cp violation in the standard model and a phenomenological … · 2020. 8. 17. · interesting eld....

CP VIOLATION IN THE STANDARD MODEL AND A PHENOMENOLOGICAL STUDY OF

THE DECAY

Wu Shuang

In partial fulfilment of the requirements for the degree of

Bachelor of Science with Honours

Department of Physics

National University of Singapore 2014/2015

Abstract

The violation of charge and parity (CP) symmetries in the weak interactionis explained by the Kobayashi-Maskawa ansatz in the Standard Model. Inthis ansatz, CP violation are characterised by complex phases in the unitaryCabbibo-Kobayahsi-Maskawa (CKM) matrix which characterises the mixingof three generations of quarks. After a general review of CP violation in theStandard Model, a phenomenological study of the B0

s ↔ B0s mixing and the

decay B0s → J/ψφ is performed to illustrate how this decay can be used to

extract a CP violation phase in the CKM matrix.

Acknowledgements

I would like to express my sincere gratitude to my supervisors Prof. Oh Choo Hiap and A/Prof.Phil Chan for their guidance and career advice.

I am also grateful to my friends Mr Kin Mimouni and Mr Tanguy Marchand for useful discus-sions.

My heartfelt appreciation goes to my family for their continuous encouragement and support.

Contents

1. Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2. CP Violation in the Framework of the Standard Model 32.1 Standard Model Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Electroweak interactions for quarks . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Transformations of Fields Under C and P Operators . . . . . . . . . . . . . . . . 82.4 Yukawa Couplings and CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Feynman Rules for Charged Current Interactions . . . . . . . . . . . . . . . . . . 10

3. The CKM Matrix 133.1 Parametrisations of the CKM Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Parametrisation with 3 Euler Angles and a Complex Phase . . . . . . . . 143.1.2 Wolfenstein Parametrisation . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Conditions for CP Violation and the Jarlskog Invariant . . . . . . . . . . . . . . 153.3 Unitarity Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Experimental Determination of the CKM Free Parameters . . . . . . . . . . . . . 18

4. CP Violation in Meson Decays 204.1 Oscillation in Neutral Flavoured Mesons . . . . . . . . . . . . . . . . . . . . . . . 204.2 Decays in Flavoured Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 CP Violation Observables and Classification . . . . . . . . . . . . . . . . . . . . . 25

5. Phenomenology in the Channel B0s → J/ψφ 27

5.1 Evaluating the Amplitude of Box Diagrams . . . . . . . . . . . . . . . . . . . . . 275.1.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 Effective Hamiltonian in the Operator Product Expansion . . . . . . . . . . . . . 345.3 Expressions for M12 and Γ12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.4 CP Violation in B0

s → J/ψφ at the Tree Level and βs . . . . . . . . . . . . . . . 395.5 Time dependent angular analysis of B0

s → J/ψφ . . . . . . . . . . . . . . . . . . 415.6 Penguin Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6. Conclusion and Outlook 45

Bibliography 46

i

List of Figures

2.1 The Standard Model of Particle Physics . . . . . . . . . . . . . . . . . . . . . . . 32.2 Charged current interactions for quarks . . . . . . . . . . . . . . . . . . . . . . . 72.3 V-A vertex for quark charged current interactions . . . . . . . . . . . . . . . . . . 112.4 Vertices for Goldstone bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Propagator for W bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Propagator for Goldstone bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Propagators for Dirac fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.8 External Dirac fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1 The db unitarity triangle relevant to B0d meson decays . . . . . . . . . . . . . . . 16

3.2 The bs unitarity triangle relevant to Bs meson decays . . . . . . . . . . . . . . . 173.3 Feynman diagrams for determination of Vud by comparing neutron β and µ− decays 183.4 The ideal results for the db unitarity triangle [10] . . . . . . . . . . . . . . . . . . 193.5 The global fit of the CKM matrix performed by the CKM fitter group [13] . . . . 19

4.1 Box diagrams arising in the lowest order perturbation calculation of off-diagonalelements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.1 Box Diagrams for the B0s ↔ B0

s oscillations . . . . . . . . . . . . . . . . . . . . . 285.2 b→ ccs decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 A QCD correction correction diagram to b→ ccs . . . . . . . . . . . . . . . . . . 365.4 B0

s → J/ψφ without mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.5 B0

s → J/ψφ with mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.6 Transversity basis for the angular analysis of B0

s → J/ψ(µ+µ−)φ(K+K−) . . . . 425.7 A gluonic penguin diagram for B0

s → J/ψφ . . . . . . . . . . . . . . . . . . . . . 445.8 Double penguin diagram for B0

s ↔ B0s mixing . . . . . . . . . . . . . . . . . . . . 44

List of Tables

2.1 Transformations of Different Fields under CP . . . . . . . . . . . . . . . . . . . . 8

5.1 Angular dependence of differential decay rate in B0s → J/ψφ . . . . . . . . . . . 43

1 Introduction

The concept of symmetries is of prime importance in physics. Precisely, a symmetry as definedin the Lagrangian formulation is a transformation that leaves the action integral of the La-grangian unchanged. Noether’s theorem states that there is an invariant quantity correspondingto each differentiable symmetry of the system. Besides differentiable (and therefore continuous)transformations, there are three important discrete transformations:

• Parity (P): Inverts all spatial coordinates

• Charge Conjugation (C): Reverses the sign of additive quantum numbers

• Time Reversal (T): Inverts time

Further details on the action of these transformations are discussed in section 2.3.

Prior to the 1960s, these three discrete transformations were generally considered to be symme-tries of Nature. After all, our classical experience tells us that physical processes should be thesame if viewed in mirror images (parity symmetry) and experimental results were consistent withintuitions. Indeed violation of these symmetries has not been observed in the electromagneticand strong interactions but does occur for the weak interaction.

In 1956, in a bold step to resolve the tau-theta puzzle in Kaons, the Chinese-American physicistsT.D. Lee and C.N. Yang hypothesised that the parity could be violated in weak interactions[30] and proposed experiments to test the violation. The Wu-experiment [43] performed byMadame Wu in the same year measured the rate of Co-60 (of opposite polarisation) undergoingβ decay and observed that parity is indeed violated. In fact, the violation is maximal; allanti-neutrinos are right-handed while neutrinos are left-handed. The Lederman experiment [21]looked at the decay π+ → µ+νµ. π+ is known to have spin 0 while µ+ and νµ have spin 1/2,therefore they must have opposite spins. This experiment observed all the neutrinos to be left-handed. Besides providing further experimental evidence for parity violation, the Ledermanexperiment also implies that charge conjugation symmetry is violated since the C-transformeddecay π− → µ−νµ cannot take place (µ− and νµ have the same spin).

The physics community was taken aback by the violation of P and C symmetries and it wascomforting that the combined action of CP in these experiments does remain a symmetry. Afurther surprise came when the Cronin-Fitch experiment [15] in 1964 found evidence of CPviolation in the neutral Kaon systems. The eigenstates of the weak interaction |KS〉 and |KL〉(S/L for shorter/longer mean lifetime) were not identical with the CP eigenstates |K+〉 and|K−〉. 1 In 1973, Kobayashi and Maskawa proposed an explanation of CP violation in the formof complex Yukawa couplings and in doing so, predicted the existence of a third generation ofquarks.

The Kobayashi and Maskawa ansatz completed the ingredients of the Standard Model and hassince enjoyed great success. The phenomenon of CP violation continues to be a vibrant andinteresting field. In recent years, CP violation is studied in the B-mesons decays at the B-factories (Belle and Babar) and the LHCb detector. In this project, I focus on the B0

s → J/ψφdecay occurring in the B0

s meson. The natural units is adopted with c = ~ = 1.

1 Unlike P or C violation which is maximal for neutrinos, the degree of CP violation is small. Here |KL〉 =1√1+ε2

(|K−〉+ ε|K+〉) with ε ≈ 2.3× 10−3.

1. Introduction 2

The structure of this report is as follows:

• For the rest of this introductory chapter, I put forward motivations to the study of CPViolation.

• Chapter 2 and 3 present a general review of how CP violation is understood in the frame-work of the Standard Model weak interactions 2 and the seminal explanation in the workof Kobayashi and Maskawa which led to the Cabibbo-Kobayashi-Maskawa (CKM) matrixthat describes generation mixing and CP violation in quarks.

• Chapter 4 focuses on CP violation in neutral meson systems for which most CP violationphenomena in the quark sector are observed.

• In Chapter 5, I employ the formalism and theory to analyse the B0s → J/ψφ decay mea-

sured at the LHCb detector which can be used to obtain the physical parameter βs in theCKM Matrix. This βs thus obtained can be compared to theoretical values and discrep-ancies could be indication of New Physics beyond the Standard Model. However sinceβs is small, the conventional experimental analysis based on tree level decays to extractthis phase might be insufficient. It is fruitful to obtain an estimate of the higher or-der corrections present at the loop or penguin level which are usually neglected at firstapproximations.

• Finally I will summarize the work and present an outlook.

1.1 Motivation

From a cosmological perspective, CP violation is necessary for providing a convention-free dis-tinction of matter and anti-matter and is therefore essential for the matter-anti asymmetry inthe universe [37].

The B-mesons system contains a rich complexity of CP violating phenomenon. In particular,the B0

s system is highly sensitive and useful for searching for New Physics beyond the StandardModel. The difficult lies in experimental precision which hopefully can be overcome at thecurrent run of the LHC. The theoretical difficulty consists in making quantitative calculationstaking into account the necessary corrections from strong interactions and penguin contributions.

The discovery of neutrino oscillations gave experimental evidence that neutrinos are massiveparticles, as opposed to previous assumptions in the Standard Model. This opened up thepossibility of observing CP violation in neutrinos, or the leptonic sector. On a personal level, Iseek to embark on a PhD involving the R&D of a liquid argon of a time projection chamber forneutrinos and rare events detection. Through this project, I hope to establish a solid theoreticalunderstanding of the Standard Model and CP violation as well as to be able to relate theinterplay of experiments and theory: how experiments determine the free parameters needed tocomplete a theoretical model and the experimental efforts probing the limits where the currenttheory may break down.

2 The strong CP problem will not be considered.

2 CP Violation in the Frameworkof the Standard Model

“ Felix, qui potuit rerum cognoscere causas 1

”Virgil, Georgica (II, v. 490)

The Standard Model (SM) is a renormalisable quantum field theory of the electromagnetic, weakand strong interactions of elementary particles. It has enjoyed great phenomenological successand has managed to explain or predict most particle physics experimental results to date. 2 atthe current energy scale but of course it is not the final word or coveted theory of everything.Physics beyond the standard model is generally referred to as New Physics and could possiblybe discovered at the energy scales of current or next generation accelerators.

An instructive summary of all the elementary particles in the SM is shown in figure 2.1.

Fig. 2.1: The Standard Model of Particle Physics

(Taken from http://commons.wikimedia.org/wiki/File:Standard_Model_Of_Particle_

Physics,_Most_Complete_Diagram.jpg Author: Latham Boyle)

1 Literally translates as ”Fortunate who was able to know the causes of things”.2 The current known exceptions are neutrino oscillation and the problem of dark matter.

http://commons.wikimedia.org/wiki/File:Standard_Model_Of_Particle_Physics,_Most_Complete_Diagram.jpg

http://commons.wikimedia.org/wiki/File:Standard_Model_Of_Particle_Physics,_Most_Complete_Diagram.jpg

2. CP Violation in the Framework of the Standard Model 4

As a quantum field theory, the Poincare group R1,3 o O(1, 3) is a symmetry of the SM andthe action integral is invariant under Poincare transformations xµ → x′µ = Λµνxν + aµ 3. Theelementary particle fields are irreducible representations of the Poincare group. Under the spinorrepresentation, the scalar fields correspond to the trivial representation (0, 0), the left and righthanded chiral components of spin 1/2 fields correspond to the representation (0, 1/2) and (1/2, 0)respectively while Dirac fields correspond to the representation (0, 1/2)⊕ (1/2, 0) and finally thespin 1 vector gauge fields correspond to the representation (1/2, 1/2).

There are 12 spin 1/2 fermions in the SM: 6 flavours of quarks subject to the electromagnetic,weak and strong interactions and 6 flavours of leptons not subject to the strong interaction. Asillustrated in figure 2.1, the quarks and leptons are classified into 3 generations, with particlesacross generations being distinguished solely by their masses. In the interaction basis, thefermion fields ψ have the following representations:

Left-handed quarks: QLi =

(ur ug ubdr dg db

)i

Right-handed u-type quarks: uRi =(ur ug ub

)i

Right-handed d-type quarks: dRi =(dr dg db

)i

(2.1)

Left-handed leptons: LLi =

(νee

)i

Right-handed leptons: lRi =(e)i

where i = 1, 2, 3 denote the fermion generation and r, g, b denote the 3 colours.

Besides space-time symmetries defined by the Poincare group, the SM also respects the gaugesymmetry defined by SU(3)C ×SU(2)L×U(1)Y . This gauge symmetry determines the interac-tion and number of spin 1 gauge bosons. SU(3)C characterises the strong interaction with thesubscript C denoting colour and all quarks have 3 colours degree of freedom. The Lie algebrasu(3) has 8 generators La (8 associated gluons Ga) with a 3 dimensional unitary representationin terms of the Gell-Mann matrices λa:

La =λa2

(a = 1, ..., 8). (2.2)

SU(2)L characterises the weak interactions with the subscript L denoting left-handed chirality.This means that the SU(2) transformations only act non-trivially on left-handed chiral compo-nents while right-handed chiral components transform as singlets. The Lie algebra su(2) has 3generators Ta (3 associated gauge bosons W a) with a 2 dimensional unitary representation interms of the Pauli matrices σa:

Ta =σa2

(a = 1, 2, 3). (2.3)

U(1)Y characterises the hypercharge interaction. The Lie algebra u(1) has 1 generator Y (1associated gauge boson B) which is related to the third component of the weak isopsin generatorby the Gell-Mann-Nishijima equation 4:

Q = Y + T3. (2.4)

This fixes the action of Y on the fermions and also implies electroweak unification. Associatedwith each gauge group is a coupling constant (a free parameter in the SM) which is denoted asgS , gW and gY corresponding to strong, weak and hypercharge interactions respectively.

