UNIVERSITY OF BRISTOL DEPARTMENT OF PHYSICS
NAME:
TROCMÉ Mathieu
DEGREE:
Erasmus Student
PROJECT/DISSERTATION NUMBER:
PA1
TITLE:
Rare B Decays
YEAR OF SUBMISSION:
2003
SUPERVISOR:
Dr Fergus Wilson
H. H. Wills Physics Laboratory University of Bristol Tyndall Avenue, Bristol BS8 1TL
Abstract
1
ABSTRACT
This report aims to describe a way to measure simultaneously the branching ratio of the meson decay B 0 → K *+ π − and its conjugate
. This way, based on the analysis of 61.6 ±
0
0
0.68 millions B 0 and B mesons, uses a cut and count method. The B 0 / B mesons were generated at SLAC between 1999 and 2002 in the PEP-II collider and detected by the BABAR detector. The K *+ / K
*−
0
were observed via their decay to a K S0π + / K S π − , which results in a
0
0
B 0 / B having the three body charmless final state K S0π + π − / K S π −π + : B 0 → K *+ π −
&
*−
0
B →K π+ 0
K S0π +
K Sπ −
The total branching ratio was found to be:
{
}
{
0
*−
BR B 0 → K *+ π − + BR B → K π +
}=
(18.6 ± 6.5 ± 1.6) × 10 −6
with a statistical significance of 3.2 σ
Acknowledgements
2
ACKNOWLEDGEMENTS
Firstly, I would like to thank Nicole Chevalier for all she has done for us throughout the year. Always available and always smiling, it was really a great pleasure to come and annoy her with some not always sensible questions. I really have appreciated this. I would also like to thank all the people that helped me from time to time when I was randomly looking for help: Robert Frazier, Nick Barlow, Jim Burke, Wahid Bhimji, Dave Newbold, Fergus Wilson. Many thanks as well go to all the Particle Physics Project Work students that have contributed to make this work more enjoyable: Meyrem, Shona, Domenico, Jamie, Chris. Finally, ‘un grand merci à David’ for having helped me with my English.
Table of contents
3
TABLE OF CONTENTS
Abstract………………………………………………………………………………………… 1 Acknoledgements……………………………………………………………………………… 2 I- Introduction…………………………………………………………………………………. 5 II- Physical Background………………………………………………………………………. 7 II.1- Symmetries in Particle Physics……………………………………………………. 7 II.1- CP violation in the Standard Model……………………………………………….. 9 II.3- B mesons and the BABAR experiment………………………………………………. 12 0
II.4- The B 0 → K *+ π − & B → K *−π + decays.……………………………………...15 III- Analysis Method…………………………………………………………………………... 18 III.1- Data Sample and Preselection…………………………………………………… 18 III.2- The Cut and Count Method (CCM).……………………………………………… 18 III.2.1- Principle of the CCM…………………………………………………… 18 III.2.2- Selection variables……………………………………………………… 19 III.2.2.1- Event shape variables…………………………………………. 19 III.2.2.2- Particle Identification………………………………………..21 III.2.2.3- The K *+ → K S0π + / K III.2.2.4- The K S0 → π −π +
*+
0
→ K S π − resonance………….…… 22 0 + − / K S → π π resonance………………… 24
III.2.2.5- Kinematic constraints: the ∆E-mES plane................................... 26 III.2.3- Computing Overview…………………………………………………… 29 III.3- Calculation of the Branching Ratio……………………………………………….. 29 III.3.1- Definition……………………………………………………………….. 29 III.3.2- MC Efficiency…………………………………………………………... 30
Table of contents
4
III.3.5- Background Characterisation and Subtraction………………………… 32 III.3.7- Errors on the branching ratio calculation……………………………… 36 III.4- Optimisation of the cuts…………………………………………………………... 37 III.4.1- Significance calculation………………………………………………… 37 III.4.2- Errors on the significance……………………………………………….. 38 III.5- Overall Process……….…………………………………………………………... 42
IV- Results……………………………………………………………………………………... 44 IV.1- Final selection criteria after optimisation………………………………………… 44 IV.2- MC Efficiency………… ………………………………………………………… 46 IV.3- Combinatoric background………………………………………………………... 47 0
IV.4- Branching ratio of the B 0 → K *+ π − & B → K *−π + decays …………………49 V- Discussion…………………………………………………………………………………... 50 VI- Conclusion…………………………………………………………………………………. 53 Appendix...……………………………………………………………………………………... 54 References…………………………………………………………………………………….. 56
I- Introduction
5
I- INTRODUCTION
We live in a matter universe. However, from the ‘Hot Big Bang’ model – the current model explaining the beginning of the universe (Gamow, 1946, [1]) – this universe began with an equal amount of matter and antimatter. This model being widely accepted – especially after the random discovery of the Cosmologic Microwave Background by Penzias and Wilson in 1965 [2] – , one question comes to mind: Where has all the antimatter gone ? Many theories generating a matter asymmetry have been proposed, even some antigravity ones. But the current one is due to Sakharov (1967, [3]) and based on the ‘Sakharov’ conditions, one of which is CP violation. Basically, to turn the properties of a particle into the ones of its antiparticle, one just needs to process a symmetry operation called CP. If this CP operation can be successfully observed in a decay, matter and antimatter are absolutely symmetric. Otherwise, it proves there is an asymmetry between matter and antimatter, which could be one explanation of the matter asymmetry in the universe. From the middle of the sixties, it has been known that CP violation is a real phenomenon that occurs in weak decays. Since then, a quantification work has been undertaken to determine whether or not CP violation processes alone can lead to an explanation of the universe matter asymmetry. Having successfully studied K meson systems first, physicists currently track CP violation in B meson systems. Two major recent collaboration are involved in that study: the BELLE collaboration at the KEK-B collider in Japan and the BABAR one at SLAC in California. These both collaborations have already observed CP violation in B mesons [4]. But there is still a lot to do…
I- Introduction
6
A way to quantify CP violation consists of measuring the branching ratio (or branching fraction) of a specific decay (i.e. the likelihood this decay happens). This is what was proposed in this computing project, namely measure the total branching fraction of the two conjugate decays: B 0 → K *+ π −
&
0
B → K *−π +
A part of the Particle Physics Group of the University of Bristol being involved in the BABAR experiment, the data to be analysed for this measurement come from California.
II- Physical Background
7
II- PHYSICAL BACKGROUND
II.1- Symmetries in Particle Physics: In Particle Physics, there are three fundamental discrete symmetry operations that can be performed on a particle to look at its behaviour. The first one is the charge conjugation operation ‘C’ which inverts all the signs of all r r the internal quantum numbers of a particle, leaving its mass, energy, momentum p , spin s r r and helicity (or ‘handedness’) h ≡ s . p unchanged: r C r r C r C p⎯ ⎯→ p , s ⎯ ⎯→ s , h ⎯ ⎯→ h The second one is the parity operation ‘P’ which reverses all the space coordinates of a r particle. Therefore, all its real vectors like its position r and its momentum are reversed, whereas all its axial or pseudo vectors like its spin are not. This implies that its helicity must change.
r P r r P r r P r P r⎯ ⎯→ − r , p ⎯ ⎯→ − p , s ⎯ ⎯→ s , h ⎯ ⎯→ −h As mentioned in the introduction, it is the combined operation CP (or PC) that changes a particle into its antiparticle. The third and last one is the time reversal operation ‘T’ which converts all the properties of a particle into those of the same particle running backwards in time, that is, moving and ‘spinning’ in the opposite direction, leaving its handedness unchanged.
r T r r T r T p⎯ ⎯→ − p , s ⎯ ⎯→ − s , h ⎯ ⎯→ h
II- Physical Background
8
These three symmetry transformations were originally thought to be exact symmetries, that is, one could not differentiate between: - a particle observed in a matter universe from its ‘antiparticle’ observed in the same antimatter universe (C conservation) - a ‘mirror-particle’ observed in a mirror universe (P conservation) - a particle moving backwards in time in a universe evolving equally backwards in time (T conservation) However, it turned out that these symmetries can all be broken or violated in weak decays. In 1956, Lee and Yang discovered that parity was not conserved via weak interaction in K mesons systems whereas it was in the electromagnetic and strong ones [5]. This was also successfully corroborated in 1957 by Ms Wu and her team [6]. Soon after, it was found that C was equally violated, especially by examining the spins of e − and e + in respectively µ + and µ − decays [7]. Another example of C violation is the nonexistence of these 2 potential particles that should only be sensitive to the weak interaction, namely the right-handed neutrino ν R
( h > 0 i.e. θ ( sr , pr ) < 90° ) and the left-handed
antineutrino ν L ( h < 0 i.e. θ ( sr , pr ) > 90° ). No experiments have ever observed one or the other. For instance, applying C to the decay of a π + emitting a ν L should give a π − emitting an ν L . But only a ν R is observed. More recently in 1998 and in K meson systems again, the CPLEAR experiment at CERN (CP standing for CP violation and LEAR for Low Energy Anti-proton Ring) observed a case of T violation [8]. Considering the previous example leading to C violation, one can nevertheless see that applying P after C would turn the ν L into ν R which is observed.
νR P
νL
does not exist => P violation
νL
C
νR
C & P
C
νL
ν R does not exist => C violation
P
Figure 1: C and P symmetry operations are violated by the weak interaction. However, the combined CP operation seems to be conserved.
