Study of the penguin-dominated decay B0 s - K 0K 0 at LHCb

UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Departamento de F´ısica de Part´ıculas Instituto Galego de F´ısica de Altas Enerx´ıas TESE DE DOUTORAMENTO Study of the penguin-dominated decay B0 s→K∗0K∗0at LHCb Presentada por Paula ´ Alvarez Cartelle Dirixida por Prof. Bernardo Adeva Andany e Prof. Abraham Antonio Gallas Torreira Santiago de Compostela, Setembro de 2014 Study of the penguin-dominated decay B0 s→K∗0K∗0at LHCb Author: Paula ´ Alvarez Cartelle Advisor: Bernardo Adeva Andany Advisor: Abraham Antonio Gallas Torreira The following web-page address contains up to date information about this dissertation and related topics: http://fpmac116.usc.es/twiki/bin/view/Main/PaulaAlvarezCartelle Text printed in Santiago de Compostela First edition, September 2014 D. BERNARDO ADEVA ANDANY, Catedr´atico de F´ısica At´omica, Molecular e Nuclear da Universidade de Santiago de Compostela e D. ABRAHAM ANTONIO GALLAS TORREIRA, Profesor Contratado Doutor de F´ısica At´omica, Molecular e Nuclear da Universidade de Santiago de Compostela, informan: Que a memoria titulada “Study of the penguin-dominated decay B0 s→K∗0K∗0 at LHCb” foi realizada por Paula ´ Alvarez Cartelle baixo a s´ua direcci´on e que constit´ue a Tese que presenta para optar ao Grado de Doutora en F´ısica. Santiago de Compostela, a 1 de Setembro de 2014. Asdo. Bernardo Adeva Andany Asdo. Abraham Antonio Gallas Torreira Asdo. Paula ´ Alvarez Cartelle Acknowledgements First of all, I would like to thank my supervisors, Bernardo Adeva and Abraham Gallas, for their support and time dedicated to my learning. Without them this thesis would not have been possible. My special gratitude goes to Diego Mart´ınez, for teaching me lots of things about LHCb and data analysis in general, and for trusting my work from the first day. To him and Xabier Cid, I also thank their support and friendship during these years. I want to show my gratitude to the people involved in the analysis of B0 s→K∗0K∗0: Laurence Carson, Xabier Cid, ´ Alvaro Dosil, Diego Mart´ınez, Antonio Romero, Juan Saborido, Brais Sanmart´ın, Cibr´an Santamarina and, specially, to Celestino Rodr´ıguez, for openning the path and making possible the discovery. Thanks to Robert Fleischer, Joaquim Mat´ıas, David London and Alakabha Datta for useful conversations about this study and to my colleagues in the BnoC Working Group, in particular to Stephane Monteil and Marc Grabalosa. I would also like to acknowledge the work of all the members of the LHCb and LHC collaborations for making possible this thesis. I thank the finantial support provided by the “Programa de Formaci´on de Personal Investigador” of the “Ministerio de Ciencia e Innovaci´on”. Finally, I would like to thank all of the people with whom I had the pleasure to work with at the University of Santiago. A special word for my fellow PhD students during these years: ´ Alvaro, Brais, Carlos and V´ıctor, and those who already passed on to a “better life”: Diego, Lucas, Xabier and Pablo. Thanks to Marcos, for his useful advice in so many different subjects, to Eliseo for his emergency seminar about silicon detectors, to M´aximo for letting me give good marks to all of his students, and to the rest of the members: Jose ´ Angel, Pablo, Antonio, Mar´ıa and Juli´an. It was a pleasure to work with such an excellent group of people. Nun ´ambito m´ais persoal, quero agradecer a todos os amigos que me aturaron durante estes anos e me animaron a seguir, en especial a Jacobo e Marta, por apoiarme sempre. Aos compa˜neiros de Santiago, en particular a Daniel, Ricardo e Jose, por alegrar a vida do doutorando. A Jorge, por seguir a´ı. Gracias a meus pais, Mar´ıa Rita e Efr´en, ´a mi˜na irm´a Marta, ´a mi˜na avoa Lola, ´a mi˜na Tata (Mar´ıa del Mar), ´a mi˜na Pepa (Mar´ıa Jos´e), a Vito e a Ester polo seu amor e apoio incondicionais, sen os cales non ter´ıa chegado ata aqu´ı. Gracias a Daniel, pola s´ua paciencia e axuda, e por facerme sorrir cada d´ıa. Por ´ultimo gustar´ıame dedicar esta tese ´a memoria de Antonio Cartelle Rodr´ıguez, de quen valent´ıa e bondade te˜no sempre presentes. Paula “Now, if you’ll only attend and not talk so much, I’ll tell you all my ideas about Looking-glass House. First, there’s the room you can see through the glass –that’s just the same as our drawing room, only the things go the other way. I can see all of it when I get upon a chair –all but the bit behind the fireplace. Oh! I do so wish I could see that bit!” Through the looking-glass, and what Alice found there, Lewis Carroll Abstract This thesis is devoted to the search for and subsequent analysis of the decay channel B0 s→K∗0K∗0, using the first data acquired by LHCb during 2010 and 2011 from the LHC proton-proton collisions at a centre-of-mass energy √s= 7 TeV, corresponding to an integrated luminosity of 37 pb−1and 1.0 fb−1respectively. As a result of the work presented here, the first observation of the B0 s→ K∗0K∗0decay was performed. With the extended data sample collected in 2011, a combined angular and mass analysis was carried out, in order to asses the longitudinal polarisation fractions of the K∗0, obtaining fL= 0.201 ± 0.057(stat.)±0.040(syst.). The branching fraction for this decay was also measured to be B(B0 s→K∗0K∗0) = (10.6 ±1.8(stat.) ±1.0(syst.) ±0.6 (fd/fs)) ×10−6, in agreement with the Standard Model prediction. All of the triple product and CP direct asymmetries measurable in the timeintegrated and flavour-untagged analysis of B0 s→(K+π−)(K−π+) were also determined. No signs of CP-violation were found, in agreement with the Standard Model expectation. 1 Introduction The Standard Model (SM) of Particle Physics is the most successful theory to explain the elementary particles composing the universe and the interactions among them, with the exception of gravity. Based on three fundamental interactions, weak, strong and electromagnetic, the SM is able to describe most of the phenomena observed in Nature. However, there are still questions it can not give an answer to. It fails to explain, for example, the nature of Dark Matter, which from cosmological observations [1] is known to account for a large fraction of the visible universe. It can neither provide a explanation for the matter-antimatter unbalance or the neutrino oscillation. Due to these issues, together with its lack of description of gravity, the SM is considered an effective theory of a more general picture which is attempted to be probed with different High Energy Physics (HEP) experiments. Currently, the most important experiments trying to discover physics beyond the SM are located at CERN, as part of the Large Hadron Collider (LHC) complex. The LHC collides protons at a nominal centre-of-mass energy of √s= 14 TeV (√s= 7 TeV during 2010 and 2011). These collisions are analysed by the four main experiments, ATLAS, CMS, LHCb and ALICE. The work of this thesis has been developed within the LHCb experiment, specialised in rare decays and CP -violation in the context of B-physics. ALICE is mainly devoted to the study of quark-gluon plasma in heavy ion collisions delivered by the LHC in special runs. ATLAS and CMS, are general purpose detectors, that, among other measurements, have performed the discovery of the last particle of the SM that remained unobserved, the Higgs boson. Some of the most remarkable LHCb results so far are in the context of CP -violation. Understanding the origin and mechanism of CP -violation is a key question in Particle Physics. In the SM, it is described by the CKM mechanism, which, although successfull in explaining the current experimental data, is known to generate insufficient CP -violation to originate the baryon asymmetry of the Universe. In the study of CP violation, Flavour Changing Neutral Current (FCNC) processes are of special interest, since new particles beyond the SM may enter the loops mediating them. In the context of B-physics two types of b→qFCNC transitions are commonly studied: neutral Bmeson mixing and loop-mediated Bdecays. B0 s→K∗0K∗0is one of the later processes. This decay into two light vector mesons (V) proceeds solely through penguin b→sdiagrams in the SM. B→V1V2transitions are actually three different decays, since the two vector mesons can have orbital angular momentum l= 0,1,2 in the final state. The B0 s→K∗0K∗0partial width arises, thus, from three helicity amplitudes that are mainly determined by the chiral structure of the quark 1 INTRODUCTION operators, both in the electroweak and QCD sectors, assuming no additional contributions from new physics (NP). The interplay between the various penguin contributions has been studied in QCD, and predictions in the framework of QCD factorization from [2] are (9.1+11.3 −6.8)×10−6for the branching ratio and fL= 0.63+0.42 −0.29 for the fraction of the total amplitude which is longitudinally polarized. According to the same paper, predictions improve to (7.9+4.3 −3.9)×10−6and fL= 0.72+0.16 −0.21, respectively, when experimental input is used (mainly from B→K∗φ). The interest in B0 s→K∗0K∗0for precision CP-violation studies has been analysed by several authors [3–6]. The main advantage of this decay stems from the fact that its exact U-spin rotated channel B0→K∗0K∗0(b→d) is also accesible to LHCb with identical experimental properties. This potentially allows full assesment of all relevant SM penguin amplitudes at lowest order and next-to-leading order, as a first step to carry out precision investigation of the CP-violating electroweak phase, φb→sd ¯ d s[6,7]. Moreover, even with an untagged and time integrated analysis of B0 s→K∗0K∗0,CP -violation and T-violation may arise from four triple products asymmetries and four direct-like asymmetries measurable from the terms corresponding to the interference among the different amplitudes [8]. The goal of this thesis is the study of B0 s→K∗0K∗0with the first two years of data taken by the LHCb experiment at the LHC. The structure of the thesis is the following. Chapter 2overviews the most important theory aspects for the search and further study of this process. These include a brief explanation of the SM and its predictions on B0 s→ K∗0K∗0. The LHCb experiment is presented in Chapter 3, where the detector and the experimental conditions during 2010 and 2011 are described. A detailed explanation on how to obtain the decay rate as a function of the decay angles and the K∗0masses for the transition B0 s→K∗0K∗0(and the scalar final state contributions) is given in Chapter 4. Chapter 5contains the analysis that lead to the discovery of the decay with the first 37 pb−1of data recorded by LHCb in 2010. The analysis of 2011 data is presented in Chapter 6, where the higher statistics (1.0 fb−1) allowed to perform a precise determination of the polarisation fractions of the decay through a mass-dependent angular analysis. This chapter also includes the search for NP perfromed by measuring the eight CP -violating quantities accessible to the untagged sample. The B(B0 s→K∗0K∗0) measurement is also reported. Finally, conclusions are drawn in Chapter 7. 2 2 Theory overview 2.1 Standard Model The Standard Model (SM) is a quantum field theory (QFT) based on strong and electroweak (EW) interactions. The SM structure is described in this section following [9–11]. The strong interactions are described by Quantum Chromodynamics (QCD) corresponding to the symmetry group SU(3)Cof color (C), while the EW interaction is described by the group SU(2)T⊗U(1)Yof weak-isospin (T) and hypercharge(Y), being then SU(3)C⊗SU(2)T⊗U(1)Ythe full group of gauge symmetry for the SM. GSM =SU(3)C⊗SU(2)T⊗U(1)Y(2.1) This symmetry is spontaneously broken into SU(3)C⊗U(1)EM by the vacuum expectation value (VEV) of (the neutral component of) a scalar isospin doublet, with hypercharge 1/2, called Higgs GSM Higgs(1,2)1/2 −−−−−−−−→ SU(3)C⊗U(1)EM As a result of the interaction with the Higgs field, EW bosons combine into the massive particles W±and Z0and the massless photon. The interaction with Higgs gives also masses to the fermions. Each fermion generation, out of a total of three, has five representations of the SM gauge symmetry QL,i (3,2)+1/6UR,i (3,1)+2/3DR,i (3,1)+1/3LL,i (1,2)−1/2ER,i (1,1)−1 The subscript number is the hypercharge, and the numbers in parenthesis indicate if it acts as a triplet or singlet in SU(3)Cand as a doublet or singlet in SU(2)T. The subscript i = 1,2,3 indicates fermion generation. The EW symmetry breaking (EWSB) and the effects induced by Higgs field such like CP violation and flavor depending processes are explained in Sect. 2.1.1. The fermion and boson content of SM is explained in more detail in Sect. 2.1.2. Thus, the SM lagrangian can be decomposed in three parts L=LKin +LHiggs +LY uk (2.2) where the kinetic part includes the corresponding covariant derivative to preserve the gauge invariance, the Higgs part includes Higgs self interactions and Yukawa part includes Higgs-fermion interactions. 3 Chapter 2. Theory overview 2.1.1 Mass generations and eigenstates A lagrangian containing only the terms of the gauge symmetry is not enough to build a model where the particles are massive. The gauge bosons are massless if the symmetry is unbroken, and masses for the fermions as self-interactions such like ΨLΨR(Dirac mass) or ΨLΨL(Majorana mass) would explicitly break the SU(2) symmetry. Non-abelian broken gauge theories are not renormalisable, thus in the SM the masses of the EW gauge bosons and the fermions are generated via spontaneous symmetry breaking. 2.1.1.1 Boson masses and EWSM The spontaneous symmetry breaking is achieved by the introduction of the Higgs, a scalar isospin doublet with hypercharge +1/2, φ=φ+ φ0(2.3) This doublet has a self interaction of the form LHiggs =µ2φ†φ−λ(φ†φ)2(2.4) The first term is similar to a mass one, but with opposite sign. Such quadratic potential does not minimise at 0 (see example in Fig. 2.1), and thus acquires a VEV ν=µ/λ. hφi0=1 √20 ν(2.5) Figure 2.1: Higgs-like potential. The minimum is not at 0, and therefore the potential has a VEV. The VEV gives masses, through the Higgs kinetic term plus the Higgs self-interaction Lagrangian, to the following boson combinations: W± µ=1 √2W(1) µ∓W(2) µ→MW=g·ν 2, Z0 µ=1 pg2+g02gW (3) µ−g0B(2) µ→MZ=qg2+g02·ν 2(2.6) 4 2.1 Standard Model From the degrees of freedom of the original Higgs field, φ=φ+ φ0=1 √2G+ 1+iG+ 2 ν+ (H0+iG0 3)(2.7) H0will be a massive scalar particle, having the massless Goldstone bosons Gi“eaten” by the gauge bosons W±and Z0, giving rise to their longitudinal polarisations and masses. The Higgs boson, H0, the last particle in the SM to be observed, has recently been discovered at the LHC with a mass of 125 GeV [12,13]. 2.1.1.2 Fermion masses and CKM matrix In order to give masses to the fermions, the corresponding couplings between them and the Higgs field are added, while keeping the Lagrangian SU(2) invariant. For example, for a single generation ∆L=−λe¯ ELφER−λd¯ QLφDR−λuab¯ QLaφ† bUR+h.c. (2.8) Substituting the VEV, the fermion masses have the form me=ν·λe √2mu=ν·λu √2md=ν·λd √2.(2.9) These λiare inputs to the SM and thus allow having very different masses for different fermions. When the three fermion generations are added to the theory, additional terms mixing quarks of different generations are possible. Alternatively, it is possible to diagonalise the Higgs couplings by switching to a different basis for the quark fields. Writing the Lagrangian in this alternative basis (hereinafter referred to as “mass basis” or “physical basis”) will of course simplify LY uk but with the cost of causing a complication in the gauge side. Calling qthe interaction eigenstates and q0the mass eigenstates, both bases are related through the unitary relations ui L=Uij uu0j Ldi L=Uij dd0j L(2.10) and thus the weak current ¯ui Lγµdi Ltransforms to u0i Lγµ(U† uUd)ij d0i L≡u0i LγµVCKM ij d0i L. Being VCKM called the CKM matrix [14,15] (from Cabibbo-Kobayashi-Maskawa). Its coefficients are usually written as VCKM =  Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb  (2.11) VCKM is not diagonal (the experimental value of the coefficients can be found in [16]) and such structure allows transitions between the different quark generations, giving rise to processes in which quarks change flavour without changing its electric charge. These processes are called Flavour Changing Neutral Currents (FCNC) and in particular include the decay B0 s→K∗0K∗0.CP violation (CPV) also arises from the non-diagonal structure of VCKM, requiring, in addition, the presence of three different generations (see Sect. 2.2.4). Equivalently, if VCKM were the identity matrix CP violation and FCNCs would not exist within the SM. 5 Chapter 2. Theory overview B0 s s B0 s bs b WW u, c, t u, c, t Vtb V∗ ts V∗ ts Vtb B0 s s B0 s bs b W W u, c, t u, c, t Vtb V∗ ts V∗ ts Vtb Figure 2.2: Diagrams contributing to B0 s-B0 soscillation. Another particular but very important example of process arising from the fact that VCKM is different from the identity matrix is the oscillation of neutral mesons composed by quarks of different generations. The off-diagonal terms of CKM matrix allow particles such like D0,K0,B0or B0 sto perform particle-antiparticle oscillations (see Fig. 2.2 for examples of diagrams involving Bs oscillation). Neutral mesons oscillation will be explained in more detail in Sect. 2.2.1. 2.1.2 Elementary particles 2.1.2.1 Fermions The Standard Model fermions can be divided in two groups, depending on whether they are affected by strong interaction (quarks) or not (leptons). Each quark has three possible color states and (at low energy) only exists in bound states of color singlets, called hadrons. Hadrons are then composed by quarks (and gluons, the gauge bosons of QCD), being the most common states quark - antiquark (mesons), and three quarks (baryons). Due to spin addition, baryons are also fermions, while mesons are bosons. Leptons are e,µ,τand a neutrino (ν) for each one. In the SM neutrinos are massless particles so that their helicity becomes equivalent to chirality. This means that there are not right-handed neutrinos in the SM and, equivalently, there are not left-handed antineutrinos1. 2.1.2.2 Bosons Apart from mesons, the SM contains the gauge bosons corresponding to strong and EW interactions, and Higgs (H0) boson, responsible for the masses of SM particles. The gauge bosons of QCD are massless particles of spin 1, called gluons, and have eight possible color states. QCD couplings have the property of becoming small at high energies (or small distances); this effect is known as “asymptotic freedom”. The gauge bosons corresponding to SU(2)T⊗U(1)Yare Wi µ(i= 1,2,3) and Bµ, for SU(2) and U(1) respectively, and the four should be massless in order to conserve the gauge symmetry. However, the symmetry breaking induced by the Higgs field changes them into W+,W−,Z0and photon (Aµ), where only the photon is massless. All of them have spin 1. 