Nikhef National Institute for Subatomic Physics University of Amsterdam
Physics Track: Particle and Astroparticle Physics
Search for Lorentz invariance violation
with B
B
B
s
s
s
0
0
0
→ J/ψφ
→ J/ψφ
→ J/ψφ decays
G. Chatzikonstantinidis Supervisors: Prof. Dr. G. Raven, Dr. J. van Tilburg, Prof. Dr. P. de JongIn this thesis we derive the differential decay rates of B0
s → J/ψφ incorporating CPT-violating effects. Subsequently we measure Lorentz and CPT-violating effects using data corresponding to 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 phenomenological description which is independent of any microscopic model, where the CPT-Lorentz violating effects 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 Lorentz violation in the context of the Standard Model Extension (SME), where CPT-Lorentz violation is parameterized through the ∆αµ coefficients, integrated in the SME Lagrangian. The results obtained 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 reported with both parameterizations are the first measurements of CPT-Lorentz violation in the Bs0 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 are found to be consistent with the latest analysis of φs [1] and are in agreement with Standard Model predictions.
1 Introduction 3
2 Theoretical motivation 4
3 The LHCb experiment 6
3.1 The Large Hadron Collider . . . 6
3.2 The LHCb detector . . . 7
4 Lorentz and CPT tests with neutral B mesons 10 4.1 Parameterizing the B0mesons effective Hamiltonian in terms of the CPT violating parameter . . . 10
4.2 The time evolution and decay of B0 mesons in the context of CPT violation . . 15
4.3 Sidereal modulations of the CPT-violating parameter . . . 19
5 CPT violation in Bs0 → J/ψφ 23 5.1 The decay of B0 s → J/ψφ . . . . 23
5.2 Time dependence and CPT violation . . . 26
5.3 Combining time-dependence including CPT-violating effects and angular depen-dence . . . 29
6 Data analysis 31 6.1 Maximum likelihood fit . . . 31
6.2 Reconstruction and selection . . . 33
6.3 Decay time and acceptance . . . 34
6.4 Angular resolution and acceptance . . . 35
6.5 Flavour tagging . . . 36
6.6 Decay rate model . . . 37
6.7 Sun-centered frame and external parameters . . . 38
7 Analysis results 42 7.1 Parameter estimates . . . 42
7.2 Binned fit . . . 44
7.3 Analysis of subsets of the data sample . . . 46
7.4 Validating the modified fitting algorithm . . . 47
7.5 Investigating the statistical errors . . . 47
7.6 Consistency test using Monte Carlo events . . . 47
7.7 Systematic uncertainties . . . 52
7.8 Overall systematic uncertainty . . . 57 7.9 Conclusion . . . 58
Appendices 60
A Signal PDF in the context of CPT violation 61
Introduction
The standard model of particle physics is a unified gauge theory which describes all know interactions, except gravity [2]. Even though it is a very successful model that is consistent with all experimental measurements, it cannot explain some fundamental phenomena, such as dark matter, dark energy and the observed asymmetry between matter and antimatter. At the same time gravity is a force that still cannot be described in this context. Hence, the standard model of particle physics is considered to be a low-energy limit of a more fundamental theory that can answer the previous fundamental questions and at the same time describe gravitational effects that 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 for an 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, searching for 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.
Theoretical motivation
A very promising approach to this problem is to search for fundamental laws that are considered to be exact in the context of the standard model, but appear to be broken at a more fundamental level. Any deviation will then be a signal for new physics. An example of a law which holds exactly in established physics is the so-called CPT symmetry. The CPT symmetry states that under the operation of C (charge conjugate), P (spatial inversion) and T (time reversal) the physics laws remain the same. Among the consequences of this theorem is that particles and antiparticles have the same mass, lifetime and opposite charge. The CPT symmetry is closely related with the Lorentz symmetry, through the CPT theorem, which briefly states that every relativistic quantum field theories exhibit also CPT invariance [4]. One straightforward question is whether it is possible to have independent violation of these symmetries. The answer to this question is given through the anti-CPT theorem which roughly states that if CPT symmetry is broken, then Lorentz invariance is also violated 1 but not vice versa [6, 7]. Thus it becomes obvious that performing searches for CPT violation can be a very powerful tool for the discovery of physics beyond the standard model.
The question that additionally must be answered is in which framework the searches for potential CPT and Lorentz violation will be conducted. From a phenomenological point of view the introduction of a parameter that quantifies CPT violation is, in many situations, quite straightforward. 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 prediction of 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 for future experiments [8]. This makes obvious the need for a microscopic model, through which a complete description of CPT and Lorentz violation can be extracted.
The framework for this work is given through the standard model extension [9], which is a very general effective field theory that contains all possible operators that break Lorentz and CPT2 symmetry. The general idea is that the underlying fundamental theory remain Lorentz and CPT invariant, but observed violations of these symmetries can be the result of spontaneous violation in the solutions of the theory. This idea is very attractive since the theory maintains the 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 Lorentz
and CPT symmetries is not valid [5].
