Study of angular distribution for B0 ˚K0 decays at -...

Study of angular distribution forB0→ φK∗0 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 72.1 B0 → φK∗0 decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Helicity basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Transversity basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Angular distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Angular distribution analysis on Monte Carlo simulated data 113.1 Implementation of the angles calculation . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Validation studies on generated data . . . . . . . . . . . . . . . . . . . 123.2 B0 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.2 Selection cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Angular distribution for MC reconstructed data . . . . . . . . . . . . . . . . . 163.3.1 Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.2 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.3 Background subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.4 Results after acceptance parametrization and background subtraction 20

3.4 Conclusion on MC reconstructed data . . . . . . . . . . . . . . . . . . . . . . 22

4 Angular distribution analysis on real data 234.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 1-D Angular distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Conclusions 27

6 Appendix A: angular distribution calculation 296.1 Angular distribution - General case . . . . . . . . . . . . . . . . . . . . . . . . 29

6.1.1 Calculation of 〈θ, φ, λ1, λ2|J,M, λ1, λ2〉 . . . . . . . . . . . . . . . . . . 296.2 Angular distribution for a scalar decaying in two vector mesons . . . . . . . . 30

6.2.1 gm functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2.2 Calculation of the distribution . . . . . . . . . . . . . . . . . . . . . . 31

6.3 B0 → K∗0φ - transversity basis . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7 Appendix B: additional figures for MC and real data 337.1 MC generated data for validation . . . . . . . . . . . . . . . . . . . . . . . . . 337.2 MC reconstructed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.3 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

When B mesons decay in two vector particles, B daughter particles can have different statesof helicities. The longitudinal fraction fL = ΓL/Γ, the fraction of states with a null helicity,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 theright term, the PDG refers to it as “naive expectation”) values does not concern only thisdecay, but most of B0 → V V decays as shown in figure 1.1. Several ideas have beensuggested to explain this phenomena, such as non-factorizable contributions to the B-decayamplitudes [1]. In order to clarify this problem, further studies are needed, in particularlyat the LHCb.

The present report focuses on determining fL and other polarization parameters for thedecay B0 → φK∗0 using the LHCb 2011 data.

Figure 1.1: Longitudinal polarization fraction fL for different B decays in two vector particles [2]

Work

This project is conducted within the work of LHCb, the detector dedicated to the study ofthe beauty quark and violation of CP symmetry. LHCb is one of the 4 main experiments atthe Large Hadron Collider (LHC) at the CERN.

5

1 Introduction

The LHCb detector

LHCb, shown in figure 1.2, is a single-arm spectrometer, composed of a vertex locatorsystem, a tracking system made of a Trigger Tracker and three tracking stations placed onboth sides of a dipole magnet, two Ring Imaging Cherenkov counters, a calorimeter systemand a muon detection system1.

The present study especially exploits information from the vertex locator and trackingsystem. The RICH are also used as they are particularly efficient to differentiate kaons frompions.

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 forthe present decay. The second step consists in the angular study on Monte Carlo simulateddata. Acceptance and resolution effects are evaluated to prepare analysis on real data. Thethird part presents preliminary measurements on real data.

1Reference [3] gives a complete description of the detector and its specification.

6

Chapter 2

Angular distribution formalism andcalculation

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 generatedby 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 φand 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, theconcept of helicity is introduced, along with the transversity basis. After a summarizedexplanation on how to obtain them, the angular distributions and their integrated forms forthe two basis are given.

2.2.1 Helicity basis

When a particle M decays in two daughter particles P1 and P2, the angular momentum con-servation states that the spin of the mother particle is equal to the total angular momentum

7

2 Angular distribution formalism and calculation

J = L + S12 of the daughter particles, where L is the orbital angular momentum, and S12

the spin of the two daughter particles system.

In the present case, the mother particle is the pseudo-scalar meson B, its spin is thenSM = 0 = J . The two daughter particles have a spin S1 = S2 = 1. The spin system S12

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: |S1, S2〉 = |1,−1〉,|0, 0〉 and | − 1, 1〉.

This can be expressed in terms of helicity. The helicity operator is the projection of the

spin in the direction of its momentum: h =−→S ·

−→p||−→p || , where

−→S is the spin of the particle

and −→p the momentum of the particle.

The authorized spin combinations correspond to the helicities (λφ, λK∗0) = (+1,+1),(0, 0) and (−1,−1). The available state can therefore be written |f+〉 = |J,M,+1,+1〉,|f0〉 = |J,M, 0, 0〉 or |f−〉 = |J,M,−1,−1〉, and form the helicity basis states (J = M = 0in the present case).

Accordingly, the amplitude of the decay are labelled : Hλ = 〈fλ|Heff |B〉, where Heff

is the effective Hamiltonian. The final state is therefore written |Ψ〉 =∑Hλ|fλ〉, where

λ = +1, 0,−1.

2.2.2 Transversity basis

Angular distributions are mainly studied to identify CP eigenvalues of final states withdifferent angular momentum configurations. It is then interesting to express the distributionin terms of parity eigenstates. From the definition of the three helicity states, it follows that:

P |f+〉 = |f−〉P |f0〉 = |f0〉P |f−〉 = |f+〉 (2.1)

Parity operator onhelicity basis vectors

|f||〉 =|f+〉+ |f−〉√

2

|f⊥〉 =|f+〉 − |f−〉√

2|f0〉 = |f0〉 (2.2)

Definition of thetransversity basis

P |f||〉 = |f||〉P |f0〉 = |f0〉P |f⊥〉 = −|f⊥〉 (2.3)

Parity operator ontransversity basisvectors

The final state is then given by:

|Ψ〉 =∑

Aλ|fλ〉

where λ = ||, 0,⊥ and A|| = H++H−√2

, A⊥ = H+−H−√2

, A0 = H0.

2.3 Angular distribution

The angular distribution d3Γ(χ, θ1, θ2) is proportional to the square of the amplitude. Thefirst step is thus to calculate the amplitude of the decay as a function of the angles of thedaughter particles.

In its center-of-mass (CM) frame, the initial particle is described by its spin configuration|i〉 = |J,M〉. The two final particles are characterized by their momentum and helicities|−→p1,−→p2, λ1, λ2〉. As we are in the CM frame, −→p1 = −−→p2 and we can express the final state

by the two angles (θ, φ) giving the direction of particle 1 and the helicities of the particles

8


|f〉 = |θ, φ, λ1, λ2〉. Their momentum are determined by energy conservation.

The amplitude of the decay is thus A = 〈f |Heff |i〉 = 〈θ, φ, λ1, λ2|Heff |JM〉 and theprobability for the particles to emerge at the angles (θ, φ) is given by |A|2. If the experimentdoes 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-basisstates is introduced :

A =∑j,m

〈θ, φ, λ1, λ2|j,m, λ1, λ2〉〈j,m, λ1, λ2|Heff |JM〉

=∑j,m

〈θ, φ, λ1, λ2|j,m, λ1, λ2〉δJj δMmHλ1λ2

= 〈θ, φ, λ1, λ2|J,M, λ1, λ2〉Hλ1λ2(2.4)

The section 6.1.1 gives the explicit calculation of 〈θ, φ, λ1, λ2|J,M, λ1, λ2〉.

The next step is to apply the equation 2.4 to the full decay B0 → φK∗0, φ → K+K−

and K∗0 → Kπ. This is related in section 6.2 and is followed by the actual derivation of theangular distribution from |A|2.

The angular distributions for helicity and transversity bases are given here after. Theirintegrated 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, θ1, θ2)

dχd cos θ1d cos θ2=

9

32π

[(|H+|2 + |H−|2) sin2 θ1 sin2 θ2 + 4|H0|2 cos2 θ1 cos2 θ2

+2{<(H+H

∗−) cos 2χ−=(H+H

∗−) sin 2χ

}sin2 θ1 sin2 θ2

+{<((H+ +H−)H∗0

)cosχ−=

((H+ −H−)H∗0

)sinχ

}sin 2θ1 sin 2θ2

](2.5)

The angular distribution is normalized such that:∫ 2π

0

dχ

∫ 1

−1

d cos θ1

∫ 1

−1

d cos θ2d3Γ(χ, θ1, θ2) = |H+|2 + |H−|2 + |H0|2.

The integration of the angular distribution over the different variables gives the followingresults :

dΓ(χ)

dχ=

1

2π

[|H+|2 + |H−|2 + |H0|2 + 2

{<(H+H

∗−) cos 2χ−=(H+H

∗−) sin 2χ

}]dΓ(θ1)

d cos θ1=

3

4

[(|H+|2 + |H−|2) sin2 θ1 + 2|H0|2 cos2 θ1

]dΓ(θ2)

d cos θ2=

3

4

[(|H+|2 + |H−|2) sin2 θ2 + 2|H0|2 cos2 θ2

](2.6)

In the transversity basis

d3Γ(φtr, θtr, θ2)

dφtrd cos θtrd cos θ2=

9

32π

[|A|||22 sin2 θtr sin2 φtr sin2 θ2

+|A⊥|22 cos2 θtr sin2 θ2 + |A0|24 sin2 θtr cos2 φtr cos2 θ2

+√

2<(A∗||A0) sin2 θtr sin 2φtr sin 2θ2 −√

2=(A∗0A⊥) sin 2θtr cosφtr sin 2θ2

−2=(A∗||A⊥) sin 2θtr sinφtrsin2θ2

](2.7)

9


By fitting the angular distribution, the norms and phases of the different amplitudesare obtained. The number of parameters to fit can be reduced by integrating the previousexpression over φtr. As seen in 2.8, there is no more dependance on combinatory amplitudes.

d2Γ(θtr, θ2)

d cos θtrd cos θ2=

9

16

[|A|||2 sin2 θtr sin2 θ2

+|A⊥|22 cos2 θtr sin2 θ2 + |A0|22 sin2 θtr cos2 θ2

](2.8)

