Study of angular distribution for B → φK decays at LHCb

41 Pages·2011·4.96 MB·English

Study of angular distribution for
0 ∗0
B → φK decays at
LHCb
July 1, 2011
Laureline Josset
Date: July 1, 2011
Master EPFL Section Physique
Master thesis
Prof. T. Tatsuya
Dr. F. Blanc

Contents
1 Introduction 5
2 Angular distribution formalism and calculation 7
2.1 B0 →φK∗0 decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Helicity basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Transversity basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Angular distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Angular distribution analysis on Monte Carlo simulated data 11
3.1 Implementation of the angles calculation . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Validation studies on generated data . . . . . . . . . . . . . . . . . . . 12
3.2 B0 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.2 Selection cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Angular distribution for MC reconstructed data . . . . . . . . . . . . . . . . . 16
3.3.1 Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.2 Resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.3 Background subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.4 Results after acceptance parametrization and background subtraction 20
3.4 Conclusion on MC reconstructed data . . . . . . . . . . . . . . . . . . . . . . 22
4 Angular distribution analysis on real data 23
4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 1-D Angular distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Conclusions 27
6 Appendix A: angular distribution calculation 29
6.1 Angular distribution - General case . . . . . . . . . . . . . . . . . . . . . . . . 29
6.1.1 Calculation of (cid:104)θ,φ,λ ,λ |J,M,λ ,λ (cid:105) . . . . . . . . . . . . . . . . . . 29
1 2 1 2
6.2 Angular distribution for a scalar decaying in two vector mesons . . . . . . . . 30
6.2.1 g functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
m
6.2.2 Calculation of the distribution . . . . . . . . . . . . . . . . . . . . . . 31
6.3 B0 →K∗0φ - transversity basis . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7 Appendix B: additional figures for MC and real data 33
7.1 MC generated data for validation . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.2 MC reconstructed data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.3 Real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3
CONTENTS
4
Chapter 1
Introduction
The goal of this project is to study angular distribution of B0 → φK∗0 decays at LHCb,
and to derive information on polarization fractions.
Motivation
WhenB mesonsdecayintwovectorparticles,B daughterparticlescanhavedifferentstates
of helicities. The longitudinal fraction f =Γ /Γ, the fraction of states with a null helicity,
L L
is expected, in the limit of helicity conservation, to have a value close to 1.
However, experimental values measured at Belle or BABAR are twice smaller for B0 →
φK∗0. This difference between experimental and theoretical (“theoretical” may not be the
right term, the PDG refers to it as “naive expectation”) values does not concern only this
decay, but most of B0 → VV decays as shown in figure 1.1. Several ideas have been
suggested to explain this phenomena, such as non-factorizable contributions to the B-decay
amplitudes [1]. In order to clarify this problem, further studies are needed, in particularly
at the LHCb.
The present report focuses on determining f and other polarization parameters for the
L
decay B0 →φK∗0 using the LHCb 2011 data.
Figure 1.1: Longitudinalpolarizationfractionf fordifferentB decaysintwovectorparticles[2]
L
Work
This project is conducted within the work of LHCb, the detector dedicated to the study of
the beauty quark and violation of CP symmetry. LHCb is one of the 4 main experiments at
the Large Hadron Collider (LHC) at the CERN.
5
1Introduction
The LHCb detector
LHCb, shown in figure 1.2, is a single-arm spectrometer, composed of a vertex locator
system, a tracking system made of a Trigger Tracker and three tracking stations placed on
both sides of a dipole magnet, two Ring Imaging Cherenkov counters, a calorimeter system
and a muon detection system1.
The present study especially exploits information from the vertex locator and tracking
system. TheRICHarealsousedastheyareparticularlyefficienttodifferentiatekaonsfrom
pions.