II- Physical Background
9
This combination of operations looking invariant reassured all the community, but a new surprise arose. In 1964, still in K mesons, Cronin and Fitch discovered the first laboratory evidence of CP violation [9]. Nowadays, only the combination of the three symmetry operations all together (the ‘CPT combination’) is believed to be invariant. That is, the backwards observation of a phenomena filmed through a mirror in an antimatter universe would be indistinguishable than the same phenomena observed in ‘natural’ conditions [10].
II.2- CP Violation in the Standard model: One of the first attempts to explain CP violation came from Wolfenstein in 1964 [11]. His theory was implying a new unknown force, the “weak superforce”. Although simple and elegant, it was abandoned, not being able to explain new phenomena. It was only in 1973 that a valid explanation, based on the work of Cabibbo [12], was proposed by Kobayashi and Maskawa [13]. Cabibbo first realised that the weak interaction does not ‘see’ the flavour of the then 4 known quarks (u/d, c/s) as the electromagnetic or the strong interactions do. Instead, it feels a mixture of quarks. To express his idea, he so created a 2x2 matrix (still only 4 quarks) comprising a real parameter θ c – today known as the Cabibbo angle – that must be found by experiment.
⎛ d ' ⎞ ⎛Vud ⎜⎜ ⎟⎟ = ⎜⎜ ⎝ s ' ⎠ ⎝ Vcd
Vus ⎞⎛ d ⎞ ⎛ cos(θ c ) sin(θ c ) ⎞⎛ d ⎞ ⎟⎜ ⎟ = ⎜ ⎟⎜ ⎟ Vcs ⎟⎠⎜⎝ s ⎟⎠ ⎜⎝ − sin(θ c ) cos(θ c ) ⎟⎠⎜⎝ s ⎟⎠
VCabibbo
d ' = (cos θ c ).d + (sin θ c ).s s ' = (− sin θ c ).d + (cos θ c ).d
where the dashed letters represent the quark eigenstates seen by the weak interaction (called flavour eigenstates), and the undashed ones the ‘normal’ quark eigenstates as felt by the electromagnetic or the strong interaction (the mass eigenstates).
II- Physical Background
10
W-
s gWVus
u Figure 2: Feynman diagram of a weak quark flavour changing process. The coupling strength at the vertex is given by the weak coupling constant gW times the corresponding element in the Cabibbo matrix.
But once again, this could not completely explain the observation of new phenomena. Some years later, Kobayashi and Maskawa realised that with the introduction of a third generation of quarks - thus leading to a 3x3 matrix with 4 independent parameters having equally to be found by experiments (3 real ones -3 other ‘mixing angles’- and one non-trivial complex phase) - these phenomena could be explained. This matrix is known as the CabibboKobayashi-Maskawa matrix (or CKM matrix), and it is the aforementioned complex phase parameter that is indeed the source of CP violation in the Standard Model of Particle Physics.
⎛ d ' ⎞ ⎛Vud ⎜ ⎟ ⎜ ⎜ s ' ⎟ = ⎜ Vcd ⎜ b' ⎟ ⎜ V ⎝ ⎠ ⎝ td
Vus Vcs Vts
Vub ⎞⎛ d ⎞ ⎟⎜ ⎟ Vcb ⎟⎜ s ⎟ Vtb ⎟⎠⎜⎝ b ⎟⎠
VCKM
Similarly, this physically translates to: W-
b gWVcb
c Figure 3: Feynman diagram of a weak quark flavour changing process. The coupling strength at the vertex is given by the weak coupling constant gW times the corresponding elements in the CKM matrix.
II- Physical Background
11
This matrix being unitary, some very useful relations can be inferred. One of special interest experimentally speaking is the following one:
Vud Vub* + Vcd Vcb* + Vtd Vtb = 0 As a matter of fact, if divided by Vcd Vcb* , it becomes: Vud Vub* V V + 1 + td tb* = 0 * Vcd Vcb Vcd Vcb Each term being complex, they can be drawn as vectors in the complex plane. Once arranged head-to-tail, they form a triangle (their sum giving 0). Since one side of this triangle is 1, it lies on the real axis and has a modulus of 1. Thus, only the coordinates of the top point need to be specified, i.e. once two of the α, β and γ angles are known, the triangle is perfectly defined. This visual help is called the “Unitarity triangle”.
Im
Vtd Vtb Vcd Vcb*
α
?
γ
Vud Vub* Vcd Vcb*
β
(0,0)
Re (1,0)
Figure 4: The unitary triangle
Basically, the goal of any CP violation experiments is to accurately measure these three angles (to assure they do sum to 180°), by as many independent means as possible. Leading to the determination of the CKM matrix elements and so to its non-trivial complex phase, the expected amount of matter in the universe could be inferred and if this does not tally with what is observed, then there should be some new physics underneath, i.e. physics beyond the Standard Model.
II- Physical Background
12
II.3- B mesons and the BABAR experiment: In order to confirm the idea of Kobayashi and Maskawa, a run for the third generation of quarks started. In 1977, the b quark was discovered in a Y resonance also called bottonium ( Υ = bb ) [14] an in 1995, the CDF collaboration (Collider Detector at Femilab) discovered the top quark [15]. The third generation of quarks was complete. Mesons with b quarks therefore appeared to be natural new candidates to investigate CP violation. B mesons, first discovered by the CLEO Collaboration in 1983 [16], are a bound state of a b quark and a light anti-quark ( B ± = ub , bu , B 0 = db ). Made up of a third generation quark and so having a large mass, they are a good environment to study CP violation. Many decays are possible, which offer many different avenues of research. The CKM matrix elements can almost be all measured, especially the third generation related ones (3rd columns and 3rd rows). This is not possible with K mesons ( K ± = us , s u and
K 0 , ( K S0 , K L0 ) = ds ) or D mesons ( D ± = cd , dc , D 0 = cu and Ds± = cs , s c ), since they are not made up of third generation quarks.
B mesons look like perfect candidates. However, unlike the C and P violation that are said to be violated ‘maximally’ –ν R and ν L do not exist – CP violation is rather a small effect in B decays. Although branching ratios of K mesons can at most be of order 10-3, B meson ones are usually at a 10-6 level. Thus, many millions of B mesons need to be produced to ensure any accuracy. To generate this massive quantity of particles, several devices has been used or designed. The two first collaborations trying to measure CP violation in B decays were: CDF at Fermilab and CLEO at CESR (Cornell Electron Storage Ring) (CLEO is not an acronym, it is just the short for Cleopatra, a suitable companion for CESR - pronounced “Caesar”) Despite not having obtained any relevant results, they paved the way for two new experiments that, as said in the introduction, have already observed CP violation in B mesons. These two current collaborations are: BELLE (B standing for B mesons, EL for electrons and LE for anti electrons EL = LE ) at the KEK-B collider (Koh-Enerugii Kasokuki kenkyu kikou - High
Energy Accelerator research organization) in Japan and BABAR (standing for B B ) at SLAC (Stanford Linear Accelerator Center) in California.
II- Physical Background
13
BABAR is also the name of the detector that has primarily been built to study CP violation in B meson decays. However, comprising all the elements of a general purpose detector, namely : - a high resolution silicon vertex tracker, - a drift chamber for general tracking and momentum measurement, - a Čerenkov detector to distinguish particles, - an electromagnetic calorimeter enclosed in a 1.5 T solenoid to measure photons and electrons energy - a kind of hadronic calorimeter to detect muons and neutral kaons it can be used for many other tasks and is so an excellent opportunity to look at many parts of the Standard Model.
Muon/Hadron Detector
Magnet Coil
EM calorimeter (Electron/Photon Detector)
Čerenkov Detector
Tracking Chamber
Silicon Vertex Detector
Figure 5: Schematic diagram of the BABAR detector
II- Physical Background
14
To create B mesons, BABAR uses the PEP-II SLAC accelerator (PEP stands for Positron Electron Project). This ‘B factory’ is an e + e − collider constructed with the express purpose of producing large quantities of B mesons. To produce these B mesons, e + and e − are collided with a centre-of-mass energy equal to that necessary to create the Y(4s) resonance (10.58 GeV). This forms a Y(4s) approximately one quarter of the time, generating a high signal to background ratio. The energy of this Y(4s) being just above the production threshold required to form a B B pair, it decays in almost 100% of the cases to produce either a B + B − pair or a 0
B 0 B pair: e + + e − → Υ (4 s ) → B + B − or B 0 B
0
If the e + and e − have the same energy, the Y(4s) is created at rest and because of the small difference of mass between this and a pair of B B , the B B pair is produced almost at rest too. This inhibits an accurate measurement of the decay length between the Bs. Thus, to allow a better resolution, e + and e − of different energy (respectively 3.1 GeV and 9.0 GeV) are collided. The PEP-II collider is indeed an asymmetric collider.
