1.4 B → K∗`+`− Angular Fits
Figure 5.4 shows the B → K∗`+`− FL and AFB fits in each s bin. The plots
on the left show the cos θK data in the mES signal region for the low and high s
bins with the fit PDFs overlaid. The plots on the right are the cos θ` fits. The
113
lθcos -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.2
)
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2
4
6
8
10AFB = 0.07074 +/- (-0.1599, 0.1233)
lθcos -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.2
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10
lθcos -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
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30
35
lθcos -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
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lθcos -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
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6
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12
AFB = 0.1910 +/- (-0.3329, 0.1531)
lθcos -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
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/ (
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lθcos -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
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lθcos -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Eve
nts
/ (
0.2
)
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5
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15
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25
30
35
40
45
Figure 5.3: Left column shows signal region B+ → K+`+`− AFB fits: total(solid blue line), signal (quasi-continuous dotted blue line), random combina-toric (10-bin dotted blue line); right column shows the binned cos θ` combina-toric PDFs taken from mES sideband. Top row is low s, bottom row is highs.
114
)Kθcos(-1 -0.5 0 0.5 1
Eve
nts
/ (
0.2
)
0
5
10
)lθcos(-1 -0.5 0 0.5 1
Eve
nts
/ (
0.2
)
5
10
)Kθcos(-1 -0.5 0 0.5 1
Eve
nts
/ (
0.2
)
0
10
20
)lθcos(-1 -0.5 0 0.5 1
Eve
nts
/ (
0.2
)5
10
15
20
Figure 5.4: Left column shows signal region B → K∗`+`− FL fits. The rightcolumn shows the signal region B → K∗`+`− AFB fits. Total (solid blueline), signal (quasi-continuous dotted blue line), random combinatoric (10-bindotted blue line); right column shows the binned cos θ` combinatoric PDFstaken from mES sideband. Top row is low s, bottom row is high s.
solid line is the sum of the various components in the fit, the 10-bin histogram
shape is the parametric step function combinatoric PDF taken from the mES
sideband and the other dotted-line curve is the signal PDF. The final results
will be put in context after a discussion of the systematic errors on this final
result.
115
5.2 Systematic Errors
Several sources of systematic uncertainty are considered in the fit of FL from
the cos θK distribution and AFB from the cos θ` distribution:
• The error on the signal yield from the mES fit is propagated into the FL
and AFB fits.
• The error on FL is propagated into the AFB fit.
• The combinatorial background shape and normalization.
• Self-crossfeed within the signal mode and feed-across from other modes.
• The Gaussian signal shape.
• The peaking backgrounds from hadronic modes and charmonium events
that escape the veto due to poor resolution or Bremsstrahlung radiation.
• The signal efficiency as a function of variations in the Wilson coefficients
Ceff7 , Ceff
9 and Ceff10
• The signal efficiency for different form factor models.
• The average fitting bias on the central values obtained from the toy
Monte Carlo studies described in § 4.3.2
• The selection of the final ∆E fit window.
116
FL systematic AFB systematicVariation low s high s low s high s−1σ ARGUS shape +0.001 +0.019 unconverged fit −0.002+1σ ARGUS shape +0.001 +0.014 −0.003 unconverged fitError 0.001 0.016 0.003 0.002
Table 5.3: Signal Yield Systematics
With the exception of the last one on this list, all these sources of
systematics are treated as additive uncertainties on the central values of the
asymmetries. They are regarded as independent and are combined in quadra-
ture. Each is discussed in this chapter.
5.2.1 Signal Yield Systematic
In the mES fit that determines the signal yield, the combinatoric background
ARGUS shape and normalization are varied. These parameters are fixed for
the angular fits to FL and AFB. To study the systematic error on the angular
asymmetries associated with these fixed parameters, the fitted ARGUS shape
parameter is varied by ±1σ from its central value. The mES fit is rerun with
the ±1σ values to determine the signal and background yields. Finally the fits
to extract FL and AFB are performed again, fixing the values of the ARGUS
shape and the yields to their ±1σ values.
The average shift of the central values of FL and AFB for the +1σ
and −1σ variations relative to the default fit are determined and used as the
systematic error. Where there is an unconverged fit, we assign the shift due
only to the converged fit. See Table 5.3.