If one wants to observe the distribution in function of the parity of the decay, rememberingthat A|| and A0 are amplitudes of the even parity states (A⊥ the amplitude of odd-parity),it is only necessary to distinguish the last state from the two first. Integrating 2.8 overθ2 leads to the distribution 2.9, where even-parity states have distribution in sin2 θtr andodd-parity in cos2 θtr.

dΓ(θtr)

d cos θtr=

3

4

[(|A|||2 + |A0|2) sin2 θtr + 2|A⊥|2 cos2 θtr

](2.9)

Other integrations lead to the following distributions :2-D distributions

d2Γ(φtr, θ2)

dφtrd cos θ2=

3

32π

[8|A|||2 sin2 φtr sin2 θ2 + 4|A⊥|2 sin2 θ2

+16|A0|2 cos2 φtr cos2 θ2 +√

2<(A∗||A0) sin 2φtr sin 2θ2

]d2Γ(φtr, θtr)

dφtrd cos θtr=

3

4π

[|A|||2 sin2 θtr sin2 φtr + |A⊥|2 cos2 θtr

+|A0|2 sin2 θtr cos2 φtr −=(A∗||A⊥) sin 2θtr sinφtr

](2.10)

1-D distributions

dΓ(θ2)

d cos θ2=

3

4

[(|A|||2 + |A⊥|2) sin2 θ2 + 2|A0|2 cos2 θ2

]dΓ(φtr)

dφtr=

1

2π

[2|A|||2 sin2 φtr + |A⊥|2 + 2|A0|2 cos2 φtr

](2.11)

10

Chapter 3

Angular distribution analysis onMonte Carlo simulated data

Analysis on MC simulated data is conducted in three steps. First, the angles calculationimplementation is verified on MC generated data. The second part focuses on B0 recon-struction, its mass distribution, selection criteria and their efficiency. In the last and mostimportant part, the angular analysis is conducted on MC reconstructed data to prepare realdata analysis.

3.1 Implementation of the angles calculation

As presented in the previous chapter, angular distribution can be expressed in either twosets of three angles, in the helicity or transversity basis. This section details how the angles,shown in figure 3.1 are calculated:

χ is the angle between the two decay planes of φ and K∗0. This is calculated using themomentum vectors of K+K− for φ and Kπ for K∗0 expressed in B0 rest frames.

θ1 is the angle between K+ momentum vector in φ rest frame and the direction of prop-agation of φ. This last step is done using the direction opposite to B0 in φ rest frame.

θ2 similarly to θ1, is the angle between K momentum vector in K∗0 rest frame and thedirection of propagation of K∗0. Again, this is done using the direction opposite toB0 in K∗0 rest frame.

θtr cos θtr is calculated using the scalar product between K+ and the direction perpen-dicular to the decay plane of K∗0, all vectors expressed in the φ rest frame.

φtr is the angle between φ direction of propagation and the projection of K+ momentumin K∗ decay plane. As for θtr, all vectors are expressed in the φ rest frame.

Convention

Several choices are made in the present study:

• θ1 is measured toward the positive kaon and not the negative one.

• θ2 is measured toward the kaon and not the pion

• χ is positive if nφ ∧ nK∗01 is in the direction of the φ.

• The transversity angles are calculated using φ→ K+K− momentum and not the K∗0.This is because φ are more easily reconstructed than K∗02.

This corresponds to what is generally done in the literature[5],[6] and is illustrated in figure3.1.

1nα is the vector director of the decay plane of the particle α, defined as nφ = pK+ ∧ pK− and nK∗0 =pK+ ∧ pπ .

2φ decays in K+K−, while K∗0 in Kπ. Pions are present at LHCb in larger quantities than kaons, theyare thus responsible for a great part of combinatory background. Moreover, the width of K∗0 is ten timeslarger than for φ, and thus more subject to background.

11

3 Angular distribution analysis on Monte Carlo simulated data

Figure 3.1: Angles definitions for the helicity (top) and transversity (bottom) basis.

3.1.1 Validation studies on generated data

Data

In order to validate the procedure to extract the amplitudes from the decay angular dis-tribution, Monte Carlo data is generated. The simulation takes into account the geometricacceptance of the detector (DecProdCut), but no other effect of the reconstruction. Simu-lations have been run to construct three sets containing between 16’000 and 30’000 signalevents each.

Decay amplitudes

Three different sets of data have been generated with different amplitudes configurationsshown in table 3.1.

Configuration H+ H0 H−n◦1 1 1.7 1n◦2 4ei 1 2e3i

n◦3 4 1 2

Table 3.1: The three configurations of complex decay amplitudes used in MC data.

Amplitudes n◦1 corresponds to the default parametrization used at LHCb. Amplitudesn◦3 has, unlike n◦1, a non-null A⊥. Amplitudes n◦2 have the same norms as for n◦3, buthave phases.

Fitting function

Angular distributions are fitted using the corresponding functions in equation 2.6, 2.9 and2.11, where all terms depending on the helicity amplitude of the decays is left as a parameter.For example, the fitting functions are:

dΓ(χ)dχ = 1

2π [p0 + 2{p1 cos 2χ − p2 sin 2χ}

], where p0 = |H+|2 + |H−|2 + |H0|2, p1 =

<(H+H∗−) and p2 = =(H+H

∗−) are the fit parameters.

12


dΓ(θ1)d cos θ1

= 34 [p0 sin2 θ1 + 2p1 cos2 θ1], where p0 = |H+|2 + |H−|2 and p1 = |H0|2

dΓ(θ2)d cos θ2

= 34 [p0 sin2 θ2 + 2p1 cos2 θ2], where p0 = |H+|2 + |H−|2 and p1 = |H0|2

dΓ(φtr)dφtr

= 12π [2p0 sin2 φtr+p1 +2p2 cos2 φtr], where p0 = |A|||2, p1 = |A⊥|2 and p2 = |A0|2

dΓ(θtr)d cos θtr

= 34

[p0 sin2 θtr + 2p1 cos2 θtr], where p0 = (|A|||2 + |A0|2) and p1 = |A⊥|2

In certain cases, additional parameters are needed for normalization purpose.

Results

Angular distributions for each angle for the configuration n◦1 are shown in figure 3.2. Dis-tributions for the configuration n◦2 and 3 are shown in appendix B in the figures 7.1 and7.2.

Entries 29134

/ ndf 2χ 76.67 / 96

norm. 0.587± 4.943

2|-

+|H2|0

+|H2|+

|H 0.588± 4.943

) *-H

+Re(H 0.1194± 0.9885

) *-H

+Im(H 0.020114± 0.001624

Chi0 1 2 3 4 5 60

50

100

150

200

250

300

350

400

450

500

Entries 29134

/ ndf 2χ 76.67 / 96

norm. 0.587± 4.943

2|-

+|H2|0

+|H2|+

|H 0.588± 4.943

) *-H

+Re(H 0.1194± 0.9885

) *-H

+Im(H 0.020114± 0.001624

Dist_Ang Entries 29134

/ ndf 2χ 81.03 / 97

2|-+|H2|0

+|H2|+

|H 0.604± 4.952

2|-+|H2|+

|H 0.248± 2.027

2|0

|H 0.357± 2.925

CosTheta_1-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

Entries 29134

/ ndf 2χ 81.03 / 97

2|-+|H2|0

+|H2|+

|H 0.604± 4.952

2|-+|H2|+

|H 0.248± 2.027

2|0

|H 0.357± 2.925

Dist_Ang

Entries 29134

/ ndf 2χ 107.4 / 97

2|-+|H2|0

+|H2|+

|H 0.604± 4.952

2|-+|H2|+

|H 0.247± 2.019

2|0

|H 0.358± 2.933

CosTheta_2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

600

Entries 29134

/ ndf 2χ 107.4 / 97

2|-+|H2|0

+|H2|+

|H 0.604± 4.952

2|-+|H2|+

|H 0.247± 2.019

2|0

|H 0.358± 2.933

Dist_Ang

Entries 29134

/ ndf 2χ 86.9 / 96

2|perp.

+|A2|0

+|A2|||

|A 0.834± 4.964

2|||

|A 0.24± 2.01

2|perp

|A 0.51454± 0.03058

2|0

|A 0.364± 2.924

Phi_tr0 1 2 3 4 5 60

50

100

150

200

250

300

350

400

Entries 29134

/ ndf 2χ 86.9 / 96

2|perp.

+|A2|0

+|A2|||

|A 0.834± 4.964

2|||

|A 0.24± 2.01

2|perp

|A 0.51454± 0.03058

2|0

|A 0.364± 2.924

Dist_Ang

Entries 29134

/ ndf 2χ 96.93 / 97

2|perp.

+|A2|0

+|A2|||

|A 0.53± 4.95

2|0

+|A2|||

|A 0.529± 4.937

2|perp

|A 0.0097± 0.0134

CosTheta_tr-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

Entries 29134

/ ndf 2χ 96.93 / 97

2|perp.

+|A2|0

+|A2|||

|A 0.53± 4.95

2|0

+|A2|||

|A 0.529± 4.937

2|perp

|A 0.0097± 0.0134

Dist_Ang

Figure 3.2: Angular distribution for χ, cos θ1, cos θ2, φtr and cos θtr (configuration n◦1). Theblack curves are the fits and the red curves the expected distribution.

The table 3.2 summarizes the results of the fits in comparison with the parameters usedin the event generation. The fits results are in good agreement with the values used in thegeneration. One can conclude that the implementation of the angles calculation is correct.

The correlation between the fit parameters can be considerably reduced by assumingnormalized amplitudes as mentioned in [1] and [5] and is done in real data (section 4.3).

