# Three-body charmless decays of B mesons

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## Charmless decays of B-mesons

The physics of charmless B-meson decays is very rich and gives access to a vast number of physical processes. The typical values of branching fractions of Bmeson decays into charmless hadronic nal states are below 10􀀀5, which means that large data samples (high integrated luminosity) and specic experimental techniques (regarding background rejection for instance) are needed in order to study them. Many observables can be measured in these decays, such as branching fractions, CP asymmetries, and CKM parameters, allowing to probe the dynamics of both weak and strong interactions. In particular, some B-meson decay modes are especially suited to measure precisely the angles of the unitarity triangle.
The B0 and B mesons were studied extensively at the B Factories, in which the energy of the e+e􀀀 collisions was chosen to be at the (4S) resonance peak, leading to the production of a large amount of B+B􀀀 and B0B0 pairs in a rather \clean » environment. Apart from a few runs at the (5S) resonance in the Belle experiment, giving access to a small sample of B0 s mesons, the B0 s sector remained largely unexplored by the rst-generation of B Factories. Colliders running at a higher energy, such as LEP, Tevatron, and the LHC, are (or were) able to produce, in addition to B0 and B mesons, more massive states such as B0 s mesons and b-baryons at the cost of a more complicated environment. In particular, the LHCb experiment is well suited to study charmless decays of both B0 and B0.
CKM parameters, such as the mixing-angle s dened in Eq. (II.61). This angle is directly related to the width dierence 􀀀s through 􀀀s = 􀀀12 cos s. Parameters of B0 and B0 s oscillations are used to constrain the unitarity triangle.
For example, the oscillation frequencies (the width dierence between the \light » and \heavy » states), which are measured to be md = 506:4 1:9 ns􀀀1 and ms = 17:757 0:021 ps􀀀1 [38], provide strong constraints on the angle .
Hadronic decays of B-mesons to charmless nal states may receive rather large contributions from loop processes. For example, the creation of s quarks in the nal state can only originate, in the SM, either from b ! sqq penguin diagrams (where q designates u, d or s) or b ! u(us) tree diagrams. The former involve the CKM matrix elements jVtbj and jVtsj whereas the latter involve jVubj and jVusj, and are thus suppressed due to the small value of jVubj. As a consequence, B0- (B0 s-)meson decays in which the nal state contains an odd (even) number of s quarks or kaons are favoured. Conversely to the charmless case, penguin diagrams in B-meson decays to charmed nal states are suppressed as they imply the jVcbj and jVubj CKM matrix elements. In light of the above, the measurement of CKM parameters in processes implying B-meson decays to charmed nal states is a powerful way to study SM-dominated eects, while similar measurements in loop-dominated B-meson decays open the possibility to probe new physics, as heavy virtual particles can contribute to the loop diagrams. Both approaches are important and complementary for a better understanding of the avour sector. An interesting example of this complementarity is the measurement of the angle that can be obtained in both tree-dominated and loop-dominated decay modes. A short review of the main tree-level based methods is given in part III.4.1, and the method using decays containing an important contribution from loop-diagram studied in this thesis is described in section III.4.2.

### Eective eld theory

The idea behind eective eld theory (EFT) [39] is somewhat common to most, if not all, elds of physics, that is to identify the scale of interest and build a theory so that the dierent scales of the problem decouple. Within this approach, large and short scales are usually suppressed by the power ratio of the dierent scales entering the problem. For example, the study of the motion of a relatively \light » and \slow » body can usually be performed with classical mechanics, and if a more precise result is needed, relativistic corrections can be computed. Another example is the multipole expansion in electrodynamics where a precise knowledge of the charge distribution is not needed to describe the eects at large distance.
In quantum eld theory the strategy is the same. After determining the domain of interest, i.e., the specic energy scale of the problem, one identies the relevant symmetries and degrees of freedom, constructs the corresponding Lagrangian, quantises the eld, and simplies the calculations (or make them possible). This picture is complicated by the fact that in loop-diagrams the integration over the momenta runs over all scales, which makes the decoupling of the dierent energy scales a priori impossible. However, this apparent complexity can be overcome by using regularisation techniques [40, 41] (for example by placing a cuto at a specic energy). As mentioned previously, very distinct energy scales appear in the physics of B-meson decays, mu;s;d QCD mb mW; QCD is the characteristic scale of the strong interaction below which perturbative calculations are no longer valid; its value is approximately 1 GeV. Weak interactions at a scale below the mass of the W boson can be described by an eective Hamiltonian1 of the form.