117
AFB systematicVariation low s high s+1σ(FL) −0.010 −0.023−1σ(FL) +0.040 +0.021Error 0.025 0.022
Table 5.4: FL Systematics
5.2.2 FL Fit Systematics
The fit to the cos θK distribution in each s bin gives a central value for FL and
a fit error. This central value is used for the default fit to the cos θ` distribution
to extract AFB in each s bin. To study the systematic error on AFB due to
FL we vary the value of FL by ±1σ from its fitted value, and redo the cos θ`
fits with the new value of FL. The systematic error is calculated by averaging
the absolute value of the shift of the central values of AFB for the +1σ and
−1σ variations of FL. See Table 5.4.
5.2.3 Combinatorial Background Systematics
The combinatorial background shape is derived from the 5.20 < mES < 5.27
sideband using a combination of lepton-flavor conserving (LFC) e+e−, µ+µ−
and wrong lepton flavor e+µ− and µ+e− events. A systematic error from
the background shape is assigned using different definitions of the sideband
sample:
• Using either the lepton flavor conserving sample or the wrong lepton
flavor sample
118
FL systematic AFB systematicVariation low s high s low s high s5.20 < mES < 5.23 +0.011 −0.008 −0.002 +0.0015.23 < mES < 5.27 −0.004 +0.007 −0.017 −0.021∆E > 0 +0.002 −0.031 −0.037 +0.003∆E < 0 +0.001 +0.017 unconverged fit −0.002LFC +0.011 −0.026 −0.037 unconverged fitwrong flavor −0.008 +0.023 +0.024 −0.003Error 0.006 0.020 0.027 0.006
Table 5.5: Combinatorial background systematics.
• Using either events with ∆E > 0 or ∆E < 0
• Using either events with 5.20 < mES < 5.23 or 5.23 < mES < 5.27
In each case, the sideband sample is split into two independent samples and the
combinatorial background shape is defined for each sub-sample. The cos θK
and cos θ` fits are redone using the shapes from the sub-samples and the shift
in the central values of AFB and FL is compared to the default fit. Note that
there is a relatively large statistical component in the determination of the
varied background shapes that propagates into the scatter of values from the
disjoint samples. See Table 5.5.
5.2.4 Crossfeed and Signal Shape Systematics
Self-crossfeed can be treated as an additional part of the signal, or it can be
treated as a mis-reconstructed background. If the ratio of the self-crossfeed
to the true signal is kept constant, the result of the fit to extract FL or AFB
should be the same whichever way the self-crossfeed is treated.
119
In the fits to the charmonium control samples described in § 4.3.1 the
ratio of self-crossfeed to truth-matched signal was allowed to float. The allowed
ranges of the ratio from these fits are used to evaluate the systematic errors
in the fits to the angular distributions.
The feed-across between different modes is fixed in shape, but the nor-
malization is again varied by the amount that is allowed by the charmonium
control samples in a similar way to the self-crossfeed.
The parameters of the Gaussian signal shape are determined from the
fit to the charmonium control samples. For the fits to the K∗`` samples,
the mean and the width of the Gaussian within the ranges allowed by the
charmonium fits is varied.
All of these studies gave errors that were within the accuracy with which
the charmonium control samples reproduce the expected values of FL = 0.56
and AFB = 0. The systematic errors from crossfeed and signal shape modeling
are then 0.010 on FL and 0.020 on AFB (Table 4.3).
5.2.5 Signal Model Systematics
To study a possible systematic error on FL and AFB as a function of their
true physical values, simulated events are generated varying the values of the
Wilson coefficients Ceff7 , Ceff
9 , Ceff10 . This gives a range of asymmetries in the
signal MC. Figures 5.5 and 5.6 graphically show the range of FL and AFB for
the variations projected into the original four disjoint fit bins of the analysis.
These datasets are used to produce signal efficiency histograms which differ
120
FL systematic AFB systematicVariation low s high s low s high s
−Ceff7 +Ceff
9 +Ceff10 +0.034 +0.032 −0.002 −0.003
+Ceff7 −2Ceff
9 +Ceff10 +0.049 +0.002 −0.018 +0.088
+Ceff7 +3Ceff
9 +Ceff10 +0.032 −0.089 −0.056 +0.040
+Ceff7 −Ceff
9 +Ceff10 +0.010 +0.023 −0.060 +0.002
+Ceff7 +Ceff
9 −Ceff10 −0.054 +0.024 +0.013 +0.059
Error 0.036 0.034 0.030 0.038
Table 5.6: Signal model systematics
from the default ones, which use the SM values of FL and AFB, allowing dif-
ferent regions of the angular distributions to contribute with different weights
depending on the angular asymmetries. Applying these alternative signal effi-
ciency histograms, the shifts in the fitted values of FL and AFB are measured.