3 Λµν is a Lorentz transformation satisfying the relation ηµνΛρµΛσν = gρσ where gµν ≡ diag(1,−1,−1,−1) is theMinkowski space-time metric. aµ is a space-time translation. The Poincare group is the group of isometries ofMinkowski space-time.

4 Using here the convention as given in figure 2.1. Another convention is Q = Y/2 + T3.


One problem with gauge theories is that gauge fields cannot admit a mass term as this breaks thegauge symmetry. In addition, since the left-handed and right-handed components of a fermionfield transforms differently under gauge transformations, mass terms ∝ ψψ = ψLψR + ψRψL

are also forbidden by gauge symmetry. This is in contradiction to the fact that massive weakgauge bosons and fermions exist. In the SM, the solution to the massive gauge bosons problemis known as the Higgs mechanism [18, 25] by introducing the Higgs field which is a scalar weakisospin doublet. Spontaneous symmetry breaking then results in a massive Higgs boson and 3massive weak bosons. In the case of fermions, they acquire mass through the Yukawa couplingsof the Higgs and the fermion fields.

2.1 Standard Model Lagrangian

The Lagrangian (density) of the SM can be formulated thus 5 [36]:

LSM = Lfermion + Lgauge + LHiggs + LYukawa + Lgauge fixing (2.5)

The fermionic portion of the Lagrangian is summed overall representations of ψ:

Lfermion = iψγµDµψ (2.6)

where ψ ≡ ψ†γ0 is the Lorentz invariant Dirac adjoint, γµ are Dirac matrices satisfying theClifford algebra γµγν + γνγµ = 2gµν and Dµ is a gauge covariant derivative:

Dµ = ∂µ + igSGaµLa + igWW

aµTa + igYBµY. (2.7)

The gauge portion of the Lagrangian is:

Lgauge = −1

4GµνG

µν − 1

4WµνW

µν − 1

4BµνB

µν (2.8)

where the bosonic tensors are given by:

Gaµν = ∂µGaν − ∂νGaµ + 2igSTr ([La, Lb]Lc)G

bµG

cν a = 1, ..., 8

W aµν = ∂µW

aν − ∂νW a

µ + 2igWTr ([Ta, Tb]Tc)WbµW

cν a = 1, 2, 3 (2.9)

Bµν = ∂µBν − ∂νBµ.

The Higgs field is introduced as a scalar weak isospin doublet [38, 39]:

φ =

(ϕ+

ϕ0

)≡

(ϕ+

v+h+iϕZ√2

)(2.10)

with the conjugate field:

φ = iσ2φ∗ =

(v+h+−ϕz√

2

−ϕ−

). (2.11)

The ϕpm and ϕZ are the Goldstone components while h correspond to the Higgs component.

The Higgs Lagrangian is:

LHiggs = (Dµφ)†(Dµφ)− µ2φ†φ− λ(φ†φ)2 (2.12)

5 Ignoring the ghost terms that arise during quantization of the gauge fields.


which is invariant under SU(2)L × U(1)Y . For λ > 0 and µ2 < 0, the potential term µ2φ†φ −λ(φ†φ)2 has minima φ†φ = −µ2

2λ ≡v2

2 with the experimental value giving v ≈ 246 GeV. Among

all these minima, a specific choice of the vacuum is 1√2

(0v

)corresponds to the v defined in

equation 2.10 and Q = 0 or 0 electric charge. Imposing the unitary gauge [40] in gauge fixingsets all the Goldstone components to 0. This results in a new parametrization of the Higgs fieldas a perturbation h about the vacuum:(

φ+

φ0

)unitary gauge−−−−−−−−−−→

1√2

(0

v + h

). (2.13)

The SU(2)L × U(1)Y symmetry is spontaneously broken by this vacuum 6. Re-expressing theHiggs Lagrangian in terms of v and h with the SU(2)L × U(1)Y covariant derivative gives:

LHiggs =1

2∂µh∂

µh− λv2h2

+1

8v2

[g2W (W 1)2 + g2

W (W 2)2 +(W 3 B

)( g2W −gW gY

−gW gY g2Y

)(W 3

B

)](2.14)

+ interaction terms.

The Higgs boson corresponds to the field h with a mass mh =√

2λv2 ≈ 125.9 GeV [1, 14]. Asgiven in figure 2.1, the mass or physical basis of the electroweak gauge bosons are related to theinteraction basis by:

W± =1√2

(W 1 ∓ iW 2)

Z = cos θWW3 − sin θWB (2.15)

γ = sin θWW3 + cos θWB

where θW is the weak mixing angle satisfying sin θW = gY√g2W+g2Y

and cos θW = gW√g2W+g2Y

. Z

and γ are eigenvectors of the matrix

(g2W −gW gY

−gW gY g2Y

)with eigenvalues g2

W + g2Y and 0.

Expressed in this basis, the second line of equation 2.13 becomes:

1

8v2[g2W (W+)2 + g2

W (W−)2 + (g2W + g2

Y )Z2]. (2.16)

This corresponds to the appearance of mass terms for for the W± and Z bosons with mW± =12vgW and mZ = 1

2v√g2W + g2

Y which are experimentally determined to be 80.4 GeV and 91.2

GeV respectively.

The Yukawa portion of the Lagrangian will be addressed in section 2.4 while the gauge fixingterms will be addressed in section 2.5.

6 U(1)EM with the generator Q is a symmetry of this vacuum, consistent with the fact that the photons aremassless.


2.2 Electroweak interactions for quarks

CP violation has only been experimentally observed in weak interactions. In this section, Iderive the interactions between the quarks and physical gauge bosons starting from equation2.6. Noting that SU(2)L only acts non-trivially on the left-handed quarks and upon substitutionof the hypercharge values given in figure 2.1, this interaction Lagrangian is:

Lquarks-electroweak = −3∑i=1

QLi γµ

(gWW

aµTa + gYBµ

1

6

)QLi

−3∑i=1

uRi γµ

(gYBµ

2

3

)uRi +

3∑i=1

dRi γµ

(gYBµ

1

3

)dRi (2.17)

= −3∑i=1

1

2

(uLi dLi

)γµ(gWW

3µ + 1

3gYBµ gW (W 1µ − iW 2

µ)

gW (W 1µ + iW 2

µ) −gWW 3µ + 1

3gYBµdL

)(uLidLi

)

− 2gY3

3∑i=1

uRi γµBµu

Ri +

gY3

3∑i=1

dRi γµBµd

Ri (2.18)

In the physical basis of the electroweak gauge bosons given in equation 2.15, the Lagrangianbecomes:

Lquarks-electroweak = −3∑i=1

gW√2

[uLi γ

µW+µ d

Li + dLi γ

µW−µ uLi

](2.19)

−3∑i=1

1

2√g2W + g2

Y

[(g2W −

2g2Y

3

)uiγ

µZµui +

(g2W +

g2Y

3

)diγ

µZµdi

](2.20)

−3∑i=1

gW gY√g2W + g2

Y

[2

3uiγ

µAµui +1

3diγ

µAµdi

](2.21)

where u ≡ uL ⊕ uR, d ≡ dL ⊕ dR and in the last line there is a notation change γ → A for thephoton to avoid confusion with the Dirac matrices.

In the standard literature terminology, equations 2.19, 2.20, 2.21 describes the charged current,neutral current and electromagnetic current interactions of the quarks respectively. Flavourchanging interactions can only occur through the charged current interactions at the tree-level.In the SM, flavour changing neutral currents can only occur at loop levels. These loop diagramsare generally highly suppressed due to the Glashow-Iliopoulos-Maiani (GIM) mechanism [23].Figure 2.2 shows the Feynman diagrams for the charged current interactions corresponding toequation 2.19.

Fig. 2.2: Charged current interactions for quarks

Left: A d-type quark changes to an u-type quark, emitting a W+ bosonRight: An u-type quark changes to a d-type quark, emitting a W− boson


2.3 Transformations of Fields Under C and P Operators

Table 2.1 summarises how the scalar, Dirac and vector fields transform under C, P and theircombined operation. η denotes the eigenvalues (±1) of the P operator for a field.

Field P C CP

φ(x0, ~x) η(φ)φ(x0,−~x) φ†(x0, ~x) η(φ)φ†(x0,−~x)

ψ(x0, ~x) γ0ψ(x0,−~x) iγ2ψ∗(x0, ~x) iγ2γ0ψ∗(x0,−~x)

ψ(x0, ~x) ψ(x0,−~x)γ0 iψT (x0, ~x)γ2γ0 iγ2γ0ψ∗(x0,−~x)

Aµ(x0, ~x) η(Aµ)Aµ(x0,−~x) A†µ(x0, ~x) η(Aµ)A†µ(x0,−~x)

Tab. 2.1: Transformations of Different Fields under CP

The Lagrangian of the quark electroweak interactions in equations 2.19, 2.20 and 2.21 transformsunder CP as:

Lquarks-electroweak CP−−→ −3∑i=1

gW√2

[dLi γ

µW−µ uLi + uLi γ

µW+µ d

Li

]−

3∑i=1

1

2√g2W + g2

Y

[(g2W −

2g2Y

3

)uiγ

µZµui +

(g2W +

g2Y

3

)diγ

µZµdi

](2.22)

−3∑i=1

gW gY√g2W + g2

Y

[2

3uiγ

µAµui +1

3diγ

µAµdi

]

evaluated at (x0,−~x). Note that in this form, the CP transformation is a symmetry of theinteraction. In section 2.4, we will see how this symmetry is broken by Yukawa couplings.

2.4 Yukawa Couplings and CP Violation

Yukawa couplings is the term in the SM Lagrangian responsible for generating the mass termsfor the fermions as well as particle oscillations due to its non-diagonal feature 7. In the notationsof equation 2.1, the Yukawa Lagrangian is:

LYukawa =

3∑i,j=1

YijψLi φψRj + h.c.

=

3∑i,j=1

[(Yd)ijQ

Li φd

Rj + (Yu)ijQLi φu

Rj + (Yl)ijL

Li φl

Rj + h.c.

]. (2.23)

Therefore Yukawa couplings as expressed in equation 2.23 describes the coupling of the Higgsfields to the left-handed and right-handed fermion fields via the matrices (Yd)ij , (Yu)ij , (Yl)ij .These matrix elements have to be determined experimentally and form the bulk of the freeparameters in the SM.

The discussion will be restricted to quarks from here. Once the Higgs field acquires a vacuumexpectation value as given in equation 2.13, the Yukawa Lagrangian for the quarks acquires the

7 The case for neutrinos is not discussed in this section.


following form:

LYukawa;quarks =3∑

i,j=1

[(Yd)ijd

Li

v√2dRj + (Yu)ijuLi

v√2uRj + h.c.

]+ interaction terms

≡ −3∑

i,j=1

[(Md)ijd

Li d

Rj + (Mu)ijuLi u

Rj + h.c.

]+ interaction terms. (2.24)

The Higgs vacuum gives rise to mass matrices for the quarks:

(Md)ij ≡ −v√2

(Yd)ij

(Mu)ij ≡ −v√2

(Yu)ij . (2.25)

Since Yukawa couplings are arbitrary, these mass matrices can be complex. Such complexYukawa couplings which cannot be removed by global phase transformations were in fact iden-tified by Kobayashi and Maskawa in their seminal paper [29] to break CP symmetry in thecharged current interactions and therefore constitutes the origin of CP violation in the SM.

The singular value decomposition theorem allows the mass matrices Md and Mu to be diago-nalized with two unitary matrices each:

ULdMdUR†d = Mdiag

d =

md 0 00 ms 00 0 mb

ULuMuU

R†u = Mdiag

u =

mu 0 00 mc 00 0 mt

. (2.26)

The mass or physical basis of the quarks are related to the interaction basis by:

(dmass)Li = (ULd )ijd

Lj ; (dmass)

Ri = (URd )ijd

Rj

(umass)Li = (ULu )iju

Lj ; (umass)

Ri = (URu )iju

Rj . (2.27)

The charged current (cc) interactions Lagrangian in equation 2.19 re-expressed in the physicalbasis of the quarks becomes:

Lquarks-cc = −3∑

i,j,k=1

gW√2

(umass)Li (ULu )ikγµW+

µ (ULd )∗kj(dmass)Lj

−3∑

i,j,k=1

gW√2

(dmass)Li (ULd )ikγµW−µ (ULu )∗kj(umass)

Lj

≡ −3∑i,j

gW√2

(umass)Li (VCKM)ijγµW+

µ (dmass)Lj (2.28)

−3∑

i,j=1

gW√2

(dmass)Li (VCKM)∗jiγµW−µ (umass)

Lj

by introducing the CKM matrix

VCKM ≡ (ULu )(ULd )†. (2.29)


Under a CP transformation, the Lagrangian in equation 2.28 becomes:

Lquarks-cc CP−−→ −3∑i,j

gW√2

(dmass)Lj (VCKM)ijγµW−µ (dmass)

Li (2.30)

−3∑

i,j=1

gW√2

(umass)Lj (VCKM)∗jiγµW+

µ (dmass)Li

which is different from the original if VCKM is complex. The complex nature of VCKM is necessarybut not sufficient for CP violation in the SM. The necessary and sufficient conditions will bediscussed in section 3.2.

On the other hand, the Lagrangian for the weak neutral current (nc) and electromagnetic currentinteractions in equations 2.20 and 2.21 are invariant under the change of basis:

Lquarks-nc = −3∑i=1

(g2W −

2g2Y3

)(umass)iγ

µZµ(umass)i +(g2W +

g2Y3

)(dmass)iγ

µZµ(dmass)i

2√g2W + g2

Y

(2.31)

Lquarks-em =

3∑i=1

gW gY√g2W + g2

Y

[2

3(umass)iγ

µAµ(umass)i +1

3(dmass)iγ

µAµ(dmass)i

]. (2.32)

This explains why there are no flavour changing neutral currents at tree levels in the SM.

