Measurement of D0 - D0bar mixing parameters and search for CP violation in D0 -> K+pi- decays with LHCb Run 2 data

(1)

Università degli studi di Pisa

Dipartimento di Fisica

Corso di Laurea Magistrale in Fisica

Master thesis

Measurement of D

0 −

D

0 mixing

parameters and search for CP

violation in D

0 →

K

+

π

−

decays with

LHCb Run 2 data

Candidate:

Advisor:

Roberto Ribatti

Dott. Michael J. Morello

(2)

(3)

Introduction

In the Standard Model the non-invariance of the weak interaction under the combined action of the charge conjugation (C) and parity (P) operators, commonly referenced as CP violation (CPV), is introduced through a single irreducible complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. The predictions of the theory are nowadays in excellent agreement with experimental data. However, the current level of accuracy, of both experiments and theory, does not allow to rule out other CP violation mechanisms beyond the Standard Model. The discovery of new sources of CP violation would be of fundamental importance, since the amount of CPV produced through the CKM mechanism is too small to explain the cosmological baryonic asymmetry observed in our Universe. This suggests that some new interactions at higher energies than those explored so far, must exist and possess additional and larger sources of CPV. They might manifest themselves also at low energies in the form of slight deviations from the SM predictions in the decay of well known particles, thanks to the contribution to their decay amplitudes from higher-order loop diagrams.

While the CKM mechanism is experimentally well established in the K- and B-meson systems, additional sources of CPV may appear in the processes involving hadrons that contain the charm quark. Charm mesons are, indeed, the only mesons containing up-type quarks where CPV can be measured. However, CPV effects are expected to be tiny within the SM, typically of the order of 10−4 to 10−3, but may be enhanced by contributions from particles beyond the SM that can leave CPV observables for down-type quarks unaffected. The LHCb collaboration has recently reported the first observation of CPV in the decay of charm mesons looking at the∆ACP observable [1]. Theoretical uncertainties on low-energy quantum-chromodynamics effects currently do not allow us to establish whether this result is consistent with the SM or indicates the presence of new dynamics in the up-quark sector [2, 3]. Tests of CPV in the mixing (and/or in the interference between the mixing and the decay), for which the SM expectations still lie one order of magnitude below the experimental uncertainty, are complementary to those of CPV in the decay and might help to clarify the picture.

The mass eigenstates of charmed neutral meson can be written as linear combinations of flavour eigenstates|D1,2i = p|D0i ±q|D0i, where p and q are complex parameters, hence it follows the D0−D0oscillations. In the limit of CP symmetry, it results that|q/p| =1 and φ≡arg(q/p) =0, and the oscillations are only characterized by the difference in mass,∆m≡m1−m2, and in decay width, ∆Γ ≡ Γ1−Γ2, between the CP-even (D2) and CP-odd (D1) mass eigenstates. These two differences are conveniently expressed in terms of dimensionless mixing parameters x≡∆m/Γ and

(6)

y ≡ ∆Γ/2Γ, where Γ is the average decay width of neutral D mesons. If CP symmetry does not hold, the oscillation probabilities for mesons produced as D0and D0can differ, further enriching the phenomenology.

Unlike other flavoured neutral mesons (K0, B0and B0s), the dynamic of the D0meson oscillations are mainly governed by the long distance amplitudes, since the contribution of the short distance ones is highly suppressed. In fact, the diagram containing the bottom quark in the loop is CKM-suppressed and the contributions from down and strange quarks cancel each other through the GIM mechanism, while in B-mesons case the top quark contribution dominates. Unfortunately, long-distance amplitudes depend on the exchange of low-energy gluons and are challenging to calculate. However, short-distance amplitudes may include contributions from a broad class of particles not described in the Standard Model, which might affect the oscillation rate or introduce a difference between the D0_{and D}0_{meson decay rates. The study of CP violation in D}0_oscillations therefore offers sensitivity to non-standard-model phenomena.

The first evidence of oscillations dates back to 2007 by the B-factories [4, 5], while the no oscillation hypothesis was ruled out in 2009 by combining measurements from different experiments [6]. The LHCb collaboration reported in 2012 the first observation from a single measurement [7], later confirmed by the CDF collaboration in 2013 [8]. Nowadays, the world average of the mixing parameters is x=0.39+0.11_−0.12% and y=0.651+0.063_−0.069% [9]. The decay width difference between the neutral charm-meson eigenstates (y parameter) is well established, and its measured value is significantly different from zero. Only very recently, an evidence for a non-zero (positive) mass difference value is emerging, although no experiment has still provided a measurement significantly different from zero. The world average for the parameters related to CP violation is: |q/p| = 0.969+0.050_−0.045 and φ(°) = −3.9+4.5_−4.6[9], and no departure from CP symmetry has been yet established.

One of the most sensitive probes for the measurement of CP-averaged and CP-violating mixing parameters in D0−D0oscillations is based on the comparison of the decay-time-dependent ratio of D0_→_K+_π− _{to D}0_→_K−

π+rates with the corresponding ratio for the charge-conjugated processes. The flavour of the D0 meson at production is determined by the charge of the low momentum associated pion πs in the strong decays D∗+ →D0π_s+ and D∗− →D0π_s−, respectively. The final states of interest are the following (to which the relative charge conjugates must be added):

• D∗+ →D0π_s+→ [K−π+]π+_s , called right-sign (RS); • D∗+ →D0_π+

s → [K+π−]π+s , called wrong-sign (WS).

The decay rate of the RS process is dominated by the amplitude of the Cabibbo-favoured D0→K−π+ decay, while the decay rate of the WS process is determined by the interference between the doubly Cabibbo-suppressed decay D0 → K+π− and the D0−D0 oscillations followed by the Cabibbo-favoured decay D0_→_K+_π−_{, which has a similar amplitude. In the limit of small mixing parameters}1_,

|x|,|y| 1, and assuming negligible CPV, the time-dependent ratio R(t)between WS and RS decay rates can be expanded as follows:

R(t) 'RD+ p RDy0t+ x 02₊_y02 4 t 2_,

where t is the decay time in D0 average lifetime unit, RD is the ratio between the decay rate of the doubly Cabibbo-suppressed decay and that one of the favoured decay. The coefficients x0and y0 linearly depend on the mixing parameters x and y through a rotation angle δ, defined as the

(7)

phase difference between the two amplitudesA(D0→K+π−)/A(D0→K+π−) = −

√

RDexp(−iδ). This angle is measured by the CLEO-c and BESIII experiments [10, 11]. If CP violation occurs, the decay-rate ratios R+(t)and R−(t)of mesons produced as D0and D0, respectively, are functions of independent sets of mixing parameters, R±_D,(x0±)2_{, and y}0±_{. The parameters R}+

Dand R −

Ddiffer if the ratio between the suppressed and favoured decay amplitudes is not CP symmetric, indicating direct CP violation. Violations of CP symmetry either in mixing,|q/p| 6=1, or in the interference between mixing and decay amplitudes, argqA(D0→K+π−)/pA(D0→K+π−)−δ6=0, are manifestations of indirect CP violation and generate differences between(x02+, y0+)and(x02−, y0−).

In 2018 the LHCb collaboration published the most precise measurement of the CP-averaged and CP-violating mixing parameters in D0₋_D0_{oscillations using the decay-time-dependent ratio} of D0→K+π− to D0→K−π+ rates [12], achieving an unprecedented precision which dominates the world averages. The analysis uses data corresponding to an integrated luminosity of 3 fb−1 from proton-proton (pp) collisions at 7 and 8 TeV centre-of-mass energies recorded with the LHCb experiment from 2011 to 2012 (LHC Run 1), and 2 fb−1from pp collisions at 13 TeV recorded from 2015 to 2016 (LHC Run 2). Since the most recent SM predictions of CPV in mixing (and in the interference between mixing and decay) are still much lower than the current experimental uncertainties and first hints of a non-zero value for the mass difference parameter, x, are emerging, it would be very important to increase the sensitivity as much as possible in order to improve the knowledge of the dynamics of the charm quark and clarify the post-∆ACPpicture.