1It is now known experimentally that neutrinos suffer flavour oscillations, which means they are not massless. This can be fitted into the SM with small modifications to it. 6 2.2 CP violation and neutral meson decays Table 2.1: SM fermions. T T3Y Q Leptons νe,νµ,ντ1 2 1 2-1 0 eL,µL,τL1 2-1 2-1 -1 eR,µR,τR0 0 -2 -1 Quarks u0L,c0L,t0L1 2 1 2 1 3 2 3 u0R,c0R,t0R0 0 4 3 2 3 d0L,s0L,b0L1 2-1 2 1 3-1 3 d0R,s0R,b0R0 0 -2 3-1 3 2.2 CP violation and neutral meson decays The CP transformation combines charge conjugation C with parity P, which means this operation changes particles into anti-particles. If CP were and exact symmetry, the laws of Nature would be the same for matter and for antimatter. It is observed that most phenomena are indeed CP -symmetric. In particular, this symmetry is respected by electromagnetic and srong interactions. The weak interactions, on the other hand, violate C, Pand also CP .CP violation was first discovered in neutral Kdecays in 1964 [17]. Today we have more evidences of CP violation in the B0-system, measured at the B-factories at the turn of the 2000’s [18,19], and the recently observation of CPV in the B0 s-system by LHCb [20]. CP symmetry is broken in the SM by complex phases in the Yukawa couplings, through a single CP -violating parameter in the CKM matrix. This description of CP violation, known as the Kobayashi-Maskawa (KM) mechanism, agrees with all measurements to date. However, the current level of experimental accuracy and the theoretical uncertainties involved in the interpretation of the various observations leave room for additional (beyond the SM) sources of CP violation. Moreover, the observed baryon asymmetry of the universe, i.e., the fact that there is much more matter than antimatter in the observed universe, could only be generated from an initial situation in which the amounts of matter and antimatter would be equal if there is CP violation. This fact was first shown by Andrei Sakharov in 1967 [21], who pointed out that baryon-number violation, Cand CP violation, and a departure from thermal equilibrium, are all necessary in order for it to be possible to generate a net baryon asymmetry in the early universe. Despite the phenomenological success of the KM mechanism, it seems to fail to accomodate the observed baryon asymmetry [22]. On the other hand, many extensions of the SM provide new sources of CP violation, so the search for physics beyond the SM is much in contact with the study of CP violation. Experimentally, a natural place to search for CP violation is meson decays. In particular, the study of neutral meson systems provides a lot of information about the nature of CP violation, to the extent that CP violation can be quantified even from CP conserving processes. In this section, the formalism and basic physics that are relevant to the measurements of CP violation in neutral meson system are explained. 7 Chapter 2. Theory overview 2.2.1 Neutral mesons systems 2.2.1.1 The effective Hamiltonian: Weisskopf-Wigner approximation Let P0refer to any neutral meson: K0,D0,B0or B0 s.P0and P0have opposite internal quantum number (flavour quantum number) F, which is conserved by strong and electromagnetic interactions, ∆F= 0, and not conserved by the weak interaction, ∆F6= 0. If only strong and electromagnetic interactions existed, P0and P0would be a stable particle-antiparticle pair with common mass m0. Because of the weak interaction, P0and P0decay. Moreover, P0↔P0transitions are also possible as a one-step process (∆F= 2) or through an intermediate state (second order ∆F= 1). The full hamiltonian for this system can be written as H=H0+HW,(2.12) where H0is the flavour invariant Hamiltonian and HW, treated as a perturbation, contains the weak interaction. Consider an inicial state which is a mixure of a P0and a P0, |ψ(t= 0)i=c|P0i+ ¯c|P0i.(2.13) At t > 0 this state will evolve in two different ways: oscillation between P0and P0and decays into lighter particles. The evolved state can be described by |ψ(t)i=c(t)|P0i+ ¯c(t)|P0i+X n cn(t)|ni(2.14) where |nirepresents any decay mode of the original mesons and tis the time measured in the P0-P0rest frame. It is possible to simplify the resolution of this problem if  only the values of c(t) and ¯c(t) are of interest, i.e., if only the time evolution of the projection of the full state, |ψ(t)i, on the two-dimensional subspace of the P0-P0 system is desired,  the considered times tare much larger than the typical strong-interaction scale. Under these assumptions, known as the Weisskopf-Wigner approximation, the system can be described by a Schr¨odinger-like equation [23] id dt |c(t)i |¯c(t)i=Hef f |c(t)i |¯c(t)i(2.15) where the effective Hamiltonian Hef f is not hermitian, else the mesons would just oscillate, they would not decay1. It is given by Hef f = hP0|Hef f |P0i hP0|Hef f |P0i hP0|Hef f |P0i hP0|Hef f |P0i!=M−i 2Γ=M11 −i 2Γ11 M12 −i 2Γ12 M∗ 12 −i 2Γ∗ 12 M22 −i 2Γ22. (2.17) 1Since Hef f is not hermitian, the probability to observe either P0or P0is not conserved, but goes down with time, d dt |c(t)|2+|¯c(t)|2=−(c(t)∗¯c(t)∗)Γ c(t) ¯c(t)!(2.16) 8 2.2 CP violation and neutral meson decays where Mand Γare 2 ×2 hermitian matrices. The explicit matrix elements for Mand Γ are given, in second-order perturbation theory by [24] (i, j = 1,2) Mij =m0δij +hi|H(∆F=2) W|ji+X n P"hi|H(∆F=1) W|nihn|H(∆F=1) W|ji m0−En#, Γij = 2πX nhi|H(∆F=1) W|nihn|H(∆F=1) W|jiδ(m0−En) (2.18) where Pstands for the principal part prescription, and the sum runs over all intermediate states n. m0and Enare the energies in the centre of mass frame defined as H0|P0i= m0|P0i,H0|P0i=m0|P0iand H0|ni=En|ni. Assuming CP T as a symmetry of HWleads to the following relations M11 =M22 ≡M Γ11 =Γ22 ≡Γ(2.19) The states |P0iand |P0iare eigenstates of H0but not of HW. Therefore these are not physical states (or mass eigenstates) with the corresponding consequence of mixing and decay. A new basis {PL, PH}which diagonalizes Hef f defines two fields that do not oscillate, just decay. As Hef f is not herminian, its eigenstates will not be ortogonal. Using the shorthand notation B=q(M12 −i 2Γ12)(M∗ 12 −i 2Γ∗ 12), the eigenvalues of Hef f can be shown to be Hef f |PLi= (M−i 2Γ+B)|PLi ≡ (ML−i 2ΓL)|PLi(2.20) Hef f |PHi= (M−i 2Γ−B)|PHi ≡ (MH−i 2ΓH)|PHi(2.21) where the real and imaginary parts have been grouped in the definition of ML,MH,ΓL and ΓHrespectively. The corresponding eigenstates are |PLi=p|P0i+q|P0i |PHi=p|P0i−q|P0i(2.22) with q p=±sM∗ 12 −i 2Γ∗ 12 M12 −i 2Γ12 (2.23) The state |PLiis the mass eigenstate with mass MLand lifetime ΓL. Similarly the mass MHand the lifetime ΓHare defined for state |PHi. The sign of q/p determines whether |PLior |PHiis heavier. It is useful to define ∆m≡MH−ML= 2<(B) ∆Γ≡ΓL−ΓH= 4=(B) (2.24) The common convention is the choice ∆m > 01, wich implies the positive sign in (2.23). Note that this choice does not imply anything for the sign of ∆Γ. 1This choice gives also meanig to the notation of Hand Lto denote the “heavy” and “light” eigenstate 9 Chapter 2. Theory overview d sl a -0.04 -0.02 0 0.0 2 s sl a -0.04 -0.02 0 0.02 LHCb D0 D0 (4S) HFAGΥ D0 d sl a -0.04 -0.02 0 0.0 2 s sl a -0.04 -0.02 0 0.02 Figure 2.5: Measurement of semileptonic Bdecay asymmetries. The bands correspond to the central values ±1 standard deviation. The solid dot indicates the SM prediction. be parameterized by one angle and three phases. However, three of those phases can be absorbed by rephasing the four quark fields. This means that for 2 fermion families, the CKM matrix can always be chosen to be real. VC=cos θCsin θC sin θCcos θC(2.60) where θCis the Cabibbo angle [14]. The conclusion is that with only two families, the SM can not account for CP violation. This argument led to the prediction of the third family of quarks, once CP violation was observed. For three generations of quarks, unitarity reduces the number of independent parametres from 18 to 91, by applying the set of constraints 3 X k=1 V∗ ki Vkj =δij .(2.61) These constraints are usually pictured as triangles in the complex plane, the unitarity triangles. For example, the first and the third columns of the CKM matrix, i.e., the band dsectors, can be used to build the unitarity relation V∗ ud Vub +V∗ cd Vcb +Vtd ∗Vtb = 0,(2.62) which can be represented as the triangle shown in Fig. 2.6, with its sides normalized to V∗ cd Vcb. 1An×ncomplex matrix, Un×n, contains 2n2parameters. If the matrix is unitary, the number of independent parameters decreases to n2. 16 2.2 CP violation and neutral meson decays Figure 2.6: Unitarity triangle defined by (2.62). From the remaining independent parameters of VCKM, five of them can be absorbed in the redefinition of the quark fields. The resulting four free parameters are tree rotation angles, the quark mixing angles θij , and one complex phase δ. Therefore, the SM with three families predicts CP violation provided this phase is not zero. In terms of these parameters, VCKM =  c12c23 ss2c13 s13e−iδ −s12c23 −c12s23s13eiδ c12c23 −s12s23s13eiδ s23c13 s12s23 −c12c23s13eiδ −c12s23 −s12c23s13eiδ c23c13  ,(2.63) where sij = sin θij and cij = cos θij . The interactions between quarks of different generations are scaled by the appropriate CKM matrix elements, which means that a certain quark transition is more or less favorable depending on the value of the CKM element involved. To clearly show this hierarchy, it is useful to use the Wolfenstein parametrization [16], which allows to write the CKM matrix as an expansion on s12 ≃0.22, VCKM =   1−λ2 2λ Aλ3(ρ−iη) −λ1−λ2 2Aλ2 Aλ3(1 −ρ−iη)−Aλ21  +O(λ4),(2.64) with λ,A,ρand ηdefined by s12 =λ=|Vus | p|Vud |2+|Vus|2, s23 =Aλ2=λ Vcb Vus  , s13eiδ =Aλ3(ρ+iη) = V∗ ub. (2.65) In this parameterization, the complex CP violating phase is γ≡arg V∗ ud Vub V∗ cd Vcb = arg(ρ+iη) + O(λ4) (2.66) which up to O(λ4) is localized in Vub =|Vub|e−iγ and Vtd =|Vtd |e−iβ, and the rest of the entries are real. βis defined as β≡arg V∗ cd Vcb V∗ td Vtb = arg(1 −ρ+iη) + O(λ4),(2.67) and it is zero if γis zero. Therefore, CP is not conserved in the SM provided γ6= 0, and the size of this violation in related to the area o the triangle shown in Fig. 2.6. 17 Chapter 2. Theory overview Figure 2.7: Current best fit for the unitarity triangle defined by (2.62) by the CKM-fitter collaboration. Experimentally, sides and angles of this triangle (and analog triangles corresponding to different unitarity relations) can be determined by measuring the decay rates and CP- violating asymmetries of processes involving those CKM elements. Fig. 2.7 shows the current experimental situation for the unitarity triangle defined by (2.62), from [35]. 2.2.4.1 Predictions for Bd,s mixing angles in the SM As it has been discussed in a previous section, the experimental data reveals that ∆Γ ∆Mfor both B meson systems. This result implies1that also |Γ12|  |M12|, and then q p=sM∗ 12 −1 2Γ∗ 12 M12 −1 2Γ12 ≃sM∗ 12 M12 ≡e−iφM(2.68) This means that in order to compute the mixing angles and the mass differences, it is enough to compute the mixing parameters Md,s 12 . As stated in Sect. 2.2.1.1,M12 contains contributions from H(∆F=2) Wand also from transitions with intermediate on-shell states at second order in H(∆F=1) W. Fortunately, in the SM the mixing of Bmesons is dominated by the ∆F= 2 box diagrams with a top quark in the loop, shown in Fig. 2.2 for the B0 s system (equivalen diagrams for B0system, exchanging each s-quark by a d-quark). 1From (2.24) and the definition of B, it can be shown that (∆M)2−1 4(∆Γ)2= 4|M12|2−|Γ12|2 ∆M·∆Γ= 4<(M12Γ∗ 12) 18 2.3 The B0 s→K∗0K∗0decay The CKM structure of these contributions is (Md 12)∗∝(Vtd V∗ tb)2 (Ms 12)∗∝(Vts V∗ tb)2. Consequently, it is easy to see that the B0-B0mixing angle in the SM is given by φSM d= 2β+O(λ4).(2.69) The analogous angle for the B0 s-system, φSM s, is zero at this level of approximation, which means that enters in terms which are suppressed by at least λ4. However, the phase itself is O(λ2). The angle βsis defined as βs≡ −arg −Vts V∗ tb Vcs V∗ cb .(2.70) Up to O(λ6), only Vts acquires a phase, so Vts =−|Vts |e−1βs. In fact, βS≃ −λ2η. Then, the B0 s-B0 smixing angle in the SM is given by φSM s= 2βs+O(λ6).(2.71) This is the way in which the mixing angles are related to the angles of the unitarity triangles in the Standard Model. 2.3 The B0 s→K∗0K∗0decay The B0 s→K∗0K∗0decay proceeds through a flavour changing neutral current (FCNC). In particular, in this process a bquark transforms into a squark. This kind of processes do not arise at tree level in the SM. The reason is that in the SM, the couplings of the quarks to the neutral Z0boson are flavour-diagonal –the terms with the neutral boson have the form ¯uLγµuLZµ–, and the FCNC transitions can only occur at higher orders in perturbation theory, in loop processes, such as penguin and box diagrams, see Fig. 2.8. These processes allow to probe physics at high energies through the virtual particles entering the loops. This feature makes them suitable for searching physics beyond the SM, which may introduce new heavy particles that affect the observables related to these transitions. In particular, time-dependent CP asymmetries in b→smodes, like B0 s→K∗0K∗0, are considered a very sensitive probe of New Physics (NP) [5]. The study of B0→φK0 s, for example, has already shown tensions with the SM predictions1[36,37]. However, the lack of a model-independent evaluation of the theoretical error is a strong limitation for a complete and meaninful test of the SM. This is not the case of the B0 s→K∗0K∗0, where the theoretical error can be controlled with high accuracy by using the U-spin mirror channel B0→K∗0K∗0[3]. In the next section, a brief introduction on the main theoretical tools used in the study of Bphysics is given. Specific predictions for the B0 s→K∗0K∗0mode based in these approaches are given in the following sections. 1CP mixing asymmetry has been measured to be S(Bd→φK0)=0.39 ±0.18, deviated from the theoretical prediction sin(2β) = 0.675 ±0.026 19 Chapter 2. Theory overview q q W bu, c, t s (a) b→sq ¯qpenguin diagram. b s u, c, t νµ W W µ µ (b) B0 s→µ+µ−box diagram Figure 2.8: Effective flavour-changing neutral current processes. 2.3.1 Factorisation and Effective field theories In general, the term “Bphysics” refers to the study of weak decays of bquarks. However, the confinement properties of QCD forbids quark level transitions, such as b→s, to be measured directly. Instead, the decays of Bmesons (or baryons) will be observed experimentally. This means that an undersanding of the conection between quark and hadron properties is required in order to determine the parameters of the weak sector of the SM. Equivalently, (at least) two different energy scales become relevant to the problem: the scale of weak interactions, given by the mass of the Wboson, MW, and the hadronic scale ΛQCD. Fortunately, in this kind of physical process, where contributions come from two (or more) widely separated energy scales, it is possible to study the low-energy (long-distance) phenomena independently of the details of the high-energy (short-distance) interactions. At this point it is also useful to introduce a simpler description of the relevant features of the full theory at a given energy scale in the form of an effective theory. In the case of Bdecays, the large scale of weak interactions with respect to the mass of the Bmesons (mBMW) motivates the use of the weak effective Hamiltonian. Additional intermediate scales can also be considered. For example, the large mass of the Bmeson compared with the low scale of the strong interaction inside hadrons allows the definition of the Heavy Quark Effective Theory [38]. 2.3.1.1 The weak effective Hamiltonian Effective field theories [39] are used to express a full, complete theory as an effective Hamiltonian constructed from a set of local operators Oiin which the high energy degrees of freedom, defined with respect to a mass scale Λ, have been integrated out. The weak effective Hamiltonian describes weak interactions at low energy, below MW, and is defined so that the amplitude of a weak process i→fis expressed as a sum of local operator amplitudes, A(i→f) = hf|Hef f |ii=GF √2X i ViCi(µ)hf|Oi|ii(2.72) where GFis the Fermi constant characterizing the strength of the underlying weak processes, Viare the suitable CKM matrix elements for the quark transition and Oiare the local operators forming a complete set for a given transition. The coeficients associated with the local operators, Ci, are called Wilson coefficients and inlcude the effects of interactions at scales higher than µ. The choice of µis arbitrary, usually chosen to be 20 2.3 The B0 s→K∗0K∗0decay O(mb) when dealing with Bdecays. The long distance interactions, on the other hand, are contained in the matrix elements of the local operators. Wilson coefficients are calculated by matching the prediction of the effective theory with the assumed full theory at high mass scale, µ∼MW; at this scale the relevant diagrams and their QCD corrections can be calculated perturbatively and evolved down to the relevant energy scale (µ∼mb) by making use of the renormalization-group equations. After renormalization, the local operators Oican be identified within the full calculation and their corresponding Wilson coefficients can be extracted. 2.3.1.2 Hadronic Matrix elements The Wilson coefficients of the effective weak Hamiltonian are process-independent and can be used directly in the description of exclusive models. The theoretical precision is thus limited by the difficulty of evaluating the, intrinsically non-perturvative, hadronic matrix elements between initial and final state, hf|Ok|ii. The concept of factorisation of matrix elements has been used to face this problem [40]. The idea being to reduce the matrix elements to products of process-independent form factors and decay constants that could be extracted from data or calculated using non-perturvatibe techniques, such as lattice simulation. In the case of Bdecays, this factorisation can be understood in terms of the heavy quark approximation [38,39]. The dynamical simplifications which occur in the heavymass limit are usually ilustrated by considering a hadron composed of a heavy quark, Q, and other light constituents. The quarks confined inside hadrons exchange momentum of magnitude ∼ΛQCD. The mass of the heavy quark in the hadron is, by definition, MQΛQCD and its Compton wavelength λQ∼1/MQis much smaller than the hadronic size Rhad∼1/ΛQCD. To resolve the quantum numbers of the heavy quark a hard probe, q2&M2 Q, is required. However, the soft gluons coupled to the light constituents of the hadron can only resolve larger distances of order Rhadand are thus blind to the flavour and spin orientation of the heavy quark, feeling only its colour field . Therefore, in the infinite MQlimit, the properties of the heavy-light hadrons are independent of the mass and spin of the heavy source of colour. Based on this limit, a whole effective theory can be constructed as an expansion in powers of ΛQCD/m, with mthe mass of the heavy quark. This effective theory is known as the Heavy Quark Effective Theory. A complementary approach to Bdecay phenomenology relies on the approximate flavour symmetries of QCD in order to find relations between observables of different processes. Both approaches are usually combined, specially in the study of non-leptonic B decays. 