2.5 Feynman Rules for Charged Current Interactions

At this point, it is useful to introduce the Feynman rules for the charged current interactions. Ingeneral, any particle interaction can be represented pictorially with a Feynman diagram and theFeynman rules assign the different elements a mathematical expression. In this way, the Feynmanrules translate a Feynman diagram to a corresponding amplitude for the interaction, makingFeynman diagrams perhaps the most powerful computational tool and intuitively appealingrepresentation in particle physics.

Gauge fixing is necessary to remove redundant degrees of freedom and to obtain a well-definedpath integral and propagators. In section 2.1, the choice of the unitary gauge removed theGoldstone bosons. However, the unitary gauge is not convenient for calculations of loops inFeynman diagrams due to difficulties in renormalising. A more general class known as the Rξgauges will be considered. Expressing the kinetic part of the Higgs field Lagrangian with thephysical W±, Z and γ bosons gives:

(Dµφ)†(Dµφ) = ∂µϕ−∂µϕ+ 1

2∂µϕ

Z∂µϕZ +1

2∂µh∂

µh+m2WW

−W+ +1

2m2ZZ

2

− imW (W−µ ∂µϕ+ −W+

µ ∂µϕ−)−mZZµ∂

µϕZ (2.33)

+ higher interaction terms.

In gauge fixing, the quadratic interaction terms arising in the second line of equation 2.33 areto be eliminated. Concentrating on the W± and ϕ± terms, the RξW gauge fixing term is [35]:

− 1

ξW(∂µW+

µ − iξWmWϕ+)(∂νW−ν + iξWmWϕ

−) (2.34)

= − 1

ξW(∂µW+

µ ∂νW−ν ) + imW (ϕ+∂νW−ν − ϕ−∂µW+

µ )− ξWm2Wϕ

+ϕ−.


The unitary gauge corresponds to ξW → ∞. The calculations in this thesis will adopt theFeynman gauge, corresponding to the case RξW = 1.

The following Feynman rules are given in [36].

The vertices for the charged current interactions is given in figures 2.3. The same rules apply

for leptons with the CKM factor removed. The factor 1−γ52 is the left-handed chiral projec-

tor so that for a fermion ψ, 1−γ52 ψ = ψL. Correspondingly, the right-handed chiral projector

is given by 1+γ5

2 , so that 1+γ5

2 ψ = ψR. γ5 is defined by iγ0γ1γ2γ3. This type of vertex isknown as the V-A vertex since the corresponding term in the Lagrangian involves the current

ψLγµψL = ψγµ(1 − γ5)ψ = ψγµψ − ψγµγ5ψ and the bilinear forms ψγµψ and ψγµγ5ψ trans-

form as vectors and axial vectors respectively. Historically, this V-A theory of charged currentinteraction accounted for the maximal P violation in weak interactions and finally developedinto the electroweak interaction theory [22, 38, 39] which replaced Fermi’s theory of beta decaywhich is not renormalisable.

Fig. 2.3: V-A vertex for quark charged current interactions

The vertex for the Goldstone bosons ϕ± is given in figure 2.4

Fig. 2.4: Vertices for Goldstone bosons


The Feynman rules for the propagators in the Feynman gauge are given in figures 2.5, 2.6 and 2.7for the W bosons, Goldstone bosons and Dirac fermions. The iε prescription in the denominatoris understood in the sense of distributions with ε→ 0 and hereafter will be dropped for notationalsimplicity.

Fig. 2.5: Propagator for W bosonsFig. 2.6: Propagator for Goldstone bosons

Fig. 2.7: Propagators for Dirac fermions

The Feynman rules for the external fermions are given in figure 2.8. u and v are associated tothe fermion and anti-fermion respectively.

Fig. 2.8: External Dirac fermions

Finally, the amplitude of an interaction Feynman diagram is obtained by writing down all theelements and integrating over the momenta of internal propagators.

3 The CKM Matrix

In this chapter, important features of the CKM matrix are studied.

From its definition given in equation 2.29: VCKM ≡ (ULu )(ULd )†, the CKM matrix is an unitarymatrix.

A result of prime importance is the number of free physical parameters in the CKM matrix. Fora general N ×N unitary matrix there are (N − 1)2 free physical parameters consisting of [7]:{

N(N−1)2 mixing angles

(N−1)(N−2)2 physical phases

. (3.1)

A physical complex phase can only arise for N ≥ 3. This in fact prompted Kobayashi andMaskawa to postulate the existence of at least 3 generations of fermions in 1973 when evenexperimental proof for the charm quark in the second generation had not been found.

I will present my derivation of the result here. A N × N unitary matrix has N2 independentreal parameters. This is because a general N × N matrix over C has 2N2 independent realparameters and the unitarity condition

∑Nj=1 UijU

∗jk = δik poses N2 constraints since (i, k) ∈

{1, ..., N} × {1, ..., N}. A mixing angle mixes any two generations and hence the number of

mixing angles for N generations is given by

(N2

)= N(N−1)

2 .

This leave a total of N(N+1)2 phases, however some can be removed by global phase transforma-

tions. There are a total of 2N global phase transformations corresponding to all quarks:

(umass)i → eiφui (umass)i

(dmass)i → eiφdi (dmass)i. (3.2)

Under the global phase transformations, the (cc) interactions Lagrangian in equation 2.28 be-comes:

Lquarks-cc = −N∑i,j

gW√2

(umass)Li e−iφui (VCKM)ije

iφdj γµW+µ (dmass)

Lj (3.3)

−N∑

i,j=1

gW√2

(dmass)Li e−iφdi (VCKM)∗jie

iφuj γµW−µ (umass)Lj .

A common re-phasing of all quark fields leaves the Lagrangian invariant. Therefore the numberof independent global phase transformations is 2N-1. To the N(N+1)

2 phases, these 2N−1 globalphase transformations have to be subtracted since they are unphysical. This leaves a total ofN(N−1)

2 physical phases as given in equation 3.1.

3. The CKM Matrix 14

3.1 Parametrisations of the CKM Matrix

The CKM matrix elements can be represented in the physical basis of the u-type and d-typequarks as:

VCKM =

Vud Vus VubVcd Vcs VcbVtd Vts Vtb

. (3.4)

The general convention is that (umass)L = uL, therefore: dmass

smass

bmass

=

Vud Vus VubVcd Vcs VcbVtd Vts Vtb

dsb

.

Four independent parameters are required to parametrise the matrix.

3.1.1 Parametrisation with 3 Euler Angles and a Complex Phase

This is the standard parametrisation with 3 Euler angles θ12, θ23, θ13 and a phase δ13. Theshort-hand sij and cij will be used to denote sin θij and cos θij respectively. In parametrisingas such, effectively a convention is already adopted by enforcing certain matrix elements to bereal. The CKM matrix is written as [34]:

VCKM =

1 0 00 c23 s23

0 −s23 c23

c13 0 s13e−iδ13

0 1 0−s13e

iδ13 0 c13

c12 s12 0−s12 c12 0

0 0 1

=

c12s13 s12c13 s13e−iδ13

−s13c23 − c12s23s13eiδ13 c12c23 − s12s23s13e

iδ13 s23c13

s13s23 − c12c23s13eiδ13 −c12s23 − s12c23s13e

iδ13 c23c13

. (3.5)

The current experimental values are [13]:θ12 = 13.04± 0.05◦, θ23 = 2.38± 0.06◦, θ13 = 0.201± 0.011◦, δ13 = 1.20± 0.08 rad.

3.1.2 Wolfenstein Parametrisation

The Wolfenstein parametrisation [42] is an approximate parametrisation taking into accountthe fact that the experimentally determined mixing angles θ12 ≥ θ23 ≥ θ13. On defining 4independent parameters λ,A, ρ, η as

sin θ12 = λ

sin θ23 = Aλ2

e−iδ13 sin θ13 = Aλ3(ρ− iη), (3.6)

the parametrisation is a series expansion about the parameter λ. To O(λ6), the CKM matrix is:

VCKM =

1− 12λ

2 − 18λ

4 λ Aλ3(ρ− iη)−λ+ 1

2A5λ5[1− 2(ρ+ iη)] 1− 1

2λ2 − 1

8λ4(1− 4A2) Aλ2

Aλ3[1− ρ

(1− 1

2λ2)− iη

(1− 1

2λ2)]−Aλ2 +Aλ4

(12 − ρ− iη

)1− 1

2A2λ4

+ O(λ6). (3.7)


The O(λ4) precision is required for the study of B0s systems since the relevant term in Vts =

−Aλ2 +Aλ4(

12 − ρ− iη

)has the phase appearing at O(λ4).

The current experimental values are [13]:λ = 0.2257+0.0009

−0.0010, A = 0.81+0.021−0.022, ρ = 0.135+0.031

−0.016, η = 0.349+0.015−0.017.

The Wolfenstein parametrisation highlights the hierarchical structure of the CKM matrix 1:

|VCKM| ∼

1 λ λ3

λ 1 λ2

λ3 λ2 1

. (3.8)

In the literature, each order of λ is commonly referred to as the degree of Cabibbo suppression.

3.2 Conditions for CP Violation and the Jarlskog Invariant

The necessary and sufficient conditions for CP violation is given in [28] as:

1. Both u-type and d-type quarks across generations must not be degenerate in mass.

2. All mixing angles 6= 0 or π/2.

3. Physical phase 6= 0 or π.

These conditions can be more elegantly expressed using the mass matrices Md and Mu definedin equation 2.25 as [27]:

detC ≡ det(−i[MuM

†u,MdM

†d

])6= 0. (3.9)

With equation 2.26, this can be expressed as:

detC = det(−i[(Mdiag

u )2, VCKM(Mdiagu )2

]V †CKM

)= −2J(m2

t −m2c)(m

2c −m2

u)(m2u −m2

t )(m2b −m2

s)(m2s −m2

d)(m2d −m2

b). (3.10)

J is known as the Jarlskog invariant and can be obtained by removing an arbitrary row andcolumn of the VCKM matrix and J is the imaginary part of the product of the diagonal elementsand complex conjugate of the off-diagonal elements of the 2×2 matrix that remains. Thus thereare 9 possible representations of J :

J ≡ Im{VuidiVujdjV∗uidj

V ∗ujdi}, (ui < uj , di < dj) (3.11)

These representations are all equivalent due to the unitarity of VCKM.

In the standard parametrisation, one representation of J gives:

J = c12c213c23s12s13s23 sin δ13. (3.12)

In the Wolfenstein parametrisation, J can be given as:

J = A2λ6η. (3.13)

J gives a measure of the strength of CP violation in the SM and the experimental value givesJ = 2.96+0.20

−0.16 × 105. This is very small compared to the theoretical maximum 16√

3that can be

calculated from equation 3.12. This is a consequence of the strong hierarchy in quark mixing.In the next section, the geometrical significance of J becomes apparent.

1 This quark charged current coupling hierarchy in the CKM matrix and the quark mass hierarchy problemhave an apparent similarity. This raises the question of the connection between the two, perhaps an answer liesin their common origins in the Yukawa couplings.


3.3 Unitarity Triangles

The unitarity of the CKM matrix gives 9 constraints on the matrix elements:

N∑j=1

(VCKM)ij(VCKM)∗jk = δik. (3.14)

Experimental efforts are required to scrutinise the validity of these unitarity constraints whicharise from local gauge invariance in the SM. Therefore deviations from unitarity will be signaturesof New Physics.

Explicitly, the i 6= k cases poses 6 orthogonal constraints:

V ∗udVcd + V ∗usVcs + V ∗ubVcb = 0 (λ, λ, λ5) (3.15)

V ∗udVtd + V ∗usVts + V ∗ubVtb = 0 (λ3, λ3, λ3) (3.16)

V ∗cdVtd + V ∗csVts + V ∗cbVtb = 0 (λ4, λ2, λ2) (3.17)

VudV∗us + VcdV

∗cs + VtdV

∗ts = 0 (λ, λ, λ5) (3.18)

VudV∗ub + VcdV

∗cb + VtdV

∗tb = 0 (λ3, λ3, λ3) (3.19)

VusV∗ub + VcsV

∗cb + VtsV

∗tb = 0 (λ4, λ2, λ2) (3.20)

where the parentheses are the order of magnitude of the corresponding (VCKM)ij(VCKM)∗jk term.These six orthogonality constraints can be represented geometrically as triangles in the complexplane, known as the unitarity triangles. All the unitarity triangles have an equal area given byArea(∆) = 1/2J where J is the Jarlskog invariant, hence conferring J a geometrical significance.

Of the 6 different unitarity triangles, the db triangle given by equation 3.19 is relevant for B0d

mesons and has been studied most extensively in the literature. It is illustrated in figure 3.1with the Wolfenstein parametrisation to O(λ2) precision.

Fig. 3.1: The db unitarity triangle relevant to B0d meson decays

The base is normalised to 1 on division by VcsV∗cb


The angles α, β and γ in the db unitarity triangle are defined to be:

α ≡ arg

[−VtdV

∗tb

VudV∗ub

](3.21)

β ≡ arg

[−VcdV

∗cb

VtdV∗tb

](3.22)

γ ≡ arg

[−VtsV

∗tb

VcsV ∗cb

]. (3.23)

It is clear that these definitions match well with the illustrated angles in figure 3.1. It is impor-tant to note that the definitions are also convention independent, since phase transformationscorrespond to rotations in the complex plane and will not affect the angles.

For this project, the sb unitarity triangle defined by equation 3.20 is crucial for the analysisof the B0

s system. In contrast to the bd unitarity triangle, the 3 sides have different order ofmagnitudes, resulting in a highly squashed triangle as illustrated in figure 3.2.