13


Ang. Extracted Configuration n◦1 Configuration n◦2 Configuration n◦3dist. parameters Gen. From fit Gen. From fit Gen. From fit

χ |H+|2 + |H0|2 + |H−|2 4.89 4.943 ± 0.588 21 20.98 ± 2.23 21 21.05 ± 4.14<(H+H∗−) 1 0.9885 ± 0.1194 −3.32 −3.336 ± 0.36 8 7.91 ± 1.56=(H+H∗−) 0 0.0016 ± 0.02 7.27 7.303 ± 0.783 0 0.124 ± 0.11

θ1 |H+|2 + |H−|2 2 2.027 ± 0.248 20 19.93 ± 2.64 20 19.95 ± 3.11|H0|2 2.89 2.925 ± 0.357 1 1.113 ± 0.165 1 1.173 ± 0.204

θ2 |H+|2 + |H−|2 2 2.019 ± 0.247 20 19.94 ± 2.64 20 19.89 ± 3.11|H0|2 2.89 2.933 ± 0.358 1 0.091 ± 0.163 1 1.173 ± 0.204

φtr |A|||2 2 2.01 ± 0.24 6.67 6.589 ± 1.948 18 17.91 ± 3.05|A0|2 0 0.03 ± 0.51 13.33 13.33 ± 2.47 2 2.067 ± 1.263|A⊥|2 2.89 2.924 ± 0.364 1 1.104 ± 1.118 1 1.07 ± 0.58

θtr |A|||2 + |A0|2 4.89 4.937 ± 0.529 7.67 7.66 ± 1.104 19 18.97 ± 3.04|A⊥|2 0 0.013 ± 0.01 13.33 13.34 ± 1.91 2 2.042 ± 0.342

Table 3.2: Results of the fits for each angular distribution and each helicities states. Plots areshown in figures 3.2 for the configuration n◦1, 7.1 for n◦2 and 7.2 for n◦3.

This should reduce the errors.

Acceptance

DecProdCut keeps signal events only within the geometric acceptance of the detector. Bytaking the ratio of the angular distribution to the expected distribution, one can study theeffect of acceptance. This does not introduce any distortion in the angular distributions asshown in appendix B in figure 7.3. No real effect can be observed, all acceptances are flat.

14


3.2 B0 Selection

3.2.1 Data

The reconstruction of B0 → φK∗0 is first studied with simulated events, for the decayamplitudes n◦3.

Signal events were generated at 7 TeV in proton-proton center of mass energy with thefull detector response, allowing comparison with real data.

3.2.2 Selection cuts

To ensure a full comparison with real data, the cuts applied on reconstructed MC data areidentical to the ones applied on real data. They are reported in table 3.3. A few comments

K & π χ2/NDF < 4PVDOCAχ2 > 16pt > 250 MeV

φ & K∗0 Mass Window 50 MeV for φMass Window 100 MeV for K∗0

vertex χ2 < 9p 10 GeVpt 1.5 GeV

B0 Mass Window 5− 5.7 GeVvertex χ2 < 9One of the grand daughter particles pt > 1.5 GeV

Table 3.3: Selection criteria applied on MC reconstructed and real data.

on the selection cuts:

• the mass range cut is quite loose to conserve the statistics.

• a tight quality cut is applied on the χ2 of the vertex fits to reject combinatorialbackground.

• the impact parameter cut (PVDOCAχ2) rejects the prompt background.

• kinematics cuts are applied to clean up the data from the background. Whereas thecut on the charged kaon and pion transverse momentum to be larger than 250 MeVdoes not affect much the kaons, it rejects a fraction of events with low pt pions.

Mass distribution

Figure 3.3 shows the reconstructed B0 mass distributions after selection cuts.The data set contains ∼ 3’860 events after selection, and 3’500 of them are MC truth.

Construction of the dataset at DaVinci level has run on 66’000 signal events. The overallefficiency for signal is then 3′860

66′000 =∼ 6%.

15


Entries 3859

/ ndf 2χ 242.5 / 95

SΣ 62.5± 3713

Sµ 0.2± 5281

Sσ 0.20± 14.25

BGΣ 14.1± 140.2

BG

slope 0.000281± -0.004256

MeV5000 5100 5200 5300 5400 5500 5600 57000

100

200

300

400

500

600

700

800

Entries 3859

/ ndf 2χ 242.5 / 95

SΣ 62.5± 3713

Sµ 0.2± 5281

Sσ 0.20± 14.25

BGΣ 14.1± 140.2

BG

slope 0.000281± -0.004256

B0_Mass

Figure 3.3: B0 mass distribution for MC reconstructed signal data for configuration n◦3 afterselection. From the fit, there are 3713 signal and 140 background events.

3.3 Angular distribution for MC reconstructed data

Before fitting the data, several effects should be studied: the acceptance, meaning the effectof the selection cuts on the angular distribution, and the resolution of the reconstructedangles, the effects due to the detection.

The both effects were studied with the simulated signal events by comparing the gener-ated distributions and reconstructed distributions after all cuts.

3.3.1 Acceptance

The acceptance is obtained from the generated angular distributions for the events after allthe selection cuts and the expected angular distribution for the same number of events, bydividing the second by the first.

Acceptances are mainly affected by the selection criteria and a little depend on thedecay amplitudes themselves. This is verified by conducting the same analysis on the threeamplitudes configuration previously studied in MC generated data.

The acceptances are then parametrized by polynomial functions to be applied on realdata.

Observations

Acceptances are shown in figure 3.4.

• χ, θ1 and θtr present a nearly flat distribution.

• θ2, on the other hand, is strongly affected by the cuts: for cos θ2 → 1, the histogramgoes down to 0. This is coherent with the observation made in figure 3.5: One canobserve that the cut at 250 MeV suppresses the pions for cos θ2 → 1. cos θ2 → 1 meansthat the kaon is going forward while the pion is going backwards in the K∗0 rest frame,thus implying that the pion has a slow velocity in the laboratory rest frames.The right bottom plot shows the acceptance before and after selection at 250 MeVtransverse momentum. The drop in acceptance is significative at cos θ2 → 1.

16


Entries 2289 / ndf 2χ 55.65 / 45

p0 0.421± 1.141 p1 0.397± -0.398 p2 0.1208± 0.2625 p3 0.02308± -0.06322 p4 0.002710± 0.004981

Acceptance Chi0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Entries 2289 / ndf 2χ 55.65 / 45

p0 0.421± 1.141 p1 0.397± -0.398 p2 0.1208± 0.2625 p3 0.02308± -0.06322 p4 0.002710± 0.004981

dist_mc_truth Entries 2689 / ndf 2χ 34.58 / 45

p0 0.259± 1.042 p1 0.6049± 0.1957 p2 1.5200± -0.1444 p3 0.9023± -0.3682 p4 1.67463± -0.09789

Acceptance CosTheta1-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Entries 2689 / ndf 2χ 34.58 / 45

p0 0.259± 1.042 p1 0.6049± 0.1957 p2 1.5200± -0.1444 p3 0.9023± -0.3682 p4 1.67463± -0.09789

dist_mc_truth

Entries 2901

/ ndf 2χ 49.32 / 46

p0 0.214± 1.145 p1 0.5055± -0.2249

p2 0.4176± -0.6442

p3 0.7103± -0.2091


0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Entries 2901

/ ndf 2χ 49.32 / 46

p0 0.214± 1.145 p1 0.5055± -0.2249

p2 0.4176± -0.6442

p3 0.7103± -0.2091

dist_mc_truth Entries 2111 / ndf 2χ 46.53 / 45

p0 0.5610± 0.5382 p1 1.3516± 0.9282 p2 0.9177± -0.5728 p3 0.2252± 0.1295 p4 0.018065± -0.009608

Acceptance PhiTr0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Entries 2111 / ndf 2χ 46.53 / 45

p0 0.5610± 0.5382 p1 1.3516± 0.9282 p2 0.9177± -0.5728 p3 0.2252± 0.1295 p4 0.018065± -0.009608

dist_mc_truth

Entries 2986

/ ndf 2χ 53.58 / 47

p0 0.213± 1.015

p1 0.24231± -0.02317

p2 0.47007± -0.06417

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Entries 2986

/ ndf 2χ 53.58 / 47

p0 0.213± 1.015

p1 0.24231± -0.02317

p2 0.47007± -0.06417

dist_mc_truth

Figure 3.4: Acceptance for χ, cos θ1, cos θ2, φtr and θtr for configuration n◦3. Plots show thedistribution divided by the expected distribution. Fit parameters correspond to thefactors in the polynomial functions p0 + p1x+ p2x

2 + ...

Acceptance parametrization

The acceptances are parametrized using polynomial functions. The flatter distribution havea linear function, while the more complicated acceptance are parametrized by degree 4polynomial. The degree of the polynomial are increased one by one until the ratio χ2/NDFis acceptable (∼< 1.2).

Figure 3.4 shows the acceptance for MC reconstructed data with the set of amplitudesn◦3. A similar work has been done on MC reconstructed data for amplitudes configurationsn◦1 and n◦2. The plots are shown in appendix B in figures 7.4 and 7.5 and parametrizationare summarized in the table 3.4.

17


CosTheta_2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pio

np

t [M

eV]

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

VSEntries 16141Mean x -0.002293Mean y 1281RMS x 0.4731RMS y 1075


VS

CosTheta_2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Kao

nP

t [M

eV]

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000



VS

Costheta2Entries 16141Mean -0.003RMS 0.4732

CosTheta_2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

50

100

150

200

250


Costheta2 Costheta2Entries 11977Mean -0.006308RMS 0.5835

Acceptance CosTheta_2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Costheta2

Figure 3.5: Upper plots show the true distribution of transverse momentum in function of cos θ2,for pions on the left and kaons on the right. One can observe that the cut at 250 MeVsuppresses the pions for cos θ2 → 1.Lower plots show the angular distribution on the left and the acceptance on theright. Red dots are the MC truth values before selection and green dots MC Truthvalues after selection at 250 MeV transverse momentum. The drop in acceptance issignificative at cos θ2 → 1.

