A new method to measure the time-integrated CP asymmetry in singly-Cabibbo suppressed D0->K+K- and D0->pi+pi- decays at LHCb

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Dipartimento di Fisica

Corso di Laurea Magistrale in Fisica

A new method to measure

the time-integrated

CP asymmetry in

singly-Cabibbo suppressed

D

0 _→K

+

K

−

and

D

0 _→π

+

π

−

decays at LHCb

Candidate:

Nico Kleijne

Advisor:

Dr. Michael J. Morello

Academic Year 2019-2020

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Introduction

In the Standard Model of Particle Physics (SM) the violation of CP symmetry (the si-multaneous application of charge conjugation C and spatial parity P ) is described by the introduction of a single complex phase in the quark-mixing matrix, also known as Cabibbo-Kobayashi-Maskawa (CKM) matrix. The CKM mechanism represents a great success for the theory, since it is the most simple and elegant way to precisely describe and predict every experimental observation of CP violation in the neutral flavoured meson systems containing strange and bottom quarks.

However, according to the SM and the CKM mechanism, the measured CP violation values do not agree with cosmological observations, which show that our universe is largely composed by matter and almost no antimatter. The evolution of our universe most certainly required CP violating effects much larger than those measured until now and explained inside the CKM mechanism [1]. For this reason, the search for new types of CP violation with an intensity much larger than those predicted by the theory is one of the primary goals of High Energy Physics in the coming years.

Some theories predict the presence of new sources of CP violation by assuming the existence of new particles beyond the Standard Model, at energy scales much larger than those directly achievable at LHC. These particles could couple with up-type quarks only, and not with down-type quarks where CP violation has been extensively and precisely studied in the last decades, and they could have escaped experimental observation until now. From this point of view, the charm quark and therefore the neutral D0 _mesons represent a favoured field of research to study CP violation.

The CP violation predicted in the Standard Model for D0 _{mesons is extremely small,} ≤ 10−3_{, even though the theoretical estimations still have large uncertainties due to the} strong interaction contribution appearing in the low energy processes [2, 3]. CP violation in the charm quark sector has been observed for the first time only recently, March 2019, by the LHCb experiment in the Cabibbo suppressed D0 _→K+_K− _{and D}0 _→π+_π− _decay channels. In particular, the observed quantity was the difference of time-integrated CP asymmetries ∆ACP := ACP(K+K−)− ACP(π+π−) = (−15.4 ± 2.9) × 10−4 [4]. Assuming that CP violation in the mixing and in the interference between mixing and decay is small, as experimentally confirmed (AΓ = (−0.29±0.20(stat)±0.06(syst))×10−3) [5, 6], the main contribution to ∆ACP comes from CP violation in the decay. However, the theoretical interpretation of the standard or non-standard nature of this result is not trivial (see for example Ref. [7, 8, 9, 10]) and it requires a great effort from the theoretical community to reduce the uncertainty on the prediction. It is, therefore, of paramount importance to provide as soon as possible a measurement of the absolute CP violation in each channel

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(either D _→K K or D _→π π ), in order to constrain the theoretical predictions. The absolute measurements of CP -violating asymmetries are, in general, extremely challenging at LHCb and they are performed by summing and subtracting to the channel of interest the asymmetries of many calibration channels1 _{which allow the cancellation of} the charge asymmetries due to the mechanisms of production and detection of D mesons. However, these strategies require to weight the kinematic distributions of the different samples and, therefore, reduce by large factors the effective statistical size of the final data sample. The current single-experiment most precise determination of absolute CP violation was obtained by LHCb [11] which measured ACP(K+K−) = (0.4± 1.2(stat) ± 1.0(syst))_{× 10}−3 _{using the full Run 1 sample (3 fb}−1

). The ACP(π+π−) asymmetry is obtained by combining the measured values of ACP(K+K−) and ∆ACP and its value in Run 1 is ACP(π+π−) = (0.7± 1.4(stat) ± 1.1(syst)) × 10−3 [11]. The combination of the current LHCb Run 1 measurement with measurements from other experiments gives a world average of ACP(K+K−) = (−0.7 ± 1.1) × 10−3 and ACP(π+π−) = (1.3± 1.4) × 10−3 _{[12]. The expected statistical uncertainty on A}

CP(K+K−) for the same analysis strategy with the LHCb Run 2 data (6 fb−1) is 0.85_{× 10}−3_{, while in the combination} of Run 1 and Run 2 data the expected value reduces to 0.7× 10−3 _{[13]. This level of} precision is likely not sufficient to observe CP -violating effects in these channels. In fact, the SU (3) flavour symmetry predicts that CP violation in the decay has equal intensity and opposite sign in the D0 _{→ K}+_K− _{and D}0 _{→ π}+_π− _{decay channels [2, 14, 15]. The} expected CP -violating asymmetry is therefore of order of ∆ACP/2 ' 0.8 × 10−3 and an uncertainty of a few units in 10−4 _{would be necessary to have a chance to observe the} effect2_.

In this thesis I developed a new method to cancel those nuisance asymmetries, by exploiting a single Cabibbo-favoured decay channel, D0 _{→ K}0

Sπ+π

−_{, with the aim of} improving the current experimental sensitivity and approaching the desired level of pre-cision. The method is fully data-driven, and much simpler since it exploits only a single calibration channel sharing many similarities with the signal decays. Subtle detection asymmetries can be kept under control, at much higher precision, with respect to the more complex previous analysis which uses more than one decay calibration channel. More-over, the use of only a single channel reduces the amount of requirements and weighting procedures necessary to equalize the kinematics of the different samples, resulting in a more optimal exploitation of the total available statistical power.

The developed analysis procedure is applied to D0 _{mesons produced in D}∗+

→ D0_π+ decays where the charge of the pion is exploited to infer the flavour of the D0_{. The data} samples were collected during the LHC Run 23 _{and they correspond to an integrated} luminosity of 5.7 fb−1. The total yields are about 56 M of D0 _{→ K}+_K−_{candidates, 18 M} of D0 _{→ π}+_π− _{candidates, and 17 M of D}0 _{→ K}0

Sπ+π

− _candidates.

1_{Usually Cabibbo-favoured processes are used since CP violation in the decay can be considered much}

smaller than the typical experimental uncertainties.

2_{This is the most unfavourable case since the SU (3) symmetry is broken, so that one of the two single}

ACP will be very likely greater than the value of 0.8× 10−3 while the other less.

3_{The integrated luminosity does not include 2015 because D}0

→ K0

Sπ

+_π− _{candidates were not}

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Chapter 1 Mixing and

CP violation in the

charm quark sector

This chapter introduces the concepts of mixing and CP violation for neutral flavoured mesons systems according to the Standard Model of Particle Physics. Particular attention is given to charmed neutral mesons as they are the subject of this thesis. The formalism for CP violation in these systems is also extensively discussed.

1.1 Introduction

The discrete transformation denoted as CP is defined as the combination of the charge conjugation C, which reverses the internal quantum numbers of all particles, and of the parity transformation P , which reverses the sign of spatial coordinates and, consequently, the handedness of all particles. So far, experimental evidence shows that the electro-magnetic and strong interactions preserve both C and P , and consequently CP is also conserved by these interactions. On the other hand, it was shown that the weak inter-action maximally violates the C and P symmetries [16], as the W boson couples only to left-handed particles and right-handed antiparticles. In 1964 the weak interaction was observed, albeit tinily, to violate CP symmetry in kaons decay as well [17].

1.2 Quark mixing in the Standard Model

In the Standard Model (SM) the charged-current interaction of quarks with the W boson is described by the following Lagrangian term

LW = −g√ 2 uL cL tL γ µ Wµ+VCKM   dL sL bL  + h.c. (1.1)

Here the Lagrangian is expressed in terms of the quark mass eigenstates uL, dL, cL, sL, tL, bL, describing the physical states. The L subscript indicates the fact that, due to the chiral nature of the weak interaction, only left-handed quarks couple to the W boson. The

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complex unitary matrix VCKM is called Cabibbo-Kobayashi-Maskawa (CKM) matrix [18, 19] and allows the definition of the weak interaction eigenstates as a combination of the quark mass eigenstates

  d0 s0 b0  =   Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb     d s b  . (1.2)

From this, it is clear that the W boson couples up-type to down-type quarks and the off-diagonal elements of VCKM allow transitions between different generations of quarks.