Figure 6: Schematic diagram of the PEP II collider showing the BABAR detector
II- Physical Background
15
*−
0
II.4- The B 0 → K *+π − & B → K π + decays: A way to measure the CKM matrix elements (or to measure the angles of the unitary triangle) is to measure the asymmetric parameter ACP . This is proportional to the difference between the branching ratio BR of a B meson decay – i.e. the likelihood the decay happens – and that of the conjugate process. For instance, if the decay studied is of the form:
B → X + Y , the conjugate process will be: B → X + Y and the asymmetric parameter will
be such that: ACP ∝ BR{B → X + Y } − BR{B → X + Y }. Formally, the asymmetric parameter
is defined as the ratio of the difference of the conjugate decays branching ratios to their sum, that is:
ACP =
BR{B → X + Y } − BR{B → X + Y } BR{B → X + Y } + BR{B → X + Y } 0
The B decay studied in this project is B 0 → K *+ π − so its conjugate is B → K *−π + . Distinguishing two conjugate neutral B mesons and ensuring they come from the same Y(4S) is not a trivial task. Actually, this would require a whole other project. Thus, no calculation of the asymmetric parameter was required. The aim of this project was only to determine a global branching ratio for the two conjugate decays simultaneously. This corresponds to the denominator of ACP . Let us consider the B 0 decay first. To detemine an accurate branching ratio for this decay, the K *+ decay into a π + and a K S0 is of special interest. Indeed, the final state produced consists of three particles and is referred to as a three body charmless state ( π − π + K S0 where π + = ud , π − = u d , and K S0 = ds , no charm quark)
B 0 → K *+ π − K S0π + But other physical processes occur, namely:
( K S0π + )π − B 0 → K *+π −
( K L0π + )π − ( K + π 0 )π −
II- Physical Background
16
As a result, the total branching ratio of the B 0 decay is given by the sum of those of these 3 possible decays:
{
BR B 0 → K *+π −
}
{
}
{
}
{
= BR B 0 → K *+π − + BR B 0 → K *+π − + BR B 0 → K *+π − K S0π +
K L0π +
}
K +π 0
As shown by the three following Feynman diagrams, all these second branching ratios are equally likely:
{
BR B 0 → K *+π −
}
{
= BR B 0 → K *+π −
K S0π +
{
BR B 0 → K *+π −
}
}
K L0π +
{
BR B 0 → K *+π −
K S0π +
{
}
K +π 0
}
{
BR B 0 → K *+π −
K L0π +
}
K +π 0
=
BR B 0 → K *+π −
{
= BR B 0 → K *+π −
=
}
{
BR B 0 → K *+π −
{
Therefore, BR B 0 → K *+π −
}
}
{
BR B 0 → K *+π −
{
1 3
=
= 3 ⋅ BR B 0 → K *+π −
}
}
K S0π + Calculating the main branching ratio amounts to measuring one of the three possible secondary ones and multiply it by a factor of 3. 0
The approach is exactly the same for the B decay. One just need to replace particles by antiparticles. Hence, the total branching ratio for the both conjugate decays is:
{
}
{
0
*−
BR B 0 → K *+π − + BR B → K π +
}=
{
}
{
}
0 0 ⎡ ⎤ 3 ⎢ BR B 0 → K S0π − + BR B → K Sπ + ⎥ ⎣ ⎦
K S0π +
0
K Sπ −
II- Physical Background
17
s d
K S0
d u
π+
s d
K L0
u
d u
π+
s
s u
K+
u
u u
π0
s u
s u
K+
u
π0
s K *+
u
s K *+
K *+
K *+
u
Figure 7: Feynman diagrams of all the possible decays coming from the decay of the K
*+
in the channel B
0
forbiddenbecause of Zweig suppression.
→ K *+ π − . The last decay is
III- Analysis Method
18
III- ANALYSIS METHOD
III.1- Data Sample and Preselection: The data sample on which this analysis is performed was collected at BABAR between 0
1999 and 2002 and is made up of 61.6 ± 0.68 millions of B 0 B events. In order to reduce the overall analysis time, the data is firstly skimmed at BABAR by discarding all the B 0 / B candidates that are irrelevant to the decay under study. For each B 0 / B
0
0
candidate,
discriminating variables are calculated. If they satisfy a loose set of selection conditions, the 0
B 0 / B candidate is integrated to a data structure called ‘Ntuple’. Over 200 variables are dumped into the Ntuple. This therefore contains all the relevant information concerning a 0
B 0 / B candidate, such as its mass, energy, momentum, …, as well as those of its daughter particles, and some event shape and particle identification information. This pre-processing of 0
data is called preselection. The Ntuple used here contains 7 893 939 B 0 / B candidates.
III.2- The Cut and Count Method (CCM): III.2.1- Principle of the CCM: At the Ntuple level, another set of selection requirements or ‘cuts’ is applied to remove 0
all the irrelevant background events. If all the conditions are satisfied by a B 0 / B candidate, this is counted. Formally, such an analysis is referred to as a ‘selection cut based counting analysis’. The discriminating variables used in this analysis can be divided in 5 categories. These are described hereafter.
III- Analysis Method
19
III.2.2- Selection variables: III.2.2.1- Event shape variables: Two event shape variables are used in this analysis: the thrust angle (or more precisely its cosine) and the Cornelius Fisher discriminant. They both aim at rejecting the continuum 0
background events coming from qq random production (with q=u,d,c,s). The B 0 / B mesons being produced almost at rest in the centre-of-mass frame, they decay isotropically (spherically) in this frame. In contrast, continuum events are very jet-like. Therefore, if a 0
B 0 / B candidate appears to have a jet-like decay, it is more likely to have been reconstructed from random continuum events. This feature provides a powerful tool to discrimate between (real) signal and continuum background events. Practically, the thrust angle is defined as the angle between the direction of thrust of the 0
three particles constituting the decayed B 0 / B and the direction of thrust of all the other events (in the centre-of-mass frame) Taking the cosine of this angle leads to a flat distribution 0
for real B 0 / B and a very peaked one near cos(θt)=±1 (θt≈±π) for dummy candidates. Concerning the Cornelius Fisher discriminant (‘fisherCrn’), its physical representation is hazier. It combines several event shape variables together. Namely it includes the summed energy of the aforementioned rest of events in nine cones around the thrust axis of the B 0 / B 0
0
candidate, as well as the cosine of the B 0 / B thrust axis with respect to the beam axis and 0
the cosine of the B 0 / B decay axis with respect to the beam axis [18]. As shown hereafter on Monte Carlo simulated data, typical values for these cuts are: .
cos(θt ) < 0.7
. fisherCrn < −0.5
III- Analysis Method
20
MC Data: | cos θt | cut 600
Number of events
500
400
300 200
100
0 0
0.1
0.2
0.3
0.4
0.5 0.6 | cos θt |
0.7
0.8
0.9
1
Figure 8: |cos(θt)| distribution before (solid line) and after (dashed line) all the cuts. Here, |cos(θt)| < 0.65
MC Data: FisherCrn cut 1800 1600
Number of events
1400 1200 1000 800 600 400 200 0 -2
-1
0
1 Fisher Crn
2
3
Figure 9: Cornelius Fisher Discriminant (‘fisherCrn’) distribution before (solid line) and after (dashed line) all the cuts. Here, fisherCrn < -0.4
III- Analysis Method
21
III.2.2.2- Particle Identification: Several PID (Particle IDentification) information are enclosed in the Ntuple. The three 0
desired decay products of the B 0 / B (the three-body final state) can thus be selected. Each of this decay product is referred to as a track. Thus, three track variables are defined: trk1, trk2 0
and trk3. The Ntuple is built in a way that the 3rd track always contains the K S0 / K S . The other tracks can only be those of either a kaon K or a pion π . 0
The three-body final state studied being π −π + K S0 / π +π − K S , a prior cut exists in order to 0
discard all the B 0 / B candidates for which the tracks 1 or/and 2 refers to a kaon (or may refer to a kaon, the preselection process leading to these variables not being perfect) The Ntuple is not built for only one secondary decay. Many secondary decays are available. In the Ntuple used, three different final states can be studied: π ±π m K S0 , π ± K m K S0 or 0
K ± K m K S0 (no distinction is made between K S0 and K S ). With permutation around track 1 and
2, this provides 8 possible combinations that are recorded in an array as shown below.
0
1
2
3
Trk1
trk2
trk3
trk1
trk2
trk3
trk1
trk2
trk3
trk1
trk2
trk3
π
π
K S0
π
K
K S0
K
π
K S0
π
π
K S0
4
5
6
7
trk1
trk2
trk3
trk1
trk2
trk3
trk1
trk2
trk3
trk1
trk2
trk3
π
K
K S0
π
π
K S0
π
K
K S0
K
K
K S0
Table 1: Final states available in the Ntuple. The 2 particles constituting one resonant state are shaded.
Second cuts can thus be applied to only look at the variables belonging to the secondary 0
decay of interest, namely in this project, the resonant state: K S0π + / K Sπ − . From the array above, only the indices 3 and 5 will therefore be used.
III- Analysis Method
22
III.2.2.3- The
K *+ → K S0π + / K
*+
0
→ K Sπ −
resonance:
Using what has been said beforehand, 2 cuts are performed on the resonant non 0
differentiated connjugate states K S0 → π −π + / K S → π + π − .
The first one concerns the mass of the K *+ / K
*−
for which the Particle Data Group 0
(PDG) [17] cites a nominal value of 0.896 GeV/c2. Any B 0 / B candidate having a K *+ / K
*−
with a mass too different from this value is therefore discarded. This mass is calculated by 0
summing those of K S0 / K S (trk3) and π ± (trk1 or trk2). The result is then put in an array variable called resMass[]. As explaimed, the cuts therefore test the value of resMass[3] and resMass[5] with a certain tolerance. A typical value for this tolerance is: m
K *+ / K
*−
−m
0
K S0 π + / K S π −
< 0.10 GeV/c2
MC Data: K* +→ K0spi +1 Mass cut
3500
Number of events
3000 2500 2000 1500 1000 500 0
0
1
2
+3 m {K* →K0spi+1}
4
5
6
+3 m {K* →K0spi+2}
4
5
6
MC Data: K* +→ K0spi +2 Mass cut
4500
Number of events
4000 3500 3000 2500 2000 1500 1000 500 0
0
1
2
Figure 10: K *+ → K S0π + / K
*+
0
→ K S π − resonance mass
distribution before (solid line) and after (dashed line) all the cuts.