Ang. Amp. Acc.dist. conf. parametrizationχ n◦1 1.057− 0.374x+ 0.3011x2 − 0.0787x3 + 0.00646x4

n◦2 0.9997 + 0.0085xn◦3 0.9975− 0.0067x

θ1 n◦1 1.08 + 0.566x− 0.544x2 − 3.054x3 + 1.476x4 + 4.514x5 − 1.238x6 − 1.943x7

n◦2 0.981 + 0.6746x+ 0.435x2 − 3.78x3 − 1.018x4 + 8.045x5 + 0.447x6 − 5.208x7

n◦3 1.042 + 0.1957x− 0.1444x2 − 0.3682x3 − 0.09789x4

θ2 n◦1 1.29− 0.24x− 0.67x2 − 0.317x3

n◦2 1.123− 0.1665x− 0.564x2 + 0.2304x3

n◦3 1.083− 0.308x− 0.1035x2 − 0.5643x3 − 0.5927x4

φtr n◦1 0.605 + 0.838x− 0.5218x2 + 0.121x3 − 0.009x4

n◦2 1.011 + 0.2526x− 0.2595x2 + 0.06935x3 − 0.00543x4

n◦3 0.538 + 0.9282x− 0.5728x2 + 0.1295x3 − 0.0096x4

θtr n◦1 1.057 + 0.001x− 0.287x2

n◦2 1.008− 0.0458xn◦3 0.994− 0.022x

Table 3.4: acceptance parametrizations for each angle and each amplitudes configuration.

3.3.2 Resolution

Acceptance focuses on the effect of selection on the angular distribution. Resolution, on theother hand, translates the effect of the detection: it compares the MC Truth values with thereconstructed ones.

Figures 3.7 and 3.8 show two plots for each angle. The upper ones plot the angle calcu-lated with the true momentums as a function of the angle calculated with the reconstructedmomentums. The lower graphics show the difference between the true and the reconstructedangles.

18


Figure 3.6: Polynomial parametrizations are plotted for each angle. One can observe a goodsimilitude between the different configurations.

Figure 3.7: Resolution plots for χ, cos θ1 and cos θ2.

Observations

All plots show that the RMS of the difference is < 0.05 radian for χ and φtr, and < 0.07 forcos θ1, cos θ2 and cos θtr.

The effect of resolution is neglected in the present study as it is much smaller in com-parison with the bin size used if fit to real data (∼ 0.78 radian for χ and φtr, and ∼ 0.125for cos θ1, cos θ2 and cos θtr.

19


Figure 3.8: Resolution plots for φtr and cos θtr.

3.3.3 Background subtraction

B0 mass distribution is described by the addition of a gaussian for the signal events and alinear function for the background. In order to obtain the angular distribution correspondingto signal events, one subtracts the background angular distribution obtained from the eventsin the sideband of the B mass region from the total one. This assumes that the backgroundangular distribution component is the same for B0 whole mass window.

After the selection, the background is limited and the subtraction has not much effect.The background subtraction is nevertheless applied as it is done on real data.

Background subtraction is implemented in the following way: B0 mass distribution isdivided in three zones.

• between 5220 and 5336 MeV, the signal zone

• between 5000 and 5220 MeV, left background zone

• between 5336 and 5556 MeV, right background zone

The two background zones are then added and rescaled to match the mass window of thesignal (by a factor of 116/440) before being subtracted from the signal zone.

Figure 7.6 in appendix B illustrates the background subtraction method.

3.3.4 Results after acceptance parametrization and background sub-traction

Angular distributions are shown in figures 3.9 and the extracted parameters with and withoutacceptance parametrizations are reported in table 3.5. The first columns correspond to theresult before taking into account the acceptance parametrization. One can observe thatnearly all parameters do not match at all the generated values.

With the parametrization of the acceptances, not only the fitted parameters agree wellwith the generated values, but the ratio χ2/NDF is also improved in comparison with thefirst columns of table 3.5.

20


Ang. Extracted Used Without acc. para. With acc. para.dist. parameters in gen. From fit Chi2/NDF From fit Chi2/NDF

χ |H+|2 + |H0|2 + |H−|2 21 14.74 ± 5.44 21.05 ± 10.02<(H+H∗−) 8 0.59.67 ± 1.7159 108.7/96 8.386 ± 3.999 53.34/46=(H+H∗−) 0 0.7982 ± 0.8142 −0.07445 ± 0.22892

θ1 |H+|2 + |H0|2 + |H−|2 21 0.6718 ± 0.237 21.01 ± 7.39|H+|2 + |H−|2 20 11.22 ± 3.96 96.84/97 19.99 ± 7.03 34.57/47

|H0|2 1 0.3457 ± 0.1478 1.006 ± 0.406θ2 |H+|2 + |H0|2 + |H−|2 21 0.5863 ± 0.0109 20.96 ± 7.31

|H+|2 + |H−|2 20 10.11 ± 0.12 447.4/97 20.04 ± 6.99 49.5/47|H0|2 1 −0.0176 ± 0.03854 0.9745 ± 0.4256

φtr |A|||2 + |A0|2 + |A⊥|2 21 0.9378 ± 0.354 20.47 ± 7.86|A|||2 18 14.74 ± 5.44 89.06/96 18.52 ± 7.06 67.04/46|A0|2 2 0.5967 ± 1.7159 2.124 ± 2.964|A⊥|2 1 0.7982 ± 0.8142 0.9527 ± 1.3767

θtr |A|||2 + |A0|2 + |A⊥|2 21 20.97 ± 7.68 20.85 ± 7.53|A|||2 + |A0|2 19 19.08 ± 6.98 132.1/97 19.22 ± 6.94 53.4/47|A⊥|2 2 1.875 ± 0.715 1.84 ± 0.7

Table 3.5: Fits results are reported in this table both before and after applying the acceptanceparametrization

Entries 3501

/ ndf 2χ 53.34 / 46

norm. 9.69± 20.38

2|-

+|H2|0

+|H2|+

|H 10.02± 21.05

) *-H

+Re(H 3.999± 8.386

) *-H

+Im(H 0.22892± -0.07445

AngDist Chi0 1 2 3 4 5 60

20

40

60

80

100

120

140

160

Entries 3501

/ ndf 2χ 53.34 / 46

norm. 9.69± 20.38

2|-

+|H2|0

+|H2|+

|H 10.02± 21.05

) *-H

+Re(H 3.999± 8.386

) *-H

+Im(H 0.22892± -0.07445

dist_mc_truth Entries 3501

/ ndf 2χ 34.57 / 47

2|-+|H2|0

+|H2|+

|H 7.39± 21.01

2|-+|H2|+

|H 7.03± 19.99

2|0

|H 0.406± 1.006

AngDist CosTheta1-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

Entries 3501

/ ndf 2χ 34.57 / 47

2|-+|H2|0

+|H2|+

|H 7.39± 21.01

2|-+|H2|+

|H 7.03± 19.99

2|0

|H 0.406± 1.006

dist_mc_truth

Entries 3501

/ ndf 2χ 49.5 / 47

2|-+|H2|0

+|H2|+

|H 7.31± 20.96

2|-+|H2|+

|H 6.99± 20.04

2|0

|H 0.4256± 0.9745

AngDist CosTheta2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

Entries 3501

/ ndf 2χ 49.5 / 47

2|-+|H2|0

+|H2|+

|H 7.31± 20.96

2|-+|H2|+

|H 6.99± 20.04

2|0

|H 0.4256± 0.9745

dist_mc_truth Entries 3501

/ ndf 2χ 45.75 / 46

2|perp.

+|A2|0

+|A2|||

|A 7.92± 20.79

2|||

|A 6.92± 18.27

2|perp

|A 2.703± 1.915

2|0

|A 1.2630± 0.8564

AngDist PhiTr0 1 2 3 4 5 60

20

40

60

80

100

120

140

160

180

Entries 3501

/ ndf 2χ 45.75 / 46

2|perp.

+|A2|0

+|A2|||

|A 7.92± 20.79

2|||

|A 6.92± 18.27

2|perp

|A 2.703± 1.915

2|0

|A 1.2630± 0.8564

dist_mc_truth

Entries 3501

/ ndf 2χ 53.4 / 47

2|perp.

+|A2|0

+|A2|||

|A 7.53± 20.85 2|

0+|A2|

|||A 6.94± 19.22

2|perp

|A 0.70± 1.84

AngDist CosThetaTr-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

Entries 3501

/ ndf 2χ 53.4 / 47

2|perp.

+|A2|0

+|A2|||

|A 7.53± 20.85 2|

0+|A2|

|||A 6.94± 19.22

2|perp

|A 0.70± 1.84

dist_mc_truth

Figure 3.9: Angular distributions for χ, cos θ1, cos θ2, φtr and θtr for the configuration n◦3, takinginto account the parametrizations of the acceptance.

21


3.4 Conclusion on MC reconstructed data

Analysis on MC data has shown that:

• calculation of angles is correctly implemented

• acceptance effect are not to be neglected and are parametrized to further used in realdata.

• resolution effects are negligible in regards to the bin size used in real data.

• normalized amplitudes should be considered in order to reduce the errors on the fittedparameters (see section 4.3 for fit on real data).

22

Chapter 4

Angular distribution analysis on realdata

4.1 Data

Analysis are conducted on 2011 (until mid-April) Stripping13 data for a luminosity of ∼60pb−1. Data is analyzed using DaVinci v28r2p21.

4.2 Selection

After the pre-selection made in the Stripping lines, all cuts are reinforced as reported intable 3.3. Selection criteria are then identical for both MC and real data for a completecomparison and in order to apply the same acceptance parametrizations.

B0 mass distribution is shown in figure 4.1. Signal is described by a gaussian of 96fitted events. The background is composed of nearly 240 events, more than two third of thesample, and follows a linear function. Background subtraction is thus necessary.