The average of the absolute value of the shift is assigned as the systematic.
See Table 5.6.
5.2.6 Fit Bias Systematics
Fit bias is measured in terms of pull distributions from toy Monte Carlo stud-
ies. Ensembles of samples using the varied Wilson coefficient Monte Carlo were
generated and fit. The results of this study are tabulated in Tables 5.7 and
5.8 for the LOW−, LOW+, MID and HIGH bins that were originally planned
for this analysis. These studies were not redone for the low s and high s bins
that were finally used. However, the bias in the fitted central values of the
ensembles compared to the input values of FL or AFB provide an estimate of
a possible systematic due to the fitting methodology. Note that the means
121
2q2 4 6 8 10 12 14 16 18 20
FL
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 5.5: Values of FL for Wilson coefficient variations by s bin: −C7
+C9,+C10 (circle), +C7 −2C9,+C10 (square), +C7 +3C9,+C10 (triangle up),+C7 −C9,+C10 (triangle down), +C7 +C9,−C10 (filled star).
quoted in these tables are in units of statistical significance σ.
The fit bias systematic is estimated from the average pull mean for
either the LOW− and LOW+ bins, or for the MID and HIGH bins, taken
over all five variations of the Wilson coefficients. The fit bias error is then the
average pull times the statistical error. The average pull means and systematic
errors are given in Table 5.9.
5.2.7 ∆E Fit Window Systematics
The ∆E selection window varies between −0.04 < ∆E < 0.04 for K (∗)µ+µ−
events in the low s region, and −0.08 < ∆E < 0.05 for e+e− events in the
122
Pull Pull Pull
Ceff7 Ceff
9 Ceff10 s bin Gen. FL Stat Mean RMS Fit Mean Sigma
−1 1 1 LOW− 0.58 −0.053 1.25 0.04 ± 0.04 0.99 ± 0.03LOW+ 0.49 0.019 1.36 −0.11± 0.04 0.78 ± 0.03MID 0.38 −0.053 1.13 −0.05± 0.04 0.89 ± 0.03HIGH 0.33 −0.099 1.10 −0.07± 0.04 0.92 ± 0.04
1 −2 1 LOW− 0.55 −0.010 1.12 0.05± 0.04 0.94 ± 0.04LOW+ 0.49 −0.065 1.40 0.01± 0.05 0.91 ± 0.04MID 0.37 0.146 1.25 0.10± 0.06 1.09 ± 0.05HIGH 0.33 −0.144 1.37 −0.29± 0.04 0.95 ± 0.03
1 3 1 LOW− 0.85 −0.100 1.29 0.16 ± 0.04 0.94 ± 0.05LOW+ 0.65 0.037 1.28 0.09 ± 0.05 0.89 ± 0.04MID 0.43 0.258 1.20 0.24 ± 0.04 0.94 ± 0.04HIGH 0.35 −0.053 1.22 −0.17± 0.05 1.01 ± 0.04
1 −1 1 LOW− 0.43 0.125 1.08 0.09 ± 0.04 0.88 ± 0.03LOW+ 0.48 −0.152 1.27 −0.22± 0.05 0.90 ± 0.04MID 0.38 0.212 1.31 0.06 ± 0.06 1.14 ± 0.04HIGH 0.33 −0.189 1.28 −0.35± 0.04 0.87 ± 0.03
1 1 −1 LOW− 0.58 0.157 1.07 0.30 ± 0.07 1.04 ± 0.06LOW+ 0.64 0.052 1.47 −0.07± 0.05 0.81 ± 0.04MID 0.45 0.099 1.19 0.10 ± 0.04 0.85 ± 0.03HIGH 0.38 −0.103 1.16 −0.15± 0.04 0.92 ± 0.03
Table 5.7: Varied Wilson Coefficient Toy FL Pulls. The numbers in the firstthree columns refer to the scaling applied to the standard values of Ceff
7 , Ceff9
or Ceff10 contained in the official signal MC.