III- Analysis Method
23
The second one deals with the helicity of this resonant state. The K *+ / K
*−
is
longitudinally polarised, which means that its helicity angle θh – defined as the angle between 0
its line of flight and its direction decay – is small. A good B 0 / B candidate will therefore have a cos(θh) greater than 0. A typical value for this cut is: cos(θ h ) > 0.4
+
MC Data: cos ( θ h{K* → K0spi +1} ) cut 2500
Number of events
2000
1500
1000
500
0
0
0.1
0.2
0.3
0.4
0.5+ 0.6 cos ( θh{K* →K0spi+1} )
0.7
0.8
0.9
1
0.5+ 0.6 cos ( θh{K* →K0spi+2} )
0.7
0.8
0.9
1
+
MC Data: cos ( θ h{K* → K0spi +2} ) cut
1800
Number of events
1600 1400 1200 1000 800 600 400 200 0
0
0.1
0.2
0.3
0.4
*+
0
Figure 11: K *+ → K S0π + / K → K S π − cos(θh) distribution before (solid line) and after (dashed line) all the cuts.
III- Analysis Method
24
III.2.2.4- The
0
K S0 → π −π + / K S → π + π −
resonance: 0
The same procedure can be performed on the resonant decay K S0 → π − π + / K S → π + π − . 0
The first cut is about the mass of the K S0 / K S for which the Particle Data Group (PDG) 0
0
[17] cites a nominal value of 0.498 GeV/c2. Any B 0 / B candidate having a K S0 / K S with a mass falling outside this value (within a certain tolerance) is therefore discarded. This mass is calculated by summing those of the π + and π − . The result is then put in a variable called gkMass (‘gk’ standing for Grand Kids, the Kids being then K *+ / K A typical value for this tolerance is: m
0
K S0 / K S
*−
).
− mπ mπ ± < 0.01 GeV/c2
MC Data: K0s→ π+π- Mass cut 3500 3000 Number of events
2500 2000 1500 1000 500 0 0.46
0.47
0.48
0.49
0.5 0.51 0 m {Ks →π +π -} 0
0.52
0.53
Figure 12: K S0 → π − π + / K S → π + π − mass distribution before (solid line) and after (dashed line) all the cuts.
0.54
III- Analysis Method
25
0
The second one concerns the measured flight length of the K S0 / K S , i.e. its decay length calculated from its decay time: ldecay = cτ decay (where c is the speed of light). In order to be dimensionless and for more efficiency, the quantity used is indeed the ratio of the decay length to its error: cτ / σ cτ . The variables associated with these values are called: gkCtau (decay length) and gkCtaue (error on the decay length). A typical value for this cut is:
cτ
σ cτ
> 5.0
MC Data: K0s→ π+π- Decay length cut 500
Number of events
400
300
200
100
0 0
2
4
6
8 10 0 12 (c τ/∆ cτ) {Ks →π + π-} 0
14
16
Figure 11: K S0 → π − π + / K S → π + π − decay length distribution before (solid line) and after (dashed line) all the cuts.
18
20
III- Analysis Method
26
III.2.2.5- Kinematic constraints, the ∆E-mES plane: 0
Due to the great efficiency of BABAR , the kinematics of the B 0 / B mesons is well defined. It is then possible, with the help of some initial information about the e + e − beam, to use this as new selection criteria. As implied by the title, 2 kinematic constraint variables are used in this final event selection. The first one, ∆E , is the difference between the reconstructed energy of the B 0 / B
0
meson E B and that expected from the beam termed “beam-energy constrained energy” Ebc :
∆E = EB − Ebc E B is derived from the momentum measurement of the three daughter particles 0
π −π + K S0 / π + π − K S (actually that of these two pions and that of the two pions coming from 0
the K S0 / K S decay since measuring a momentum requires charged particles) and a hypothesised mass associated with each momentum. This mass hypothesis is necessary because only the momentum of each daughter particle is known. Thus, here, one momentum 0
must be associated with a K S0 / K S mass and the other two with a pion one. In natural units: i =3 ⎧ EB = ∑ ⎨ i =1 ⎩
2 ⎫ pi2 + mhyp i ⎬ ⎭
As for Ebc , it is calculated from the 4-momentum of the beam ( E beam , p beam ) and the 0 r momentum of the B 0 / B meson pB , which is equal to the sum of those of its three daughter particles. Still in natural units, Ebc is given by: 2 r r 2 − p beam − 2( pbeam . p B ) Ebeam Ebc = 2 Ebeam
If the chosen mass hypothesis is correct, ∆E should be centered around 0.
III- Analysis Method
27
The second kinematic variable used, m ES , known as the “beam-energy substituted mass”, is, with the same notation, defined as: m ES = Ebc2 − p B2 0
This therefore tests whether the momentum of the reconstructed B 0 / B fits with the expected energy for the beam to give the correct mass. In that case, the mass found should be 0
approximately that of a B 0 / B meson, namely: 5.279 GeV/c2. If this is not the case, either 0
0
the B 0 / B candidate was not a real B 0 / B , or it has been reconstructed incorrectly, that is, from random particles. It is worth noting that contrary to ∆E , m ES , since using Ebc and not E B , does not depend on the mass hypothesis.
Usually, these 2 kinematic variables are used in pairs. A very common visual construction in Particle Physics is the ∆E - m ES plane in which ∆E is plotted versus m ES . After having applied all the previous cuts, the surviving events are plotted in that plane and only those which lie around ∆E = 0.0 and mES = 5.279 GeV/c2 are finally counted. All the others are discarded. Practically, this is achieved by defining a box in the plane termed ‘signal region’ (SR). The size of that box is generally defined in such a way that 99% of the events generated with Monte Carlo simulated data fall inside. Typical values for these cuts (i.e. typical size for the SR) are: . ∆E = 0.0 ± 0.1 . mES = 5.279 ± 0.010 GeV/c2 As one can see hereafter, another box is defined in the ∆E - m ES plane. This box, termed the Grand Side Band (GSB) is bigger than the SR and lies on its left. Its interest is the following. Once all the selection criteria have been applied, several non-signal events still remain in the SR as one can see by the fact that all the events do not fall inside the SR. There are two sources of parasitic events. The first one is background from other decay channels. This is negligible in the decay studied. The second one comes from the fact that a B 0 / B
0
meson can be reconstructed in more than one way as shown by the sketch hereafter. This kind of background is called random combinatoric background.
III- Analysis Method
28
B0 ?
π+ B
B0 ?
0
π
−
K
0 S
B0 ?
π−
π+
B0 ? B0
K S0
π−
K S0
π−
π+
B0 ? Figure 12:
Illustration of random combinatoric background. A B 0 / B
0
can be
reconstructed from 3 random particles (major source of combinatoric background) or a mixing of random particle(s) and daughter particle(s) or a mixing of daughter particles 0
belonging to different B 0 / B .
In order to subtract this background events from the final counting, a statistical assessment on their density is performed on a neighbouring region: the GSB. MC Data: DeltaE-MES Plane
0.4 0.2
∆E
0 -0.2 -0.4 -0.6 -0.8 5.18
5.2
5.22
5.24 mES
5.26
5.28
5.3
5.22
5.24 mES
5.26
5.28
5.3
MC Data: DeltaE-MES Plane
0.4 0.2
∆E
0 -0.2 -0.4 -0.6 -0.8 5.18
5.2
Figure 13: The
∆E - m ES plane before (on top) and after (on bottom) all the cuts.
The SR is the small box on the right, the GSB, its left neighbour.
III- Analysis Method
29
III.2.3- Computing overview: This analysis has been performed within the CERN software to process particle physics data, namely ROOT. ROOT is a C++ interpreter with many inbuilt functions and classes that allow an easy analysis of such data. As on could have guessed, the basis of the programme is just a big loop within which several conditional structures (‘if’ statements) lie:
potential_real_B0 = 0 ; for i=1 to total_Number_Of_B0_Candidates { if ( cut1 ok ) then if ( cut2 ok ) then ... potential_real_B0 = potential_real_B0 + 1 ; }
For further details, please refer to the Appendix (p.54), in which this loop is fully given.
III.3- Calculation of the Branching Ratio: III.3.1- Definition: As everything said above may suggest, the branching ratio BR of a decay, i.e. its occurring likelihood, is given by: BR =
real N sig
NB0
0
real where N B 0 is the total number of B 0 / B events and N sig is the final number of events
counted (i.e. after the aforementioned statistical subtraction). The subscript real is just here to remind that this branching ratio must be calculated using real data (or on-line resonance data) MC and not MC simulated one for which this final number will therefore be noted: N sig .