2Contains also operators that preserve CPT symmetry. It must become clear that Lorentz violation is allowed
without necessary CPT violation but not the opposite.
the standard model with the Higgs mechanism, with the difference that now the vacuum is, also saturated 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 as the gauge structure and power coupling renormalizability. Energy and momentum are of course also conserved. The Lagrangian of the SME is given by:
LSM E = LSM+ δL (2.1)
where LSM E represent the SME Lagrangian,3 LSM the SM Lagrangian and the δL represents CPT and Lorentz violating terms, that are controlled by a set of tensorial coefficients that are to be determined experimentally [6]. These coefficients describe the expectation values of fields produced by type of interactions that destabilize the vacuum; the presence of these terms is responsible for the breakdown of Lorentz and CPT symmetry. In the context of certain string theories it is shown that spontaneous symmetry violation can trigger vacuum expectation values for non-scalar fields; these values can then be identified with the tensorial coefficients described before. In the limit where the CPT-and Lorentz-violating terms are zero the SME Lagrangian will recover to the SM Lagrangian. Furthermore, the SME Lagrangian is still invariant under Lorentz 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 the experimental 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 promising measurements 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.
The LHCb experiment
In this chapter the experimental setup that is used for this analysis is briefly discussed.
3.1
The Large Hadron Collider
The Large Hadron Collider (LHC) is a circular collider that accelerates two beams of protons or lead ions up to few TeV of energy. The two proton beams, before being accelerated in the LHC ring, pass through a series of pre-accelerators (see figure 3.1). First, hydrogen atoms are stripped from the electrons and then accelerated up to 50 MeV by a linear accelerator (LINAC2). From there, they are fed to the Proton Synchrotron Booster (BOOSTER) and then to the Proton Synchrotron (PS). In the last step of this chain, they are accelerated by the Super Proton Synchrotron (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
experiments at the LHC are CMS, ATLAS, ALICE and LHCb. The ATLAS and CMS detectors are general purpose detectors, where the Higgs particle has been discovered. ALICE is designed to 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 perform studies for CPT-violating phenomena.
3.2
The LHCb detector
The LHCb detector is focused on studying the properties of b-and c-hadrons at the LHC and the design of the detector is optimized for this purpose. The LHCb detector is designed as a single-arm spectrometer which covers a pseudorapidity region of 2 < η < 5. The motivation behind this choice is that at high collision energies the b¯b pairs mainly travel (together) in a
forward or backward direction and are highly collimated along the beam line. The geometrical acceptance of the detector is between 10 − 300 mrad in the bending plane (x − y) of the magnet and 10 −250 mrad in the non-being (y −z) one (see figure 3.2). Furthermore, the LHCb detector has very good vertex resolution in order to deal with secondary vertices (particularly important for the fast oscillations of the Bs0 − ¯Bs0 system) and finally the trigger system is specifically designed to search for signatures originating from b-and c-hadron decays [11]. An outline of the detector 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 more details see [11].
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 precision that 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 placed between the Velo and the magnet. The TT improves the momentum estimate of the charged decay products and at the same time provides important information for the reconstruction 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 the charged 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 inner region of the T-stations. The outer tracker (OT) is a gas straw detector that covers the remaining area. Both are placed after the magnet, with a purpose to measure the 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 angle of the emitted light from charged particles. Combining this information with the momentum, the mass of the particles can be determined. This resolves the identity of 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 that are placed before the electromagnetic calorimeter to provide additional information about the identity of the particles. The SPD provides good discrimination between neutral and charged particles, and it also provides an estimate of the total number of charged particles. After the SPD there is a thin plate of a lead that will initiate a shower; next the PS use this to discriminate between electromagnetic and hadronic showers. This information is used by the electromagnetic and hadronic calorimeters to improve their performance.
(c) Calorimeters:
The electromagnetic (ECAL) and the hadronic (HCAL) calorimeters are placed after the SPD and PS detectors. The ECAL is a sampling calorimeter, which measures the energy of electromagnetically interacting particles. It is composed of a lead absorber and a scintillator of about 25X0. The sufficient thickness of the ECAL ensures that the electromagnetic interacting particles will be well confined. On the other hand, the hadronic calorimeter (HCAL) measures the energy of hadronically interacting
particles. It consists of layers of iron and scintillator that are placed parallel to the
z-axis. The thickness of the HCAL is 5.6λint, which is not enough to fully contain the hadronic showers.
(d) Muon Chambers:
The muon chambers (M1-M5) are placed at the end of the detector. The M2-M3 chambers are equipped with multiwire proportional chambers, while M1 is equipped with a Triple-GEM detector, which has a better radiation hardness. The muon stations provide good muon identification and muon triggers, which is crucial for many B decays.
Lorentz and CPT tests with neutral B
mesons
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 are present in the effective Hamiltonian, describing the evolution of the B meson system. These parameters 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, a complete 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 that are integrated in the SME Lagrangian and result in CPT and consequently Lorentz violation.
4.1
Parameterizing the B
0mesons effective Hamiltonian
in 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. In the case of the B0 mesons this is the so-called beauty (B). The Hamiltonian that describes the evolution of the system is expressed as:
H = Hstrong + HQED+ Hweak = H∆B=0+ H∆B=0+ H∆B6=0 . (4.1) The fact that weak interactions do not preserve the internal quantum number B is manifested in the Hamiltonian by the presence of off-diagonal elements1, which will drive the neutral B mesons to oscillate.