Entries 332

/ ndf 2χ 43.4 / 30

SΣ 11.80± 96.21

Sµ 2.5± 5270

Sσ 2.19± 17.43

BGΣ 16.7± 235.8

BG

slope 0.000312± -0.001092

MeV5000 5100 5200 5300 5400 5500 5600 57000

10

20

30

40

50

Entries 332

/ ndf 2χ 43.4 / 30

SΣ 11.80± 96.21

Sµ 2.5± 5270

Sσ 2.19± 17.43

BGΣ 16.7± 235.8

BG

slope 0.000312± -0.001092

B0_Mass

Figure 4.1: Real data B0 mass distribution after cuts reported in figure 3.3

4.3 1-D Angular distribution

Angular distributions are separately fitted. It is done after background subtraction andtakes into account acceptance effects by multiplying the fitting function by the acceptanceparametrization reported in table 3.6.

1A full description of LHCb software can be found in [3]

23

4 Angular distribution analysis on real data

In addition, normalized amplitudes are considered, leaving two independent squaredamplitudes and two independent phases2.

|A|||2 + |A0|2 + |A⊥|2 = 1arg(H0) = arg(A0) = 0

⇒ 4 free parameters: |A⊥|, |A0|, arg(A||), arg(A⊥)

Angular distributions are shown in figure 4.2 and results of the fits to the 1-dimensionaldistributions are compiled in table 4.2.

4.3.1 Results

After background subtraction, the dataset contains only about ninety events. Acceptanceparametrization obtained from configuration n◦1 and n◦2 are used separately.

RD

/ ndf 2χ 5.288 / 6

) *-H

+Re(H 0.08264± 0.07876

) *-H

+Im(H 0.08348± 0.08267

Chi0 1 2 3 4 5 60

5

10

15

20

25

30RD

/ ndf 2χ 5.288 / 6

) *-H

+Re(H 0.08264± 0.07876

) *-H

+Im(H 0.08348± 0.08267

with_acc_para_1 RD

/ ndf 2χ 5.792 / 7

2|-+|H2|+

|H 0.0879± 0.4369

CosTheta_1-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30RD

/ ndf 2χ 5.792 / 7

2|-+|H2|+

|H 0.0879± 0.4369

with_acc_para_1

RD

/ ndf 2χ 7.205 / 7

2|-+|H2|+

|H 0.0833± 0.4193

CosTheta_2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30RD

/ ndf 2χ 7.205 / 7

2|-+|H2|+

|H 0.0833± 0.4193

with_acc_para_1 RD

/ ndf 2χ 3.389 / 5

2|||

|A 0.9070± 0.2871

2|perp

|A 1.8083± 0.2405

2|0

|A 0.9095± 0.4719

Phi_tr0 1 2 3 4 5 60

5

10

15

20

25

30RD

/ ndf 2χ 3.389 / 5

2|||

|A 0.9070± 0.2871

2|perp

|A 1.8083± 0.2405

2|0

|A 0.9095± 0.4719

with_acc_para_1

RD

/ ndf 2χ 6.223 / 7

2|0

+|A2|||

|A 0.0859± 0.7701

CosTheta_tr-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

RD

/ ndf 2χ 6.223 / 7

2|0

+|A2|||

|A 0.0859± 0.7701

with_acc_para_1

Figure 4.2: Angular distribution χ, cos θ1, cos θ2, φtr and cos θtr for real data. The acceptanceparametrizations from the configuration n◦1 have been used. The same plots with theconfiguration n◦2 are shown in appendix B in figure 7.7.

Observations

χ As seen from the fitted values and their errors, fitting parameters are not very muchsensitive to the χ distribution. More statistics are needed.

2this procedure is also applied in Refs [1] and [5]

24


Ang. Extracted PDG From fit From fitdist. parameters values with acc. n◦1 with acc. n◦2

χ <(H+H∗−) ± 0.09924 ± 0.08242 0.09924 ± 0.08242=(H+H∗−) ± 0.09249 ± 0.08238 0.09249 ± 0.08238

θ1 |H+|2 + |H−|2 0.52 ± 0.03 0.4369 ± 0.0879 0.4498 ± 0.886

|H0|2.

0.48 ± 0.03 0.5631 ± 0.0879 0.5502 ± 0.0886θ2 |H+|2 + |H−|2 0.52 ± 0.03 0.4193 ± 0.0833 0.4458 ± 0.0857

|H0|2.

0.48 ± 0.03 0.5807 ± 0.0833 0.5542 ± 0.0857φtr |A|||2 0.28 ± 0.08 0.2871 ± 0.907 0.3098 ± 0.9107

|A⊥|2 0.24 ± 0.05 0.2405 ± 1.8083 0.2462 ± 1.815|A0|2 0.48 ± 0.03 0.4719 ± 0.9095 0.4381 ± 0.9121

θtr |A|||2 + |A0|2 0.76 ± 0.05 0.7701 ± 0.0859 0.8337 ± 0.0776

|A⊥|2.

0.24 ± 0.05 0.2299 ± 0.0859 0.1663 ± 0.0776

Table 4.1: Real data results in comparison with values reported in the PDG[4]. The extractedparameters marked by . are deduced from the normalization of the norms, i.e. θ2second parameter is calculated from the first as |H0|2 = 1− |H+|2 + |H−|2.

φtr Fits for φtr give relatively good result but have large errors. This is probably linked tothe correlation between the parameters, which are fitted without normalized parame-ters. The low statistics prevent the fit to converge otherwise.

θ1,2,tr θ angles, on the other hand, give consistent and compatible with PDG values.

Calculation of polarisation fraction

Observations have conclude that only the information from the cos θ distribution are ex-ploitable.

As θ1 and θtr use the same information (momentum from φ and its daughter parti-cles), only one of them is kept. Moreover, tabulated values are generally expressed in thetransversity basis, θtr is then preferred. θ2 and θtr results are kept to calculate polarizationfractions.

Polarization Definition PDG From fitfraction values with acc. n◦1

fL |A0|2/Σ|Aλ|2 0.48 ± 0.03 0.58 ± 0.08f⊥ |A⊥|2/Σ|Aλ|2 0.24 ± 0.05 0.23 ± 0.09

φ|| − π arg(A||/A0)− π −0.74 ± 0.13 NA

φ⊥ − π arg(A⊥/A0)− π −0.75 ± 0.13 NA

Table 4.2: Comparison of polarization fraction from fit results and values reported in the PDG[4].

Polarization fraction calculated from real data fits reported in table 4.2 are within statis-tics errors of the PDG values, despite the low statistics.

The increase of statistics could considerably improve the results. Indeed, real data col-lected in 2011 only until April has been used for the present analysis for an integratedluminosity of ∼ 60pb−1, while it reached ∼ 330pb−1 in June and ∼ 1fb−1 are expected bythe end of the year. The sample size would increase by a factor ∼ 16 and the statisticalerrors by a factor ∼

√16 = 4. Systematics errors could soon dominate the statistical ones.

Systematic errors

Systematic errors have been neglected in the present study in comparison with statisticalerrors. However, systematic effects due to the background subtraction and the acceptanceparametrization should be investigated if this analysis is conducted on higher statistics.

The assumption made to perform background subtraction is that the background is lin-ear and that its angular distribution is the same in the signal and side band regions. This

25


should be studied in depth and one should consider including the B mass distribution in thefit for a more efficient signal-background distinction.

The acceptance is parametrized using polynomial functions. Despite their similitudes,the acceptance parametrizations for the different configuration are not equivalent. Table4.2 shows the fit results for real data using two different parametrizations. One can noticethat the difference between the results is smaller than the statistical errors. If the statisticsis increased, the systematic errors could become dominant. Two options could reduce thesystematics errors: first, periodic functions could be considered for a better parametrizationsthan with polynomial functions. Secondly, the process could be iterated: the polarizationfractions obtained could be used as the amplitudes configuration for the generation of MCdata, leading to new polarization fractions.

26

Chapter 5

Conclusions

The goal of this project was to study angular distribution of B0 → φK∗0 decays at LHCb,and to derive information on polarization fraction.

Prior to conduct the analysis on real data, several observation have been made on MCdata. A study of the acceptance effects has shown their crucial importance and that theyshould not be neglected. Resolution, on the other hand, is sufficiently good; its effects on theangular distributions are small relative to the bin size used in real data. One should how-ever include resolution effects in the fit when statistics increases (and the bin size is reduced).

Strong of those observations, angular distribution analysis has been conducted on realdata. After event selection and background subtraction, the sample contained only ninetysignal events. This allowed nevertheless the angular distributions to be individually fitted.The extracted parameters reported in table 4.2 are in agreement with the PDG valuesand the statistical error less than a factor 3 larger. LHCb will soon be competitive for themeasurement of the polarization parameters. Concerning the phases of the decay amplitudes,however, no meaningful information could be retrieved from the fits.

Suggested improvements

Real data collected in 2011 only until April has been used for the analysis. The increase ofstatistics would not only lead to better statistical results, but also allow multidimensionalunbinned maximum likelihood fits to extract simultaneously the four independent parame-ters. Further improvement could be obtained by including the B mass distribution in thefit for a more efficient signal extraction. Information on the polarization fraction concerningthe phases (φ|| and φ⊥) could then be obtained.

Longer term prospects

On a larger time scale, with large statistics, a comparison between the B0 and B0 decaypolarization parameters will be possible. Direct CP violation could be measured in theasymmetry between polarization fractions, e.g. A0

CP = (fL − fL)/(fL + fL).

27

5 Conclusions

28

Chapter 6

Appendix A: angular distributioncalculation

6.1 Angular distribution - General case

This development follows section 2.3.

6.1.1 Calculation of 〈θ, φ, λ1, λ2|J,M, λ1, λ2〉For this subsection as for the rest of chapter, quantization axis for the orbital momentum ischosen along the axis z.