In general, a complex n_{× n unitary matrix can be described by n}2 _{independent real} parameters. It is possible to redefine all the quark fields by changing their phases

U _{→ e}iξU _{and D}_{→ e}iξD_,

where U represents a generic up-type field and D a generic down-type field. In this way the elements of VCKM transform as

VU D → eiξUVU De−iξD.

However, changing all the quark fields by the same global phase leaves VCKM unchanged, thus a total of 2n_{−1 unphysical phases can be removed by redefining the fields. This leaves} (n_{− 1)}2 _{independent parameters; out of these, n(n}_{− 1)/2 parameters correspond to the} independent rotation angles on the n basis vectors, while the remaining (n− 1)(n − 2)/2 parameters are complex phases.

In the case of two generations of quarks (n = 2) there is only one independent mixing angle θC and no complex phase. The mixing between two generations of quarks can thus be described by a 2× 2 rotation matrix, the Cabibbo matrix

VC = cos θC sin θC − sin θC cos θC . (1.3)

Since no complex phase is present, CP violation is not possible with n = 2.

In the Standard Model three generation of quarks exist (n = 3) and VCKM can be de-scribed by three independent mixing angles θ12, θ13, θ23, and a complex phase δ, which is responsible for CP violation. In this way the CKM matrix can be written as [20]:

VCKM =   Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb  =   1 0 0 0 c23 s23 0 _−s23 c23     c13 0 s13e−iδ 0 1 0 −s13eiδ 0 c13     c12 s12 0 −s12 c12 0 0 0 1   =   c12c13 s12c13 s13e−iδ −s12c23− c12s23s13eiδ c12c23− s12s23s13eiδ s23c13 s12s23− c12c23s13eiδ −c12s23− s12c23s13eiδ c23c13  , (1.4) where sij = sin θij and cij = cos θij.

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1.2. Quark mixing in the Standard Model

• Direct measurements that look at tree-level interactions to directly extract the value of |Vij|. In some cases the elements have a poor resolution, such as |Vtb| and |Vcs|, or are too suppressed to be measured, such as |Vtb| and |Vts|.

• Indirect measurements that look at higher-order processes and extract the product of more than one Vij, such as |VtbVtd|.

By requesting VCKM to be unitary further constraints can be placed on its elements, especially on the less precisely measured, such as |Vtb|. These constraints are given by

|Vud|2+|Vus|2+|Vub|2 =1, (1.5) |Vcd|2+|Vcs|2+|Vcb|2 =1, (1.6) |Vtd|2+|Vts|2+|Vtb|2 =1. (1.7) When all the measurements and constraints are combined, the magnitudes of the elements of VCKM are given as [21]:   |Vud| |Vus| |Vub| |Vcd| |Vcs| |Vcb| |Vtd| |Vts| |Vtb|  =   0.974390+0.000014 −0.000058 0.224834+0.000252−0.000059 0.003683+0.000075−0.000061 0.224701+0.000254 −0.000058 0.973539+0.000038−0.000060 0.04162+0.00026−0.00080 0.008545+0.000075−0.000157 0.040900.00026−0.00076 0.9991270.000032−0.000012  . (1.8) From the measured values it can be seen that the mixing matrix is almost diagonal, with relatively small off-diagonal parameters. In fact, the rotation angles between the quark mass and weak eigenstates are small θ12 = 13◦, θ23 = 2.3◦ and θ13 = 0.2◦. Consequently, the interactions of quarks between different generations is suppressed relative to those in the same generation, and in particular the coupling between first and third genera-tion is strongly suppressed. This hierarchical structure in the couplings between different generations of quarks can be made clear by using the so-called Wolfenstein parametriza-tion [22]. In this way, the elements of the CKM matrix are expressed as an expansion in the relatively small parameter

λ := s12= |V us| p|Vud|2+|Vus|2

∼ 0.22.

The other three real parameters describing the CKM matrix are A, ρ and η and are defined as Aλ2 _{:= s} 23 = λ Vcb Vus and Aλ3_(ρ − iη) := s13e−iδ = Vub∗,

In this parametrization the CP violating phase is thus represented by η. The mixing matrix is then written up to order λ4 _as

VCKM =  

1_{− λ}2_/2 _λ _Aλ3_(ρ_{− iη)}

−λ 1− λ2 _Aλ2

Aλ3₍₁_{− ρ − iη) −Aλ}2 ₁



+O(λ4) (1.9) The CKM matrix written in this way clearly shows the hierarchy in the interactions between quark of different generations, expanded in terms of the parameter λ. This hierarchy is also sketched in Fig. 1.1.

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Figure 1.1: Graphical representation of the values of the CKM matrix.

It is also possible to define two new generalized Wolfenstein parameters that take into account higher order λ terms as:

¯ ρ = ρ(1₋ 1 2λ 2_{) and} _{η = η(1}_¯ − 1 2λ 2_). _(1.10)

Even if several parametrizations exist for the CKM matrix, it is possible to define a quantity proportional to the CP -violating effects that is independent from the particular parametrization. For example the Jarlskog invariant JCP is defined as [23]

JCP X m,n ikmjln := Im(VijVklVil∗V ∗ kj). (1.11)

Going back to the parametrization of Eq. (1.4) the Jarlskog parameter can be expressed as

JCP = s12s13s23c12c23c213sin δ13. (1.12) For CP violation to be observable the following must hold

(m2 t − m 2 c)(m 2 t − m 2 u)(m 2 c− m 2 u)(m 2 b − m 2 s)(m 2 b − m 2 d)(m 2 s− m 2 d)× JCP 6= 0. (1.13) Hence, for CP violation to be possible, the quark masses must not be degenerate, the three mixing angles must be different from 0 or π/2 and the complex phase must be different from 0 or π.

The unitarity of the CKM matrix implies the orthogonality of its rows and columns. These orthogonality conditions lead to a set of six equations

X i VijVik∗ = δjk and X j VijVkj∗ = δik. (1.14) These six constraints can be visualized as the so-called unitarity triangles in the complex plane. For example from Eq. (1.14) one can write

VudVub∗ + VcdVcb∗ + VtdVtb∗ = 0 (1.15) and dividing by VcdVcb∗ one gets

VudVub∗ VcdVcb∗ + 1 + VtdV ∗ tb VcdVcb∗ = 0. (1.16)

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1.2. Quark mixing in the Standard Model

It can be shown [24] that this relation is represented by a triangle in the complex plane with vertices (0, 0), (1, 0) and (¯ρ, ¯η) and this triangle is shown in Fig. 1.2. In the same way, other five unitarity triangles can be defined, one for each orthogonality constraint, all having equal area A∆= JCP/2.

The current experimental measurements of the CKM parameters can be represented in the (¯ρ, ¯η) plane and confronted with the constraints given by the unitarity triangle of Eq. (1.16). This is done in Fig. 1.3 where the 95% CL regions overlap consistently and no deviation from the Standard Model is visible.

Figure 1.2: Graphical representation of the unitarity triangle.

Figure 1.3: Current experimental status of the global fit to all available experimental measurements related to the unitarity triangle phenomenology. The shaded areas have 95% CL. Figure taken from Ref. [21].

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1.3 Mixing in the flavoured neutral mesons

In the Standard Model, flavoured neutral mesons present a unique phenomenon: the oscillation or mixing of a particle into its own antiparticle. There are valid reasons why these are the only particle that can oscillate. In order to conserve the electromagnetic charge and the baryonic number the particle-antiparticle oscillations can only occur for neutral mesons which have null charge and null baryonic number. With flavoured we mean that they have a non-null flavour quantum number, such as strangeness, charmness or bottomness. This request excludes neutral mesons which are their own antiparticles such as π0 _{(uu-dd), J/ψ (cc) or Υ (bb). Finally, the lifetime of the top quark is too short} and it cannot form bound states. Thus only four mesons can mix between particle and antiparticle states, they are: K0 _{(ds), D}0 _{(cu), B}0 _{(db) and B}0

s (sb).