#### Dalitz-plane amplitude analysis

A Dalitz-plane amplitude analysis is a very powerful tool to study three-body decays. It allows to establish the resonant structure of the decay and its dynamics, to characterise intermediate resonances by measuring their masses, widths, and spins. An amplitude analysis in the Dalitz plane also gives access to a variety of CP violating observables, such as CP asymmetries. These CP asymmetries can be measured in dierent regions of the CP or integrated over the phase space. A full time-dependent Dalitz-plot analysis with tagging information (i.e. information on the avour of the decaying particle) can also be performed if the dataset is large enough. This type of analysis provides more information than a quasi-two-body analysis: the phases of the dierent contributions can be directly determined, without trigonometric ambiguities. Neutral-meson mixing can also be studied and the Dalitz-plot analysis .

Limitations and alternatives to the Isobar model

The isobar model is a convenient and powerful approximation. However, it is not completely satisfactory. First of all, the description of the Dalitz plane in terms of isobars is model dependent, and thus there is an irreducible, non-trivial, uncertainty due to the modelling. This model is also unsatisfactory from the theory point of view: indeed, the addition of more than one Breit-Wigner propagator in a channel violates the unitarity of the S-matrix, resulting in the non-conservation of the probability currents.
The accuracy of the model also depends on the knowledge of the hadronic parameters that enter the functions describing the strong dynamics (cf. the terms Fj(m2 12;m2 23) in Eq. (III.21)), which are not always well assessed by the theory or accessible from the data. Furthermore, the description of resonances is based on a quantum-mechanical approach, leading to barrier factors and angular functions that are non-relativistic. This may prove to be insucient even if relativistic corrections of these terms exist.
As mentioned before, the Breit-Wigner-like propagators are well suited fo narrow resonances, so the P- and D-waves, which usually proceed via narrowand isolated resonances, are in general rather well described by the isobar model. On the other hand, the S-wave is often more intricate and contains many overlapping broad states for which the Breit-Wigner approach does not give a satisfactory description. In addition to this, the BW parameters, such as the mass and the width, are usually reaction dependent and thus can vary between dierent decay modes.
In principle, three-body decays do not exclusively proceed via quasi-twobody decays, and long-distance eects, such as nal-state interactions (e.g. rescattering), are not (or hardly) taken into account by the isobar model. Moreover, unlike the case of charm decays, where the contribution from nonresonant (NR) component is generally rather small, NR eects dominate in many decay modes of the B meson. This can be explained by the rather large phase space accessible to the B meson. For three-body charmless B decays the maximum value for the invariant mass is of order 5 GeV whereas most of the resonances are localised on the boundaries of the Dalitz plane, between 0.5 and 2 GeV, which explains the large size of the NR component. Such a large phasespace actually contains dierent kinematic regions that correspond to distinct QCD regimes. This indicates that a unique description for the whole Dalitz plane may not be sucient.
To tackle some of these issues, several attempts have been made to improve the model, such as the addition of relativistic corrections to the lineshapes and to the angular distributions. In a recent LHCb analysis of the B ! K+K􀀀 decay mode [72], the use of a lineshape describing the \$ KK rescattering [73], together with a NR form factor accounting for partonic interaction in the nal state [74], result in a good description of the data and of the large CP asymmetry observed in the low invariant mass region.
Other approaches than the isobar model have been developed, such as the K-matrix approximation [75, 76]. The K-matrix is hermitian, real and symmetric. It is dened in terms of the T-matrix introduced in Eq. (III.5) as K (T􀀀1 + i1)􀀀1. This formalism is still model dependent and is based on approximations (the S-wave is supposed not to interact with the rest of the nal state particles) but it has the advantage of preserving unitarity. Since the K-matrix approximation is more complex than the isobar model, it is generally used to parameterise the S-wave only, especially when it contains many broad, overlapping resonances; the other resonances being described using isobars. The improvement of Dalitz-plot analysis techniques can only be achieved via a joint eort between the theory and experimental communities. Previous LHCb results showing a large CP violation at low invariant mass of B ! hh0h00 decays [77], whichcannot be explained only by resonant eects, generated a motivation for reaching a better understanding of the underlying physics. Another interesting recent result is the Dalitz plane analysis of B+ ! ++􀀀 performed by LHCb [78, 79], where dierent but complementary approaches to describe the S-wave were compared: the isobar model, the K-matrix formalism, and a quasi-model-independent approach based on a binning of the phase space. The three methods proved to be consistent with each other showing the robustness of the dierent descriptions.