The time evolution for the generic state of the B0 ↔ ¯B0 system, is represented by a vector in Hilbert space:
|Ψ (t)i = a(t)|B0i + b(t)| ¯B0i +X
k
ck(t)|fki (4.2)
where |fki stands for all possible final states that the B0 mesons can decay. The Schr¨odinger
1The off-diagonal elements appear in the Hamiltonian, since the flavour eigenstates of H = H
strong+ HQED
are not at the same time eigenstates of the Hweak
equation, satisfied by |Ψ (t)i is:
i∂
∂tΨ (t) = HΨ (t) , (4.3)
where H is an infinite-dimensional Hermitian matrix in the Hilbert space. The exact solution to this problem is very complicated [13]. However, the problem can be significantly simplified using 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 state will be: |Ψ (0)i = a(0)|B0i + b(0)| ¯B0i.
• we are interesting in computing only the coefficients a(t) and b(t), and not the coefficients
ck(t).
• we restrict the description to t much larger than the typical strong interaction scale.
With these assumptions, the Schr¨odinger equation describing the evolution of the B0− ¯B0, can be expressed as: i∂ ∂t a(t) b(t) ! = H11 H12 H21 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 Hermitian and 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 = 1
2(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 |fki represents all the intermediate states:
Mij = m0δij+ hi|Hw|ji +
X
k
Phi|Hw|fkihfk|Hw|ji
m0− Ek
, (4.7)
Γij = 2π
X
k
δ(m0− Ek)hi|Hw|fkihfk|Hw|ji . (4.8)
Hence, the elements of the effective Hamiltonian can be written as: Hij = Mij −
i
2Γij . (4.9)
The mixing of the B0and ¯B0is 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), and is in the standard model dominated by the top-quark contribution [15], whose large mass is responsible for the large observed mixing between the flavour eigenstates.
(a) (b)
Figure 4.1: Box diagrams for B0
s − ¯Bs0 mixing. [16] Taking into account the hermiticity of M and Γ , it follows that:
Mij = Mji∗ and Γij = Γji∗, with i 6= j .
However, this does not imply any dependence between H12and H21, unless additional conditions are 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 a b ! . (4.10)
Since B0 mesons decay, the left side of equation 4.10 must be negative, which implies that Γ is a 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 by equation 4.7 and knowing how the CP, T and CPT operators acts on a ket, we can show that several conditions are imposed from the invariance of these discrete symmetries on the elements of the effective Hamiltonian. An example is given for CP symmetry, where we use:
CP |B0i = eiξcp|B¯0i and CP ¯|B0i = e−iξcp|B0i and that CP is an exact symmetry, which implies:
CP Hw(CP )† = Hw. Γ11= 2π X k δ(m0− Ek)hB0|CP (CP )†HwCP (CP )†|fkihfk|CP (CP )†HwCP (CP )†|B0i = 2πX k δ(m0− Ek)e−iξcph ¯B0|CP Hw(CP )†|fkieiξcphfk|CP Hw(CP )†| ¯B0i = 2πX k δ(m0− Ek)h ¯B0|CP Hw(CP )†| ¯fkih ¯fk|CP Hw(CP )†|B¯0i = Γ22 .
Performing similar operations with the other discrete symmetries to the elements of the effective Hamiltonian, 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, all the matrix elements of the latter matrices being real, except of a global phase [17].
Using the previous conditions imposed for the indirect (mixing) invariance of the CP, T and CPT 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 of these. But first we will continue with the phenomenological description of neutral B meson oscillations.
The physicals states |Bai and |Bbi are the eigenstates of the effective Hamiltonian, with complex 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 mass of 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− iΓ12/2 M21− 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 = 1 2[ H11+ H22± q (H11− H22)2+ 4H12H21 ] = m − i 2Γ ± s (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 ≡ 1
2(M11+ M22); Γ ≡ 1
2(Γ11+ Γ22); δm ≡ M11− M22; δΓ ≡ Γ11− Γ22 . (4.14) Using equation 4.13 we define:
∆µ ≡ µH − µL = q 4H12H21+ (H22− H11)2 = ∆m + i 2∆Γ , (4.15) ∆m ≡ mH − mL= 2Re s (M12− i 2Γ12)(M ∗ 12− i 2Γ ∗ 12) + 1 4(δm − i 2δΓ ) 2 , (4.16) ∆Γ ≡ ΓL− ΓH = 4Im s (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 lighter eigenstate, it will be, by definition, a positive quantity. However, the sign of ∆Γ , must be determined experimentally [17].