For these four neutral flavoured mesons the interaction eigenstates in which they are produced are different from the mass ones, which are the eigenstates of the free Hamil-tonian driving their time evolution. Thus if we indicate the interaction eigenstates of a generic meson as _|M0_{i and |M}0_{i we can describe the initial state of a meson |ψ(0)i at} time t = 0 as a superposition of the two states:

|ψ(0)i = a(0)|M0

i + b(0)|M0i. (1.17)

If we also consider all possible decays to the final states _|fii, this state will evolve in time as

|ψ(t)i = a(t)|M0

i + b(t)|M0i +X i

ci(t)|fii. (1.18) Using the Weisskopf-Wigner approximation [12, 25], the evolution of the state can be restricted to the subspace spanned by |M0_{i and |M}0_{i. In this mixing subspace, the time} evolution can be described by a non-Hermitian effective Hamiltonian H = M_{− iΓ/2,} where both M and Γ are Hermitian. The Schr¨odinger equation then becomes

i∂ ∂t a(t) b(t) = M₋ i 2Γ a(t) b(t) . (1.19)

The elements of the mass matrix M are defined as Mij = m0δi,j+hMi|Hw|Mji +

P

khMi|Hw|fkihfk|Hw|Mji m0− Efk

, (1.20)

while the elements of the decay matrix Γ are defined as Γij = 2π X k δ(m0− Efk)hMi|Hw|fkihfk|Hw|Mji, (1.21) where M1 = M0, M2 = M 0

, _Hw is the weak Hamiltonian and m0 is the M0 mass. The Γ term allows the meson decay, as the time evolution in the mixing subspace becomes non-unitary. In fact, using Eq. 1.19 one gets

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1.3. Mixing in the flavoured neutral mesons

where we want the last term to be negative in order to represent a decay rate, hence Γ is positive defined. Since M and Γ are both Hermitian their diagonal elements must be real, while, for the off-diagonal elements, M12 = M21∗ and Γ12 = Γ∗21 must hold. From this we conclude that H has eight free parameters. However, it is possible to add further constraints on these free parameters by imposing invariance under discrete symmetries, such as CP T , CP or T . These possible constraints are listed in Tab. 1.1. In particular, if from here on we assume CP T invariance, we can use M := M11= M22and Γ := Γ11= Γ22.

Symmetry Constraints

CP T M11= M22 Γ11= Γ22

CP M11= M22 Γ11= Γ22 Im(Γ12/M12) = 0 T Im(Γ12/M12) = 0

Table 1.1: Constraints on M and Γ by imposing invariance under different discrete sym-metries [26].

The eigenvectors of H are called |MHi and |MLi and have defined masses (mH and mL) and decay widths (ΓH and ΓL). The eigenvectors can be written as

|MHi = p|M0i − q|M 0 i, |MLi = p|M0i + q|M 0 i, (1.23)

with p and q complex parameter such that_|p2_{| + |q}2_{| = 1. Hence, the two eigenstates are} normalized but not necessarily orthogonal, since H is not Hermitian. We also have

q p = s M∗ 12− iΓ ∗ 12/2 M12− iΓ12/2 . (1.24)

and the corresponding eigenvalues are given by λH,L := mH,L− i 2ΓH,L = M− i 2Γ± q p M12− i 2Γ12 . (1.25)

From this, one can see that

M = mH + mL

2 and Γ =

ΓM + ΓL

2 . (1.26)

The time evolution of an eigenstates is given by

|MH,L(t)i = exp(−imH,Lt) exp(−ΓH,Lt/2)|MH,L(0)i. (1.27) By using Eq. (1.23) and Eq. (1.27) one can obtain the time evolution of a meson produced in a flavour eigenstate at t = 0 |M0_(t)_{i = g} +(t)|M0i + q pg−(t)|M 0 i, |M0(t)_{i = g}+(t)|M 0 i + p qg−(t)|M 0 i, (1.28)

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where _|M0_(t)_{i is the time-evolution at time t of a meson produced as a |M}0_{i at t = 0,} and similarly for a_|M0_{i. The coefficients g}+(t) and g−(t) are given by

g±(t) =

e−iλHt± e−λLt

2 . (1.29)

Once the time evolution of Eq. (1.28) is known, it is possible to compute the probability of no-oscillation as P(M0₍₀₎ → M0_{(t)) =} |hM0_(t) |M0 i|2 ₌ |g+(t)|2, P(M0(0)_{→ M}0(t)) = _|hM0(t)_|M0_i|2 ₌ |g+(t)|2, (1.30)

and the probability of oscillation as P(M0₍₀₎ → M0(t)) =_|hM0_(t) |M0i|2 ₌ q p 2 |g−(t)|2, P(M0(0)_{→ M}0_{(t)) =} |hM0(t)_|M0 i|2 ₌ p q 2 |g−(t)|2. (1.31)

The probability of no-oscillation is the same for both mesons, while, if _{|p/q| 6= 1, the} probability of oscillation is different for M0_{(t) and M}0_{(t). Two new variables known as} the mixing parameters x and y can be defined as:

x := ∆m Γ = mH − mL Γ and y := ∆Γ 2Γ = ΓH − ΓL 2Γ , (1.32)

and, when inserted in Eq. (1.29), we obtain |g±(t)|2 = 1 2e −Γt [cosh(yΓt)± cos(xΓt)]. (1.33)

1.4 Mixing phenomenology

From Eq. (1.33) we can see that the two mixing parameters determine the behaviour of the oscillations of neutral flavoured mesons. The x parameter appears in the cosine term, hence the rate of the oscillations is proportional to the mass difference between the two physical eigenstates. The y parameter appears in the hyperbolic cosine term and determines the difference in the exponential decay trend. The parameters relevant to the mixing for the four neutral flavoured mesons are listed in Tab. 1.2. In Fig. 1.4 the time-dependent probabilities of Eqs. (1.30) and (1.31) are plotted using these parameters. The effective Hamiltonian H determines the values of the mixing parameters, which largely vary between different flavours, resulting in extremely different phenomenologies.

• K0 _mesons _{are the only one to have y} _{≈ 1, which means that the two physical} eigenstates have lifetimes differing by nearly three orders of magnitude. These two eigenstates are named based on this lifetime difference as K-short (K0

S) and K-long (K0

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1.4. Mixing phenomenology

K0

S when considering decays close to the production point, and a very pure sample of K0

L when considering decays far from the production point, where almost all of the K0

S have already decayed. There is also a sizeable mass difference between KS0 and K0

L and, since x≈ 1, the probability of flavour change presents one large sinusoidal oscillation.

• B0 _and _B0

s mesons have a very small y ∼ 10−2 − 10−3, which results in almost identical lifetimes for the two physical eigenstates. However, x has a non negligible value and oscillations are possible. In particular, the large value of x for the B0 s implies fast oscillations and a high spatial resolution is needed to resolve them. For this reason the observation of oscillation was achieved only recently [27].

• D0 _mesons _{have both very small x and y. This results in an extremely small} fraction of mesons changing their flavour and large data samples are needed to observe the phenomenon of oscillation. The first observation by a single experiment was achieved by LHCb in 2012 [28].

Meson _{hMassi M [ MeV ] Lifetime [ ps ]} x y

K0 _497.6 89.5 _0.95 _0.997 52900 D0 ₁₈₆₄ _0.410 _0.004 _0.007 B0 ₅₂₈₀ _1.52 _0.77 _-0.001 B0 s 5367 1.51 26.7 0.06

Table 1.2: Approximate masses, lifetimes and mixing parameters of the four neutral flavoured mesons. The masses are the average of the two physical eigenstates. The values are taken from Ref. [12] and Ref. [29].

In general, there are two types of contributions in the amplitudes used in the calculation of x and y: the short distance (or high energy) and the long distance (or low energy) contributions.

The short distance contributions are those involving four weak interaction vertices as shown in the box diagram of Fig. 1.5. These amplitudes dominate the mixing in the K0_, B0_{and B}0

s systems. In the D0system this contribution is instead strongly suppressed: the box diagrams involving b quarks are CKM-suppressed by a factor of_|VubVcb∗|2/|VusVus∗|2 ≈ 10−5 _{while the diagrams involving u and s quarks are strongly suppressed by the GIM} mechanism in the limit of SU(3) flavour symmetry [30, 31]. Taking into account the next-to-leading order terms, the short distance contributions to the value of x and y for the D0 _{meson still remain as small as}_O(10−6_{) [32]. The experimentally measured values of x} and y ([28, 33, 34, 35, 36]) are much larger <_{∼ O(10}−2_{), thus the D}0_-D0 _{mixing amplitude} must be dominated by the long distance contributions.

