Master Thesis

Nikhef National Institute for Subatomic PhysicsUniversity of Amsterdam

Physics Track: Particle and Astroparticle Physics

Search for Lorentz invariance violationwith B0

s → J/ψφB0s → J/ψφB0s → J/ψφ decays

G. Chatzikonstantinidis

Supervisors:Prof. Dr. G. Raven,Dr. J. van Tilburg,

Prof. Dr. P. de Jong

Abstract

In this thesis we derive the differential decay rates of B0s → J/ψφ incorporating CPT-violating

effects. Subsequently we measure Lorentz and CPT-violating effects using data correspondingto 3 fb−1 of pp collision data, collected with the LHCb detector at center-of-mass energy of

√s =

7 and 8 TeV. To parameterize Lorentz and CPT violation we use a general phenomenologicaldescription which is independent of any microscopic model, where the CPT-Lorentz violatingeffects are quantified with the complex parameter z. The values obtained are Re(z) = −0.022±0.033 ± 0.003 and Im(z) = 0.004 ± 0.011 ± 0.002. Also we parameterize CPT and Lorentzviolation in the context of the Standard Model Extension (SME), where CPT-Lorentz violationis parameterized through the ∆αµ coefficients, integrated in the SME Lagrangian. The resultsobtained are ∆α0 − 0.38∆αZ = −0.987 ± 1.382 ± 0.166 × 10−14 GeV, ∆αX = 0.921 ± 2.106 ±0.188 × 10−14 GeV and ∆αY = −3.817 ± 2.106 ± 0.188 × 10−14 GeV. The results reportedwith both parameterizations are the first measurements of CPT-Lorentz violation in the B0

s

system. In both parameterizations the results are consistent with no CPT and Lorentz violation.Additionally for both of the parameterizations we measure φs, |λs|, Γs = (ΓL + ΓH)/2, ∆Γs =ΓL − ΓH and ∆ms = mH −mL. The results obtained for the CP and lifetime parameters arefound to be consistent with the latest analysis of φs [1] and are in agreement with StandardModel predictions.

Contents

1 Introduction 3

2 Theoretical motivation 4

3 The LHCb experiment 63.1 The Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 The LHCb detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Lorentz and CPT tests with neutral B mesons 104.1 Parameterizing theB0 mesons effective Hamiltonian in terms of the CPT violating

parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 The time evolution and decay of B0 mesons in the context of CPT violation . . 154.3 Sidereal modulations of the CPT-violating parameter . . . . . . . . . . . . . . . 19

5 CPT violation in B0s → J/ψφ 23

5.1 The decay of B0s → J/ψφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Time dependence and CPT violation . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Combining time-dependence including CPT-violating effects and angular depen-

dence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6 Data analysis 316.1 Maximum likelihood fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2 Reconstruction and selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.3 Decay time and acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.4 Angular resolution and acceptance . . . . . . . . . . . . . . . . . . . . . . . . . 356.5 Flavour tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.6 Decay rate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.7 Sun-centered frame and external parameters . . . . . . . . . . . . . . . . . . . . 38

7 Analysis results 427.1 Parameter estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.2 Binned fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.3 Analysis of subsets of the data sample . . . . . . . . . . . . . . . . . . . . . . . 467.4 Validating the modified fitting algorithm . . . . . . . . . . . . . . . . . . . . . . 477.5 Investigating the statistical errors . . . . . . . . . . . . . . . . . . . . . . . . . . 477.6 Consistency test using Monte Carlo events . . . . . . . . . . . . . . . . . . . . . 477.7 Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1

CONTENTS 2

7.8 Overall systematic uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Appendices 60

A Signal PDF in the context of CPT violation 61

B Non-collimated analysis results 64

Chapter 1

Introduction

The standard model of particle physics is a unified gauge theory which describes all knowinteractions, except gravity [2]. Even though it is a very successful model that is consistent withall experimental measurements, it cannot explain some fundamental phenomena, such as darkmatter, dark energy and the observed asymmetry between matter and antimatter. At the sametime gravity is a force that still cannot be described in this context. Hence, the standard modelof particle physics is considered to be a low-energy limit of a more fundamental theory that cananswer the previous fundamental questions and at the same time describe gravitational effectsthat are expected to have a significant contribution at Planck energies (EP ≈ 1.22 · 1019 GeV).

At present, a considerable amount of theoretical work is conducted in the framework of find-ing a fundamental theory that can incorporate both gravity and quantum mechanics. However,from an experimental point of view all the theoretical models that are proposed as candidates foran underlying theory face a huge obstacle, because at low energies, due to the Planck suppres-sion, any gravitational effect becomes almost impossible to be observed [3]. Therefore, searchingfor quantum signatures from gravitational effects that are accessible at present energies is chal-lenging, but might give access to the physics of high-energy scales.

3

Chapter 2

Theoretical motivation

A very promising approach to this problem is to search for fundamental laws that are consideredto be exact in the context of the standard model, but appear to be broken at a more fundamentallevel. Any deviation will then be a signal for new physics. An example of a law which holdsexactly in established physics is the so-called CPT symmetry. The CPT symmetry states thatunder the operation of C (charge conjugate), P (spatial inversion) and T (time reversal) thephysics laws remain the same. Among the consequences of this theorem is that particles andantiparticles have the same mass, lifetime and opposite charge. The CPT symmetry is closelyrelated with the Lorentz symmetry, through the CPT theorem, which briefly states that everyrelativistic quantum field theories exhibit also CPT invariance [4]. One straightforward questionis whether it is possible to have independent violation of these symmetries. The answer to thisquestion is given through the anti-CPT theorem which roughly states that if CPT symmetryis broken, then Lorentz invariance is also violated 1 but not vice versa [6, 7]. Thus it becomesobvious that performing searches for CPT violation can be a very powerful tool for the discoveryof physics beyond the standard model.

The question that additionally must be answered is in which framework the searches forpotential CPT and Lorentz violation will be conducted. From a phenomenological point ofview the introduction of a parameter that quantifies CPT violation is, in many situations, quitestraightforward. This approach has the advantage that it is not related to any theoretical model.However, without an underlying theory we will not be able to make any theoretical predictionof this parameter and at the same time it is impossible to connect the results of different CPT-violating parameters between different measurements. Consequently, no bounds can be set forfuture experiments [8]. This makes obvious the need for a microscopic model, through which acomplete description of CPT and Lorentz violation can be extracted.

The framework for this work is given through the standard model extension [9], which is avery general effective field theory that contains all possible operators that break Lorentz andCPT2 symmetry. The general idea is that the underlying fundamental theory remain Lorentzand CPT invariant, but observed violations of these symmetries can be the result of spontaneousviolation in the solutions of the theory. This idea is very attractive since the theory maintainsthe full symmetry, which is only violated by the vacuum. The same concept is introduced in

1There are situations where the anti-CPT theorem regarding the connection between the violation of Lorentzand CPT symmetries is not valid [5].

2Contains also operators that preserve CPT symmetry. It must become clear that Lorentz violation is allowedwithout necessary CPT violation but not the opposite.

4

CHAPTER 2. THEORETICAL MOTIVATION 5

the standard model with the Higgs mechanism, with the difference that now the vacuum is, alsosaturated by tensor or vector fields [10].

The standard model extension (SME) is constructed as general as possible from the under-lying fundamental theory, while at the same time preserving all the desirable features, such asthe gauge structure and power coupling renormalizability. Energy and momentum are of coursealso conserved. The Lagrangian of the SME is given by:

LSME = LSM + δL (2.1)

where LSME represent the SME Lagrangian,3 LSM the SM Lagrangian and the δL representsCPT and Lorentz violating terms, that are controlled by a set of tensorial coefficients that areto be determined experimentally [6]. These coefficients describe the expectation values of fieldsproduced by type of interactions that destabilize the vacuum; the presence of these terms isresponsible for the breakdown of Lorentz and CPT symmetry. In the context of certain stringtheories it is shown that spontaneous symmetry violation can trigger vacuum expectation valuesfor non-scalar fields; these values can then be identified with the tensorial coefficients describedbefore. In the limit where the CPT-and Lorentz-violating terms are zero the SME Lagrangianwill recover to the SM Lagrangian. Furthermore, the SME Lagrangian is still invariant underLorentz transformations of the observer frame, but it is not invariant under Lorentz transfor-mations of the particle frame. In other words, the violation of Lorentz symmetry makes theexperimental results depend on the direction and the velocity of the experiment [9].

Even though there is no evidence yet for CPT or Lorentz violation, there are very promisingmeasurements that can access with present energies possible CPT-and-Lorentz violating effects.This thesis describes a search for CPT and Lorentz violation using decays of neutral B mesons.

3The complete SME Lagrangian contains also a gravity sector.

Chapter 3

The LHCb experiment

In this chapter the experimental setup that is used for this analysis is briefly discussed.

3.1 The Large Hadron ColliderThe Large Hadron Collider (LHC) is a circular collider that accelerates two beams of protons orlead ions up to few TeV of energy. The two proton beams, before being accelerated in the LHCring, pass through a series of pre-accelerators (see figure 3.1). First, hydrogen atoms are strippedfrom the electrons and then accelerated up to 50 MeV by a linear accelerator (LINAC2). Fromthere, they are fed to the Proton Synchrotron Booster (BOOSTER) and then to the ProtonSynchrotron (PS). In the last step of this chain, they are accelerated by the Super ProtonSynchrotron (SPS) and then injected in the LHC with an energy of 450 GeV.

Figure 3.1: Schematic picture of CERN’s accelerator complex.

In LHC the two beams are accelerated in opposite directions, until they reach their final en-ergies. Then, they intersect in four points (interaction points), where detectors are constructed.In 2011 and 2012 each beam reached an energy of 3.5 TeV and 4 TeV respectively. The main

6

CHAPTER 3. THE LHCB EXPERIMENT 7

experiments at the LHC are CMS, ATLAS, ALICE and LHCb. The ATLAS and CMS detectorsare general purpose detectors, where the Higgs particle has been discovered. ALICE is designedto investigate the properties of the quark gluon plasma, created by the collision of lead ions.Finally, the LHCb detector is optimized to study CP asymmetries and rare decays of b-and c-hadrons. In this thesis we take advantage of the LHCb properties (see text) in order to performstudies for CPT-violating phenomena.

3.2 The LHCb detectorThe LHCb detector is focused on studying the properties of b-and c-hadrons at the LHC andthe design of the detector is optimized for this purpose. The LHCb detector is designed as asingle-arm spectrometer which covers a pseudorapidity region of 2 < η < 5. The motivationbehind this choice is that at high collision energies the bb pairs mainly travel (together) in aforward or backward direction and are highly collimated along the beam line. The geometricalacceptance of the detector is between 10− 300 mrad in the bending plane (x− y) of the magnetand 10−250 mrad in the non-being (y−z) one (see figure 3.2). Furthermore, the LHCb detectorhas very good vertex resolution in order to deal with secondary vertices (particularly importantfor the fast oscillations of the B0

s − B0s system) and finally the trigger system is specifically

designed to search for signatures originating from b-and c-hadron decays [11]. An outline of thedetector is given in figure 3.2.

Figure 3.2: A y − z projection of the LHCb detector and its sub-detectors.

The subdetectors of LHCb detector, illustrated in figure 3.2, are briefly discussed. For moredetails see [11].

CHAPTER 3. THE LHCB EXPERIMENT 8

1. Tracking system

(a) Vertex Locator:The Vertex Locator (Velo) is a silicon microstrip detector, providing precise measure-ments of the track positions of charged decay particles close to the interaction point.This is used to identify the primary and secondary vertices. To reach the precisionthat is required, the Velo is placed as close as possible to the interaction point.

(b) The Tracker Turicensis:The Tracker Turicensis (TT) consists of four layers of silicon microstrips and is placedbetween the Velo and the magnet. The TT improves the momentum estimate of thecharged decay products and at the same time provides important information for thereconstruction of long-living particles, such as K0

S.(c) Magnet:

The magnet in LHCb is a normal-conductive dipole magnet which generates an in-tegrated magnetic field of 4.2 Tm. The magnetic field bends the trajectories of thecharged particles, enabling the determination of their momentum.

(d) The Inner and Outer Tracker:The inner tracker (IT) is a silicon microstrip detector which is enclosed in the innerregion of the T-stations. The outer tracker (OT) is a gas straw detector that coversthe remaining area. Both are placed after the magnet, with a purpose to measurethe momentum of the charged particles that fly through the magnetic field.

2. Particle identification

(a) Ring Imaging Detectors:The ring imaging detectors (RICH) are Cherenkov detectors that measure the angleof the emitted light from charged particles. Combining this information with themomentum, the mass of the particles can be determined. This resolves the identityof the particles.

(b) The Scintillator Pad and the Pre-Shower Detectors: The scintillator pad(SPD) and the pre-shower detectors (PS) are two scintillating pad detectors thatare placed before the electromagnetic calorimeter to provide additional informationabout the identity of the particles. The SPD provides good discrimination betweenneutral and charged particles, and it also provides an estimate of the total numberof charged particles. After the SPD there is a thin plate of a lead that will initiate ashower; next the PS use this to discriminate between electromagnetic and hadronicshowers. This information is used by the electromagnetic and hadronic calorimetersto improve their performance.

(c) Calorimeters:The electromagnetic (ECAL) and the hadronic (HCAL) calorimeters are placed afterthe SPD and PS detectors. The ECAL is a sampling calorimeter, which measures theenergy of electromagnetically interacting particles. It is composed of a lead absorberand a scintillator of about 25X0. The sufficient thickness of the ECAL ensures thatthe electromagnetic interacting particles will be well confined. On the other hand,the hadronic calorimeter (HCAL) measures the energy of hadronically interacting

CHAPTER 3. THE LHCB EXPERIMENT 9

particles. It consists of layers of iron and scintillator that are placed parallel to thez-axis. The thickness of the HCAL is 5.6λint, which is not enough to fully containthe hadronic showers.

(d) Muon Chambers:The muon chambers (M1-M5) are placed at the end of the detector. The M2-M3chambers are equipped with multiwire proportional chambers, while M1 is equippedwith a Triple-GEM detector, which has a better radiation hardness. The muonstations provide good muon identification and muon triggers, which is crucial formany B decays.

Chapter 4

Lorentz and CPT tests with neutral Bmesons

The neutral B meson interferometry is a very powerful tool, enabling sensitive tests for CPT-violating phenomena [12]. The CPT violation can be quantified in terms of parameters that arepresent in the effective Hamiltonian, describing the evolution of the B meson system. Theseparameters are then re-expressed in terms of quantities that appear in the SME Lagrangian.This will enable us to set tight constraints on the CPT-violating parameters. In the following, acomplete description of how the CPT-violating parameter appears in the decay rates of the B0

mesons is presented, together with the description of this parameter in terms of quantities thatare integrated in the SME Lagrangian and result in CPT and consequently Lorentz violation.

4.1 Parameterizing the B0 mesons effective Hamiltonianin terms of the CPT violating parameter

Here we will deal with the general phenomenology that is relevant for any neutral meson system,where, unlike with π0, neutral mesons can be distinguished by an internal quantum number. Inthe case of the B0 mesons this is the so-called beauty (B). The Hamiltonian that describes theevolution of the system is expressed as:

H = Hstrong +HQED +Hweak = H∆B=0 +H∆B=0 +H∆B 6=0 . (4.1)

The fact that weak interactions do not preserve the internal quantum number B is manifestedin the Hamiltonian by the presence of off-diagonal elements1, which will drive the neutral Bmesons to oscillate.

The time evolution for the generic state of the B0 ↔ B0 system, is represented by a vectorin Hilbert space:

|Ψ(t)〉 = a(t)|B0〉+ b(t)|B0〉+∑k

ck(t)|fk〉 (4.2)

where |fk〉 stands for all possible final states that the B0 mesons can decay. The Schrodinger1The off-diagonal elements appear in the Hamiltonian, since the flavour eigenstates of H = Hstrong +HQED

are not at the same time eigenstates of the Hweak

10

CHAPTER 4. LORENTZ AND CPT TESTS WITH NEUTRAL B MESONS 11

equation, satisfied by |Ψ(t)〉 is:i∂

∂tΨ(t) = HΨ(t) , (4.3)

where H is an infinite-dimensional Hermitian matrix in the Hilbert space. The exact solutionto this problem is very complicated [13]. However, the problem can be significantly simplifiedusing the Weisskopf-Wigner approximation, where the following assumptions are made:

• for t = 0, a(0) and b(0) are non-zero, while all the other coefficients ck = 0, so the initial statewill be: |Ψ(0)〉 = a(0)|B0〉+ b(0)|B0〉.

• we are interesting in computing only the coefficients a(t) and b(t), and not the coefficientsck(t).

• we restrict the description to t much larger than the typical strong interaction scale.

With these assumptions, the Schrodinger equation describing the evolution of the B0− B0, canbe expressed as:

i∂

∂t

(a(t)b(t)

)=(H11 H12H21 H22

)(a(t)b(t)

), (4.4)

where the evolution of the generic state is described in the B0 - B0 sub-space by the effective,non-Hermitian Hamiltonian H. The effective Hamiltonian can always be split into a Hermitianand anti-Hermitian part:

H = M − i

2Γ , (4.5)

where both the M (mass matrix) and the Γ (decay matrix) are by construction Hermitian.

M = 12(H +H†) = M † ; Γ = i(H−H†) = Γ † . (4.6)

The elements of the effective Hamiltonian can be expressed in second order-perturbation theory[14], where |fk〉 represents all the intermediate states:

Mij = m0δij + 〈i|Hw|j〉+∑k

P 〈i|Hw|fk〉〈fk|Hw|j〉m0 − Ek

, (4.7)

Γij = 2π∑k

δ(m0 − Ek)〈i|Hw|fk〉〈fk|Hw|j〉 . (4.8)

Hence, the elements of the effective Hamiltonian can be written as:

Hij = Mij −i

2Γij . (4.9)

The mixing of theB0 and B0 is governed by the off-diagonal elements of the effective-Hamiltonian.The Γij quantifies contributions of real, on-shell states to which the flavour eigenstates can de-cay, while the Mij quantifies the contribution via virtual, off-shell states (box-diagram), andis in the standard model dominated by the top-quark contribution [15], whose large mass isresponsible for the large observed mixing between the flavour eigenstates.