Fig. 3.2: The bs unitarity triangle relevant to Bs meson decays

The base is normalised to 1 on division by VcsV∗cb

The relevant angle βs is defined as:

βs ≡ arg

[−VtsV

∗tb

VcsV ∗cb

]. (3.24)

One more angle βK arises from the ds unitarity triangle given by equation 3.18:

βK ≡ arg

[−VcsV

∗cd

VusV ∗ud

]. (3.25)

The phase invariant angles γ, β, βs and βK correspond to the 4 CP violating terms in the CKMmatrix and can be expressed with the Wolfenstein parameters as:

γ = arg [−ρ+ iη)] ≈ η (3.26)

β = arg [−1 + ρ+ iη)] ≈ −η (3.27)

βs = arg[1− λ2(1/2− ρ− iη) +O(λ4)

]≈ λ2η (3.28)

βK = arg[1−A2λ4(1/2− ρ− iη) +O(λ6)

]≈ λ4η. (3.29)

The CKM matrix can thus be expressed with these angles as:

VCKM =

|Vud| |Vus| |Vub|e−iγ−|Vcd|eiβK |Vcs| |Vcb||Vtd|e−iβ −|Vts|eiβs |Vtb|

+O(λ6). (3.30)


3.4 Experimental Determination of the CKM Free Parameters

The magnitudes of a matrix element Vij can be determined by comparing the decay rate of aprocess involving the Vij vertex to the decay rate of muons. To illustrate this, the determinationof Vud is by measuring the β decay rate. The relevant tree level Feynman diagrams are given infigure 3.3.

Fig. 3.3: Feynman diagrams for determination of Vud by comparing neutron β and µ− decays

The vertex factor for muon decays is 1 by convention.

The expressions for the decay amplitudesM can be written with Feynman rules given in section2.5. In the low energy limit k2 � m2

W , the W boson propagator simplifies as:

−i gµν

k2 −m2W

−−−−−→k2�m2

W

igµν

m2W

. (3.31)

The above approximation essentially reduces the problem to Fermi’s theory of weak decay with4 point vertices. The amplitude in the tree-level decay diagram is:

Md→ue−νe =g2W

8M2W

Vud[uuγd(gV − gAγ5)ud][ue−γµ(1− γ5)vνe ], (3.32)

Mµ−→e−νµνe =g2W

8M2W

[uνµγµ(1− γ5)uµ− ][ue−γµ(1− γ5)vνe ]. (3.33)

The presence of spectator quarks in the neutron undergoing β decay changes the vector andaxial vector coupling strength and introduces additional coefficients gV and gA. Consequently,the neutron beta decay rate is difficult to calculate theoretically. The differential decay rate canbe calculated from the decay amplitudes using Fermi’s golden rule. For the muon decay:

dΓµ−→e−νµνe =< |M|2 >

2mµ

d3 ~Pe−

(2π)3Ee−

d3 ~Pνµ(2π)3Eνµ

d3 ~Pνe(2π)3Eνe

(2π)4δ4(Pµ− − Pe− − Pνµ − Pνe) (3.34)

where < |M|2 can be calculated in the rest frame of the muon:

< |M|2 = 2g4W

M4W

(k1k3)(k2k4) =g4W

M4W

m2µ−E3(mµ− − 2E3). (3.35)

On the other hand, the CP violating phase is more difficult to determine. Since CP violation aredue to complex phases which cannot be observed for a single states, they have to be searched ininterference phenomena. The main strategy is to look for asymmetries between meson decaysand the CP conjugated process. The formalism for analysing meson oscillation and decays will bedeveloped in Chapter 4 to obtain expressions for the asymmetries and an in-depth investigationof the extraction of the CP violating phase βs will be done specifically in Chapter 5 for the B0

s

system.


Current experimental results do already allow for the determination for all 4 free parameters inthe CKM matrix. Different results are also consistent with the unitarity of the CKM matrixwithin experimental errors as illustrated by recent results.

Fig. 3.4: The ideal results for the db unitarity triangle [10]

Measurements of CP violation in the B0d , K0 and K+ systems converges to the

(ρ, η) = (ρ(1− λ2/2), η(1− λ2/2)) vertex for consistency with unitarity constraints

Fig. 3.5: The global fit of the CKM matrix performed by the CKM fitter group [13]

Continuous experimental effort is however important to improve the precision and the B-physicsexperiments offer multiple possibilities to measure different unitarity triangles, thus obtainingfurther overconstrain. Possible deviations from the SM and New Physics may arise in the formof disagreement of angles or sides of the unitarity triangles. The current LHCb detector couldattain precisions necessary to measure the βs angle for the Bs

0 system. The small magnitude ofβs makes it particularly sensitive to the search for New Physics.

4 CP Violation in Meson Decays

“ If I could remember the names of all these particles, I’d be a botanist. ”Enrico Fermi, as quoted in Hyperspace by Michio Kaku

All current experimental proof of CP violation are observed in meson decays. A meson iscomposed of one quark and one anti-quark and all mesons decay via weak interactions. Inthis chapter, the Hamiltonian formalism of meson decays will be reviewed. The main source ofreference for this chapter is [34]. This will proceed in three stages:

1. The oscillation in neutral flavoured mesons is first considered. In fact, most experimentalobservations of CP violation have been limited to the decay of such systems with the onlyexceptions coming in recent years through the B± mesons. Furthermore, the B0

s systemwhich will be the focus of Chapter 5 is an example of a neutral meson system.

2. The decay of meson systems will then then be developed to obtain the master equationsdescribing the decay.

3. Finally a review of how CP violation may arise in the different mesons and the classifica-tions of different types of CP violation is made.

4.1 Oscillation in Neutral Flavoured Mesons

A neutral flavoured meson consists of a pair of quark and antiquark of the same type (u or d)but across different generations. CP violation has been experimentally observed for all membersof this class of mesons, including: K0(ds) & K0(ds), D0(uc) & D0(uc), B0

d(db) & B0d(db), B0

s (sb)

& B0s (sb) 1.

A neutral flavoured meson and its antiparticle are distinguished only by flavour quantum num-bers and a natural description of such a meson and its antiparticle is the flavour eigenstateswhich will be denoted |M0〉 & |M0〉 with CP |M0〉 = −|M0〉. Here, M represents a generic me-son and the analysis which follows holds for the any of K, D, B or Bs systems. The Hamiltonianfor the system can be given as:

H = HS +HEM +HW = H0 +HW (4.1)

where the subscripts S,EM,W correspond to the strong, electromagnetic and weak interactionsrespectively. The flavour eigenstate is an eigenstate of the strong and electromagnetic interactiondescribed by the HamiltonianH0. However, as was shown in Chapter 2, the weak charged currentinteractions change quark flavours and hence the flavour eigenstate is not an eigenstate of theweak interaction. The weak interaction Hamiltonian HW will lead to |M0〉 ↔ |M0〉 oscillationsas well as decays.

1 The T 0(ut) & T 0(ut) and T 0c (ct) & T 0

c (ct) have not yet been observed.

4. CP Violation in Meson Decays 21

The most general state under the evolution of such a Hamiltonian assumes the form:

|ψ(t)〉 = a(t)|M0〉+ b(t)|M0〉+∑f

cf (t)|f〉 (4.2)

where |f〉 are all the possible intermediate decay states of |M0〉 & |M0〉 and t is the proper timemeasured in the rest frame of the |M0〉 ↔ |M0〉 system. The presence of decay terms complicatesthe problem. A simplification can be effectuated under the Weisskopf-Wigner approximation[41] which consists of:

• looking at time scales � strong interaction or QCD time scales,

• the weak interaction has negligible effects on the decay states: 〈f |HW |f〉 ≈ 0,

• the initial state involves only linear combinations of |M0〉 & |M0〉.

Under this approximation, the evolution of the entire system is simplified to the study of theevolution in the |M0〉 & |M0〉 subspace described by an effective Hamiltonian consisting of ananti-Hermitian portion that quantifies the decay. The effective Hamiltonian has the form [34]:

Heff = M− i

2Γ (4.3)

where both M (mass) and Γ (decay) are Hermitian. The specific elements of the effectiveHamiltonian will be calculated in Chapter 5.

In the |M0〉, |M0〉 basis, the effective Hamiltonian for mixing has the representation:

Heff =

(M11 − i

2Γ11 M12 − i2Γ12

M∗12 − i2Γ∗12 M22 − i

2Γ22

). (4.4)

To supplement the discussion in [34], the matrix elements can be expressed in perturbationtheory, treating the weak Hamiltonian HW as a perturbation to H0. To a second order pertur-bation:

M11 = E0 + 〈M0|HW |M0〉+∑f

∣∣〈f |HW |M0〉∣∣2

E0 − Ef(4.5)

M12 = 〈M0|HW |M0〉+∑f

〈M0|HW |f〉〈f |HW |M0〉E0 − Ef

(4.6)

M22 = E0 + 〈M0|HW |M0〉+∑f

∣∣∣〈f |HW |M0〉∣∣∣2

E0 − Ef(4.7)

Γ11 = 2π∑f

δ (E0 − Ef )∣∣〈f |HW |M0〉

∣∣2 (4.8)

Γ12 = 2π∑f

δ (E0 − Ef ) 〈M0|HW |f〉〈f |HW |M0〉 (4.9)

Γ22 = 2π∑f

δ (E0 − Ef )∣∣∣〈f |HW |M0〉

∣∣∣2 (4.10)

where E0 = 〈M0|H0|M0〉 = 〈M0|H0|M0〉 = 〈M0|H0|M0〉 and Ef = 〈f |H0|f〉 and the matrixelements of M is evaluated for the principal branch.


Expressed as above, the diagonal matrix elements correspond to flavour conserving interactionswhile the off-diagonal matrix elements correspond to flavour changing interactions.

The M12 or dispersive term quantifies theM0 ↔M0 oscillations due to the off-shell contributionsvia virtual states. The Γ12 or absorptive term contains a Dirac delta energy conservation termand quantifies the M0 ↔ M0 oscillations due to on-shell contributions via physical states |f〉.The principal contributions towards M12 and Γ12 can be represented by box Feynman diagramswhich is illustrated in figure 4.1 for the B0

s example.

Fig. 4.1: Box diagrams arising in the lowest order perturbation calculation of off-diagonal elements

Another simplification to the effective Hamiltonian can be made if Lorentz invariance is assumed.In a recent work, [24] has shown that Lorentz invariance implies CPT symmetry and the CPTtheorem states that particles and antiparticles have identical masses and mean lifetimes. Assuch the matrix elements can be simplified to: M11 = M22 ≡ M and Γ11 = Γ22 ≡ Γ.

The effective Hamiltonian is therefore:

Heff =

(M− i

2Γ M12 − i2Γ12

M∗12 − i2Γ∗12 M− i

2Γ

). (4.11)

Diagonalising this matrix gives the physical eigenstates. The eigenvalues are:

λ± = M− i

2Γ±

√(M12 −

i

2Γ12

)(M∗12 −

i

2Γ∗12

)

= M± Re

{√(M12 −

i

2Γ12

)(M∗12 −

i

2Γ∗12

)}(4.12)

− i

2

[Γ∓ 2Im

{√(M12 −

i

2Γ12

)(M∗12 −

i

2Γ∗12

)}],

λ+ ≡ λH ≡ MH −i

2ΓH, (4.13)

λ− ≡ λL ≡ ML −i

2ΓL. (4.14)

The subscripts H and L denote heavy and light respectively. It would also be useful to introducethe mass difference and decay constant difference:

∆M ≡ MH −ML = 2Re

{√(M12 −

i

2Γ12

)(M∗12 −

i

2Γ∗12

)}, (4.15)

∆Γ ≡ ΓH − ΓL = −4Im

{√(M12 −

i

2Γ12

)(M∗12 −

i

2Γ∗12

)}(4.16)


and the convention here is that ∆M > 0, justifying the use of H and L above. However the signof ∆Γ has to be determined in experiments by measuring the decay lifetimes of the heavy andlight eigenstates.

The corresponding physical eigenstates |MH〉 and |ML〉 are given by:

|MH〉 = p|M0〉 − q|M0〉, (4.17)

|ML〉 = p|M0〉+ q|M0〉, (4.18)

withp

q=

√M12 − i

2Γ12

M∗12 − i2Γ∗12

. (4.19)

Thus the time evolution of a state ψ(t) is:

|ψ(t)〉 = e−iλH t|MH(0)〉〈MH(0)|ψ(0)〉+ e−iλLt|ML(0)〉〈ML(0)|ψ(0)〉. (4.20)

In particular, for |ψ(0)〉 = |M0〉, on defining |M0physical(t)〉 ≡ |ψ(t)〉:

|M0physical(t)〉 = g+(t)|M0〉+

q

pg−(t)|M0〉, (4.21)

similarly for |ψ(0)〉 = |M0〉, on defining |M0physical(t)〉 ≡ |ψ(t)〉:

|M0physical(t)〉 =

p

qg−(t)|M0〉+ g+(t)|M0〉, (4.22)

where

g±(t) ≡ 1

2

[e−i(MH− i

2ΓH)t ± e−i(ML− i

2ΓL)t

]. (4.23)

4.2 Decays in Flavoured Mesons

Given an effective Hamiltonian (different from that governing neutral meson mixing), consideringthe decay of a flavoured meson |M〉 (charged or neutral) to a state |f〉, there are four associateddecay amplitudes on taking into account the CP conjugated antiparticle states |M〉 and |f〉:

AM→f ≡ 〈f |Heff|M〉 (4.24)

AM→f ≡ 〈f |Heff|M〉 (4.25)

AM→f ≡ 〈f |Heff|M〉 (4.26)

AM→f ≡ 〈f |Heff|M〉. (4.27)

There are two types of phases that may arise in the decay amplitudes [34]:

• The weak phase φ is due to contributions from the weak charged current interactionsduring decay. The CP transformation will reverse the sign of the phase, in other words,the weak phase for AM→f and its CP conjugate AM→f have opposite signs. The phasedifference constitutes a convention independent physical phase.

• The strong phase δ is associated with the scattering of intermediate on-shell states thatarise during decay. These scatterings are primarily due to strong interactions which con-serve CP.