The long distance contributions are given by the right diagram of Fig. 1.5. In this case the oscillation can be interpreted as the decay of the meson to a common final state, which then recombines to form the opposite flavour meson. For this reason the size of this contribution is proportional to the phase space shared by the meson and the anti-meson.

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Figure 1.4: Probabilities of no oscillation (blue) and of oscillation (red) for the four neutral flavoured meson systems

In the case of the K0 _{there is a small number of possible final states and they are almost} all shared by the K0_{. In the case of B}0 _{and B}0

s mesons there is a large number of possible final states and only a few are shared by their anti-mesons. There are two possible kinds of approximations that can be used to calculate long distance contributions. The inclusive approaches, such as heavy quark effective field theory, exploit power series expansion in the inverse of the quark mass: this approximation is very well-suited for mesons containing the heavy b quark. On the other hand, exclusive approaches use experimental inputs to account for all the possible final states. Unfortunately, the D0 _{meson sits in between} the ranges of application of these two methods. The c quark is not heavy enough to make a reliable power expansion [37, 38], while the D0 _{meson is not light enough to} have few final states, and the calculations of long distance contributions rely on several assumptions [39, 40]. Hence, in the literature, the theoretical predictions for mixing (and CP violation) parameters span several order of magnitudes [41]. In this situation it is crucial to provide very precise measurements in the charm sector in order to improve the inputs used by the theoretical computations.

Figure 1.5: Graphical representation of the two contributions to the mixing amplitudes: short distance box diagram (left) and long distance re-scattering diagram (right).

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1.5. CP violation

1.5 CP violation

We define the decay amplitudes of a flavoured meson M , (which could be charged or neutral) and of its CP conjugate M into a multi-particle final state f or its CP conjugate f as

Af =hf|H|Mi, Af =hf|H|Mi, Af =hf|H|Mi, Af =hf|H|Mi.

(1.34) There are three possible types of CP violation in neutral-mesons systems, they are:

• CP violation in the decay. This kind of CP violation is the only one possible also for charged particles decays. This happens when

Rf := |Af/A_f| 6= 1. (1.35) This means that the probability for a M0 _{to decay to a final state f without} oscil-lating is different from the CP -conjugate process, where a M0 decays into a final state f . In order to have CP violation at least two decay amplitudes must exist. In fact, one can express the total decay amplitudes as the sum of several amplitudes ai: Af = X i |ai|ei(φi+δi) and Af = X i |ai|ei(−φi+δi). (1.36) where φi are weak phases originating from the CKM matrix and they change sign under CP transformation since the weak interaction does not conserve CP . On the other hand, since the strong interaction is CP conserving, the strong phases δi originating from final state interactions in the hadronic systems remain unchanged under CP transformation. A golden observable for the CP violation in the decay is the time integrated CP asymmetry defined as

ACP(f ) :=

Γ(M0 _{→ f) − Γ(M}0 _{→ f)}

Γ(M0 _{→ f) + Γ(M}0 _{→ f)}. (1.37) Neglecting oscillations, we have Γ(M0 _{→ f) ∝ |A}

f|2 and Γ(M 0

→ f) ∝ |Af|2 and using Eq. (1.36) we obtain

ACP(f ) =−

2P

i,j|ai||aj| sin(φi− φj) sin(δi− δj) P

i,j|ai|2+|aj|2 + 2|ai||aj| cos(φi− φj) cos(δi− δj)

. (1.38) Hence, it is clear that at least two different amplitudes ai with different strong and weak phases must exist in order to observe CP violation in the decay.

• CP violation in the mixing. This phenomenon is observed when the probability for a M0 _{to oscillate into a M}0 _{is different from the probability for the opposite} process

P(M0₍₀₎

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and, using Eq. (1.31), this implies

Rm := |p/q| 6= 1. (1.40)

For CP violation in the mixing a golden variable is defined by

ACP(f, t) :=

Γ(M0_(t)_{→ f) − Γ(M}0_(t)_{→ f)}

Γ(M0_(t)_{→ f) + Γ(M}0_(t)_{→ f)}. (1.41) • CP violation in the interference between mixing and decay. This can occur only if M0 _{and M}0 _{share the same final state f . The other condition necessary to} observe this kind of CP violation is that

Im(λf) +Im(λf)6= 0, (1.42) where λf := q p Af Af . (1.43)

If the final state is a CP eigenstates the condition of Eq. (1.42) simplifies to

Im(λf)6= 0. (1.44)

With this condition λf can be written as

λf =−ηCP(f )RmRfeiφf, (1.45) where ηCP(f ) is the CP -eigenvalue of the final state and the phase difference between the decay with and without oscillation φf is defined as

φf := arg −ηCP(f ) q p Af Af . (1.46)

Hence, Eq. (1.44) implies φf 6= {0, π}.

CP violation has been observed at more than 5 standard deviations for the K0_{, B}0 _and B0

s. In the case of the K0 all three types of CP violation have been observed in the K0 _{→ π}+_π− _{decay. In the case of the B}0_{, CP violation in the decay has been observed} in the B0 _{→ K}+_π− _{channel and CP violation in interference between decay and mixing} in the B0 _{→ J/ψK}0

S channel. Regarding the Bs0, CP violation has been observed in the B0

s → K+π− decay. Finally, the measurements of CP violation in the D0 system will be the subject of the next chapter.

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1.6. D0 _{decays to} _K+_K− _and _π+_π−

1.6 D

0

decays to

K

+

K

−

and

π

+

π

−

As the goal of this thesis is to develop a new method to measure CP violation in the D0 _{→ K}+_K− _{(and D}0 _{→ π}+_π−_{) decay, we briefly discuss the theoretical framework} of CP violation in the two-body decays of D0 _{mesons. The decay rate at time t of an} originally pure D0 _{state into a final state f (either K}+_K− _{or π}+_π−_{) can be computed} using Eq. (1.28) as Γ D0(t)→ f = hf|H|D0 (t)i = Nf g+(t)Af + q pg−(t)Af 2 = Nf 2 e −Γt |Af|2+ q p |A f|2 cosh(yΓt) + |Af|2− q p |A f|2 cos(xΓt) +2Re q pA ∗ fAf sinh(yΓt)− 2Im q pA ∗ fAf sin(xΓt) , (1.47) while for the decay of a D0 _{into f we have}

Γ D0(t)→ f = hf|H|D0 (t)i = Nf p qg−(t)Af + g+(t)Af 2 = Nf 2 e −Γt |Af|2+ p q |Af| 2 cosh(yΓt) + |Af|2− p q |Af| 2 cos(xΓt) +2Re p qA ∗ fAf sinh(yΓt)− 2Im p qA ∗ fAf sin(xΓt) , (1.48) where_Nf is a time-independent normalisation factor. In Eqs. (1.47) and (1.48) the various terms correspond to different amplitudes: the terms proportional to _|Af|2 and |Af|2 are associated with the decay without any oscillations, the terms proportional to|q/p|2_|A

f|2 and _|p/q|2_|A

f|2 are associated to the decay after the oscillation and the remaining terms proportional to sinh(yΓt) and sin(xΓt) are associated to the interference of the other two cases. One can expand Eqs. (1.47) and (1.48) at first order in xΓt, yΓt 1 (this approximation is valid for Γt_{∼ O(1 − 10) since x, y < 10}−2_{) as}

Γ(D0_(t) → f) ≈Nfe−Γt|Af|2[1− ηCP(f )RmRf(y cos φf − x sin φf)Γt] , Γ(D0(t)→ f) ≈Nfe−Γt|Af|21 − ηCP(f )R−1m R −1 f (y cos φf + x sin φf)Γt . (1.49)

By inserting Eqs. (1.49) into Eq. (1.41) and approximating again at the linear order in xΓt, yΓt 1 the time-dependent CP -asymmetry becomes

ACP(f, t) = AdecCP(f )− AΓ(f ) t τD0

, (1.50)

where τD0 = 1/Γ is the lifetime of the meson measured in the Cabibbo-favoured D0 →

K−_π+ _{decay channel and}

AdecCP(f ) := 1− R2 f 1 + R2 f . (1.51)