CHAPTER 4. LORENTZ AND CPT TESTS WITH NEUTRAL B MESONS 12

(a) (b)

Figure 4.1: Box diagrams for B0s − B0

s mixing. [16]

Taking into account the hermiticity of M and Γ , it follows that:

Mij = M∗ji and Γij = Γ ∗ji, with i 6= j .

However, this does not imply any dependence betweenH12 andH21, unless additional conditionsare imposed.2

Furthermore, using equations 4.5 and 4.6 it is trivial to show that:

d

dt(|a|2 + |b|2) = −(a∗b∗)Γ

(ab

). (4.10)

Since B0 mesons decay, the left side of equation 4.10 must be negative, which implies that Γ isa positive definite matrix. This implies, amongst others, that Γ11 and Γ22 must be positive.

Finally, using the definition of the matrix elements of the effective Hamiltonian given byequation 4.7 and knowing how the CP, T and CPT operators acts on a ket, we can showthat several conditions are imposed from the invariance of these discrete symmetries on theelements of the effective Hamiltonian. An example is given for CP symmetry, where we use:CP |B0〉 = eiξcp ¯|B0〉 and CP ¯|B0〉 = e−iξcp|B0〉 and that CP is an exact symmetry, which implies:CPHw(CP )† = Hw.

Γ11 = 2π∑k

δ(m0 − Ek)〈B0|CP (CP )†HwCP (CP )†|fk〉〈fk|CP (CP )†HwCP (CP )†|B0〉

= 2π∑k

δ(m0 − Ek)e−iξcp〈B0|CPHw(CP )†|fk〉eiξcp〈fk|CPHw(CP )†|B0〉

= 2π∑k

δ(m0 − Ek)〈B0|CPHw(CP )†|fk〉〈fk|CPHw(CP )† ¯|B0〉

= Γ22 .

Performing similar operations with the other discrete symmetries to the elements of the effectiveHamiltonian, it can be shown that H has the following properties:

• if CPT or CP is conserved, then Γ11 = Γ22 and M11 = M22 or, in other words, H11 = H22.

• if T or CP is conserved, the Γ12 = e2iξcpΓ21 and M12 = e2iξcpM21 or, in other words, |H12| =|H21|.

2In the case of the T-symmetry the condition that is imposed is Im(M∗12Γ12) = 0 or |H12| = |H21|, in other

words the phases that appear in the off-diagonal matrix elements of M and Γ are the same. This results in, allthe matrix elements of the latter matrices being real, except of a global phase [17].

CHAPTER 4. LORENTZ AND CPT TESTS WITH NEUTRAL B MESONS 13

Using the previous conditions imposed for the indirect (mixing) invariance of the CP, T andCPT symmetries, we will show later that we can define parameters that quantify these viola-tions/invariances and we will use them in order to parametrize our physical states in terms ofthese. But first we will continue with the phenomenological description of neutral B mesonoscillations.

The physicals states |Ba〉 and |Bb〉 are the eigenstates of the effective Hamiltonian, withcomplex eigenvalues that can be expressed as:

µa = ma −i

2Γa ; µb = mb −i

2Γb . (4.11)

It is a general rule to label the physical states in the case of the B mesons by choosing the massof the eigenstates as a label: a = L and b = H referring to the light and heavy eigenstates,respectively [18]. The eigenvalues can be found by solving the following determinant equation:

det[M11 − iΓ11/2− µ M12 − iΓ12/2M21 − iΓ21/2 M22 − iΓ22/2− µ

]= 0 . (4.12)

We find that the eigenvalues3 in terms of the elements of the effective Hamiltonian are [14,19]:

µH,L = 12[ H11 +H22 ±

√(H11 −H22)2 + 4H12H21 ]

= m− i

2Γ ±√

(M12 −i

2Γ12)(M∗12 −

i

2Γ∗12) + 1

4(δm− i

2δΓ )2 , (4.13)

where in the previous expression for the eigenvalues we have defined that:

m ≡ 12(M11 +M22); Γ ≡ 1

2(Γ11 + Γ22); δm ≡M11 −M22; δΓ ≡ Γ11 − Γ22 . (4.14)

Using equation 4.13 we define:

∆µ ≡ µH − µL =√

4H12H21 + (H22 −H11)2 = ∆m+ i

2∆Γ , (4.15)

∆m ≡ mH −mL = 2Re√

(M12 −i

2Γ12)(M∗12 −

i

2Γ∗12) + 1

4(δm− i

2δΓ )2 , (4.16)

∆Γ ≡ ΓL − ΓH = 4Im√

(M12 −i

2Γ12)(M∗12 −

i

2Γ∗12) + 1

4(δm− i

2δΓ )2 . (4.17)

Since we have taken ∆m to be the mass of the heavier eigenstate minus the mass of the lightereigenstate, it will be, by definition, a positive quantity. However, the sign of ∆Γ , must bedetermined experimentally [17].

Using the previous definitions and the properties of the effective Hamiltonian regarding Tand CPT symmetries, we will define the following CPT-and-T violating parameters:

δT ≡|H12| − |H21||H12|+ |H21|

; δCPT ≡H11 −H22

∆µ, (4.18)

3It must be noticed that the real and imaginary part of the eigenvalues do not correspond to the eigenvaluesof the M and Γ matrices respectively.

CHAPTER 4. LORENTZ AND CPT TESTS WITH NEUTRAL B MESONS 14

where in terms of the elements of the mass and decay matrix the previous CPT-and-T violatingparameters can be written respectively as:

δCPT ≡δm− i

2δΓ

2√

(M12 − i2Γ12)(M∗

12 − i2Γ∗12) + 1

4(δm− i2δΓ )2

=δm− i

2δΓ

∆m+ i2∆Γ

, (4.19)

δT ≡|M12 − i

2Γ12| − |M∗12 − i

2Γ∗12|

|M12 − i2Γ12|+ |M∗

12 − i2Γ∗12|

. (4.20)

From the previous definitions it can be deduced that, in order CPT symmetry to be conserved,δCPT = 0, while in order T symmetry to be invariant, δT = 0 and finally in order for CPsymmetry to be conserved then both δCPT and δT must be zero. This implies that both δT andδCPT are CP violating quantities [14]. In the following we will define δCPT as z and we willfocus, only on the CPT symmetry. Hence, with the previous definitions, the eigenstates of theeffective Hamiltonian H, can be written in the most general4 form as :

|BL〉 = pL|B0〉+ qL|B0〉 , (4.21)|BH〉 = pH |B0〉 − qH |B0〉 , (4.22)

where after diagonalizing the H we find that:qLpL

= 2H21

∆µ(1− z) ; qHpH

= 2H21

∆µ(1 + z) , (4.23)

yielding:qHqLpHpL

= M∗12 − iΓ ∗12/2

M12 − iΓ12/2, (4.24)

where we can define:q

p≡ −

√qHqLpHpL

= −

√√√√M∗12 − iΓ ∗12/2

M12 − iΓ12/2. (4.25)

Now the eigenstates of the effective Hamiltonian can be written as:

|BL〉 = p√

1− z|B0〉+ q√

1 + z|B0〉 , (4.26)|BH〉 = p

√1 + z|B0〉 − q

√1− z|B0〉 , (4.27)

and the flavour eigenstates as:

|B0〉 =√

1 + z

2p |BH〉+√

1− z2p |BL〉 , (4.28)

|B0〉 =√

1 + z

2q |BL〉 −√

1− z2q |BH〉 . (4.29)

From equation 4.28 it can be deduced that if z 6= 0, the composition of the flavour eigenstatesis asymmetric in terms of the physical states [14]. We must note that even though the flavourand the CP eigenstates are orthogonal:

4None of the constrains implied by the CPT, T or CP symmetry are taken into account.

CHAPTER 4. LORENTZ AND CPT TESTS WITH NEUTRAL B MESONS 15

〈B0|B0〉 = 0 ; 〈B+|B−〉 = 0 ,

the same in not true in general for the physical states:

〈BH |BL〉 = p∗HpL − q∗HqL . (4.30)

In case CP is a conserved symmetry then the right-hand side of equation 4.30 is zero, but it canbe also zero when δT = 0 and Im(z) = 0 [18]. As the following section is dedicated on how theCPT-violating parameter z appears in the decay rates of the B0 mesons, is crucial to make thefollowing remark: If z is aimed to be an observable that quantifies CPT violation, it should beinvariant under re-phasing of the physical states |BH,L〉. Using equations 4.23 and the definitionfor the z and ∆µ, we can re-write the CPT-violating parameter as:

z =1− qH/pH

qL/pL

1 + qH/pHqL/pL

, (4.31)

where, since in the expression of the z parameter we have only ratios of qH/pH over qL/pL,the phases will cancel, and the z parameter will remain invariant. Hence, the z parameter is ameasurable, complex, CPT-violating quantity [14].

4.2 The time evolution and decay of B0 mesons in thecontext of CPT violation

Since the states |BH〉 and |BL〉 are eigenstates of the Schrodinger equation, the time evolutionof these states is obtained as:

|BH〉 = e−imH t−12ΓH t|BH(0)〉 , (4.32)

|BL〉 = e−imLt−12ΓLt |BL(0)〉 . (4.33)

By combining equations 4.26, 4.28 and 4.32 we get:

|B0(t)〉 = (g+(t)− zg−(t))|B0〉+√

1− z2 q

pg−(t)|B0〉 ,

|B0(t)〉 = (g+(t) + zg−(t))|B0〉+√

1− z2p

qg−(t)|B0〉 ,

(4.34)

where

g±(t) ≡ 12(e−imLt− 1

2ΓLt ± e−imH t−12ΓH t)

= 12e−iMte−iΓ t/2(e+i∆mt/2e−∆Γt/4 ± e−i∆mt/2e+∆Γt/4) . (4.35)

In the previous equation we have defined that: M ≡ MH+ML

2 and Γ ≡ ΓH+ΓL2 . However, in the

following chapters describing the decay rates of B0s → J/ψφ, we will denote M and Γ with Ms

and Γs respectively.Using the previous equations we can deduce that, if we start with a |B0〉 state at t = 0, the

probability after t = t0, to find a |B0〉 or a |B0〉 state and vice versa, is given by:

CHAPTER 4. LORENTZ AND CPT TESTS WITH NEUTRAL B MESONS 16

|〈B0|B0(t)〉|2 = |1− z2||qp|2|g−(t)|2 , (4.36)

|〈B0|B0(t)〉|2 = |g+(t)|2 + |z|2|g−(t)|2 − 2Re[ zg−(t)g+(t)∗ ] , (4.37)

|〈B0|B0(t)〉|2 = |1− z2||pq|2|g−(t)|2 , (4.38)

|〈B0|B0(t)〉|2 = |g+(t)|2 + |z|2|g−(t)|2 + 2Re[ zg−(t)g+(t)∗ ] . (4.39)

From the previous equations we note that |〈B0|B0(t)〉|2 6= |〈B0|B0(t)〉|2 only when z 6= 0;meaning that the probability of a P (B0 → B0) 6= P (B0 → B0). Additionally, we can deducethat when | q

p| = 1 (T invariance) then P (B0 → B0) = P (B0 → B0), independently of the CPT

symmetry [14,18].In order for oscillations of B0 and the subsequent decays into a final state f to be described,

the general formalism that was presented before will be extended. We denote the followingdecay amplitudes as:

Af = 〈f |B0〉; Af = 〈f |B0〉; Af = 〈f |B0〉; Af = 〈f |B0〉 (4.40)

and also define the following complex parameters:

λf = q

p

AfAf

; λf = 1λf

; λf = q

p

AfAf

; λf = 1λf

(4.41)

What must be noticed is that the choice of the previous definitions is connected with theinvariance of the observables under a re-phasing of the states. As an example we can see that ifwe re-phase the final state |f〉 → eiφf |f〉 and the physical state B0 as |B0〉 → eiφ|B0〉 the quantityAf does not remain invariant. In contrast it can be shown that the quantities, λf , λf , λf , λftogether with the z parameter and the |Af |, |Af |, |Af |, |Af |, | qp | remain invariant, and can thusbe used as observables [18].

Using equation 5.2 and the definitions given before for the decay amplitudes we can deducethat:

A(B0 → f) = (g+ − zg−)Af + q

p

√1− z2g−Af , (4.42)

A(B0 → f) = (g+ + zg−)Af + p

q

√1− z2g−Af , (4.43)

A(B0 → f) = (g+ − zg−)Af + q

p

√1− z2g−Af , (4.44)

A(B0 → f) = (g+ + zg−)Af + p

q

√1− z2g−Af . (4.45)

The generic expression for the time-dependent decay rates (ΓB0→f (t) = |〈f |T |B0(t)|2), that givethe probability for an initial neutral B meson to decay to a general state f at a given time willbe obtained by squaring the previous amplitudes; yielding the following expressions:

CHAPTER 4. LORENTZ AND CPT TESTS WITH NEUTRAL B MESONS 17

ΓBd→f = 12 |Af |

2|pq|(1−qf)e−Γst

cosh(∆Γst/2)[ 1 + |λf |2 − 2qfRe(z)Re(λf )− 2Im(z)Im(λf ) ]+ cos(∆mst)[ qf(1− |λf |2) + 2qfRe(z)Re(λf ) + 2Im(z)Im(λf ) ]+2 sinh(∆Γst/2)[ qfRe(z)|λf |(1−qf) − Re(λf ) ]+2 sin(∆mst)[ qfIm(z)|λf |(1−qf) − qfIm(λf ) ] , (4.46)

ΓBd→f = 12 |Af |

2|qp|(1+qf)e−Γst

cosh(∆Γst/2)[ 1 + |λf |2 − 2qfRe(z)Re(λf ) + 2Im(z)Im(λf ) ]+ cos(∆mst)[ qf(−1 + |λf |2) + 2qfRe(z)Re(λf )− 2Im(z)Im(λf ) ]+2 sinh(∆Γst/2)[ qfRe(z)|λf |(1+qf) − Re(λf ) ]+2 sin(∆mst)[ qfIm(z)|λf |(1+qf) + qfIm(λf ) ] . (4.47)

Where in the previous equations we have approximate that z2 = 0, since we expect a priori that|z| is a very small number. The parameter qf represents the tagging, taking values 1,−1 and0. Where qf = 1 represents a B0, qf = −1 a B0 and qf = 0 the untagged case, where we donot have any information about the flavour of the B0 meson and hence only the terms that arecommon for the B0 and B0 are kept. The previous equations can be written in a more elegantform if we approximate z ≈ 2z

1+|λf |2and define the following quantities:

Df ≡−2Re(λf )1 + |λf |2

; Sf ≡2Im(λf )1 + |λf |2

; Cf ≡1− |λf |21 + |λf |2

; (4.48)

Df ≡−2Re(λf )1 + |λf |2

; Sf ≡2Im(λf )1 + |λf |2

; Cf ≡1− |λf |2

1 + |λf |2. (4.49)

These quantities are linked together by the following condition:

D2f + S2

f + C2f = D2

f + S2f + C2

f = 1 . (4.50)

Substitution of these into 4.46 and 4.47 yields:

ΓBqf→f = 12 |Af |

2|pq|(1−qf)e−Γst(1 + |λf |2)

cosh(∆Γst/2)[ 1 + qfRe(z)Df − Im(z)Sf ] + cos(∆mst)[ qfCf − dRe(z)Df + Im(z)Sf ]+ sinh(∆Γst/2)[ qfRe(z) +Df ] + sin(∆mst)[ qfIm(z)− qfSf ] ,

(4.51)

ΓBqf→f = 12 |Af |

2|qp|(1+qf)e−Γst(1 + |λf |2)

cosh(∆Γst/2)[ 1 + qfRe(z)Df + Im(z)Sf ] + cos(∆mst)[ −qfCf − qfRe(z)Df − Im(z)Sf ]+ sinh(∆Γst/2)[ qfRe(z) +Df ] + sin(∆mst)[ qfIm(z) + qfSf ] .

(4.52)

CHAPTER 4. LORENTZ AND CPT TESTS WITH NEUTRAL B MESONS 18

A special case is when the final state |f〉 is a CP eigenstate, such as the |J/ψφ〉. Usingequations 4.41 and the fact that |f〉 is proportional to |f〉 since they are CP eigenstates, resultsin Cf = −Cf , Df = Df , Sf = −Sf , |Af | = |Af | and λf = 1

λf[16]. Making the decay rates

given by the equations 4.51 and 4.52 to be identical. For the case of the decay of Bs → J/ψφthe previous decay rates need to be modified, as this decay is governed by several intermediatestates. The previous formalism for the decay rates, includes CPT-violating phenomena onlyin the mixing process, while direct CPT violation in the decay amplitude of the B0 mesons isexpected to be negligible.