Figure 1.2: LHCb detector, scheme from [3]
Report structure
This report is structured in the following way:
The first part introduces the helicity formalism and establishes the angular distribution for
the present decay. The second step consists in the angular study on Monte Carlo simulated
data. Acceptance and resolution effects are evaluated to prepare analysis on real data. The
third part presents preliminary measurements on real data.
1Reference[3]givesacompletedescriptionofthedetectoranditsspecification.
6
Chapter 2
Angular distribution formalism and
calculation
2.1 B0 → φK∗0 decay
B0 is a pseudo-scalar meson composed of a quark and an antiquark ¯bd. The φ and K∗0
particles are also mesons but of spin 1. The decay B0 →φK∗0 is predominantly generated
by a¯b→s¯gluonic penguin diagram.
Figure 2.1: B0 →φK∗0 Feynman diagram
Followed by the decays of φ and K∗0. In the present study, the channel K+K− for φ
and Kπ for K∗0 are chosen. The branching fractions[4] are:
B0 → φK∗0 (9.8±0.6)·10−6
φ → K+K− (48.9±0.5)%
K∗0 → K+π− 2/3
(→ K0π0 1/3)
2.2 Formalism
This section details the formalism for the calculation of angular distribution. First, the
concept of helicity is introduced, along with the transversity basis. After a summarized
explanation on how to obtain them, the angular distributions and their integrated forms for
the two basis are given.
2.2.1 Helicity basis
WhenaparticleM decaysintwodaughterparticlesP andP ,theangularmomentumcon-
1 2
servationstatesthatthespinofthemotherparticleisequaltothetotalangularmomentum
7
2Angulardistributionformalismandcalculation
J = L+S of the daughter particles, where L is the orbital angular momentum, and S
12 12
the spin of the two daughter particles system.
In the present case, the mother particle is the pseudo-scalar meson B, its spin is then
S = 0 = J. The two daughter particles have a spin S = S = 1. The spin system S
M 1 2 12
can take the value 0, 1 or 2, leaving L equal to 0, 1 or 2.
Invoking the symmetry of wave functions under bosons exchange and parity conserva-
tion, only S = L = 0 is left. Only three combinations of spin remains: |S ,S (cid:105) = |1,−1(cid:105),
1 2
|0,0(cid:105) and |−1,1(cid:105).
This can be expressed in terms of helicity. The helicity operator is the projection of the
spin in the direction of its momentum: hˆ = →−S · →−p , where →−S is the spin of the particle
→−
||p||
→−
and p the momentum of the particle.
The authorized spin combinations correspond to the helicities (λ ,λ ) = (+1,+1),
φ K∗0
(0,0) and (−1,−1). The available state can therefore be written |f (cid:105) = |J,M,+1,+1(cid:105),
+
|f (cid:105) = |J,M,0,0(cid:105) or |f (cid:105) = |J,M,−1,−1(cid:105), and form the helicity basis states (J = M = 0
0 −
in the present case).
Accordingly, the amplitude of the decay are labelled : H = (cid:104)f |H |B(cid:105), where H
λ λ eff eff
(cid:80)
is the effective Hamiltonian. The final state is therefore written |Ψ(cid:105) = H |f (cid:105), where
λ λ
λ=+1,0,−1.
2.2.2 Transversity basis
Angular distributions are mainly studied to identify CP eigenvalues of final states with
differentangularmomentumconfigurations. Itistheninterestingtoexpressthedistribution
intermsofparityeigenstates. Fromthedefinitionofthethreehelicitystates,itfollowsthat:
|f (cid:105)+|f (cid:105)
|f (cid:105)= + √ − P|f (cid:105)=|f (cid:105)
P|f+(cid:105)=|f−(cid:105) || 2 || ||
P|f (cid:105)=|f (cid:105)
P|f0(cid:105)=|f0(cid:105) |f (cid:105)= |f+(cid:105)√−|f−(cid:105) 0 0
⊥ P|f (cid:105)=−|f (cid:105) (2.3)
P|f (cid:105)=|f (cid:105) (2.1) 2 ⊥ ⊥
− +
|f (cid:105)=|f (cid:105) (2.2)
0 0 Parity operator on
Parity operator on Definition of the transversity basis
helicity basis vectors transversity basis vectors
The final state is then given by:
(cid:88)
|Ψ(cid:105)= A |f (cid:105)
λ λ
where λ=||,0,⊥ and A|| = H+√+2H−, A⊥ = H+√−2H−, A0 =H0.