The LHCb experiment continued to take data until the end of the LHC Run 2, in 2017 and 2018, collecting an additional sample corresponding to an integrated luminosity of 4 fb−1from pp collisions at 13 TeV, more than doubling the signal yields. The work described in this thesis has therefore the main purpose of improving the statistical precision on CP-averaged and CP-violating mixing parameters extending its measurement to the total currently available LHCb data sample, including the new data, collected during the second half of LHC Run 2. Furthermore, it aims at a significant reduction of the dominant systematic uncertainties, paving the way to the future measurements, in the LHCb-Upgrade era (LHC Run 3 and beyond), where still higher statistical precision, comparable or even lesser than the current systematic one, is expected.

During the LHC Run 2, the LHCb experiment collected a huge sample of extremely clean D0 → K−π+ RS decays, about 443 million, and a huge sample of suppressed D0 → K+π− WS decays, about 1.8 million). This sample is much larger than the one collected during the LHC Run 12, therefore the work of this thesis is focused at the moment on the extension of the analysis using only the data collected in Run 2. During this period running conditions (including trigger configurations) remained almost unchanged making the analysis work easier. Data collected during the LHC Run 1 will be added later, before the publication of the final result on the journal, allowing a further gain in statistical precision of about 7% with respect to the preliminary results reported in this thesis.

The global strategy of the analysis is already well established owing to the work performed during the last years by the LHCb collaborators. However, the precision regime obtainable with both the current available data sample and the one of the future measurements in the LHCb-Upgrade era, required a revisitation of all the aspects of the analysis with the aim of reducing as much as possible the size of dominant systematic uncertainties. The selection of the data sample has been re-optimized in order to reduce the impact of the requirements that could make difficult a reduction of the systematic uncertainties. This has been done without significant loss of statistical power. The simultaneous fit to the invariant D∗-mass, used to determine the signal yields, has been improved in order to properly account for the statistical correlations between WS and RS mass shapes. Particular

(8)

attention has been dedicated in reducing the size of the secondary systematic uncertainty, resulting to be the dominant one in the previous measurement. It consists in a contamination of D∗ mesons originating from weak decays of b-hadrons (secondary decays) and not in the pp primary vertex. Since the reconstructed decay time of secondary D0 mesons is biased towards higher values, the value of the time-dependent ratio R(t) can be artificially reduced, biasing (underestimating) the determination of x0and y0. The effect of this residual contamination has been deeply studied, and a new methodology has been developed in order to remove all the approximations adopted in the past, that are not accurate for the current level of precision. It results in a drastic reduction of the associated systematic uncertainty, about a factor 30 less than the previous one assessed in the published measurement.

In conclusion, the thesis presents a preliminary measurement of the CP-averaged and CP-violating mixing parameters in D0−D0oscillations using the decay-time-dependent ratio of D0→K+π−to D0→K−π+ rates. The measurement uses the whole data sample recorded by the LHCb experiment during the LHC Run 2, from 2015 to 2018, corresponding to an integrated luminosity of about 6 fb−1 at a centre-of-mass energy of 13 TeV. The returned central values of mixing parameters are kept "blind", to avoid any experimenter bias until the final approval of the measurement by the collaboration, while the error estimation is determined, showing an improvement of about 30% with respect to the previous published analysis.

(9)

1

Mixing and CPV in the SM

1.1. The Standard Model of Particle Physics

The Standard Model of particle physics (SM) is a quantum field theory that describes all known elementary particles and their interactions via three of the four fundamental known forces: strong, electromagnetic and weak. It is defined by three fundamental features1_:

• the symmetries of the Lagrangian;

• the representations of fermions and scalars; • the pattern of spontaneous symmetry breaking.

Lagrangian symmetry

The Standard Model is a local gauge-invariant theory with a non-abelian gauge group of symmetry

GSM=SU(3)C⊗SU(2)L⊗U(1)Y. (1.1)

SU(3)_C is the symmetry of Quantum Chromodynamics (QCD), the strong force theory, and the subscript C stands for the color charge. SU(2)L⊗U(1)Yis the symmetry of electroweak interactions, as in the Glashow-Weinberg-Salam theory [14, 15, 16]; here L refers to the chirality of weak interactions and Y indicates the hypercharge.

Imposing the local gauge invariance leads to the arise of a boson gauge field (in the SM they are all vector fields) for each of the group generator. So there are eight gluon boson fields Gilinked to the SU(3)Cgenerators, that are the strong force mediators. The three weak interaction boson fields Wi, linked to the SU(2)Lgenerators, and the single hypercharge boson field B, from the U(1)Ygenerator, do not correspond to physical observed particles: these ones emerge after the spontaneous symmetry breaking.

(10)

CHAPTER 1. MIXING AND CPV IN THE SM

Particle representations

There are three fermions generations and for each of them five representations of the GSMgroup: Q_LiI (3, 2)_+1/6, u_RiI (3, 1)_+2/3, dI_Ri(3, 1)_−1/3, L_LiI (1, 2)_−1/2, `I_Ri(1, 1)₋₁, (1.2) where i is the generation index, L and R stand for left-handed and right-handed, respectively, and refer to chirality, I denotes the interaction eigenstate. In this notation each field is followed by two numbers in parenthesis that are the dimensions of the representation of the SU(3)C and SU(2)L groups, and a subscript that is the carried hypercharge.

The SU(2)Ldoublets can be decomposed into their components

Q_LiI =u I Li dI Li , L_LiI = ν_LiI `I Li .

A schematic view of these representations is illustrated in Tab. 1.1. U(1)emcharge is Q= I3+1₂Y, where I3is the weak isospin and Y is the hypercharge.

Field SU(3)C SU(2)L U(1)Y U(1)em QI Li= uI Li dI Li 3 2 1/6 2/3 −1/3 uI_Ri 3 1 2/3 2/3 d_RiI 3 1 -1/3 -1/3 LI_Li= ν_LiI `I_Li 1 2 -1/2 0 −1 `_RiI 1 1 -1 -1

Table 1.1: Schematic view of Standard Model particles representations and their relative dimensions.

Particles that are singlets in a specific representation are uncharged for the corresponding interac-tion and do not interact with its mediators. Usually, we identify two groups:

• quarks (QI_Li, uI_Ri, d_RiI ) that are SU(3)Ctriplets and interact with the strong force; • leptons (LI_Li,`_RiI ) that are SU(3)Csinglets.

Moreover it is possible to classify:

• left-handed particles (QI_Li, LI_Li) that are SU(2)Ldoublets and interact with Wibosons; • right-handed particles (u_RiI , dI_Ri,`I

Ri) that are SU(2)L singlets, therefore they do not interact with Wibosons.

W3and B bosons mix into the physical observed bosons Z0and A after the spontaneous symmetry breaking. B boson couples to right-handed particles and, as a consequence, Z0couples to right-handed particles too. Thus, this classification is not so meaningful. In addition to fermions representation there is a single scalar (0 spin) multiplet:

(11)

1.1. THE STANDARD MODEL OF PARTICLE PHYSICS

Spontaneous symmetry breaking

This additional scalar has a non-null vacuum expectation value:

hφi = √1 2 0 v (1.3)

that leads to a spontaneous symmetry breaking of the gauge group: GSM→SU(3)C⊗U(1)EM,

where U(1)EM is the symmetry group of electromagnetism. The field φ can be usefully parameterized by φ(x) = √1 2 0 v+H(x) (1.4)

where H(x)is a neutral scalar field known as the Higgs boson.