The decay amplitudes can be calculated from the relevant Feynman diagrams. Such technicalitieswill be be treated specifically for the B0

s ↔ B0s system in Chapter 5 and 6.

Up to an overall constant that arises after from integration over the decay phase space, thedecay rates are given by the amplitudes modulo squared :

ΓM→f (t) = |〈f |Heff|M(t)〉|2 (4.28)

ΓM→f (t) =∣∣〈f |Heff|M(t)〉

∣∣2 (4.29)


∣∣2 (4.30)


∣∣2 . (4.31)

In the case of charged flavoured mesons, the decay probabilities are time independent and arejust given by the absolute square of the corresponding decay amplitudes. For the case for neutralflavoured mesons, it is necessary to take into account the oscillations described by equations 4.21and 4.22. The time dependent decay probability equations for the neutral flavoured mesons are:

ΓM0→f (t) =∣∣∣〈f |Heff|g+(t)M0〉+ 〈f |Heff| qpg−(t)M0〉

∣∣∣2 (4.32)

=∣∣AM0→f

∣∣2 |g+(t)|2 +

∣∣∣∣qpAM0→f

∣∣∣∣2 |g−(t)|2 + 2Re

{A∗M0→fg

∗+(t)

q

pAM0→fg−(t)

}ΓM0→f (t) =

∣∣∣〈f |Heff|g+(t)M0〉+ 〈f |Heff| qpg−(t)M0〉∣∣∣2 (4.33)

=∣∣∣AM0→f

∣∣∣2 |g+(t)|2 +

∣∣∣∣qpAM0→f

∣∣∣∣2 |g−(t)|2 + 2Re

{A∗M0→fg

∗+(t)

q

pAM0→fg−(t)

}ΓM0→f (t) =

∣∣∣〈f |Heff|pq g−(t)M0〉+ 〈f |Heff|g−(t)M0〉∣∣∣2 (4.34)

=∣∣∣AM0→f

∣∣∣2 |g+(t)|2 +∣∣AM0→f

∣∣2 ∣∣∣∣pq g−(t)

∣∣∣∣2 + 2Re

{p

qA∗M0→fg

∗−(t)A

M0→fg+(t)

}ΓM0→f (t) =

∣∣∣〈f |Heff|pq g−(t)M0〉+ 〈f |Heff|g−(t)M0〉∣∣∣2 (4.35)

=∣∣∣AM0→f

∣∣∣2 |g+(t)|2 +∣∣∣AM0→f

∣∣∣2 ∣∣∣∣pq g−(t)

∣∣∣∣2 + 2Re

{p

qA∗M0→fg

∗−(t)A

M0→fg+(t)

}.

In each of the above equations, the first term is associated with direct decay, the second termis associated with decay after a net oscillation to the CP conjugated antiparticle and the thirdterm is associated with interference between decay and oscillation. CP violating effects willmanifest in these terms as will be elaborated in the next section.

On defining Γ ≡ Γ1+Γ22 , the g+(t) and g−(t) terms can be expressed as:

|g±(t)|2 =e−Γt

2

(cosh

∆Γt

2± cos ∆mt

)(4.36)

g∗±(t)g∓(t) =e−Γt

2

(sinh

∆Γt

2∓ i sin ∆mt

). (4.37)


4.3 CP Violation Observables and Classification

For meson decays, the CP violation observable is an asymmetry in the decay rates of the meson|M〉 and its CP conjugated antiparticle |M〉. The CP asymmetry is defined as:

ACP (t) ≡ΓM→f (t)− ΓM→f (t)

ΓM→f (t) + ΓM→f (t). (4.38)

Following [34], such asymmetries and hence the CP violating effects can be broadly classifiedinto three types:

CP Violation in Decay: This type is defined by∣∣∣∣AM→fAM→f

∣∣∣∣ 6= 1. (4.39)

The interpretation is that the decay rates of the meson to a final state |M〉 → |f〉 is differentfrom the CP conjugated decay |M〉 → |f〉, that is:

ΓM→f 6= ΓM→f . (4.40)

This can arise when 2 or more amplitudes with different weak phases φ and strong phases δcontribute:

AM→f =∑i

Aieiδi+φi (4.41)

AM→f =∑i

Aieiδi−φi . (4.42)

The CP asymmetry is time independent:

ACP =

∣∣∣AM→fAM→f

∣∣∣2 − 1∣∣∣AM→fAM→f

∣∣∣2 + 1. (4.43)

Since charged mesons do not exhibit oscillations, this is the only type of CP violating effect thatmay be present for charged meson decays. An example is the decay B± → DK± [3].

CP Violation in Mixing: This type is defined by∣∣∣∣qp∣∣∣∣ 6= 1. (4.44)

The interpretation of this is that the probability of oscillation of a meson to its CP conjugateantimeson is different from the oscillation of the antimeson to meson. The CP asymmetry hasthe expression:

ACP =1−

∣∣∣ qp ∣∣∣41 +

∣∣∣ qp ∣∣∣4 . (4.45)

An example of this type of CP violation is the |K0〉 ↔ |K0〉 oscillation where CP violation wasfirst discovered.


CP violation in interference between a decay without mixing and a decay withmixing: The decay state |f〉 is a CP eigenstate:

|f〉 = CP |f〉 = ηCP (f)|f〉. (4.46)

The decay without mixing |M0〉 → |f〉 and decay with mixing |M0〉 → |M0〉 → |f〉 results intwo decay amplitudes that would result in an interference if:

Im

{q

p

AM0→f

AM0→f

}6= 0. (4.47)

This type of CP violation is important in the study of B0 and B0s decays.

On introducing the parameter

λf ≡q

p

AM0→f

AM0→f= ηCP (f)

q

p

AM0→f

AM0→f, (4.48)

equations 4.32 and 4.34 can be recast as:

ΓM0→f =e−Γt

2

∣∣AM0→f∣∣2 (4.49)[

(1 + |λf |2) cosh ∆Γt/2± 2Re{λf} sinh ∆Γt/2 + (1− |λf |2) cos ∆mt∓ 2Im{λf} sin ∆mt]

ΓM0→f =

e−Γt

2

∣∣AM0→f∣∣2 ∣∣∣∣pq

∣∣∣∣2 (4.50)[(1 + |λf |2) cosh ∆Γt/2± 2Re{λf} sinh ∆Γt/2− (1− |λf |2) cos ∆mt± 2Im{λf} sin ∆mt

]where the + sign is for CP even eigenstates |f〉 and the − sign is for CP odd eigenstates |f〉.Making the approximation that the CP violation in mixing is negligible, that is

∣∣∣ qp ∣∣∣ ≈ 1, the CP

asymmetry will have the form:

ACP (t) =2Im{λf} sin ∆mt− (1− |λf |2) cos ∆mt

2Re{λf} sinh ∆Γt/2 + (1 + |λf |2) cosh ∆Γt/2. (4.51)

5 Phenomenology in the Channel

B0s → J/ψφ

After the historical discovery of CP violation in kaon mixing, the importance of B-mesons forstudying CP violating phenomenon was highlighted by [12]. In their seminal paper, Carter andSanda identified the decay B0 → J/ψKs as the golden channel to measure the CP violating angleβ. The first experimental observation of CP violation outside the kaon systems was thereafterachieved in 2001 in the Babar [5] and Belle [4] experiments and allowed the determination ofβ. This completed the determination of the free parameters for the CKM matrix. The primaryfocus of current B-physics experiments is to determine the unitarity angles α, γ, β and βsthrough different modes of decays so as to overconstrain the unitarity triangle and to look forpossible discrepancies which signal New Physics.

Among the multiplicity of B-physics experiments, an important mode of decay is the B0s → J/ψφ

decay for the Bs system, the direct analogue of B0 → J/ψKs for the Bd system with a spectatorstrange quark replacing the spectator down quark.

In the previous chapter, the formalism of CP violation for mesons was developed with an effectiveHamiltonian. In the subsequent sections, the expressions for the matrix elements of the effectiveHamiltonian are derived in the framework of the operator product expansion. I start with acalculation the amplitude of the box diagrams for B0

s ↔ B0s . The B0

s → J/ψφ decay at the treelevel is analysed to illustrate the interference of the mixing and decay and how this interferencecan be used to extract the B0

s CP violating phase βs. Finally, a brief discussion is includedon the complications that arise both experimentally (such as the necessity of a time dependentangular analysis) and theoretically (penguin diagrams beyond the tree level whose branchingratios and amplitudes pose great theoretical difficulties, thus making it difficult to assess thedegree of penguin contributions for given decays).

5.1 Evaluating the Amplitude of Box Diagrams

To obtain an expression for M12 and Γ12 of the effective Hamiltonian describing the B0s ↔ B0

s

oscillation, it is necessary to calculate the amplitude M≡ 〈B0s |Heff|B0

s 〉. This actually consistsof both weak interactions and QCD corrections. The lowest order transition amplitudes aredescribed by the box diagrams 5.1. The result for the amplitude can be found in the literature[9, 26]. However, most calculation details are missing. Therefore in this section, I will providean extensive calculation of the amplitude employing the Feynman rules given in section 2.5.

There are four different types of diagrams involving intermediate quarks i,j as well as W andGoldstone bosons φ. For each type of diagram, there exists two possible topologies which arereferred to as the Mandelstam s and t channels in analogy to 4 particle scattering. In the 5.1, thediagrams on the left are s channel diagrams, where the intermediate quark j has 4-momentumk−k1−k2 while the diagrams on the right are t channel diagrams where the intermediate quarkj has 4-momentum k− k1− k3. The 4-momentum k2 and k3 are that of the strange quark in B0

s

and the antistrange quark in B0s . In the following calculations, the 4-momenta of the quarks can

be approximated by their rest energy, therefore k2 ≈ k3 � k1 by virtue of the mass hierarchyms � mb.


(a) 2 W propagators: s channel (b) 2 W propagators: t channel

(c) W and ϕ propagators: s channel (d) W and ϕ propagators: t channel

(e) ϕ and W propagators: s channel (f) ϕ and W propagators: t channel

(g) 2 ϕ propagators: s Channel (h) 2 ϕ propagators: t channel

Fig. 5.1: Box Diagrams for the B0s ↔ B0

s oscillations

Left: s channel diagram where the intermediate quark j has momentum k − k1 − k2

Right: t channel diagrams where the intermediate quark j has momentum k − k1 − k3


Concentrating on the s channel, the amplitudes for diagrams 5.1a, 5.1c, 5.1e and 5.1g are:

M(5.1a) = i∑

i,j=u,c,t

∫d4k

(2π)4

−igµν

(k − k1)2 −m2W

−igρσ

k2 −m2W

(5.1)

[b(k1)

−igW√2V ∗ibγµ

1− γ5

2iγαkα +mi

k2 −m2i

−igW√2Visγρ

1− γ5

2s(k3)

][b(k2)

−igW√2V ∗ibγσ

1− γ5

2iγβ(k − k1)β +mj

(k − k1)2 −m2j

−igW√2Visγν

1− γ5

2s(k4)

]

M(5.1c) = i∑

i,j=u,c,t

∫d4k

(2π)4

−igµν

(k − k1)2 −m2W

i

k2 −m2W

(5.2)

[b(k1)

−igW√2V ∗ibγµ

1− γ5

2iγαkα +mi

k2 −m2i

−igW√2Vis

mi

mW

1− γ5

2s(k3)

][b(k2)

−igW√2V ∗ib

mj

mW

1 + γ5


(k − k1)2 −m2j

−igW√2Visγν

1− γ5

2s(k4)

]

M(5.1e) = i∑

i,j=u,c,t

∫d4k

(2π)4

i

(k − k1)2 −m2W

−igρσ

k2 −m2W

(5.3)

[b(k1)

−igW√2V ∗ib

mi

mW

1 + γ5

2iγαkα +mi

k2 −m2i

−igW√2Visγ

ρ 1− γ5

2s(k3)

][b(k2)

−igW√2V ∗ibγ

σ 1− γ5


(k − k1)2 −m2j

−igW√2Vis

mj

mW

1− γ5

2s(k4)

]

M(5.1g) = i∑

i,j=u,c,t

∫d4k

(2π)4

i

(k − k1)2 −m2W

i

k2 −m2W

(5.4)

[b(k1)

−igW√2V ∗ib

mi

mW

1 + γ5

2iγαkα +mi

k2 −m2i

−igW√2Vis

mi

mW

1− γ5

2s(k3)

][b(k2)

−igW√2V ∗ib

mj

mW

1 + γ5


(k − k1)2 −m2j

−igW√2Vis

mj

mW

1− γ5

2s(k4)

].

Here γµ = gµνγν . Using the property that γ5γµ + γµγ5 = 0 and γ5γ5 = 1, summing up the

above and simplifying gives:

M(s) = i∑

i,j=u,c,t

g4W

256π4VisV

∗ibVjsV

∗jb (5.5)∫

d4k1[

(k − k1)2 −m2W

] [k2 −m2

W

] [k2 −m2

i

] [(k − k1)2 −m2

j

]{kα(k − k1)β

[b(k1)γµγαγρ(1− γ5)s(k3)

] [b(k2)γργ

βγµ(1− γ5)s(k4)]

−2m2im

2j

m2W

[b(k1)γα(1− γ5)s(k3)

] [b(k2)γβ(1− γ5)s(k4)

]+m2im

2j

m4W

kα(k − k1)β

[b(k1)γα(1− γ5)s(k3)

] [b(k2)γβ(1− γ5)s(k4)

]}.