III- Analysis Method
30
III.3.2- MC Efficiency: In fact, this last formula is not exactly true. The preselection and cuts being rather tight, 0
real many real B 0 / B mesons are not counted in N sig . Therefore, the branching ratio must be
scaled up by a corrective factor k to take this matter into account. In order to assess this, a MC 0
simulation is performed on events which are exclusively B 0 / B mesons. The simulated mesons thus created pass the same preselection and cuts as the real ones. The probability of a 0
B 0 / B falling inside the SR, in other words, the efficiency of the computing process, is consequently given by:
ε MC =
MC N sig
NB0
where ε MC stands for ‘MC Efficiency’ Note that no statistical subtraction of the combinatoric background is done. This feature can be taken into account with simulated data for which almost everything can be known (particles can be tagged, …) This can also be seen on the previous schemes (figure 13) on which there are no events in the GSB. The scale-up factor k is so given by: k=
1
ε MC
=
N BMC O MC N sig
As a result,
BR =
real N sig × 1
ε MC
N
real B0
However, it is not so simple. The MC Efficiency thus calculated must be corrected to take into account any inconsistencies with real data. The major source of correction lies in a poor modelling of the detector. It is therefore this corrected MC Efficiency ε MC , corr and not the prior calculated one ε MC that must finally be used in the calculation of branching ratio: real × 1 N sig
BR =
ε MC ,corr
N
real B0
III- Analysis Method
31
The corrected MC Efficiency ε MC , corr is given by:
ε MC , corr = kcorrε MC where the correction kcorr depends on many other factors that are associated with the cuts applied : kcorr = kevtShp × k PID × ktrk × k K 0 × k∆E × km S
ES
The k evtShp factor is a correction on the event shape modelling process, the k PID one on the PID process, the k trk one on the tracking process, the k K 0 one on the K S0 selection and the k ∆E S
and the k mES ones on the ∆E and m ES calculations. Except for k trk which must be calculated from the data used, all the other correction factors have been previously calculated, especially by N. Chevalier [18].
Two kinds of error therefore come from the corrected MC efficiency: a statistical one MC due to the counting method used to calculate ε MC ( N sig is just a counter) and a systematical
one due to the above corrections (the error on the k factors being only systematical) used to calculate the ‘real’ MC Efficiency (the corrected one). This mathematically leads to the following errors:
ε MC , corr = kcorrε MC
σε
stat
σε
MC , corr
syst MC , corr
2
⎛ σ kstat × ⎜ corr ⎜ k corr ⎝
⎞ ⎛ σ εstat ⎟ + ⎜ MC ⎟ ⎜ ε MC ⎠ ⎝
⎛ σ ksyst = ε MC , corr × ⎜ corr ⎜ kcorr ⎝
⎞ ⎛ σ εsyst ⎟ + ⎜ MC ⎟ ⎜ ε MC ⎠ ⎝
= ε MC ,corr
Since as aforementioned
2
σ kstat = 0.0 corr
programmer ( N B O = 50,000 ± 0 ± 0 ), MC
2
⎞ ⎟ =ε MC ,corr ⎟ ⎠
⎛ σ εstat × ⎜ MC ⎜ ε MC ⎝
⎞ ⎟ ⎟ ⎠
⎞ ⎛ σ ksyst ⎟ =ε ⎜ corr MC , corr × ⎟ ⎜ kcorr ⎝ ⎠
⎞ ⎟ ⎟ ⎠
2
MC
and since N B O
σ εsyst = 0.0 MC
(since
being fixed by the MC MC / NB ε MC = N sig O
)
III- Analysis Method
32
With,
1) ε MC =
N N
MC sig MC B0
⇒ σ εstat = ε MC MC
(since N BMC O
2) kcorr = kevtShp × k PID × ktrk × k K 0 × k∆E × km
⇒ σ ksyst
corr
⎛ σ ksyst = kcorr × ⎜ evtShp ⎜k ⎝ evtShp
2
stat 2 ⎞ ⎛⎜ σ N BMC0 ⎞⎟ ε ⎟ + ⎟ = MCMC ⎟ ⎜ N MC 0 N sig ⎠ ⎜⎝ B ⎟⎠ = 50,000 ± 0 ± 0 )
⎛ σ NstatMC sig × ⎜ MC ⎜N ⎝ sig
S
ES
2
2
⎞ ⎛ σ syst ⎟ + ⎜ k PID ⎟ ⎜k ⎠ ⎝ PID
⎞ ⎛σ ⎟ +⎜ ⎟ ⎜ ktrk ⎠ ⎝
syst k trk
⎞ ⎟ ⎟ ⎠
2
⎛ σ ksyst ⎜ K0 +⎜ S ⎜ k K S0 ⎝
2
⎞ ⎛ σ syst ⎟ ⎜ k ∆E ⎟ +⎜ ⎟ ⎝ k ∆E ⎠
⎞ ⎟ ⎟ ⎠
2
⎛ σ ksyst ⎜ m + ⎜ ES ⎜ k m ES ⎝
⎞ ⎟ ⎟ ⎟ ⎠
2
III.3.3- Background Characterisation and Subtraction: As previously mentioned, the background events remaining in the whole ∆E - m ES plane once all the cuts have been applied, are mainly of a random combinatoric nature, i.e. they are 0
in fact B 0 / B candidates reconstructed from completely random particles. Thus, for one real 0
0
B 0 / B , many B 0 / B candidates exist. To avoid multi-counting, a special variable called 0
nevent is used. This is associated with all the B 0 / B candidates assumed to come from the
same real decay:
0
B 0 / B candidate #
0
1
2
3
4
5
6
7
8
9
nevent
1
1
2
2
2
2
3
4
5
5
Table 2: To avoid multi-counting, the ‘nevent’ variable is used.
0
0
Amongst all these B 0 / B candidates, only one or zero can be a real B 0 / B . In order to only keep one candidate and not to underestimate the number of combinatoric background events in the GSB (this will be of great help later), a candidate is usually chosen completely at random.
III- Analysis Method
33
This process being unfortunately quite cumbersome and time-consuming, it has not been applied here. Instead, only the first candidate of each nevent sequence is kept (e.g. in the example above, the candidates 0,2,6,7,8), which remains somewhat random if the data ranking is itself random. Nevertheless, combinatoric background events can still remain in the SR. As already said, this background must be subtracted from the number of events observed in the SR to correctly calculate the BR. A basic way to estimate this background is to characterise the background distribution in both the GSB and the SR using ‘off-line resonance’ data, i.e. with 0
data collected during a phase where no B 0 / B mesons can be created, the energy of the beam being too small to form the Y(4s) necessary to their creation. This can sound odd, but the combinatoric background is mainly due to completely random particles from continuum events (such as massive massive qq production) rather than rare different daughter particles association. The ∆E and m ES background distributions thus generated being almost independent, they can reasonably be studied separately. The ∆E background distribution is (weakly) quadratic: N ∆E = a(∆E ) 2 + b(∆E ) + c
(where N is the number of counts) Whereas the m ES one has the shape of an Argus function [19]: Nm
ES
⎛ m = C ⋅ ⎜⎜ ES ⎝ mMAX
⎞ ⎛ m ⎟⎟ × 1 − ⎜⎜ ES ⎠ ⎝ mMAX
2 ⎧⎪ ⎡ ⎛ m ⎞ ⎟⎟ × exp⎨ − ξ .⎢1 − ⎜⎜ ES ⎢⎣ ⎝ mMAX ⎠ ⎪⎩
⎞ ⎟⎟ ⎠
2
⎤ ⎥ ⎥⎦
⎫⎪ ⎬ ⎪⎭
(where mMAX is the maximum possible value of m ES and ξ and C are respectively the “Argus background shape parameter” and the “scale factor”. mMAX is actually the same as that of the on-line resonance data (real data) i.e. 5.29 GeV/c2 which corresponds to half the energy of the beam (10.58 GeV), the B 0 and the B 0 sharing it equally. The off-line resonance data being obtained with a beam energy of 40 MeV less (that is, 20 MeV less by B mesons), the
m ES values must so be plotted shifted of 20 MeV/c2 to keep the same SR and GSB size values.
III- Analysis Method
34
By integrating the fitted function of these distributions with respect to the size of the SR and the GSB, the ratio R of the number of combinatoric background events in the SR to that in the GSB, can be calculated:
R=
nbg _ SR nbg _ GSB
=
∫N
∆E
∫N
d (∆E )
SR
∫ N ∆E d (∆E )
×
SR
∫N
GSB
GSB
m ES
m ES
d (mES ) d (mES )
A visual insight of all this characterisation process is given below.
Number of counts
GSB SR
mES (GeV/c2) 5.279
5.29
Figure 14: m ES Argus-shaped background distribution used to assess the proportion of combinatoric background in the SR.
Number of counts GSB
SR
∆E -0.2
-0.1
0.0
+0.1
+0.2
Figure 15: ∆E quadratic background distribution used to assess the proportion of combinatoric background in the SR.
III- Analysis Method
35
The final number of signal in the SR is therefore given by: real N sig = ntot _ SR − nbg _ SR
where ntot _ SR is the total number of events falling down the SR as counted once all the cuts have been applied and nbg _ SR the estimated number of combinatoric background defined from what has been said above by: nbg _ SR = R × nbg _ GSB = R × ntot _ GSB
where nbg _ GSB is the number of combinatoric background in the GSB, that is simply the number of events counted in the GSB once all the cuts have been applied ntot _ GSB .