Using the previous definitions and the properties of the effective Hamiltonian regarding T and CPT symmetries, we will define the following CPT-and-T violating parameters:
δT ≡ |H12| − |H21| |H12| + |H21| ; δCP T ≡ H11− H22 ∆µ , (4.18)
3It must be noticed that the real and imaginary part of the eigenvalues do not correspond to the eigenvalues
where in terms of the elements of the mass and decay matrix the previous CPT-and-T violating parameters can be written respectively as:
δCP T ≡ δm −2iδΓ 2q(M12− 2iΓ12)(M12∗ −2iΓ ∗ 12) + 14(δm − i 2δΓ )2 = δm − i 2δΓ ∆m +2i∆Γ , (4.19) δT ≡ |M12−2iΓ12| − |M12∗ − 2iΓ ∗ 12| |M12− 2iΓ12| + |M12∗ − 2iΓ ∗ 12| . (4.20)
From the previous definitions it can be deduced that, in order CPT symmetry to be conserved,
δCP T = 0, while in order T symmetry to be invariant, δT = 0 and finally in order for CP symmetry to be conserved then both δCP T and δT must be zero. This implies that both δT and
δCP T are CP violating quantities [14]. In the following we will define δCP T as z and we will focus, only on the CPT symmetry. Hence, with the previous definitions, the eigenstates of the effective Hamiltonian H, can be written in the most general4 form as :
|BLi = pL|B0i + qL| ¯B0i , (4.21) |BHi = pH|B0i − qH| ¯B0i , (4.22) where after diagonalizing the H we find that:
qL pL = 2H21 ∆µ(1 − z); qH pH = 2H21 ∆µ(1 + z) , (4.23) yielding: qHqL pHpL = M ∗ 12− iΓ12∗/2 M12− iΓ12/2 , (4.24)
where we can define:
q p ≡ − s qHqL pHpL = − v u u t M12∗ − iΓ∗ 12/2 M12− iΓ12/2 . (4.25)
Now the eigenstates of the effective Hamiltonian can be written as: |BLi = p √ 1 − z|B0i + q√1 + z| ¯B0i , (4.26) |BHi = p √ 1 + z|B0i − q√1 − z| ¯B0i , (4.27) and the flavour eigenstates as:
|B0i = √ 1 + z 2p |BHi + √ 1 − z 2p |BLi , (4.28) | ¯B0i = √ 1 + z 2q |BLi − √ 1 − z 2q |BHi . (4.29)
From equation 4.28 it can be deduced that if z 6= 0, the composition of the flavour eigenstates is asymmetric in terms of the physical states [14]. We must note that even though the flavour and the CP eigenstates are orthogonal:
hB0| ¯B0i = 0 ; hB
+| ¯B−i = 0 , the same in not true in general for the physical states:
hBH|BLi = p∗HpL− qH∗ qL . (4.30) In case CP is a conserved symmetry then the right-hand side of equation 4.30 is zero, but it can be also zero when δT = 0 and Im(z) = 0 [18]. As the following section is dedicated on how the CPT-violating parameter z appears in the decay rates of the B0 mesons, is crucial to make the following remark: If z is aimed to be an observable that quantifies CPT violation, it should be invariant under re-phasing of the physical states |BH,Li. Using equations 4.23 and the definition for the z and ∆µ, we can re-write the CPT-violating parameter as:
z = 1 − qH/pH qL/pL 1 + qH/pH qL/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 a measurable, complex, CPT-violating quantity [14].
4.2
The time evolution and decay of B
0mesons in the
context of CPT violation
Since the states |BHi and |BLi are eigenstates of the Schr¨odinger equation, the time evolution of these states is obtained as:
|BHi = e−imHt− 1 2ΓHt|B H(0)i , (4.32) |BLi = e−imLt− 1 2ΓLt |BL(0)i . (4.33)
By combining equations 4.26, 4.28 and 4.32 we get: |B0(t)i = (g +(t) − zg−(t))|B0i + √ 1 − z2q pg−(t)| ¯B 0i , | ¯B0(t)i = (g+(t) + zg−(t))| ¯B0i + √ 1 − z2p qg−(t)|B 0i , (4.34) where g±(t) ≡ 1 2(e −imLt−12ΓLt± e−imHt−12ΓHt) = 1 2e −i ¯M t
e−i ¯Γ t/2(e+i∆mt/2e−∆Γ t/4± e−i∆mt/2e+∆Γ t/4) . (4.35) In the previous equation we have defined that: ¯M ≡ MH+ML
2 and ¯Γ ≡
ΓH+ΓL
2 . However, in the following chapters describing the decay rates of B0
s → 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 |B0i state at t = 0, the probability after t = t0, to find a | ¯B0i or a |B0i state and vice versa, is given by:
|h ¯B0|B0(t)i|2 = |1 − z2||q p| 2|g −(t)|2 , (4.36) |hB0|B0(t)i|2 = |g +(t)|2+ |z|2|g−(t)|2− 2Re[ zg−(t)g+(t)∗ ] , (4.37) |hB0| ¯B0(t)i|2 = |1 − z2||p q| 2|g −(t)|2 , (4.38) |h ¯B0| ¯B0(t)i|2 = |g+(t)|2+ |z|2|g−(t)|2+ 2Re[ zg−(t)g+(t)∗ ] . (4.39) From the previous equations we note that |h ¯B0| ¯B0(t)i|2 6= |hB0|B0(t)i|2 only when z 6= 0; meaning that the probability of a P (B0 → B0) 6= P ( ¯B0 → ¯B0). Additionally, we can deduce that when |pq| = 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 following decay amplitudes as:
Af = hf |B0i; A¯f = hf | ¯B0i; Af¯= h ¯f |B0i; A¯f¯= h ¯f | ¯B0i (4.40) and also define the following complex parameters:
λf = q p ¯ Af Af ; λ¯f = 1 λf ; λf¯= q p ¯ Af¯ Af¯ ; ¯λf¯= 1 λf¯ (4.41) What must be noticed is that the choice of the previous definitions is connected with the invariance of the observables under a re-phasing of the states. As an example we can see that if we re-phase the final state |f i → eiφf|f i and the physical state B0as |B0i → eiφ|B0i the quantity
Af does not remain invariant. In contrast it can be shown that the quantities, λf, ¯λf, λf¯, ¯λf¯ together with the z parameter and the |Af|, | ¯Af|, |Af¯|, | ¯Af¯|, |qp| remain invariant, and can thus be used as observables [18].