Once fixed the gauge symmetry as in Eq. (1.1), the representations as in Eq. (1.2) and the pattern of spontaneous symmetry breaking, the Standard Model LagrangianLSM is derived as the most general renormalizable Lagrangian with these requirements.

SM Lagrangian

The Standard Model Lagrangian can be written as sum of three terms:

L_SM = L_kinetic+ L_Higgs+ L_Yukawa.

Lkinetic term is the sum of all kinetic terms (i ¯ψγµ∂µψ) of fermions, where the standard derivative is

replaced by the covariant derivative Dµ₌_∂µ₊_ig

sGµaLa+igW_bµTb+ig0BµY where:

• La are SU(3)C generators in each representation: 12λa for triplets (λa are 3×3 Gell-Mann matrices) and 0 for singlets;

• Tbare SU(2)L generators in each representation: 12σbfor doublets (σb are 2×2 Pauli matrices) and 0 for singlets;

• Y is the the U(1)_Ycharge.

LHiggsterm is the scalar self interaction and it is written as

L_Higgs= −µ2φ†φ+λ(φ†φ)2.

BothL_kinetic andL_Higgsin the Standard Model are CP symmetric.

LYukawa term describes the coupling between fermions and the scalar field. This part of the La-grangian is, in general, CP violating and this is the only source of CP violation predicted by the Standard Model.

(12)

1.2. The CKM matrix and CP violation in the SM

The Yukawa sector

The masses and mixings of quarks arise from the Yukawa interactions of fermions with the scalar field:

L_Yukawa= −Y_ijdQ_LiI φ d_RjI − Y_ijuQ_LiI φ u˜ I_Rj −Y_ij`L_LiI φ`_RjI + h.c. (1.5) where Y_iju,d,`are 3×3 complex matrices (i and j are generation labels) and ˜φ=iσ2φ†. Equation 1.5 gets a mass term for the fermions when φ acquires a vacuum expectation value as in (1.3). Using Eq. (1.4) and neglecting interaction term, one can write the mass term:

LM= −(Md)ijdILidIRj− (Mu)ijuILiuIRj− (M`)ij`ILi`IRj where Mf = v √ 2Y f_.

Mass eigenstate The physical observed states are obtained by diagonalizing Mu,d,`, because the

mass basis correspond, by definition, to diagonal mass matrices. So we need to find pairs of unitary matrix V_Lf and V_Rf that perform the transformation

Vf LMfVf R† =M diag

f . (1.6)

In this way we obtain the mass eigenstates as a rotation of the weak interaction basis (as illustrated in the left panel of Fig. 1.1):

dLi= (VdL)ijdLjI , dRi= (VdR)ijdIRj, (1.7) uLi= (VuL)ijuILj, uRi= (VuR)ijuRjI ,

`Li= (V`L)ij`ILj, `Ri= (V`R)ij`IRj, νLi= (VνL)ijν_LjI .

Mixing in the Standard Model

Mixing in weak charged current Let’s take, from theLkinetic term, the interaction between quarks

and charged SU(2)_L gauge bosons Wµ± =

1 √

2(W 1

µ∓iWµ2), expressed as a function of the weak

interaction eigenstates: L_W± = −√g 2Q I LiγµWµ+Q I Li+h.c. = −g √ 2u I LiγµWµ+d I Li+h.c. .

Here only left-handed quarks appear because the charged weak bosons W± preserve the SU(2)L symmetry, differently from Z0, and couple only with left-handed particles. Switching to the mass eigenstates as in Eq. (1.7) we obtain

L_W± = −√g 2uLiVuLγ µ_W+ µV † dLdILi+h.c.

(13)

1.2. THE CKM MATRIX AND CP VIOLATION IN THE SM

and fixing the order of quarks in generations by their masses: ui→ (u, c, t)and di → (d, s, b), we can write: L_W± = √−g 2(ul, cL, tL)γ µ_W+ µVCKM   dl sL bL  +h.c., with VCKM≡VLuV d† L =   Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb  , where VCKMis a 3×3 unitary matrix (as product of unitary matrices), also known as the Cabibbo-Kobayashi-Maskawa (CKM) matrix [17, 18]. In general, a 3×3 unitary matrix has 9 free parameters. However there is arbitrary freedom in VCKM definition. Let’s take two arbitrary diagonal unitary matrices Puand Pd Pu=   eiα1 ₀ ₀ 0 eiα2 ₀ 0 0 eiα3  , P_d=   eiβ1 ₀ ₀ 0 eiβ2 ₀ 0 0 eiβ3  , using them to transform VuL and VdL:

e

VuL=PuVuL, Ve_dL=PuVdL.

This process does not change mass eigenstates as in Eq. (1.6) PfVf LMfVf R† =PfM

diag f ,

because Pf is diagonal and what is happening is just an arbitrary phase shifting of each quark field. However VCKM changes form:

˜

VCKM=PuVCKMP_d†=   

Vudei(α1−β1) Vusei(α1−β2) Vubei(α1−β3) Vcdei(α2−β1) Vcsei(α2−β2) Vcbei(α2−β3) Vtdei(α3−β1) Vtsei(α3−β2) Vtbei(α3−β3)

   .

Only five of these nine phase differences are independent, thus by fixing these ones we end up with four free parameters: three mixing angles and a single CP-violating phase. Among the possible parametrizations, a standard choice is [19]:

VCKM=   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   (1.8) =   c12c13 s12c13 s13e−iδ −s12c23−c12s23s13eiδ c12c23−s12s23s13eiδ s23c13 s12s23−c12c23s13eiδ −c12s23−s12c23s13eiδ c23c13  , (1.9)

where sij=sin θij, cij =cos θij, and δ is the CP violating phase. We chose angles θijto lie in the first quadrant, so that sij, cij >0.

Mixing in lepton flavours Among the seven V_L,Ru,d,`,ν, we considered only V_Lu and V_Ld, that enter in

the CKM matrix and rule quarks mixing and CP violation in charged current interactions. Just like we did before, we can define a mixing matrix for left-handed neutrinos and leptons. Nevertheless in Standard Model neutrinos are massless and VνLcan be chosen such that VνL =V`L, hence the mixing

matrix is now the unit matrix. In the Standard Model extension that allows neutrinos to have mass, the lepton mixing matrix VPMNS(known as Pontecorvo–Maki–Nakagawa–Sakata matrix [20]) leads to mixing and CP violation in the lepton interaction with W±.

(14)

Mixing in weak neutral current The neutral weak boson Z0is the result of the mix between W3

and B bosons. The Z0boson couples to right-handed particles, so one can expect V_Ru,d to come into play in this interaction, however Z0boson couples u with u, d with d, left-handed with left-handed and right-handed with right-handed, e.g.

L_Z0 ∝ dLiγµ(V_dL†VdL)dLjZµ+. . . ,

hence V matrices simplify and weak neutral interaction do not show mixing or CP violation phenom-ena.

Other source of CP violation Another natural possible source for CP violation in the Standard

Model is related to strong interaction through the θQCD parameter. However experiments on the electric dipole moment of the neutron [21] have set a strong constraint on its value: θQCD<10−9and nowadays, it is compatible with 0.

Summing up, VCKMdescribes the only source of flavour changing and CP violation mechanism in the Standard Model.

Figure 1.1:Graphical representation of CKM mechanism.