The invariance or non-invariance of the discrete symmetries T, CPT and CP will have afundamental effect on the decay rates, where we must note that we consider only CPT-violatingphenomena in the mixing process. In the case that | q

p| 6= 0, which means that we have indirect T

violation, will have as a result that the magnitudes of λf and λf to be different. In case we haveCP violation in the decay, i.e, |Af | 6= |Af |, will result again in λf 6= λf [16]. A very interestingcase arises when the final state is a CP eigenstate, such as for the decay of the B0

s → J/ψφ.Even without T and CPT violation in the mixing or CP violation in the decay a difference inΓB0→f and ΓB0→f can be caused by the interference of B0(→ B0) → f and B0(→ B0) → f .This can be seen more clearly if we write the asymmetry for the decay rates expressed as (see4.63):

ACP =ΓB0→f − ΓB0→f

ΓB0→f + ΓB0→f(4.53)

In general this asymmetry is named CP asymmetry but this is because T and CP asymmetry areequivalent in the case that CPT invariance is assumed. There are many different possibilitiesthat can cause non-zero value for this asymmetry, such as T, CPT violation in mixing andCP violation in the decay. We will express the asymmetry in the most general form, where weinclude CPT-violating effects with the exception of the decay amplitudes. Defining the followingexpressions:

Ach± = [ (1∓ |p/q|2)[ 1− Im(z)Sf ] + (1± |p/q|2)Re(z)Df ] , (4.54)Ash± = [ (1± |p/q|2)Re(z) + (1∓ |p/q|2)Df ] , (4.55)Ac ± = [ (1± |p/q|2)[ Cf − Re(z)Df ] + (1∓ |p/q|2)Im(z)Sf ] , (4.56)As ± = [(1± |p/q|2)Im(z)− (1± |p/q|2)Sf ] . (4.57)

Yields:

ACP/CPT/T = Ach+ cosh(∆Γst/2) + Ac+ cos(∆mst) + Ash+ sinh(∆Γst/2) + As+ sin(∆mst)Ach− cosh(∆Γst/2) + Ac− cos(∆mst) + Ash− sinh(∆Γst/2) + As− sin(∆mst)

.

(4.58)In the following we will take different combinations of the invariances of the discrete symmetriesand see how the asymmetry ACP/CPT/T will be effected. In the case that |q/p| = 1 correspondingto T invariance in mixing, then ACP/CPT/T → ACP/CPT and the coefficients defined in equations

CHAPTER 4. LORENTZ AND CPT TESTS WITH NEUTRAL B MESONS 19

4.59-4.62 are modified to5:

Ach± = [ (1∓ 1)[ 1− Im(z)Sf ] + (1± 1)Re(z)Df ] , (4.59)Ash± = [ (1± 1)Re(z) + (1∓ 1)Df ] , (4.60)Ac ± = [ (1± 1)[ Cf − Re(z)Df ] + (1∓ 1)Im(z)Sf ] , (4.61)As ± = [(1± 1)Im(z)− (1± 1)Sf ] . (4.62)

where if we furthermore, neglect higher order topologies and restrict the description to tree-leveldecay amplitudes, then |Af | = |Af |, i.e, we have CP invariance in the decay, resulting in Cf = 0.Assuming additionally CPT invariance in mixing, then the asymmetry is simplified significantly:

ACP = −Sf sin(∆mst)cosh(∆Γst/2) +Df sinh(∆Γst/2) . (4.63)

From the previous equation it can be deduced that ACP 6= 0 if Im(λf ) 6= 0. Since in this thesiswe are interested in the effects of CPT violation, we continue by focusing on the asymmetrywhich originates in a possible CPT violation. In the context of the SME extension Im(z) canbe approximated to zero for B0 mesons (see next chapter). Hence, the asymmetry, in the casethat we have T invariance in the mixing and additionally CP invariance in the decay can beexpressed, respectively, as:

ACP/CPT = cos(∆mst)Cf − sin(∆mst)Sfcosh(∆Γst/2) +Df sinh(∆Γst/2)+

Re(z)Df [ cosh(∆Γst/2)− cos(∆mst) ] + sinh(∆Γst/2)cosh(∆Γst/2) +Df sinh(∆Γst/2) ,

(4.64)

ACP/CPT = − sin(∆mst)Sfcosh(∆Γst/2) +Df sinh(∆Γst/2)+

Re(z)Df [ cosh(∆Γst/2)− cos(∆mst) ] + sinh(∆Γst/2)cosh(∆Γst/2) +Df sinh(∆Γst/2) .

(4.65)

In this case we can factorize the asymmetry in two terms, representing the asymmetry originatingfrom the interference and the one from CPT violation. In case that Im(λf ) = 0, then Df = ±1.Hence the CPT asymmetry can be expressed as:

ACPT = Re(z)±1∓ 0.5e∓∆Γst/2[ e∆mst + e−∆mst ]. (4.66)

4.3 Sidereal modulations of the CPT-violating parame-ter

In the previous section we described how the CPT violation can be quantified in the contextof the neutral B meson oscillations, by starting from the effective Hamiltonian and afterwards

5We must note that we neglect CPT violation in the decay. Any asymmetries originating from the decayare refereed as CP violation. In principle CPT, CP and T violating effects can also be quantified in the decayprocess.

CHAPTER 4. LORENTZ AND CPT TESTS WITH NEUTRAL B MESONS 20

allowing for CPT violation, we finally conclude that CPT violation can be manifest through thecomplex parameter z:

z =δm− i

2δΓ

∆m+ i2∆Γ

. (4.67)

In the context of standard model extension (SME), it can be shown6 that δΓ = 0, whichrelates the real and imaginary part of the z parameter, through the following expression:

Re(z)∆Γ = −2Im(z)∆m . (4.68)

Given the values [1] of ∆Γs = 0.0805± 0.0096 ps−1 and ∆ms = 17.768± 0.024 ps−1 [20] for theB0s system, equation 4.68 results in:

Im(z)Re(z) = − ∆Γs

2∆ms

= −0.0022± 0.0003 . (4.69)

Where it become obvious that, neglecting the Im(z) from the description of the CPT violationis a very good approximation.

As discussed before, a microscopic model that can incorporate CPT and Lorentz violation,while at the same time preserve many desirable features exists. In the context of this model,named Standard Model Extension, the numerator of the CPT violating parameter can be ex-pressed in terms of quantities that are integrated in the SME Lagrangian and results in bothCPT and Lorentz violation. The CPT violating parameter can then be written as:

z = βµ∆αµ∆m+ i

2∆Γ, (4.70)

where βµ = γ(1,−→β ) is the four-vector velocity of the B0 meson in the observer frame. The factthat ∆m and ∆Γ are extremely small (see text) increases the sensitivity of the ∆αµ coefficients,resulting on B0 mesons to be considered as ideal interferometers. In the previous equation, ∆αµis defined as : ∆αµ = rq1α

q1µ −rq2α

q2µ , where αq1

µ and αq2µ are the CPT violating coupling constants

(mass dimensions) associated with terms −αqµqγµq, integrated in the SME Lagrangian, whereq is a quark field of a specific flavour, while the parameters rq1 and rq2 , quantify normalizationand quark-binding effects [3]. Neglecting the small effects of the latter parameters we can write:

∆αµ ≈ αq1µ − αq2

µ . (4.71)Using the equation 4.71 the following approximate7 equation for the neutral mesons can bewritten:

∆αK0

µ −∆αB0d

µ +∆αB0s

µ ≈ 0 . (4.72)From equation 4.70 it is obvious that amongst the consequences of the violation of CPT and

Lorentz symmetry, in the context of the SME, is the four-momentum dependence of the observ-ables, which results in the dependence of the observables on the magnitude and the orientationof the B0 mesons momentum. This dependence result in profound experimental consequences,

6The numerator of the CPT-violating parameter is the product of the vacuum expectation value for the tensorfield that breaks the CPT invariance and the momentum of the neutral meson; this represents a real number,resulting in δΓ = 0.

7This is an approximate equation, since the normalization and quark-bind effects have been ignored.

CHAPTER 4. LORENTZ AND CPT TESTS WITH NEUTRAL B MESONS 21

since the relative direction of the lab-frame with the constant background vacuum expectationvalues of the CPT-violating fields that span the universe, effects directly the determination ofthe CPT-violating coefficients. The previous statement becomes even more clear if we re-writethe CPT-violating parameter in the following manner.

z ∝ γ(∆α0 − ~β∆~α)∝ γ(∆α0 − |β||∆α| cos θ) .

(4.73)

Where θ represent the angle between the B0 mesons and the constant vector fields. Fromequation 4.73 we can deduce that:

• Increasing the boost of the B0 mesons, will enhance the CPT-violating effects; in theLHCb detector, the B0 mesons are produced with 〈γ〉 ≈ 17.48.

• Using B0 mesons that are produced with 〈γ〉 ≈ 1, gives direct access to ∆α0. Eventhough producing B0 mesons at rest is not possible, this is a good approximation forCLEO experiment where B mesons are produced with β ≈ 0.06.

• Since |β| cos θ < 1, the spatial component of the constant vector fields is always suppressedcompared to the ∆α0 [21].

From equation 4.73, it become obvious that the CPT-violating parameters, depend directlyon the relative angle between the B0 meson momentum and the constant background fields.One immediate implication of this is that the CPT-violating parameter will exhibit siderealmodulation, as a consequence of the Earth’s rotation.

To display the explicit sidereal dependence of the z parameter, we will express the previousparameter in terms of a non-rotating frame. We adopt the basis (X, Y , Z) for the non-rotatingSun-centered frame and (x, y, z) for the lab-frame.

Figure 4.2: Bases in the non-rotating frame and in the laboratory frame [21].

The (X, Y , Z) frame is defined in celestial equatorial coordinates, with Z aligned along to theEarth’s rotation axis. X and Y have declination 00 and right ascension 00 and 900 respectively;

CHAPTER 4. LORENTZ AND CPT TESTS WITH NEUTRAL B MESONS 22

forming a right-handed base [21]. In the lab-frame it is convenient to take the z axis alignedwith the direction of the beam, see Figure 6.1. In order to be able to observe sidereal variations,the angle χ that is defined by cosχ = zZ must be non zero. Provided the previous conditionis fulfilled, the z axis will rotate around the Z axis with sidereal frequency Ω. By defining(∆α1, ∆α2, ∆α3) as the CPT and Lorentz breakdown components of the coefficients ∆−→α inthe lab-frame, and as (∆αX , ∆αY , ∆αZ) are defined in the non-rotating frame, performing tworotations from the lab-frame to the Sun-centered frame, results in:

∆α1 = ∆αX cosχ cosΩt+∆αY cosχ sinΩt−∆αZ sinχ , (4.74)∆α2 = ∆αY cosΩt−∆αX sinΩt , (4.75)∆α3 = ∆αX sinχ cosΩt+∆αY sinχ sinΩt+∆αZ cosχ . (4.76)

Combining the previous expressions for the CPT-violating coefficients with equation 4.67 andusing the fact that ∆αX,Y,Z = −∆αX,Y,Z [22], we obtain for the CPT-violating parameters thefollowing expression:

z = z(t, θ, φ, p)

= γ(p)∆m+ i

2∆Γ∆α0 + β∆αZ(cos θ cosχ− sin θ cosφ sinχ)

+ β[∆αY (cos θ sinχ+ sin θ cosφ cosχ)−∆αX sin θ sinφ] sinΩt+ β[∆αX(cos θ sinχ+ sin θ cosφ cosχ) +∆αY sin θ sinφ] cosΩt ,

(4.77)

which for highly collimated B0, reduces to:

z = z(t, p)

= γ(p)∆m+ i

2∆Γ∆α0 + β∆αZ cosχ+ β∆αY sinχ sinΩt

+ β∆αX sinχ cosΩt ,

(4.78)

where −→β = β(sinθcosφ, sinθsinφ, cosθ) is the velocity of the B0 meson in the lab-frame, p =βmBγ(p) the magnitude of the momentum and t the sidereal time. Since we are interested onlyin the real part, we can write:

Re(z) = Re(z)(t, p)

= γ(p)∆m∆m2 + 1/4∆Γ 2∆α0 + β∆αZcosχ+ β∆αY sinχsinΩt

+ β∆αXsinχcosΩt .

(4.79)

Since B0 mesons in LHCb are produced with a high boost, the variation of the magnitude of βis extremely small. For this reason β will be set to be one. This result in the ∆α0 and ∆αZ tobe combined and measured as one parameter, defined as: ∆α0 + cosχ∆αZ .

Chapter 5

CPT violation in B0s → J/ψφ

In this chapter we will briefly discuss the angular part of the probability density function (PDF)of the B0

s → J/ψφ decay and we will derive the modified time-dependent terms including CPT-violating effects.

5.1 The decay of B0s → J/ψφ

The decay of the B0s → J/ψ(µ+µ−)φ(K+K−), where the most relevant Feynman diagrams for

the decay are illustrated in Figure 5.1, has a very clean signature, making this decay experimen-tally very attractive. This decay is the analogue of B0

d → J/ψK0S, with the spectator d-quark

replaced with an s-quark. However, this does result in major quantitative differences. An im-portant impact is that, since the mixing in the B0

s system is governed by the Vts element of theCKM matrix1, CP violation is expected to be very small, making the B0

s → J/ψφ decay sen-sitive to non-standard amplitudes that can contribute in the mixing process or in higher-orderdecay topologies, so-called penguin diagrams.

(a) b (b) b

Figure 5.1: (a) The dominant (tree) and (b) suppressed (penguin) decay diagrams; the curledline represents the contribution from gluons.

1From the Wolfenstein parametrization, it can be noticed that, Vts is real up to O(λ3), while Vtd is alreadyimaginary, resulting in βs to be much smaller compared to β [23].

23

CHAPTER 5. CPT VIOLATION IN B0S → J/ψφ 24

As it was described before, the ACP/T asymmetry between the decays of B0s/B

0s → J/ψφ,

originate from the presence of the interference term Im(λf ), arising from the contribution oftwo different amplitudes, with and without mixing, in the decay process. The evolution of themixing process is dominated by the top quark, so it is proportional to (VtsV ∗tb)

2 . The amplitudesfor the tree-level decays that occur with and without mixing are proportional to (V ∗csVcb) and(VcsV ∗cb) respectively. This yields for the phase of λJ/ψφ:

arg[q

p

AJ/ψφ

AJ/ψφ

]= arg[(VtsV ∗tb)

2] + arg[(V ∗csVcb)]− arg[(VcsV ∗cb)]

= 2 arg[−VtsV

∗tb

VcsV ∗cb

]≡ 2βs ,

where βs represents one of the angles appearing in the Unitary Triangle of the B0s system. In

the derivation of the previous expression for the weak phase we have neglected real, on shellcontributions which originate from the off-diagonal elements of the decay matrix, since areestimated to be very small [24] and higher-order topologies in the decay process. The quantitythat is measurable is φs, which, considering the previous assumptions, is predicted to be equalto −2βs. What should be noticed is that since φs is determined by the relative phase of thedecay amplitudes, where CPT violation is neglected and the term q/p that is determined fromthe off-diagonal elements of the M and Γ matrices; CPT violation will non effect directly φs.However, a model that cannot describe CPT-violating phenomena, in case that CPT symmetryis not conserved, will result in a not accurate prediction of φs. The λf can now be expressedin terms of the φs, that will appear consequently in the decay rates, enabling to perform ameasurement for this phase.

The decay of B0s → J/ψφ involves the decay of a pseudoscalar mesons (spin zero) to two

vector mesons (P → V V ). Since the total angular momentum of the decay must be conserved(~L + ~S = 0), the spins must be compensated by the orbital angular momentum, resulting inthe following possible values L ∈ (0, 1, 2). This leads to the following possibilities for the CPeigenvalue of the final state:

CP |Jψφ〉 = ηf |J/ψφ〉 = (−1)L|J/ψφ〉 , (5.1)

where from the previous equation it can be deduced that the eigenvalues of the final state areodd or even depending on the relative orbital momentum of the two vector mesons, giving riseto three possible intermediate angular momentum states. The amplitudes can be decomposedinto three components; describing the decay in terms of spin-polarization states. These are thelongitudinal (0) and the transverse to the direction of motion of the vector mesons. Where in thelast case the polarization vectors are either parallel (||), or perpendicular (⊥) to each other [25].The polarization states with L = 0 and L = 2 are even under a CP operator, while the statewith L = 1 is odd under the CP operator. Using the previous definitions, the differential decayrate of the Bs → J/ψφ decay can be expressed as:

d4Γ

dtd ~Ω=

∑6i=1 Ti(t)fi( ~Ω)∫ ∫ ∑6

j=1 Tj(t)fj( ~Ω)dtd ~Ω, (5.2)

where Ti(t) represents the time-dependent components (see next section), expressed in terms ofthe physical parameters. fi are angular-dependent functions (see Tables 5.1 and 5.2), derivedusing the helicity formalism [26], where ~Ω = (θµ, θK , φh).

CHAPTER 5. CPT VIOLATION IN B0S → J/ψφ 25

Figure 5.2: Definition of the so-called helicity angles. The angles θµ and θK are defined in therest frame of J/ψ and K−K+, while the φh represents the angle between the two frames, takingas an origin the rest frame of the B0

s [16, 27].

i Ti(t) fi( ~Ω)1 |A0(t)|2 2 cos2 θK sin2 θµ2 |A||(t)|2 sin2 θK(1− sin2 θµ cos2 φh)3 |A⊥(t)|2 sin2 θK(1− sin2 θµ sin2 φh)4 Im(A∗||(t)A⊥(t)) sin2 θK sin2 θµ sin 2φh5 Re(A∗0(t)A||(t)) 1

2

√2 sin 2θK sin 2θµ cosφh

6 Im(A∗0(t)A⊥(t)) −12

√2 sin 2θK sin 2θµ sinφh

Table 5.1: The sum over the products: Ti(t)fi( ~Ω) represents the PDF of the B0s → J/ψK+K−

decay; considering only K+K− meson pairs in an L = 1 state (P-wave).