2.3 Angular distribution
The angular distribution d3Γ(χ,θ ,θ ) is proportional to the square of the amplitude. The
1 2
first step is thus to calculate the amplitude of the decay as a function of the angles of the
daughter particles.
Initscenter-of-mass(CM)frame,theinitialparticleisdescribedbyitsspinconfiguration
|i(cid:105) = |J,M(cid:105). The two final particles are characterized by their momentum and helicities
→− →− →− →−
|p ,p ,λ ,λ (cid:105). As we are in the CM frame, p = −p and we can express the final state
1 2 1 2 1 2
by the two angles (θ,φ) giving the direction of particle 1 and the helicities of the particles
8
2Angulardistributionformalismandcalculation
|f(cid:105)=|θ,φ,λ ,λ (cid:105). Their momentum are determined by energy conservation.
1 2
The amplitude of the decay is thus A = (cid:104)f|H |i(cid:105) = (cid:104)θ,φ,λ ,λ |H |JM(cid:105) and the
eff 1 2 eff
probabilityfortheparticlestoemergeattheangles(θ,φ)isgivenby|A|2. Iftheexperiment
does not measure the helicities of the particles, they must be summed over.
In order to exploit angular momentum conservation, a complete set of the helicity-basis
states is introduced :
(cid:88)
A= (cid:104)θ,φ,λ ,λ |j,m,λ ,λ (cid:105)(cid:104)j,m,λ ,λ |H |JM(cid:105)
1 2 1 2 1 2 eff
j,m
(cid:88)
= (cid:104)θ,φ,λ ,λ |j,m,λ ,λ (cid:105)δJδMH
1 2 1 2 j m λ1λ2
j,m
=(cid:104)θ,φ,λ ,λ |J,M,λ ,λ (cid:105)H (2.4)
1 2 1 2 λ1λ2
The section 6.1.1 gives the explicit calculation of (cid:104)θ,φ,λ ,λ |J,M,λ ,λ (cid:105).
1 2 1 2
The next step is to apply the equation 2.4 to the full decay B0 → φK∗0, φ → K+K−
andK∗0 →Kπ. Thisisrelatedinsection6.2andisfollowedbytheactualderivationofthe
angular distribution from |A|2.
The angular distributions for helicity and transversity bases are given here after. Their
integrated forms are also reported as the core of the study focuses on 1-D angular distribu-
tion. Angles are defined in the next section.
In the helicity basis
d3Γ(χ,θ ,θ ) 9 (cid:104)
1 2 = (|H |2+|H |2)sin2θ sin2θ +4|H |2cos2θ cos2θ
dχdcosθ dcosθ 32π + − 1 2 0 1 2
1 2
+2(cid:8)(cid:60)(H H∗)cos2χ−(cid:61)(H H∗)sin2χ(cid:9)sin2θ sin2θ
+ − + − 1 2
+(cid:8)(cid:60)(cid:0)(H +H )H∗(cid:1)cosχ−(cid:61)(cid:0)(H −H )H∗(cid:1)sinχ(cid:9)sin2θ sin2θ (cid:105)(2.5)
+ − 0 + − 0 1 2
The angular distribution is normalized such that:
(cid:90) 2π (cid:90) 1 (cid:90) 1
dχ dcosθ dcosθ d3Γ(χ,θ ,θ )=|H |2+|H |2+|H |2.