(
real N sig = ntot _ SR − R × ntot _ GSB
So finally,
)
The systematical error on ntot _ SR and ntot _ GSB is zero (these are just counters) as well as the statistical one on R. The systematical error on this is also obtained by playing with the different parameters offered by the fit function of ROOT. The value of R is taken as the best fit value with an error equal to the half of the maximum deviation to this value:
R = Rbest _ fit ±
1 ⎧ ⎫ MAX ⎨ Rbest _ fit − Rmax_ fit , Rbest _ fit − Rmin_ fit ⎬ 2 ⎩ ⎭
real Concerning N sig ,
σ Nstat = (σ nstat ) 2 + (σ nstat ) 2 = nSR + (σ nstat ) 2 real sig
tot _ SR
bg _ SR
bg _ SR
σ Nsyst = (σ nsyst ) 2 + (σ nsyst ) 2 = σ nsyst real sig
SR
With,
bg _ SR
bg _ SR
2
σ nstat
⎛ σ stat = nbg_SR × ⎜⎜ R ⎝ R
stat 2 ⎞ ⎛⎜ σ ntot _ GSB ⎞⎟ ⎟⎟ + = ⎜ ⎟ ⎠ ⎝ ntot _ GSB ⎠
σ nsyst
⎛ σ syst = nbg _ SR × ⎜⎜ R ⎝ R
syst 2 ⎞ ⎛⎜ σ ntot _ GSB ⎞⎟ ⎛ σ syst ⎟⎟ + = nbg _ SR × ⎜⎜ R ⎠ ⎜⎝ ntot _ GSB ⎟⎠ ⎝ R
bg_SR
bg _ SR
2
nbg_SR ntot _ GSB
= R ntot _ GSB
⎞ ⎟⎟ = ntot _ GSB × σ Rsyst ⎠
III- Analysis Method
36
III.3.4- Errors on the branching ratio: As previously seen, the calculation of the branching ratio requires the following stages:
⎛ real 1 BR = ⎜ N sig × ⎜ ε MC ,corr ⎝
⎞ ⎟ ⎟ ⎠
N Breal 0
ε MC ,corr = k corr ε MC
real N sig = ntot _ SR − nbg _ SR
nbg _ SR = R × ntot _ GSB
ε MC =
MC N sig
N BMC 0
Hence, the errors on that branching ratio are such that:
stat σ BR
⎛σ = BR × ⎜ real ⎜N ⎝ sig stat real N sig
⎛σ
syst = BR × ⎜ σ BR ⎜
⎝
syst N sig
N sig
2
⎞ ⎛σε ⎟ + ⎜ MC , corr ⎟ ⎜ε ⎠ ⎝ MC ,corr 2
stat
⎞ ⎛σε ⎟ + ⎜ MC ,corr ⎟ ⎜ε ⎠ ⎝ MC ,corr syst
⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠
2
2
2
⎛ σ Nstatreal ⎜ B0 + ⎜ real ⎜ N B0 ⎝
⎞ ⎟ ⎟ = BR × ⎟ ⎠
⎛ σ Nsystreal ⎜ B0 + ⎜ real ⎜ N B0 ⎝
⎞ ⎟ ⎟ ⎟ ⎠
⎛ σ Nstat ⎜ sig ⎜N ⎝ sig
2
⎞ ⎛ σ εstat ⎟ + ⎜ MC ,corr ⎟ ⎜ε ⎠ ⎝ MC ,corr
⎞ ⎟ ⎟ ⎠
2
( since N Breal 0 = 61 600 000 ± 0 ± 680 000 given )
Where all the intermediate calculations to arrive to the final results can be found in looking at the previous parts. It is nonetheless worth noting that the statistical error comes from ntot _ SR and ntot _ GSB whereas the systematical one comes from the background charaterisation ratio R and the corrective factor for the MC Efficiency kcorr (and to a lesser extent from N Breal 0 ).
2
III- Analysis Method
37
III.4- Optimisation of the cuts: III.4.1- Significance calculation: All the cuts values, except for ∆E and m ES , are optimised by maximising an approximation of the statistical significance of the branching ratio σ defined by:
σ =
s s+b
where b is the number of background in the signal region (by analogy from above, it corresponds to nbg _ SR ) b = R × ntot _ GSB
and s is the estimated final number of events in the signal region (by analogy from above, it real ) corresponds to N sig
s = N Breal × BR ' × ε MC ,corr 0 s× 1 (since still by analogy BR = '
ε MC , corr
N Breal 0
)
s is said estimated since the branching ratio BR’ used to calculate it must be assumed (for
reason given later – Cf III.5, p.42) Although this may not completely reflect the true branching ratio for the channel under study, the optimisation is usually not heavily affected. A sensible value for it is usually the previous value found by someone else. The last person having studied this decay being N. Chevalier, the BR she found was used. Practically, each time a cut varies, 2 values change: the MC Efficiency appearing in s first, the number of events in the GSB leading to b secondly. Thus, by recalculating the corrected MC Efficiency and recounting the number of events in the GSB for each different value of one cut, the significance can be plotted versus these changing values and the maximum then found.
III- Analysis Method
38
All the selection cuts are thus varied independently in order of their use. Unfortunatelly, the cuts being not completely independent (or orthonormal), changing the value of one can change the others. For example, imagine the optimisation of cos(θt) leads to a value of 0.6 for a fisherCrn value of -0.5. Using this new value of 0.6 for cos(θt) within the fisherCrn optimisation gives a fisherCrn of -0.2. But reusing this new value of -0.2 for the fisherCrn within the cos(θt) optimisation will not yield a cos(θt) of 0.6 but, say, 0.7. The cuts
are not orthonormal. It can be difficult to deal with such a process but it is usually (rapidly) convergent . At last, this may provide several sets of cuts values, the ‘winning’ one being that yielding the best sensitivity.
III.4.2- Errors on the significance: The calculation of the statistical and systematical errors on the branching ratio statistical significance is given below.
σ =
s s+b
By definition,
s = N Breal × BR ' × ε MC ,corr 0
with
b = R × ntot _ GSB
dσ = (
∂σ ∂σ )ds + ( )db ∂s ∂b 2
⎛ ∂σ ⎞ ⎛ ∂σ ⎞ σ σ = ⎜⎜ σ s ⎟⎟ + ⎜⎜ σ b ⎟⎟ ⎝ ∂s ⎠ ⎝ ∂b ⎠
where
s/2+b ∂σ = ∂s ( s + b)3 / 2 s/2 ∂σ =− ∂b ( s + b) 3 / 2
2
III- Analysis Method
39
Hence,
With,
1
σσ =
1 ⎛s ⎞ 2 ⎛s⎞ 2 ⎜ + b ⎟σ s + ⎜ ⎟σ b 3/ 2 ( s + b) ⎝2 ⎠ ⎝2⎠
s = N Breal × BR ' × ε MC ,corr 0 2
σ sstat
⎛ σ Nstatreal ⎜ B0 = s × ⎜ real ⎜ N B0 ⎝
⎞ ⎛ σ stat ⎞ 2 ⎛ σ syst ⎟ ⎜ BR ' ⎟ ⎜ ε MC , corr +⎜ ⎟ +⎜ ' ⎟ ⎟ ⎝ BR ⎠ ⎝ ε MC ,corr ⎠
σ ssyst
⎛ σ Nsystreal ⎜ B0 = s × ⎜ real ⎜ N B0 ⎝
⎞ ⎛ σ syst ⎞ 2 ⎛ σ syst ⎟ ⎜ BR ' ⎟ ⎜ ε MC , corr + ⎟ +⎜ ' ⎟ ⎜ ⎟ ⎝ BR ⎠ ⎝ ε MC ,corr ⎠
2
2
⎞ ⎟ = s× ⎟ ⎠
⎞ ⎟ ⎟ ⎠
2
2
⎞ ⎛ σ εstat ⎟ + ⎜ MC , corr ⎟ ⎜ε ⎠ ⎝ MC ,corr
⎞ ⎟ ⎟ ⎠
2
2
MC ⎛ N sig And where ε MC ,corr = k corr ε MC = k corr ⎜ MC ⎜N 0 ⎝ B
And,
stat ⎛ σ BR ' ⎜ ⎜ BR ' ⎝
⎞ ⎟ ⎟ ⎠
( See previous parts for errors calculation )
b = R × ntot _ GSB 2
σ bstat
⎛ σ stat = b × ⎜⎜ R ⎝ R
stat 2 ⎞ ⎛⎜ σ ntot _ GSB ⎟⎟ + ⎜ ⎠ ⎜⎝ ntot _ GSB
⎞ ⎟ = ⎟⎟ ⎠
σ bsyst
⎛ σ Rsyst = b × ⎜⎜ ⎝ R
syst 2 ⎞ ⎛⎜ σ ntot _ GSB ⎟⎟ + ⎜ ⎠ ⎜⎝ ntot _ GSB
⎞ syst ⎟ = b × ⎛⎜ σ R ⎜ R ⎟⎟ ⎝ ⎠
2
b ntot _ GSB
= R ntot _ GSB ⎞ ⎟⎟ = ntot _ GSB × σ Rsyst ⎠
This gives the exact error on the statistical significance σ. Nonetheless, the interest of the optimisation process is not to accurately calculate this value. Its interest is, via this σ calculation, to be able to say which of 2 following values is the best if any differences between them. Thus, the error must be taken into account and especially the variation of error between these 2 points instead of the total error itself in which this variation may be (is) drowned.
III- Analysis Method
40
For example in the first case of the figure below, the points 1, 2 and 3, for which the total error is plotted, cannot really be differentiated, whereas in the second case, in which only the error variation is plotted, only the points 2 and 3 can be seen as similar.
. . . 2
. . .
3
2
1
3
1
Figure 16: Illustration of the optimisation interest via the error calculation of the statistical significance σ.