2.3.2 The B0 s→K∗0K∗0in the Standard Model The decay of a B0 smeson into two light K∗0vector mesons proceeds dominantly in the SM through the penguin diagrams shown Fig. 2.9. This process has been studied in the framework of QCD factorisation, and SM predictions of its expected branching ratio (B) and longitudinal polarization fraction (fL, defined in the following section) have been provided [2]. Additionally, the use of flavour symmetries has been propossed in order to 21 Chapter 2. Theory overview K∗0 K∗0 B0 s b W+s d u, c, t d s s B0 s b W+u, c, t ssK∗0 K∗0 s d d Figure 2.9: Diagrams contributing to the decay B0 s→K∗0K∗0in the SM. Left: The coloursuppressed penguin diagram, which is expected to be the dominant contribution. Right: Penguin anihilation contribution. relate this decay to its B0counterpart for the study of CP -violation [4,5]. Next sections summarize the theoretical work related to B0 s→K∗0K∗0. 2.3.2.1 B→V V decays: Amplitudes and Observables The amplitude of a Bmeson decaying into two vector mesons V1and V2can be written, using the notation B(p)→V1(k1, 1)V2(k2, 2), as [41] AB→V V =aε∗ 1·ε∗ 2+b m2 B (p·ε∗ 1)(p·ε∗ 2) + ic m2 B µνρσpµqνε∗ρ 1ε∗σ 2(2.73) where k1,2and ε1,2are the momentum and polarization for the vector meson 1 and 2 respectively and q≡k1−k2. The amplitudes aand bare linear combinations of the amplitudes describing final states with relative angular momentum between V1and V2of L= 0,2. The amplitue ccorresponds to L= 1. Alternatively, a basis of amplitudes describing decays to final state particles with definite helicity is given by A0=k1·k2 m1m2−a+bp2 k1·k2A±=a±2p mB c.(2.74) In experimental analyses, the transversity basis, where A±are replaced by Ak= (A++ A−)/√2 and A⊥= (A+−A−)/√2, corresponding to linearly polarised final states, is preferably used . Experimentally, the magnitudes and relative phases of the various amplitudes can be extracted from the analysis of the angular distributions of the vector resonances decay products. The full angular dependence of the cascade where both vector mesons decay into a pair of pseudoscalar particles will be given in Chapter 4. The decay rate can be written as dΓ dΩ ∝A0F0(Ω) + AkFk(Ω) + A⊥F⊥Ω)|2(2.75) where Fi(Ω) are functions describing the angular distribution of the Bdaughters. Therefore, a given B→V V decay allows the definition of five observables corresponding to the three magnitudes and two relative phases of the helicity amplitudes or the five angular coefficients obtained from the expansion of (2.75). A typical set of observables consist of the branching fraction, two out of the three polarisation fractions fL,fk,f⊥, and two 22 2.3 The B0 s→K∗0K∗0decay phases δk,δ⊥, where fL,k,⊥=|A0,k,⊥|2 |A0|2+|Ak|2+|A⊥|2, δk,⊥= arg Ak,⊥ A0 (2.76) It is conventional to combine the five observables of the B→V V decay with those of its CP conjugate Bdecay. Denoting the Bdecay helicity amplitudes as ¯ Ahand the corresponding transversity amplitudes as ¯ Ak,⊥= ( ¯ A+±¯ A−)/√2, the equality Ak,⊥=¯ Ak,⊥ is true in the absence of CP violation. The CP average observables and asymmetries are thus given by fh=1 2(fB h+f¯ B h),Ah CP =f¯ B h−fB h f¯ B h+fB h (2.77) (h=L, k,⊥) for the polarisation fractions and δh=δ¯ B h−∆δh=δB h+ ∆δh(2.78) for the phase observables. 2.3.2.2 The B0 s→K∗0K∗0amplitude Following the convention by Beneke, Rohrer and Yang [2], the SM weak effective Hamiltonian mediating B0 s→K∗0K∗0transitions is given by Hef f =GF √2X p=u,c VpbV∗ ps    C1Op 1+C2Op 2+X i=3,...10,7γ,8g CiOi   +h.c. (2.79) where Op 1,2are the left-handed current-current operators arising from W-boson exchange, O3,...6and O7,...10 are QCD and electroweak penguin operators, and O7γand Q8gare the electromagnetic and chromomagnetic dipole operators. They are given by Op 1= (¯pb)V−A(¯sp)V−A, Op 2= (¯pibj)V−A(¯sjpi)V−A, O3= (¯sb)V−APq(¯qq)V−A, O4= (¯sibj)V−APq(¯qjqi)V−A, O5= (¯sb)V−APq(¯qq)V+A, O6= (¯sibj)V−APq(¯qjqi)V+A, O7= (¯sb)V−APq3 2eq(¯qq)V+A, O8= (¯sibj)V−APq3 2eq(¯qjqi)V+A, O9= (¯sb)V−APq3 2eq(¯qq)V−A, O10 = (¯sibj)V−APq3 2eq(¯qjqi)V−A, O7γ=−e 8π2mb¯sσµν(1 + γ5)Fµνb, O8g=−gs 8π2mb¯sσµν(1 + γ5)Gµνb (2.80) where ( ¯q1q2)V±A= ¯q1γµ(1 ±γ5)q2,i, j are colour indices, eqare the electric charges of the quarks in units of |e|, and the sum runs over all active quark flavours in the effective theory, i.e. q=u, d, s, c, b. In general, the matrix elements hV1V2|Oi|¯ Bi, can be calculated using the QCD factorisation approach [2,42]. In this framework they can be expressed, at leading order in the ΛQCD/mbexpansion, as hV1V2|Oi|¯ Bi=FB→V1TI i∗fV2ΦV2+ [V1↔V2]+TII i∗fBΦB∗fV1ΦV1∗fV2ΦV2,(2.81) 23 Chapter 2. Theory overview Table 2.2: CP -averaged branching fraction and longitudinal polarization fraction calculated in the framework of QCD factorisation [2]. Using experimental input from B0→φK∗0 decays the column labeled “ˆαc,− 4f. d.” is obtained. The first uncertainty is the uncertainty associated with the CKM parameters and the second one is the theoretical uncertainty. Decay B/10−6fL/% default ˆαc,− 4f. d. default ˆαc,− 4f. d B0 s→K∗0K∗09.1+0.5+11.3 −0.4−6.87.9+0.4+4.3 −0.4−3.963+0+42 −0−29 72+0+16 −0−21 B0→K∗0K∗00.6+0.1+0.5 −0.1−0.30.6+0.1+0.3 −0.1−0.269+1+34 −1−27 69+1+16 −1−20 where non-perturbative effects are cotained in the form factors FB→Vi, the decay constants fViand the meson light-cone distribution amplitudes, Φ(the star products imply an integration over light-cone momentum fractions of the constituent quarks inside the meson). The hard scattering kernels TI,II iinclude only short distance effects and can be calculated through perturbation theory. The result of computing this hard-scattering kernels for the various operators in the effective weak Hamiltonian is usually presented in terms of “factorised operators” A([¯qV1qV1][¯qV1qV1]) with coefficients αp i(V1, V2) [42]. The coeficients αp icontain all dynamical information, while the arguments of Aiencode the flavour composition of the final state (V1V2) to which a given term can contribute. The matrix elements of the factorized operators, AV1V2, are simply proportional to a form factor times a decay constant. The decay amplitude for the B0 s→K∗0K∗0decay, can thus be written as (h= 0,+,−) Ah B0 s→K∗0K∗0=X p=u,c VpbV∗ ps nAh K∗0K∗0βp,h 4+Ah K∗0K∗0[ˆαp,h 4+βp,h 4]o(2.82) where βp,h iare the coefficients corresponding to a second set of factorised operators describing weak annihilation amplitudes. From this expression, predictions for the branching ratio, B, and the longitudinal polarization fraction, fLof B0 s→K∗0K∗0have been provided [2]. These results are summarized in Table 2.2, together with the equivalent results for B0→K∗0K∗0. 2.3.2.3 CP violation in B0 s→K∗0K∗0 Consider the formalism presented in Sect. 2.2 for B0 smesons, with f=K∗¯ K∗. As the final state is a mixture of CP eigenstates, this decay is sensitive to the three types of CP violation. The decay amplitude for a certain helicity (or polarisation) in the final state can be expressed as [40] A(B0 s→K∗0K∗0) = −V∗ tbVts Ps−V∗ ubVus PGIM s.(2.83) where Pscontains the hadronic matrix elements corresponding to b→spenguins containing a t-quark loop, and PGIM srepresents the GIM-suppressed difference of contributions coming from charm and up quarks loops. 24 2.3 The B0 s→K∗0K∗0decay Neglecting the contribution of PGIM s[5], the B0 s→K∗0K∗0is mediated by a single amplitude, so that under this asumption not CP violation in the decay can be produced, ¯ A A=VtbV∗ ts V∗ tbVts =e−iφDwith φD=−2βs.(2.84) As was explained in Sect. 2.2.4.1, within the SM, neither CP violation in mixing is expected for the B0 s-system, q p=e−i(2βs).(2.85) Consequently, there is a cancellation between the phases arising from mixing and decay1, and the total CP -violating weak phase, φs, is negligibly small in the SM for this channel, as λ(B0 s→K∗0K∗0) = q p A(B0 s→K∗0K∗0) A(B0 s→K∗0K∗0)= 1.(2.86) The study of CP -violation in this decay is then a null test of the SM. This is a very atractive feature in favour of B0 s→K∗0K∗0, but there is another one. As the precision in the measurement of the CP -violation parameters will increase, an estimate of the error induced by neglecting PGIM swill be needed. However, the polluting contribution, PGIM s, is an hadronic object, in general, difficult to compute. The advantage of this mode is the possibility of calculating the theoretical error in a model independent way extracting the size of PGIM sfrom the analysis of B0→K∗0K∗0. This will be explained with more detail in the following section. 2.3.2.4 U-spin symmetry The prediction given in the previous section for φB0 s→K∗0K∗0 srelies on the fact that the second amplitude PGIM scan be neglected to a good approximation. The theoretical error induced by this assumption has been studied by various authors [3,5,6,43]. As already mentioned, the main advantage of B0 s→K∗0K∗0over other decay modes is that the size of the polluting amplitude can be estimated from the study of the U-spin-conjugate process B0→K∗0K∗0. This is a pure b→dpenguin decay, whose amplitude can be written as A(B0→K∗0K∗0) = −V∗ tbVtd Pd−V∗ ubVud PGIM d(2.87) which is equivalent to (2.83), except that in this case the two combinations of CKM matrix elements are of the same order of magnitude. As a consequence, the sensitivity to the second amplitude, PGIM d, is maximal in this case. Assuming that no new physics contributes in an appreciable way to B0→K∗0K∗0, the measurement of its branching fraction (B) and time-dependent CP asymmetry, allows the determination |Pd|,|PGIM d|and their relative strong phase, δd. In the SU(3)-symmetry limit, PGIM d=PGIM sand δd=δs. Imposing these relations, introduces an error that is related to the size of the SU(3) breaking. Fig. 2.10 for example, shows the precision on the theoretical prediction expected when 100% SU(3) breaking effects are assumed [5]. 1Other authors [8] argue that if PGIM is neglected, every O(λ4) term needs to be neglected too. Since φM∝arg(V∗ tbVts ) and =(V∗ tbVts )∼O(λ4), φMand φB0 s→K∗0K∗0 svanish in this approximation. 25 Chapter 3. The LHC beauty experiment Figure 3.1: CERN accelerator complex with LHC at its end. Figure 3.2: Cross–section of a LHC cryodipole. 32 3.2 The LHCb experiment (a) ALICE (b) ATLAS (c) CMS (d) LHCb Figure 3.3: Four main detectors in LHC 3.2 The LHCb experiment The Large Hadron Collider beauty (LHCb) experiment was designed to look for indirect evidence of physics beyond the SM in CP –violation and rare decays of beauty and charm hadrons [55]. The dedication to Bphysics constrained the design of the dectector and the choice of its work enviroment. The forward geometry of the detector takes advantage of the large bb cross–section at low angle at the considered energies, see Fig. 3.4. Another essencial requirement for the physics in the experiment is the ability to identify the point where the proton–proton collision took place (primary vertex or PV) and the point where other short–lived but flying particles decayed (secondary vertex or SV). In order to ease this task, LHCb was designed to work at an instantaneous luminosity of 2–5×1032 cm−2·s−1, smaller than LHC nominal 1034 cm−2·s−1. This is achieved by changing the beam focus at the LHCb interaction point independently from the other interaction points. The detector is located in the Intersection Point 8 of the LHC, former emplacement of LEP experiment DELPHI. A modification to the LHC optics, displacing the interaction point by 11.25 m from the centre, has permitted maximum use to be made of the existing cavern for the LHCb detector components [56]. In the next sections the LHCb detector and its different subdetectors are described in more detail. 33 Chapter 3. The LHC beauty experiment Figure 3.4: Angular distribution of bb pairs produced in pp collisions at √s= 7 TeV. 3.3 LHCb detector The LHCb detector is a single–arm spectrometer with a forward angular coverage from approximately 10 mrad to 300 (250) mrad in the bending (non-bending) plane, with respect to the beam axis. In terms of pseudorapidity (η), the LHCb acceptance covers the range 2 < η < 5. The layout of the LHCb spectrometer is shown in Fig. 3.5. The right–handed coordinate system adopted has the z axis along the beam, and the y axis along the vertical. In the following sections, each of the LHCb subsystems will be described in detail. 3.3.1 Tracking System An efficient and precise reconstruction of the trajectories of charged particles is of fundamental importance for LHCb. from a reconstructed track it is possible to determine the bending of the charged particle in the magnetic field and thus to measure its momentum and electric charge. Combinations of tracks allow to define vertices, measuring their precise location and their charged track multiplicity. Moreover, a reconstructed track can be interpolated and extrapolated in space, permitting information gathered in different subdetectors to caracterize the charged particle. The LHCb tracking system consists of the Vertex Locator (VELO) and the Tracking Turicensis (TT) placed upstream of the dipole magnet, and T1-T3 tracking stations downstream of the magnet. 3.3.1.1 Vertex Locator The VELO [57] measures coordinates of tracks close to the interaction point, allowing to separate the decay vertex of the b- or c-hadron from the primary pp interaction. It 34 3.3 LHCb detector Figure 3.5: View of the LHCb detector. is made of 2×21 stations arranged along and perpendicular to the beam direction. Each station contains two semicircular silicon strip sensors mounted back–to–back. One of them measures the rcoordinate, with circular strips centered around the beam axis, and the other one measures the φcoordinate with straight, almost radial strips. Fig. 3.6 (left) shows one of the r-sensors. The minimum pitch in the sensors, at the innermost radius, is 38 µm, with a linear increase up to 101.6 µm, ensuring that the first points on the track are measured with the finest pitch available. The VELO acceptance covers a pseudo–rapidity range of 1.6< η < 4.9 for particles coming from primary vertices in the range −10.6< z < 10.6 cm. Two dedicated stations, containing only r–sensors, placed upstream the VELO constitute the pile–up veto system that was conceived to veto events with large number of primary vertices. To improve the resolution on the primary vertex, the first measurement of the track should be as close to the primary interaction as possible. The sensitive area of the modules starts thus at 8mm of the beam axis. This radius is much smaller than the aperture required by the LHC during the injection. In order to prevent severe radiation damage in the detector, the sensors must be retracted by 3cm when LHC is being filled. Consequently, the VELO was designed so that the two halves of the detector can be moved away from the beam in the horizontal direction. Fig. 3.6 (right) shows the layout of one half of the VELO detector. 3.3.1.2 TT and Tracking Stations Together with the VELO, the LHCb tracking system is composed by the TT, right upstream the magnet, and by the tree tracking stations between the magnet and RICH2. 35 Chapter 3. The LHC beauty experiment Figure 3.6: Left, layout of one half of the LHCb Vertex Locator. Right, one of r sensors. Figure 3.7: Left, layout of the four TT layers. Right, layout of one of the three IT stations. Two different technologies are employed in these detectors. The TT and the innermost region of the tracking stations, called usually Inner Tracker (IT), is covered by silicon microstrip detectors [58], for this reason the system TT-IT is also called Silicon Tracker (ST). The outer region of the tracking stations, or Outer Tracker (OT), is made of straw–tube drift chambers [59]. In order to get a 3D reconstruction, each of the ST stations consists of four detection layers with vertical strips in the first and last layers and strips rotated by +5 º (-5 º ) for the second (third) layer. The strip pitch in this modules is 200 µm, which gives a single hit resolution of 50 µm. The TT covers the full acceptance of the detector. The IT covers a small cross–shaped region around the beam pipe that constitutes only the 1.3% of a tracking station. However, approximately 20% of the charged tracks produced close to the primary vertex and going through the tracking stations pass through its area. Fig. 3.7 shows the layout of TT and IT subdetectors. The OT modules are arranged in three stations, each one consisting of four detection layers. As in the case of the ST, the first and the last layers are oriented vertically, while those in the center are rotated by ±5 º with respect to the others. A mixture of Argon and CO2is chosen as counting gas. It provides a drift–time below 50 ns and a drift–coordinate resolution of 200 µm. 3.3.1.3 Magnet In order to measure the momentum of the charged particles produced in LHCb, the experiment uses a dipole magnet [60] that provides an integrated magnetic field of 4Tm. 36 3.3 LHCb detector Figure 3.8: Left, cross section of one OT module. Right, TT, IT and OT systems layout. Figure 3.9: Bycomponent of the magnetic field as a function of the z coordinate. The measurement covers the forward acceptance of ±300 mrad horizontally (bending plane) and of ±250 mrad vertically. The dipole is composed of two pure Al coils of conical saddle shape placed mirror– symmetrically to each other in a window–frame Fe yoke. The magnetic field provided by the dipole is measured with a precision of a few times 10−4to achieve the required momentum resolution. Fig. 3.9 shows the measured magnetic field (Bycomponent) along the z axis. The direction of the magnetic field is changed periodically, in order to reduce systematic uncertainties that might affect CP violation studies. 