1 2 1 2 + − 0
0 −1 −1
The integration of the angular distribution over the different variables gives the following
results :
dΓ(χ) = 1 (cid:104)|H |2+|H |2+|H |2+2(cid:8)(cid:60)(H H∗)cos2χ−(cid:61)(H H∗)sin2χ(cid:9)(cid:105)
dχ 2π + − 0 + − + −
dΓ(θ ) 3(cid:104) (cid:105)
1 = (|H |2+|H |2)sin2θ +2|H |2cos2θ
dcosθ 4 + − 1 0 1
1
dΓ(θ ) 3(cid:104) (cid:105)
2 = (|H |2+|H |2)sin2θ +2|H |2cos2θ (2.6)
dcosθ 4 + − 2 0 2
2
In the transversity basis
d3Γ(φ ,θ ,θ ) 9 (cid:104)
tr tr 2 = |A |22sin2θ sin2φ sin2θ
dφ dcosθ dcosθ 32π || tr tr 2
tr tr 2
+|A |22cos2θ sin2θ +|A |24sin2θ cos2φ cos2θ
√⊥ tr 2 0 tr√ tr 2
+ 2(cid:60)(A∗A )sin2θ sin2φ sin2θ − 2(cid:61)(A∗A )sin2θ cosφ sin2θ
|| 0 tr tr 2 0 ⊥ tr tr 2
(cid:105)
−2(cid:61)(A∗A )sin2θ sinφ sin2θ (2.7)
|| ⊥ tr tr 2
9
2Angulardistributionformalismandcalculation
By fitting the angular distribution, the norms and phases of the different amplitudes
are obtained. The number of parameters to fit can be reduced by integrating the previous
expressionoverφ . Asseenin2.8,thereisnomoredependanceoncombinatoryamplitudes.
tr
d2Γ(θ ,θ ) 9 (cid:104)
tr 2 = |A |2sin2θ sin2θ
dcosθ dcosθ 16 || tr 2
tr 2
(cid:105)
+|A |22cos2θ sin2θ +|A |22sin2θ cos2θ (2.8)
⊥ tr 2 0 tr 2
If one wants to observe the distribution in function of the parity of the decay, remembering
that A and A are amplitudes of the even parity states (A the amplitude of odd-parity),
|| 0 ⊥
it is only necessary to distinguish the last state from the two first. Integrating 2.8 over
θ leads to the distribution 2.9, where even-parity states have distribution in sin2θ and
2 tr
odd-parity in cos2θ .
tr
dΓ(θ ) 3(cid:104) (cid:105)
tr = (|A |2+|A |2)sin2θ +2|A |2cos2θ (2.9)
dcosθ 4 || 0 tr ⊥ tr
tr
Other integrations lead to the following distributions :
2-D distributions
d2Γ(φ ,θ ) 3 (cid:104)
tr 2 = 8|A |2sin2φ sin2θ +4|A |2sin2θ
dφ dcosθ 32π || tr 2 ⊥ 2
tr 2
√ (cid:105)
+16|A |2cos2φ cos2θ + 2(cid:60)(A∗A )sin2φ sin2θ
0 tr 2 || 0 tr 2
d2Γ(φ ,θ ) 3 (cid:104)
tr tr = |A |2sin2θ sin2φ +|A |2cos2θ
dφ dcosθ 4π || tr tr ⊥ tr
tr tr
(cid:105)
+|A |2sin2θ cos2φ −(cid:61)(A∗A )sin2θ sinφ (2.10)
0 tr tr || ⊥ tr tr
1-D distributions
dΓ(θ ) 3(cid:104) (cid:105)
2 = (|A |2+|A |2)sin2θ +2|A |2cos2θ
dcosθ 4 || ⊥ 2 0 2
2
dΓ(φ ) 1 (cid:104) (cid:105)
tr = 2|A |2sin2φ +|A |2+2|A |2cos2φ (2.11)
dφ 2π || tr ⊥ 0 tr
tr
10