CKM matrix parametrizations

Wolfenstein parametrization It is experimentally known that in Eq. (1.8) s13 s23 s12 1

so it is convenient to switch to the Wolfenstein parametrization that instead of the parameters (s12, s23, s13, δ) uses four new parameters (λ, A, ρ, η), or alternatively (λ, A, ¯ρ, ¯η), defined as follows:

s12=λ= p |Vus| |Vud|2+ |Vus|2 , s23= Aλ2=λ Vcb Vus , (1.10)

s13eiδ=Vub∗ =Aλ3(ρ+iη) =

Aλ3(¯ρ+i¯η)√1−A2_λ4

√

1−λ2[1−A2λ4(¯ρ+i¯η)] ,

where λ= |Vus| '0.22 is used as an expansion parameter and η represents the CP violation phase. One can than write VCKM toO(λ4)in terms of this new parametrization [22]:

VCKM =  

1−λ2/2 λ Aλ3(ρ−iη)

−λ 1−λ2/2 Aλ2 Aλ3(1−ρ−iη) −Aλ2 1



(15)

1.2. THE CKM MATRIX AND CP VIOLATION IN THE SM

A graphical representation of λ dependence in mixing parameters is shown in the right panel of Fig. 1.1.

Jarlskog invariant It is possible to define a CP violating quantity that is independent from

parametrization, known as Jarlskog invariant [23], J, defined by

Im[VijVklVil∗V ∗

kj] = J

∑

m,n

eikmejln.

Figure 1.2:Sketch of the unitarity triangle.

Unitarity triangles The unitarity of the CKM matrix imposes orthogonality among rows and

columns:

∑

i VijVik∗ =δjk,

∑

j VijVkj∗ =δik. (1.12)

This six constraints are represented as triangles in a complex plane. The scalar products of rows or columns next to each other are nearly degenerate. All the triangles have the same area of|J|/2. The most common unitarity triangle comes from

VudVub∗ +VcdVcb∗ +VtdVtb∗ =0,

dividing each term by VcdV_cb∗, that is the best experimentally known term, one gets VudVub∗ VcdV_cb∗ +1+ VtdV ∗ tb VcdV_cb∗ =0,

where one can show vertices are exactly(0, 0),(1, 0), and(¯ρ, ¯η). This unitarity triangle is shown in Fig. 1.2.

Current experimental status

Experimental measurement of CKM parameters CKM parameters are of fundamental importance

for Standard Model to make predictions and their precise measurements can put strong constraints on beyond Standard Model theories. There are two main ways to constraint CKM parameters:

(16)

Figure 1.3:Current experimental status of the global fit to all available experimental measurements related to the unitarity triangle phenomenology [24]. The shaded areas have 95% CL.

• Direct measurements that look at tree level processes in order to directly extract the value of

|Vij|. Some of these elements however have poor precision, such as|Vtb|and|Vcs|, or are too suppressed to be measured, such as|Vtd|and|Vts|.

• Indirect measurements that look at high-order processes to extract the value of product of Vij, such as|VtbVtd|.

Many measurements are conveniently displayed and compared in the ¯ρ, ¯η plane in Fig. 1.3. The shaded 95% CL regions overlap consistently around the global fit region.

Combining the information from both direct and indirect measurements and imposing the CKM unitarity constraint in Eq. (1.12), one can over-constraint the parameters, significantly reducing the allowed range of possible values for CKM elements. The global fit for the Wolfenstein parameters gives [24]:

λ=0.224747+0.000254_−0.000059, A=0.8403+0.0060_−0.0201, ¯ρ=0.1577+0.0096_−0.0074, ¯η=0.3493+0.0095_−0.0071. The fit results for the magnitudes of all nine CKM elements are

|VCKM| =    0.974410+0.000014_−0.000058 0.224745+0.000254_−0.000059 0.003746+0.000090_−0.000062 0.224608+0.000254_−0.000060 0.973526+0.000050_−0.000061 0.04240+0.00030_−0.00115 0.008710+0.000086_−0.000246 0.04169+0.00028_−0.00108 0.999093+0.000049_−0.000013    . (1.13)

(17)

1.3. FLAVOURED NEUTRAL MESONS MIXING

1.3. Flavoured neutral mesons mixing

In the Standard Model there are exactly four flavoured neutral mesons that cannot decay into lighter particles through a strong or electromagnetic interaction: K0_{(d ¯s), D}0_{(c ¯}_{u), B}0_{(d¯b) and B}0

s (s¯b). With flavoured we mean they posses a non-null flavour quantum number (strangeness, charmness or bottomness), thus we are excluding neutral mesons such as π0, η0, J/Ψ and so on: these particles decay via electromagnetic or strong interaction.

As previously seen, for those four flavoured neutral mesons, the interaction eigenstate in which they are produced is different from the mass one, that is the eigenstate of the free Hamiltonian, that drives their time evolution. As a consequence, it is possible for these mesons to be produced with a certain flavour and than to oscillate into their antiparticles, changing its flavour quantum number by two units. This phenomenon is known as "mixing".

Time evolution of flavour eigenstates

In general, the initial state is a linear combination of the interaction eigenstates M0and M0(where M0stands for K0, D0, B0or B0s):

|ψ(0)i =a(0) |M0i +b(0) |M0i.

Schrödinger equation describes the time evolution of the state: i¯h∂

∂t|ψi =HM|ψ(t)i, (1.14)

where HMis the free Hamiltonian and ψ(t)is a superposition of|M0i,|M 0

iand all the possible final state|fkiin which the two mesons can decay:

|ψ(t)i =a(t) |M0i +b(t) |M0i +

∑

k

ck(t) |fki.

Since we are interested only in mixing, and the time scale under consideration is much larger than strong interaction one, it is useful to apply the Weisskopf-Wigner approximation [25, 26] in order to describe the M0−M0subspace evolution writing down a 2×2 effective HamiltonianH:

i∂ ∂t M0(t) M0(t) ! = _H 11 H12 H21 H22 _M0₍_t₎ M0(t) ! .

We can split this Hamiltonian into its Hermitian and an anti-Hermitian terms

H = M − i

2Γ,

whereM = (H + H†)/2 is also known as the mass matrix andΓ=i(H − H†)as the decay matrix.

Γ rules the decay rate of the state from the M0₋_M0_{subspace. Let’s call}_|_ψ

(M0₎ithe projection of|ψi on this subspace. From Eq. (1.14) it can be obtained that

∂

∂thψ(M0)|ψ(M0)i =ihψ(M0)| (H

†_{− H) |}_ψ

(18)

The right term must be negative, since it corresponds to the decay rate, henceΓ is positive defined.

MandΓ are Hermitian, soMij= M∗_jiandΓij=Γ∗_ji, henceHhas eight free parameters. However,

it’s possible to fix some of these free parameters imposing invariance under discrete transformations, such as CPT, CP or T, as shown in Tab. 1.2.

Invariance Constraints

CPT M₁₁ = M22 Γ11 =Γ22

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

Table 1.2:Constraints onMandΓ depending on interaction invariance under different discrete transformations.

From here on, CPT invariance is assumed, hence we defineM:= M₁₁= M₂₂andΓ :=Γ₁₁ =Γ₂₂. Let’s define theHeigenstates to be

|M1i:=p|M0i +q|M0i, (1.15)

|M2i:=p|M0i −q|M 0

i,

they are normalized imposing|p|2_{+ |}_q_|2₌_{1. As the matrix}_H_{is not Hermitian, the two eigenstates} are not necessarily orthogonal. The relative eigenvalues are

λ1,2:=m1,2− i 2Γ1,2= M − i 2Γ± q p M₁₂− i 2Γ12 (1.16) with q p = ± v u u t M∗ 12−2iΓ12∗ M₁₂− i 2Γ12 , (1.17) so that |M1,2(t)i =exp(−im1,2t)exp(−Γ1,2t/2) |M1,2(0)i. (1.18) The sign of the square root in the Eq. (1.17) can be taken to be positive without loss of generality thanks to the arbitrariness of the sign of q in Eq. (1.15). From Eq. (1.16), it can be seen that the following relations hold:

M = m1+m2

2 , Γ=

Γ1+Γ2 2 .