Equation 5.5 contains two integrals:

I1(i, j) ≡∫d4k

1[(k − k1)2 −m2

W

] [k2 −m2

W

] [k2 −m2

i

] [(k − k1)2 −m2

j

] (5.6)

I2(i, j) ≡∫d4k

kα(k − k1)β[(k − k1)2 −m2

W

] [k2 −m2

W

] [k2 −m2

i

] [(k − k1)2 −m2

j

] . (5.7)

Both integrals I1 and I2 have no problems of ultra-violet divergence. To evaluate them, I firstre-expressed the denominator D with a partial fraction decomposition:

D =1[

(k − k1)2 −m2W

] [k2 −m2

W

] [k2 −m2

i

] [(k − k1)2 −m2

j

] (5.8)

=1

(m2W −m2

i )(m2W −m2

j )

{1

[k2 −m2W ][(k − k1)2 −m2

W ](5.9)

+1

[k2 −m2i ][(k − k1)2 −m2

j ](5.10)

− 1

[k2 −m2W ][(k − k1)2 −m2

j ](5.11)

− 1

[k2 −m2i ][(k − k1)2 −m2

W ]

}(5.12)

Note now that each individual term in the above decomposition results in a divergent integralbut the divergence will cancel each other. The standard technique of dimensional regularisationcan be used to calculate each integral.

The denominators can be combined with the Feynman parametrisation:

1

AB=

∫ 1

0dx

1

xA+ (1− x)B. (5.13)

This gives:

1

[k2 −m2W ][(k − k1)2 −m2

W ]=

∫ 1

0dx

1

k2 + 2k[k1(x− 1)]− [m2W + k2

1(x− 1)](5.14)

1

[k2 −m2i ][(k − k1)2 −m2

j ]=

∫ 1

0dx

1

k2 + 2k[k1(x− 1)]− [(m2i + k2

1)(x− 1) +m2jx]

(5.15)

1

[k2 −m2W ][(k − k1)2 −m2

j ]=

∫ 1

0dx

1

k2 + 2k[k1(x− 1)]− [(m2W + k2

1)(x− 1) +m2jx]

(5.16)

1

[k2 −m2i ][(k − k1)2 −m2

W ]=

∫ 1

0dx

1

k2 + 2k[k1(x− 1)]− [(m2i + k2

1)(x− 1) +m2Wx]

.(5.17)

In fact, the generalized Feynman parametrisation:

1

A1...An= (n− 1)!

∫ 1

0...

∫ 1

0dx1...dxn

δ(1− x1 − ...− xn)

(x1A1 + ...+ xnAn)n(5.18)

can be used to combine the denominator D directly but the resultant integrand have to be inte-grated over 3 Feynman parametrising variables and the computation is more messy. Thereforethe partial fraction decomposition is effectuated so that the resultant integrand need only beintegrated over 1 Feynman parametrising variable. Furthermore, this will recover similar resultsto that presented in [9].


To calculate the integral over the 4-momentum, the following dimensional regularisation inte-gration formulae are used:∫

ddk1

(k2 + 2kq −∆)n= iπd/2

Γ(n− d/2)

Γ(n)

1

(−q2 −∆)n−d/2(5.19)∫

ddkkα

(k2 + 2kq −∆)n= −iπd/2 Γ(n− d/2)

Γ(n)

qα

(−q2 −∆)n−d/2(5.20)∫

ddkkαkβ

(k2 + 2kq −∆)n= iπd/2

qαqβ

(−q2 −∆)n−d/2(5.21)[

Γ(n− d/2)

Γ(n)+

Γ(n− 1− d/2)

2Γ(n)gαβ(−q2 −∆)

]where the integral is performed in d dimensions and Γ is the gamma function. Defining ε ≡ 4−d,the asymptotic expression of the gamma function near non-positive integers:

Γ(ε

2) =

2

ε− γ +O(ε) (5.22)

Γ(−1 +ε

2) = −2

ε− 1 + γ +O(ε). (5.23)

where γ = limn→∞

∑ni (1/i)− lnn ≈ 0.577 is the Euler-Mascheroni constant. For n = 2, the above

dimensional regularisation integrals become:∫ddk

1

(k2 + 2kq −∆)2= iπd/2

[2

ε− γ +O(ε)

]m−εW

(−q2 −∆

m2W

)−ε/2= iπd/2m−εW

[− ln

(−q2 −∆

m2W

)+

2

ε+ γ

]+O(ε) (5.24)∫

ddkkα

(k2 + 2kq −∆)n= −iπd/2m−εW qα

[− ln

(−q2 −∆

m2W

)+

2

ε− γ]

+O(ε) (5.25)∫ddk

kαkβ(k2 + 2kq −∆)n

= iπd/2m−εW qαqβ

[− ln

(−q2 −∆

m2W

)+

2

ε− γ]

(5.26)

+ iπd/2[ln

(−q2 −∆

m2W

)− 2

ε− 1 + γ

]gαβ2

(−q2 −∆

)+O(ε)

where in the first line, the m2W term is introduced to give a dimensionless expression which can

then be expanded as aε = exp(ε ln a) = 1 + ε ln a+O(ε2).

To ease the notation, the following functions are introduced corresponding to the expressions of(q2 + ∆) for equations 5.14 to 5.17:

Equation 5.14 : f1(x) ≡ 1

m2W

[k21(x− 1)2 +m2

W + k21(x− 1)] (5.27)

limk1→mb

1 + x(x− 1)xb


m2W

[k21(x− 1)2 + (m2

i + k21)(x− 1) +m2

jx] (5.28)

limk1→mb

xxi + (1− x)xj + x(x− 1)xb


m2W

[k21(x− 1)2 + (m2

W + k21)(x− 1) +m2

jx] (5.29)

limk1→mb

x+ (1− x)xj + x(x− 1)xb


m2W

[k21(x− 1)2 + (m2

i + k21)(x− 1) +m2

Wx] (5.30)

limk1→mb

1− x+ xxi + x(x− 1)xb


where xi ≡m2i

m2W

and the static limit for the beauty quarks is taken k21 ≈ m2

b .

Equations 5.9 to 5.12 and 5.24 to 5.26 are used to evaluate the integral I1(i, j) and I2(i, j). Thepoles involving 1

ε cancel out as expected and taking the limit ε→ 0:

I1(i, j) = −iπ2 1

m4W (1− xi)(1− xj)

∫ 1

0dx ln |f1(x)|+ ln |f2(x)| − ln |f3(x)| − ln |f4(x)|(5.31)

I2(i, j) = −iπ2 1

m4W (1− xi)(1− xj)

∫ 1

0dx (5.32)

(k1)α(k1)βx(x− 1)[ln |f1(x)|+ ln |f2(x)| − ln |f3(x)| − ln |f4(x)|]

+ m2W

gαβ2

[f1(x) ln |f1(x)|+ f2(x) ln |f2(x)| − f3(x) ln |f3(x)| − f4(x) ln |f4(x)|].

Back to equation 5.5, using the following property of the gamma matrices:

γµγαγρ = gµαγρ + gαργµ − gµργα − iεσµαργσγ5 (5.33)

where εσµαρ is the 4 dimensional Levi-Civita tensor, the following simplification is obtained:[b(k1)γµγαγρ(1− γ5)s(k3)

] [b(k2)γργ

βγµ(1− γ5)s(k4)]

= 4[b(k1)γα(1− γ5)s(k3)

] [b(k4)γβ(1− γ5)s(k2)

], (5.34)

thus giving:

M(s) = i∑

i,j=u,c,t

g4W

256π4VisV

∗ibVjsV

∗jb (5.35)

[(4 + xixj)I2(i, j)− 2m2WxixjI1(i, j)]

[b(k1)γα(1− γ5)s(k3)

] [b(k4)γβ(1− γ5)s(k2)

].

Contracting over α and β and using the Dirac equation b(γαkα −mb) = 0, the final result is:

M(s) =∑

i,j=u,c,t

G2Fm

2W

8π2VisV

∗ibVjsV

∗jb (5.36){

F V−A(i, j)[b(k1)γα(1− γ5)s(k3)

] [b(k4)γα(1− γ5)s(k2)

]+ FS−P (i, j)

[b(k1)(1− γ5)s(k3)

] [b(k4)(1− γ5)s(k2)

]}where GF =

√2g2W

8m2W

is the Fermi constant and F V−A(i, j) and FS−P (i, j) are the integrals:

F V−A(i, j) ≡ 1

(1− xi)(1− xj)

∫ 1

0dx

(2∑

k=1

−4∑

k=3

)(5.37)

[(2 +1

2xixj)fk(x)− 2xixj ] ln |fk(x)|

FS−P (i, j) ≡ 1

(1− xi)(1− xj)

∫ 1

0dx

(2∑

k=1

−4∑

k=3

)xb(4 + xixj)x(1− x) ln |fk(x)|.(5.38)

The superscript V −A suggests a V −A⊗ V −A operator and S −P suggests a S −P ⊗S −Poperator where V , A, S and P denote vector, axial vector, scalar and pseudo scalar type ofbilinear forms.


To take into account the t channel contribution, the difference is the exchange of the two strangequarks and the Fierz transformations give in particular for the V −A⊗ V −A operator:[b(k1)γα(1− γ5)s(k3)

] [b(k4)γα(1− γ5)s(k2)

]=[b(k1)γα(1− γ5)s(k2)

] [b(k4)γα(1− γ5)s(k3)

].

Since the box amplitude in equation 5.5 only involves V − A⊗ V − A operators, the t channelgives exactly the same contribution as the s channel. Therefore, not taking into account QCDcorrections:

〈B0s |Heff|B0

s 〉 =∑

i,j=u,c,t

G2Fm

2W

16π2VisV

∗ibVjsV

∗jb (5.39){

F V−A(i, j)[b(k1)γα(1− γ5)s(k3)

] [b(k4)γα(1− γ5)s(k2)

]+ FS−P (i, j)

[b(k1)(1− γ5)s(k3)

] [b(k4)(1− γ5)s(k2)

]}

5.1.1 Interpretation

The same procedure above applies to the case for the analysis of the B0d system or the K0

systems. In the K0 systems, it is usual to totally neglect 4-momenta of the external quarks.

The first importance of equation 5.36 is that it can be used to interpret the claim in section3.2 that quark masses cannot be degenerate for CP violation. From the unitarity of the CKMmatrix

∑i V∗isVib = 0, there will be complete cancellation of the box amplitudes in the case of

quark mass degeneracy across the quarks i. In this case the vanishing off-diagonal elements ofthe effective Hamiltonian means that CP is a symmetry of the effective Hamiltonian. Of course,this is just a heuristic picture and not a rigorous demonstration since degeneracy involving justtwo quarks will not result in a complete cancellation of the amplitude.

The Glashow-Iliopoulos-Maiani (GIM) mechanism can also be understood in view of equation5.36 and the mass hierarchy of quarks. The GIM mechanism states that flavour changingneutral currents must involve loops that are suppressed. However, the analytic expression forM is very long when fully integrated and for the purpose of the following discussion, it issufficient to interpret the physics in the approximation mb → 0. In this case, the FS−P (i, j) andS − P ⊗ S − P terms disappears and upon integration of F V−A(i, j), equation 5.39 reduces tothe result derived in [26]:

〈B0s |Heff|B0

s 〉 = −∑

i,j=u,c,t

G2Fm

2W

4π2VisV

∗ibVjsV

∗jbE(i, j)

[bγα(1− γ5)s

] [bγα(1− γ5)s

]. (5.40)

where the E(i, j)s are referred to as the Inami-Lim functions in the literature

E(i, j) ≡ − 3xixj4(1− xi)(1− xj)

+xixj ln(xi)

xi − xj

[1

4+

3

2(1− xi)− 3

4(1− xi)2

](5.41)

+xixj ln(xj)

xj − xi

[1

4+

3

2(1− xj)− 3

4(1− xj)2

].

In particular, taking the limit j → i,

E(i, i) =4xi − 11x2

i + x3i

4(1− xi)2− 3x3

i ln(xi)

2(1− xi)3. (5.42)

The GIM suppression can be understood with equation 5.40 in light of the ratio of the mass ofthe intermediate quark to that of the W boson. For both the up and charm quarks, xu � xc � 1.


Equation 5.42 implies O(x2u) and O(x2

c) contribution of the amplitude and therefore the up andcharm quark contributions are highly GIM suppressed. The truth quark however has a mass∼ 173.3 GeV [34], more than that of the W boson mass of mW = 80.4 GeV.

The numerical values of the relevant E(i, j) are:

E(t, t) ≈ 2.4 (5.43)

E(c, t) ≈ 2.3× 10−3 (5.44)

E(c, c) ≈ 2.7× 10−4. (5.45)

The magnitudes of the corresponding CKM factor are:

|VtsVtb|2 ∼ λ4 (5.46)

|VcsVtbVcbVts| ∼ λ4 (5.47)

|VcsVcb|2 ∼ λ4. (5.48)

This implies that for the B0s system, the total box amplitude is dominated by truth quarks

contributions, with the leading corrections being O(10−3).

5.2 Effective Hamiltonian in the Operator Product Expansion

The expression for the box diagrams amplitude only takes into account the charged currentweak interaction and not the Quantum Chromodynamics (QCD) corrections such as gluonicinteractions and colour degrees of freedom. To incorporate QCD corrections, a general frameworkis to formulate the effective Hamiltonian using the operator product expansion [31]:

Heff =GF√

2

∑i

(VCKM)iCi(µ)Qi(µ) (5.49)

where GF =√

2g2W8m2

W= 1.16639 × 10−5 GeV−2~3c3 is the Fermi constant, (VCKM)i = (V ∗isVib) are

the CKM factors, Ci(µ) are known as Wilson coefficients depending on the energy scale µ andQi are the local quark operators.

The physical motivation for the operator product expansion is the separation of long distance(strong coupling regime) and short distance (weak coupling regime) of the QCD interaction.The dependence of the QCD running coupling constant gS on the energy scale µ is given by thefollowing β function [35] which crucially has a negative sign:

β(gS) ≡ µ∂gs∂µ

= −b0g

3S

16π2(5.50)

where b0 = 7 for QCD. Integrating the β function, the QCD analogue of the fine structureconstant αS is given by:

αS(µ) ≡g2S(µ)

4π=

αS(µ0)

1 + b0 ln(µ/µ0)αS/2π(5.51)

where µ0 is the arbitrary renormalisation point. From equation 5.51, it is immediately seenthat QCD is an asymptotically free quantum field theory and the coupling strength decreasesat higher energy scales. QCD exhibits colour confinement for the strong coupling regime atlong distances whereas at short distances, coupling is weak and perturbative calculations can beperformed using the Feynman diagrams technology.