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This term is non-null if and only if Rf 6= 1, hence it corresponds to CP violation in the decay. The slope is given by

AΓ(f ) := 2ηCP(f )R2f (1 + R2 f)2 [(RmRf − R−1m R −1 f )y cos φf − (RmRf + Rm−1R −1 f )x sin φf] (1.52) and is non-null when Rf 6= 1 (CP violation in the decay), Rm 6= 1 (CP violation in the mixing) or φf 6= 0, π (CP violation in the interference between mixing and decay). By neglecting terms of order higher than one in the CP violation parameters Adec

CP(f ), (Rm−1) and φf, using ηCP(π+π−) = ηCP(K+K−) = 1, AΓ becomes

AΓ(f )≈ −xφf + y(Rm− 1) − yAdecCP(f ). (1.53) Using the fact that y < 10−2 _{and A}dec

CP(f ) ∼ 10−3 [4], the expression for AΓ can be approximated by neglecting the term yAdec

CP(f ) < 10

−5_{, which is associated with CP} violation in the decay, and the phase φf can be assumed to be φD, a decay independent number. In this way AΓ becomes a universal number for all two-body decays

AΓ(f ) ≈ −xφD+ y(Rm− 1). (1.54) It is worth mentioning that the above equation is often written in literature [12, 29] as

AΓ := 1

2[(Rm− R −1

m )y cos φD− (Rm+ Rm−1)x sin φD], (1.55) where no expansion is made for small values of the phase φD and of the Rm−1 parameter.

The violation of the CP symmetry has been observed in the charm quark sector, only very recently, on March 2019, by the LHCb Collaboration [4], through the analysis of the two-body singly Cabibbo-suppressed D0 _{→ K}+_K− _{and D}0 _{→ π}+_π− _{decays. Part of the} next chapter is, therefore, dedicated to the description of such a measurement, along with a short overview of the experimental state-of-the-art of the most relevant CP -violating observables.

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Chapter 2 Measurement of

CP violation in

D

0

_{→ h}

+

h

−

decays

This chapter is specifically dedicated to the current experimental state-of-the-art of the CP violation observables in the system of the D0 _{→ h}+_h− _{decays. The first observation} of CP violation in the charm sector performed by LHCb using the ∆ACP observable is then discussed. The consequent search for CP violation in a single decay channel (either D0 _{→ K}+_K− _or _D0 _{→ π}+_π−_{) is presented together with its experimental limits. Finally,} the motivation for this thesis which develops a new and more precise method to estimate the single channel asymmetries is given.

2.1 Production of charmed mesons

Several experiments study the physics of charmed mesons. These experiment can be classified into two main groups based on their production mechanism: e+_e− _{colliders and} hadrons colliders.

The e+_e− _{colliders have a series of advantages:}

• The center-of-mass energy corresponds to the total energy of the interaction and can be tuned to resonantly produce strong-interacting bound-states with precise properties and quark content. This results in an high ratio between signal and background cross-section, of order σsig/σbkg ∼ 0.2 − 0.3.

• The interaction rate is of order 10 Hz, which allows recording the majority of the produced events, without trigger-level selections. The average number of vertices per interaction is close to one and the mean charge multiplicity is of order of 10 particles per events. This results in low combinatorial backgrounds.

• The acceptance is almost hermetic, allowing the full reconstruction of the final state. This includes final states containing neutral particles such as π0_{s and photons. Even} the invisible neutrinos can be indirectly reconstructed by kinematically constraining the final state 4-momentum to coincide with the initial 4-momentum.

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• D-factories have their center-of-mass energy set slightly above 3770 MeV. This allows the production of a resonant ψ(3770) (cc) which then decays to a pair of quantum correlated D0_D0 _{or D}+_D−_{. At this energy, the total cross-section for the} production of a DD pair is 8 nb. In the laboratory frame, the e+ _{and e}− _{have the} same energy, the resulting D mesons are produced almost at rest and their decay-time cannot be measured. This set-up was used by three experiments: MARK III, CLEO-c and BESIII.

• B-factories have their center-of-mass energy set to 10.6 GeV. This allows the res-onant production of Υ (4S)(bb) which then decays to a pair of quantum correlated B0_B0 _{or B}+_B−_{. The e}+ _{and e}− _{are set to different energies in the laboratory frame} and consequently the resulting B mesons are produced with a sizeable momentum. This net momentum allows the B meson to travel for a finite distance and its decay time can be reconstructed. The knowledge of the decay time of the meson is of key importance to study the mixing phenomenology. The main goal of B-factories is to study B mesons, but these BB pairs can also decay to DD pairs: the total cross section for producing at least one D0 _{at this energy is 1.45 nb. Even though the} cross section is smaller, experiments such as BaBar and Belle were able to produce a large quantity of D mesons thanks to their extremely high instantaneous luminosity, which resulted in a two order of magnitudes larger total integrated luminosity with respect to CLEO-c and BESIII.

The main advantage of hadron colliders is that more heavy flavour particles can be pro-duced with five or six orders of magnitude higher cross sections with respect to e+_e− collid-ers. This huge gain in cross section is counterbalanced by a smaller signal-to-background ratio σcharm/σtot ∼ O(10−2− 10−3). The large backgrounds and the high rate of collisions (O(10 MHz)) require an extremely selective real time trigger which discards the majority of light-flavoured uninteresting events. Nevertheless, hadron collider experiments such as CDF and LHCb have demonstrated to be able to compete in precision with the e+_e− collider experiments.

Experiment Year beam √s σacc(D0) R Ldt ∼ n(D0)

CLEO-c 2003-2008 e+e− 3.77 GeV 8 nb 0.5 fb−1 4.0× 106 BESIII 2010-2011 e+e− 3.77 GeV 8 nb 3 fb−1 2.4_{× 10}7 BaBar 1999-2008 e+_e− _{10.6 GeV} _{1.45 nb} _{500 fb}−1 7.3× 108 Belle 1999-2010 e+e− 10.6 GeV 1.45 nb 1000 fb−1 1.5_{× 10}9 Belle II∗ _2019-2025 _e+_e− _{10.6 GeV} _{1.45 nb} _{50 ab}−1 _7.5 × 1010 CDF 2001-2011 pp 2 TeV 13 µb 10 fb−1 1.3× 1011 LHCb Run 1 2011-2012 pp 7-8 TeV 1.4-1.6 mb 1.1+2.1 fb−1 4.6_{× 10}12 LHCb Run 2 2015-2018 pp 13 TeV 2.7 mb 5.9 fb−1 1.6× 1013 LHCb Run 3∗ _2021-2024 _pp _{13 TeV} _{2.7 mb} _{5.9 fb}−1 _4.6 × 1013

Table 2.1: List of the main experiments studying the quark charm. The number of D0 produced in the acceptance is used as a term of comparison [42]. The ∗ _{indicates running} or future experiments together with their expected yields.

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2.2. Flavour tagging

2.2 Flavour tagging

In the study of mixing and CP violation in D0 _{mesons it is necessary to reconstruct the} flavour of the mesons at production time. This can be done in three different ways.

• D∗ _tagging _{exploits the strong decay D}∗+ _→D0_π+ _{and its charge-conjugate D}∗− →D0_π−_{, where the charge of the pion determines the flavour of the D}0_{. Since the} mass difference between the mother particle (mD∗ ≈ 2010 MeV) and the products

(mD0 + m_π ≈ 2004 MeV) is very small, the pion is produced with a relatively low

momentum and is therefore commonly referred as ”soft” pion (πs). Since the soft pion has a low momentum the magnetic field of the detector can curve its trajectory, until the pion goes out of acceptance, reducing the detection efficiency of the entire decay. If this momentum-dependent detection efficiency is slightly different between a π+_{and a π}−_{it can lead to fake physics asymmetries. The small mass difference has} also the advantage that a high mass resolution can be achieved and, consequently, the combinatorial backgrounds can be efficiently rejected.

• Semi-leptonic tagging uses the weak decay B →D0_µ− _{X (where X stands for} any non-reconstructed final state). In this case the charge of the muon is used to tag the flavour of the D0 _{meson. Even if the branching-ratio of semi-leptonic decays} is smaller, the presence of the muon allows for a clear trigger signature with very high efficiency. A difference in detection efficiency between a µ− _{and a µ}+ _{can lead} to fake physical asymmetries.