The time evolution of a particle that was created in its flavour eigenstate at t = 0 can be easily obtained substituting Eq. (1.18) in Eq. (1.15):

(19)

where|M0(t)iindicates the time-evolution at time t of a state that a t=0 corresponded to the flavour eigenstate|M0i, and similarly for|M0(t)i. The coefficients g±(t)are so defined:

|g±(t)|2= 1 2 e −Γt_cosh∆Γt 2 ±cos(∆m t) , (1.20) ∆m :=m2−m1= −2Re p H12H21 , ∆Γ :=Γ₂−Γ1=4Im p H₁₂H₂₁.

Equation 1.20 is often rewritten in terms of two adimensional mixing parameters x :=∆m/Γ, y :=∆Γ/2Γ,

then it becomes

|g±(t)|2= 1 2 e

−Γt_[_cosh₍_yΓt_{) ±}_cos₍_xΓt_)]_. _(1.21)

From Eq. (1.19), it follows that the probability of no oscillation is

P (M0(0) →M0(t)) = | hM0(t)|M0i |2= |g+(t)|2,

P (M0(0) →M0(t)) = | hM0(t)|M0i |2_{= |}_g +(t)|2, whereas the probability of oscillation is

P (M0(0) →M0(t)) = | hM0(t)|M0i |2= q p 2 · |g−(t)|2, (1.22) P (M0(0) →M0(t)) = | hM0(t)|M0i |2₌ p q 2 · |g−(t)|2,

The probability of the M0and M0mesons to not oscillate is the same for both mesons, whereas the probability to oscillate in their antiparticle can be different, provided that|q/p| 6=1.

Mixing phenomenology

From Eq. (1.21) we see that the rate of oscillation, that is the cosine term, is determined by x, the mass difference between the two physical eigenstate, while the difference in decay width (y) appears in the hyperbolic cosine term, that reflects a change in the exponential decay trend. These two parameters alone define the characteristic behaviour of neutral meson mixing and for this reason are also known as mixing parameters.

Meson hMassi M[MeV] hWidthiΓ [ps−1_] _{Lifetime [ps]} _x _y

K0 497.6 0.00559 89.5 52900 0.95 0.997 D0 1864 2.43 0.410 0.004 0.007 B0 5280 0.658 1.52 0.77 -0.001 B0 s 5367 0.662 1.51 26.7 0.06

Table 1.3:Overview of approximate parameters relevant to meson mixing. Masses and widths are the average of the two physical eigenstates. Values taken from Ref. [26] and [9].

(20)

Figure 1.4:Flavour-changing (red) and flavour-unchanging (blue) PDFs for the four neutral meson systems.

The key features of the four flavoured neutral mesons are summed up in Tab. 1.3, while Figs. 1.4 and 1.5 shows the four different oscillations behaviors. As it is clear D mesons mixing parameters are very different compared to those of K or B mesons.

K system The kaon system is the only one to have y≈1, hence two mass eigenstates with drastically

different lifetimes, so different that at the time of their discovery it was believed they were two completely different particles. Today they are named K-short (K0_S) and K-long (K0_L). Moreover, there is also a big mass difference: in fact x≈1, which results in a sizeable sinusoidal oscillation as shown in Fig. 1.4. Thanks to the big lifetime difference one can study relatively pure samples of just one of K0_Sor K0_L, for example looking at decays close to interaction point where K0_Ldecays are just a small fraction, or far away where most K0_S have already decayed.

B systems Either the two B-meson have small width difference, still they have sizeable values for x.

In B0s system, in particular, the big x value leads to fast oscillation that need high spatial resolution to be resolved: for this reason B0s oscillation has been accurately measured only in recent years [27].

D system In the charm meson system x and y are both very small, thus very large data samples are

necessary to gain a sufficient statistical precision for their measurement. That is why it takes till 2007 for a first evidence of charm mixing [4, 5] and only in 2012 LHCb performed a single measurement with high statistical significance [7].

(21)

Figure 1.5:The widths and mass differences of the physical states of the flavoured neutral mesons. The width corresponds to the inverse lifetime while the mass difference determines the oscillation frequency.

Figure 1.6:The two main diagrams contributing to the D0−D0 mixing: (left) the box diagram and (right) re-scattering diagram.

Mixing amplitudes are governed by two kinds of contributions, known as short and long distance contributions.

Short distance contribution The short distance contributions are due to the box diagram,

repre-sented in the left panel of Fig. 1.6 for the case of D meson. Being entirely due to weak interaction (there are four weak interaction vertex) the typical length scale is much smaller than strong in-teractions, hence the name of the contribution. In K and B mesons system the amplitude due to this box diagram is the dominant term, while it is suppressed in the D system. The contribution

(22)

of bottom quark in the loop is CKM-suppressed by a factor|VubVcb∗|2/|VusVcs∗|2≈10−5, while the contributions from down and strange quark cancel each other through the GIM mechanism, indeed V_cd∗Vud+Vcs∗Vus≈10−5. Accounting only for short distance contributions the predicted value for x and y in the D system is aboutO(10−6), drastically smaller than the experimentally measured one of the order ofO(10−2). Thus we expect long distance contributions to be the main ingredient in the D mixing amplitudes.

Long distance contribution The other way for mixing to proceed is through intermediate on-shell

states common to both mesons and anti-meson, as represented in the right panel of Fig 1.6 for the case of D meson. This is the so called long distance contribution. It can be seen as the decay of the neutral meson in a common final state, e.g. K+K−or π+π− in the D case, followed by a recombination of the final state particles to the anti-meson. This contribution in the charm mesons mixing does not suffer of the suppression as the short distance effect, so it dominates the mixing amplitude.

March 31, 2018 17:23 WSPC/INSTRUCTION FILE

petrov

CHARM MIXING IN THE STANDARD MODEL AND BEYOND

3

Standard Model mixing predictions

1.00E-09 1.00E-08 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 Reference Index |x | o r |y |

Fig. 1.

Standard Model predictions for |x| (open triangles) and |y| (open squares).

are determined by a set of selection rules according to the initial virtual photon

quantum numbers J

P C

_{= 1}

−−

3. Since we know whether this D(k

2

) state is tagged

as a (CP-eigenstate) D

±

from the decay of D(k

1

) to a final state S

σ

of definite

CP-parity σ = ±, we can easily determine y in terms of the semileptonic branching

ratios of D

±

, which we denote B

ℓ±

. Neglecting small CP-violating effects,

y =

1

4 B

ℓ +

(D)

B

ℓ −

(D)

−

B

ℓ −

(D)

B

ℓ +

(D)

.

(5)

A more sophisticated version of this formula as well as studies of feasibility of this

method can be found in Ref. [3].

The current experimental bounds on y and x are 4

y < 0.008 ± 0.005 ,

x < 0.029 (95% C.L.) .

3. Charm mixing predictions in the Standard Model

The current experimental upper bounds on x and y are on the order of a few times

10

−3

, and are expected to improve in the coming years. To regard a future discovery

of nonzero x or y as a signal for new physics, we would need high confidence that

the Standard Model predictions lie well below the present limits. As was shown in

[5], in the Standard Model, x and y are generated only at second order in SU(3)

F

breaking,

x , y ∼ sin

2

θ

C

× [SU(3) breaking]

2

,

(6)

where θ

C

is the Cabibbo angle. Therefore, predicting the Standard Model values of

x and y depends crucially on estimating the size of SU(3)

F

breaking.

Theoretical predictions of x and y within the Standard Model span several orders

of magnitude

a

(see Fig. 1). Roughly, there are two approaches, neither of which give

Figure 1.7:Standard Model predictions for|x|(open triangles) and|y|(open squared). Horizontal line references are tabulated in Tab. 5 of Ref. [28]. Plot taken from Ref. [29].