Using equation 5.2 and the definitions given before for the decay amplitudes we can deduce that: A(B0 → f ) = (g+− zg−)Af + q p √ 1 − z2g −A¯f , (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 −A¯f¯, (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) = |hf |T |B0(t)|2), that give the probability for an initial neutral B meson to decay to a general state f at a given time will be obtained by squaring the previous amplitudes; yielding the following expressions:
ΓBd→f = 1 2|Af| 2|p q| (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 = 1 2| ¯Af¯| 2|q p| (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 and 0. Where qf = 1 represents a B0, qf = −1 a ¯B0 and qf = 0 the untagged case, where we do not have any information about the flavour of the B0 meson and hence only the terms that are common for the B0 and ¯B0 are kept. The previous equations can be written in a more elegant form if we approximate z ≈ 1+|λ2z
f|2 and define the following quantities:
Df ≡ −2Re(λf) 1 + |λf|2 ; Sf ≡ 2Im(λf) 1 + |λf|2 ; Cf ≡ 1 − |λf|2 1 + |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:
Df2 + Sf2+ Cf2 = Df2¯+ Sf2¯+ Cf2¯= 1 . (4.50) Substitution of these into 4.46 and 4.47 yields:
ΓBqf→f = 1 2|Af| 2|p q| (1−qf)e−Γst(1 + |λ f|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 = 1 2| ¯Af¯| 2|q p| (1+qf)e−Γst(1 + |¯λ ¯ f|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¯ ]} .
A special case is when the final state |f i is a CP eigenstate, such as the |J/ψφi. Using equations 4.41 and the fact that | ¯f i is proportional to |f i since they are CP eigenstates, results
in Cf = −Cf¯, Df = Df¯, Sf = −Sf¯, |Af| = |Af¯| and ¯λf¯ = λ1
f [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/ψφ the previous decay rates need to be modified, as this decay is governed by several intermediate states. The previous formalism for the decay rates, includes CPT-violating phenomena only in the mixing process, while direct CPT violation in the decay amplitude of the B0 mesons is expected to be negligible.
The invariance or non-invariance of the discrete symmetries T, CPT and CP will have a fundamental effect on the decay rates, where we must note that we consider only CPT-violating phenomena in the mixing process. In the case that |qp| 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 have CP violation in the decay, i.e, |Af| 6= | ¯Af¯|, will result again in λf 6= ¯λf¯[16]. A very interesting case 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 in
ΓB0→f and ΓB¯0→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 (see 4.63):
ACP =
ΓB0→f− ΓB¯0→f
ΓB0→f + ΓB¯0→f
(4.53) In general this asymmetry is named CP asymmetry but this is because T and CP asymmetry are equivalent in the case that CPT invariance is assumed. There are many different possibilities that can cause non-zero value for this asymmetry, such as T, CPT violation in mixing and CP violation in the decay. We will express the asymmetry in the most general form, where we include CPT-violating effects with the exception of the decay amplitudes. Defining the following expressions: 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/CP T /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 symmetries and see how the asymmetry ACP/CP T /T will be effected. In the case that |q/p| = 1 corresponding to T invariance in mixing, then ACP/CP T /T → ACP/CP T and the coefficients defined in equations
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-level decay 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 =
−Sfsin(∆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 thesis we are interested in the effects of CPT violation, we continue by focusing on the asymmetry which originates in a possible CPT violation. In the context of the SME extension Im(z) can be approximated to zero for B0 mesons (see next chapter). Hence, the asymmetry, in the case that we have T invariance in the mixing and additionally CP invariance in the decay can be expressed, respectively, as:
ACP/CP T =
cos(∆mst)Cf − sin(∆mst)Sf cosh(∆Γst/2) + Dfsinh(∆Γst/2)
+
Re(z)Df[ cosh(∆Γst/2) − cos(∆mst) ] + sinh(∆Γst/2) cosh(∆Γst/2) + Dfsinh(∆Γst/2) , (4.64) ACP/CP T = − sin(∆mst)Sf cosh(∆Γst/2) + Dfsinh(∆Γst/2) +
Re(z)Df[ cosh(∆Γst/2) − cos(∆mst) ] + sinh(∆Γst/2) cosh(∆Γst/2) + Dfsinh(∆Γst/2)
.
(4.65)
In this case we can factorize the asymmetry in two terms, representing the asymmetry originating from 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:
ACP T = 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 context of 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 decay
are refereed as CP violation. In principle CPT, CP and T violating effects can also be quantified in the decay process.
allowing for CPT violation, we finally conclude that CPT violation can be manifest through the complex parameter z: z = δm − i 2δΓ ∆m +2i∆Γ . (4.67)
In the context of standard model extension (SME), it can be shown6 that δΓ = 0, which relates 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 the
B0
s 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 violation is 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 both CPT and Lorentz violation. The CPT violating parameter can then be written as:
z = β
µ∆α µ
∆m +2i∆Γ , (4.70)
where βµ= γ(1,−→β ) is the four-vector velocity of the B0 meson in the observer frame. The fact that ∆m and ∆Γ are extremely small (see text) increases the sensitivity of the ∆αµcoefficients, resulting on B0 mesons to be considered as ideal interferometers. In the previous equation, ∆α
µ is defined as : ∆αµ= rq1α
q1 µ −rq2α
q2
µ, where αqµ1 and αqµ2 are the CPT violating coupling constants (mass dimensions) associated with terms −αq
µqγ¯ µq, integrated in the SME Lagrangian, where
q is a quark field of a specific flavour, while the parameters rq1 and rq2, quantify normalization and quark-binding effects [3]. Neglecting the small effects of the latter parameters we can write:
∆αµ ≈ αqµ1 − αqµ2 . (4.71)
Using the equation 4.71 the following approximate7 equation for the neutral mesons can be written: ∆αKµ0 − ∆αB0d µ + ∆αB 0 s 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 orientation of 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 tensor
field that breaks the CPT invariance and the momentum of the neutral meson; this represents a real number, resulting in δΓ = 0.