• Opposite side tagging is mainly used at e+_e− _{colliders. In this case the fact that} quarks are produced in pairs is exploited: whenever a charmed meson is produced a hadron with the opposite flavour is also produced. By measuring the flavour of the opposite hadron, the flavour of the considered meson is determined. In D-factories the D0_D0 _{or D}+_D− _{are produced in a quantum-entagled state and more properties} such as relative strong phases can be measured.

2.3 CP violation in the D

0

mixing

Several experiments are able to measure the relevant D0 _{mixing parameters either} forbid-ding or allowing for CP violation. In particular, four experimental sources dominates the world average precision fit [29] as they measure quantities that are sensitive to the values of x, y,_{|p/q| and φ}D defined in Sec. 1.3 and Sec. 1.5.

The first useful quantity is AΓdefined in Sec. 1.6 and its value is extracted by studying the time-dependent CP violation in D0 _{→ K}+_K− _{and D}0 _{→ π}+_π− _{decays. The world} average of this quantity AΓ= (−0.31 ± 0.20) × 10−3 is dominated by the LHCb measure-ment [5, 6]. The second important decay channel sensitive to the mixing phenomenology is the D0 _{→ K}0

Sπ+π−decay in which the distribution of events across the Dalitz plot [43, 44] is fitted as a function of the D0 _{decay time and the values of the following variables are}

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obtained: xCP := 1 2 x cos φD q p + p q + y sin φD q p − p q , yCP := 1 2 y cos φD q p + p q − x sin φD q p − p q , ∆x := 1 2 x cos φD q p − p q + y sin φD q p + p q , ∆y := 1 2 y cos φD q p − p q − x sin φD q p + p q , (2.1)

where in case of no CP violation in the mixing or in the interference between mixing and decay (|q/p| = 1 and φD = 0) they simply become xCP = x, yCP = y and ∆x = ∆y = 0. In this case the world precision is dominated by the measurements of BaBar [45], Belle [46] and LHCb [47]. The formalism developed by Ref. [47] to describe D0 _{→ K}0

Sπ+π

− _decays will become useful in Sec. 2.7. The variable yCP can also be measured by comparing the width Γ of D0 _{flavour specific decays to the final state K}−_π+ _{with the width Γ}

CP + of D0 _{mesons to CP -even final states (either K}+_K− _{or π}+_π−_{) as y}

CP = ΓCP +/Γ− 1. The world average of this measurement is dominated by BaBar [48], Belle [49] and LHCb [50]. Finally, the decay channel in which D0_-D0 _{oscillation were first observed [33, 34, 51] is} exploited. The D0 _{decay time dependent ratio of doubly-Cabibbo-suppressed to Cabibbo} favoured D0 _{→ K}−_π+_{decays is fitted and the values of the following variables are obtained}

x0± := q p ± (x0cos φD ± y0sin φD), y0± := q p ± (y0cos φD∓ x0sin φD), (2.2)

where x0 _{and y}0 _{are given by the rotation of x and y by a strong phase δ}

Kπ. The most precise measurement in this channel is dominated by LHCb [52].

When all the information coming from these and other experiments is combined in the single fit of Ref. [29] the most precise measurements of the four mixing parameters are obtained and they are given in Tab. 2.2. The two-dimensional contour plot for the same mixing parameters are shown in Fig. 2.1.

Parameter No CPV in DCS decays CPV-allowed

x (%) 0.43+0.10_−0.11 0.37± 0.12

y (%) 0.62± 0.06 0.68+0.06_−0.07

|q/p| 0.997_{± 0.008} 0.951+0.053_−0.042

φD (◦) 0.14± 0.32 −5.3+4.9−4.5

Table 2.2: Results of the global fit of Ref. [29] for the most relevant D0 _{mixing parameters.} Two scenarios are shown, in the first CP violation is not allowed in doubly-Cabibbo-suppressed decays, in the second CP violation is fully allowed.

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2.4. Measurement of the golden observable ∆ACP 0.2 − 0 0.2 0.4 0.6 0.8 1 x (%) 0.2 − 0 0.2 0.4 0.6 0.8 1 y (%) CPV allowed σ 1 σ 2 σ 3 σ 4 σ 5 !"#$% &'!() *+*+ 0.4 − −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 |q/p|-1 40 − 30 − 20 − 10 − 0 10 20 30 40 Arg(q/p) [degrees] σ 1 σ 2 σ 3 σ 4 σ 5 !"#$% &'!() *+*+

Figure 2.1: Confidence intervals for the D0 _{mixing parameters x and y (left) and the CP} violation parameters q/p and φD as the result of the global fit of Ref. [29].

2.4 Measurement of the golden observable

∆A

CP

CP violation in the D0 _{meson decay was recently observed for the first time with more than} 5 standard deviations by the LHCb experiment [4]. This first observation was achieved by measuring a golden observable defined as

∆ACP := ACP(K+K−)− ACP(π+π−), (2.3) where ACP(f ) is the time-integrated CP asymmetry to the final state f

ACP(f ) =

Γ(D0 _{→ f) − Γ(D}0 _{→ f)}

Γ(D0 _{→ f) + Γ(D}0 _{→ f)} (2.4)

By integrating Eq. (1.50) we get ACP(f ) =

Z

ACP(f, t)dt = AdecCP(f )− AΓhti τD0

, (2.5)

and by substituting this expression for the time-integrated CP asymmetry into Eq. (2.3) we get

∆ACP = ∆AdecCP − AΓ ∆hti

τD0

, (2.6)

where ∆Adec

CP := AdecCP(K+K −₎

− Adec CP(π+π

−_{) and ∆}

hti := htiK+_K− − hti_π+_π−. In

exper-iments where no time-dependent reconstruction efficiencies are present, such as BaBar and Belle, hti is equal to the lifetime of the D0 _{meson for all decays and ∆}_{hti = 0. On} the contrary, at LHCb time-dependent efficiencies are introduced by the trigger selections and the average effective decay times _hti_K+_K− and hti_π+_π− must be taken into account.

If a non-null value of ∆Adec

CP is observed it implies that at least one between AdecCP(K+K−) and Adec

CP(π+π

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Experimentally one cannot directly measure ACP(f ) but instead one can define a raw asymmetry computed using the event yields N (D0 _{→ f) and N(D}0 _{→ f) as}

Araw(D0 → f) =

N (D0 _{→ f) − N(D}0 _{→ f)}

N (D0 _{→ f) + N(D}0 _{→ f)}. (2.7) In general, in the Araw(D0 → f) observable there are other contributions besides ACP(f ), which are caused by the production and detection mechanisms of the experiment. In the LHCb analysis of Ref. [4] both D0 _{mesons produced in prompt D}∗ _{decays and in the} B semileptonic decays were used. Here we discuss the additional asymmetries and its cancellation in the prompt sample only, as it is the basis of our analysis.

When considering pp collisions an asymmetry for the production of charmed mesons is present:

AP(D∗+) :=

σ(D∗+₎_{− σ(D}∗−₎

σ(D∗+_{) + σ(D}∗−₎, (2.8)

where σ(D∗_{) indicates the cross-section to produce a D}∗ _{in a pp collision.}

There is also an asymmetry associated with the different efficiency to detect a positive soft pion with respect to a negative soft pion. Particles of opposite sign are bent by the magnetic field into opposite sides of the detector and, if differences in efficiency between different parts of the detector are present due to misalignment or dead regions, a net detection asymmetry is created. In particular, since low momentum particle are bent more strongly, this asymmetry is higher for low momentum soft pions. The detection asymmetry is given by AD(πs+) := (π+ s)− (π − s) (π+ s ) + (πs−) , (2.9) where (π±

s) is the probability to detect a π ±_.