It was discussed whether the measured size of the mixing parameters could be interpreted as a hint for physics beyond the Standard Model [30, 31, 32]. The biggest problem in answering this question is the non-existence of a precise SM prediction. In fact, accurate calculations of long distance effects for the D0meson are difficult and characterized by large uncertainties, since the value of the mass of the charm quark is placed somewhat halfway between the heavy and the light quark systems. As a consequence, theoretical predictions of the D0−D0mixing (and of CPV) parameters are very challenging, and several orders of magnitude are spanned in the literature [29], as shown in Fig. 1.7. Therefore, it is crucial to measure mixing parameters very precisely in order to use them as input in theoretical computation, improving our knowledge of the dynamics of the charm sector.

(23)

1.4. CP VIOLATION

1.4. CP violation

Experimentally, there are three kinds of CP violation [33]: • CP violation in the decay;

• CP violation in the mixing;

• CP violation in the interference between decay and mixing.

The first one is also called direct violation, while the others are referred as indirect CP violation.

CP violation in the decay

The decay amplitudes of a generic (either charged or neutral) particle X into a final state f is defined as

A(X→ f):= hf| H |Xi (1.23) and consequently for the anti-particle X into the C conjugated final state ¯f,

A(X→ f):= hf| H |Xi. (1.24) whereHis the decay Hamiltonian. CP violation in the decay is observed if the probability of the particle X to decay into final state f is different from the probability of the anti-particle X to decay into the C-conjugated final state ¯f. In general, two types of phases can enter these amplitudes.

Weak phases These phases originate from weak interaction through CKM matrix phase (the only

source of CP violation in the Standard Model) hence they change sign under CP transformation.

Strong phases The so called scattering phases due to final state interaction are generated by the

strong interaction whenever there are hadrons (hence the name strong phases). Thanks to their origin from the strong interaction these phases do not change sign under CP transformation.

Let’s take a decay process which can proceed through several amplitudes ai,

A(X→ f) =

∑

i

|ai|ei(φi+δi), A(X→ f) =

∑

i

|ai|ei(−φi+δi),

where φiare the weak phases, and δi are the strong ones. The difference between the two squared amplitude is: |A(X→ f)|2− A(X→ f) 2 = −2

∑

i,j |ai||aj|sin(φi−φj)sin(δi−δj).

In order to observe|A(X→ f)| 6= |A(X→ f)|, that definitely states the existence of weak phase, hence CP violation, one needs at least two amplitudes with different weak and strong phases. A golden observable, sensitive to CP violation in the decay, is the CP asymmetry defined as

Adecay_CP (f) = Γ(X→ f) −Γ(X→ f)

(24)

whereΓ is the time-integrated decay width. Γ(X→ f)∝|A(X→ f)|2andΓ(X→ f)∝|A(X→ f)|2_, thus Adecay_CP (f) = |A(X→ f)| 2_{− |A(}_X_→ _f_)||2 |A(X→ f)|2_{+ |A(}_X_→ _f_)|2 = 1−R2_f 1+R2_f, where we define Rf Rf = A(X→ f) A(X→ f) , (1.25)

hence Rf 6=1 is a consequence of CP violation. This is the only kind of CP violation observable in particles different from the four flavoured neutral meson.

CP violation in the mixing

CP violation in the mixing is observable if the probability of the M0meson to oscillate into M0after a time t differs from the same probability for the CP-conjugate process, i.e. M0oscillates into a M0. From Eq. (1.22) we known this is possible if and only if the magnitude of the ratio between the coefficient in the superposition of|M0iand|M0iin the mass eigenstates of Eq. (1.15) differs from 1:

Rm := q p 6=1.

CP violation in the interference

Whenever a final state is achievable from both M0 _{and M}0 _{mesons then CP symmetry can be} violated in the interference between the decay without mixing, M0→ f , and that with the mixing, M0→M0→ f . This condition occurs when

Im(λf) + Im(λ_f) 6=0 where λf := qA(M0→ f) pA(M0_→ _f₎ and λf := qA(M0→ f) pA(M0_→ _f₎. (1.26)

If the final state f is CP-symmetric, we can write λf as

λf = q p A(M0→ f) A(M0_→ _f₎ exp " i arg qA(M 0 → f) pA(M0_→ _f₎ !# =RmRfeiφf, where φf :=arg qA(M0→ f) pA(M0_→ _f₎ ! .

(25)

1.4. CP VIOLATION

CP violation in the D

0

system

Charm quark is the third most massive of the six quarks with a bare mass of about 1.3 GeV. Along with the strange quark it is part of the second generation. It has an electric charge of+2₃, just like up and top quarks and these particles are often referred as up-type quarks. It is never observed free, but only in bounded states with other quarks/anti-quarks. Hadrons containing charm quark can exist as: • open charm mesons or baryons, that contain one or more (in baryons case) charm quark linked

to different quarks;

• charmonium state, that is a c ¯c bounded state with no naked charm.

As previously said, according to Standard Model, strong and electromagnetic interactions are flavour diagonal (i.e. the flavour is conserved in these interactions). Therefore the only way to decay for a charm quark is through annihilation with an anti-charm quark, if it is present, like in the charmonium case, or through a weak decay mediated by a W±boson, in open charm particles cases. Thus, open charm particles are the only systems where it is possible to study the decay of a up-type quark in a down type quark in a bounded state. That is because the up quark is the lightest of quarks, hence it is stable, while top quark is too heavy and it decays much faster than hadronization time scale.

Figure 1.8:Sketch of the unitary triangle for charm meson decays. The figure is out of scale as the vertical direction is enlarged by a factor of twenty.

Charm unitary triangle Studying charmed meson decays, the elements of interest in the CKM

matrix are those of the first two rows, so the corresponding unitary constraint is V_cd∗Vud+Vcs∗Vus+V_cb∗Vub=0

where we can develop each term in the Wolfenstein parametrization:

V_cd∗Vud= −λ+ λ 3 2 + λ5 8 (1+4A 2_{) −} λ5A2(ρ+iη) + O(λ7), (1.27) Vcs∗Vus=λ−λ 3 2 − λ5 8 (1+4A 2_{) + O(} λ7), (1.28) V_cb∗Vub=λ5A2(ρ+iη) + O(λ11), (1.29) resulting into a mashed triangle with a side of the order of λ5as shown in Fig. 1.8, whereΛf =Vc f∗Vu f. Because of the presence of low-energy strong-interaction effects, as explained in Sect. 1.3, theoretical predictions of the size of CP violation in charm decays are difficult to compute reliably, and the asymmetries are expected to be of the order of 10−3−10−4in magnitude for direct CPV [34, 2, 3], and even lesser for CPV in mixing and in the interference between the mixing and the decay [35].

(26)

(27)

2

Measurement of the D

0 −

D

0 oscillations

2.1. Charm production

The study of charm physics takes place in a wide variety of different experimental set-ups, different accelerators, energies and production mechanisms, with a consequently wide range of spanned cross sections. The first big classification can be done between e+e−and hadrons collider, where the distinction can be synthesized as a trade-off between quality and quantity:

• the e+e− colliders have a much cleaner environment with known interaction energy, few interaction vertices per event, almost hermetic detectors, very low background and very high efficiency;

• the hadrons colliders can reach higher energies and consequently much higher cross section, so they compensate to lower quality events with bigger statistic.

For what concerns e+e−colliders, there are two main experimental set-ups at use.