3.3.1.4 Tracking and Vertexing Event reconstruction relies on the determination of the trajectories of all the charged particles (tracks) and the position where they were generated (vertices). Tracking algorithms combine hits in VELO, TT and Tracking stations to reconstruct the trajectory and measure the momentum of charged particles. Depending on the origin of the hits used to define the track, these can be classified as (see Fig. 3.10):  Long tracks: traverse the full tracking system from the VELO to the T–stations. 37 Chapter 3. The LHC beauty experiment VELO TT T1 T2 T3 VELO Track Long Track Downstream Track T Track Upstream Track Figure 3.10: Illustration of the five different track types in LHCb. They provide the most precise momentum measurement, and therefore are the most important tracks in LHCb analyses.  Upstream tracks: traverse only the VELO and the TT. They correspond in general to low momentum particles that were bent out of the detector acceptance by the magnetic field.  Downstream tracks: go through the TT and T stations. They are usually produced by long-living particles decaying outside the VELO, like K0 Sor Λ.  VELO tracks: are just measured in the VELO. They are typically large angle or backward tracks, useful to reconstruct the primary vertex.  T tracks: have only hits in the T–stations and are tipically produced by particles generated in secondary interactions. Different algorithms are used to reconstruct different types of tracks. In the case of long tracks, the algorithms look first for almost aligned track seeds in the VELO, where the magnetic field is low. These seeds are then complemented with hits from the other tracking subdetectors to form tracks [61]. Once the track is found, it is refitted using Kalman fitter algorithm [62], that accounts for multiple scattering and energy loss caused by crossed materials. This procedure provides also a χ2related with the probability that the track corresponds to a real particle instead of a mixture of hits left by different particles (ghosts). There is also the posibility of reconstructing the same track through different algorithms (clones); in this case only the best out of the two is kept. The efficiency to reconstruct long tracks in lhcb has been evaluated using muons from J/ψdecays [63,64]. The efficiency for 2011 data and and simulation as a function of momentum, rapidity and the number of primary vertices is shown in Fig. 3.11 The global efficiency is measured to be larger than 96%. Hadronic interactions not taken into account in this calculation are reflected in a larger systematic uncertainty. The precision in the determination of the trajectory of the particle inside the magnetic field, is directly related with the momentum resolution, and a good momentum resolution translates in good mass resolution, a key ingredient for the physics in the experiment. 38 3.3 LHCb detector Figure 3.11: Tracking efficiency for the 2011 data and simulation as a function of the momentum, p(left), the pseudorapidity, η(center) and the number of reconstructed primary vertices, NP V (right). Figure 3.12: Invariant mass distribution of K+π−[20] (left) and K+K−K−π+[65] (right). The mass resolution is ∼22 MeV/c2for the two body decay and ∼15 MeV/c2for the four body decay. LHCb momentum resolution goes from dp/p = 0.4% for tracks with pT= 5 GeV/c to dp/p = 0.6% for tracks with pT= 100 GeV/c. This results in a mass resolution of ∼22 MeV/c2(∼15 MeV/c2) at the B–meson mass for two body (four body) Bdecays, see Fig. 3.12. VELO tracks are used to reconstruct primary vertices [66]. The resolution improves significantly with the number of tracks used to reconstruct the vertex. This number ranges from 5 (minimum required) up to 100. Fig. 3.13 shows the PV resolution for the x, y and z coordinates in 2011 data, as a function of the number of tracks entering the PV reconstruction. A distinctive feature of events containing Bor Ddecays, is the existence of a secondary vertex (SV) separated from the PV, due to the large lifetime of these particles. Equivalently, particles created in these SV’s will have a sizable impact parameter 1(IP) with respect to the primary interaction. A natural way of showing the power of the LHCb track and vertex reconstruction is through the precision achieved in the determination of the IP and the proper–time. Fig. 3.14 shows the IPx(xprojection of the IP) and IPy(y projection of the IP) resolutions as a function of 1/pTfor 2011 data and their comparison with those for MC simulation. The proper–time resolution is at the level of 50 fs. This excellent precision allows the fast B0 s–B0 soscillation to be resolved, with an oscillation frequency of ∆ms= 17.768 ±0.023(stat) ±0.006(syst)ps−1, as measured by LHCb [67] (see Fig. 3.15). 1The impact parameter is defined as the geometrical distance between a track and a certain vertex, normally the PV. 39 Chapter 3. The LHC beauty experiment N 5 10 15 20 25 30 35 40 m]µResolution [ 0 5 10 15 20 25 30 35 40 LHCb preliminary = 7 TeVs m)µ (A B m)µ (C 97.7 0.61 -1.0 0.9± 0.01± 0.2± x{ m)µ (A B m)µ (C 92.7 0.59 -1.5 0.8± 0.01± 0.2± y{ N 5 10 15 20 25 30 35 40 m]µResolution [ 0 50 100 150 200 250 300 LHCb preliminary = 7 TeVs m)µ (A B m)µ (C 923 0.69 -16 7± 0.01± 1± z{ Figure 3.13: PV resolution in x, y (left) and z (right) coordinates in 2011 data as a fucntion of the number of tracks entering the PV reconstruction. Data points are fitted with the parametrization: Res =A/NB+C. [c/GeV] T 1/p 0 0.5 1 1.5 2 2.5 3 m]µ) [ X (IPσ 0 10 20 30 40 50 60 70 80 90 100 LHCb VELO Preliminary T = 12.0 + 24.3/pσ2011 Data: T = 11.2 + 22.2/p σSimulation: LHCb VELO Preliminary T = 12.0 + 24.3/pσ2011 Data: T = 11.2 + 22.2/p σSimulation: = 7 TeVs 2011 Data Simulation [c/GeV] T 1/p 0 0.5 1 1.5 2 2.5 3 m]µ) [ Y (IPσ 0 10 20 30 40 50 60 70 80 90 100 LHCb VELO Preliminary T = 11.7 + 23.8/p σ2011 Data: T = 11.1 + 21.8/p σSimulation: LHCb VELO Preliminary T = 11.7 + 23.8/p σ2011 Data: T = 11.1 + 21.8/p σSimulation: = 7 TeVs 2011 Data Simulation Figure 3.14: IPx(left) and IPy(right) resolution as a function of 1/pTfor 2011 data and MC simulation. Data points have been fittes with a linear fucntion. Figure 3.15: Decay time distribution for B0 s→D− sπ+candidates tagged as mixed (different flavour at decay and production; red, continous line) or unmixed (same flavour at decay and production; blue, dotted line), used in the LHCb measurement of ∆ms. 40 3.3 LHCb detector 3.3.2 Particle Identification Detectors Key features for the physics at LHCb, such as the ability to tag the flavour of B mesons or to reject backgrounds that are kinetically and topologically similar to the signal, are only possible if particle identification is available. Distinction between different species of long lived particles is achieved at LHCb with  two Ring Imaging Cherenkov (RICH) detectors, for K–πseparation;  the Calorimeter System, made of a Scintillator Pad Detector and Preshower (SPD/PS), and electromagnetic calorimeter (ECAL) and a hadronic calorimeter (HCAL), that provides identification of electrons, photons and hadrons with measurements of position and energy;  the Muon detection System, made of five stations labelled M1 to M5, used to identify muons that have passed the calorimeters. 3.3.2.1 RICH detectors LHCb has two RICH detectors, covering different momentum ranges. RICH1, located upstream the dipole magnet, is optimised for low momentum particles, from 1 to 60 GeV/c, and uses aerogel and C4F10 as radiators. RICH2, placed downstream the tracking stations, covers larger momenta, from 15 up to and beyond 100 GeV/c, using a CF4radi- ator. RICH1 covers the full LHCb acceptance from ±25 mrad to ±300 mrad (horizontal) and ±250 mrad (vertical). RICH2 has a limited angular acceptance from ∼ ±15 mrad to ±120 mrad (horizontal) and ±100 mrad (vertical), where high momentum particles are abundant. The Cherenkov light produced by particles traversing the radiators is reflected out of the spectrometer acceptance by a combination of spherical and flat mirrors. Hybrid Photon Detectors (HPDs) are used to detect Cherenkov photons in the wavelength range 200–600 nm. Fig. 3.16 shows the layout of both RICH detectors. 3.3.2.2 Calorimeter system The calorimeter system is used to select transverse energy (ET) hadron, electron and photon candidates for the first level of the trigger (L0) and to provide the indentification of electrons, photons and hadrons as well as the measurement of their energies and positions. LHCb uses an electromagnetic calorimeter (ECAL) followed by a hadron calorimeter (HCAL) to identify electromagnetic and hadronic showers, respectively. They are both sampling devices composed of alternating layers of scintillator and absorber, lead in the case of the ECAL and iron in the HCAL. Fig. 3.17 (right) shows, as an example, the internal structure of one HCAL cell. Longitudinal segmentation in the electromagnetic shower, needed to distinguish e±from the overwhelming background of neutral and charged pions, is achieved with the installation of a Scintillator Pad Detector(SPD) and a Preshower (PS) detector before the ECAL. A thin lead converter is installed between PS and SPD. In all calorimeters scintillation light is transmitted to Photo–Multipliers (PMTs) by wavelength–shifting (WLS) fibres. 41 Chapter 3. The LHC beauty experiment 1200 1300 1400 µ 0 0.5 1 1.5 2 2.5 Fill number 1200 1300 1400 ) -1 Int. Lumi. (pb 0 2 4 1800 2000 2200 µ 0 0.5 1 1.5 2 2.5 3 3.5 Fill number 1800 2000 2200 ) -1 Int. Lumi. (pb 0 20 Figure 3.23: Average number of visible pp interaccions as a function of the fill number during 2010 (left) and 2011 (right), compared with the design value (red dashed line). two charged tracks in the final state and a displaced secondary vertex [79]. Exclusive selections require all the decay products in a specific decay chain to be reconstructed. After HLT2 50% of the rate consist of inclusive hadronic triggers, 25% are triggers on leptons and the remaining rate come from exclusive triggers. 3.3.3.3 Trigger performance The LHCb trigger was able to cope remarkably well with the non–standard conditions imposed by the LHC machine during 2010 and 2011, when the average number of visible pp interactions per bunch crossing was ∼2.5 and ∼1.5 respectively. The trigger efficiency for several representative physics channels was calculated using 2011 data [80]. LHCb is planning to upgrade the detector in 2018 [81,82], which will feature a fully software based trigger. 3.4 LHCb running conditions during 2010 and 2011 The study presented in this thesis is based on the data recorded by LHCb during 2010 and 2011 from the proton-proton collisions at a centre-of-mass energy of √s= 7 TeV provided by the LHC. The first data taking period, during 2010, was devoted to comissioning and establishing confidence in the operation of the LHC, and so the running conditions changed continuously. A total integrated luminosity of 37 pb−1was delivered during this period. Furthermore, although the LHC operated with only 10% of the nominal number of bunches per beam, ∼80% of the design instantaneous luminosity was achieved at LHCb by chaging the focussing of the beams at the interaction point. This lead to an increase in the average number of visible interactions per bunch crossing, µ, with respect to the nominal number, see Fig. 3.23. A higher µimplies a rise of the readout rate per bunch crossing, the event size and the processing time. LHCb, and in particular its trigger system, showed a great flexibility to cope with this continuously changing non-standard running conditions. 48 3.4 LHCb running conditions during 2010 and 2011 Figure 3.24: Integrated luminosity as a function of time during 2011. In 2011, the LHC running conditions were more stable. During the early data taking period, the number of bunches colliding at LHCb steadily increased from 180 to 1296, which is about half of the final number of bunches in the LHC machine. From then on, peak luminosities at LHCb were leveled in order not to exceed 3-3.5×1032 cm−2s−1, which corresponds to an average µ∼1.5, almost a factor four higher than by design. Fig. 3.24 shows the integrated luminosity as a function of time for the 2011 data taking period, during which a total integrated luminosity of 1.0 fb−1was collected by LHCb. During both periods, luminosity leveling allowed LHCb to mantain constant the instantaneous luminosity during each fill, see Fig. 3.25. The orientation of the magnetic field was also frequently changed, in order to record approximately the same amount of data with positive (By>0) and negative (By<0) magnet polarity. Figure 3.25: Instantaneous luminosity at ATLAS, CMS and LHCb vs. time during a typical LHC fill. The luminosity leveling yields a constant luminosity for LHCb. 49 4 Phenomenology of B0 s→K∗0(K+π−)K∗0(K−π+) Among the numerous two-body charmless decays of the B0 smesons, the decay B0 s→ K∗0K∗0is of special interest. In one hand, the final state of this decay is a CP -eigenstate, what makes it particularly suitable for the study of CP -violation. On the other hand, the presence of two vector resonances in the final state means that this process in fact represents three different decays, one for each of the three possible helicities of the vector mesons. Therefore, a larger number of observables are available compared to the decays into P P or P V final states (P≡pseudo-scalar, V≡vector). Moreover, when a neutral vector meson is detected via its decay V→P P 0, there is usually an indistinguishable contribution coming from the decay of a scalar resonace S→P P 0or from the scalar non resonant P P 0production [83,84]. It is necessary thus to take into account this additional contributions, which in the case of B0 s→K∗0K∗0extends the total number of amplitudes to six 1. These configurations, commonly referred to as “S–wave amplitudes”, will be shown to be also linear combinations of CP -eigenstates, able to generate additional CP -violating quantities in the interference with the “P–wave amplitudes”. This CP -violating observables can be measured without knowledge of the decay time or flavour (B0 sor B0 s) of the decaying B0 smeson, and can thus always be determined, even from samples with modest statistics. 4.1 The B→V V angular distribution Consider the Bmeson decay into two neutral vector mesons, B→V1V2, where V1and V2undergo two-body strong decays into pseudoscalar particles: V1→h1H1and V2→ h2H2. As already mentioned, this process is usually described by three helicity amplitudes according to the three possible helicity configuration of the vectors in the final state. Using the helicity formalism [85], the total amplitude can be written as M(B→V1V2)∝X λ=0,±1 Aλ(m1, m2)eiλφd1 λ,0(θ1)d1 −λ,0(θ2) (4.1) where λis the helicity of V1and Aλ(m1, m2) is a helicity amplitude depending on the mass of the two vectors: m1≡M(h1, H1) and m2≡M(h2, H2). The angles describing the 1Other B→VV decays might not need as many new amplitudes. For example, in B0→φφ the most general description is achieved with only five amplitudes due to the presence of identical particles in the final state [8] 51 Chapter 4. Phenomenology of B0 s→K∗0(K+π−)K∗0(K−π+) Figure 4.1: Definition of the angles involved in the analysis for the B0 s→K∗0K∗0. decay are shown in 4.1,θ1(2) is the angle between the direction of h1(2) and the direction opposite to the Bmeson momentum in the rest frame of V1(2) and ϕis the angle between the decay planes of the two vectors in the Brest frame. The Wigner d-matrix elements involved can be expressed as dl m,0(θ) = s(l−m)! (l+m)!Pm l(cos θ) (4.2) with Pm l(cos θ) representing the Associated Legendre Polinomials. It is useful to rewrite the total amplitude in terms of the transversity amplitudes, A0=A0, Ak=1 √2(A++A−), A⊥=1 √2(A+−A−),(4.3) since they correspond to final states with definite CP -eigenvalue (ηk=η0= 1 and η⊥=−1). In the particular case of B0 s→K∗0(892)K∗0(892) (V1≡K∗0and V2≡K∗0), with h1=K+, H1=π−, h2=K−, H2=π+. However, the Kπ mass spectrum is not that corresponding to only a K∗0(892) meson. Instead, the spectrum is a mixture of resonances with different spin (J= 0,1,2), their interferences and nonresonant production. To take these contributions into account, additional amplitudes should be included in (4.1). In the next section, the decay rate for B0 s→(K+π−)J1(K−π+)J2is calculated. Since the contribution from J= 2 resonances in the mass region of the K∗0(892) is expected to be very small, only the contributions with J1,2= 0,1 will be considered. 4.2 The B0 s→(K+π−)(K−π+) model The most general description of the decay Bs→(K+π−)J1(K−π+)J2when the Kπ pairs are required to be in a narrow mass region around the K∗0(892), i.e. only the S−(Ji= 0) and P−wave (Ji= 1) production of the Kπ pairs are considered, is given by six decay amplitudes. In addition to the usual three amplitudes corresponding to the decay into two 52 4.2 The B0 s→(K+π−)(K−π+) model vectors –commonly called P–wave amplitudes–, the S–wave amplitudes: AV S :B→V1(h2H2)0 ASV :B→(h1H1)0V2 ASS :B→(h1H1)0(h2H2)0, need to be taken into account 1. Extending (4.1) to include these contributions and using the transversity amplitudes to describe the P–wave component, the differential decay rate can be expressed as d5Γ dΩdm1dm2dt=NA0(t) cos θ1cos θ2+Ak(t) √2sin θ1sin θ2cos ϕ +iA⊥(t) √2sin θ1sin θ2sin ϕM1(m1)M1(m2) −AV S(t) √3cos θ1M1(m1)M0(m2) +ASV (t) √3cos θ2M0(m1)M1(m2) −ASS(t) 3M0(m1)M0(m2) 2(4.4) where the dependence of each amplitude with the two body effective masses mi≡M(hiHi) has been made explicit in terms of the P–wave and S–wave mass propagators, M1,0(m). These propagators will be described in more detail in Sect. 4.3.1.1. The time evolution induced by B0 s-B0 smixing is encoded in the time-dependence of the amplitudes Ak(t). Note that, unlike the K∗0K∗0final state, the S-wave configurations SV and V S defined in (4.4) do not build CP -eigenstates. Nevertheless, one may consider the following quantum final states: |s+i=1 √2|K∗0(K−π+)0i+|(K+π−)0K∗0i |s−i=1 √2|K∗0(K−π+)0i−|(K+π−)0K∗0i(4.5) which are indeed CP -eigenstates with opposite CP -parities (η+=−1 and η−= +1). Therefore, by introducing A+ s=1 √2(AV S +ASV ), A− s=1 √2(AV S −ASV ),(4.6) the decay rate (4.4) can be reformulated in terms of CP-odd and CP -even amplitudes 1The subscript ( )Jdenotes the relative orbital angular momentum, J, of the pair. 53 Chapter 4. Phenomenology of B0 s→K∗0(K+π−)K∗0(K−π+) (the SS final configuration is a CP -eigenstate with ηSS = 1) as follows, d5Γ ΓdΩdm1dm2dt=NA0(t) cos θ1cos θ2+Ak(t) √2sin θ1sin θ2cos ϕ +iA⊥(t) √2sin θ1sin θ2sin ϕM1(m1)M1(m2) −A+ s(t) √6(cos θ1M1(m1)M0(m2)−cos θ2M0(m1)M1(m2)) −A− s(t) √6(cos θ1M1(m1)M0(m2) + cos θ2M0(m1)M1(m2)) −Ass 3M0(m1)M0(m2) 2 =N 21 X n=1 Kn(t, m1, m2)Fn(Ω).