In order to only take into account this variational error, all the constant errors must be removed, that is:
(σ )
stat ' s
stat ⎛ σ BR ' ⎜ = s× ' ⎜ BR ⎝
(σ )
syst ' s
(σ ) (σ )
⎛ σ Nsystreal ⎜ B0 = s × ⎜ real ⎜ NB0 ⎝
2
⎞ ⎛ σ εstat ⎟ + ⎜ MC , corr ⎟ ⎜ ε MC , corr ⎠ ⎝
2
⎛ σ stat ⎞ ⎞ ⎟ → s × ⎜ ε MC , corr ⎟ ⎜ε ⎟ ⎟ ⎝ MC , corr ⎠ ⎠
2
⎞ ⎛ σ syst ⎞ 2 ⎛ σ syst ⎞ 2 ⎟ ⎜ BR ' ⎟ ⎜ ε MC , corr ⎟ → 0 .0 + ⎟ +⎜ ' ⎟ ⎜ ⎟ ⎟ ⎝ BR ⎠ ⎝ ε MC , corr ⎠ ⎠
stat ' b
= R ntot _ GSB → R ntot _ GSB unchanged
syst ' b
= ntot _ GSB × σ Rsyst → 0.0
III- Analysis Method
41
Practically, this leads to the following schemes:
s/ s+b vs |cosθt| 4 3.5 3
s / s+b
2.5 2 1.5 1 0.5 0 0
0.2
0.4
0.6 | cos θt |
0.8
1
0.4
0.6 | cos θt |
0.8
1
s/ s+b vs |cos θt| 3
2.5
s / s+b
2
1.5
1
0.5
0 0
0.2
Figure 17: Real example of the optimisation interest via the error calculation of the statistical significance σ. The second scheme obviously contains more relevant information than the first one. A relevant value for the cut can thus be determined.
III- Analysis Method
42
III.5- Overall process: The policy of the BABAR collaboration states that such an analysis must be blind, which means that the number of candidates within the SR must remain unknown until the selection cuts are optimized and frozen: once the signal region is exposed, the analysis cannot be modified in any way (hence the use of an assumed branching ratio in the optimisation process). This is done to prevent any biases in the analysis. As a matter of fact, one could be tempted to use the branching ratio first obtained (BR1) as a new seed for the optimisation and so generate a new set of cuts that could be more appropriate to the decay studied. Like the assumed branching ratio (BR0) used to generate the first branching ratio (BR1), this branching ratio (BR1) could in turn be used to create a new set of cuts and then a new branching ratio (BR2). And so on, BRn giving BRn+1 that would give BRn+2 that would give… However, such an iterative procedure is very dangerous. If the optimisation acts in a region of large fluctuation, this feed-back process will tend to misidentify fluctuations as maxima, which will completely warp the analysis. Nonetheless, the optimisation process does introduce biases into the estimation of the number of background events in the SR. As a matter of fact, maximising σ =s/(s+b)1/2 is equivalent to minimising b, that is, minimising the background in the GSB and so its estimate in the SR. As a result, any fluctuation of the number of background events will be minimised, thus leading to a biased analysis since tending to reduce the effect of this fluctuation by the use of a ‘special’ set of cuts. To avoid this, the set of background events counted in the GSB (on which the calculation of background events in the SR is based) is not the same during the optimisation phase and the BR calculation phase. Only the even-numbered events are taken into account during the optimisation phase. For the BR calculation phase, the number of background events in the GSB is counted by using these optimised cuts on the other half of the events, the odd-numbered events. This number should therefore be scaled by two to estimate the number of background in the SR. The point being to not favour any one of the 2 sets, all the events are used to count the number of events in the SR.
III- Analysis Method
43
b
Even-numbered events
s
ª Even-numbered events
Optimisation
BR Calculation
ntot _ GSB
Odd-numbered events
ntot _ SR
Every event
Table 3: Counting processes used during the optimisation and the BR calculation phases. This is done to avoid any fluctuation to be underestimated.
Consequently, the final branching ratio is given by: real N sig × 1
BR =
where
ε MC ,corr
N
real B0
[
]
real N sig = ntot _ SR − R × (2 × ntot _ GSB )
IV- Results
44
IV- RESULTS
IV.1- Final selection criteria after optimisation: As previously said, the optimisation has been performed using the branching ratio found by N. Chevalier [18], namely:
{
{
}
0
*−
BR B 0 → K *+ π − + BR B → K π +
}=
(16.1 ± 8.5 ± 3.0) × 10 −6
So, the branching ratio BR’ used in the optimisation procedure is:
{
}
{
0
*−
BR ' = BR B 0 → K *+ π − + BR B → K π + K S0π +
}=
1 × (16.1 ± 8.5 ± 3.0 ) × 10 −6 3
0
K Sπ −
This leads to the following set of cuts:
Selection Criteria
Cut Value
cos (θthrust)
cos(θ t ) < 0.65
Cornelius Fisher Discriminant
fisherCrn < −0.4
K *+ → K S0π + / K K *+ → K S0π + / K
*+
*+
0
→ K S π − resonance mass
m
K *+ → K S0π + / K
*−
0
= 0.896 ± 0.075 GeV/c2
cos(θ h ) < 0.45
0
→ K S π − resonance cos (θhelicity)
K S0 → π −π + / K S → π +π − resonance mass
0
→K Sπ −
m
0
K S0 →π + π − / K S →π − π +
0
K S0 → π −π + / K S → π +π − resonance ‘decay length’ Table 4: Final optimised selection criteria.
= 0.498 ± 0.010 GeV/c2
cτ
σ cτ
> 4.0
IV- Results
45
s/ s+b vs fisherCrn
3
3
2.5
2.5
2
2
s / s+b
s / s+b
s/ s+b vs |cos θt|
1.5
1.5
1
1
0.5
0.5
0
0 0
0.2
0.4
0.6 | cos θt |
0.8
1
-1
2.5
2.5
2
2
s / s+b
s / s+b
3
1.5
1
0.5
0.5
0.04
0.06 0.08 0.1 ∆ resMass (GeV)
0.12
0.14
0 0
0.16
1
0.2
0.4
0.6 | cos θB |
0.8
1
s/ s+b vs gkCtau/gkCtaue 3
2.5
2.5
2
2
s / s+b
s / s+b
s/ s+b vs ∆ gkMass 3
1.5
1.5
1
1
0.5
0.5
0 0
0.5
1.5
1
0.02
0 fisherCrn
s/ s+b vs |cosθB|
s/ s+b vs ∆ resMass 3
0 0
-0.5
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 ∆ gkMass (GeV)
0 0
2
4 6 gkCtau / gkCtaue
8
10
Figure 18: Optimisation cuts schemes
Concerning the size of the SR (in which 99% of the MC signal does fall inside) and that of the GSB, they have been chosen as follows:
∆E = 0.0 ± 0.1
SR
GSB
mES = 5.279 ± 0.010 GeV/c2 ∆E = 0.0 ± 0.2
mES = 5.240 ± 0.020 GeV/c2
Table 5: Size of the SR and the GSB.
IV- Results
46
IV.2- MC Efficiency correction:
ε MC , corr = kcorrε MC with
ε MC =
MC N sig
N BMC 0
kcorr = kevtShp × k PID × ktrk × k K 0 × k∆E × km S
ES
MC = 3104 ± 56 ± 0 . Using the previous cuts, it is found that: N sig
So, as N BMC = 50000 ± 0 ± 0 fixed, 0
ε MC = (6.34 ± 0.11 ± 0.00 ) % The values of the k correction factors are:
k evtShp
= 0.928 ± 0.000 ± 0.051
k PID
= 0.963 ± 0.000 ± 0.045
k ∆E km
= 1.000 ± 0.000 ± 0.025
kK0 S
= 1.080 ± 0.000 ± 0.026
[20]
k trk
= 0.979 ± 0.000 ± 0.030
(*)
k corr
= 0.945 ± 0.000 ± 0.081
= 1.000 ± 0.000 ± 0.010
ES
(*) k trk = k trk _ 1 × k trk _ 2 with
σk
trk _ i
k trk _ i
= 1.5% [18] ⇒
σk
trk
ktrk
Ntuple variables, ktrk is the mean of the Gaussian fit to the ktrk_1 ktrk_2 distribution
Consequently,
[18]
ε MC , corr = (5.99 ± 0.11 ± 0.52 ) %
=
σk
trk _ 1
ktrk _ 1
+
σk
trk _ 2
ktrk _ 2
= 3 .0 %
IV- Results
47
IV.3- Combinatoric background: . ∆E background distribution:
N ∆E = a(∆E ) 2 + b(∆E ) + c a = 4339 ± 0 ± 186 GeV-2
b = −9300 ± 0 ± 39 GeV-1
with
c = 7011 ± 0 ± 13
. m ES background distribution:
Nm
ES
⎛ m = C ⋅ ⎜⎜ ES ⎝ mMAX
⎞ ⎛ m ⎟⎟ × 1 − ⎜⎜ ES ⎠ ⎝ mMAX
2 ⎧⎪ ⎡ ⎛ m ⎞ ⎟⎟ × exp⎨ − ξ .⎢1 − ⎜⎜ ES ⎢⎣ ⎝ mMAX ⎠ ⎪⎩
C = (57.99 ± 0.00 ± 2.50 ) × 10 4
ξ = −24.84 ± 0.00 ± 0.11
with
mMAX = 5.29 GeV/c2
Hence, as
R=
∫N
SR
∆E
d (∆E )
∫ N ∆E d (∆E )
GSB
×
∫N
m ES
d (mES )
SR
∫N
m ES
d (mES )
GSB
R = 0.162 ± 0.000 ± 0.001
⎞ ⎟⎟ ⎠
2
⎤ ⎥ ⎥⎦
⎫⎪ ⎬ ⎪⎭
IV- Results
48
Another common notation is to express R as a function of the size of the SR (ASR) and that of the GSB (AGSB). As a result, as here AGS=4.ASR ⎛ A ⎞ R = (0.647 ± 0.000 ± 0.003) × ⎜⎜ SR ⎟⎟ ⎝ AGSB ⎠
DeltaE Fit to 2nd Order Polynomial using OffRes Data 12000
deltaEfit Nent = 897274 Mean = -0.1173 RMS = 0.2238 x^2 parameter = 4339 x parameter = -9300 Constant term = 7011
10000
No of events
8000
6000
4000
2000
0
-0.4
-0.3
-0.2
-0.1 -0 DeltaE (GeV)
0.1
0.2
0.3
0.4
Figure 19: ∆E quadratic background distribution used to assess the proportion of combinatoric background in the SR.