since the relative direction of the lab-frame with the constant background vacuum expectation values of the CPT-violating fields that span the universe, effects directly the determination of the CPT-violating coefficients. The previous statement becomes even more clear if we re-write the 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. From equation 4.73 we can deduce that:
• Increasing the boost of the B0 mesons, will enhance the CPT-violating effects; in the LHCb detector, the B0 mesons are produced with hγi ≈ 17.48.
• Using B0 mesons that are produced with hγi ≈ 1, gives direct access to ∆α
0. Even though producing B0 mesons at rest is not possible, this is a good approximation for CLEO 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 suppressed compared to the ∆α0 [21].
From equation 4.73, it become obvious that the CPT-violating parameters, depend directly on 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 sidereal modulation, as a consequence of the Earth’s rotation.
To display the explicit sidereal dependence of the z parameter, we will express the previous parameter in terms of a non-rotating frame. We adopt the basis (X,c Y ,b Z) for the non-rotatingb
Sun-centered frame and (x,b y,b z) for the lab-frame.b
Figure 4.2: Bases in the non-rotating frame and in the laboratory frame [21].
The (X,c Y ,b Z) frame is defined in celestial equatorial coordinates, withb Z aligned along to theb
forming a right-handed base [21]. In the lab-frame it is convenient to take the z axis alignedb
with 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 cos χ = ˆz ˆZ must be non zero. Provided the previous condition
is fulfilled, the z axis will rotate around theb Z axis with sidereal frequency Ω. By definingb
(∆α1, ∆α2, ∆α3) as the CPT and Lorentz breakdown components of the coefficients ∆−→α in the lab-frame, and as (∆αX, ∆αY, ∆αZ) are defined in the non-rotating frame, performing two rotations from the lab-frame to the Sun-centered frame, results in:
∆α1 = ∆αXcos χ cos Ωbt + ∆αY cos χ sin Ωbt − ∆αZsin χ , (4.74)
∆α2 = ∆αY cos Ωt − ∆αb Xsin Ωt ,b (4.75)
∆α3 = ∆αXsin χ cos Ωbt + ∆αY sin χ sin Ωt + ∆αb Zcos χ . (4.76)
Combining the previous expressions for the CPT-violating coefficients with equation 4.67 and using the fact that ∆αX,Y,Z = −∆α
X,Y,Z [22], we obtain for the CPT-violating parameters the following expression:
z = z(bt, θ, φ, p)
= γ(p)
∆m +2i∆Γ{∆α0+ β∆αZ(cos θ cos χ − sin θ cos φ sin χ)
+ β[∆αY(cos θ sin χ + sin θ cos φ cos χ) − ∆αXsin θ sin φ] sin Ωtb
+ β[∆αX(cos θ sin χ + sin θ cos φ cos χ) + ∆αY sin θ sin φ] cos Ωt} ,b
(4.77)
which for highly collimated B0, reduces to:
z = z(bt, p)
= γ(p)
∆m +2i∆Γ{∆α0+ β∆αZcos χ + β∆αY sin χ sin Ωbt
+ β∆αXsin χ cos Ωt} ,b
(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 andbt the sidereal time. Since we are interested only
in the real part, we can write: Re(z) = Re(z)(t, p)b
= γ(p)∆m
∆m2+ 1/4∆Γ2{∆α0+ β∆αZcosχ + β∆αYsinχsinΩbt + β∆αXsinχcosΩbt} .
(4.79)
Since B0 mesons in LHCb are produced with a high boost, the variation of the magnitude of β is extremely small. For this reason β will be set to be one. This result in the ∆α0 and ∆αZ to be combined and measured as one parameter, defined as: ∆α0 + cos χ∆αZ.
CPT violation in B
s
0
→ 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 B
s0→ J/ψφ
The decay of the B0
s → 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 Bd0 → J/ψK0
S, 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 the CKM matrix1, CP violation is expected to be very small, making the Bs0 → J/ψφ decay sen-sitive to non-standard amplitudes that can contribute in the mixing process or in higher-order decay topologies, so-called penguin diagrams.
(a) b (b) b
Figure 5.1: (a) The dominant (tree) and (b) suppressed (penguin) decay diagrams; the curled line represents the contribution from gluons.
1From the Wolfenstein parametrization, it can be noticed that, V
ts is real up to O(λ3), while Vtd is already
imaginary, resulting in βsto be much smaller compared to β [23].