"D-factories" Setting the interaction energy slightly over 3770 MeV,Ψ(3770)is resonantly produced

and decays to quantum correlated D0D0or D−D+ pairs. The cross section for DD pair production atΨ(3700)resonance is about 8 nb. The downside is thatΨ(3770)is produced at rest, not allowing to measure D mesons decay time and all decay-time related observables. This set-up was at use at MARK III, CLEO-c and BESIII experiments.

B-factories The alternative is running at an higher centre-of-mass energy of 10.6 GeV in order to resonantly produceΥ(4S)that decays in quantum correlated B0B0or B+B−pairs, that are the main objects of study of these experiments, and that is why they are known as B-factories. BB pairs decay into all possible charm particles and not only DD pairs, still the cross section for producing at least one D0is smaller than at theΨ(3700)resonance: 1.45 nb. Nevertheless a comparable fraction of D mesons is produced in the continuum e+e−→c¯c.

(28)

CHAPTER 2. MEASUREMENT OF THE D0−D0OSCILLATIONS

This production mechanism is used by BaBar and Belle experiments at asymmetric colliders that allow a better decay-time resolution. B-factories deal with smaller cross sections by reaching higher instantaneous luminosity (∼ 1−2×1034 cm−2s−1) that allows to collect over the course of their experimental life-time about 500 fb−1 at the BaBar experiment, and about 1000 fb−1 at the Belle experiment, compared to the much lower 0.5 fb−1and 3 fb−1respectively collected by the CLEO-c and BESIII 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 2.4×107 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 1.5×109 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 2011-2011 pp 7-8 TeV 1.4-1.6 mb 1.1+2.1 fb−1 4.6×1012 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 17 fb−1 4.6×1013

Table 2.1:Charm production values for different experiments taking as a term of comparison the D0_production in the respective detector acceptances [36]. Future experiments are marked with * and the expected values are reported.

Hadrons colliders Charm production cross section at hadron colliders is many order of magnitude

bigger than at e+e− colliders. Moreover at hadron colliders charm production occurs in very asymmetric collisions which leads to a good time resolution thanks to the boost.

At the Tevatron accelerator, that is a proton-antiproton collider with√s=2 TeV, the CDF experiment measured a D0production cross section of 13 µb [37] in the detector acceptance and collected an integrated luminosity of 10 fb−1.

While at the LHC accelerator, a proton-proton collider, the LHCb experiment measured (in the detector acceptance) 1.4 mb with√7 TeV [38] (1.1 fb−1collected in 2011), 1.6 mb with√8 TeV (2.1 fb−1collected in 2012) and finally 2.7 mb with√13 TeV [39] (5.9 fb−1collected during all Run 2 from 2015 to 2018). Ultimately, we are talking about a 106factor of difference in cross section compared to B factories.

In Tab. 2.1 it is synthesized the key feature of the main charm oriented experiments. It is clear that LHCb produced and will produce in the future the biggest charm data-set ever collected. The large accumulated statistics allowed the experiment to measure the mixing in D0system with high statistical significance in a single measurement and, for the first time in history, on the 21stof March 2019 the collaboration managed to measure the CP violation in charm particle decays [1].

2.2. Flavour tagging

There are many different ways to measure mixing in D0mesons. Most of them need to identify the flavour of the D0at production and the decay time.

(29)

2.3. MEASUREMENT OF D0−D0MIXING PARAMETERS

D∗ tagging In order to tag the flavour at production, it is typically exploited the strong decay

D∗+ →D0π+and its charge-conjugate D∗− →D0π−, where the charge of the pion determines the flavour of the D0_{. The pion from the D}∗+ _{decay has little kinetic energy and it is, thus, commonly} referred as "soft" pion (πs). The magnetic field can easily curve the trajectories of low momentum particles, such as the soft pions, until swept them out of the detector acceptance, reducing detection efficiency. For this same reason, low momentum particles show big detection asymmetries that can lead to fake physics asymmetries if not properly treated. However, the small amount of free energy in the D∗decays allow a better mass resolution and, consequently, a good background rejection.

Semileptonic tagging Another way to tag the flavour of D0meson uses the flavour-specific decays

of the B meson. In particular, the process B → D0µ−X (where X stands for a non-reconstructed fraction of the final state) is a good choice thanks to the high branching ratio of semi-leptonic decays, and the high trigger efficiency for muons that compensates the low production rate if compared to hadrons. Moreover, with this tagging approach the D0reconstruction efficiency is almost independent from its flight distance, while with the D∗tagging technique, D0with decay vertex near the interaction point has very lower trigger efficiency. This is commonly used in combination with the D∗tagging to improve the decay-time coverage, because it provides a bigger lever arm for measuring time-dependent effects.

Opposite side tagging In particular at e+e− colliders, a third interesting option concerns the

reconstruction of the opposite side charm meson in a flavour specific decay. The idea behind this technique is that quarks are always produced in pairs, thus every charm meson has to be produced with an hadron containing an anti-charm quark. Moreover e+e− colliders operating at Ψ(3770)

resonant production energy produce quantum-entangled D0D0or D−D+ pairs, so the measurement of one of the D mesons gives information about the other one.

2.3. Measurement of D

0 −

D

0 mixing parameters

Theoretically, the most straight-forward mixing measurement exploits the rate of the forbidden decay D0→K+µ−νµwhich is accessible only through D0−D

0

mixing. The ratio of the time-integrated rate of these forbidden decays to their allowed counterparts, D0→K−µ+νµis equals to:

Γ(D0→K+µ−νµ) Γ(D0_→_K−_µ+_ν µ) ' x 2₊_y2 2 :=Rm

Very large samples of D0mesons are required for this measurement and this observable is disad-vantaged at hadronic colliders, due to the missing neutrino energy. Therefore there are still no evidence for D0mixing in this observable. The most precise measurement is Rm= (1.3±2.2(stat.) ± 2.0(syst.)) ×10−4from Belle collaboration [40].

Another probe for D0−D0mixing is the so called yCPobservable that is obtained measuring the effective lifetimes,Γ1andΓ2, of a CP eigenstate (K+K− or π+π−) with respect to the lifetime,Γ, of a flavour-specific state (K−π+):

yCP= Γ1

+Γ2 2Γ −1.

(30)

In the limit of CP conservation yCP is equal to the mixing parameter y. The world average for this observable is dominated by the LHCb measurement: yCP= (0.57±0.13(stat.) ±0.09(syst.))% that use LHC Run 1 data [41].

The D0→K0

Sπ+π−and D0→KS0K+K−decays provide a direct access to the mixing parameters x and y through the simultaneous measurement of the decay-time evolution and resonance amplitudes in the Dalitz plot. The LHCb collaboration has recently published a measurement of the mixing pa-rameters that exploit the D0→Ks0π+π−decay channel [42]: x= (−0.86±0.53(stat.) ±0.17(syst.))% and y= (0.03±0.46(stat.) ±0.13(syst.))%, assuming CP conservation.

The measurement that historically leads to the observation of D0−D0mixing [4, 43] consists in the study of the decay-time-dependence of the ratio between doubly Cabibbo-suppressed D0→K+_π− decays and Cabibbo-favoured D0→K−π+decays. This is one of the most sensitive observable for D0mixing and it is the subject of this thesis.

Mixing in WS decays

Figure 2.1:Leading tree-level diagrams for the doubly Cabibbo-suppressed D0_→_K+

π−decay (left) and the Cabibbo-favoured D0→K−π+decay (right).

The D0 → K+π− process is also known as wrong-sign (WS) decay because in the D∗ tagging approach, the soft pion charge is opposite to the one from the D0decay. This process can take place in two different ways schematized in the top panel of Fig. 2.2:

• via direct decay D0 →K+π− (wiht a branching fraction B = (0.137±0.003) ·10−3 [26]) as shown in the left panel of Fig. 2.1. This amplitude is CKM-suppressed by a factor|VcdVus∗| 'λ2 and it is called doubly Cabibbo-suppressed (DCS) because it takes a λ suppression factor in both vertices.