In the previous discussion it became obvious that the K+K− system produced with the decayof the spin one φ resonance must have orbital angular momentum L = 1. However there mightbe contributions from B0

s → J/ψK+K− decays, where the invariant mass of the K+K− systemis close to the φ(1020) meson. Such a situation arises when the two kaons originate from thescalar mesons α0 and f 0 [26], both with on invariant mass of 980 MeV/c2, resulting in theorbital angular momentum of the K+K− system to be L = 0 (S-wave). Since the B0

s is spinless,in that case the orbital angular momentum of the J/ψK+K− system needs to be L = 1 [16].By taking into account the S-wave contribution, the differential decay rate for the decay ofBs → J/ψK+K− is given by equation 5.2, with the indexes i, j to run from one to ten, wherethe additional four terms that quantify the S-wave contribution are given by Table 5.2.

i Ti(t) fi( ~Ω)7 |As(t)|2 2

3 sin2 θµ8 Re(A∗0(t)As(t)) 4

3

√3 cos θK sin2 θµ

9 Re(A∗||(t)As(t)) 13

√6 sin θK sin 2θµ cosφh

10 Im(A∗⊥(t)As(t)) −13

√6 sin θK sin 2θµ sinφh

Table 5.2: Four additional terms that originate from the S-wave contribution.

CHAPTER 5. CPT VIOLATION IN B0S → J/ψφ 26

5.2 Time dependence and CPT violationIn the previous section we briefly discussed the angular dependence of the differential decayrates of the Bs → J/ψK+K−, where the different components of the PDF were summarized inTables 5.1 and 5.2. In this section we will derive the expressions of the Ti(t)(time dependentcomponents), including CPT-violating phenomena. In order to derive the time-dependent com-ponents of the PDF in terms of the intermediate states, it is necessary to go back to equations4.42-4.45 and replace the A(t) and A(t), with Ai(t) and Ai(t) respectively, resulting on:

Ai(B0 → f) = Ai[ (g+ − zg−) + λi√

1− z2g− ] , (5.3)

Ai(B0 → f) = Ai[ (g+ + zg−) + 1λi

√1− z2g− ] , (5.4)

whereλi ≡

q

p

AiAi

. (5.5)

Since, the final state is a CP eigenstate, equation 5.5 will be re-written as:

λi = ηi|λi|e−iφis , (5.6)

where the CP eigenvalues are given by:

ηi =

+1 if 0, ||−1 if ⊥, S

In equation 5.6, |λi| and φis are two quantities that are different for each intermediate state.Different approximations can be chosen; we can approximate | q

p| ≈ 1 and furthermore neglect

higher-order contributions in the decay, resulting on |λi| = 1 and φis to be the same for allintermediate states. However in this thesis we choose not to constrain |λi| but we assume thatall λi are the same. This implies that there is either no, or equal amount of CP violation in thedecay process for all the intermediate states. This choice will lead to the φis being independentof the polarization state. Additionally, T violation in mixing | q

p| 6= 1 is still allowed, but

all the |λi| would be affected in the same manner. Hence, from now on we will use that:λi = ηiλ = ηi|λ|e−iφs .

Using the identities defined in equation 4.48 and taking into account the previous expressionfor λ in terms of the intermediate angular momentum states, the identities can be re-written as:

Dλi ≡ ηi−2Re(λ)1 + |λ|2 ; Sλi ≡ ηi

2Im(λ)1 + |λ|2 ; Cλ ≡

1− |λ|21 + |λ|2 , (5.7)

where it follows that:

ηiλ = −Dλi + iSλi1 + C

; ηiλ

= −Dλi − iSλi1− C ; |λ2| = 1− C

1 + C, (5.8)

CHAPTER 5. CPT VIOLATION IN B0S → J/ψφ 27

Ai(t) = [ (g+ − zg−) + ηiλg−√

1− z2 ]αi

=[

(g+ − zg−) + g−

[−Dλi + iSλi1 + C

√1− z2

] ]αi ,

(5.9)

Ai(t) = [ (g+ + zg−) + g−niλ

√1− z2 ]αi

=[

(g+ + zg−)− g−[Dλi + iSλi

1− C√

1− z2] ]

αi ,(5.10)

where, since αi = Ai(0), it follows that αi = ηiαi [25]. Additionally, by ignoring second-ordercontributions from the CPT-violating parameter z, and defining2 Dλi = ηiD and Sλi = ηiS, thedecay amplitudes can be expressed as:

Ai(t) =[

(g+ − zg−) + g−ηi

[−D + iS

1 + C

] ]αi , (5.11)

Ai(t) =[ηi(g+ + zg−) −

[D + iS

1− C

] ]ηiαi . (5.12)

Squaring the amplitudes, we obtain the ten time-dependent terms that appear in the first columnof Tables 5.1 and 5.2:

A∗i (t)Aj(t) = α∗iαj1 + C

[ g2+(1 + C)− zg∗+g−(1 + C) + g∗+g−(−D + iS)ηj

− z∗g∗−g+(1 + C)− z∗g2−(−D + iS)ηj − g∗−g+(D + iS) + g2

−(D + iS) + g2−(1− C)ηiηj ] ,

(5.13)

A∗i (t)Aj(t) = α∗iαjηiηj1− C [g2

+(1− C) + zg∗+g−(1− C)− g−g∗+(D + iS)ηj

+ z∗g∗−g+(1− C)− z∗g2−(D + iS)ηj − g∗−g+(D − iS)ηi − zg2

−(D − iS)ηi + g2−(1 + C)ηiηj] .

(5.14)

Using the following expressions that follow from the definitions of the g+ and g−:

|g±|2 = e−Γst

2 [ cosh(∆Γst/2)± cos(∆mst) ] , (5.15)

g∗+g− = e−Γst

2 [ − sinh(∆Γst/2) + i sin(∆mst) ] , (5.16)

and furthermore defining the following quantities:

R+ij ≡

1 + ηiηj2 ; R−ij ≡

1− ηiηj2 ; RIm

ij ≡ iηi − ηj

2 ; RReij ≡

ηi + ηj2 , (5.17)

2The ηi and ηj , will be absorbed in the time-dependent part and will not appear explicitly in S,D and Cterms. These coefficients can be written in terms of the |λ| and φs.

CHAPTER 5. CPT VIOLATION IN B0S → J/ψφ 28

equations 5.13 and 5.14 can be written as:

A∗i (t)Aj(t) = α∗iαje−Γst

1 + C[

cosh(∆Γst/2)[(R+ij + R−ijC) + Re(z)(RRe

ij D + RImij S) + Im(z)(RIm

ij D − RReij S)]

+ cos(∆mst)[(R−ij + R+ijC)− Re(z)(RRe

ij D + RImij S)− lm(z)(RIm

ij D − RReij S)]

+ sinh(∆Γst/2)[(RReij D + RIm

ij S) + Re(z)(1 + C)]+ sin(∆mst)[(RIm

ij D − RReij S) + Im(z)(1 + C)] ] ,

(5.18)

A∗i (t)Aj(t) = α∗iαjηiηje−Γst

1− C [

cosh(∆Γst/2)[(R+ij − R−ijC) + Re(z)(−RRe

ij D + RImij S)− Im(z)(RIm

ij D + RReij S)]

+ cos(∆mst)[(R−ij − R+ijC)− Re(z)(−RRe

ij D + RImij S) + Im(z)(RIm

ij D + RReij S)]

+ sinh(∆Γst/2)[(RReij D − RIm

ij S) + Re(z)(C − 1)]+ sin(∆mst)[(RIm

ij D + RReij S) + Im(z)(C − 1)] ] .

(5.19)

The previous equations can be combined in a more compact form in the following manner:

A∗qfi (t)Aqf

j (t) = α∗iαje−Γst

1 + qfC[

cosh(∆Γst/2)[(R+ij + R−ijC) + qfRe(z)(RRe

ij D + RImij S) + Im(z)(RIm

ij D − RReij S)]

+ cos(∆mst)[qf(R−ij + R+ijC)− qfRe(z)(RRe

ij D + RImij S)− Im(z)(RIm

ij D − RReij S)]

+ sinh(∆Γst/2)[(RReij D + RIm

ij S) + qf+ijRe(z) + qf

−ijRe(z)C]

+ sin(∆mst)[qf(RImij D − RRe

ij S) + qf+ijIm(z) + qf

−ijIm(z)C] .

(5.20)

where:qf

+ij =

qf if ηiηj > 01 if ηiηj < 0

qf−ij =

1 if ηiηj > 0

qf if ηiηj < 0

A complication arises in the description of the time-dependent part of the B0s → J/ψK+K−

decays if CPT is violated, manifested by the presence of qf+ij, qf

−ij. These terms are odd (tagged)

or even (untagged) depending the combination of the polarization amplitudes. The differen-tial decay rates, described in equation 5.20, quantifies CPT-violating effects in a general phe-nomenological description. In case that we take into account the SME condition, resulting in

CHAPTER 5. CPT VIOLATION IN B0S → J/ψφ 29

the imaginary part to be approximately zero, the previous expression will be simplified to:

A∗qfi (t)Aqf

j (t) = α∗iαje−Γst

1 + qfC[

cosh(∆Γst/2)[(R+ij + R−ijC) + qfRe(z)(RRe

ij D + RImij S)]

+ cos(∆mst)[qf(R−ij + R+ijC)− qfRe(z)(RRe

ij D + RImij S)]

+ sinh(∆Γst/2)[(RReij D + RIm

ij S) + qf+ijRe(z) + qf

−ijRe(z)C]

+ sin(∆mst)qf [RImij D − RRe

ij S]

(5.21)

Comparing equations 5.20 and 5.21, we notice that CPT violation does not affect the sin term,when describing CPT-violating phenomena in the context of SME. Finally, in the case that wehave CPT invariance, the previous expression will be significantly simplified to:

A∗qfi (t)Aqf

j (t) = α∗iαje−Γst

1 + qfC[

cosh(∆Γst/2)[R+ij + R−ijC] + cos(∆mst)qf [R−ij + R+

ijC]+ sinh(∆Γst/2)[RRe

ij D + RImij S] + sin(∆mst)qf [RIm

ij D − RReij S]

(5.22)

5.3 Combining time-dependence including CPT-violatingeffects and angular dependence

In the same manner as with the angular PDF, we construct a tabular representation (see Table5.3) of the time dependent parts of the PDF. As an example the fifth term of the PDF forqf = 1, is illustrated bellow:

T5(t)f5( ~Ω) = |α0||α|||e−Γst cos(δ|| − δ0)× cosh(∆Γst/2)[(1 + |λ|2)/2− Re(z)|λ|cosφs + Im(z)|λ| sinφs]+ cos(∆mst) [(1− |λ|2)/2 + Re(z)|λ| cosφs − Im(z)|λ| sinφs]+ sinh(∆Γst/2)[−|λ| cosφs + Re(z)] + sin(∆mst)[|λ| sinφs + Im(z)]

×12√

2 sin 2θK sin 2θµ cosφh .

(5.23)

Where we express the products α∗iαj in terms of the strong phases, originating from the hadronicmatrix elements that quantify the non-pertubative strong physics (see equations below) and also,rewriting S, D and C in terms of |λ| and φs.

Re(α∗iαj) = Re(|αi||αj|ei(δj−δi)) = |αi||αj| cos(δj − δi) , (5.24)Im(α∗iαj) = Im(|αi||αj|ei(δj−δi)) = |αi||αj| sin(δj − δi) . (5.25)

CHAPTER 5. CPT VIOLATION IN B0S → J/ψφ 30

i Ti(t) Fi αi βi ci di

1 |A0|2 |a0|2e−Γst1+qfC

1 + qfRe(z)D − Im(z)S qfC − qfRe(z)D + Im(z)S D + Re(z)(qf + C) −qfS + Im(z)(qf + C)2 |A|||2

|a|||2e−Γst1+qfC

1 + qfRe(z)D − Im(z)S qfC − qfRe(z)D + Im(z)S D + Re(z)(qf + C) −qfS + Im(z)(qf + C)3 |A⊥|2 |a⊥|2e−Γst

1+qfC1− qfRe(z)D + Im(z)S qfC + qfRe(z)D − Im(z)S −D + Re(z)(qf + C) qfS + Im(z)(qf + C)

4 |AS |2 |aS |2e−Γst1+qfC

1− qfRe(z)D + Im(z)S qfC + qfRe(z)D − Im(z)S −D + Re(z)(qf + C) qfS + Im(z)(qf + C)5 Re(A∗0(t)A||(t))

Re(α∗0α||)eΓst1+qfC

1 + qfRe(z)D − Im(z)S qfC − qfRe(z)D + Im(z)S D + Re(z)(qf + C) −qfS + Im(z)(qf + C)Im(α∗0α||)eΓst

1+qfC 0 0 0 0

6 Im(A∗0(t)A⊥(t)) Re(α∗0α⊥)eΓst1+qfC

qfRe(z)S + Im(z)D −qfRe(z)S − Im(z)D S qfDIm(α∗0α⊥)eΓst

1+qfCC qf Re(z)(1 + qfC) Im(z)(1 + qfC)

7 Im(A∗||(t)A⊥(t))Re(α∗||α⊥)eΓst

1+qfCqfRe(z)S + Im(z)D −qfRe(z)S − Im(z)D S qfD

Im(α∗||α⊥)eΓst

1+qfCC qf Re(z)(1 + qfC) Im(z)(1 + qfC)

8 Re(A∗0(t)AS(t)) Re(α∗0αS)eΓst1+qfC

C qf Re(z)(1 + qfC) Im(z)(1 + qfC)Im(α∗0αS)eΓst

1+qfCqfRe(z)S + Im(z)D −qfRe(z)S − Im(z)D S qfD

9 Re(A∗||(t)AS(t))Re(α∗||αS)eΓst

1+qfCC qf Re(z)(1 + qfC) Im(z)(1 + qfC)

Im(α∗||αS)eΓst

1+dC qfRe(z)S + Im(z)D −qfRe(z)S − Im(z)D S qfD

10 Im(A∗⊥(t)AS(t)) Re(α∗⊥αS)eΓst1+qfC

0 0 0 0Im(α∗⊥αS)eΓst

1+qfC1− qfRe(z)D + Im(z)S qfC + qfRe(z)D− Im(z)S −D + Re(z)(qf + C) qfS + Im(z)(qf + C)

Table 5.3: The tabular representation of the time dependent terms, where: Ti(t) =Fi(αi cosh(∆Γst/2)+βi cos(∆mst)+ci sinh(∆Γst/2)+di sin(∆mst/2)). By combining this tablewith the tables containing the angular parts of the different combinations of the polarizationamplitudes, we can construct the signal PDF. Since the product of two time-dependent polar-ization amplitudes is a complex object, the interference terms are split into the sum of a realand imaginary part.

Decay Time [ps]0 1 2 3 4 5

Dec

ay P

rob.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Figure 5.3: The figure represents the PDF, for qf = 1 and without considering the variousexperimental effects, e.g resolution or acceptance effects, we include however an average mistageffect. The blue solid line represents, the PDF without taking CPT-violating phenomena intoaccount, while for the red dotted line we set Re(z) = Im(z) = 0.2. The angles are set to specificvalues. It can be noticed that the deference between the PDFs with and without CPT violation,is the change in the amplitude and phase of the oscillation.

Chapter 6

Data analysis

The modified PDF that incorporates CPT-violating effects, was described in the previous chap-ters. Taking into account various experimental effects that distort the distribution of the ob-servables will result in the signal PDF. Since in this thesis we only modified the time-dependentcomponents; the experimental effects, such as acceptance effects, that result in the signal PDFwill be briefly discussed, together with the fitting strategy we follow for estimating the param-eters of interest.

6.1 Maximum likelihood fitThe signal PDF depends on several parameters that we are interested in, including the CPT-violating parameters. The modified PDF together with the data, will be used to construct thelikelihood function, where by using the maximum likelihood theorem [28] the parameters ofinterests are extracted. Since the method that is used to estimate the parameters of interest isthe likelihood maximization, a brief description of the method is given. For a set of independentmeasurements (data), where ~O are the observables and a set of parameters indicated by ~Π, thelikelihood function for the normalized PDF can be written as:

L = L1 × L2 × ...× LN= PDF (O1; ~Π)× PDF (O2; ~Π)× ...× PDF (ON ; ~Π)=∏i=N

PDF ( ~O; ~Π) .(6.1)

The set of parameters ~Π that maximizes equation 6.1, consists of the estimations of the pa-rameters of interest. Maximizing the joint likelihood is equivalent of either maximizing thelog-likelihood or minimizing the negative log-likelihood. The following sets of parameters areestimated for different parameterizations of the PDF :• In the phenomenological description;

~Π = (|A0|2, |A|||2, |A⊥|2, |As|2, δ0, δ||, δ⊥, δS, Γs, ∆Γs, ∆ms, φs,Re(z), Im(z)).

• In the phenomenological description, when setting Im(z) to be zero;~Π = (|A0|2, |A|||2, |A⊥|2, |As|2, δ0, δ||, δ⊥, δS, Γs, ∆Γs, ∆ms, φs,Re(z)).

• In the context of the SME;~Π = (|A0|2, |A|||2, |A⊥|2, |As|2, δ0, δ||, δ⊥, δS, Γs, ∆Γs, ∆ms, φs, ∆α0− 0.38∆αZ , ∆αX , ∆αY ).

31

CHAPTER 6. DATA ANALYSIS 32

In practice, we minimize the negative log-likelihood (NLL) and consequently we find the valuesfor the parameters of interest that maximize the likelihood. The negative log-likelihood, is thesum of the individual NLL’s for each event:

w(Oi; ~Π) = − lnPDF (O; ~Π)= −

∑i=N

lnPDF (Oi; ~Π) . (6.2)

The set of parameters that minimizes the negative log-likelihood is found by solving the followingequation:

dw

dΠi

= −∑i=N

ln PDF(Oi; ~Π)∫PDF( ~O; ~Π)

= 0 . (6.3)

Where we denote with PDF and PDF the unnormalized and normalized PDF respectively.