Djm′m(αβγ) functions are the matrix elements of the rotation operator R(αβγ) in terms

of angular momentum eigenstates, where α, β and γ are Euler angles.

R(αβγ)|jm〉 =

j∑m′=−j

Djm′m(αβγ)|jm′〉 (6.1)

Using the fact that a rotation about a given axis n is generated by the angular momentumoperator

−→J · n, the rotation operator can be decomposed as [7]:

R(αβγ) = e−iαJz · e−iβJy · e−iγJz (6.2)

Going back to Djm′m(αβγ) definition,

Djm′m(αβγ) = 〈jm′|R(αβγ)|jm〉 (6.3)

= 〈jm′|e−iαJz · e−iβJy · e−iγJz |jm〉 (6.4)

As the quantization axis for the orbital momentum is along z,

Djm′m(αβγ) = e−iαm

′· djm′m(β) · e−iγm (6.5)

where

djm′m(β) = 〈jm′|e−iβJy |jm〉

This matrix element is given by the Wigner formula [8]:

djm′m(β) =∑n

{ (−1)n((j +m)!(j −m)!(j +m′)!(j −m′)!)1/2

(j −m′ − n)!(j +m− n)!(n+m′ −m)!n!

·(cos 12β)2j+m−m′−2n(− sin 1

2β)m′−m+2n

}(6.6)

where the sum is over all integers n for which the factorials are defined.

Finally:

|θ, φ, λ1, λ2〉 = R(φθ0)|j,m, λ1, λ2〉

=

j∑m′=−j

Djm′m(φθ0)|j,m′, λ1, λ2〉

29

6 Appendix A: angular distribution calculation

〈J,M, λ1, λ2|φ, θ, λ1, λ2〉 =

j∑m′=−j

Djm′m(φθ0)〈J,M, λ1, λ2|j,m′, λ1, λ2〉

=

j∑m′=−j

Djm′m(φθ0)δJj δ

Mm′

= DJMm(φθ0) (6.7)

The third angle does not appear in the equation. The choice made here is to set γ to 0.This corresponds to the phase choice first made by Dunietz [9]. Ref [7] discussed this choice.

6.2 Angular distribution for a scalar decaying in twovector mesons

We consider the case in which each vector meson decays in two particles that have nullhelicity and null J.

JK+ = J11 =0= J12 = JK−

JK = J21 =0= J22 = Jπ

sK+ = s11 =0= s12 = sK−

sK = s21 =0= s22 = sπ

The amplitude of the complete disintegration is then:

AB0→(1)(2) = 〈φ(1)K∗0(2)|H|B0〉

= DJMm

∗(φθ0)Hλ1λ2

= D00|λ1−λ2|

∗(φθ0)Hλ1λ2

= Hλ1λ2

A(1)→(11)(12) = 〈(11)(12)|H|φ(1)〉= DJ1

m1m′∗(φ1θ10)Hλ11λ12

= Ds1m1|λ11−λ12|

∗(φ1θ10)Hλ11λ12

= Ds1m10∗(φ1θ10)Hλ11λ12

A(2)→(21)(22) = 〈(21)(22)|H|K∗0(2)〉= DJ2

m2m′∗(φ2θ20)Hλ21λ22

= Ds2m2|λ21−λ22|

∗(φ2θ20)Hλ21λ22

= Ds2m20∗(φ2θ20)Hλ21λ22

Atot =∑

m=+,0,−HmD

s1m0∗(φ1θ10)Ds2

m0∗(φ2θ20) (6.8)

The ∗ in DJMm

∗refers to the complex conjugate of the function.

The summation over the available states of helicity is needed as λK∗0 and λφ are notmeasured.

30


Using 6.6 and the complex conjugate of 6.5 in 6.8,

Atot =∑

m=+,0,−Hme

imφ1ds1m0(θ1)eimφ2ds2m0(θ2)

=∑

m=+,0,−Hme

imχds1m0(θ1)ds2m0(θ2) (6.9)

=∑

m=+,0,−Hmgm

where χ = φ1 + φ2, the azimuthal angle between the two decay planes of the sub-particles.The sign“+” finds its explanation in the illustration 6.1

Figure 6.1: Scheme from [8], illustrating χ = φ1 + φ2.

6.2.1 gm functions

From 6.9, gm = eimχds1m0(θ1)ds2m0(θ2).As s1 = s2 = s = 1, only three cases of d1

m0 are explicited. From the formula 6.6:

d1+10(θ) = −

√2 cos θ2 sin θ

2 = − 1√2

sin θ

d100(θ) = cos2 θ

2 − sin2 θ2 = cos θ

d1−10(θ) = −

√2 sin θ

2 cos θ2 = − 1√2

sin θ

Leading to the three following functions for gm:

g+1 = 12eiχ sin θ1 sin θ2

g0 = cos θ1 cos θ2

g−1 = 12e−iχ sin θ1 sin θ2 (6.10)

6.2.2 Calculation of the distribution

The angular distribution Γ(χ, θ1, θ2) is proportional to the square of the amplitude and isnormalized.

31


d3Γ(χ, θ1, θ2)

dχd cos θ1d cos θ2∝ |A|2

|A|2 =(∑m

Hmgm)∗(∑

n

Hngn)

|A|2 =∑m

|Hm|2|gm|2 + 2∑m>n

(<(H∗mHn)<(g∗mgn)−=(H∗mHn)=(g∗mgn)

)(6.11)

d3Γ(χ, θ1, θ2)

dχd cos θ1d cos θ2=

9

32π

[(|H+|2 + |H−|2) sin2 θ1 sin2 θ2 + 4|H0|2 cos2 θ1 cos2 θ2

+2{<(H+H

∗−) cos 2χ−=(H+H

∗−) sin 2χ

}sin2 θ1 sin2 θ2

+{<((H+ +H−)H∗0

)cosχ−=

((H+ −H−)H∗0

)sinχ

}sin 2θ1 sin 2θ2

](6.12)

The angular distribution 6.12 is obtained by using the explicit gm functions 6.10 in 6.11 andby adding a normalization factor such that∫ 2π

0

dχ

∫ 1

−1

d cos θ1

∫ 1

−1

d cos θ2d3Γ(χ, θ1, θ2) = |H+|2 + |H−|2 + |H0|2.

The integration of the angular distribution over the different variables gives the followingresults :

dΓ(χ)

dχ=

1

2π

[|H+|2 + |H−|2 + |H0|2 + 2

{<(H+H

∗−) cos 2χ−=(H+H

∗−) sin 2χ

}]dΓ(θ1)

d cos θ1=

3

4

[(|H+|2 + |H−|2) sin2 θ1 + 2|H0|2 cos2 θ1

]dΓ(θ2)

d cos θ2=

3

4

[(|H+|2 + |H−|2) sin2 θ2 + 2|H0|2 cos2 θ2

]

6.3 B0 → K∗0φ - transversity basis

Instead of starting from the beginning, one can simply pass from the helicity basis to thetransversity basis using the following change of coordinates:

x′ = sin θtr cosφtr = cos θ1 = z

y′ = sin θtr sinφtr = sin θ1 cosχ = x

z′ = cos θtr = sin θ1 sinχ = y

Making the change of coordinates in 6.12, the expression 6.13 is obtained. It is notnecessary to normalize this distribution as the change of coordinates is unitary.

d3Γ(φtr, θtr, θ2)

dφtrd cos θtrd cos θ2=

9

32π

[|A|||22 sin2 θtr sin2 φtr sin2 θ2

+|A⊥|22 cos2 θtr sin2 θ2 + |A0|24 sin2 θtr cos2 φtr cos2 θ2

+√

2<(A∗||A0) sin2 θtr sin 2φtr sin 2θ2 −√

2=(A∗0A⊥) sin 2θtr cosφtr sin 2θ2

−2=(A∗||A⊥) sin 2θtr sinφtrsin2θ2

](6.13)

32

Chapter 7

Appendix B: additional figures for MCand real data

7.1 MC generated data for validation

Angular distribution for the configuration n◦2

Dist_AngEntries 21583Mean 3.512RMS 1.739

/ ndf 2χ 108.3 / 96Prob 0.1847p0 2.23± 20.98 p1 2.23± 20.98 p2 0.360± -3.336 p3 0.783± 7.303

Chi0 1 2 3 4 5 6

0

50

100

150

200

250

300

350

400

450


/ ndf 2χ 108.3 / 96Prob 0.1847p0 2.23± 20.98 p1 2.23± 20.98 p2 0.360± -3.336 p3 0.783± 7.303

Dist_Ang Dist_AngEntries 21583Mean 0.004303RMS 0.4696

/ ndf 2χ 90.66 / 97Prob 0.6619p0 2.79± 21.04 p1 2.64± 19.93 p2 0.165± 1.113

CosTheta_1-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250

300

350


/ ndf 2χ 90.66 / 97Prob 0.6619p0 2.79± 21.04 p1 2.64± 19.93 p2 0.165± 1.113

Dist_Ang

Dist_AngEntries 21583Mean -0.009RMS 0.4704

/ ndf 2χ 138.9 / 97Prob 0.003391p0 2.79± 21.03 p1 2.64± 19.94 p2 0.163± 1.091

CosTheta_2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250

300

350


/ ndf 2χ 138.9 / 97Prob 0.003391p0 2.79± 21.03 p1 2.64± 19.94 p2 0.163± 1.091

Dist_Ang


/ ndf 2χ 98.46 / 96Prob 0.4112p0 4.09± 21.02 p1 1.948± 6.589 p2 2.47± 13.33 p3 1.118± 1.104

Phi_tr0 1 2 3 4 5 6

0

50

100

150

200

250

300


/ ndf 2χ 98.46 / 96Prob 0.4112p0 4.09± 21.02 p1 1.948± 6.589 p2 2.47± 13.33 p3 1.118± 1.104