(4.7) In the last equality, a more compact formulation of this decay rate has been introduced. The functions Kncontain the dependence with the different amplitudes entering the decay and Fngive the angular distribution associated with each amplitude combination. These functions are shown in Tables 4.1 and 4.2. In order to write down the rate corresponding to the CP -conjugated decay, B0 s→ (K−π+)J1(K+π−)J2, the amplitudes Akneed to be substituted by the corresponding ¯ Ak in (4.7). Also, the decay angles for the conjugated process ¯ Ω:{¯ θ1,¯ θ2,¯ϕ}can be related to Ω:{θ1, θ2, ϕ}by ¯ θ1=θ2 ¯ θ2=θ1 ¯ϕ=−ϕ, (4.8) and the two body invariant mass ¯m1(2) verify ¯m1(2) =m2(1).(4.9) This is easily understood by considering B0 s→K∗0K∗0. As noted in the previous section, in the decay B→V1V2, the helicity angles are defined with respect to the momenta of V1and V2. In B0 s→K∗0K∗0,V1=K∗0and V2=K∗0. On the other hand, in the CP -conjugated decay B0 s→K∗0K∗0,V1=K∗0and V2=K∗0. That is, the indices 1 and 2 have been exchanged and the decay variables become those in (4.8) and (4.9). Note that this transformation is equivalent to the exchange AK(t)→ηk¯ AK(t), where ηkis the CP -eigenvalue associated to the amplitude Ak, and so the decay rate for B0 sdecays can be written as d5¯ Γ dΩdm1dm2 =N 21 X n=1 ¯ Kn(t, m1, m2)Fn(Ω) (4.10) where Fn(Ω) are the same function as in (4.7). 54 4.2 The B0 s→(K+π−)(K−π+) model Table 4.1: Values of the Knand Fnfucntions listed in (4.7). The time dependence is encoded in the amplitudes, i.e. Ak=Ak(t). nKn(t, m1, m2)Fn(Ω) 1|A0|2|M1(m1)|2|M1(m2)|2cos2θ1cos2θ2 2|Ak|2|M1(m1)|2|M1(m2)|21 2sin2θ1sin2θ2cos2ϕ 3|A⊥|2|M1(m1)|2|M1(m2)|21 2sin2θ1sin2θ2sin2ϕ 4<(AkA∗ 0)|M1(m1)|2|M1(m2)|21 2√2sin 2θ1sin 2θ2cos ϕ 5=(A⊥A∗ 0)|M1(m1)|2|M1(m2)|2−1 2√2sin 2θ1sin 2θ2sin ϕ 6=(A⊥A∗ k)|M1(m1)|2|M1(m2)|2−1 2sin2θ1sin2θ2sin 2ϕ 71 2|A+ s+A− s|2|M1(m1)|2|M0(m2)|21 3cos2θ1 8 1 √2<A+ sA∗ 0M∗ 1(m2)M0(m2) +A− sA∗ 0M∗ 1(m2)M0(m2)|M1(m1)|2−2 √3cos2θ1cos θ2 9 1 √2<A+ sA∗ kM∗ 1(m2)M0(m2) +A− sA∗ kM∗ 1(m2)M0(m2)|M1(m1)|2−1 √6sin 2θ1sin θ2cos ϕ 10 1 √2=(A+ s)∗A⊥M∗ 0(m2)M0(m2) +(A− s)∗A⊥M∗ 0(m2)M0(m2)|M1(m1)|2 1 √6sin 2θ1sin θ2sin ϕ 11 1 √2<A+ sA∗ SSM∗ 0(m1)M1(m1) +A− sA∗ SSM∗ 0(m1)M1(m1)|M0(m2)|2 2 3√3cos θ1 55 Chapter 4. Phenomenology of B0 s→K∗0(K+π−)K∗0(K−π+) Table 4.2: Values of the Knand Fnfucntions listed in (4.7), (cont).The time dependence is encoded in the amplitudes, i.e. Ak=Ak(t). The mass function in K17 is defined as ζ(m1, m2)≡M∗ 1(m1)M∗ 0(m2)M0(m1)M1(m2). nKn(t, m1, m2)Fn(Ω) 12 1 2|A+ s−|A− s|2|M0(m1)|2|M1(m2)|21 3cos2θ2 13 1 √2<A+ sA∗ 0M∗ 1(m2)M0(m2) −A− sA∗ 0M∗ 1(m2)M0(m2)|M1(m2)|2 2 √3cos θ1cos2θ2 14 1 √2<A+ sA∗ kM∗ 1(m2)M0(m2) −A− sA∗ kM∗ 1(m2)M0(m2)|M1(m2)|2 1 √6sin θ1sin 2θ2cos ϕ 15 1 √2=−(A+ s)∗A⊥M∗ 0(m2)M0(m2) −(A− s)∗A⊥M∗ 0(m2)M0(m2)|M1(m2)|2−1 √6sin θ1sin 2θ2sin ϕ 16 1 √2<A+ sA∗ SSM∗ 0(m1)M1(m1) −A− sA∗ SSM∗ 0(m1)M1(m1)|M0(m1)|2−2 3√3cos θ2 17 |A+ s|2−|A− s|2<ζ(m1, m2) +2=(A+ s)∗A− s=ζ(m1, m2)−1 3cos θ1cos θ2 18 |Ass|2|M0(m1)|2|M0(m2)|21 9 19 <AssA∗ 0M∗ 1(m1)M∗ 1(m2)M0(m1)M0(m2)−2 3cos θ1cos θ2 20 <AssA∗ kM∗ 1(m1)M∗ 1(m2)M0(m1)M0(m2)−√2 3sin θ1sin θ2cos ϕ 21 =AssA∗ ⊥M∗ 1(m1)M∗ 1(m2)M0(m1)M0(m2)) √2 3sin θ1sin θ2sin ϕ 56 4.2 The B0 s→(K+π−)(K−π+) model 4.2.1 Time evolution The time evolution of the states |B0 s(t)iand |B0 s(t)iinduced by the B0 s-B0 smixing was described in section Sect. 2.2.1.2. As a consequence of the oscillation, the helicity amplitudes in the angular distribution are also time-dependent and follow Ak(t) = hk|HW|B0 s(t)i=g+(t)Ak+ηk q pg−(t)¯ Ak ¯ Ak(t) = hk|HW|B0 s(t)i=p qg−(t)Ak+ηkg+(t)¯ Ak(4.11) where Ak≡ hk|HW|B0 siand ¯ Ak≡ hk|HW|B0 siare the amplitudes at t= 0. As noted above, all the final states considered are CP -eigenstates, i.e. |¯ ki=CP |ki=ηf|ki, with CP -eigenvalue ηk= 1 for k= 0,k, S−, SS and ηk=−1 for k=⊥, S+1 If the time dependence of the different amplitudes is made explicit, every function Kn(t, m1, m2) can be written as Kn(t, m1, m2) = 1 2eΓsthan(m1, m2) cosh ∆Γ t 2+bn(m1, m2) sinh ∆Γ t 2 +cn(m1, m2) cos (∆mst) + dn(m1, m2) sin (∆mst)i(4.12) where the coefficients an,bn,cnand dncontain combinations between the amplitudes at t= 0 and the mass-dependent propagators. The values of these coefficients are given in Tables A.3 to A.6 in Appendix A. It is easy to identify CP -violating quantities when taking a closer look at the coefficients an,bn,cnand dn. There are four kinds of such observables:  direct CP asymmetries: <[AkA∗ k0−¯ Ak¯ A∗ k0] (for k=k0, they become the familiar |Ak|2−|¯ Ak|2), present in the coefficients an(n= 8−11) and cn(n= 1−4,7,13− 16,18,20,21);  indirect or mixing-induced CP asymmetries: =[(A∗ k¯ Ak0+¯ AkAk0∗)e−iφM], present in the coefficients bn(n= 8 −11) and dn(n= 1 −4,7,13 −16,18,20,21);  triple product asymmetries: =[A⊥A∗ k−¯ A⊥¯ Ak], present in an(n= 5,6,17,19) and c12;  mixing-induced triple product asymmetries: =[( ¯ A⊥A∗ k+Aperp∗¯ Ak)e−iφM], present in bn(n= 5,6,17,19) and d12. An equivalent expression to (4.12) can be found for the ¯ Knfunctions, ¯ Kn(t, m1, m2) = 1 2eΓsth¯an(m1, m2) cosh ∆Γ t 2+¯ bn(m1, m2) sinh ∆Γ t 2 +¯cn(m1, m2) cos (∆mst) + ¯ dn(m1, m2) sin (∆mst)i(4.13) 1The CP -eigenvalues can be understood in terms of the total angular momentum of the final state. States with L= 0, (SS, S−and a comnination of 0 and k) or L= 2 (a different combination of 0 and k) are CP -even, while those with L= 1 (⊥, S+) are CP -odd. 57 Chapter 4. Phenomenology of B0 s→K∗0(K+π−)K∗0(K−π+) ] 2 Mass [MeV/cπK 800 900 1000 FB A -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ] 2 Mass [MeV/cπK 800 900 1000 FB A -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 4.2: Asymmetry on cos θas a function of the Kπ mass for different values of A− s (left, with a fixed value of δs= 0) and δs(right, with a fixed value of A− s= 0.20) where |AP|2has been used to denote the sum |A0|2+|Ak|2+|A⊥|2and the denominator, D(mi), is given by D(mi) = |AP|2+1 2|A+ s|2 ΓH +|A− s|2 ΓL|M1(mi)|2 +1 2|A+ s|2 ΓH +|A− s|2 ΓL+|ASS|2|M0(mi)|2.(4.35) The shape of AF B is illustrated in Fig. 4.2 for different values of the amplitude A− s (left) and its phase difference with respect to the longitudinal polarization A0(right). The magnitude of the later amplitudes are responsible of the strength of the asymmetry as function of mKπ. Also δ− saffects to the shape of AF B by changing its mKπ derivative. 4.3.2 Triple products and Direct CP asymmetries As introduced in Sect. 2.3.2.5, two CP -violating TP asymmetries arise in B→V V decays that are proportional to the interference terms between the CP -odd amplitude A⊥and the two CP -even A0and Ak. In order to determine these CP -violating quantities a full angular analysis is not needed. Instead, it can be shown that they can be obtained from the following asymmetries in the azimuthal distribution, ϕ, [41] A(1) T=Γ(sign(cos θ1cos θ2) sin ϕ > 0) −Γ(sign(cos θ1cos θ2) sin ϕ < 0) Γ(sign(cos θ1cos θ2) sin ϕ > 0) + Γ(sign(cos θ1cos θ2) sin ϕ < 0) =−2√2 π=(A⊥A∗ 0−¯ A⊥¯ A∗ 0) |A0|2+|Ak|2+|A⊥|2+|A+ s|2+|A− s|2+|ASS|2(4.36) A(2) T=Γ(sin 2ϕ > 0) −Γ(sin 2ϕ < 0) Γ(sin 2ϕ > 0) + Γ(sin 2ϕ < 0) =−4 π=(A⊥A∗ k−¯ A⊥¯ A∗ k) |A0|2+|Ak|2+|A⊥|2+|A+ s|2+|A− s|2+|ASS|2(4.37) 64 4.3 Untagged analysis When the S–wave contribution is taken into account, two more CP -even amplitudes, A− s and Ass, are subject of interference with A⊥. Asymmetric integration of the decay rate, analogous to 4.36 and 4.37, can be worked out to measure these two new quantities. Namely, A(3) T=Γ(sign(cos θ1+ cos θ2) sin ϕ > 0) −Γ(sign(cos θ1+ cos θ2) sin ϕ < 0) Γ(sign(cos θ1+ cos θ2) sin ϕ > 0) + Γ(sign(cos θ1+ cos θ2) sin ϕ < 0) =32 5π√3R=A⊥A−∗ s−¯ A⊥¯ A−∗ sM1(m)M∗ 0(m)dm |A0|2+|Ak|2+|A⊥|2+|A+ s|2+|A− s|2+|ASS|2(4.38) A(4) T=Γ(sin ϕ > 0) −Γ(sin ϕ < 0) Γ(sin ϕ > 0) + Γ(sin ϕ < 0) =3π 4√2R=A⊥A∗ SS −¯ A⊥¯ A∗ SSM1(m)M∗ 0(m)dm |A0|2+|Ak|2+|A⊥|2+|A+ s|2+|A− s|2+|ASS|2(4.39) Since A+ sis also CP –odd, its interference terms with the CP –even amplitudes change sign when going from B0 sto B0 sdecay rate. In particular, these terms have the form <(A+ sA∗ k), with k= 0,k, s−, ss. Consequently, new CP -violating quantities arise in the the untagged decay rate in the form Re(A+ sA∗ k−¯ A+ s¯ A∗ k). These term has the structure of a direct CP asymmetry. There are actually four of them, accessible from Bs→K+π−K−π+ decays, which will from now on be designated as A(i) D(i= 1,4). Parameterising each amplitude as Ak=Piai keiδi keiφi k, these terms can be written as follows Re(A+ sA∗ k−¯ A+ s¯ A∗ k) = −2X ij ai s+aj ksin(δi s+−δj k)sin(φi s+−φj k).(4.40) and are only nonzero if the weak phase difference between two amplitudes is not zero. As in the case of the TP asymmetries, these quantities are very small in the Standard Model (O(λ2)) and the measurement of a large value for any A(i) Dwould imply the presence of New Physics. These quantities can also be determined from the following asymmetries in the angular distribution A(1) D=Γ(cos θ1cos θ2(cos θ1−cos θ2)>0) −Γ(cos θ1cos θ2(cos θ1−cos θ2)<0) Γ(cos θ1cos θ2(cos θ1−cos θ2)>0) + Γ(cos θ1cos θ2(cos θ1−cos θ2)<0) =√2 5√3"9R<(A+ sA∗ 0−¯ A+ s¯ A∗ 0)M0(m)M∗ 1(m)dm |A0|2+|Ak|2+|A⊥|2+|A+ s|2+|A− s|2+|ASS|2 +5R<(A+ sA∗ SS −¯ A+ s¯ A∗ SS)M1(m)M∗ 0(m)dm |A0|2+|Ak|2+|A⊥|2+|A+ s|2+|A− s|2+|ASS|2#(4.41) A(2) D=Γ((cos θ1−cos θ2) cos ϕ > 0) −Γ((cos θ1−cos θ2) cos ϕ < 0) Γ((cos θ1−cos θ2) cos ϕ > 0) + Γ((cos θ1−cos θ2) cos ϕ < 0) =−32 5π√3R<(A+ sA∗ k−¯ A+ s¯ A∗ k)M0(m)M∗ 1(m)dm |A0|2+|Ak|2+|A⊥|2+|A+ s|2+|A− s|2+|ASS|2(4.42) 65 Chapter 4. Phenomenology of B0 s→K∗0(K+π−)K∗0(K−π+) A(3) D=Γ((cos θ1−cos θ2)>0) −Γ((cos θ1−cos θ2)<0) Γ((cos θ1−cos θ2)>0) + Γ((cos θ1−cos θ2)<0) =2√2 5√3"3R<(A+ sA∗ 0−¯ A+ s¯ A∗ 0)M0(m)M∗ 1(m)dm |A0|2+|Ak|2+|A⊥|2+|A+ s|2+|A− s|2+|ASS|2 +5R<(A+ sA∗ SS −¯ A+ s¯ A∗ SS)M1(m)M∗ 0(m)dm |A0|2+|Ak|2+|A⊥|2+|A+ s|2+|A− s|2+|ASS|2#(4.43) A(4) D=Γ((cos2θ1−cos2θ2)>0) −Γ((cos2θ1−cos2θ2)<0) Γ((cos2θ1−cos2θ2)>0) + Γ((cos2θ1−cos2θ2)<0) =<A+ sA−∗ s−¯ A+ s¯ A−∗ s |A0|2+|Ak|2+|A⊥|2+|A+ s|2+|A− s|2+|ASS|2(4.44) 4.3.3 Summary of the analysis strategy The analyses presented in this thesis focussed in the determination of the observables accesible to the untagged sample. In Chapter 5, the study of the first LHCb data (37 pb−1) that lead to the discovery of the B0 s→K∗0K∗0mode is described. Chapter 6sumarizes the update of those results with a larger data sample (1.0 fb−1). The CP -averaged branching fraction and polarization fractions for B0 s→K∗0K∗0were determined through the study of the angular distribution of the four body (K+π−)(K−π+) final state, under the assumption of CP -conservation, following the Standard Model prediction. A model independent search for New Physics was also performed by measuring the four (CP -violating) TP asymmetries and four direct CP asymmetries associated to the interference between B0 s→K∗0K∗0and the different S–wave contributions. 66 5 First observation of B0 s→K∗0K∗0 During 2010, LHCb collected the first pp collisions delivered by the LHC at a center of mass energy of √s= 7 TeV. A data sample of corresponding to 37 pb−1of integrated luminosity was recorded. In this chapter, the analysis of this first data, which lead to the first observation of the B0 s→K∗0K∗0decay, is presented. 5.1 Introduction Before the start of the LHC, the decay channel B0 s→K∗0K∗0had never been observed. Only an upper limit for the its branching fraction of 1.68 ×10−3at 90% confidence level (CL) had been reported by the SLD experiment [93]. On the U-spin related channel, the b→dtransition B0→K∗0K∗0, there is still some controversy. Whilst BaBar reported its discovery (with a 6σstatistical significance) and a measurement of its branching fraction of (1.28+0.35 −0.30 ±0.11) ×10−6[94], Belle set, a few years later, the upper limit B(B0→K∗0K∗0)<0.8×10−6at 90% CL [95]. In the same paper, BaBar reported a measurement of the longitudinal polarization fraction for B0→K∗0K∗0of fL= 0.80+0.10 −0.12 ±0.06. The strategy of the analysis detailed in this chapter can be summarized as follows. First, a set of selection requirements was defined to identify the signal candidates, i.e. B0 s→K∗0K∗0decays where the K∗0(K∗0) resonances decay subsequently into K+π− (K−π+). This selection was optimized to reject most of the background while keeping the efficiency for the signal as high as possible. Then, the four-body (K+π−K−π+) invariant mass spectrum of the signal candidates was analysed in order to extract the number of events corresponding to B0 sdecays and determine the statistical significance of such a signal. The branching ratio of the signal can then be computed by comparing the observed number of signal candidates to the number of candidates for a reference decay channel with known branching fraction. In this case, the B0 d→J/ψK∗0decay was chosen as the reference mode. Additionally, a simplified version of the angular analysis presented in Chapter 4was performed, using only candidates with a mass in a narrow window around the B0 smeson mass, with the objective of measuring the polarization fractions of the B0 s→K∗0K∗0 decay. 67 Chapter 5. First observation of B0 s→K∗0K∗0 5.2 Data sample and Event selection 5.2.1 Data and Monte Carlo samples As explained, the present analysis is performed using the 2010 LHCb dataset, which includes ∼37 pb−1of integrated luminosity taken at √s= 7 TeV. The data belong to the Reco08-Stripping12 campaign, and have been reconstructed with Brunel v37r8p6 [96] and analysed with DaVinci v26r3 [97]. Two Monte Carlo samples were used in this analysis, corresponding to the decays B0 s→K∗0K∗0and B0 d→J/ψK∗0, that where modeled using EvtGen [98]. Both samples belong to the Monte Carlo production MC10, and were generated with the software corresponding to the Gauss release v39r0 [99], which uses GEANT4 v92r4 [100] for the full detector simulation. Then, they were reconstructed using Boole v21r9 [101] and Brunel v37r8p5. The pp interactions have been simulated assuming a beam energy of 3.5 TeV and an average number of interactions per crossing ν= 2.5, which corresponds to an average number of visible interactions per crossing µ= 1.75. 5.2.2 Event selection In order to search for the decay process B0 s→K∗0(K+π−)K∗0(K−π+) a number of offline selection criteria were applied. When a four-track secondary vertex is found, the reconstructed momentum of the B0 scandidate is used to calculate the smallest impact parameter with respect to all primary vertices in the event. Tracks are required to have pT>500 MeV/c, and a large impact parameter (IPχ2>9) with respect to the PV. The difference in the natural logarithm of the likelihoods of the kaon and pion hypotheses must be greater than 2 for K+and K−candidates, and less than 0 for π+and π− candidates. In addition, the Kπ combinations1must form an acceptable quality common vertex (χ2/ndf <9, where ndf is the number of degrees of freedom in the vertex fit) and must have an invariant mass within ±150 MeV/c2of the nominal K∗0mass (this is around ±3 times its physical width [16]). The K∗0and K∗0candidates must have pT>900 MeV/c and the distance of closest approach (DOCA) between their trajectories must be less than 0.3 mm. The secondary vertex must be well fitted (χ2/ndf<5). Finally, the B0 scandidate momentum is required to point to the PV. To improve the signal significance, a multivariate discriminant is defined that takes into account the properties of the B0 s→K∗0(K+π−)K∗0(K−π+) signal, as well as those of the background. This discriminant, in particular a geometrical likelihood (GL) [11,102], takes the following set of variables as input:  B0 scandidate impact parameter with respect to the closest primary vertex.  Decay time of the B0 scandidate.  Minimum IPχ2of the four tracks with respect to all primary vertices in the event.  DOCA between the two K∗0trajectories reconstructed from the pion and kaon tracks. 1This expression refers hereafter to both charge combinations: K+π−and K−π+. 68 5.3 The B0 s→K∗0K∗0signal GL cut 0 0.2 0.4 0.6 0.8 1 S+BS / 0 1 2 3 4 5 6 SoRSB GL 0 0.5 1 ) 2 ) (MeV/c + π - K - π + M(K 4800 5000 5200 5400 5600 5800 Figure 5.1: Left: S √S+Bas a function of the GL cut. The optimal GK cut was found to be ∼0.24. Right: Invariant mass versus GL scatter plot for B0 s→K∗0K∗0candidates in data before any GL cut.  pTof the B0 scandidate. For a given input sample, the above distributions are converted into a set of uncorrelated, Gaussian-distributed variables. Two vectors are defined for each event indicating its distance to the signal {Si}and to the background {Bi}hypotheses by means of χ2 S=PS2 i and χ2 B=PB2 i, where the index iruns over the five discriminating variables indicated above. The quantity ∆χ2=χ2 S−χ2 Bis found to be a good discriminant between the two hypotheses and is used to construct the GL function in such a way that it is uniformly distributed in the range [0,1] for signal events and tends to have low values for the background. The GL was trained using a fully reconstructed B0 s→K∗0K∗0simulation sample for the signal, and a selected background sample from the first 2 pb−1of data (Stripping09), which is not used in the analysis. The GL selection requirement was determined by maximising the figure of merit S √S+B, where Sand Bare the number of events from the testing signal and background samples that survive each cut in the GL. The optimal cut was found to be GL>0.24 (see Fig. 5.1 left). The GL requirement together with the above selection criteria, resulted in the mass spectrum in Fig. 5.2 for the selected K+π−K−π+candidates. Fig. 5.1 shows that the events with masses below the signal region have on average slightly higher GL values than those with masses above. This indicates the presence of a background from partially reconstructed Bdecays. 5.3 The B0 s→K∗0K∗0signal 5.3.1 Four-body invariant mass fit The invariant mass M(K+, π−, K−, π+) of the selected canditates was then analysed. The model used to describe the data includes four different components. The signals from Bs(d)→K+π−K−π+decay modes are described by two Gaussian probability density functions (PDF) centered at the B0and B0 smasses respectively and sharing a common width. A decreasing exponential models the combinatorial background. Finally, the 69 Chapter 5. First observation of B0 s→K∗0K∗0 ) 2 c) (MeV/ + π - K - π + M(K 5000 5200 5400 5600 5800 ) 2 cEvents / (30 MeV/ 0 5 10 15 20 25 30 35 40 LHCb Figure 5.2: Fit to the K+π−K−π+mass distribution of selected candidates. The fit model (dashed pink curve) includes a signal component that has two Gaussian components corresponding to the B0 sand B0decays. The background is described as an exponential component (dotted blue) plus the parametrization described in (5.1) in the text (dash-dotted green). background coming from partially reconstructed B-decays is parameterised as follows: P hysBkg(M) = M01−M02 M2 pΘ(Mp−M0)e−kp·M0⊗G(M−M0;σp),(5.1) where Θis the Heaviside-step function, ⊗represents the convolution, M0is the variable over which the convolution integral is calculated, G(M−M0;σp) is a Gaussian PDF with standard deviation σpand Mpand kpare free parameters. With this model, I(M) = NB0 sG(M−mB0 s, σ) + NB0G(M−mB0, σ) +NBkg fpP hysBkg(M) + (1 −fp)e−cbM(5.2) an extended maximum likelihood fit was performed to the four-body mass spectrum of the selected candidates. The fit results are given in Table 5.1. The measured B0 ssignal yield in a window of ±50 MeV/c2around the B0 smass is NB0 s= 49.8±7.5(stat.). The width of the B0 speak is in good agreement with the LHCb resolution measured in decays with similar kinematics such as B0 s→J/ψ φ. In order to calculate the significance of the B0 ssignal, the fit was repeated excluding the B0 ssignal component1. According to Wilks’ theorem [103], the variation in the negative log-likelihood between both fits follows a χ2(∆ndof) distribution, where ∆ndof = 1 is the difference in number of free parameters between the model with and without the 1For this test, the mass and width of the Bs(d)signals were fixed to those obtained from independent LHCb measurements in B0 s→J/ψφ and B0 d→J/ψK∗0respectively: mB0 s= (5362.88 ±0.84) MeV/c2, mB0= (5275.75 ±0.47) MeV/c2and σ= (18.80 ±0.73) MeV/c2. 