• via mixing in D0_{followed by the Cabibbo-favoured (CF) decay D}0_→_K+_π−_{(wiht a branching} fractionB = (38.9±0.4) ·10−3 [26]) as shown in the right panel of Fig. 2.1, where no CKM suppression is present.

Assuming CP conserving and expanding for small value of x and y (xΓt1 and yΓt), as exper-imentally confirmed, the time-dependent decay rate of the WS decay can be written as [44, 45]:

Γ(D0(t) →K+π−) 'e−Γt|A(D0→K−π+)|2 RD+ p RDy0Γt+ x 02₊_y02 4 (Γt) 2 , (2.1) where RD is the ratio of the DCS to the CF rate

RD= A(D0→K+π−) A(D0→K+_π−₎ 2

(31)

2.3. MEASUREMENT OF D0−D0MIXING PARAMETERS

and x and y appear rotated by an angle δKπ x0=x cos δKπ+y sin δKπ

y0=x sin δKπ−y cos δKπ.

The angle δKπis the strong phase difference between DCS and CF amplitude

A(D0→K+π−)

A(D0→K+_π−₎ = − p

RDe−iδKπ.

This parameter is not accessible with this measurement, and it is taken from measurements performed using quantum-correlated D0−D0pairs produced at threshold. Such measurements are available from CLEO-c and BESIII [10, 11]. In addition, further constraints on these strong phase differences can be obtained from the combination of several measurements that share the underlying mixing parameters but are subject to different strong phase differences. From the global fit we get a small value for phase, δKπ= (12.1+8.6_−10.2)◦, and this reduces the sensitivity on the mass difference parameter x, with respect to that one on the decay width difference y. Moreover, the uncertainty on the knowledge of the δKπphase also limits the sensitivity on x and y.

Figure 2.2:A sketch of wrong-sign (top) and right-sign (bottom) decays.

From an experimental point of view, the measurement of mixing parameters performed with only the WS sample is extremely challenging. The behaviour shown in Eq. (2.1) is highly modified by several experimental effects that are very difficult to accounted for with very high precision, as the experimental lifetime acceptance and the detection charge asymmetries.

The D0→K−π+ decay is known as right-sign (RS) because the charge of the soft pion has the same sign of the charge of the pion produced in the D0decay. This process occurs in two different ways, as shown in the bottom panel of Fig. 2.2:

• direct Cabibbo-favoured decay D0→K−π+;

• mixing in D0followed by doubly Cabibbo-suppressed decay D0→K+π+.

The second amplitude is highly suppressed compared to the first one, hence the right-sign decay has a negligible contribution from mixing and mainly proceeds exponentially:

(32)

whereA(D0 →K−π+)is the CF amplitude and assuming no CPV A(D0 →K−π+) = A(D0 → K+π−). Experimentally, the measured quantity is the ratio R between WS and RS rates:

R(t) = Γ(D 0₍_t_{) →}_K+ π−) Γ(D0₍_t_{) →}_K−_π+₎ =RD+ p RDy0Γt+ x02+y02 4 (Γt) 2_. _(2.2)

The measurement of a ratio of similar decay modes is much more robust with respect to a measure-ment of an observable decay rate where all very subtle lifetime detector acceptance effect must be accounted for with a high level of precision. Moreover, the size of the RS sample is much larger than that one of the WS sample, and it can be used as a calibration sample in order to extract from data itself all the experimental feature of the WS signal sample with a very limited usage of external inputs, as the full simulation.

2.4. Search for CP violation in charm decays

One of the most sensitive probe for CPV manifestations in the charm decays is the comparison of the time-dependent decay rates of the D0and D0mesons decaying to a CP eigenstate, such as π+π−and K+_K−_{. The time-dependent CP asymmetry can be expanded for small values of mixing parameters} (xΓt1 and yΓt1), obtaining the following well-known equation

ACP(f ; t) = Γ (D0(t) → f) −Γ(D0(t) → f) Γ(D0₍_t_{) →} _f_{) +}_Γ₍_D0₍_t_{) →} _f₎ ' A decay CP (f) −AΓ(f) · t τ_D0 ,

where f stands for π+π− or K+K−. The constant term, Adecay_CP , measures the CP violation in the decay, and its experimental determination, with a very high precision, requires to overcome several difficulties1, as a precise knowledge of production and detection asymmetries. The AΓparameter is, instead, defined as [46]:

A_Γ(f) ' −xφf +y(|q/p| −1) −yAdecayCP (f),

where the phase φf is equal to the mixing phase φ :=arg(q/p)with a very good approximation. The A_Γ parameter, therefore, encloses all the three different types of CPV, and its experimental determination is less problematic of that one of Adecay_CP , being a time slope. The world average for the A_Γmeasurement is dominated by the measurement of the LHCb collaboration A_Γ(K+_K−₊

π+π−) =

(0.9±2.1(stat.) ±0.7(syst.)) ×10−4[46], hence showing no evidence for indirect CP violation. However, from an experimental point of view it is possible to find some golden observables, which allow to overcome the experimental challenges, in order to achieve the maximum possible precision. One of them is certainly the∆ACPobservable, defined as

∆ACP =ACP(K−K+) −ACP(π+π−),

where the difference provides a full cancellation of both production and detection asymmetries. The LHCb collaboration has recently reported, on March 21stthe first observation of CPV in the decay (or direct) by measuring, with the full data sample (LHC Run 1 and Run 2), a significant nonzero value for the∆ACP observable [1]:

∆ACP = (−15.4±2.9) ×10−4.

(33)

2.4. SEARCH FOR CP VIOLATION IN CHARM DECAYS

This is the first observation of the CP violation in the charm decays. Individual asymmetries are measured using only 3 fb−1of data (LHC Run 1) with a worse precision, which does not yet allow to observe the absolute effect [47]

ACP(K−K+) = (0.04±0.12(stat.) ±0.10(syst.))%, ACP(π+π−) = (0.07±0.14(stat.) ±0.11(syst.))%.

Alternative processes to search for CP violation, with a high sensitivity, are the D0→K0_Sπ+π− and D0→K_S0K+K− decays. They provide a direct access to the mixing parameters x and y, and to the CPV parameters q p and φ [42].

Lastly, another powerful probe for CP-violating effects lies in the D0_→_K+_π−_{decay, that provide} information on all types of CPV manifestations: in the decay, in the mixing, and in the interference. This is one of the subjects of this thesis, and, therefore, additional details, about how these decay modes allow searching for CPV effects, are discussed in the following sections, along with an overview of the current experimental status.

CP asymmetries in D

0

→

K

+

π

−

decays

The measurement of the mixing parameters, RD, x, and y, with the WS decays can be separately repeated for D0 and D0decays, and two independent sets of parameters can be defined: R±_D, y0± and x02±, where the symbol+(−)stands for D0(D0) decays. These parameters satisfy the following equations R(t)± 'R±_D+ q R±_Dy0±Γt+x 02±₊_y02± 4 (Γt) 2_, _(2.3) with x0±= q p ± (x0cos φ±y0sin φ), y0±= q p ± (y0cos φ∓x0sin φ).

If the ratio between suppressed and favoured amplitude were not CP symmetrical, we would have a difference between R+_D and R−_D, showing CP violation in the decay (or direct). CP violation in mixing and in the interference would manifest themselves with a difference between(x2+_{, y}0+₎_and

(x2−, y0−), that would correspond to q p 6= −1, φ6=0 =⇒ arg h qA(D0→K+π−)/pA(D0→K+π−) i −δKπ 6=0.

If φ is small, as experimentally confirmed, the observables x0±and y0±gives direct access to the CPV parameters q p and φ.