As it was described before, the ACP/T asymmetry between the decays of Bs0/ ¯Bs0 → J/ψφ, originate from the presence of the interference term Im(λf), arising from the contribution of two different amplitudes, with and without mixing, in the decay process. The evolution of the mixing process is dominated by the top quark, so it is proportional to (VtsVtb∗)
2
. The amplitudes
for the tree-level decays that occur with and without mixing are proportional to (Vcs∗Vcb) and (VcsVcb∗) respectively. This yields for the phase of λJ/ψφ:
arg " q p AJ/ψφ ¯ AJ/ψφ # = arg[(VtsVtb∗) 2 ] + arg[(Vcs∗Vcb)] − arg[(VcsVcb∗)] = 2 arg " −VtsV ∗ tb VcsVcb∗ # ≡ 2βs ,
where βs represents one of the angles appearing in the Unitary Triangle of the Bs0 system. In the derivation of the previous expression for the weak phase we have neglected real, on shell contributions which originate from the off-diagonal elements of the decay matrix, since are estimated to be very small [24] and higher-order topologies in the decay process. The quantity that is measurable is φs, which, considering the previous assumptions, is predicted to be equal to −2βs. What should be noticed is that since φs is determined by the relative phase of the decay amplitudes, where CPT violation is neglected and the term q/p that is determined from the 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 symmetry is not conserved, will result in a not accurate prediction of φs. The λf can now be expressed in terms of the φs, that will appear consequently in the decay rates, enabling to perform a measurement for this phase.
The decay of B0
s → 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 in
the following possible values L ∈ (0, 1, 2). This leads to the following possibilities for the CP eigenvalue of the final state:
CP |J ψφi = ηf|J/ψφi = (−1)L|J/ψφi , (5.1) where from the previous equation it can be deduced that the eigenvalues of the final state are odd or even depending on the relative orbital momentum of the two vector mesons, giving rise to three possible intermediate angular momentum states. The amplitudes can be decomposed into three components; describing the decay in terms of spin-polarization states. These are the longitudinal (0) and the transverse to the direction of motion of the vector mesons. Where in the last 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 state with L = 1 is odd under the CP operator. Using the previous definitions, the differential decay rate of the Bs → J/ψφ decay can be expressed as:
d4Γ dtd ~Ω = P6 i=1Ti(t)fi( ~Ω) R R P6 j=1Tj(t)fj( ~Ω)dtd ~Ω , (5.2)
where Ti(t) represents the time-dependent components (see next section), expressed in terms of the physical parameters. fi are angular-dependent functions (see Tables 5.1 and 5.2), derived using the helicity formalism [26], where ~Ω = (θµ, θK, φh).
Figure 5.2: Definition of the so-called helicity angles. The angles θµ and θK are defined in the rest frame of J/ψ and K−K+, while the φ
h represents the angle between the two frames, taking as an origin the rest frame of the B0s [16, 27].
i Ti(t) fi( ~Ω)
1 |A0(t)|2 2 cos2θKsin2θµ 2 |A||(t)|2 sin2θK(1 − sin2θµcos2φh) 3 |A⊥(t)|2 sin2θK(1 − sin2θµsin2φh) 4 Im(A∗
||(t)A⊥(t)) sin2θKsin2θµsin 2φh 5 Re(A∗0(t)A||(t)) 12
√
2 sin 2θKsin 2θµcos φh 6 Im(A∗0(t)A⊥(t)) −12
√
2 sin 2θKsin 2θµsin φh
Table 5.1: The sum over the products: Ti(t)fi( ~Ω) represents the PDF of the Bs0 → 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 decay of the spin one φ resonance must have orbital angular momentum L = 1. However there might be contributions from B0
s → J/ψK+K
− decays, where the invariant mass of the K+K− system is close to the φ(1020) meson. Such a situation arises when the two kaons originate from the scalar mesons α0 and f0 [26], both with on invariant mass of 980 MeV/c2, resulting in the orbital 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 of
Bs → J/ψK+K− is given by equation 5.2, with the indexes i, j to run from one to ten, where the additional four terms that quantify the S-wave contribution are given by Table 5.2.
i Ti(t) fi( ~Ω) 7 |As(t)|2 23sin2θµ 8 Re(A∗0(t)As(t)) 43 √ 3 cos θKsin2θµ 9 Re(A∗||(t)As(t)) 13 √
6 sin θKsin 2θµcos φh 10 Im(A∗⊥(t)As(t)) −13
√
6 sin θKsin 2θµsin φh
5.2
Time dependence and CPT violation
In the previous section we briefly discussed the angular dependence of the differential decay rates of the Bs → J/ψK+K−, where the different components of the PDF were summarized in Tables 5.1 and 5.2. In this section we will derive the expressions of the Ti(t)(time dependent components), 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 equations 4.42-4.45 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−) + 1 λi √ 1 − z2g − ] , (5.4) where λi ≡ q p ¯ Ai Ai . (5.5)