PDF (Oi; ~Π) = PDF(Oi; ~Π)∫PDF( ~O; ~Π)

. (6.4)

A notable situation arises when the unnormalized PDF is multiplied with a function that doesnot depend on the parameters of interest but depends on the data. An example would be aknown efficiency function, such as the angular efficiency. In that case the normalized PDF canbe written as:

PDF (Oi; ~Π) = PDF(Oi; ~Π)f(Oi)∫PDF( ~O; ~Π)f( ~O)d~O

, (6.5)

and equation 6.6, can be re-written as:

dw

dΠi

= − d

dΠi

∑i=N

ln PDF(Oi; ~Π)∫PDF( ~O; ~Π)f( ~O)d~O

− d

dΠi

∑i=N

ln f(Oi)∫PDF( ~O; ~Π)f( ~O)d~O

. (6.6)

Since f( ~O) is independent of the parameters of interest, the second term in equation 6.6 is zero.We can deduce that f( ~O) affects only the normalization.

In order to find the standard deviation of the parameters of interest we have to computethe variance-covariance matrix. The standard errors are defined to be the square roots of thediagonal elements. The variance-covariance matrix is given by [29]:

var(Π) =(−E[∂ lnL

∂Πi

∂ lnL∂Πj

])−1

. (6.7)

The second derivative of the likelihood function represent the curvature. Hence, the standarderrors give an estimate of the shape of the negative log-likelihood around the minimum. Amongthe conditions that the maximum likelihood theorem requires is that the shape of the likelihoodfunction around the minimum is normally distributed. Hence, the shape of the negative log-likelihood around the minimum should be a parabola.

NLL ≈ 12σ2

Πi

(Πi −Πi)2 + C , (6.8)

where Πi represents the expectation value of Πi, C is the value of the likelihood in the minimumand σΠi is the width of the parabola. It become obvious that the more narrow the parabola, the

CHAPTER 6. DATA ANALYSIS 33

smaller the standard deviation. So, we can interpret the results for the parameters estimates interms of confidence intervals [29]; where the interval which is contained in −σΠi < Πi−Πi < σΠi ,defines a 68% confidence level. However, what we assumed is that the shape of the likelihoodaround the minimum has a parabolic shape, resulting in symmetric errors around the minimum.In case the parameters are highly correlated or the data sample in not sufficiently large, theparabolic shape of the likelihood can be distorted, resulting in non-symmetric errors. Studiesfor the behavior of the likelihood for the CPT-violating parameters is done and presented in theanalysis result chapter.

6.2 Reconstruction and selectionBackground events are removed as much as possible from the data sample, before we fit with themodified PDF. An event selection of B0

s → J/ψK+K− is done, to remove as many backgroundevents as possible.

The first step in the extraction of B0s → J/ψK+K− candidates is the trigger selection.

The level-zero (L0) hardware trigger selects events containing muons with sufficient transversemomentum pT . Subsequently, the high-level triggers (HLT) select events into two steps: HLT1and HLT2. In both stages there are two types of selections. A selection type that is using asa criterion the distance the B0

s travel before its decay (biased), which affects the decay-timedistribution and the one that does not use this information (unbiased), which, of course, doesnot introduce any time-decay acceptance effects.

1. HLT1

(a) The unbiased HLT1, after reconstruction of the µ+µ− pair, requires the two candi-dates to have an invariant mass of at least 2.7 GeV/c2 and the two muon trajectoriesto be significantly close in order to originate from one vertex [16]. That will result ineliminating a significant amount of combinatorial background [27].

(b) The biased HLT1, does not use the invariant mass, but selects single tracks that aresignificantly detached from any primary vertex.

2. HLT2

(a) The biased and unbiased HLT2, require the reconstructed J/ψ → µ+µ− candidatesto have an invariant m(µ+µ−) mass between 2.86 GeV/c2 and 3.34 GeV/c2.

(b) In addition to the previous requirement the biased HLT2 requires a minimum decay-length significance (DLL) of three [16].

After the offline reconstruction process where the di-muon candidates are matched with K+K−

pairs, a so-called stripping selection is imposed, in order to obtain the final dataset that containsthe B0

s → J/ψK+K− candidates used for the determination of the CPT-violating parameters[16,27].

After the selection, the data sample still contains some background events. For the sub-traction of the combinatorial background the events are weighted (sWeights) by using the sF ittechnique [30] in order to cancel the combinatorial background in the sidebands and the signalregion.

CHAPTER 6. DATA ANALYSIS 34

]2) [MeV / c-K+ Kψm(J/

5300 5350 5400 5450

)2Si

gnal

Eve

nts

/ (1.

17 M

eV /

c

500

1000

1500

2000

2500

3000

Figure 6.1: Invariant mass of the signal events.

Before computing the sWeights, the presence of other non-combinatorial background decaysmust be treated. Two additional background decays are taken into account, where both of themoriginate from particles misidentification. These decays are the B0

s → J/ψK0∗(→ K+π−) andthe Λ0

b → J/ψpK−. The invariance mass of these decays peaks underneath the invariant mass ofB0s and for this reason are called peaking background decays (also called resonant or reflection

background). The peaking background decays are subtracted from the data sample by injectingMC peaking background events with negative weights1, in order to cancel out the likelihoodcontribution of the equivalent peaking background events that are present in the data sample.

6.3 Decay time and acceptanceThe decay time of the B0

s is determined from the measured distance between the primary vertex(PV) and the secondary vertex (SV) and the measured momentum. The uncertainty on themeasurement of the decay-time distort the theoretical PDF. The uncertainty on the decay-time(0.05 ps), is taken into account by convolving the theoretical PDF of the B0

s → J/ψK+K−,with a resolution model:

PDF (t, ~Ω) = R(t|σt)⊗ PDFphys(t, ~Ω) , (6.9)

where the PDFphys represents the theoretical PDF and is given by equation 5.2. The decay-time resolution model is taken to be a double Gaussian. The widths of which are taken to be afunction of a per-event estimated decay-time uncertainty [27].

An additional effect that distort the time-dependent part of the theoretical model, is thedecay time efficiency. As it was described before in the selection process, both the HLT1 andHLT2 use as information the decay time for the selection. This introduces a time-dependent

1The sum of the normalized weights is equal to the number of the peaking background decays, which areestimated to be approximately seven thousand candidates.

CHAPTER 6. DATA ANALYSIS 35

acceptance effect. The decay-time acceptance is taken into account by multiplying the timedependent terms with a decay-time acceptance function:

PDF (t, ~Ω) ∝ ε(t) · PDFphys(t, ~Ω) , (6.10)

where ε(t) is the time dependent acceptance. The time acceptance is a product of two functionsthat quantify different experimental effects. Monte Carlo studies have shown that the efficiencyof the reconstruction is reduced for large decay times, this will introduce an effect [16] that isparametrized as:

ε(t) = 1 + βt ≈ eβt , (6.11)The ε(t) then can be absorbed by the exponential, that is present in the decay rates (see equation5.20).

Γs → Γs + β ≡ Γ effs . (6.12)

The parameter β is evaluated separately for the 2011 and 2012 runs and is found to beβ2011 = −0.0090 ± 0.0022 ps−1 and β2012 = −0.0124 ± 0.0019 ps−1. These parameters areGaussian constrained in the nominal fit. An additional acceptance is introduced by the trigger-requirements. The acceptance functions for the events that are also selected by the bias triggerselection have a non-uniform shape. This shape is determined by overlapping biased and unbi-ased events. From this process histograms for the different trigger categories are made, repre-senting the acceptance in small decay times. Finally, this acceptance functions are multipliedwith the theoretical PDF.

6.4 Angular resolution and acceptanceImperfections in the tracking will lead to a difference between the true and measured valuesfor the angles that describe the angular distribution of the B0

s → J/ψK+K−. The angularresolution has been studied by using fully simulated MC events and is found to be approximately0.01 rad [16]. By ignoring the angular resolution it is found that the bias on the parameters ofinterests is negligible and for this reason it is not taken into account in this analysis. Additionally,the geometry of the LHCb spectrometer and the selection criteria introduce an acceptance effecton the decay angles, that must be taken into account [26].

The angular PDF can be re-written in terms of associated Legendre polynomials P ji (cos θK)

and spherical harmonics Yl,m(cos θµ, φh) [25] and consequently taking advantage of the orthogo-nality properties of these functions. As an example the fifth term of the PDF for qf = 1 will bere-written as:

T5(t)f5( ~Ω) =Re(α∗0α||)e−Γst

1 + C×

cosh(∆Γst/2)[1 + Re(z)D − Im(z)S] + cos(∆mst)[C − Re(z)D + Im(z)S]+ sinh(∆Γst/2)[D + Re(z)(1 + C)] + sin(∆mst)[−S + Im(z)(1 + C)]

× 2√

2√

35P

12 Y2,1 .

(6.13)

The angular acceptance is taken into account, by multiplying the theoretical PDF with theangular acceptance function:

PDF (t, cos θK , cos θµ, φh) = ε(cos θK , cos θµ, φh) · PDFphys(t, cos θK , cos θµ, φh) , (6.14)

CHAPTER 6. DATA ANALYSIS 36

The angular acceptance function is also expressed in terms of associated Legendre polynomialsand spherical harmonics, multiplied by equivalent weights:

ε(cos θK , cos θµ, φh) =∑i,l,m

cil,mPi(cos θK)Yl,m(cos θµ, φh) . (6.15)

The weights are determined by the Monte Carlo events, and describe the shape of the angularacceptance function.

6.5 Flavour taggingIn a previous chapter the theoretical decay rates that describe the time evolution of the signalsample of the B0

s → J/ψK+K− decays, including CPT-violating effects are derived. Thedifferential decay rates that corresponds to a B meson being originally B0

s , B0s or untagged, are

distinguished by the variable qf that takes the values 1,−1 and 0 respectively. Two algorithmsthat determine the initial flavour of the B0

s mesons are used for this analysis: the opposite-sidetagging (OS) and the same-side tagging (SS).

The OS tagging algorithm determines the initial flavour of the B meson that correspondsto the signal event by the flavour of the spectator b-hadron. There are several methods thatare used to for this purpose, such as using the charge of the leptons that are produced in asemileptonic B decays, or the charge of the kaons produced in b → c → s transitions and thetotal charge of the decay vertex [27]. These methods are combined to optimize the taggingdecision. On the other hand, the SS tagging algorithm uses the information regarding thehadronization of the spectator s quark, which is usually hadronized into a charged kaon. Sincethe charge of the kaon is correlated with the initial flavour of the B0

s , the flavour of the signal-Bmeson can be identified.

Even though two different tagging algorithms are used for the determination of the initialflavour of the signal B mesons, there is a probability that this estimation is wrong (see Figures6.2a and 6.2b). For this reason the estimate of the initial flavour is accompanied with a per-event mistag probability, denoted as ηalg, where alg stands for the tagging algorithm. Eventsare classified as tagged with 0 ≤ ηalg < 0.5 and untagged with ηalg = 0.5.

Both of the tagging algorithms are calibrated using self-tagging decays, e.g B+ → J/ψK+.The following relation between the estimated and the actual mistag probability is assumed, forB0s and B0

s :

ωalg = (palg0 + ∆palg02 ) + (palg1 + ∆palg1

2 )(ηalg − 〈ηalg〉) , (6.16)

ωalg = (palg0 −∆palg0

2 ) + (palg1 −∆palg1

2 )(ηalg − 〈ηalg〉) , (6.17)

where palgi (i = 0, 1) are averaged calibration parameters for the B0s and B0

s and ∆palgi arethe differences between the tagging calibration parameters. The 〈ηalg〉, is the average mistagprobability of the B0

s → J/ψK+K− and appears in order to minimize the correlation betweenthe tagging calibration parameters.

CHAPTER 6. DATA ANALYSIS 37

OSη0 0.1 0.2 0.3 0.4 0.5

Sign

al E

vent

s / (

0.0

011

)

0

100

200

300

400

500

600

(a)

SSη0 0.1 0.2 0.3 0.4 0.5

Sign

al E

vent

s / (

0.0

011

)

0

500

1000

1500

2000

2500

(b)

Figure 6.2: Estimated wrong-tag probability for the (a) opposite-side tagging and (b) same-sidetagging algorithms in the decay of B0

s → J/ψK+K−.

The data sample is split into four categories, according to the tagging decision: exclusiveOS tagged events (qOSf 6= 0 and qSSf = 0), exclusively SS tagged events (qSSf 6= 0 and qOSf = 0),double tagged events (qSSf 6= 0 and qOSf 6= 0) and untagged events (qOSf = 0 and qSSf = 0). Thelatter represents 33% of the total sample [27].

The time-dependent decay rate, without taking into account resolution or acceptance effects,but including the wrong tag probability can be written in the following manner:

PDF (t, ~Ω|qfOS, qf

SS, ηOS, ηSS) ∝(1 + qf

OS(1− 2ωOS))(1 + qfSS(1− 2ωSS))PDFphys(t, Ω|B0

s )+(1− qf

OS(1− 2ωOS))(1− qfSS(1− 2ωSS))PDFphys(t, Ω|B0

s ) .(6.18)

The previous equation describes the PDF in case that CPT violation is incorporated in thedecay rates as a constant parameter. In the parametrization where the CPT-violating parameterexhibits sidereal modulation, we substitute Re(z)→ Re(z)(tGPS, PB0

s), see equation 4.78. Hence

the time dependent decay rate is expressed as:

PDF (t, ~Ω|qfOS , qf

SS , ηOS , ηSS , tGPS , PB0s) ∝

(1 + qfOS(1− 2ωOS))(1 + qf

SS(1− 2ωSS))PDFphys(t, Ω|B0s )+

(1− qfOS(1− 2ωOS))(1− qf

SS(1− 2ωSS))PDFphys(t, Ω|B0s ) .

(6.19)

Where two additional conditional observables are added. The PDF is normalized separately forB0s and B0

s , which ensures that small normalization asymmetries, such as a possible productionor detection asymmetry between B0

s/B0s , can be neglected.

6.6 Decay rate model

To construct the final signal PDF that incorporates CPT-violating effects, the decay-time res-olution, decay-time acceptance, the angular acceptance and the flavour tagging must be taken

CHAPTER 6. DATA ANALYSIS 38

into account. So, equations 6.18 and 6.19, would be extended to:

PDF (t, ~Ω|qfOS, qf

SS, ηOS, ηSS, σt) ∝(1 + qf

OS(1− 2ωOS))(1 + qfSS(1− 2ωSS))ε( ~Ω)ε(t)[R(t|σt)⊗ PDFphys(t, Ω|B0

s )]+(1− qf

OS(1− 2ωOS))(1− qfSS(1− 2ωSS))ε( ~Ω)ε(t)[R(t|σt)⊗ PDFphys(t, Ω|B0

s )] .(6.20)

PDF (t, ~Ω|qfOS, qf

SS, ηOS, ηSS, σt, tGPS, PB0s) ∝

(1 + qfOS(1− 2ωOS))(1 + qf

SS(1− 2ωSS))ε( ~Ω)ε(t)[R(t|σt)⊗ PDFphys(t, Ω|B0s )]+

(1− qfOS(1− 2ωOS))(1− qf

SS(1− 2ωSS))ε( ~Ω)ε(t)[R(t|σt)⊗ PDFphys(t, Ω|B0s )] .

(6.21)

where ε(t) and ε( ~Ω) represent the angular and time acceptance respectively and R(t|σt) is thetime resolution.

6.7 Sun-centered frame and external parametersAs it was noticed in a previous chapter, regarding the CPT violation in the context of the SME,the CPT-violating parameter Re(z), will vary in time with a periodicity that depends on theEarth’s sidereal frequency.

Re(z) ∝ βµ∆aµ . (6.22)

Since a fixed laboratory frame and a non-rotating frame, named Sun-Centered Frame (SCF)are used for the derivation of equation 6.22, the expression for the Re(z) from which the ∆αµcoefficients are going to be determined, will depend on the intrinsic characteristics of the lab-frame, which in this analysis, is the LHCb detector. The exact position and direction of theLHCb detector, the direction of the beam and the exact orientation of the LHCb detector inthe geodetic plane are required.

AngleAngleAngle radradradlongitude λ 0.807064± 0.000002latitude l 0.106401± 0.000002azimuth θ 4.12414± 0.00005tilt α 0.0036010± 0.0000005

Table 6.1: Angles that define the position and direction of the LHCb beam. The tilt accounts forthe non-perfect alignment of the LHCb coordinate system with respect to the geodetic plane [31].

In Figure 6.3 the Sun-Centred Frame is illustrated.

CHAPTER 6. DATA ANALYSIS 39

Figure 6.3: Sun-Centered Frame defined with the right handed base X, Y and Z. Where Zpointing along the axis of Earth’s rotation, X points to the vernal equinox and Y = Z× X [32].

.The vernal equinox (V) is one of the two points on the celestial sphere where the celestial

equator and the plane defined by X and Y intersect. For the definition of X and Y see Figure6.3. Since V is a fixed point on the celestial sphere, as Earth’s rotates, V will rotate in theopposite direction. Due to Earth’s precession, V moves eastwards with a rate of approximatelyone degree per 72 years. So to be accurate the first point of Aries should be accompaniedwith the date, of the measurement from which its position was defined. Therefore, the SCF isdefined at a specific moment in time; that is the J2000 epoch, defined as 1/1/2000 at 12h UT1(conceptually is the mean solar time at 00 longitude, where distant stars are used for the precisemeasurement of the Sun’s position). Finally the sidereal phase (Ωt), or the time dependentangle of the LHCb beam with respect to the X axis (vernal equinox) of the SCF, has to bedetermined. The sidereal frequency can be express as:

Ω = 2πTsid

= 7.29211515 · 10−5rad · sec−1 . (6.23)

The time keeping in LHCb is done by a GPS receiver. Each event is characterized by a time-stamp (tGPS), which gives the number of seconds in UTC, (UTC≈UT1) that elapsed fromthe 1/1/1970 at 00:00:00 UTC (Unix epoch), to the time that the event is collected. Using thetime-stamp of an event, the corresponding sidereal phase can be found using the following lineartransformation [33]:

Ωt = ΩtGPS + t0 (6.24)

The previous transformation can be interpreted in the following manner. The first term ΩtGPScorresponds to the number of Earth’s rotations around its axis in a time interval equal to tGPS.However, since the Unix time is just the sum of seconds since 1/1/1970 at 00:00:00 UTC, it isindependent of the geodetic position, meaning that if two events, one in LHCb and one in SLAChave the same time-stamp, the term ΩtGPS for both of them is the same. However, what is notthe same is the angle of their beam axis with respect to the X axis, at the time that the Unixepoch started. So the only difference between these two events is the angle of their beam withthe vernal equinox at the start of the Unix epoch. As a result the second term becomes obvious.