Extraction of the CKM angle

The expression of the angle in terms of CKM matrix elements does not involve the top quark, meaning that can be extracted from tree-level decays, which are unlikely to be aected by new physics 7. Loop processes can also be used, which yield, in general, a less precise value of . On the other hand, these processes are more sensitive to new physics. It is then possible to compare the measurements of from \tree-level » and \loop-level » processes as a probe for new physics. It has to be noticed here that, in the case of new physics, the \loop-level » values of are likely to be process-dependent and thus they are expected to dier from one decay mode to another. In this part, we will suppressed B􀀀 ! D0K􀀀 decay (right).

I Introduction
II General theoretical framework: Standard Model and CP violation
II.1 Symmetries
II.1.1 Gauge symmetries
II.1.2 Discrete symmetries
II.1.3 Realisation of symmetries and spontaneous symmetry breaking
II.2 Weak interaction
II.3 CP violation
II.3.2 CKM matrix
II.3.3 Neutral meson mixing
II.3.4 Classication of CP violating eects
II.3.5 Experimental constraints on the CKM matrix
III Three-body charmless decays of B mesons
III.1 Charmless decays of B-mesons
III.2 Eective eld theory
III.3 3-body decays
III.3.1 Three-body kinematics
III.3.2 Dalitz-plane amplitude analysis
III.3.3 Isobar model
III.3.4 Limitations and alternatives to the Isobar model
III.4 Extraction of the CKM angle
III.4.1 Extraction of the CKM angle from tree decays
III.4.2 Extraction of the CKM angle from loop decays
IV Extraction of the CKM phase using charmless 3-body decays of B mesons
IV.1 Introduction
IV.2 Description of the method
IV.3 Practical implementation of the method
IV.3.1 Implementation of the decay modes
IV.3.2 Error propagation
IV.3.3 Fitting procedure
IV.3.4 Algorithm to extract the minima from a scan
IV.3.5 Choice of points on the Dalitz plane
IV.4 Baseline results
IV.5 Systematic uncertainties
IV.5.1 Results allowing for SU(3) breaking
IV.6 Studies of SU(3) breaking
IV.6.1 Comparison of the amplitudes of B0 ! KSK+K􀀀 and B+ ! K++􀀀
IV.6.2 Fitted value of SU(3) over the Dalitz Plane
IV.7 Summary and conclusion
IV.8 Perspectives
V The LHCb experiment
V.1 The LHC
V.2 The LHCb detector
V.3 Track reconstruction
V.3.1 The vertex locator
V.3.2 The Tracker Turicensis
V.3.3 The Dipole Magnet
V.3.4 The Inner Tracker
V.3.5 The Outer Tracker
V.3.6 Track and vertex reconstruction
V.4 Particle identication
V.4.1 The Ring-Imaging Cherenkov detectors
V.4.2 The calorimeters
V.4.3 The muon stations
V.4.4 Particle identication methods
V.5 Trigger
V.6 Simulation
VI Update of the branching fraction measurements of B0 (s) ! K0 Shh0 modes
VI.1 Analysis strategy
VI.2 Analysis tools
VI.2.1 Maximum likelihood estimator
VI.2.2 Multivariate Analysis
VI.2.3 The sPlot method
VI.2.4 The TISTOS method
VI.3 Dataset, Montecarlo samples, and reconstruction
VI.3.1 Trigger selection
VI.3.2 Stripping selection
VI.4 Selection
VI.4.1 Preselection
VI.4.2 Peaking backgrounds
VI.4.3 Multivariate Analysis
VI.4.4 Particle identication
VI.5 Eciencies
VI.5.1 Generator level cuts
VI.5.2 Reconstruction and stripping eciencies
VI.5.3 Trigger eciency
VI.5.4 Preselection, vetoes and MVA eciencies
VI.5.5 PID eciency
VI.5.6 Summary of the eciencies
VI.6 Fit model
VI.7 Fit results
VI.8 Systematic uncertainties
VI.8.1 Selection
VI.8.2 Tracking
VI.8.3 PID
VI.8.4 L0 trigger
VI.8.5 B transverse momentum and pseudo-rapidity: agreement between data and MC
VI.8.6 Fit model
VI.9 Combination of the results and extraction of the relative branching fraction of B0 s ! K0 SK+K􀀀
VI.10 Summary and conclusion
VIIConclusion
Appendices

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