By inverting Eq. (2.4) one can calculate the partial decay width as Γ(D0 → f) = 1 + A₂CP(f )(Γ(D0 → f) + Γ(D0 → f)), Γ(D0 → f) = 1− A₂CP(f )(Γ(D0 → f) + Γ(D0 → f)). (2.10)

In the same way one can invert Eqs. (2.8) and (2.9) to obtain the production cross-section and the detection efficiency. The observed yields are then given by

N (D0 → f) ∝ σ(D∗+)(π+ s )Γ(D 0 → f) ∝ (1 + AP(D∗+))(1 + AD(πs+))(1 + ACP(f )), N (D0 → f) ∝ σ(D∗− )(πs−)Γ(D 0 → f) ∝ (1 − AP(D∗+))(1− AD(πs+))(1− ACP(f )). (2.11) From Eq. (2.11) we can express the raw asymmetry of Eq. (2.7) as

Araw(f ) =

ACP(f ) + AP(D∗+) + AD(πs+) + ACP(f )AP(D∗+)AD(πs+) 1 + ACP(f )AP(D∗+) + ACP(f )AD(πs+) + AP(D∗+)AD(π+s)

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2.4. Measurement of the golden observable ∆ACP Both the production asymmetry AP(D∗+) and the detection asymmetry AD(πs+) are ex-pected to be of order 1% [53, 54, 55, 56], hence Eq. (2.12) can be written, up toO(10−6_), as

Araw(f ) = ACP(f ) + AP(D∗+) + AD(πs+). (2.13) From this last expression, one can calculate the difference between the raw asymmetries in the D0 _→K+_K− _{sample and in the D}0 _→π+_π− _{sample as}

∆Araw :=Araw(D0 → K+K−)− Araw(D0 → π+π−) =ACP(K+K−)− ACP(π+π−) = ∆ACP,

(2.14) where the production and detection asymmetries cancel out in the difference. However, these nuisance asymmetries can depend on the decay kinematic, as previously explained. Hence, in order to fully cancel the nuisance asymmetries, a weighting technique is applied to equalise the kinematic distributions of the two samples. The final result for the prompt decays is

∆Aπ−tagged_CP = (_{−18.2 ± 3.2(stat) ± 0.9(syst)) × 10}−4. (2.15) In analogous manner to the prompt sample, the CP asymmetry in semileptonic decays is obtained by canceling the nuisance production asymmetry of the B mesons and the detection asymmetry of the tagging muon. The final result for this sample is

∆Aµ−tagged_CP = (_{−8.8 ± 7.7(stat) ± 5.0(syst)) × 10}−4. (2.16) Finally, the combination of these two values with those measured in LHCb Run 1 gives

∆ACP = (−15.4 ± 2.9) × 10−4, (2.17) which is the first observation at more than 5σ of CP violation in the charm sector by a single experiment. The measured values of ∆ACP from all relevant experiments are given in Tab. 2.3. B-factiories such as BaBar and Belle have the advantage that the time

Experiment ∆ACP[×10−4]

BaBar [57] 24± 62(stat) ± 26(syst) CDF [58] _{−62 ± 21(stat) ± 10(syst)} LHCb Run 1 µ-tagged [59] 14_{± 16(stat) ± 8(syst)} LHCb Run 1 π-tagged [60] −10 ± 8(stat) ± 3(syst) LHCb Run 2 µ-tagged [4] _{−9 ± 8(stat) ± 5(syst)} LHCb Run 2 π-tagged [4] _{−18 ± 3(stat) ± 1(syst)}

Table 2.3: Summary of the measurements of ∆ACP by the different experiments. acceptance for the decay of a D0 _{is constant and the average observed decay time of the} meson is equal to its lifetime for all decay channels. For this reason, ∆hti in Eq. (2.6) has a null value and a measurement of ∆ACP is equivalent to a measurement of ∆AdecCP. At LHCb, instead, ∆_{hti 6= 0 and the residual contribution of the time dependent CP} asymmetry must be removed. To do so the average decay time of the D0 _{→ K}+_K− and D0 _{→ π}+_π− _{is measured while the value of A}

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analyses. The world averaged value of AΓ is dominated by the LHCb measurement and its value is AΓ = (−0.31 ± 0.20) × 10−3 [5, 6]. After subtracting the time dependent contribution, the final result for the difference of CP asymmetry in the decay for the combined LHCb Run 1 and Run 2 measurements is

∆AdecCP = (−15.6 ± 2.9) × 10 −4

. (2.18)

The current status of all measurements of ∆Adec

CP and AΓ is summarized in Fig. 2.2.

Figure 2.2: Combination of all measurements of ∆Adec

CP (∆adirCP) and AΓ (aindCP). The bands indicate the ±1σ intervals. The filled circle indicates the no CP violation point (0,0) while the ellipses indicate the two-dimensional 68% CL, 99.7% CL and 99.99997% CL regions [29].

2.5 Current measurement of

A

CP

(K

+

K

−

) and A

CP

(π

+

π

−

)

The analysis described in the previous section was able to observe CP violation in the decays of charmed mesons. However, with that strategy, it is not possible to separate the two contribution given to ∆ACP by ACP(K+K−) and ACP(π+π−) and a direct mea-surement of one of the two quantities is necessary. In Ref. [11] the current most precise determination of ACP(K+K−), and consequently of ACP(π+π−), is presented. The anal-ysis was performed using an integrated luminosity corresponding to 3 fb−1 at the energy of 7 and 8 TeV, collected by the LHCb experiment in Run I.

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2.5. Current measurement of ACP(K+K−) and ACP(π+π−) The analysis described in Ref. [11] is similar to the one of Ref. [4]. First, as in the previous section, the raw asymmetry in the K+_K− _{sample is given by}

Araw(D0 → K+K−) = ACP(K+K−) + AP(D∗+) + AD(πs+). (2.19) Now, to cancel the nuisance asymmetries the control channel D∗+

→D0 ₍_→K−_π+_)π+ s is used. Since this new decay is Cabibbo-favoured the CP asymmetry of this channel is assumed to be negligible. However, the final state is not a CP -eigenstates and therefore a detection asymmetry of the final state exists. The raw asymmetry of this channel is

Araw(D0 → K−π+) = AP(D∗+) + AD(πs+) + AD(K−π+), (2.20) where AD(K−π+) is the detection asymmetry for the final state, which is mainly caused by the different interaction cross-sections of the kaons with the detector material. To cancel this new asymmetry a second control channel is given by the Cabibbo-favoured decay D+ _→K−_π+_π+_{. The raw asymmetry of this channel can be written as}

Araw(D+ → K−π+π+) = AP(D+) + AD(K−πl+) + AD(π+h), (2.21) where π+

l indicates the lowest momentum pion and π +

h the highest momentum one. The new asymmetries AP(D+) and AD(π+h) are canceled with the last control channel of this analysis, the Cabibbo-favoured D+_→K0_π+_{decay. In this last channel the raw asymmetry} is given by

Araw(D+→ K0π+) = AP(D+) + AD(π+) + AD(K0). (2.22) The association of the momenta of the two pions in the decay D+ _→K−_π+_π+ _{is made to} best match the kinematic distributions of the pions with which they are combined to cancel the asymmetries. The AD(K0) term is due to the mixing, interference and interaction with the detector of neutral kaons, and it is usually determined in LHCb through the usage of the Geant4 based realistic simulation of the detector [11, 61], which prevents the exploitation of the part of the samples where the K0

S is decayed outside the acceptance of the vertex detector. This causes a penalty in statistics of about two thirds, in addition to the need of having a reliable description of the detector material.

Before combining all the raw asymmetries together to obtain ACP(K+K−) a complex kinematic weighting is performed. This is the most complex part of the analysis and it is necessary to make sure that the kinematic distribution of all corresponding particles be-tween different channels are equalised. In particular, the kinematics of the D+_→K−_π+_π+ decay is extremely different from the one of the D+ _→K0

Sπ+ decay and about 90% of the total statistical power is lost in the equalisation cuts. In this way all the nuisance asym-metries, which depend on the kinematics of the decays, can be cancelled through a series of subtractions

ACP(K+K−) = Araw(D0 → K−K+)− Araw(D0 → K−π+) + Araw(D+→ K−π+π+)− Araw(D+ → K0π+) + AD(K0),

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where in the subtraction it is assumed that the CP violation in the three Cabibbo-favoured calibration decays is much smaller than the experimental resolution. The final result was found to be

ACP(K+K−) = (14± 15(stat) ± 10(syst)) × 10−4. (2.24) Here the statistical uncertainty is limited by the selection and the weighting procedure which reduce an original sample of around 9.6 M D0 _→K+_K− _{candidates to an effective} sample of 1.6 M candidates. Moreover, the smallest sample in the analysis is the D+ →K0_π+ _{decay, which contains roughly 1.1 M candidates and thus dominates the} statis-tical uncertainty. By combining the knowledge of ∆ACP and ACP(K+K−) the value of ACP(π+π−) can also be extracted. The combination with previous LHCb measurements gives

ACP(K+K−) = (4± 12(stat) ± 10(syst)) × 10−4, ACP(π+π−) = (7± 14(stat) ± 11(syst)) × 10−4.