123
Pull Pull Pull
Ceff7 Ceff
9 Ceff10 s bin Gen. AFB Stat Mean RMS Fit Mean Sigma
−1 1 1 LOW− 0.17 −0.06 1.02 0.02 ± 0.04 0.94 ± 0.03LOW+ 0.33 0.01 1.05 −0.24 ± 0.06 0.94 ± 0.05MID 0.42 −0.09 1.21 0.31 ± 0.05 0.99 ± 0.04HIGH 0.42 0.01 1.16 0.05 ± 0.05 0.89 ± 0.04
1 −2 1 LOW− −0.12 −0.07 1.01 0.08 ± 0.05 0.98 ± 0.04LOW+ −0.25 0.07 1.03 −0.10 ± 0.05 0.91 ± 0.04MID −0.33 0.19 1.15 −0.14 ± 0.04 0.92 ± 0.04HIGH −0.32 −0.23 1.22 0.26 ± 0.05 1.01 ± 0.05
1 3 1 LOW− 0.03 −0.24 1.07 0.16 ± 0.04 0.94 ± 0.04LOW+ 0.17 0.17 1.14 −0.11 ± 0.06 0.96 ± 0.05MID 0.26 0.08 1.04 −0.01 ± 0.04 0.91 ± 0.03HIGH 0.26 −0.05 1.08 0.06 ± 0.05 1.09 ± 0.04
1 −1 1 LOW− −0.18 0.03 1.09 0.05 ± 0.04 0.94 ± 0.04LOW+ −0.33 0.15 1.08 −0.27 ± 0.06 1.06 ± 0.06MID −0.43 0.25 1.12 −0.18 ± 0.04 0.95 ± 0.03HIGH −0.41 0.02 1.33 0.20 ± 0.06 1.13 ± 0.05
1 1 −1 LOW− 0.08 −0.12 1.04 0.15 ± 0.05 1.00 ± 0.04LOW+ −0.18 0.18 1.18 −0.13 ± 0.06 1.07 ± 0.05MID −0.38 −0.22 1.45 0.17 ± 0.05 1.01 ± 0.04HIGH −0.37 −0.13 1.06 0.18 ± 0.04 0.94 ± 0.04
Table 5.8: Varied Wilson Coefficient Toy AFB Pulls. The numbers in the firstthree columns refer to the scaling applied to the standard values of Ceff
7 , Ceff9
or Ceff10 contained in the official signal MC.
FL systematic AFB systematiclow s high s low s high s
Average pull 0.077 0.136 0.112 0.130Fit bias Error 0.012 0.020 0.023 0.052
Table 5.9: Fit bias systematic error.
124
2q2 4 6 8 10 12 14 16 18 20
AF
B
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Figure 5.6: Values of AFB for Wilson coefficient variations by s bin: −C7
+C9,+C10 (circle), +C7 −2C9,+C10 (square), +C7 +3C9,+C10 (triangle up),+C7 −C9,+C10 (triangle down), +C7 +C9,−C10 (filled star). The squarepoints in the vetoed charmonium regions are a plotting artifact and shouldbe ignored.
high s region. To study possible systematic effects associated with the choice
of this window, common windows are applied to all modes and varied between
−0.04 < ∆E < 0.04 and −0.10 < ∆E < 0.10. Beyond ∆E < −0.10 other
types of crossfeed begin to enter the sample (events in which one additional
pion is missing from a reconstructed signal candidate).
With these different ∆E windows the complete analysis is performed.
The combinatorial background shape, the peaking background contributions,
as well as the signal efficiencies are all redone. New mES fits are performed
to determine the signal and background yields for each ∆E window. Finally,
125
FL systematic AFB systematicVariation low s high s low s high s|∆E| < 0.04 −0.030 0.000 −0.044 0.000|∆E| < 0.06 −0.017 −0.099 −0.097 +0.010|∆E| < 0.08 −0.021 −0.137 −0.089 +0.014|∆E| < 0.10 −0.016 −0.077 −0.109 −0.218Error 0.021 0.078 0.085 0.061
Table 5.10: ∆E systematics
new angular fits are performed to determine FL and AFB. See Table 5.10 for
the results.
It was decided that these variations should not be included as a system-
atic error. Increasing the ∆E window in size does add some more signal events
particularly for the K(∗)e+e− modes, but it introduces a lot of additional back-
ground which leads to larger statistical variations of the central values. The
different selections also have strong correlations between them, with only the
additional events in the broader window accounting for changes in the fits
Therefore this study is treated as a cross-check that there are no hidden
systematics associated with the rather tight ∆E selection, e.g. that events in
the bremsstrahlung tail of the e+e− sample have different angular asymmetries
from events in the Gaussian peak.