Dist_Ang


/ ndf 2χ 103.9 / 97Prob 0.2964p0 3.0± 21 p1 1.104± 7.659 p2 1.91± 13.34

CosTheta_tr-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250

300

350

400

450


/ ndf 2χ 103.9 / 97Prob 0.2964p0 3.0± 21 p1 1.104± 7.659 p2 1.91± 13.34

Dist_Ang

Figure 7.1: Angular distribution for χ, cos θ1, cos θ2, φtr and cos θtr (configuration n◦2). Fitparameters for the angular distribution are equivalent to the ones in figure 3.2

33

7 Appendix B: additional figures for MC and real data

Angular distribution for the configuration n◦3


/ ndf 2χ 100.7 / 96Prob 0.3525p0 4.14± 21.05 p1 4.14± 21.05 p2 1.56± 7.91 p3 0.1101± 0.1235

Chi0 1 2 3 4 5 6

0

50

100

150

200

250

300

350


/ ndf 2χ 100.7 / 96Prob 0.3525p0 4.14± 21.05 p1 4.14± 21.05 p2 1.56± 7.91 p3 0.1101± 0.1235

Dist_Ang Dist_AngEntries 16141Mean 0.0008903RMS 0.4703

/ ndf 2χ 123 / 97Prob 0.0384p0 3.28± 21.04 p1 3.11± 19.95 p2 0.192± 1.095

CosTheta_1-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250

300


/ ndf 2χ 123 / 97Prob 0.0384p0 3.28± 21.04 p1 3.11± 19.95 p2 0.192± 1.095

Dist_Ang


/ ndf 2χ 102 / 97Prob 0.3433p0 3.29± 21.06 p1 3.11± 19.89 p2 0.204± 1.173

CosTheta_2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250


/ ndf 2χ 102 / 97Prob 0.3433p0 3.29± 21.06 p1 3.11± 19.89 p2 0.204± 1.173

Dist_Ang Dist_AngEntries 16141

Mean 3.126

RMS 1.699

/ ndf 2χ 111.5 / 96

Prob 0.1338

p0 3.62± 21.05

p1 3.05± 17.91

p2 1.263± 2.067

p3 0.58± 1.07

Phi_tr0 1 2 3 4 5 6

0

50

100

150

200

250

300

Dist_AngEntries 16141

Mean 3.126

RMS 1.699

/ ndf 2χ 111.5 / 96

Prob 0.1338

p0 3.62± 21.05

p1 3.05± 17.91

p2 1.263± 2.067

p3 0.58± 1.07

Dist_Ang


Mean 0.00556

RMS 0.4893 / ndf 2χ 120.1 / 97

Prob 0.05615p0 3.36± 21.01 p1 3.04± 18.97 p2 0.342± 2.042

CosTheta_tr-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250


Mean 0.00556

RMS 0.4893 / ndf 2χ 120.1 / 97

Prob 0.05615p0 3.36± 21.01 p1 3.04± 18.97 p2 0.342± 2.042

Dist_Ang

Figure 7.2: Angular distribution for χ, cos θ1, cos θ2, φtr and cos θtr (configuration n◦3). Fitparameters for the angular distribution are equivalent to the ones in figure 3.2

34


Acceptance for MC generated data

AcceptanceEntries 26569

Mean 3.135

RMS 1.813

Acceptance Chi0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Mean 3.135

RMS 1.813

Acceptance AcceptanceEntries 26075

Mean -0.002175

RMS 0.5772


0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Mean -0.002175

RMS 0.5772

Acceptance


Mean -0.005668

RMS 0.5777


0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Mean -0.005668

RMS 0.5777

Acceptance AcceptanceEntries 28644

Mean 3.137

RMS 1.812

Acceptance PhiTr0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Mean 3.137

RMS 1.812

Acceptance

AcceptanceEntries 12505Mean 0.0009973RMS 0.5827

Acceptance CosThetaTr-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

AcceptanceEntries 12505Mean 0.0009973RMS 0.5827

Acceptance

Figure 7.3: Acceptance for χ, cos θ1, cos θ2, φtr and cos θtr (configuration n◦1).

35


7.2 MC reconstructed data

Acceptances and angular distribution for configuration n◦1

dist_mc_truth_4Entries 1178Mean 3.158RMS 1.802

/ ndf 2χ 25.96 / 25Prob 0.4098p0 0.533± 1.057 p1 0.516± -0.374 p2 0.1578± 0.3011 p3 0.03011± -0.07874 p4 0.003548± 0.006467

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


/ ndf 2χ 25.96 / 25Prob 0.4098p0 0.533± 1.057 p1 0.516± -0.374 p2 0.1578± 0.3011 p3 0.03011± -0.07874 p4 0.003548± 0.006467

dist_mc_truth

dist_mc_truthEntries 1284Mean 3.164RMS 1.85

/ ndf 2χ 23.87 / 26Prob 0.5832p0 3.109± 4.833 p1 3.145± 4.887 p2 0.741± 1.141 p3 0.10826± -0.08848

0 1 2 3 4 5 6

20

30

40

50

60

70

80


/ ndf 2χ 23.87 / 26Prob 0.5832p0 3.109± 4.833 p1 3.145± 4.887 p2 0.741± 1.141 p3 0.10826± -0.08848

dist_mc_truth


/ ndf 2χ 37.07 / 42Prob 0.6868p0 0.25± 1.08 p1 0.8719± 0.5663 p2 1.6214± -0.5441 p3 3.377± -3.054 p4 3.593± 1.476 p5 5.740± 4.514 p6 2.801± -1.238 p7 4.150± -1.943

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


/ ndf 2χ 37.07 / 42Prob 0.6868p0 0.25± 1.08 p1 0.8719± 0.5663 p2 1.6214± -0.5441 p3 3.377± -3.054 p4 3.593± 1.476 p5 5.740± 4.514 p6 2.801± -1.238 p7 4.150± -1.943

dist_mc_truth


/ ndf 2χ 37.07 / 47Prob 0.8501p0 3.414± 4.891 p1 1.403± 2.003 p2 2.018± 2.886

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

10

15

20

25

30

35

40

45

50

55


/ ndf 2χ 37.07 / 47Prob 0.8501p0 3.414± 4.891 p1 1.403± 2.003 p2 2.018± 2.886

dist_mc_truth

dist_mc_truth_4Entries 1163Mean -0.1338RMS 0.5102

/ ndf 2χ 114.9 / 46Prob 8.074e-08p0 0.225± 1.291 p1 0.5417± -0.2399 p2 0.4361± -0.6693 p3 0.7493± -0.3165

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


/ ndf 2χ 114.9 / 46Prob 8.074e-08p0 0.225± 1.291 p1 0.5417± -0.2399 p2 0.4361± -0.6693 p3 0.7493± -0.3165

dist_mc_truth

dist_mc_truthEntries 1284Mean -0.1945RMS 0.5776

/ ndf 2χ 114.8 / 47Prob 1.336e-07p0 3.416± 4.887 p1 1.396± 1.992 p2 2.034± 2.905

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60


/ ndf 2χ 114.8 / 47Prob 1.336e-07p0 3.416± 4.887 p1 1.396± 1.992 p2 2.034± 2.905

dist_mc_truth


/ ndf 2χ 56.6 / 45Prob 0.1151p0 0.388± 0.605 p1 0.3906± 0.8379 p2 0.1281± -0.5218 p3 0.0249± 0.1208 p4 0.002872± -0.009156

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


/ ndf 2χ 56.6 / 45Prob 0.1151p0 0.388± 0.605 p1 0.3906± 0.8379 p2 0.1281± -0.5218 p3 0.0249± 0.1208 p4 0.002872± -0.009156

dist_mc_truth


/ ndf 2χ 62.59 / 46Prob 0.0521p0 4.838± 4.978 p1 1.382± 2.041 p2 2.81621± -0.06942 p3 1.896± 2.716

0 1 2 3 4 5 6

10

15

20

25

30

35

40

45


/ ndf 2χ 62.59 / 46Prob 0.0521p0 4.838± 4.978 p1 1.382± 2.041 p2 2.81621± -0.06942 p3 1.896± 2.716

dist_mc_truth


/ ndf 2χ 38.2 / 47Prob 0.8164p0 0.213± 1.057 p1 0.22965± 0.00126 p2 0.4486± -0.2872

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


/ ndf 2χ 38.2 / 47Prob 0.8164p0 0.213± 1.057 p1 0.22965± 0.00126 p2 0.4486± -0.2872

dist_mc_truth


/ ndf 2χ 38.2 / 47Prob 0.8163p0 2.953± 4.889 p1 2.953± 4.891 p2 4.848e-02± -8.715e-05

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

10

20

30

40

50


/ ndf 2χ 38.2 / 47Prob 0.8163p0 2.953± 4.889 p1 2.953± 4.891 p2 4.848e-02± -8.715e-05

dist_mc_truth

Figure 7.4: Angular distribution with acceptance parametrizations for χ, cos θ1, cos θ2, φtr andcos θtr (configuration n◦1). Fit parameters for the acceptance correspond to the factorsin the polynomial functions p0 + p1x + p2x

2 + ... Fit parameters for the angulardistribution are equivalent to the ones in figure 3.2

36


Acceptances and angular distribution for configuration n◦2


/ ndf 2χ 41.99 / 28Prob 0.04348p0 0.3666± 0.9997 p1 0.10165± 0.00851

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


/ ndf 2χ 41.99 / 28Prob 0.04348p0 0.3666± 0.9997 p1 0.10165± 0.00851

dist_mc_truth


/ ndf 2χ 39.76 / 26Prob 0.04116p0 11.59± 20.67 p1 11.95± 21.32 p2 1.819± -3.203 p3 4.026± 7.144