70 5.3 The B0 s→K∗0K∗0signal Table 5.1: Fitted values of the model parameters for the mass spectrum, as described in the text. NB0 sand NB0are the number of events for the B0 sand B0signals, mB0 sis the fitted B0 s meson mass and σis the Gaussian width. The mass difference between B0 sand B0was fixed to its nominal value [16]. NBkg is the number of background events in the full mass range (4900-5800 MeV/c2), and cbis the exponential parameter of the combinatorial background model. Mp,σpand kpare the parameters of Eq. (5.1). Finally, fpis the fraction of the background associated with Eq. (5.1). Parameter Value NB0 s50.1/7.5 NB011.2/4.3 mB0 s( MeV/c2)5362.5/4.8 σ( MeV/c2)21.2/3.3 NBkg 90/10 cb(10−3( MeV/c2)−1)-3.37/0.55 kp(10−2( MeV/c2)−1)5.5/5.3 fp0.06 + 0.24 −0.05 Mp( MeV/c2)5170/170 σp( MeV/c2)37/23 B0 ssignal. Therefore, it is possible to turn this number into a probability according to χ2-statistics, and from there to a gaussian standard deviation. The obatined significance was 10.9σ. The peak at the B0mass, though not significant, is compatible with the B0→K∗0K∗0branching fraction measured by BaBar [94]. 5.3.2 B0 s→K∗0K∗0purity As previously explained, among the B0 s→(K+π−)(K−π+) candidates identified in the previous section, not all of them correspond to B0 s→K∗0K∗0events. In particular, scalar resonances (and non resonant production) in the Kπ spectrum can not be distinguished from the vector-vector decay without further analysis. The Kπ mass combinations of the candidates with a four-body invariant mass within a±50 MeV/c2window around the B0 ssignal were studied. A maximum likelihood fit in the (mK+π−, mK−π+) plane was perfomed. Three components were included in the signal model, namely a double Breit-Wigner distribution describing B0 s→K∗0K∗0production, a symmetrized product of a Breit-Wigner and a nonresonant linear model adjusted for phase-space in the Kπ mass, and a double nonresonant component. The non-B0 scomponent under the peak, which is essentially combinatorial background, was included in the fit with a fixed yield determined from the results in Table 5.1. The shape of its mass distribution was extracted from a fit to the Kπ mass spectrum observed in two 400 MeV/c2wide sidebands below and above the B0 smass. The sizeable K∗0 contribution present in this background was taken into account. The fit result, as shown in Fig. 5.3, gives (62±18)% K∗0K∗0production. A model for B0 s→K∗0K∗0(1430), representing a broad scalar state interfering with B0 s→K∗0K∗0 was also studied. The small number of events made it impossible to measure precisely 71 Chapter 5. First observation of B0 s→K∗0K∗0 ) 2 c) (MeV/πM(K 800 900 1000 ) 2 cEntries / ( 30 MeV/ 0 10 20 30 LHCb Figure 5.3: Background subtracted K+π−and K−π+combinations for selected candidates within a ±50 MeV/c2window of the B0 smass. The solid blue line shows the projection of the 2D fit model described in the text, indicating the K∗0K∗0yield (dashed-dotted red line) and a nonresonant component (blue dotted line), assumed to be a linear function times the two-body phase space. The dashed red line indicates the overall B0 s→K∗0Xcontribution. the size of such a contribution for all values of the interfering phase. However, for values of the phase away from π/2 and 3π/2 it was determined to be below 12%. Further study of this issue requires a larger data sample and was postponed until the 2011 data was available, see Chapter 6. Additionally, other four-body Bdecays that could possibly fake a B0 s→K∗0K∗0signal were also searched for; in particular, decays into charmed mesons like B0 s→D− s(→ K+K−π−)π+. However, as the K∗0meson is light compared to the B0 smeson, the invariant masses of the three-body systems K+K−π±and K+π−π±are rather high, above those of the charmed hadrons. This kinematically excludes the possibility of contamination from b→cdecays with very short charm flight distance. 5.4 Analysis of K∗0polarization Due to the small number of events available, a full mass-dependent angular analysis as the one propossed in Sect. 4.3.1 was not attempted. Instead, a mass integrated study of the agular distribution of the decay products, assuming no contamination from the S– wave amplitudes, was performed. Under these assumptions, the four-particle K+π−K−π+ distribution in the three helicity angles, θ1,θ2and ϕ(defined in Fig. 4.1), is described by the three transversity amplitudes A0,Akand A⊥. In a time-integrated and flavouraveraged analysis, and assuming no CP -violation arises in this decay as the Standard Model predicts, the angular distribution is given by 72 5.4 Analysis of K∗0polarization P DF (θ1, θ2, ϕ) = 1 ΓL|A0|2cos2θ1cos2θ2 +1 ΓL|Ak|21 2sin2θ1sin2θ2cos2ϕ +1 ΓH|A⊥|21 2sin2θ1sin2θ2sin2ϕ +1 ΓL 1 2√2|AL||Ak|cos δksin 2θ1sin 2θ2cos ϕ. (5.3) The measurable parameters of this PDF are the relative fraction of each of the amplitudes, usually referred to as polarisation fractions, fL,k =|A0,k |2 |AL|2+|Ak|2+|A⊥|2, k =k,⊥,(5.4) and δk, the phase difference between A0and Ak. The definition (5.4) implies that fL+fk+ f⊥= 1. The constants ΓL,H are the total widths of the light and heavy mass eigenstates of the B0 s-system, respectively, and their values were fixed to those obtained from the total B0 sdecay width, Γs= (ΓL+ΓH)/2, and the width difference, ∆Γs=ΓL−ΓH, reported in [16]. The effects induced by the detector geometrical acceptance, and the reconstrucction and selection processes need to be taken into account before comparing (5.3) to data. These effects were determined using simulated B0 s→K∗0K∗0events and were described in terms of the acceptance function ε, as a function fo the three decay angles. The acceptance function was found compatible with being constant in ϕ. In contrast, it has a significant dependence on the K∗0polarization angle θ1. The two-dimensional angular acceptance function, ε(cos θ1,cos θ2), drops asymmetrically as cos θ1,2becomes close to ±1, as a consequence of the minimum pand pTof the tracks imposed by the reconstruction and selection . This effect is more important for the limit cos θ→+1, i.e. when the πmeson is emitted backwards with respect to the K∗0momentum. This acceptance function is shown in Fig. 5.4. The Monte Carlo simulation of the K∗0acceptance was extensively cross-checked using the B0 d→J/ψK∗0control channel, taking advantage of the fact that the K∗0 polarization in this channel was measured at the B-factory experiments [104,105]. This acceptance shows no appreciable difference between K∗0and K∗0, and a small average correlation, given the size of the simulated sample. Consequently, the one-dimensional acceptance θ(cos θ) has been used as the basis of the analysis, and it has been determined it in five bins of cos θ. Since the longitudinal polarization fraction for the B0 d→J/ψK∗0 channel is well measured, a comparison between data and simulation is possible. Agreement was found including variations of the angular distribution with longitudinal and transverse K∗0momentum. In the region cos θ > 0.6 these variations were four times larger than for lower values of cos θ. The background cos θdistribution was studied in two 200 MeV/c2sidebands, defined below and above the B0 ssignal region. Like the signal, it showed a dip close to cos θ= +1 1This notation refers to a generic θangle, and will be followed from now on unless differences between θ1and θ2become relevant for the discussion. 73 Chapter 5. First observation of B0 s→K∗0K∗0 between B0 s→K∗0K∗0(fL= 0.31 ±0.12(stat.)±0.04(syst.)) and B0→K∗0K∗0 (fL= 0.80+0.10 −0.12(stat.)±0.06(syst.)), despite the fact that the two decays are related by a U-spin rotation. However, the ratio of the branching ratios of B0 sand B0decays is consistent with 1/λ2where λis the Wolfenstein parameter, as expected. 80 6 Time-integrated angular analysis of B0 s→K∗0K∗0 After the discovery of the B0 s→K∗0K∗0decay channel was stablished with the analysis described in the previous chapter, a more detailed study of this process was performed with a higher luminosity data sample taken by LHCb during 2011. With this larger dataset, the full angular analysis is feasible allowing an accurate determination of the S–wave contributions in the B0 s→(K+π−)(K−π+) final state. Also, a search for physics beyond the Standard Model can be performed through the measurement of the eight CP -violating quantities accessible to this decay. 6.1 Introduction One of the principal motivation of this work has been the search for possible new physics components to electroweak phases in the amplitudes describing the decay Bs→K+π−K−π+ in a mass window of ±150MeV/c2around the K∗0(892) resonance for both K+π−and K−π+systems. As explained in Chapter 4, six amplitudes are needed to describe such transition, which is dominated by the B0 s→K∗0(892)K∗0(892) (vector-vector or VV) and the B0 s→K∗ 0(800)K∗0(892)1(scalar-vector or SV) final states. In an untagged and time integrated analysis, to which this study with only 1f b−1 has been restricted, CP -violation and T-violation may arise from 4 triple products and 4 direct-like CP asymmetries measurable from the interference terms [8]. These terms involve the two CP -odd amplitudes A⊥(VV) and A+ s(SV). Additionally, the full angular and mass analysis of the 4-body final state is presented. The objective is to determine all magnitudes and measurable phases of the amplitudes A0, Ak, A⊥, A+ s, A− s, and Ass , under the assumption, supported by the triple product analysis in Sect. 6.7, that the CP -violating terms are negligible. Takin into account these results, a determination of the branching fraction of the VV mode is also provided. Particularly relevant to this analysis has been the measurement of the longitudinal fraction of the K∗0(892) polarization, and the magnitude of the overall S-wave contribution. Important theoretical activity has been generated in relation with Bs→K∗0¯ K∗0decays 1This notation refers to both CP -conjugated final states: B0 s→K∗ 0(800)K∗0(892) and B0 s→ K∗0(892)K∗ 0(800). Also, throughout this chapter the scalar contribution will be referred to as K∗ 0(800), which is nonetheless calculated as a superposition of a broad low mass structure and a K∗ 0(1430) relativistic amplitude. 81 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 and their potentiality for CP-violation analysis [4–6,44], and additional studies have been issued more recently that include the scalar final states [113]. 6.2 Data sample and Monte Carlo simulation The analysis presented in this chapter is based on the data taken by LHCb during 2011. This data sample corresponds to 1f b−1of pp collisions at a centre-of-mass energy of √s= 7 TeV. Events have been reconstructed using Brunel v41r1 and analysed using DaVinci v29r2, as corresponds to the Reco12-Stripping17 campaign. The Monte Carlo samples used in this analysis belong to the MC11a generation. They correspond to the decay channels B0 s→K∗0K∗0and B0→φK∗0. For this simulation, the pp interactions per crossing ν= 2.0, which corresponds to an average number of visible interactions per crossing µ= 1.4. The samples were generated using the Gauss release v41r2 (which uses GEANT4 v94r2p1.p02 for the detector simulation), and reconstructed using Boole v23r1 and Brunel v41r1p1. 6.3 Event selection and Signal Yield In this section, the selection requirements used to discriminate the B0 s→K∗0K∗0signal from the background are presented, together with the study of specific Bdecays that could contaminate the selected sample due to its similarities with the signal. Finally, the study of the invariant mass of the four particles in the final state is described and the number of B0 s→(K+π−)(K−π+) candidates is measured. 6.3.1 Event selection Events fulfilling the requirements in the StrippingBs2Kst0Kst0Line selection, coming from any physics trigger line1, were considered. Then, an offline selection very similar to the one used in Sect. 5.2.2 was applied. Table 6.1 shows the relevant decisions in each step of the trigger. Stripping and offline selections requirements are summarized in Table 6.2. Table 6.1: Trigger lines used for B0 s→K∗0K∗0selection. Trigger level Trigger lines L0 L0Global Decision Hlt1 Hlt1Phys Decision Hlt2 Hlt2Phys Decision To improve the signal significance a Geometrical Likelihood (GL) was introduced after the cuts indicated above. The GL was trained using truth-matched B0 s→K∗0K∗0MC events as signal, from the MC11a generation. As background, a sample of ∼2 pb−1 of 2010 data selected through the same stripping line was used. The signal region is excluded from the background sample by imposing |M(KπKπ)−mB0 s|>30 MeV/c2. The variables combined into the GL are: 1This means that the candidates do not need to follow any specific trigger path. 82 6.3 Event selection and Signal Yield Table 6.2: The signal selection requirements are indicated. The IPχ2is defined as the variation in the fit χ2for a given vertex (in this case it refers to the PV) reconstructed with and without the considered track. The B0 smeson candidate flight distance significance (FDS), is defined as FD/σFD, where F D is the distance between the PV and the B0 sdecay vertex and σFD is the uncertainty in the determination of that distance. B0 sDOCA is the distance of closest approach between the K∗0and the K∗0trajectories. DLLa−bdenotes the logarithm of the ratio between the probabilities of hypothesis a and b. The last two columns indicates the offline selection cuts, those on the right were applied after the GL definition. Stripping selection Offline selection All tracks pT>500 MeV All tracks IPχ2>9 All tracks χ2<5 K±DLLKπ >−5>2>10 K±DLLp−K<10 π±DLLK−π<10 <0 K∗0mass window ±150 MeV K∗0pt>900 MeV K∗0vertex χ2<9 Bsmass window ±500 MeV BsDOCA <0.3 mm Bsvertex χ2/ndof <15 <5 BsFDS >15 BsIPχ2<25 B0 s→K∗0K∗0GL >0.14 83 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 B_ctauh_b Entries 26114 Mean 0.7703 RMS 0.8699 0 1 2 3 4 5 6 0 500 1000 1500 2000 2500 3000 B_ctauh_b Entries 26114 Mean 0.7703 RMS 0.8699 τB c lessIPSh_b Entries 26114 Mean 2.352 RMS 2.126 0 10 20 30 40 50 60 70 80 0 2000 4000 6000 8000 10000 lessIPSh_b Entries 26114 Mean 2.352 RMS 2.126 less IPS B_DOCAh_b Entries 26114 Mean 0.04714 RMS 0.04177 0 0.05 0.1 0.15 0.2 0.25 0.3 0 500 1000 1500 2000 2500 3000 3500 B_DOCAh_b Entries 26114 Mean 0.04714 RMS 0.04177 B DOCA B_pth_b Entries 26114 Mean 1304 RMS 1074 0 5000 1000015000 20000 25000 30000 35000 40000 0 1000 2000 3000 4000 5000 B_pth_b Entries 26114 Mean 1304 RMS 1074 T B p B_IPh_b Entries 26114 Mean 0.06774 RMS 0.03894 0 0.05 0.1 0.15 0.2 0.25 0.3 0 500 1000 1500 2000 2500 3000 B_IPh_b Entries 26114 Mean 0.06774 RMS 0.03894 B IP GLK2012csb.h_b Entries 26114 Mean 0.002962 RMS 0.014 0 0.05 0.1 0.15 0.2 0.25 0.3 1 10 2 10 3 10 4 10 GLK2012csb.h_b Entries 26114 Mean 0.002962 RMS 0.014 GL Figure 6.1: Signal (red) and background (blue) samples distribution in the different variables entering the definition of the Geometrical Likelihood discriminator and the GL itself.  Lifetime of the B0 scandidate,  Minimum IP χ2of the four daughters track with respect to any the primary vertex,  B0 simpact parameter,  DOCA between the two K∗0candidates,  pTof the B. Fig. 6.1 shows the signal and background distribution in these variables together with the response of the calculated GL for signal and background. As expected the GL distribution is flat for the signal and clusters near 0 for the background. The optimal GL cut was obtained by maximizing S √S+Bfor 500 B0 s→K∗0K∗0expected events and 5000 background events, quantities chosen to approximate the S/B ratio expected after the rectangular-cut selection. The obtained result is ∼0.14, see Fig. 6.2. After the full selection, multiple candidates per event are very suppressed. Only one event in the final sample show two different candidates. For the subsequent analysis one of them was randomly discarded. 6.3.2 Specific backgrounds The need to keep a relatively wide mass window around the K∗0resonance, could allow peaking contribution from specific modes in the selected sample, mainly coming through 84 6.3 Event selection and Signal Yield GL 0 0.2 0.4 0.6 0.8 1 S+BS/ -1 10 1 10 SoRSB Figure 6.2: S/√S+Bas a function of the GL cut. The optimal GL cut was found to be ∼0.14 and is idicated by the blue arrow in the plot. the mis-identification of one of the four particles in the final state. Harder PID cuts have been applied in order to veto these crossfeeds, since in most cases there is no precise knowledge about their angular distribution. B0→ρK∗0decays are likely to be selected when a pion from the ρdecay is misidentified as a kaon. Fig. 6.3 shows the four body invariant mass (in the KπKπ hypothesis) of the events selected with different requirements in the DLLk−πof the kaon candidates. Events with smaller values of this discriminant tend to accumulate in the region between the B0 sand B0nominal masses, where the contribution from B0→ρK∗0is expected. A sample of ∼2 million Monte Carlo simulated B0→ρK∗0events was analysed in order to estimate the expected size of this contribution in our dataset for different values of the DLLk−πcut. Requiring the DLLk−πof the kaons in the event to be greater than 10, only 10 events survived. By normalising to B0→φK∗0(following the same procedure that will be discussed in Sect. 6.6 for signal), the number of B0→ρK∗0events expected in the signal region was estimated to be 3.5±1.6 (8.9±3.5 in the full mass range). Another possible peaking contamination comes from B0→φK∗0decays when a kaon from the φdecay is identified as a pion. No specific PID cut is applied to reject this kind of events, since their contribution is expected in the low mass sideband, far from the B0 s signal region. Fig. 6.6(b) in the next section shows the invariant mass distribution under the KπKπ mass hypotheses, of B0→φK∗0Monte Carlo simulated events selected using the requirements in Table 6.2 (with the exception of the PID cuts in the mis-identified kaon). Finally, a specific background coming from Λb→pπKπ decays has also been identified. Although this mode has not yet been discovered, if a proton from such a decay were misidentified as a kaon, these events would accumulate between the mass of the B0 sand the mass of the Λb. Fig. 6.4 shows the scatter plot of the four body invariant mass evaluated under the proton (antiproton) mass hypothesis or the K+(K−) mass hypothesis. The contaminating signal becomes evident when the RICH detector is used. In order to reject this source of background, the difference between the probabilities of proton and kaon hypothesis is required to be DLLp−K<10. 