Since, the final state is a CP eigenstate, equation 5.5 will be re-written as:
λi = ηi|λi|e−iφ
i
s , (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 |pq| ≈ 1 and furthermore neglect higher-order contributions in the decay, resulting on |λi| = 1 and φis to be the same for all intermediate states. However in this thesis we choose not to constrain |λi| but we assume that all λi are the same. This implies that there is either no, or equal amount of CP violation in the decay process for all the intermediate states. This choice will lead to the φi
s being independent of the polarization state. Additionally, T violation in mixing |qp| 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 expression for 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|2 1 + |λ|2 , (5.7) where it follows that:
ηiλ = −Dλi+ iSλi 1 + C ; ηi λ = −Dλi− iSλi 1 − C ; |λ 2| = 1 − C 1 + C , (5.8)
Ai(t) = [ (g+− zg−) + ηiλg− √ 1 − z2 ]α i = (g+− zg−) + g− −D λi + iSλi 1 + C √ 1 − z2 αi , (5.9) ¯ Ai(t) = [ (g++ zg−) + g− ni λ √ 1 − z2 ] ¯α i = (g++ zg−) − g− D λi+ iSλi 1 − C √ 1 − z2 ¯ αi , (5.10)
where, since αi = Ai(0), it follows that ¯αi = ηiαi [25]. Additionally, by ignoring second-order contributions from the CPT-violating parameter z, and defining2 D
λi = ηiD and Sλi = ηiS, the
decay 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 column of Tables 5.1 and 5.2: A∗i(t)Aj(t) = α∗iαj 1 + C[ g 2 +(1 + C) − zg ∗ +g−(1 + C) + g ∗ +g−(−D + iS)ηj
− z∗g−∗g+(1 + C) − z∗g−2(−D + iS)ηj − g−∗g+(D + iS) + g−2(D + iS) + g−2(1 − C)ηiηj ] , (5.13) ¯ A∗i(t) ¯Aj(t) = α∗iαjηiηj 1 − C [g 2 +(1 − C) + zg ∗ +g−(1 − C) − g−g+∗(D + iS)ηj
+ z∗g∗−g+(1 − C) − z∗g−2(D + iS)ηj − g−∗g+(D − iS)ηi− zg−2(D − iS)ηi+ g−2(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 + ηiηj 2 ; R − ij ≡ 1 − ηiηj 2 ; R Im ij ≡ i ηi− ηj 2 ; R Re ij ≡ ηi+ ηj 2 , (5.17) 2The η
i and ηj, will be absorbed in the time-dependent part and will not appear explicitly in S,D and C
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)(R Re ij D + R Im ij S) + Im(z)(R Im ij D − R Re ij S)] + cos(∆mst)[(Rij−+ R+ijC) − Re(z)(RijReD + RImij S) − lm(z)(RijImD − RReij S)] + sinh(∆Γst/2)[(RijReD + R Im ij S) + Re(z)(1 + C)] + sin(∆mst)[(RImij D − R Re ij 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)(−R Re ij D + R Im ij S) − Im(z)(R Im ij D + R Re ij S)] + cos(∆mst)[(R−ij − R+ijC) − Re(z)(−RijReD + RijImS) + Im(z)(RijImD + RijReS)] + sinh(∆Γst/2)[(RReij D − R Im ij S) + Re(z)(C − 1)] + sin(∆mst)[(RImij D + R Re ij S) + Im(z)(C − 1)] ] . (5.19) The previous equations can be combined in a more compact form in the following manner:
A∗qf i (t)A qf j (t) = α∗iαje−Γst 1 + qfC [ cosh(∆Γst/2)[(R+ij + R − ijC) + qfRe(z)(RReij D + R Im ij S) + Im(z)(R Im ij D − R Re ij S)] + cos(∆mst)[qf(Rij−+ R+ijC) − qfRe(z)(RijReD + RijImS) − Im(z)(RImij D − RReij S)] + sinh(∆Γst/2)[(RijReD + R Im ij S) + qf+ijRe(z) + qf−ijRe(z)C] + sin(∆mst)[qf(RijImD − R Re ij S) + qf+ijIm(z) + qf−ijIm(z)C] . (5.20) where: qf+ij = ( qf if ηiηj > 0 1 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 B0
s → 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
the imaginary part to be approximately zero, the previous expression will be simplified to: A∗qf i (t)A qf j (t) = α∗iαje−Γst 1 + qfC [ cosh(∆Γst/2)[(R+ij + R − ijC) + qfRe(z)(RReij D + R Im ij S)] + cos(∆mst)[qf(Rij−+ Rij+C) − qfRe(z)(RReij D + RijImS)] + sinh(∆Γst/2)[(RReij D + R Im ij S) + qf+ijRe(z) + qf−ijRe(z)C] + sin(∆mst)qf[RImij D − R Re 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 we have CPT invariance, the previous expression will be significantly simplified to:
A∗qf i (t)A qf j (t) = α∗iαje−Γst 1 + qfC [ cosh(∆Γst/2)[Rij++ R − ijC] + cos(∆mst)qf[R−ij + R + ijC] + sinh(∆Γst/2)[RReij D + RImij S] + sin(∆mst)qf[RijImD − RReij S] (5.22)
5.3
Combining time-dependence including CPT-violating
effects and angular dependence
In the same manner as with the angular PDF, we construct a tabular representation (see Table 5.3) of the time dependent parts of the PDF. As an example the fifth term of the PDF for qf = 1, is illustrated bellow:
T5(t)f5( ~Ω) = |α0||α|||e−Γstcos(δ||− δ0)×
{ cosh(∆Γst/2)[(1 + |λ|2)/2 − Re(z)|λ|cosφs+ Im(z)|λ| sin φs] + cos(∆mst) [(1 − |λ|2)/2 + Re(z)|λ| cos φs− Im(z)|λ| sin φs] + sinh(∆Γst/2)[−|λ| cos φs+ Re(z)] + sin(∆mst)[|λ| sin φs+ Im(z)]} ×1
2 √
2 sin 2θKsin 2θµcos φh .
(5.23)
Where we express the products α∗iαj in terms of the strong phases, originating from the hadronic matrix 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)