5.2.8 Peaking Background Systematics
The peaking background contributions have been determined from control
samples in the data. For the charmonium leakage, the backgrounds are scaled
according to the approximate 10% uncertainty on the charmonium branching
126
Source FL systematic AFB systematicof Error low s high s low s high smES fit yields 0.001 0.016 0.003 0.002FL fit error N/A N/A 0.025 0.022Background shape 0.006 0.020 0.027 0.006Signal model 0.036 0.034 0.030 0.038Fit bias 0.012 0.020 0.023 0.052Efficiency/cross-feed 0.010 0.010 0.020 0.020Total 0.04 0.09 0.10 0.08
Table 5.11: Total Systematic Errors
fractions. For the hadronic peaking backgrounds, the background is scaled
according to the errors from the hadronic control samples given in Table 4.1.
Since these backgrounds are only at the level of 0.1 events, the systematic
errors on the angular fits are negligible.
5.2.9 Total Systematic Error
As stated in the introduction to this section, the individual systematic errors
are combined in quadrature to obtain the total systematic errors on the fit
quantities. The final systematics are listed in Table 5.11.
5.3 Conclusion and Outlook
This analysis provides the first direct measurement of FL and AFB in the
low s region. This region is particularly interesting for theorists because the
theoretical predictions are dominated by perturbative contributions and a the-
oretical precision of order 10% is possible. Above this region, long distance
contributions from the cc resonances (charmonium) dominate the theoretical
127
uncertainties [59].
Figure 5.7 graphically shows the results for FL and AFB, respectively,
overlaid on the expected SM and various non-SM distributions. In summary,
the forward-backward asymmetry AFB and longitudinal K∗ polarization FL
of the rare decays B → K∗`+`− has been measured in two bins of the di-
lepton mass. In the low s region the expected values of AFB and FL are
AFB = −0.03 and FL = 0.64 from the SM. The values obtained from this
analysis are AFB = 0.24+0.18−0.23 ± 0.103 and FL = 0.35 ± 0.16 ± 0.05, where the
first error is statistical and the second is systematic. The agreement with the
SM is not particularly good, with both measurements being more consistent
with “flipped-sign” C7 = −C7. In the high s region the expected values
are AFB = 0.44 and FL = 0.38 and measure AFB = 0.76+0.49−0.30 ± 0.078 and
FL = 0.69+0.21−0.23 ± 0.10. The large positive AFB result in the high s region
rules out flipped-sign C9C10 at more than 3σ significance. These results are
consistent with measurements by Belle, and replace the earlier BABAR results
in which only a limit on AFB was set in the low s region. These results and
the previous BABAR result are consistent in the high s region.
The current measurements show large AFB in all regions of s. It is,
however, difficult to draw strong conclusions regarding the deviation from SM
predictions due to the low statistics. All of the measurements presented here
are based on 349 fb−1 of data and are limited by statistical uncertainties. The
final BABAR dataset contains 432 fb−1 which will provide a modest contribution
to reduction of the statistical error. There will be an improvement in the muon
128
Figure 5.7: FL and AFB results and theory expectations.
129
detector performance in the extra 80 fb−1 due to the LST upgrade. However,
this will be a small contribution to the reduction of statistical errors. A future
(and the final) BABAR B → K(∗)`+`− analysis will attempt to improve the
measurement by modeling correlations in the cos θK and cos θ` distributions
and thus fitting for FL and AFB simultaneously. This method, along with the
increase in statistics, could be enough to perform the analysis in the four s
bins initially planned for this analysis.
In order to obtain a significant reduction in statistical error and an
increase in the number of s bins in which this analysis is performed will require
data from the LHCb experiment and/or the so-called SuperB factory. LHCb
will only be sensitive to di-muon decays in which there is a charged kaon.
Analogous to BABAR’s “golden mode” (B → J/ψK0S), the B → K(∗)`+`−
decay will be a golden mode at a SuperB factory capable of delivering on the
order of 100 times the luminosity of the current B-factories. In this scenario,
it should be possible to make precise direct measurements of AFB in multiple
bins of s. This measurement will allow a determination of the AFB 0-crossing
in s. The absence of such would be a sure sign of new physics.
130
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Vita
Chris James Schilling was born in Tuscon, Arizona on 22 July 1980. He grad-
uated from the Arkansas School for Mathematics and Sciences in 1998. He
attended the University of Arkansas and graduated with degrees in Mathe-
matics and Physics with Honors in 2002. He then attended the University of
Texas in Austin, Texas where he received his Doctor of Philosophy in Physics
in 2008.
Permanent address: 851 Roble Ave. #1Menlo Park, CA
This dissertation was typeset with LATEX† by the author.
†LATEX is a document preparation system developed by Leslie Lamport as a specialversion of Donald Knuth’s TEX Program.
136