CHAPTER 6. DATA ANALYSIS 40

The term t0 is a phase shift that represents the sidereal phase of the Greenwich meridian at theUnix epoch plus the angle of the beam with respect to the Greenwich meridian [31].

AngleAngleAngle radradradt0 0.7363± 0.0004χ 1.961872± 0.0000005cos(χ) −0.38118± 0.00005sin(χ) 0.92450± 0.00005

Table 6.2: Angles that define the position and direction of the LHCb beam in the SCF [33].

Using the values for the parameters that define the position of the direction of the LHCb beam,equation 4.78 will be re-written as:

Re(z) = γ(pz)∆ms

∆m2s + 1/4∆Γ 2

s

∆α0 − 0.38 · β∆αZ + 0.92 · β∆αY sin(ΩtGPS + t0)

+∆αXcos(ΩtGPS + t0)(6.25)

]°[labθ5 10 15

Sign

al E

vent

s / (

0.0

4 )

0

500

1000

1500

2000

2500

(a)

γ50 100 150

Sign

al E

vent

s / (

0.4

)

0

1000

2000

3000

4000

5000

6000

(b)

Figure 6.4: (a) Distribution of the azimuthal angle of the B0s mesons with respect to the LHCb

Z-axis, with 〈θLab〉 = 3.0120, (b) distribution of the Lorentz factor.

From Figure 6.4, it becomes obvious that B0s are, in very good approximation, collimated

along the Z axis. Additionally, since the B0s are very boosted, with 〈γ〉 = 17.48, the β factor

can be set to one. The average B0s momentum in the Z axis is 〈Pz〉 = 93.6 GeV/c2, so the

γ =√

(1 + ( Pzms

)2 ≈ Pzms

. Finally equation 6.25 can be expressed as2:

Re(z) = ∆msPzms[∆m2

s + 1/4∆Γ 2s ]∆α0 − 0.38 ·∆αZ + 0.92 · ∆αY sin(ΩtGPS + t0)

+∆αXcos(ΩtGPS + t0)(6.26)

2In the collimated analysis we assume that the B mesons are aligned to the Z axis. However, since this is notexactly true we use as a B mesons momentum the Pz. The same fit is performed by using the total momentuminstead. The difference in the parameters of interest is found to be negligible, which was expected (see text).

CHAPTER 6. DATA ANALYSIS 41

Where ms = 5366.77 MeV/c2 the B0s mass. As it was also done in the φs analysis [1], ∆ms is

unconstrained in the fit. Finally, what must be noticed is that since sinχ = 0.98, the siderealoscillation of Re(z) is close to maximum. This make the LHCb detector ideal for searchingCPT-violating effects in the context of the SME.

Chapter 7

Analysis resultsThe modified fitting algorithm for B0

s → J/ψφ decays that now incorporates CPT-violatingphenomena, is applied to the data, after background subtraction, yielding the parameters ofinterest. The statistical uncertainties are estimated from the shape of the negative log-likelihoodaround the minimum. The general parametrization of the time-dependent components of thePDF is chosen in a way that CP violation in the decay is identical among all the intermediatestates i ∈ 0,⊥, ‖, S and that CPT-violating phenomena are introduced only in the mixingprocess (i.e. indirect CPT violation). The CPT violation in the mixing process is parametrizedin three different ways.

The first parametrization is introduced from a purely phenomenological point of view andis not connected with any microscopic model, describing the evolution of the Bs-Bs system byan additional complex parameter z in the decay rates, represented by Re(z) and Im(z).

The second parametrization involves the measurement of CPT violation in the context ofthe SME extension. As described in chapter 2, it can be shown that in the B0

s system theimaginary part of the z parameter can be neglected; this results in the z parameter to be real.Furthermore, since in the context of this model the z parameter would exhibit sidereal variationsand momentum dependence, it is expressed as a function of the sidereal time, the B0

s mesonmomentum and the CPT-violating coefficients ∆α0, ∆αX , ∆αY , ∆αZ , that are integrated in theSME Lagrangian. Since we perform a collimated analysis, coefficients ∆α0, ∆αZ are combinedas: ∆α0 − 0.38∆αZ .

Finally, for the parameterization of the phenomenological model we set Im(z) = 0 (SMEconstrain) and measure only Re(z). We choose to perform this measurement as well, in order toestimate the effective CPT violation in the context of SME. In that case Re(z) is approximatelythe value of the ∆α0− 0.38∆αz, which depends obviously from the characteristics of the LHCbbeam and must be carefully related to results from other experiments.

7.1 Parameter estimatesThe parameter estimates are presented in Tables 7.1, 7.2 and 7.3. The first column containsthe values and the statistical uncertainties of the parameter estimates for the CPT invariantcase, the second column contains the values and the statistical uncertainties of the estimatedparameters if we take into account the possibility for CPT violation. The third column containsthe difference (pull) of the parameters of interest in the two cases. Correlations between theparameters in all the different parameterizations are also presented in Tables 7.4, 7.5 and 7.6,where correlations higher than 30% are highlighted.

42

CHAPTER 7. ANALYSIS RESULTS 43

Table 7.1: Parameter estimates for the phenomenological description.

parameter No CPT With CPT difference (pull)Im(z) −− 0.004± 0.011 −−Re(z) −− −0.022± 0.033 −−Γs[ps−1] 0.6592± 0.0031 0.6590± 0.0031 −0.00010 (−0.033)∆Γs [ps−1] 0.0785± 0.0092 0.0790± 0.0092 0.00053 ( 0.058)∆ms[ps−1] 17.723± 0.057 17.730± 0.061 0.0068 ( 0.119)φs[rad] −0.057± 0.050 −0.070± 0.056 −0.0124 (−0.250)|λs| 0.963± 0.019 0.958± 0.020 −0.0045 (−0.238)

Table 7.2: Parameter estimates when setting Im(z) = 0 for the phenomenological description.

parameter No CPT With CPT difference (pull)Re(z) −−−− −0.022± 0.033 −−−−Γs[ps−1] 0.6592± 0.0031 0.6591± 0.0031 −0.00006 (−0.021)∆Γs [ps−1] 0.0785± 0.0092 0.0787± 0.0091 0.00020 ( 0.022)∆ms[ps−1] 17.723± 0.057 17.721± 0.056 −0.0025 (−0.045)φs[rad] −0.057± 0.050 −0.060± 0.050 −0.0024 (−0.049)|λs| 0.963± 0.019 0.959± 0.019 −0.0033 (−0.177)

Table 7.3: Parameter estimates in the context of SME.

parameter No CPT With CPT difference (pull)∆αX [10−14 GeV] −− 0.921± 2.106 −−∆αY [10−14 GeV] −− −3.817± 2.106 −−

∆α0 − 0.38∆αZ [10−14 GeV] −− −0.987± 1.382 −−Γs [ps−1] 0.6592± 0.0031 0.6592± 0.0031 −0.00008 (−0.025)∆Γs [ps−1] 0.0785± 0.0092 0.0784± 0.0091 0.00005 ( 0.005)∆ms[ps−1] 17.723± 0.057 17.720± 0.056 0.0030 ( 0.053)φs[rad] −0.057± 0.050 −0.065± 0.050 −0.0081 (−0.162)|λs| 0.963± 0.019 0.961± 0.019 −0.0021 (−0.112)

Table 7.4: Correlation matrix for the phenomenological description.

Re(z) Im(z) φs |λs| Γs ∆Γs ∆msRe(z) 1 −0.008 0.058 0.266 0.028 −0.023 0.049Im(z) − 1 −0.463 −0.043 −0.146 0.095 0.384φs − − 1 0.048 0.031 −0.115 −0.100|λs| − − − 1 0.016 −0.003 −0.235Γs − − − − 1 −0.402 −0.01∆Γs − − − − − 1 0.001∆ms − − − − − − 1

CHAPTER 7. ANALYSIS RESULTS 44

Table 7.5: Correlation matrix when setting Im(z) = 0 for the phenomenological description.

Re(z) φs |λs| Γs ∆Γs ∆msRe(z) 1 0.063 0.27 0.027 −0.024 0.056φs − 1 −0.002 0.01 −0.073 0.085|λs| − − 1 0.011 0.004 −0.179Γs − − − 1 −0.401 0.005∆Γs − − − − 1 −0.033∆ms − − − − − 1

Table 7.6: Correlation matrix in the context of SME.

∆αX ∆αY ∆α0 − 0.38∆αZ φs |λs| Γs ∆Γs ∆ms∆αX 1 −0.006 0.032 0.019 0.012 −0.002 0.015 −0.001∆αY − 1 0.042 0.073 −0.010 −0.009 0.019 0.007

∆α0 − 0.38∆Z − − 1 0.077 0.236 0.022 −0.027 0.038φs − − − 1 0.000 0.005 −0.064 0.070|λs| − − − − 1 0.007 0.004 −0.179Γs − − − − − 1 −0.40 0.001∆Γs − − − − − − 1 −0.219∆ms − − − − − − − 1

7.2 Binned fit

A different way to estimate the CPT-violating quantities ∆αX , ∆αY and ∆α0 − 0.38∆Z is bysplitting the data-sample in twenty four bins of sidereal phase and fitting for Re(z). For eachbin we perform an unbinned likelihood fit for Re(z) with the modified PDF that we used in thenominal fit. Finally, we fit Re(z) as a function of the sidereal phase using the expression givenin equation 6.25. In this fit we approximate the average boost factor, 〈γ(p)〉 = γ(〈p〉) ≈ 17.23,β = 1 and for ∆ms and ∆Γs we set the values, obtained by the nominal fit.

)πSidereal Phase / ( 20 0.2 0.4 0.6 0.8 1

Sign

al E

vent

s / (

0.0

022

)

460

480

500

520

540

560

580

Figure 7.1: Distribution of the B0s → J/ψK+K− signals decays in the sidereal phase. The signal

events are uniformly distributed.

CHAPTER 7. ANALYSIS RESULTS 45

)πSidereal Phase / ( 20 0.5 1

Re(

z)

0.6−

0.4−

0.2−

0

0.2

0.4

Figure 7.2: The sidereal oscillation of Re(z) as it is obtained from the binned (blue solid line)and the unbinned (dashed green line) fit, while the dotted red line represents the effective valueof Re(z).

The resulting distribution illustrated in Figure 7.2 represented by the blue solid line is afit using equation 6.25. The estimated values for the CPT-violating coefficients obtained withthis strategy, are illustrated in Table 7.7. The first and second columns contain the valuesobtained from the binned and unbinned (nominal) fit respectively. The third column containsthe difference of the estimated parameters between the two different fitting strategies.

The difference between the two estimates for the CPT-violating parameters is defined (forstrongly correlated parameters) as [34]:

∆± σ∆ = α1 − α2 ±√|σ2

1 − σ22| (7.1)

where the parameters α1, α2 and σ1, σ2 correspond to the parameters estimates and the relevantstatistical errors obtained by the two different fitting strategies. Since ∆ ± σ∆ is compatiblewith zero, we can conclude that the results obtained with both strategies agree.

Table 7.7: Parameter estimates for the CPT-violating coefficients with the binned and theunbinned fit.

parameter binned fit unbinned fit difference (∆± σ∆)∆αX [10−14GeV] −0.526± 3.949 0.921± 2.106 0.394± 3.356∆αY [10−14GeV] −7.108± 3.883 −3.817± 2.106 2.764± 3.356

∆α0 − 0.38∆αZ [10−14GeV] −3.159± 2.501 −0.987± 1.382 2.172± 2.084

CHAPTER 7. ANALYSIS RESULTS 46

7.3 Analysis of subsets of the data sampleThe data sample is split into different subcategories. The modified PDF is fitted on the resultingsubsets. The data sample is split into two random samples, into two samples containing oddand even number of the event index, into two subsets depending on the polarity of the magnetand finally in periods of 2011 and 2012. The results of the previous validations are presented inFigures 7.3a - 7.3f for all the different parameterizations.

Nom

inal

Split

1

Split

2

Split

Odd

Split

Eve

n

+ P

olar

ity

- P

olar

ity

2012

dat

a

2011

dat

a

Re(

z)

0.1−0.08−0.06−0.04−0.02−

0

0.02

0.04

0.06

(a)

Nom

inal

Split

1

Split

2

Split

Odd

Split

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n

+ P

olar

ity

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olar

ity

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dat

a

2011

dat

a

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)

0.02−

0.01−

0

0.01

0.02

0.03

(b)

Nom

inal

Split

1

Split

2

Split

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+ P

olar

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olar

ity

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dat

a

2011

dat

a

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z)

0.1−0.08−0.06−0.04−0.02−

0

0.02

0.04

0.06

(c)

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inal

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dat

a

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a

GeV

]-1

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0Xα∆

8−6−4−2−0

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6

8

(d) b

Nom

inal

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dat

a

2011

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a

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]-1

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0Yα∆

10−

8−

6−

4−

2−

0

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(e)

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inal

Split

1

Split

2

Split

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+ P

olar

ity

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olar

ity

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a

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dat

a

GeV

]-1

4[1

0Zα∆

-0.3

80α∆

5−4−3−2−1−0

12

3

(f)

Figure 7.3: Results of the fit on the different subsets of the data sample for the phenomenologicaldescription: (a), (b), (f) and in the context of SME: (c), (d), (e).

CHAPTER 7. ANALYSIS RESULTS 47

The results of the previous tests show that there are no significant discrepancies for the CPT-violating parameters, estimated by the different subsets. This means that the modified PDFproduces stable results.

7.4 Validating the modified fitting algorithmIn order to validate the modified fitting algorithm that includes CPT violation, we performed2000 pseudo-experiments, where we studied the distribution of the fitted parameters and thepull distributions for the parameters of interest. In case that there is no bias in the modi-fied algorithm, the distribution for the parameters of interest, Re(z), Im(z), ∆αX , ∆αY and∆α0− 0.38∆Z , should have a mean equal to the value for which the pseudo-data are generated.Furthermore, if the values of the parameters of interest and the uncertainties are estimatedcorrectly then the pull distributions should have a Gaussian shape, with zero mean and widthof one. A non-zero mean will signify a bias, and deviations of the width from one, will indicatean overestimate or underestimate of the statistical uncertainties. The mean and the width ofthe pull distributions are calculated by performing an unbinned maximum likelihood fit with aGaussian PDF. As it can be seen from Figures 7.4a - 7.4f and 7.5a - 7.5f, the toy studies do notshow any bias in our modified PDF in any of the different parametrization that can quantifyCPT-violating phenomena in the mixing process.

7.5 Investigating the statistical errorsSince the parabolic shape of the negative log-likelihood (NLL) around the minimum is only anapproximation, a likelihood scan for the Re(z), Im(z), ∆αX , ∆αY and ∆αZ is performed. Fig-ures 7.6a - 7.6f shows the log-likelihood scans for the CPT-violating parameters. The blue/greensolid line represents the profiled NLL (parameter of interest fixed and floating all the other pa-rameters) minus the NLL value obtained from the nominal fit. A parabola with center at theminimum and a second derivative equal to that obtained by the nominal fit, is shown withthe dotted black line (ideal likelihood). If the negative log-likelihood is parabolic around theminimum the solid and the dotted lines would match.

The errors in all the cases are very symmetric. Small deviations from the parabolic shapeare observed in the likelihood scans of Im(z), ∆αY and ∆αX . However, these deviations are notsignificant; so the assumption of the asymptotic normality is valid. It is also very interesting tosee what is the distribution of the errors, obtain from the fits on the pseudo-data (see Figures7.7a - 7.7f).

7.6 Consistency test using Monte Carlo eventsAs an additional way to validate the modified fitted algorithm we use 25000 Monte Carlo events,generated without CPT violation. We fit these events with the modified PDF that contains theCPT-violating parameters, as a consistency test. If there is no bias in the PDF, the resultsshould be consistent with zero CPT violation. The results are listed in Tables 7.8, 7.9 and 7.10.The results are consistent with no CPT violation, as it is expected, indicating that the modifiedPDF that incorporates CPT-violating effects is well behaved.

CHAPTER 7. ANALYSIS RESULTS 48

Pull Re(z)2− 0 2

Ent

ries

/ 0.

23

020406080

100120140160180200 µ = 0.001± 0.026µ = 0.001± 0.026µ = 0.001± 0.026

σ = 1.008± 0.016σ = 1.008± 0.016σ = 1.008± 0.016

(a)Pull Im(z)

2− 0 2

Ent

ries

/ 0.

23

020406080

100120140160180200 µ = −0.051± 0.023µ = −0.051± 0.023µ = −0.051± 0.023

σ = 1.015± 0.016σ = 1.015± 0.016σ = 1.015± 0.016

(b)

Zα∆-0.380α∆Pull 2− 0 2

Ent

ries

/ 0.

21

020406080

100120140160180200

µ = −0.0003± 0.026µ = −0.0003± 0.026µ = −0.0003± 0.026σ = 1.006± 0.016σ = 1.006± 0.016σ = 1.006± 0.016

(c)Xα∆Pull

2− 0 2

Ent

ries

/ 0.

23

020406080

100120140160180200

µ = 0.005± 0.023µ = 0.005± 0.023µ = 0.005± 0.023σ = 1.037± 0.017σ = 1.037± 0.017σ = 1.037± 0.017

(d)

Yα∆Pull 2− 0 2

Ent

ries

/ 0.