mes Fit to Argus Function using OffRes Data
10000
mesfit Nent = 897274 Mean = 5.237 RMS = 0.02662 Constant C = 2.449e+04 Xi parameter = -24.84
No of events
8000
6000
4000
2000
0
5.2
5.22
5.24 5.26 mes (GeV)
5.28
5.3
Figure 20: m ES Argus-shaped background distribution used to assess the proportion of combinatoric background in the SR.
IV- Results
49
IV.4- Branching ratio of the
B 0 → K ∗+ π −
&
∗−
0
B → K π+
Quantity
Value
ntot _ SR
53 ± 7 ± 0
ntot _ GSB (x2)
186 ± 20 ± 0
nbg _ SR
30.1 ± 3.1 ± 0.1
real N sig
22.9 ± 7.9 ± 0.1
Statistical Significance
3.2σ
{
}
{
*−
0
BR B 0 → K *+ π − + BR B → K π +
}
(6.20 ± 2.15 ± 0.54) × 106
0 S
K S0π +
{
}
K π−
{
0
*−
BR B 0 → K *+ π − + BR B → K π +
}
(18.6 ± 6.5 ± 1.6) × 106
Table 6: Final Findings
On Line Resonance Data: DeltaE-MES Plane 0.4 0.2
∆E
0 -0.2 -0.4 -0.6 -0.8 5.18
5.2
decays:
5.22
5.24 m ES
5.26
5.28
Figure 21: Final ∆E - m ES events distribution.
5.3
V- Discussion
50
V- DISCUSSION
As detailed in the previous chapter, the total branching ratio found for the neutral *−
0
conjugate B decays B 0 → K *+π − and B → K π + via the K *+ / K
*−
decay into a
0
0
K S0π + / K S π − using 61.6 ± 0.68 millions B 0 and B mesons is:
{
}
{
*−
0
BR B 0 → K *+ π − + BR B → K π +
}=
(18.6 ± 6.5 ± 1.6) × 10 −6
with a statistical significance of 3.2 σ
This measurement is, within errors, in good agreement with the previous ones, namely:
(x106)
Secondary Decay (conjugate implied)
Branching Ratio (x10-6)
Statistical Significance
CLEO, 1999 [21]
5.8
K *+ → K S0π +
22.0 ± 7.0 ± 5.0
5.2 σ
BELLE, 2001 [22]
22.8
K *+ → K +π 0
26.0 ± 8.3 ± 3.5
4.3 σ
BABAR, 2002 [18]
22.7
K *+ → K +π 0
16.1 ± 8.5 ± 3.0
2.2 σ
BABAR, 2003
61.6
K *+ → K S0π +
18.6 ± 6.5 ± 1.6
3.2 σ
Experiment
N
B0 / B
0
Table 7: Previous measurements of the B → K 0
∗+
π−
0
+ B →K
*−
π+
branching ratio.
V- Discussion
51
As one can see hereafter, it also agrees with the nominal value of the theoretical branching ratio found by Cottingham et al. [23] which is: 15.3 × 10−6 .
CLEO, 1999 [21] BELLE, 2001 [22] BABAR, 2002 [18] BABAR, 2003
0
5
10
15
20
Figure 22: Previous measurements of the
25
30
35 0
40
( x10-6 )
*−
B 0 → K ∗+ π − + B → K π + branching ratio.
(The half length of the uncertaintiy bars is the sum of the statistical and systematical errors).
As expected from the use of a significantly larger data set, the statistical error is lower than those found by the previous experiments. On the other hand, a systematical error almost twice smaller than the more recent calculated ones seems rather dubious. The calculations having been checked several times, this must be due to a too simplistic way to calculate R (the other source of statistical error being k corr which was here almost independent of the computing treatment performed). Many other more accurate ways (such as using several statistical bands over both on-line and off-line resonance data) exist to estimate this. Two other points could also be improved. First, it could be of interest to verify that the part of background coming from other B decays (e.g. K *+ → K +π ° ) is truly negligible. Second, it could be necessary to use a real random treatment for the multiple candidates (nevent) instead of the ‘pseudo-random’ one actually used. Although this result is consistent with the prior ones, it is worth noting that it is still besmirched with a quite large total error of about 45%. This error should decreases with years, the quantity of data collected by BABAR becoming more and more important. It then could be of interest to use an ellipse instead of a rectangle to define the signal region, the distribution in the ∆E-mES plane assumed to be ellipsoidal as modelled in the MC data (this has not been performed in this project, the number of signal detected being too small)
V- Discussion
52
Concerning the optimisation process, no real problems have been encountered. The cuts appeared to be rather orthonormal, that is, changing the value of one did not change much the values of the others. The convergent process have thus been achieved in two steps, the first optimisation yielding values that have been used instead of the prior ones in a second optimisation run which lead in turn to some new values that finally appear to remain the same after a last run. As mentioned, the cuts values found were independent of that of the branching ratio used. Whatever the one used (the CLEO one, the BABAR 2002 one, and even the one found here – simply used out of curiosity – ), they remained identical. Although the difference between these branching ratios is small, the astonishing stability of the optimisation process remains suspicious. Once again, it may come from the aforementioned too simplistic approach used.
VI- Conclusion
53
VI- CONCLUSION
The total branching ratio measured for the conjugate neutral B decays B 0 → K *+π − and 0
*−
B → K π + is consistent with both previous experimental results and theoretical prediction.
To the best of the author’s knowledge, this result should be the most accurate yet achieved, since being derived from the larger data set ever. In time, its accuracy should improve, the data gathered at BABAR increasing day after day. A study of both channels separetely with respect to their mother particle Y(4s) is now expected. The asymmetric parameter ACP could therefore be calculated and lead to the value of some of the CKM matrix elements. This will contribute to the appraisal of the Standard Model by checking whether or not it could predict the matter-antimatter asymmetry in the universe.
Currently, it is believed that CP violation as predicted by the Standard Model is not large enough to create the amount of matter observed in the universe today. Thus, continuing to study CP violation is very important, since it may pave the way for exciting new physics whether this model is not the whole story.
Appendix
54
APPENDIX
This appendix displays a skimmed version of the main loop of the code used to count the number of events falling down the Signal Region and the Grand Side Band with respect to the cuts discussed beforehand.
Int_t nSR Int_t nGSB
= 0 = 0
; ;
lastEvt
= -1 ;
// To avoid double counting since each event can refer // to the same original BB decay
nbytes
= 0
// buffer variable
;
// Number of events falling down the Signal Region // Number of events falling down the Grand Side Band
for ( Int_t i=0 ; i GetEntry(i) ;
// Loads the next entry from the ntuple // and records the number of bytes loaded
if ( TMath::Abs(cosTTB) < cut_cosTTB ) { if ( fisherCrn < cut_fisherCrn ) { if ( trk1K==0 && trk2K==0 ) {
// cos theta thrust CUT // Cornelius fisher CUT // first PID CUT
// second PID CUT & mass of the Ks0-pi+ resonance checking if ( ( resMass[3] > cut_resMassMin ) && ( resMass[3] < cut_resMassMax ) ) { // second PID CUT & helicity angle of the Ks0-pi+ resonance checking if ( TMath::Abs(resCosB[3]) > cut_resCosB ) { // Ks0->pi+pi- mass checking if ( ( gkMass[0] > cut_gkMassMin ) && ( gkMass[0] < cut_gkMassMax ) { // Ks0->pi+pi- decay length checking if ( gkCtau[0]/gkCtaue[0] > cut_gkCtauCtaue ) { // nGSB counting: candidate in the GSB box? // (Embc corresponds to mES & des[0] corresponds to deltaE) // and no double counting checking (lastEvt!=nevent) if ( (Embc>mesGSBMin && EmbcdeltaEGSBMin && des[0]mesSRMin && EmbcdeltaESRMin && des[0] cut_resMassMin ) && ( resMass[5] < cut_resMassMax ) ) { if ( TMath::Abs(resCosB[5]) > cut_resCosB ) { if ( ( gkMass[0] > cut_gkMassMin ) && ( gkMass[0] < cut_gkMassMax ) { if ( gkCtau[0]/gkCtaue[0] > cut_gkCtauCtaue ) { if (
(Embc>mesGSBMin && EmbcdeltaEGSBMin && des[0]mesSRMin && EmbcdeltaESRMin && des[0]