85 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 ) 2 ) (MeV/c + π, - , K - π, + M(K 5000 5500 0 50 100 150 200 250 300 > 0 πK- PID ± K >2 πK PID ± K < 2 πK- PID ± 0 < K Figure 6.3: Invariant mass of the four particles in the final state for different cuts in the P IDK−πof the kaon candidates. ) 2 ) (GeV/cπ K πM(p 5 6 7 8 ) 2 ) (GeV/cπ K πM(K 4.8 5 5.2 5.4 5.6 5.8 6 ) 2 ) (GeV/cπ K πM(p 5 6 7 8 ) 2 ) (GeV/cπ K πM(K 4.8 5 5.2 5.4 5.6 5.8 6 Figure 6.4: Left: Invariant mass of the four particles in the final state evaluated in the proton (antiproton) hypothesis versus the same mass in the K+(K−) hypothesis when no p/K separation is attempted in the RICH detector. Right: The same scatter plot under the requirement P IDp−K>10. 86 6.3 Event selection and Signal Yield 6.3.3 Four body mass fit After the selection explained above an unbinned maximum likelihood fit was performed to the mass spectrum of the selected K+π−K−π+candidates. The signal is modeled by a combination of two Crystal Ball distributions [106] that share a common mean and width. Their relative fraction and the parameters describing both tails (below and above a certain threshold) are extracted from a fit to B0 s→K∗0K∗0simulated data, see Fig. 6.5, and fixed in the fit to the data. The low mass tail of the distribution accounts for events that undergo final state radiation while the high mass tail is present due to events reconstructed with lower resolution. We use this parametrization to describe both B0 sand B0signals. The mass difference between B0 sand B0is fixed to the value calculated from [16]. The remaining two parameters (the mass of B0 sand a common width for B0 sand B0mesons) are determined from the fit. ) 2 ) (MeV/cπ K πM(K 5200 5300 5400 5500 5600 5700 ) 2 Events / ( 6 MeV/c 10 2 10 3 10 ) 2 ) (MeV/cπ K πM(K 5200 5300 5400 5500 5600 5700 pull -5 0 5 Figure 6.5: Fit to the four–body invariant mass spectrum of B0 s→K∗0K∗0simulated data using the model described in the text, which contains a radiative component (red) and a high mass tail accounting for events reconstructed with low resolution (green). Even though the contribution from the different crossfeed backgrounds considered in the previous section is suppressed by the PID cuts, a parameterization for each of them has been included in the fit and the fraction of each mode with respect to the total number of background events, Nbkg, is allowed to float in the minimization:  B0→ρK∗0events are parameterized using a Crystal Ball distribution. The parameters of the distribution are extracted from a fit to data. A very tight selection was applied to isolate B0→ρK∗0signal (see Appendix B). Fig. 6.6(a) shows the invariant mass of the four particles in the final state under the KπKπ mass hypothesis for this sample.  B0→φK∗0contribution is modeled using a combination of two Crystal Ball distributions with parameters obtained from a fit to B0→φK∗0simulated events, see Fig. 6.6(b). 87 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 Table 6.3: Parameter values of the models describing the signal and the different peaking contributions taken into account in the invariant mass fit. The parameters αand nrepresent, respectively, the threshold and the order of the power law tail of the corresponding Crystal Ball distribution. Parameter B0 s→K∗0K∗0B0→ρK∗0B0→φK∗0Λ0 b→pπ−K∗0 µ5367.08 ±0.11 5349.2±1.6 5214.77 ±0.69 5496.2±1.6 σ14.418 ±0.093 28.7±1.3 19.70 ±0.52 31.4±1.0 α11.71 ±0.14 −0.86 ±0.10 0.463 ±0.025 0.306 ±0.017 n11.87 ±0.12 8.8±3.7 8.0±1.1 3.82 ±0.30 fCB10.60 ±0.14 1 0.987 ±0.013 1 α2−2.00 ±0.20 - −0.49 ±0.62 - n22.67 ±0.61 - 4.0±3.4 -  Since no MC sample for Λ0 b→pπKπ was available, a simplified four-momentum simulation for Λ0 b→(pπ)K∗0was used [114]. In it, the Λbmomentum spectrum is taken from the one observed for Bs in the full MC, and the 2-body phase-space is used to perform Λbdecay into K∗0and a pπ system with the invariant mass observed in data. The resulting M(KπKπ) distribution, shown in figure Fig. 6.6(c), is modeled using a Crystal Ball distribution. The parameters of these models are summarized in Table 6.3. Additionaly, a modified ARGUS shape, i.e. a convolution of the ARGUS distribution [115] and a Gaussian, accounts for partially reconstructed B decays, and is described by: fP(m)∝ ·m01−m02 m2 0Θ(mP hysBkg −m0)e−kP hysBkg ·m0⊗G(m−m0;σ) (6.1) where Θis the Heaviside-step function, ⊗stands for the convolution product, m0is the variable over which the convolution integral is calculated, G(m−m0;σP hysBkg) is a Gaussian p.d.f. with standard deviation σP hysBkg representing the experimental resolution, which is forced to be the same as the signal one (σ). mP hysBkg and kP hysBkg determine the sape of the partially reconstructed background and are allowed to float during the minimization. Finally, the combinatorial background is parameterized by a decreasing exponential with its slope (ccomb) floating in the fit. By fitting this model to the mass spectrum of the B0 s→K∗0K∗0selected candidates, the preferred value for the fraction of B0→ρK∗0events hits the lower physical limit (by definition fB0→ρK∗0≥0). For the final result, the fraction of B0→ρK∗0events is fixed to zero and the impact of a non-zero contribution from this decay is taken into account as a systematic uncertainty (see Sect. 6.6.5.1). The results of the fit to the four body mass spectrum are shown in Table 6.4. The fitted model is compared with data in Fig. 6.7. 88 6.3 Event selection and Signal Yield ) 2 ) (MeV/c + π, - ,K - π, + M(K 5300 5400 5500 5600 5700 ) 2 Events / ( 8.6 MeV/c 0 20 40 60 80 100 120 140 ) 2 ) (MeV/c + π, - ,K - π, + M(K 5300 5400 5500 5600 5700 pull -5 0 5 (a) ) 2 ) (MeV/cπ K πM(K 4800 4900 5000 5100 5200 5300 ) 2 Events / ( 5.5 MeV/c 0 100 200 300 400 500 600 700 ) 2 ) (MeV/cπ K πM(K 4800 4900 5000 5100 5200 5300 pull -5 0 5 (b) ) 2 ) (MeV/c + π, - ,K - π, + M(K 4600 4800 5000 5200 5400 5600 ) 2 Events / ( 14 MeV/c 0 100 200 300 400 500 600 ) 2 ) (MeV/c + π, - ,K - π, + M(K 4600 4800 5000 5200 5400 5600 pull -5 0 5 (c) Figure 6.6: Parameterisation of the peaking backgrounds: (a) B0→ρK∗0candidates selected from data when the mass hypothesis of one of the pions from the ρdecay is changed to the kaon hypothesis. (b) B0→φK∗0Monte Carlo simulated events when the mass hypothesis of one of the kaons from the φdecay is changed to the pion hypothesis. (c) ToyMC generated Λb→K∗0pπ events when the mass hypothesis of the proton is changed into the kaon hypothesis. 89 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 Kaon Momentum [MeV/c] 20 40 60 80 3 10× Kaon pseudorapidity 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Pion Momentum [MeV/c] 20 40 60 80 3 10× Pion pseudorapidity 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 6.13: Efficiency of the PID selection of kaons and pions in B0 s→K∗0K∗0candidates as a function of pseudorapidity. 1 θcos -1 -0.5 0 0.5 1 2 θ cos -1 -0.5 0 0.5 1 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 θcos -1 -0.5 0 0.5 1 MC ε / weighted ε 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 θcos 2 θcos Figure 6.14: Effect of reweighting the MC sample according to the PID selection efficiencies calculated in data. 1 θ cos -1 -0.5 0 0.5 1 (GeV/c) T p - π 1 2 3 4 5 6 7 0 100 200 300 400 500 2 θ cos -1 -0.5 0 0.5 1 (GeV/c) T p + π 1 2 3 4 5 6 7 0 100 200 300 400 500 Figure 6.15: cos θdistribution of the MC simulated data as a function of the transverse momentum of the pion. 96 6.4 Acceptance effects 1 θcos -1 -0.5 0 0.5 1 2 θcos -1 -0.5 0 0.5 1 0.5 1 1.5 2 2.5 3 -3 10× 1 θcos -1 -0.5 0 0.5 1 2 θcos -1 -0.5 0 0.5 1 0.5 1 1.5 2 2.5 -3 10× 1 θcos -1 -0.5 0 0.5 1 )θ(cosε 0 0.01 0.02 0.03 TOS non-TOS Figure 6.16: Top: Angular acceptance functions for TOS (left) and non-TOS (right) events calculated from B0 s→K∗0K∗0Monte Carlo simulated data after reweighting with PID efficiencies observed in data. Bottom: cosθ1projection of the TOS (blue) and non-TOS (red) acceptance functions. differences between data and MC in some kinematic variables, for instance, the pTof the daughter tracks. This could occur, for example, through the presence of S-wave in the data which is not present in the simulation. Still, genuine discrepancies between data and MC should be taken into account. In order not to overestimate these effects an iterative procedure is applied and a systematic uncertainty is assigned in Sect. 6.5.4.3. To take into account the different proportion of TOS and non-TOS events in data and MC we will perform a simultaneous fit to TOS and non-TOS data aplying a different acceptance correction to each sample. Both acceptance functions are shown in Fig. 6.16. 97 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 6.5 Amplitude Analysis The magnitude and phase of the different amplitudes contributing to the B0 s→K∗0K∗0 decay are determined using a 5D fit to the three helicity angles (Ω: cos θ1, cos θ2,ϕ) and the invariant mass of the two K-πpairs (m1≡M(K+π−), m2≡M(K−π+)) of all the candidates with a four-body invariant mass |M(K+, π−, K−, π+)−mB0 s|<30 MeV/c2. 6.5.1 The 5-D model The model used to describe the distribution of data in this five variables is given by F(Ω, m1, m2) = (1 −fbkg)P DF (Ω, m1, m2) ×ε(Ω, m1, m2) + fbkgP DFbkg(Ω, m1, m2) (6.3) where P DF (Ω, m1, m2) is the probability density function given by (4.18), ε(Ω, m1, m2) is the acceptance function describing the effects introduced by the data reconstruction, selection and triggering, and P DFbkg(Ω, m1, m2) describes de distribution of the background, which is a fraction fbkg of the full dataset. In order to avoid non-physical values of the parameters during the minimization, some of them have been rewritten as follows fk=xk·(1 −fL) |A+ s|2=x+ s·(1 −|A− s|2) |Ass|2=xss ·(1 −|A− s|2−|A+ s|2) (6.4) where x(fk), x(|A+ s|2) and x(|Ass|2) are free parameters allowed to float within (0,1), ensuring that the sum of all the squared amplitudes is never greater than 1. Consequently, the free parameters in the fit to the data are: fL,xk,|A− s|2,x+ s,xss,δk, (δ⊥−δ+ s), δ− s, δss; where the usual definition of the polarization fractions in B0 s→K∗0K∗0, fL=|A0|2 |A0|2+|Ak|2+|A⊥|2fk,⊥=|Ak,⊥|2 |A0|2+|Ak|2+|A⊥|2(6.5) has been assumed. Note that only the phase difference (δ⊥−δ+ s) is accesible to the untagged analysis. 6.5.1.1 Background The expected background fraction in the ±30 MeV/c2mass window around the B0 smass is estimated from the mass fit to be (2.64±0.27)% for the TOS sample and (4.53±0.52)% in the non-TOS sample. The distribution of the background in the three angles and two masses is extracted from the events in the right-hand B0 smass sideband ([5550,5700] MeV/c2) with lower GL values (GL>0.01), see Fig. 6.17. We parametrize the background component as the factorized product p.d.f .Bkg(Ω, m1, m2) = MBkg(m1) ×MBkg(m2) ×FBkg(θ1)×FBkg(θ2) (6.6) 98 6.5 Amplitude Analysis 1 θcos -1 -0.5 0 0.5 1 Events / ( 0.2 ) 0 20 40 60 80 100 2 θcos -1 -0.5 0 0.5 1 Events / ( 0.2 ) 0 20 40 60 80 100 120 ϕ 0 2 4 6 Events / ( 0.628319 ) 0 10 20 30 40 50 60 70 80 ) 2 ) (MeV/c - π + M(K 0.8 0.9 1 3 10× ) 2 Events / ( 30 MeV/c 0 20 40 60 80 100 ) 2 ) (MeV/c + π - M(K 0.8 0.9 1 3 10× ) 2 Events / ( 30 MeV/c 0 10 20 30 40 50 60 70 80 90 Figure 6.17: Angular and masses distributions of events in the sideband defined in the text. The blue line is the projection of the fitted background parametrization. whith MBkg(m) = fB|M1(m)|2+ (1 −fB) log(λbkgm) FBkg(cos θ) = 1 + P5 1cbkg i(cos θ)iif cos θ < 0.8 0if cos θ > 0.8(6.7) where M(m1) is the spin-1 Breit-Wigner propagator defined in Sect. 4.3.1.1, and describe background candidates containing real K∗0mesons. The fraction of these events with respect to the total is represented by fB. The background distribution in ϕis compatible with being flat. The result of this fit is shown in Table 6.5. Table 6.5: Values of the parameters in the p.d.f. of the background component obtained from the fit to the events in the sideband. Parameter Value fB0.182+0.032 −0.059 λbkg (−2.3±4.2) ×10−4( MeV/c2)−1 cbkg 10.18 ±0.20 cbkg 2−1.05+0.31 −0.27 cbkg 3−2.30 ±0.89 cbkg 40.68 ±0.42 cbkg 51.28 ±0.76 Additionally, the sF it formalism was also tried to describe the background. The results can be found in Appendix 6.5.3.4 and are compatible with those obtained using the cF it approach. 99 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 6.5.2 Fit results Using the model described above, an unbinned maximun likelihood fit has been performed simultaneously for TOS and non-TOS B0 s→K∗0K∗0candidates with |M(K+, π−, K−, π+)− mB0 s|<30 MeV/c2. As already mentioned, the background fraction has been fixed to the one obtained from the fit to the KπKπ invariant mass spectrum. The results of the fit are sumarized in table 6.6. Fig. 6.18 shows the different projections of the fit. The evolution of the forward-backward asymmetry with the Kπ invariant mass becomes more clear in Fig. 6.19, where the cos θ1(cos θ2) distribution is shown for different bins of m1(m2). A low value of the longitudinal porlarization fraction of the vector-vector component is measured. This results is compatible with that obtained in Sect. 5.4, and therefore still significantly smaller than the longitudinal polarization fraction measured by BaBar for the U-spin rotated decay, B0→K∗0K∗0. Furthermore, the overall contribution of the S–wave amplitudes is found to be large, |A0|2+|Ak|2+|A⊥|2= 0.665 ±0.067. In Fig. 6.20, the likelihood profile1for the parameter fLis given, showing parabolic behaviour around the minimum. Additionally, the (1-6 σ) contour between |A− s|2and fLis shown in Fig. 6.21. No additional minimum with inverted values of |A− s|2and fL, that would also describe the evolution of the forward backward asymmetry with the mass observed in data, is found. Table 6.6: Results given by the simultaneous fit to B0 s→(K+π−)(K−π+) TOS and non- TOS candidates with |M(K+, π−, K−, π+)−mB0 s|<30 MeV/c2. The values of fk,|A+ s|2and |Ass |2are calculated, following (6.4), from the free parameters in the PDF, whose correlations have been taken into account in the error calculation. Parameter Value fL0.201 ±0.057 xk0.269 ±0.055 |A− s|20.485 ±0.051 x+ s0.222 ±0.058 xss 0.164 ±0.050 δk5.31 ±0.24 δ⊥−δ+ s1.95 ±0.21 δ− s1.79 ±0.19 δss 1.06 ±0.27 fk0.215 ±0.046 |A+ s|20.114 ±0.037 |Ass|20.066 ±0.022 1The likelihood profile for a particular parameter is obtained by minimising the likelihood with respect to the rest of the parameters, for each (fixed) value of the parameter of interest. 100 6.5 Amplitude Analysis 1 θcos -1 -0.5 0 0.5 1 Entries/(0.2) 0 50 100 150 2 θcos -1 -0.5 0 0.5 1 Entries/(0.2) 0 50 100 150 (rad)ϕ 0 2 4 6 Entries/(0.62) 0 20 40 60 80 100 ) 2 ) (MeV/c - π + M(K 800 900 1000 ) 2 Events / ( 20 MeV/c 0 50 100 150 ) 2 ) (MeV/c + π - M(K 800 900 1000 ) 2 Events / ( 20 MeV/c 0 50 100 150 ) 2 ) (MeV/c - π + M(K 800 900 1000 FB Asymmetry -0.5 0 0.5 ) 2 ) (MeV/c + π - M(K 800 900 1000 FB Asymmetry -0.5 0 0.5 Figure 6.18: Projections of the model fitted to B0 s→(K+π−)(K−π+) data (blue solid line). The solid dots represent the selected data after the background component has been subtracted following (6.6) and the acceptance effect has been corrected. The red dashed line is the P-wave component, the green dashed line is the S-wave component and the light-blue dashed line represents the A+ SA0interference term. 101 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 1 θcos -1 -0.5 0 0.5 1 Entries/(0.2) 0 20 40 60 80 2 [746.0,886.0] MeV/c∈) - π + m(K 1 θcos -1 -0.5 0 0.5 1 Entries/(0.2) 0 20 40 60 80 2 [886.0,926.0] MeV/c∈) - π + m(K 1 θcos -1 -0.5 0 0.5 1 Entries/(0.2) 0 20 40 60 80 2 [926.0,1046.0] MeV/c∈) - π + m(K 2 θcos -1 -0.5 0 0.5 1 Entries/(0.2) 0 20 40 60 80 2 [746.0,886.0] MeV/c∈) + π - m(K 2 θcos -1 -0.5 0 0.5 1 Entries/(0.2) 0 20 40 60 80 2 [886.0,926.0] MeV/c∈) + π - m(K 2 θcos -1 -0.5 0 0.5 1 Entries/(0.2) 0 20 40 60 80 2 [926.0,1046.0] MeV/c∈) + π - m(K Figure 6.19: cos θ1(top) and cos θ2(bottom) distributions for different bins of m1and m2 respectively. The solid dots represent the selected data after background subtraction and acceptance correction. The red dashed line is the P–wave component, the green dashed line is the S–wave component and the light-blue dashed line represents the A+ SA0interference term. L f 0 0.2 0.4 0.6 0.8 1 Projection of Profile of -log(likelihood) 0 50 100 150 L f 0 0.2 0.4 0.6 0.8 1 Projection of Profile of -log(likelihood) 0 2 4 6 8 10 Figure 6.20: Negative ∆ log likelihood profile for the parameter fL(left) and zoom around the minimum (right). Only the statistical uncertainty is included. 102 6.5 Amplitude Analysis L f 0 0.2 0.4 0.6 0.8 1 2 | - s |A 0 0.2 0.4 0.6 0.8 1Minimum σ1 σ2 σ3 σ4 σ5 σ6 Figure 6.21: Likelihood confidence regions in the |A− s|-fLplane. The best fit result is represented by the black solid dot. 103 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 6.5.3 Additional cross-checks The result obtained in the previous section confirms the low longitudinal polarization fraction in the B0 s→K∗0K∗0decay measured in Sect. 5.4 and reveals that the S–wave contribution is larger than those observed in similar decays, such as B0→φφ [116] and B0 s→φK∗0[117]. A series of crosschecks have been carried out in order to test the validity of this result. 