22

020406080

100120140160180200 µ = 0.015± 0.023µ = 0.015± 0.023µ = 0.015± 0.023

σ = 1.051± 0.016σ = 1.051± 0.016σ = 1.051± 0.016

(e)Zα∆-0.380α∆Pull

2− 0 2

Ent

ries

/ 0.

21

020406080

100120140160180200

µ = 0.003± 0.022µ = 0.003± 0.022µ = 0.003± 0.022σ = 0.993± 0.016σ = 0.993± 0.016σ = 0.993± 0.016

(f)

Figure 7.4: Distribution of the pull, for the various CPT-violating parameters. The blue linerepresents the shape of the best Gaussian that can be fitted on the pseudo-data, while thered/green dotted one represents a Gaussian with µ = 0 and σ = 1. In the context of thephenomenological description: (a), (b), (c) and in the context of SME: (d), (e), (f).

CHAPTER 7. ANALYSIS RESULTS 49

Fit value of Re(z) 0.05− 0 0.05

Ent

ries

/ 0.

0048

020406080

100120140160180200 µ = 0.00003± 0.00048µ = 0.00003± 0.00048µ = 0.00003± 0.00048

σ = 0.02155± 0.00012σ = 0.02155± 0.00012σ = 0.02155± 0.00012

(a)Fit value of Im(z)

0.01− 0 0.01

Ent

ries

/ 0.

0012

020406080

100120140160180200220 µ = 0.00024± 0.00012µ = 0.00024± 0.00012µ = 0.00024± 0.00012

σ = 0.00535± 0.00086σ = 0.00535± 0.00086σ = 0.00535± 0.00086

(b)

Fit value of Re(z) 0.05− 0 0.05

Ent

ries

/ 0.

0044

020406080

100120140160180200

µ = 0.00006± 0.00048µ = 0.00006± 0.00048µ = 0.00006± 0.00048σ = 0.02156± 0.00035σ = 0.02156± 0.00035σ = 0.02156± 0.00035

(c)GeV] -14[10Xα∆Fit value of

5− 0

Ent

ries

/ 0.

29

0

20

40

60

80

100

120

140

160 µ = 0.01022± 0.03512µ = 0.01022± 0.03512µ = 0.01022± 0.03512σ = 1.55886± 0.02574σ = 1.55886± 0.02574σ = 1.55886± 0.02574

(d)

GeV] -14[10Yα∆Fit value of 0 5

Ent

ries

/ 0.

29

0

20

40

60

80

100

120

140

160

180µ = −0.02487± 0.03538µ = −0.02487± 0.03538µ = −0.02487± 0.03538σ = 1.57047± 0.02593σ = 1.57047± 0.02593σ = 1.57047± 0.02593

(e)GeV] -14[10Zα∆-0.380α∆Fit value of

2− 0 2

Ent

ries

/ 0.

2

020406080

100120140160180 µ = 0.00454± 0.02260µ = 0.00454± 0.02260µ = 0.00454± 0.02260

σ = 1.0071± 0.01631σ = 1.0071± 0.01631σ = 1.0071± 0.01631

(f)

Figure 7.5: Distribution of the estimates for the CPT-violating parameters. The red/green linerepresent the value for which the pseudo-data are generated. In the context of the phenomeno-logical description: (a), (b), (c) and in the context of SME: (d), (e), (f).

CHAPTER 7. ANALYSIS RESULTS 50

Re(z) 0.2− 0.1− 0 0.1 0.2

log(

L)

∆-

0

2

4

6

8

10

12

14

(a)

Im(z) 0.2− 0.1− 0 0.1 0.2

log(

L)

∆-

0

2

4

6

8

10

12

14

(b)

Re(z) 0.2− 0.1− 0 0.1 0.2

log(

L)

∆-

0

2

4

6

8

10

12

14

(c)

GeV]-14[10Xα∆10− 5− 0 5 10

log(

L)

∆-

0

2

4

6

8

10

12

14

(d)

GeV]-14[10Yα∆15− 10− 5− 0 5 10

log(

L)

∆-

0

2

4

6

8

10

12

14

(e)

GeV]-14[10Zα∆-0.380α∆10− 5− 0 5 10

log(

L)

∆-

0

2

4

6

8

10

12

14

(f)

Figure 7.6: The likelihood scans for the CPT-violating parameters in the phenomenologicaldescription (a), (b), (c) and in the context of the SME (d), (e) ,(f).

CHAPTER 7. ANALYSIS RESULTS 51

Error of Re(z)0.021 0.0215

Ent

ries

/ 3.

7e-0

5

020406080

100120140160180200220

µ = 0.021363± 0.000003µ = 0.021363± 0.000003µ = 0.021363± 0.000003σ = 0.000154± 0.000005σ = 0.000154± 0.000005σ = 0.000154± 0.000005

(a)Error of Im(z)

0.005 0.0055

Ent

ries

/ 3.

8e-0

5

020406080

100120140160180200

µ = 0.005267± 0.000004µ = 0.005267± 0.000004µ = 0.005267± 0.000004σ = 0.000179± 0.000003σ = 0.000179± 0.000003σ = 0.000179± 0.000003

(b)

Error of Re(z) 0.021 0.0215

Ent

ries

/ 3.

7e-0

5

020406080

100120140160180200220

µ = 0.021361± 0.000003µ = 0.021361± 0.000003µ = 0.021361± 0.000003σ = 0.000153± 0.000002σ = 0.000153± 0.000002σ = 0.000153± 0.000002

(c)GeV]-14 [10Xα∆Error of

1.45 1.5 1.55

Ent

ries

/ 0.

0051

020406080

100

120140160180 µ = 1.5028± 0.0005µ = 1.5028± 0.0005µ = 1.5028± 0.0005

σ = 0.0248± 0.0004σ = 0.0248± 0.0004σ = 0.0248± 0.0004

(d)

GeV]-14 [10Yα∆Error of 1.45 1.5 1.55 1.6

Ent

ries

/ 0.

0052

020406080

100120140160180 µ = 1.5319± 0.0006µ = 1.5319± 0.0006µ = 1.5319± 0.0006

σ = 0.0249± 0.0004σ = 0.0249± 0.0004σ = 0.0249± 0.0004

(e)GeV]-14 [10Zα∆-0.380α∆Error of

1 1.05

Ent

ries

/ 0.

0032

020406080

100120140160180200 µ = 1.0150± 0.0003µ = 1.0150± 0.0003µ = 1.0150± 0.0003

σ = 0.0104± 0.0002σ = 0.0104± 0.0002σ = 0.0104± 0.0002

(f)

Figure 7.7: Distribution of the errors of the CPT-violating parameters, obtained from the fitsto the pseudo-data. In the context of the phenomenological description: (a), (b), (c) and in thecontext of SME: (d), (e), (f).

CHAPTER 7. ANALYSIS RESULTS 52

Table 7.8: Parameter estimates for the phenomenological description.

parameter 2011 Events 2012 EventsIm(z) 0.007± 0.019 0.012± 0.019Re(z) 0.041± 0.052 0.018± 0.058

Table 7.9: Parameter estimates when setting Im(z) = 0 for the phenomenological description.

parameter 2011 Events 2012 EventsRe(z) 0.040± 0.051 0.014± 0.057

Table 7.10: Parameter estimates in the context of SME.

parameter 2011 Events 2012 Events∆αX [10−14GeV] −1.514± 3.357 −4.212± 3.225∆αY [10−14GeV] 0.132± 3.554 −0.066± 3.225

∆α0 − 0.38∆αZ [10−14GeV] 1.975± 2.501 0.526± 2.238

7.7 Systematic uncertainties

Decay angle model and the acceptance: simulationIn order to account for differences between the data and the simulated events, the latter areweighted, before the angular acceptance weights are calculated. In order to obtain the system-atic uncertainties from the reweighing process, the fit on the data is performed using angularacceptance weights calculated with corrected (nominal fit) and uncorrected simulated events,and the difference is quoted as a systematic error (see Table 7.11).

Table 7.11: Systematic uncertainties arising from the acceptance simulation.

parameter syst. error,Re(z) 0.0015Im(z) 0.0006

Γs 0.0001∆Γs −∆ms 0.0036φs 0.0005|λs| 0.0054

parameter syst. errorRe(z) 0.0016

Γs 0.0001∆Γs −∆ms 0.0014φs 0.0002|λs| 0.0051

parameter syst. error∆αX 0.0103∆αY 0.0103

∆α0 − 0.38∆αZ 0.0592Γs 0.0001∆Γs −∆ms 0.0018φs 0.0005|λs| 0.0052

CHAPTER 7. ANALYSIS RESULTS 53

Decay angle model and the acceptance: statisticsThe shape of the acceptance function is fully determined from the acceptance weights, thatare calculated from a simulated sample. The values of the acceptance weights are affected bystatistical uncertainties (finite simulated sample). To estimate the effect of these uncertainties,we perform the fit for one thousand sets of acceptance weights, generated from a multivariateGaussian [16]. The average difference between the mean of the distribution of the parametersof interest and the values obtained from the nominal fit, is quoted as a systematic error (seeTable 7.12).

Table 7.12: Systematic uncertainties arising from statistical uncertainties in the angular accep-tance.

parameter syst. errorRe(z) 0.0001Im(z) 0.0001

Γs −∆Γs −∆ms 0.0007φs 0.0001|λs| 0.0001

parameter syst. errorRe(z) −

Γs −∆Γs −∆ms 0.0008φs 0.0007|λs| 0.0003

parameter syst. error∆αX 0.0092∆αY 0.0092

∆α0 − 0.38∆αZ 0.0020Γs −∆Γs −∆ms 0.0001φs −|λs| 0.0001

Tagging calibration parametersThe equations that describe the estimated mistag probabilities for both OS and SS taggingalgorithms contain in total eight tagging calibration parameters. The values of the calibrationparameters are estimated from a calibration sample. For the calibration of these parameters, selftagging decays, such as B+ → J/ψK+ are used. These calibration parameters, in the previousanalysis for the φs [1], are Gaussian constrained; so any systematic uncertainty was alreadyincluded in the statistical errors. In this analysis, these parameters are set to be constant.

Table 7.13: Tagging calibration parameters. For more details of how are connected with theequation 6.16 see [35].

parameter calibration value12(pOS0 + pOS0 ) +0.379± 0.00412(pSS0 + pSS0 ) +0.445± 0.00512(pOS1 + pOS1 ) +1.00± 0.0412(pSS1 + pSS1 ) +1.00± 0.09

parameter calibration valuepOS0 − pOS0 +0.0140± 0.0012pSS0 − pSS0 −0.0158± 0.0014pOS1 − pOS1 +0.066± 0.012pSS1 − pSS1 +0.008± 0.022

Previous studies had already shown that even if these parameters are set to be constant theydo not effect significantly the parameters of interest [16]. In this context we performed approx-imately one thousand preudo-experiments where the pseudo-data are generated with randomvalues of calibration parameters obtained from equivalent Gaussians with mean and width givenby Table 7.13. The fit bias (see text) is quoted as a systematic uncertainty.

CHAPTER 7. ANALYSIS RESULTS 54

Table 7.14: Systematic uncertainties arising from the tagging calibration parameters.

parameter syst. errorRe(z) 0.0001Im(z) 0.0001

Γs −∆Γs 0.0002∆ms 0.0001φs 0.0003|λs| 0.0009

parameter syst. errorRe(z) 0.0001

Γs 0.0001∆Γs 0.0003∆ms 0.0001φs −|λs| 0.0005

parameter syst. error∆αX 0.0294∆αY 0.0294

∆α0 − 0.38∆αZ 0.0408Γs −∆Γs 0.0003∆ms 0.0002φs 0.0004|λs| 0.0004

Peaking backgroundResonant backgrounds originating from the decays Λ0

b → J/ψpK− and B0 → J/ψK0∗ aresubtracted, by using simulated peaking background events with negative weights, such thatafter combining these events with the B0

s → J/ψK+K− data sample, the equivalent peakingbackground events are canceled. From the treatment of the peaking background two typesof systematics uncertainties arise. The first has to do with the uncertainty of the peakingbackground yields and the second with uncertainties related with the distribution of the relevantparameters, such as the estimated mistag probability. The systematic uncertainty originatingfrom the yields of the peaking backgrounds is estimated by varying the number of the resonantdecays between ±1σ and recomputing the peaking background weights. The new weights areused to perform the fit for the parameters of interest. The average difference between thenominal fit results and the results obtained by a ±1σ fluctuation of the yields is quoted assystematic uncertainty (see Table 7.15).

Table 7.15: Systematic uncertainties arising from the yields of the peaking background.

parameter syst. errorRe(z) 0.0008Im(z) −

Γs −∆Γs 0.0001∆ms 0.0009φs 0.0009|λs| 0.0004

parameter syst. errorRe(z) 0.0008

Γs −∆Γs 0.0001∆ms 0.0008φs 0.0009|λs| 0.0003

parameter syst. error∆αX 0.0581∆αY 0.0581

∆α0 − 0.38∆αZ 0.0652Γs 0.0001∆Γs 0.0002∆ms 0.0011φs 0.0004|λs| 0.0006

The distributions of the decay angles and the tagging variables that we get from the simulationdo not agree well with the one in data. So, the simulated events for the resonant backgroundare re-weighted to match the data better. However, this process is not perfect. Hence, thedifference between the parameters of interests obtained using re-weighted and not re-weighted

CHAPTER 7. ANALYSIS RESULTS 55

distributions for the resonant backgrounds events is taken as a systematic uncertainty (see Table7.16) [16].

Table 7.16: Systematic uncertainties arising from peaking background re-weighting.

parameter syst. errorRe(z) 0.0024Im(z) 0.0004

Γs 0.0001∆Γs 0.0005∆ms 0.0023φs 0.0003|λs| 0.0032

parameter syst. errorRe(z) 0.0025

Γs −∆Γs 0.0004∆ms 0.0022φs 0.0003|λs| 0.0032

parameter syst. error∆αX 0.1202∆αY 0.1202

∆α0 − 0.38∆αZ 0.1303Γs 0.0002∆Γs 0.0003∆ms 0.0029φs −|λs| 0.0034

Mass factorizationThe sFit method that is used to subtract the background from the data sample make theassumption that the mass is independent of the decay time and angles. However, in the φsanalysis it is found that there is a correlation between the m(J/ψK+K−) model and the cos θµ[16]. We study the impact of this effect by computing the sWeights in bins of cos θµ. Finally wecombine the sub-samples and we perform the fit. The difference in the parameters of interestbetween this fit and the nominal one, is quoted as a systematic error (see Table 7.17).

Table 7.17: Systematic uncertainties arising from the mass factorization.

parameter syst. errorRe(z) 0.0009Im(z) 0.0015

Γs −∆Γs 0.0008∆ms 0.0031φs 0.0051|λs| 0.0021

parameter syst. errorRe(z) 0.0004

Γs −∆Γs 0.0007∆ms 0.0065φs 0.0031|λs| 0.0021

parameter syst. error∆αX 0.1237∆αY 0.1237

∆α0 − 0.38∆αZ 0.0112Γs −∆Γs 0.0006∆ms 0.0056φs 0.0009|λs| 0.0023

Approximation of Im(z) being zeroIn the parametrization of the CPT violation in the context of the SME, Im(z) is set to be zero.This choice is justified by the SME constraint, where for the B0

s Im(z) is 450 times smallerthan Re(z). Performing the fit for Re(z) with Im(z) = 0 yields a difference between the esti-mates of Re(z) with and without using the SME condition equal to ∆Re(z) = 0.00054. Hence,the resulting shift of the ∆αµ is negligible. The same holds for the other parameters of interests.

CHAPTER 7. ANALYSIS RESULTS 56

Momentum scale uncertaintyAs discussed in chapter 3.3, the boost factor plays a crucial role in the measurement of theCPT-violating coefficients. Hence, a possible uncertainty in the momentum of the B0

s mesonscan affect the estimates for the parameters of interest. The momentum scale uncertainty isestimated to be 0.02% [36].

Re(z) ∝ γ(∆α0 − ~β∆~α) (7.2)

From equation 7.2, it can be deduced that the uncertainty in the momentum scale of the B0s ,

will scale ∆αµ by the same 0.02%. The effects of this uncertainty on the CP and life-timeparameters is found to be negligible [1]. Meanwhile, the correlation between the CPT and theCP, lifetime parameters is small and the absolute shift in the ∆αµ parameters is completelynegligible compared to the statistical uncertainty; hence, small effects of the momentum scale inthe CPT-violating coefficients will not affect significantly the estimates for the other parametersof interest.

Sun-centred frame parametersThe SFC parameters that are used in the formula that describes the sidereal variation of Re(z)are known very accurately (see table 6.2 and [31]). The most significant uncertainty in the SFCparameters, originates from the sidereal phase (t0), which is ±0.0004 rad. Scaling the siderealphase by ±1σ and performing the fit, we find that the effect on the parameters of interests isnegligible.

Peaking background GPSIn the collimated analysis in order to estimate the CPT-violating coefficient, we use the GPStime of the events, from which we calculate the sidereal phase. As discussed before, in orderto eliminate the contribution of peaking background decays from the data sample we injectsimulated peaking background events with negative weights. However, simulated events of-course are not associated with a GPS time. Hence, for each of the MC peaking backgroundevents, we associate randomly a GPS time obtained from the B0

s → J/ψK+K−, data sample.In principle this process can introduce a small systematic error, however since the number ofthe peaking background decays is small, approximately 7000 candidates, compared to the 95000signal candidates, it can be neglected.

Fit biasTo estimate any possible bias originating from the fitting procedure and due to the finite sample,toy studies are performed. We generate 2000 toys of equivalent size to the data sample, wherethe values for the parameters of interest are set close to the estimated values. For the parameterswhere the pull is not compatible with zero within three standard deviations the observed fit biasis quoted as a systematic uncertainty. The fit bias is defined as pfit − pgenerate, where pgenerateis the value of the parameter for which the toys are generated and pfit is the value of the fittedparameter. The resulting bias is small and for almost all the parameters is negligible.