In the operator product expansion, the short distance or perturbative contributions are encodedin the Wilson coefficients and the long distance or non-perturbative contributions are representedas the local quark operators [35]. The distinction of short and long distances is set by thearbitrary energy scale µ, typically dependent on the renormalisation scheme.

The Wilson coefficients are calculated from the Feynman diagrams. In addition to the amplitudescalculated from the electroweak Feynman diagrams, the Wilson coefficients also incorporatesshort distance QCD corrections. The perturbative calculations of the QCD corrections havebeen carried out to next leading order (NLO) [31]. The energy scale dependence of the Wilsoncoefficients arises at this point due to renormalisation. However this energy scale dependence hasto cancel with that from long distance contributions so that the final expression is not dependenton the renormalisation scheme.

The long distance contributions involving the local quark operators on the other hand cannotbe evaluated non-perturbatively in general and presents the largest theoretical uncertainties.Another issue is to match the µ dependence with that in the Wilson coefficients. Some methodsof evaluating the long distance contributions include numerical approaches such as lattice QCDwhich performs Monte Carlo integration over a finite lattice and theoretical approaches such asQCD sum rules and heavy quark effective theory which approximates the beauty quark as staticsources of colour [10].

The operator product expansion not only has the merit of separating the short and long distancecontributions but is also of great convenience. A decay amplitude to a final state |f〉 with theeffective Hamiltonian is simply given by:

AB0S→f

= 〈f |Heff|B0S〉 =

GF√2

∑i

(VCKM)iCi(µ)〈f |Qi|B0S〉. (5.52)

For the analysis of decay amplitudes, the Feynman diagrams involve tree diagrams as well asanother class of diagrams known as the penguin diagrams. The operator product expansionof the effective Hamiltonian for the B0

s system is typically written as a sum of current-current(i = 1, 2), QCD penguins (i = 3, 4, 5, 6) and electroweak penguins (i = 7, 8, 9, 10) operators [16]:

Heff =GF√

2

10∑i=1

(VCKM)iCi(µ)Qi (5.53)

Current-Current Operators

Q1 = [siγµ(1− γ5)cβ][cjγ

µ(1− γ5)bi] (5.54)

Q2 = [siγµ(1− γ5)ci][cjγ

µ(1− γ5)bj ] (5.55)

QCD Penguin Operators

Q3 = [siγµ(1− γ5)cj ]

∑q=u,c,d,s,b

[qjγµ(1− γ5)qi] (5.56)

Q4 = [siγµ(1− γ5)ci]

∑q=u,c,d,s,b

[qjγµ(1− γ5)qj ] (5.57)

Q5 = [siγµ(1− γ5)cj ]

∑q=u,c,d,s,b

[qjγµ(1 + γ5)qi] (5.58)

Q6 = [siγµ(1− γ5)ci]

∑q=u,c,d,s,b

[qjγµ(1 + γ5)qj ] (5.59)


Electroweak Penguin Operators

Q7 =3

2[siγ

µ(1− γ5)cj ]∑

q=u,c,d,s,b

eq[qjγµ(1− γ5)qi] (5.60)

Q8 =3

2[siγ

µ(1− γ5)ci]∑

q=u,c,d,s,b

eq[qjγµ(1− γ5)qj ] (5.61)

Q9 =3

2[siγ

µ(1− γ5)cj ]∑

q=u,c,d,s,b

eq[qjγµ(1 + γ5)qi] (5.62)

Q10 =3

2[siγ

µ(1− γ5)ci]∑

q=u,c,d,s,b

eq[qjγµ(1 + γ5)qj ]. (5.63)

where i and j are colour indices and eq denotes the electric charge of the quark q.

The current-current operators are for decays at the tree level whereas the penguin operatorsarise in penguin diagrams to be discussed in section 5.6. The important decay mode for B0

s isb→ ccs which arises in the analysis of the B0

s → J/ψφ decay.

Fig. 5.2: b→ ccs decay Fig. 5.3: A QCD correction correction diagram to b→ ccs

The issue of QCD corrections in weak interactions is not addressed in this thesis but for the sakeof completeness, a brief discussion is presented on the calculation procedure. Figure 5.3 is oneof the diagrams that arise at the leading order QCD corrections. There are 6 diagrams of thistype, by connecting any two external quarks with a gluon. The gluon-quark vertex and gluonpropagator are given by [36]:

gluon-quark vertex = −igSγµLaδ(3∑i=1

ki) (5.64)

gluon propagator =i

k2(5.65)

where La are the generators of SU(3)C and can be represented using the Gell-Mann matricesLa

λa2 . In the approximation of zero external momenta, the amplitude for figure 5.3 is:

M(5.3) = V ∗csVcbg2W g

2S

2

∫d4k

(2π)4

k2

k2(k2 −m2c)

2(k2 −m2W )

(5.66)

[sγµLaγαγν

1− γ5

2c][cγνLaγαγµ

1− γ5

2b]

= −V ∗csVcbg2WαS8π

ln

(m2W

m2B0s

)(5.67)

[sLaγα(1− γ5)c][cLaγα(1− γ5)b].

The appearance of the m2B0s

in the logarithm term is introduced as the energy scale for the

correction [35]. This logarithm term characterises the µ dependence of the Wilson coefficients.


The La satisfies the following identity:

(La)ij(La)kl =1

2δilδkj −

1

6δijδkl. (5.68)

Inserting this expression into equation 5.68 will give the two current operators with the colourdependence.

The phenomenon of neutral meson oscillations can also be studied in the framework of theoperator product expansion. The box diagrams are the perturbative calculations at the lowestorder, giving rise to effective vertices consisting of a Wilson coefficient and the local quarkoperators given by:

〈B0s |[bγα(1− γ5)s

] [bγα(1− γ5)s

]|B0

s 〉 (5.69)

〈B0s |[b(1− γ5)s

] [b(1− γ5)s

]|B0

s 〉. (5.70)

The importance of the B0s system is due to the small CP violation phase βs, hence making

it sensitive for the search of hypothetical particles such as supersymmetric particles. Suchparticles can appear in the box diagrams at sufficient energy scales and modify the value of M12,contributing to observable deviations from the SM. In some extensions beyond the SM, theremay also be New Physics involving flavour changing neutral currents at the tree level [7].

5.3 Expressions for M12 and Γ12

The operator product expansion for the effective Hamiltonian makes it possible to obtain theexpressions for M12 and Γ12. From equation 4.4, one obtain:

M12 −i

2Γ12 = 〈B0

s |Heff|B0s 〉. (5.71)

The calculation of 〈B0s |Heff|B0

s 〉 was done in section 5.1 and QCD corrections can be appendedto equation 5.39 by introducing the correction factors ηQCD(i, j).

〈B0s |Heff|B0

s 〉 =∑

i,j=u,c,t

G2F

16π2VisV

∗ibVjsV

∗jbηQCD(i, j) (5.72){

F V−A(i, j)〈B0s |[bγα(1− γ5)s

] [bγα(1− γ5)s

]|B0

s 〉

+ FS−P (i, j)〈B0s |[b(1− γ5)s

] [b(1− γ5)s

]|B0

s 〉}. (5.73)

As discussed in sections 4.1 and 5.1.1, the M12 term gives the off-shell contributions which isdominated by the truth quark contribution. Hence M12 can be expressed as the Hermitianconjugate of equation 5.72 for i = t, j = t:

M12 =G2Fm

2W

16π2(VtbV

∗ts)

2ηQCD(t, t) (5.74){F V−A(t, t)〈B0

s |[sγα(1− γ5)b

] [sγα(1− γ5)b

]|B0

s 〉

+ FS−P (t, t)〈B0s |[s(1− γ5)b

] [s(1− γ5)b

]|B0

s 〉}. (5.75)

The matrix elements 〈B0s |[sγα(1− γ5)b

] [sγα(1− γ5)b

]|B0

s 〉 and 〈B0s |[s(1− γ5)b

] [s(1− γ5)b

]|B0

s 〉are given as its vacuum insertion approximation multiplied by a bag parameter B(B0

s ) [31]:

〈B0s |[sγα(1− γ5)b

] [sγα(1− γ5)b

]|B0

s 〉 = −8

6B(B0

s )f2(B0s )mB0

s(5.76)

〈B0s |[s(1− γ5)b

] [s(1− γ5)b

]|B0

s 〉 = −5

6B(B0

s )f2(B0s )mB0

s. (5.77)


where f2(B0s ) is the decay parameter for B0

s . The QCD correction factor ηQCD(t, t) is approxi-mately 0.837 at an energy scale of mb in the dimensional regularisation scheme [31]. In fact allthe QCD correction factors in the present discussion are of order unity.

In the case of M12, since mb � mW , the F V−A(i, j) can be approximated by the Inami-Limfunction E(i, j) ≈ 2.7 defined in equation 5.40 and FS−P (j, j) can be approximated by 0. Thisgives:

M12 ≈G2Fm

2W

12π2B(B0

s )f2(B0s )mB0

sηQCD(t, t)(VtbV

∗ts)

2E(t, t). (5.78)

The Γ12 term gives the on-shell contributions and satisfies the condition mi +mj < mb. For thecalculation of Γ12, the external momenta of the quarks cannot be ignored and the expressioncannot be simply expressed in terms of the Inami-Lim functions. The prescription to evaluateΓ12 follows from equation 4.9:

1. Replacing the ln |fi(x)| terms in F V−A(i, j) and FS−P (i, j) with 2π.

2. Integration over the domain x ∈ [0, 1]fi(x) < 0|.

Denoting the modified functions with F V−A(i, j) and FS−P (i, j), this gives the expression forΓ12:

Γ12 =∑i,j=u,c

G2Fm

2W

8π(VibV

∗is)

2ηQCD(i, j) (5.79)[8

6F V−A(i, j)B(B0

s )f2(B0s )mB0

s− 5

6FS−P (t, t)B(B0

s )f2(B0s )mB0

s

].

On evaluating F V−A(i, j) and FS−P (i, j) in Mathematica, the first significant term gives of

x2b = m2

b(VtbV∗ts) while the corrections are quadratic expressions in O

(m2c

m2bm

2W

). The approximate

expression for Γ12 is:

Γ12 =G2Fm

2b

8π

[ηQCD(t, t)(VtbV

∗ts)

2 (5.80)

+ηQCD(t, c)VtbV∗tsVcbV

∗csO

(m2c

m2b

)+ ηQCD(c, c)(VcbV

∗cs)

2O(m4c

m4b

)].

As mentioned in the previous section, calculation of the bag parameter B(B0s ) and the decay

parameter f2(B0s ) poses serious theoretical difficulties. This makes a theoretical calculation of

∆m and ∆Γ difficult and with large uncertainties. For example in [31], the value of f(B0s ) is

estimated as 240±40 MeV with B(B0s ) ranging from 0.85 to 1.41 from lattice QCD calculations.

These different inputs of f(B0s ) and the corresponding estimate of B(B0

s ) can result in morethan a 100% change in the calculated value of ∆Γ

Γ .

Nonetheless, since B(B0s ) and f(B0

s ) appear in both the expressions for M12 and Γ12, importantphysical conclusions can be drawn for the ratio of M12 and Γ12. Their dependence on the massof different intermediate quark translates to:∣∣∣∣ Γ12

M12

∣∣∣∣ ∼ O(m2c

m2t

)∼ 10−4. (5.81)

which gives an extremely small CP violation in mixing.


In addition, the following approximations can be made to excellent precision:

∆m ≈ 2|M12| (5.82)

∆Γ ≈ |Γ12| (5.83)

q

p≡

√M∗12 − Γ∗12

M12 − Γ12≈√

M∗12

M12=V ∗tbVtsVtbV

∗ts

. (5.84)

Some relevant experimental values are given in [34] as:

∆m = 17.69± 0.08 ps−1 (5.85)

∆Γ = 0.100± 0.013 ps−1 (5.86)

1

Γ= 1.497± 0.015 ps (5.87)

mB0s

= 5.37MeV. (5.88)

5.4 CP Violation in B0s → J/ψφ at the Tree Level and βs

The B0s → J/ψφ is an example of CP violation due to interference between decay without mixing

and decay with mixing. The decay amplitude can be written as:

AB0s→J/ψφ = Ano mixing

B0s→J/ψφ

+AmixingB0s→J/ψφ

(5.89)

= 〈J/ψφ|Heff|Bs0〉+ 〈J/ψφ|Heff|B0

s 〉〈B0s |Heff|B0

s 〉.

A phase difference between the two amplitudes will give rise to an observable CP asymmetry.Figures 5.4 and 5.5 show the B0

s decay without mixing and decay after transition to the B0s via

box diagram contributions.

Fig. 5.4: B0s → J/ψφ without mixing Fig. 5.5: B0

s → J/ψφ with mixing

From the diagrams, the CP violation in decay is given by

AB0s→J/ψφ

AB0s→J/ψφ

=V ∗csVcbVcsV ∗cb

. (5.90)

Together with the expression for CP violation in mixing qp =

VtsV ∗tbV ∗tsVtb

derived in equation 5.84, the

λf parameter defined in equation 4.48 is:

λJ/ψφ = ηCP (J/ψφ)q

p

AB0s→J/ψφ

AB0s→J/ψφ

(5.91)

= ηCP (J/ψφ)VtsV

∗tb

V ∗tsVtb

V ∗csVcbVcsV ∗cb

. (5.92)


The relation of λJ/ψφ with βs ≡ arg[− VtsV ∗tbVcsV ∗cb

]defined in equation 3.24 is readily identified:

λJ/ψφ = ηCP (J/ψφ)ei2βs

= ηCP (J/ψφ) [cos 2βs + i sin 2βs] . (5.93)

Therefore the decay rates in equation 4.50 and 4.51 assume the expression:

ΓB0s→J/ψφ = e−Γt

∣∣AB0s→J/ψφ

∣∣2 (5.94)

[cosh ∆Γt/2 + ηCP (J/ψφ) cos 2βs sinh ∆Γt/2− ηCP (J/ψφ) sin 2βs sin ∆mt]

ΓB0s→J/ψφ

= e−Γt∣∣AB0

s→J/ψφ∣∣2 (5.95)

[cosh ∆Γt/2 + ηCP (J/ψφ) cos 2βs sinh ∆Γt/2 + ηCP (J/ψφ) sin 2βs sin ∆mt] .