(2.25)

The statistical uncertainty on ACP(K+K−) for the same analysis with only Run 2 data is predicted to be 8.5_×10−4_{, while in the combination of Run 1 and Run 2 data the prediction} reduces to 7× 10−4 _{[13]. On the other hand, the systematic uncertainties on the Run 1} measurement are dominated by effects that are intrinsically related to the presence of the three control channels and their size is comparable to that of the statistical uncertainty. Hence, the reduction of the systematic uncertainties in Run 2 is a non-trivial task and it could concretely become the limiting factor in the final precision. For comparison, the measurements of ACP(K+K+) and ACP(π+π−) from all relevant experiments are given in Tab. 2.4.

Experiment ACP(K+K−)[×10−4] ACP(π+π−)[×10−4] BaBar [57] 0_{± 34(stat) ± 13(syst)} _{−24 ± 52(stat) ± 22(syst)} Belle [62] −43 ± 30(stat) ± 11(syst) 43± 52(stat) ± 12(syst)

CDF [58] _{−32 ± 21} 31_{± 22}

LHCb Run 1 µ-tagged [63] _{−6 ± 15(stat) ± 10(syst)} _{−20 ± 19(stat) ± 10(syst)} LHCb Run 1 π-tagged [11] 4± 12(stat) ± 10(syst) 7± 14(stat) ± 11(syst) Table 2.4: Summary of the measurements of ACP(K+K−) ACP(π+π−) by the different experiments.

Despite achieving such an high accuracy, LHCb will most likely not be able to observe any significant signal of CP violation, in the single decays, using the full available data sample. In fact, the SU (3) flavour symmetry predicts that CP violation in the decay has equal intensity and opposite sign in the D0 _→K+_K− _{and D}0 _→π+_π− _{decay channels [2,} 14, 15]. The expected CP violating effect is of the order of ∆ACP/2 ' 8 × 10−4 and an uncertainty of a few units of 10−4 _{is, therefore, necessary to significantly observe any} effect.

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2.6. This thesis: a new approach

2.6 This thesis: a new approach

In this thesis we develop a new method for measuring ACP(K+K−) and ACP(π+π−) by canceling the production and detection nuisance asymmetries using a single calibration channel with the aim of improving the experimental sensitivity. This new approach over-comes all the weaknesses of the previous analysis methodology, being fully data-driven and, therefore, more robust. The considered control channel is the Cabibbo-favoured D0 _{→ K}0

Sπ+π

− _{decay. For this decay, the raw observed asymmetry can be expressed as} Araw(D0 → KS0π + π−) = ACP(KS0π + π−) + AP(D∗+) + AD(π+s) + AD(π+π−) + AD(K0), (2.26) where the second and third terms are the production and detection asymmetries already described in the previous sections. The term AD(π+π−) is related to the fact that this is a three-body decay which can proceed through several intermediate resonances. This leads to a non-uniform distribution of events across the Dalitz plane: the D0 _{and D}0 _decays populate opposite regions of the plane and a non-symmetric efficiency across the plane can generate artificial asymmetries. The size of this asymmetry is about 10−3 _{and, therefore,} extremely dangerous in a measurement which aims a precision of few units in 10−4_{. The} source of this asymmetry will be exhaustively described in Sec. 5.4, along with the new fully data-driven method, developed to correct for it. This is one of the main results of this thesis. The last term of Eq. (2.26), instead, has the same nature of the detection asymmetry AD(K0) described in the previous section. This asymmetry depends on the decay length of the K0

S and, therefore, on its proper decay time t. Such an asymmetry is null for time value of t = 0, while at higher values, t/τK0

S ≈ 2.5, its size can reach

values up to 2%. Also in this case, we develop a fully data-driven method, overcoming all the limitations of the previous adopted strategy [11]. The removal of this asymmetry is therefore achieved at very high precision. The method will be discussed in Sec. 5.5. After correcting for AD(π+π−), equalising the kinematics of the two decays and the removal of AD(K0) we obtain the measurements of interest as

∆Aππ−Kππ_CP := ACP(π+π−)− ACP(KS0π + π−)≈ ACP(π+π−), ∆AKK−Kππ_CP := ACP(K+K−)− ACP(KS0π + π−)≈ ACP(K+K−). (2.27)

2.7 D

0

decays to

K

_S0

π

+

π

−

Since we exploit the D0 _→K0

Sπ+π−decay as a calibration channel, we discuss the possible CP violation effects in this channel. Inspired by Ref. [64], we develop for the first time an expression for the time-dependent CP asymmetry of D0 _{→ K}0

Sπ+π

−_{. In this} three-body decay the dynamics is more complex than the two-three-body decays D0 _{→ K}+_K− _and D0 _{→ π}+_π− _{and the CP asymmetry must be studied as a function of the Dalitz variables.} In fact, we parametrize the phase-space using the Dalitz formalism by defining the flavour-dependent invariant square masses:

m2±:= m2_(K0 Sπ ±₎ _{for D}0 _{→ K}0 Sπ+π − _decays m2_(K0 Sπ∓) for D0 → KS0π+π− decays . (2.28)

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Therefore, we indicate the final state as a function of its position on the Dalitz plane as f (m2

+, m2−). The decay amplitudes of Eq. (1.34) also become a function of the Dalitz plane position Af → Af(m2+, m2−). Eqs. (1.47) and (1.48) can be rewritten in this more general form as Γ D0_(t) → f(m2 +, m 2 −) = hf(m2+, m 2 −)|H|D0(t)i =Nf g+(t)Af(m2+, m 2 −) + q pg−(t) ¯Af(m 2 −, m 2 +) 2 , (2.29) Γ D0_(t)_{→ f(m}2 +, m2−) = hf(m2+, m2−)|H|D0(t)i =_Nf p qg−(t)Af(m 2 +, m 2 −) + g+(t) ¯Af(m2−, m2+) 2 . (2.30) In order to study the time dependent asymmetry ACP(t) as a function of the final state kinematics, we divide the Dalitz-plot into two sets of n bins each, symmetric about the principal bisector m2

+ = m2−. The bins with m2+ > m2− (lower region) are labeled with index b = 1, ..., n and their symmetric bins (upper region) with index b =_{−1, ..., −n. We} integrate Eqs. (2.29) and (2.30) over each bin of the Dalitz-plot to obtain the event yields as Nb(t) := Z b dm2 +dm 2 −Γ D0(t)→ f(m2+, m 2 −) = Fb|g+(t)|2+ q p ¯ F−b|g−(t)|2+ 2 q FbF¯−bRe q pXbg ∗ +(t)g−(t) , ¯ Nb(t) := Z b dm2+dm 2 −Γ D 0 (t)→ f(m2 +, m 2 −) = ¯Fb|g+(t)|2+ p q F−b|g−(t)|2+ 2 q ¯ FbF−bRe p qX¯bg ∗ +(t)g−(t) , (2.31)

where the following definitions are introduced: Fb := Z b dm2 +dm 2 −|Af(m2+, m 2 −)|2, F¯b := Z b dm2 +dm 2 −| ¯Af(m2+, m 2 −)|2, (2.32) Xb := 1 pFbF¯−b Z b dm2 +dm2−A ∗ f(m2+, m2−) ¯Af(m2−, m2+), ¯ Xb := 1 p_¯ FbF−b Z b dm2 +dm 2 −A¯∗f(m 2 +, m 2 −)Af(m2−, m2+). (2.33) The D0 _{→ K}0 Sπ+π

− _{decay proceeds through several intermediate resonances. The} weak interaction is responsible for the decay of the D0 _{to a resonance, which then} im-mediately decays into two mesons. The weak part of the decay amplitude can only be produced by the two tree-level diagrams of Fig. 2.3: one is Cabibbo-favored and the other is doubly-Cabibbo-suppressed. In both cases the CKM elements involved in the transition (V∗

A new method to measure the time-integrated CP asymmetry in singly-Cabibbo suppressed D0->K+K- and D0->pi+pi- decays at LHCb