CHAPTER 7. ANALYSIS RESULTS 57

7.8 Overall systematic uncertaintyThe list of all the sources for systematic uncertainties for the different parametrization is illus-trated in tables 7.18, 7.19 and 7.20.

Table 7.18: Systematic uncertainties for the phenomenological description.

Source Re(z) Im(z) Γs ∆Γs ∆ms φs |λs|[ps−1] [ps−1] [ps−1] [rad]

Ang. acc. Sim. 0.0015 0.0006 0.0001 − 0.0036 0.0005 0.0054Ang. acc. Stat. 0.0001 0.0001 − − 0.0007 0.0001 0.0001Tagging calib. 0.0001 0.0001 − 0.0002 0.0001 0.0003 0.0009P. background Yields 0.0008 − − 0.0001 0.0009 0.0009 0.0004P. background Stat. 0.0024 0.0004 0.0001 0.0005 0.0023 0.0003 0.0032Mass factorization 0.0009 0.0015 − 0.0008 0.0031 0.0051 0.0021Fit bias − − 0.0001 − − − −Quadratic sum of syst. 0.0031 0.0016 0.0001 0.0010 0.0054 0.0052 0.0066

Table 7.19: Systematic uncertainties when setting Im(z) = 0 for the phenomenological descrip-tion.

Source Re(z) Γs ∆Γs ∆ms φs |λs|[ps−1] [ps−1] [ps−1] [rad]

Ang. acc. Sim. 0.0016 0.0001 − 0.0014 0.0002 0.0051Ang. acc. Stat. − − − 0.0008 0.0007 0.0003Tagging calib. 0.0001 0.0001 0.0003 0.0001 − 0.0005P. background Yields 0.0008 − 0.0001 0.0008 0.0009 0.0003P. background Stat. 0.0025 − 0.0004 0.0022 0.0021 0.0032Mass factorization 0.0004 − 0.0001 0.0065 0.0003 0.0021Assumption on Im(z) 0.0005 − 0.0003 0.0067 0.0092 0.0013Fit bias − 0.0001 − − − −Quadratic sum of syst. 0.0031 0.0001 0.0006 0.0097 0.0095 0.0065

Table 7.20: Systematic uncertainties in the context of SME.

Source ∆α0 − 0.38∆αZ ∆αX ∆αY Γs ∆Γs ∆ms φs |λs|[10−14GeV] [10−14GeV] [10−14GeV] [ps−1] [ps−1] [ps−1] [rad]

Ang. acc. Sim. 0.0592 0.0103 0.0103 0.0001 − 0.0018 0.0005 0.0052Ang. acc. Stat. 0.0020 0.0092 0.0092 − − 0.0001 − 0.0001Tagging calib. 0.0408 0.0321 0.0265 − 0.0003 0.0002 0.0004 0.0004P. background Yields 0.0652 0.0581 0.0581 0.0001 0.0002 0.0011 0.0004 0.0006P. background Stat. 0.1303 0.1202 0.1202 0.0002 0.0003 0.0029 − 0.0034Mass factorization 0.0112 0.1237 0.1237 − 0.0006 0.0056 0.0009 0.0023Assumption on Im(z) 0.0329 0.0329 0.0329 0.0002 0.0003 0.0097 0.0049 0.0009M. scale uncertainty 0.0001 0.0005 0.0005 − − 0.0029 0.0001 0.0002P. background GPS − − − − − − − −SFC parameters − 0.0001 0.0001 − − − − −Fit bias − − − 0.0001 − − − −Quadratic sum of syst. 0.1661 0.1888 0.1888 0.0003 0.0008 0.0121 0.0050 0.0067

CHAPTER 7. ANALYSIS RESULTS 58

7.9 ConclusionThe goal of this analysis is to search for violation of the Lorentz symmetry by measuring CPT-violating effects using B0

s → J/ψφ decays. These are the first measurements of ∆αµ in theSME and of Re(z) and Im(z) in the phenomenological description for the B0

s system. We mustnotice again that CPT violation in the SME implies automatically Lorentz violation, while CPTviolation in the phenomenological description does not guarantee Lorentz violation. At the sametime we report measurements for φs, |λs|, Γs, ∆Γs and ∆ms that are in agreement with thelatest B0

s → J/ψφ analysis. The results for the parameters of interest are listed in Tables 7.21,7.22, 7.23.

Table 7.21: Results for the phenomenological description.

parameter Value Statistical uncertainty Systematic uncertaintyIm(z) −0.022 ±0.033 ±0.003Re(z) 0.004 ±0.011 ±0.002Γs[ps−1] 0.6590 ±0.0031 ±0.0001∆Γs [ps−1] 0.0790 ±0.0092 ±0.0010∆ms[ps−1] 17.730 ±0.061 ±0.005φs[rad] −0.070 ±0.056 ±0.005|λs| 0.958 ±0.020 ±0.007

Table 7.22: Results when setting Im(z) = 0 for the phenomenological description.

parameter Value Statistical uncertainty Systematic uncertaintyRe(z) −0.022 ±0.033 ±0.003Γs[ps−1] 0.6591 ±0.0031 ±0.0001∆Γs [ps−1] 0.0787 ±0.0091 ±0.0006∆ms[ps−1] 17.721 ±0.056 ±0.010φs[rad] −0.060 ±0.050 ±0.010|λs| 0.959 ±0.019 ±0.007

Table 7.23: Results in the context of SME.

parameter Value Statistical uncertainty Systematic uncertainty∆αX [10−14GeV] 0.921 ±2.106 ±0.188∆αY [10−14GeV] −3.817 ±2.106 ±0.188

∆α0 − 0.38∆αZ [10−14GeV] −0.987 ±1.382 ±0.166Γs[ps−1] 0.6592 ±0.0031 ±0.0003∆Γs [ps−1] 0.0784 ±0.0091 ±0.0008∆ms[ps−1] 17.720 ±0.056 ±0.012φs[rad] −0.065 ±0.050 ±0.005|λs| 0.961 ±0.019 ±0.007

CHAPTER 7. ANALYSIS RESULTS 59

For the results of all the parameters that we report the uncertainties are dominated by thestatistical uncertainties. However, the study for the systematic uncertainties is not yet complete,since more sources of systematic errors, have to be taken into account, as it is done in the nominalanalysis for the φs [1]. Additionally, we have to investigate possible momentum dependenceeffects, such as dependence of the sWeights and the tagging calibration parameters on themomentum. Regarding the effects of these systematics on the CP and lifetime parameters, weexpect, based on the study that is done for the φs analysis, that the systematic error will not besignificantly affected and will still remain smaller than the statistical uncertainties. Concerningthe systematic effect of these sources on the CPT-violating parameters, to first order we do notexpect to be directly affected or at least to contribute in a way that the systematic uncertaintybecome important.

The results for the CPT violation in the context of SME, show that there is no evidence forCPT violation and consequently no Lorentz violation, while the results in the context of thephenomenological description, shows that there is no CPT violation. Furthermore, concerningCP violation in the B0

s → J/ψφ, the results of the φs and |λs|, show that there is no evidencefor CP violation.

A further justification regarding the choice to fit also with the phenomenological model butby setting Im(z) = 0 should be made. Except of the motivation explained at the beginning ofthis chapter; in general there are numerous motivations behind this choice. As an example wecan imagine a model in the phenomenological description with Im(z) = 0 by definition, or asituation where the Im(z) is measured with flavour-specific decays which are very sensitive toIm(z) [22] and is found to be zero.

In the new run the statistical uncertainty for all the parameters will be halved, just byquadrupling the statistics. Also for the CPT-violating parameters in the context of the phe-nomenological description and in the context of the SME with Re(z) being constant the sta-tistical uncertainties will be still dominant. However an interesting situation arise, for theCPT-violating parameters ∆αµ; since Re(z) ∝ γ, the CPT reach will be increased, resulting intotal the statistical uncertainties for the CPT-violating coefficients to be improved. However,the systematical uncertainties for the CPT-violating coefficients ∆αX and ∆α0 − 0.38∆αZ willbe still larger than the systematic uncertainties. It is after the upgrade of LHCb that the controlof the systematic uncertainties may become important.

Appendices

60

Appendix A

Signal PDF in the context of CPTviolation

The full signal PDF of B0s → J/ψK+K−, where qf represents the tagging.

T1(t)f1( ~Ω) = |α0|2e−Γst

1 + qfC

cosh(∆Γst/2)[1 + qfRe(z)D − Im(z)S] + cos(∆mst)[qfC − qfRe(z)D + Im(z)S]+ sinh(∆Γst/2)[D + Re(z)(qf + C)] + sin(∆mst)[−qfS + Im(z)(qf + C)]×2 cos2 θK sin2 θµ

(A.1)

T2(t)f2( ~Ω) = |α|||2e−Γst

1 + qfC

cosh(∆Γst/2)[1 + qfRe(z)D − Im(z)S] + cos(∆mst)[qfC − qfRe(z)D + Im(z)S]+ sinh(∆Γst/2)[D + Re(z)(qf + C)] + sin(∆mst)[−qfS + Im(z)(qf + C)]× sin2 θK(1− sin2 θµ cos2 φh)

(A.2)

T3(t)f3( ~Ω) = |α⊥|2e−Γst

1 + qfC

cosh(∆Γst/2)[1− qfRe(z)D + Im(z)S] + cos(∆mst)[qfC + qfRe(z)D − Im(z)S]+ sinh(∆Γst/2)[−D + Re(z)(qf + C)] + sin(∆mst)[qfS + Im(z)(qf + C)]× sin2 θK(1− sin2 θµ sin2 φh)

(A.3)

T4(t)f4( ~Ω) = |αS|2e−Γst

1 + qfC

cosh(∆Γst/2)[1− qfRe(z)D + Im(z)S] + cos(∆mst)[qfC + qfRe(z)D − Im(z)S]+ sinh(∆Γst/2)[−D + Re(z)(qf + C)] + sin(∆mst)[qfS + Im(z)(qf + C)]

×23 sin2 θµ

(A.4)

61

APPENDIX A. SIGNAL PDF IN THE CONTEXT OF CPT VIOLATION 62

T5(t)f5( ~Ω) = |α0||α|||e−Γst

1 + qfCcos(δ|| − δ0)

cosh(∆Γst/2)[1 + qfRe(z)D − Im(z)S] + cos(∆mst)[qfC − qfRe(z)D + Im(z)S]+ sinh(∆Γst/2)[D + Re(z)(qf + C)] + sin(∆mst)[−qfS + Im(z)(qf + C)]

×12√

2 sin 2θK sin 2θµ cosφh(A.5)

T6(t)f6( ~Ω) = |α0||α⊥|e−Γst

1 + qfC

cosh(∆Γst/2)[[qfRe(z)S + Im(z)D] cos(δ⊥ − δ0) + C sin(δ⊥ − δ0)]+ cos(∆mst)[[−qfRe(z)S − Im(z)D] cos(δ⊥ − δ0) + qf sin(δ⊥ − δ0)]+ sinh(∆Γst/2)[S cos(δ⊥ − δ0) + [Re(z)(1 + qfC)] sin(δ⊥ − δ0)]+ sin(∆mst)[qfD cos(δ⊥ − δ0) + [Im(z)(1 + qfC)] sin(δ⊥ − δ0)]

×−12√

2 sin 2θK sin 2θµ sinφh(A.6)

T7(t)f7( ~Ω) = |α||||α⊥|e−Γst

1 + qfC

cosh(∆Γst/2)[[qfRe(z)S + Im(z)D] cos(δ⊥ − δ||) + C sin(δ⊥ − δ||)]+ cos(∆mst)[[−qfRe(z)S − Im(z)D]cos(δ⊥ − δ||) + qf sin(δ⊥ − δ||)]+ sinh(∆Γst/2)[S cos(δ⊥ − δ||) + [Re(z)(1 + qfC)] sin(δ⊥ − δ||)]+ sin(∆mst)[qfD cos(δ⊥ − δ||) + [Im(z)(1 + qfC)] sin(δ⊥ − δ||)]× sin2 θK sin2 θµ sin 2φh

(A.7)

T8(t)f8( ~Ω) = |α0||αS|e−Γst

1 + qfC

cosh(∆Γst/2)[C cos(δS − δ0) + [qfRe(z)S + Im(z)D] sin(δS − δ0)]+ cos(∆mst)[[qf cos(δS − δ0) + [−qfRe(z)S − Im(z)D] sin(δS − δ0)]+ sinh(∆Γst/2)[[Re(z)(1 + qfC)]cos(δS − δ0) + S sin(δS − δ0)]+ sin(∆mst)[[Im(z)(1 + qfC)]D cos(δS − δ0) + qfD sin(δS − δ0)]

×43√

3 cos θK sin2 θµ

(A.8)

T9(t)f9( ~Ω) = |α||||αS|e−Γst

1 + qfC

cosh(∆Γst/2)[C cos(δS − δ||) + [qfRe(z)S + Im(z)D] sin(δS − δ||)]+ cos(∆mst)[[qf cos(δS − δ||) + [−qfRe(z)S − Im(z)D] sin(δS − δ||)]+ sinh(∆Γst/2)[[Re(z)(1 + qfC)] cos(δS − δ||) + S sin(δS − δ||)]+ sin(∆mst)[[Im(z)(1 + qfC)]D cos(δS − δ||) + qfD sin(δS − δ||)]

×13√

6 sin θK sin 2θµ cosφh(A.9)

APPENDIX A. SIGNAL PDF IN THE CONTEXT OF CPT VIOLATION 63

T10(t)f10( ~Ω) = |α⊥||αS|e−Γst

1 + qfCsin(δS − δ⊥)

cosh(∆Γst/2)[1− qfRe(z)D + Im(z)S] + cos(∆mst)[qfC + qfRe(z)D − Im(z)S]+ sinh(∆Γst/2)[−D + Re(z)(qf + C)] + sin(∆mst)[qfS + Im(z)(qf + C)]

×−13√

6 sin θK sin 2θµ sinφh(A.10)

Appendix B

Non-collimated analysis results

As it was described in chapter 4, the formula that describes Re(z) in the context of a non-collimated analysis is given by equation B.1:

Re(z) = Re(z)(t, θ, φ, PB0s)

=∆msPB0

s

ms[∆m2s + 1/4∆Γ 2

s ]∆α0 +∆αZ(cos θ cosχ− sin θ cosφ sinχ)

+ [∆αY (cos θ sinχ+ sin θ cosφ cosχ)−∆αX sin θ sinφ] sinΩt+ [∆αX(cos θ sinχ+ sin θ cosφ cosχ) +∆αY sin θ sinφ] cosΩt ,

(B.1)

where we have set β = 1. However, since B0s mesons are in a very good approximation colli-

mated along the z axis (see Figure 6.4a), the highlighted terms in equation B.1, will not varysignificantly, resulting in the parameters ∆α0 and ∆αZ to be highly correlated. In order toperform the non-collimated analysis and avoid the correlation with the latter parameters, were-parametrize equation B.1 in terms of ∆αX , ∆αY and two new parameters defined below:

∆α1 = ∆α0 + cosχ∆αZ , (B.2)∆α2 = − cosχ∆α0 +∆αZ . (B.3)

Using equations B.2 and B.3, the constant part (non-sidereal time dependent) of equation B.1can be re-written as:

∆α0 +∆αZ(cosθcosχ− sinθcosφsinχ) =∆α1

1 + cos2 χ[ 1 + cos2 χ cos θ − sin θ cosχ sinχ cosφ ]

+ ∆α2

1 + cos2 χ[ − cosχ+ cosχ cos θ − sin θ cosφ sinχ ]

(B.4)

We perform in the same manner as before an unbinned maximum likelihood fit, from whichwe estimate the CPT violating parameters: ∆α1, ∆α2 and ∆αx and ∆αY . The values for theparameter estimates together with the correlation matrix are given in tables B.1 and B.2, wherecorrelations higher than 30% are highlighted. As it was expected, the difference between the

64

APPENDIX B. NON-COLLIMATED ANALYSIS RESULTS 65

parameters ∆α0 − 0.38∆αZ (∆α0 + cosχ∆αZ), ∆αX and ∆αY obtained from the collimatedand the non-collimated analysis is negligible.

Table B.1: Parameter estimates with the non-collimated analysis in the context of SME.

parameter nominal value value difference (sigma)∆α1[10−14GeV] −− −0.922± 1.382 −−∆α2[10−14GeV] −− −26.986± 38.834 −−∆αX [10−14GeV] −− 1.053± 2.106 −−∆αY [10−14GeV] −− −3.818± 2.106 −−Γs [ps−1] 0.6592± 0.0031 0.6592± 0.0031 0.00007 ( 0.024)∆Γs [ps−1] 0.0785± 0.0092 0.0785± 0.0091 0.00000 ( 0.000)∆ms[ps−1] 17.723± 0.057 17.722± 0.056 −0.0009 (−0.017)|λs| 0.963± 0.019 0.960± 0.019 −0.0023 (−0.125)

φs[rad] −0.057± 0.050 −0.064± 0.050 −0.0068 (−0.136)

Table B.2: Correlation matrix in the context of SME.

∆α1 ∆α2 ∆αX ∆αY φs |λs| Γs ∆Γs ∆ms∆α1 1 −0.056 0.029 0.035 0.079 0.236 0.022 −0.026 0.041∆α2 − 1 0.029 0.026 −0.027 0.014 0.002 −0.007 −0.049∆αX − − 1 0.000 0.020 0.014 −0.002 0.016 −0.004∆αY − − − 1 0.071 −0.011 −0.010 0.021 0.007φs − − − − 1 −0.002 0.005 −0.063 0.077|λs| − − − − − 1 0.007 0.004 −0.172Γs − − − − − − 1 −0.401 0.000∆Γs − − − − − − − 1 −0.017∆ms − − − − − − − − 1

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