The CP asymmetry is given by:

ACP (t) =sin 2βs sin ∆mt

cos 2βs sinh ∆Γt/2 + cosh ∆Γt/2. (5.96)

From equations 5.94 and 5.95, knowledge of Γ, ∆m and ∆Γ, a decay rate measurement of B0s →

J/ψφ and B0s → J/ψφ will give an experimental determination of the angle βs. Experimentally,

this would require tagging or identification of the decaying meson [2]. The method of taggingfollows from the hadronisation or production of the B0

s or B0s meson. In the event of a B0

s ,a strange quark from a meson M hadronises with the antibeauty quark and the remainingantiquark in the meson M will hadronise with another quark with a large probability of givinga kaon. Observation of this kaon would be signature of a B0

s . Of course, tagging proceduresare by no means perfect, leading to statistical uncertainties. Another issue comes in due to thefact that the Jψ and φ are vector mesons and can result in both CP even and odd eigenstates,further complicating the experimental analysis. This will addressed in the next section.

The determination of angle βs based on the SM prediction based on the previously determinedCKM parameters and the experimental measurement [2] give:

βs(SM) = −0.036± 0.002 rad (5.97)

βs(B0s → J/ψφ) = 0.15± 0.18( stat)± 0.06( sys) rad. (5.98)

If a significant deviation is present, this signals the possible discovery of new particles. Withinthe experimental errors, the two results are consistent with each other. The large statistical andsystematic uncertainties arise in part due to the fast oscillation rate in B0

s ↔ B0s mixing and in

part due to small statistics: 50 µb bb cross section at a centre of mass frame energy 2 TeV andyearly luminosity of 1 fb−1 only yielded ∼ 3000 B0

s → J/ψφ events from 2002 to 2008. Afterthe LHC upgrade, the uncertainty is expected to be improved to ±0.008 [6]. This substantialimprovement in precision will have significant implications for the measurement of βs.


5.5 Time dependent angular analysis of B0s → J/ψφ

The B0s meson, composed of a strange and antibeauty quark is a pseudoscalar particle. Both

the J/ψ meson 1 (composed of a charm and anticharm quark) and the φ meson (composed of astrange and antistrange quark) are pseudovector particles with the representation JCP = 1−−2. Angular momentum conservation gives 3 possibilities for the states of J/ψφ:

• L = 0: Polarisation of vector mesons longitudinal with respect to momentum, hereafterlabelled polarisation state L.

• L = 1: Polarisation of vector mesons transverse with respect to momentum and perpen-dicular among themselves, hereafter labelled polarisation state ⊥.

• L = 2: Polarisation of vector mesons transverse with respect to momentum and parallelamong themselves, hereafter labelled polarisation state //.

The CP eigenvalue is given by:

CP (J/ψφ) = CP (J/ψ)CP (φ)(−1)L = (−1)L. (5.99)

Therefore the L and // polarisation states correspond to CP even eigenstates while the ⊥polarisation state correspond to a CP odd eigenstate. This admixture of CP even and CPodd eigenstates results in an experimental complication which needs to disentangle the differentpolarisation states via a time dependent angular analysis.

The full decay chain is given by:

B0s → J/ψ(µ+µ−)φ(K+k−). (5.100)

where the vector mesons J/ψ and φ decay to the final decay states µ+µ− and K+K− respec-tively. The 4 final decay particles has 9 degrees of freedom since there are 16 components of4-momentum which are subject to 7 constraints due to the masses of the 7 particles involvedin the decay. Of these 9 degrees of freedom, 3 are related to the momentum in the centre ofmass frame and 3 are related to the orientation of the coordinate system. Since the decay hastranslational invariance and rotational invariance due to the B0

S being spinless, these 6 degreesof freedom will not affect the angular analysis. The last 3 degrees of freedom can be viewed asthe decay angles of the final decay states with respect to each other and to the B0

s .

The angular analysis is typically performed in the transversity basis such as in LHCb [2] 3. Thetransversity basis is illustrated in figure 5.6.

1 The J/ψ meson was discovered independently in 1974 by Samuel C.C. Ting who named it J and Burt Richterwho named it ψ. Recognising the independent contribution of both physicists, it is now listed as J/ψ.

2 J gives the spin, C gives the eigenvalue under C and P gives the eigenvalue under P of the particle.3 Another basis is the helicity basis adopted by the CDF collaboration


Fig. 5.6: Transversity basis for the angular analysis of B0s → J/ψ(µ+µ−)φ(K+K−)

In the B0s rest frame, the x-axis is defined to be in the direction of flight of the φ meson. The

K+K− system defines the x-y plane with the the y-axis chosen such that py(K+) > 0. In the

rest frame of the φ meson, ψ is defined to be the angle subtended by the K+ meson and thex-axis. In the rest frame of the J/ψ meson, θ is the polar angle and ϕ is the azimuthal angle ofthe µ+.

From the above definitions, the decay angles in the transversity basis have the expressions:

cosψ = p(K+) · x (5.101)

cos θ = p(µ+) · z (5.102)

cosϕ sin θ = p(µ+) · z (5.103)

sinϕ sin θ = p(µ+) · y. (5.104)

For J/ψ in its rest frame, the polarisation state for L, ⊥ and// are in direction x, z and yrespectively. In analogy to photon polarisation correlations in neutral pion decays, the decayamplitude can be expressed as [17]:

AB0s→J/ψφ =

mφALEφ

(εJψ · x)(εφ · x)−A//√

2[εJψ · εφ − (εJψ · x)(εφ · x)]− iA⊥√

2εJψ × εφ · x

= AL cosψ −A//√

2sinψ +

iA⊥√2

sinψ. (5.105)

In the approximation of small muon mass, the muon pairs will only couple to transverse polarisa-tion states of the J/ψ meson. Denoting the unit vector of µ+ by n = (cosϕ sin θ, sinϕ sin θ, cos θ),the differential decay rate is proportional to [17]:

|A|2 =∑i,j

AiAj [δij − ninj ] (5.106)

where Ai ≡ AL cosψδi1 −A//√

2sinψδi2 + iA⊥√

2sinψδi3 and the tensor [δij − ninj ] expresses the

coupling of the muon pairs to transverse polarisation states of the J/ψ.


From equation 5.106, the differential decay rate in the transversity basis is given by:

d3ΓB0s→J/ψ/φ

d cos θd cosψdϕ∝

6∑i=1

hi(t)Θ(θ, ψ, ϕ) (5.107)

d3ΓB0s→J/ψ/φ

d cos θd cosψdϕ∝

6∑i=1

hi(t)Θ(θ, ψ, ϕ). (5.108)

where the functions h(t), h(t) and Θ(θ, ψ, ϕ) are summarised in table 5.1

i h(t) h(t) Θ(θ, ψ, ϕ)

1 |AL(t)|2 |AL(t)|2 2 cos2 ψ(1− sin2 θ cos2 ϕ)

2 |A⊥(t)|2 |A⊥(t)|2 sin2 θ sin2 ϕ

3 |A//(t)|2 |A//(t)|2 sin2 ψ(1− sin2 θ sin2 ϕ)

4 Im{A∗//A⊥}(t) Im{A//∗A⊥}(t) sin2 ψ sin 2θ sinϕ

5 Re{A∗LA//}(t) Re{AL∗A//}(t) − 1√

2sin 2ψ sin2 θ sin 2ϕ

6 Im{A∗LA⊥}(t) Re{AL∗A⊥}(t) 1√

2sin 2ψ sin 2θ cosϕ

Tab. 5.1: Angular dependence of differential decay rate in B0s → J/ψφ

The h(t) and h(t) are calculated using equations 4.21 and 4.22:

|AL(t)|2 = e−Γt|AL(0)|2 [cosh ∆Γt/2 + cos 2βs sinh ∆Γt/2− sin 2βs sin ∆mt](5.109)

|A⊥(t)|2 = e−Γt|A⊥(0)|2 [cosh ∆Γt/2− cos 2βs sinh ∆Γt/2 + sin 2βs sin ∆mt](5.110)

|A//(t)|2 = e−Γt|A//(0)|2 [cosh ∆Γt/2 + cos 2βs sinh ∆Γt/2− sin 2βs sin ∆mt](5.111)

Im{A∗//A⊥}(t) = e−Γt|A//(0)A⊥(0)|[− cos(δ⊥ − δ//) sin 2βs sinh ∆Γt/2 (5.112)

− cos(δ⊥ − δ//) cos 2βs sin ∆mt+ sin(δ⊥ − δ//) cos(∆mt)]

Re{A∗LA//}(t) = e−Γt|AL(0)A//(0)|[− cos(δ// − δL) cosh ∆Γt/2 (5.113)

− cos(δ// − δL) cos 2βs sinh ∆Γt/2− cos(δ// − δL) sin 2βs sin ∆mt]

Im{A∗LA⊥}(t) = e−Γt|AL(0)A⊥(0)| [− cos(δ⊥ − δL) cos 2βs sinh ∆Γt/2 (5.114)

− cos(δ⊥ − δ//) cos 2βs sin ∆mt+ sin(δ⊥ − δL) cos ∆mt]

where δ is the strong or CP even phase for the different decay modes.


5.6 Penguin Diagrams

“ (T)he answer to every problem involved penguins ”Rick Riordan, The Throne of Fire

The analysis of the B0s → J/ψφ decay amplitude for the extraction of βs only took into account

tree level decays. However, another class of Feynman diagram contributes to the decay ampli-tude, referred to as the penguin diagrams. For example, in the B0

s → J/ψφ channel, the decaymay also proceed via the gluonic penguin diagram in figure 5.7. In some decay channels, forexample B0

s → K0K0, the decay cannot proceed through tree level diagrams and constitutespure penguin decays.

Fig. 5.7: A gluonic penguin diagram for B0s → J/ψφ

From figure 5.7, it can be seen that the penguin decay consists of a sum of the CKM elements∑i VisV

∗ib as compared to that of the tree level decay consisting only VcsV

∗cb. This means that the

experimental determined phase in the B0s → J/ψφ channel is not entirely due to βs but contains

contributions from the decay via penguin diagrams. Such are typically considered to be muchsmaller than the tree level contribution and ignored. However, this might not be justified for thecase of βs which is only −0.036 rad in the SM. In a recent work adopting a perturbative QCDfactorisation approach to study the amplitude and branching ratio of the penguin diagrams [33],it was shown that the correction to βs can be of the order O(10−3). Likewise, in [19], adoptinga different approach based on SU(3)C symmetry, the penguin contribution was estimated tomodify βs up to O(10%). It must be stressed however that a complete and reliable estimate ofthe QCD corrections has not yet been achieved. In view of the expected significant improvementin precision at LHCb, it is essential to take into account the contribution of penguin diagrams,lest deviations from the SM value of βs be misinterpreted as new particles induced CP violations.

Replacing the gluon by the Z boson or photon will give rise to the electroweak penguin diagrams.Penguin diagrams also occur in B0

s ↔ B0s mixing via the double penguin diagram in figure 5.8.

Fig. 5.8: Double penguin diagram for B0s ↔ B0

s mixing

6 Conclusion and Outlook

“ Whatever Nature has in store for mankind, unpleasant as it may be, menmust accept, for ignorance is never better than knowledge. ”

Enrico Fermi, Source unknown

Since its discovery in the neutral kaon oscillations, CP violation has been an intriguing subjectfield. The Kobayashi-Maskawa model for CP violation was reviewed in the framework of theStandard Model, explicitly deriving the CKM matrix arising from Yukawa coupling of fermionsand the Higgs field. A complex phase in the CKM matrix was identified as responsible for CPviolation in charged current interactions.

Although all 4 free parameters of the CKM matrix have been experimentally determined, it isessential to overconstrain the model and search for deviations that will signal New Physics. Aparticularly interesting platform to study CP violation is the B0

s systems because its unitaritytriangle is very poorly constrained and also its high sensitivity for probing New Physics due tothe angle βs ≈ −0.036 being small in the SM.

A phenomenological study of the golden mode B0s → J/ψφ for the extraction of βs was thus

performed. A detailed calculation of the charged current interaction induced B0s and B0

s mixingwas made. The operator product expansion formalism is also used to show how QCD correc-tions may be incorporated. This formalism was thus used to obtain expressions of the effectiveHamiltonian governing mixing and governing decay. Furthermore, analysis of the B0

s → J/ψφwas explicitly shown to exhibit interference effects due to phase difference between decay andmixing and this asymmetry can be used to experimentally extract βs.

Some complications regarding the B0s → J/ψφ decay was also discussed, such as the need to

perform a time dependent angular analysis to disentangle the CP odd and CP even final statesas well as the possible non-negligible contributions of penguin diagrams for this decay.

The completion of the LHC upgrade and operations at a centre of mass frame energy of 14 TeVand yearly luminosity of 2 fb−1 will yield ∼ 100000 B0

s → J/ψφ events per year. The optimiseddetector design of the LHCb detector also contributes the drastic improvement in statistics andprecision expected at the LHCb. Therefore, the different modes of decay channels B0

s may provesignificant in the search for New Physics via their manifestations in modifying CP violationpredictions, complementary to the direct search of new particles at the Atlas and CMS detector.

A logical continuation is to work on the understanding of penguin diagram contributions andhow their effects may be isolated through the comparison of different decay channels. A solidtheoretical grasp of penguin effects is inescapable for a full interpretation of the experimentresults generated by the LHCb measurements with much better precision.

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