0 1 2 3 4 5 6

10

20

30

40

50

60

70

80

90


/ ndf 2χ 39.76 / 26Prob 0.04116p0 11.59± 20.67 p1 11.95± 21.32 p2 1.819± -3.203 p3 4.026± 7.144

dist_mc_truth


/ ndf 2χ 50.99 / 42Prob 0.161p0 0.3067± 0.9811 p1 1.6035± 0.6746 p2 3.8131± 0.4353 p3 11.60± -3.78 p4 10.585± -1.018 p5 24.262± 8.045 p6 7.6931± 0.4471 p7 14.985± -5.208

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


/ ndf 2χ 50.99 / 42Prob 0.161p0 0.3067± 0.9811 p1 1.6035± 0.6746 p2 3.8131± 0.4353 p3 11.60± -3.78 p4 10.585± -1.018 p5 24.262± 8.045 p6 7.6931± 0.4471 p7 14.985± -5.208

dist_mc_truth


/ ndf 2χ 51.01 / 47Prob 0.3187p0 14.7± 21 p1 14.0± 20 p2 0.782± 1.002

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

10

20

30

40

50


/ ndf 2χ 51.01 / 47Prob 0.3187p0 14.7± 21 p1 14.0± 20 p2 0.782± 1.002

dist_mc_truth


/ ndf 2χ 80.92 / 46Prob 0.001124p0 0.213± 1.123 p1 0.5408± -0.1665 p2 0.4185± -0.5639 p3 0.7595± -0.2304

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


/ ndf 2χ 80.92 / 46Prob 0.001124p0 0.213± 1.123 p1 0.5408± -0.1665 p2 0.4185± -0.5639 p3 0.7595± -0.2304

dist_mc_truth


/ ndf 2χ 80.89 / 47Prob 0.001543p0 14.47± 20.99 p1 13.79± 20.01 p2 0.7844± 0.9836

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

10

20

30

40

50


/ ndf 2χ 80.89 / 47Prob 0.001543p0 14.47± 20.99 p1 13.79± 20.01 p2 0.7844± 0.9836

dist_mc_truth


/ ndf 2χ 41.06 / 45Prob 0.6397p0 0.432± 1.011 p1 0.4111± 0.2526 p2 0.1290± -0.2595 p3 0.02513± 0.06935 p4 0.002895± -0.005429

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


/ ndf 2χ 41.06 / 45Prob 0.6397p0 0.432± 1.011 p1 0.4111± 0.2526 p2 0.1290± -0.2595 p3 0.02513± 0.06935 p4 0.002895± -0.005429

dist_mc_truth


/ ndf 2χ 40.83 / 46Prob 0.6879p0 21.63± 21.05 p1 10.350± 6.646 p2 12.92± 13.27 p3 5.7881± 0.9475

0 1 2 3 4 5 65

10

15

20

25

30

35

40

45


/ ndf 2χ 40.83 / 46Prob 0.6879p0 21.63± 21.05 p1 10.350± 6.646 p2 12.92± 13.27 p3 5.7881± 0.9475

dist_mc_truth


/ ndf 2χ 43.39 / 48Prob 0.662p0 0.142± 1.008 p1 0.24829± -0.04583

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


/ ndf 2χ 43.39 / 48Prob 0.662p0 0.142± 1.008 p1 0.24829± -0.04583

dist_mc_truth


/ ndf 2χ 43.19 / 47Prob 0.6312p0 16.18± 21.09 p1 6.137± 7.972 p2 9.95± 12.96

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

10

20

30

40

50


/ ndf 2χ 43.19 / 47Prob 0.6312p0 16.18± 21.09 p1 6.137± 7.972 p2 9.95± 12.96

dist_mc_truth

Figure 7.5: Angular distribution with acceptance parametrizations for χ, cos θ1, cos θ2, φtr andcos θtr (configuration n◦2). Fit parameters for the acceptance correspond to the factorsin the polynomial functions p0 + p1x + p2x

2 + ... Fit parameters for the angulardistribution are equivalent to the ones in figure 3.2

37


Background subtraction illustration

The background subtraction is illustrated in figure 7.6. It is shown on a sample for theconfiguration n◦3. The cuts are loosened to illustrate the method detailed in section 3.3.3.

Dist_Ang_z1

Entries 15783

Mean 0.02551

RMS 0.5331

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

160

180

200

220

Dist_Ang_z1

Entries 15783

Mean 0.02551

RMS 0.5331

Dist_Ang_z1 Dist_Ang_finalEntries 5523Mean -0.04503RMS 0.4538

/ ndf 2χ 164.7 / 97Prob 2.176e-05p0 2.25± 20.91 p1 2.25± 21.17 p2 0.7071± -0.2154

CosTheta_2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

160

180

200

220

Dist_Ang_finalEntries 5523Mean -0.04503RMS 0.4538

/ ndf 2χ 164.7 / 97Prob 2.176e-05p0 2.25± 20.91 p1 2.25± 21.17 p2 0.7071± -0.2154

Dist_Ang_final

Dist_Ang_bgl

Entries 6862

Mean 0.1918

RMS 0.634

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

Dist_Ang_bgl

Entries 6862

Mean 0.1918

RMS 0.634

Dist_Ang_bgl Dist_Ang_bgr

Entries 4562

Mean 0.1201

RMS 0.6345

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

Dist_Ang_bgr

Entries 4562

Mean 0.1201

RMS 0.6345

Dist_Ang_bgr

Figure 7.6: The blue distribution is the angular distribution before the background subtraction inthe central zone. The bottom plots are the angular distributions for the backgroundbetween 5000 and 5220 MeV on the left and between 5336 and 5556 MeV on the right.The right top plot shows the initial distribution in blue, the subtracted backgroundin green and the resulting distribution in black. The black curve is the fit results andthe gray curve the expected distribution.

38


7.3 Real data

Angular distribution with acceptance from configuration n◦2

RD

/ ndf 2χ 5.852 / 6

) *-H

+Re(H 0.08242± 0.09924

) *-H

+Im(H 0.08238± 0.09249

Chi0 1 2 3 4 5 60

5

10

15

20

25

30RD

/ ndf 2χ 5.852 / 6

) *-H

+Re(H 0.08242± 0.09924

) *-H

+Im(H 0.08238± 0.09249

with_acc_para_2 RD

/ ndf 2χ 4.151 / 7

2|-+|H2|+

|H 0.0886± 0.4498

CosTheta_1-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30RD

/ ndf 2χ 4.151 / 7

2|-+|H2|+

|H 0.0886± 0.4498

with_acc_para_2

RD

/ ndf 2χ 8.124 / 7

2|-+|H2|+

|H 0.0857± 0.4458

CosTheta_2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30RD

/ ndf 2χ 8.124 / 7

2|-+|H2|+

|H 0.0857± 0.4458

with_acc_para_2 RD

/ ndf 2χ 1.894 / 5

2|||

|A 0.9107± 0.3098

2|perp

|A 1.8150± 0.2462

2|0

|A 0.9121± 0.4381

Phi_tr0 1 2 3 4 5 60

5

10

15

20

25

30RD

/ ndf 2χ 1.894 / 5

2|||

|A 0.9107± 0.3098

2|perp

|A 1.8150± 0.2462

2|0

|A 0.9121± 0.4381

with_acc_para_2

RD

/ ndf 2χ 5.643 / 7

2|0

+|A2|||

|A 0.0776± 0.8337

CosTheta_tr-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30RD

/ ndf 2χ 5.643 / 7

2|0

+|A2|||

|A 0.0776± 0.8337

with_acc_para_2

Figure 7.7: Angular distribution χ, cos θ1, cos θ2, φtr and cos θtr for real data. The acceptanceparametrizations from the configuration n◦2 have been used.

39


40

Bibliography

[1] K. Nakamura et al. [Particle Data Group], “Polarization in B decays”, JP G 37 075021(2010).

[2] HFAG website http://slac.stanford.edu/xorg/hfag/ Charmless B Decay Polarizations(August 2010).

[3] The LHCb Collaboration, A.A. Alves Jr. et al. [Particle Data Group], “The LHCbDetector at the LHC’, 2008 JINST 3 S08005.

[4] K. Nakamura et al. [Particle Data Group], pages 133-135, JP G 37 075021 (2010).

[5] K.-F. Chen et al., “Measurement of Polarization and Triple-Product Correlations inB0 → φK∗0 Decays”, Phys. Rev. Lett. 94, 22184 (2005).

[6] B. Aubert et al., “Time-dependent and time-integrated angular analysis of B → ϕK0sπ

0

and ϕK±π∓”, Phys. Rev. D 78, 092008 (2008).

[7] J. D. Richman, “An Experimenter’s Guide to the Helicity Formalism”, CALT-68-1148(1984) California Institute of Technology, Pasadena, California 91125.

[8] K. Abe, M. Satpathy and H. Yamamoto, “Time-dependent Angular Analysis of BDecays”, UH-511-982-01 BELLE note n◦419.

[9] I. Dunietz et al., “How to extract CP -violating asymmetries from angular correlations”,Phys. Rev. D 43, N◦7 (1991).

41

http://pdg.lbl.gov

http://pdg.lbl.gov

http://slac.stanford.edu/xorg/hfag/rare/winter10/polar/OUTPUT/PLOTS/polar_plain.pdf

http://slac.stanford.edu/xorg/hfag/rare/winter10/polar/OUTPUT/PLOTS/polar_plain.pdf

http://iopscience.iop.org/1748-0221/3/08/S08005/pdf/jinst8_08_s08005.pdf

http://pdg.lbl.gov

http://charm.physics.ucsb.edu/people/richman/ExperimentersGuideToTheHelicityFormalism.pdf

http://charm.physics.ucsb.edu/people/richman/ExperimentersGuideToTheHelicityFormalism.pdf

http://arxiv.org/pdf/hep-ex/0103002

Study of angular distribution for B0 ˚K0 decays at -...

Documents

Transcript of Study of angular distribution for B0 ˚K0 decays at -...