6.5.3.1 M(K+π−)×M(K−π+) analysis in the wide mass window Although the baseline analysis has focused on the ±150 MeV/c2window around the nominal K∗0(892) mass, an additional invariant mass analysis has been performed in an extended window, spanning the interval [740,1700] MeV/c2, in order to assess the presence of higher partial waves. In the neighbourhood of M(Kπ)∼1430 MeV/c2a resonance corresponding to the triplet states K∗ J(1430) with J= 0,1,2, from B0 s→K∗ J(1430)K∗0(892), is expected. A full mass-dependent angular analysis including all possible amplitudes with J= 0,1,2 and their corresponding interference terms, which would extend, with a similar level of precision, the one performed in the previous section, has not been attempted. Nonetheless a simplified version of such analysis is presented here, where only the invariant masses m1≡M(K+π−) and m2≡M(K−π+) are used, in the m1×m2plane. This allows to assess the relative contributions of the S, P, and D partial waves and to perform a rough estimate of their extrapolation into the region |m1,2−mK∗0(892)|<150 MeV/c2. It is particularly important to verify that the contribution of the D-wave is indeed negligible in the region allowed by our main analysis. The data sample used in this section has been selected through the StrippingBs2Kst - 0Kst 0Line stripping line from the same data sample described in Sect. 6.2. The requierements of this line, together with the offline cuts applied to the candidates are shown in Table 6.7. All physic trigger lines were considered in this study. Fig. 6.22 shows the scatter plot of the K+π−pair invariant mass versus the K−π+ pair invariant mass. The background has been subtracted in each 60 ×60 ( MeV/c2)2 bin through a fit to the KπKπ invariant mass, like the one described in Sect. 6.3.1. A multichannel analysis has been performed on the background subtracted data, using only the information of the invariant masses, in order to determine the contributions from the various partial waves in the Kπ system. In this check the effect of the asymmetric acceptance in the angular integration is neglected, so the interference terms between P–wave and S–wave cancel out. The full model to describe the Kπ invariant mass spectrum contains:  S–wave component. The K∗ 0(1430) is combined with a non-resonant term using the LASS parametrization described in Sect. 4.3.1.1. Alternatively, the S–wave component has been parameterised using the K–matrix formalism [118], see Appendix C.  P–wave. The P–wave resonances are combined in a single propagator following the expressions given in Sect. 4.3.1.1 TP=TK∗0+γ1TH,1+γ2TH,2(6.8) 104 6.5 Amplitude Analysis ) 2 c) (MeV/ - π + M(K 800 1000 1200 1400 1600 ) 2 c) (MeV/ + π - M(K 800 1000 1200 1400 1600 0 20 40 60 80 100 Figure 6.22: Background subtracted scatter plot of the K+π−pair invariant mass versus the K−π+pair invariant mass. Table 6.7: Stripping and offline requierements used to select the wide mass window dataset. The GL discriminator is defined in Sect. 6.3.1 Stripping selection Offline selection All tracks pT>500 MeV All tracks IPχ2>9 All tracks ProbNNghost <0.8 All tracks χ2<5 K±DLLKπ >2>4 K±DLLp−K<15 π±DLLK−π<0<−2 K∗0mass window [740,2100] MeV/c2[740,1700] MeV/c2 K∗0pt>900 MeV K∗0vertex χ2<9 Bsmass window ±500 MeV Bsdaughters PpTi>5000 MeV/c BsDOCA <0.3mm BsDIRA >0.99 Bsvertex χ2/ndof <15 <5 BsFDS >9>15 BsIPχ2<25 ΛbVETO (offline) ¬|M(pπKπ)−mΛb|<50 & KDLLp−K>0 B0 s→K∗0K∗0GL >0.14 105 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 ) 2 ) (MeV/c + π - , K - π, + M(K 5200 5400 5600 5800 ) 2 Events / ( 14 MeV/c 0 20 40 60 80 100 120 140 160 180 200 220 240 ) 2 ) (MeV/c + π - , K - π, + M(K 5200 5400 5600 5800 Bs weights 0 50 100 150 200 250 ) 2 ) (MeV/c + π - , K - π, + M(K 5200 5400 5600 5800 Bd weights 0 10 20 30 40 50 ) 2 ) (MeV/c + π - , K - π, + M(K 5200 5400 5600 5800 Bkg weights -10 0 10 20 30 40 Figure 6.26: Result of the fit to the 4 body mass spectrum, from which we obtain the signal weights (top left). B0 ssignal (top right), B0signal (bottom left) and background (bottom right) weights as a function of the invariant mass. Table 6.10: Comparison between the results obtained with the sF it method and the cF it method described in Sect. 6.5.1.1. The statistical uncertainty obtained with the sF it method for each of the parameters is also provided. Parameter ∆ sF it σ(stat) fL0.008 0.050 fk0.024 0.044 |A− s|20.006 0.046 |A+ s|20.004 0.031 |Ass|20.007 0.016 δk0.053 0.196 δ⊥−δ+ s0.017 0.182 δ− s0.060 0.173 δss 0.169 0.210 112 6.5 Amplitude Analysis Figure 6.27: Projections of the sF it in the angular variables, the two Kπ pairs masses and the FB asymmetries. 1 θcos -1 -0.5 0 0.5 1 Events / ( 0.2 ) 0 20 40 60 80 100 120 140 2 θcos -1 -0.5 0 0.5 1 Events / ( 0.2 ) 0 20 40 60 80 100 120 (rad)ϕ 0 2 4 6 Events / ( 0.628319 rad ) 0 20 40 60 80 100 ) 2 ) (MeV/c - π + M(K 800 900 1000 ) 2 Events / ( 27.2727 MeV/c 0 20 40 60 80 100 120 140 160 180 200 220 ) 2 ) (MeV/c - π + M(K 800 900 1000 ) 2 Events / ( 27.2727 MeV/c 0 20 40 60 80 100 120 140 160 180 200 ) 2 ) (MeV/c - π + M(K 800 900 1000 FB Asymmetry -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 ) 2 ) (MeV/c - π + M(K 800 900 1000 FB Asymmetry -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 113 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 6.5.4 Systematic uncertainties 6.5.4.1 Fit bias In order to study possible fit biases we have performed a simplified simulation study. Samples of the same size as the one found in data are generated from the PDF described in the previous sections, with values of the physics parameters similar to those obtained from the nominal fit to the data. These toy samples are then fitted using the same PDF, and the obtained parameters are compared with those used in generation to check that no bias is introduced in the analysis. In total, 500 experiments are generated and fitted. The results of the gaussian fits to the pull distributions of the fitted parameters are shown in Fig. D.1 in Appendix D.1. In Table 6.11 the mean and width of these gaussian distributions are summarised. The maximum between the fit bias and its uncertainty is taken as a systematic uncertainty. Table 6.11: Pull mean and width for the parameters obtained from the angular fit. The expected bias and systematic uncertainty are also quoted. Parameter Pull mean Pull width σ(stat) Bias Syst. fL0.030 ±0.033 1.000 ±0.026 0.057 0.002 ±0.002 0.002 fk-0.094 ±0.036 1.068 ±0.031 0.046 -0.004 ±0.002 0.004 |A+ s|2-0.082 ±0.038 1.070 ±0.030 0.037 -0.003 ±0.001 0.003 |A− s|2-0.064 ±0.035 1.036 ±0.029 0.051 -0.003 ±0.002 0.003 |Ass|2-0.064 ±0.039 1.004 ±0.032 0.022 -0.001 ±0.001 0.001 δk0.035 ±0.032 0.958 ±0.024 0.240 0.008 ±0.008 0.008 δ⊥−δ+ s-0.113 ±0.032 0.967 ±0.027 0.210 -0.024 ±0.007 0.024 δ− s0.112 ±0.034 0.999 ±0.029 0.190 0.021 ±0.006 0.021 δss 0.044 ±0.031 0.941 ±0.023 0.270 0.012 ±0.008 0.012 6.5.4.2 MC statistics In order to estimate the systematic error in the fit parameters induced by the limited statistics available in the MC, the data were fitted one thousand times after performing random variations of the acceptance function according to its statistical uncertainty. The results can be seen in Appendix D.2. The width of a gaussian fit to the pull obtained for each parameter was taken as the systematic uncertainty, see Table 6.12. 6.5.4.3 Data & Monte Carlo discrepancies Appendix D.3 shows a comparison between the data and MC for the main variables entering the selection. From the point of view of the angular analysis, the discrepancies in the pTspectrum of the B0 smeson and its daughters need to be taken into account, since the pTselection cut is responsible for most of the acceptance effect. As it has been previously explained, part of these discrepancies can be related to the different set of polarisation amplitudes generated in the simulation and measured in data. The discrepancy coming from any different source needs to be taken into account when 114 6.5 Amplitude Analysis Table 6.12: Systematic uncertainties of the parameters in the fit associated to the statistical uncertainty in the determintation of the detector angular acceptance. Parameter Pull width fL0.0095 fk0.0083 |A− s|20.0072 |A+ s|20.0050 |Ass|20.0007 δk0.0368 δ⊥−δ+ s0.0192 δ− s0.0357 δss 0.0759 calculating the angular acceptance. In order to assess this effect an iterative procedure is applied. This procedure reweights the MC in helicity angles before comparing the B0 sand π±pTspectra (the K±spectra are compatible between data and MC) with those in data to extract a correction. This correction is applied in the calculation of the new acceptance function and the full fit to the data is repeated. The procedure goes as follows: 1. Fit to data using the acceptance function calculated from the nominal MC simulation. 2. Reweight the Monte Carlo in the angular variables according to the result of the fit in step 1. 3. Compare the B0 sand π±pTdistributions between MC and data and extract a correction function for these variables. 4. Using the previous correction, recalculate the angular acceptance from Monte Carlo. 5. Use this new acceptance to fit to data. 6. Go back to step number 1, and repeat until the fit result converges. This procedure is applied separately for TOS and non-TOS data sets. The variation of the parameter values in each iteration are shown in Table 6.13. The result of the fit converges after 4-5 iterations. The variations corresponding to the last iteration are applied as a correction to the analysis, and also taken as an estimate of the systematic error arising from incorrect description of the pTspectra in the MC. The effect of the iterative procedure on the cos θacceptance function is illustrated in Fig. 6.28. Additionally, it has been checked that the MC describes well the differences between TOS and non-TOS events. The ratio TOS/non-TOS found in data with the corresponding ratio predicted by simulation have been compared, see Fig. 6.29, and were found to be compatible with the available statistics. Thus, no additional systematic uncertainty is considered. 115 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 θcos -1 -0.5 0 0.5 1 MC ε/ iter-5 ε 0 0.5 1 1.5 2 1 θcos 2 θcos TOS θcos -1 -0.5 0 0.5 1 MC ε/ iter-5 ε 0 0.5 1 1.5 2 1 θcos 2 θcos no-TOS Figure 6.28: Ratio between the acceptance functions calculated from MC before and after the correction in the pTspectra obtained with the iterative method described in the text. Left: TOS sample. Right: non-TOS sample. θcos -1 -0.5 0 0.5 1 TOS ε / no-TOS ε 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Data MC Figure 6.29: Comparison of the ratio between TOS and non-TOS angular acceptance functions for data and MC. Table 6.13: Variation in the fit result for different acceptance functions calculated through the iterative procedure explained in the text. The variation corresponding to the fifth iteration, after which the result is stable, is taken as a systematic uncertainty. ∆B0 spTIter 1 Iter 2 Iter 3 Iter 4 Iter 5 Syst. fL0.0023 0.0094 0.0100 0.0101 0.0101 0.0101 0.0101 fk-0.0006 -0.0034 -0.0037 -0.0038 -0.0038 -0.0038 -0.0038 |A+ s|2-0.0005 -0.0009 -0.0018 -0.0020 -0.0021 -0.0021 -0.0021 |A− s|20.0031 0.0085 0.0098 0.0101 0.0102 0.0102 0.0102 |Ass|20.0001 0.0002 0.0003 0.0003 0.0003 0.0003 0.0003 δk-0.0058 -0.0269 -0.0379 -0.0408 -0.0415 -0.0417 -0.0417 δ⊥−δ+ s-0.0024 0.0002 -0.0002 -0.0004 -0.0004 -0.0004 -0.0004 δ− s-0.0185 -0.0580 -0.0709 -0.0742 -0.0750 -0.0753 -0.0753 δss -0.0260 -0.1269 -0.1718 -0.1838 -0.1869 -0.1877 -0.1877 116 6.5 Amplitude Analysis 6.5.4.4 Acceptance model As explained in section Sect. 6.4.1, the acceptance function was assumed to be a factorizable function of the helicity angles and the invariant mass of the Kπ pairs. To test the validity of this assumption, different models were tried for the acceptance function. The small statistics in the MC1and the strong acceptance effect in cos θ1,2makes a 5D treatment of the acceptance very difficult. As an alternative, a generic cos θ-mKπ model was tried. The acceptance was described by ε(m, cos θ) = X i,j cij Pi(m0)Pj(cos θ) (6.12) where Piare Legendre polynomials of order iand m0= 2(m−mmin)/(mmax −mmin)−1 is a renormalization of the Kπ invariant mass. The values of mmin and mmax are the boundaries of the considered Kπ invariant mass range, mmin =mK∗0−150 MeV/c2and mmax =mK∗0+ 150 MeV/c2. In order to calculate the cij coefficients from the Monte Carlo sample, one can take advantage of the general averaging procedure for a generic function f(X) of the observables X:{m1, m2, Ω}, 1 Ngen X accepted f(X) = 1 Ngen X generated f(X)(X) ≈ZP DF (X)(X)f(X)dX (6.13) where (X) represents the efficiency of accepting an event and P DF (X) is the probability density function used to generate the MC sample in dX≡dm1dm2d(cos θ1)d(cos θ2)dϕ. In this particular case, the acceptance is assumed to factorize as (X) = ε(m1,cos θ1)× ε(m2,cos θ2) (and be flat in φ). By choosing fij (X) = 2i+ 1 2 2j+ 1 2 Pi(m0 1)Pj(cos θ1) P DF (X),(6.14) the average provides the desired cij coefficients 1 Ngen X accepted fij (Xn)≈2i+ 1 22j+ 1 2ZP DF (X)(X)Pi(m0 1)Pj(cos θ1) P DF (X)dX =Cnorm 2i+ 1 22j+ 1 2× ×X ab cabZPa(m0 1)Pb(cos θ1)Pi(m0 1)Pj(cos θ1)dm0 1d(cos θ1) =Cnorm cij ,(6.15) where the ortogonality properties of the Legendre polinomials have been used in the last step. The factor Cnorm = 2πmmax −mmin 22Zε(m0,cos θ)dm0d(cos θ),(6.16) 1From ∼2×106events generated, ∼20×103are selected. The sample is then split in TOS (60%) and non-TOS (40%). However, these events are not evenly distributed in the decay variables. The MC sample was generated with high polarization which means that regions where cos θ1,2→0 are less populated. The same occurs in the tails of the Kπ mass distribution. 117 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 Table 6.14: Acceptance coefficients calculated using the MC simulated TOS (left) and non-TOS (right) events as described in the text. Coefficient TOS Value c00 0.250 c01 -0.091 ±0.010 c02 -0.091 ±0.013 c03 -0.049 ±0.016 c04 -0.027 ±0.019 c10 0.015 ±0.014 c11 -0.020 ±0.020 c12 0.054 ±0.027 c13 -0.027 ±0.031 c14 -0.043 ±0.039 c20 -0.065 ±0.017 c21 0.054 ±0.022 c22 -0.012 ±0.036 c23 0.056 ±0.037 c24 0.035 ±0.045 Coefficient non-TOS Value c00 0.250 ±0.000 c01 -0.182 ±0.024 c02 -0.132 ±0.038 c03 0.082 ±0.046 c04 -0.001 ±0.046 c10 -0.018 ±0.049 c11 0.025 ±0.088 c12 0.139 ±0.089 c13 -0.180 ±0.104 c14 0.039 ±0.111 c20 0.070 ±0.056 c21 -0.136 ±0.118 c22 0.107 ±0.130 c23 0.051 ±0.126 c24 -0.124 ±0.122 is just a normalization constant for the full acceptance and can be ignored. Fig. 6.30 shows the acceptance for TOS events and Fig. 6.31 for non-TOS events. The data points correspond to B0 s→K∗0K∗0MC simulated data which were divided by the generator PDF on an event by event basis. The curve is the projection of the acceptance calculated using the procedure described before. The values obtained for cij are shown in Table 6.14. The fit to the data has been repeated using the acceptance calculated above. The difference in the fit parameters between the result obtained and the nominal one is shown in Table 6.15, and it is considered as an estimate of the systematic uncertainty. Table 6.15: Variation in the fit result when the analytical parameterization explained in the text is used as acceptance function. Parameter ∆ fL0.031 fk-0.008 |A− s|20.007 |A+ s|20.019 |Ass|2-0.003 δk-0.13 δ⊥−δ+ s0.016 δ− s-0.16 δss -0.096 total S–wave 0.022 118 6.5 Amplitude Analysis 1 θcos -1 -0.5 0 0.5 1 Acceptance 0 0.5 1 1.5 2 2.5 9 10× 2 θcos -1 -0.5 0 0.5 1 Acceptance 0 0.5 1 1.5 2 9 10× ) 2 ) (MeV/c - π + m(K 0.8 0.9 1 3 10× Acceptance 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 9 10× ) 2 ) (MeV/c + π - m(K 0.8 0.9 1 3 10× Acceptance 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 9 10× Figure 6.30: Acceptance of the TOS events as a function of the helicity angle and mKπ. The points correspond to MC simulated data and the blue curve is the projection of the acceptance calculated using the method described in the text. 1 θcos -1 -0.5 0 0.5 1 Acceptance 0 1 2 3 4 5 9 10× 2 θcos -1 -0.5 0 0.5 1 Acceptance 0 1 2 3 4 5 9 10× ) 2 ) (MeV/c - π + m(K 0.8 0.9 1 3 10× Acceptance 0 1 2 3 4 5 9 10× ) 2 ) (MeV/c + π - m(K 0.8 0.9 1 3 10× Acceptance 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 9 10× Figure 6.31: Acceptance of the non-TOS events as a function of the helicity angle and mKπ. The points correspond to MC simulated data and the blue curve is the projection of the acceptance calculated using the method described in the text. 119 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 6.5.4.5 Mass resolution In order to estimate the effect of neglecting the mass resolution in the model of the Kπ mass spectrum, a set of 1000 independent toy experiments was performed. The Kπ mass was smeared according to a gaussian resolution of 5 MeV/c2, similar to the one estimated from Monte Carlo simulation (see Appendix D.4), and the smeared data were fitted to the same model used at the generator level. The pull distributions of the fit parameters are shown in Fig. 6.32, while the mean and width from a gaussian fit to those distributions are summarized in Table 6.16. As expected, the overall contributions of the S–wave components, represented dominantly by the |A− s|2parameter, decreases as a consequence of the flatter K∗0line shape. But the effect is small in absolute terms of the S–wave fraction: -0.01 (22% of the statistical error). A small bias in the phases of the amplitudes was also found. These variations are taken as the systematic uncertainties associated to the invariant mass resolution. Table 6.16: Pull mean and width for the parameters obtained from the angular fit. The expected bias and systematic uncertainty are also quoted. Parameter Pull mean Pull width σ(stat) Bias Syst. fL-0.027 ±0.035 0.998 ±0.029 0.057 -0.002 ±0.002 - fk-0.036 ±0.034 1.019 ±0.026 0.046 -0.002 ±0.002 - |A+ s|20.030 ±0.037 1.014 ±0.031 0.037 0.001 ±0.001 - |A− s|2-0.218 ±0.033 0.987 ±0.025 0.051 -0.011 ±0.002 0.011 |Ass|2-0.028 ±0.041 0.965 ±0.035 0.022 -0.001 ±0.001 - δk-0.102 ±0.031 0.944 ±0.023 0.240 -0.024 ±0.007 0.024 δ⊥−δ+ s0.052 ±0.034 1.032 ±0.027 0.210 0.011 ±0.007 0.011 δ− s-0.117 ±0.034 0.950 ±0.026 0.190 -0.022 ±0.006 0.022 δss 0.161 ±0.032 0.953 ±0.026 0.270 0.043 ±0.009 0.043 6.5.4.6 S-wave mass model As described in Sect. 4.3.1.1, the S–wave component of the Kπ spectrum is described by a combination of a relativistic Breit–Wigner amplitude and a non–resonant amplitude following [37]. Different models were tried to check the consistency of the result. First variation consisted of parameterising the S–wave component only with a relativistic spin-0 Breit–Wigner propagator at the mass of K∗ 0(1430). A combination of two Breit–Wigner propagators (BW) at the poles of κ(800) and K∗ 0(1430) following the Isobar model was also tried, M0(m) = α BW (mκ, Γκ) + BW(mK∗0 0(1430), ΓK∗0 0(1430)) (6.17) where the magnitude and phase of the constant αwere both floating during the minimisation. Three different values of the mass and width of the κstate were tested: (A) : mκ= 682 ±29 MeV/c2;Γκ= 547 ±24 MeV/c2 (B) : mκ= 658 ±13 MeV/c2;Γκ= 557 ±24 MeV/c2 (C) : mκ= 700 ±80 MeV/c2;Γκ= 650 ±120 MeV/c2 following [16], [91] and [90] respectively. 120 6.5 Amplitude Analysis pull L f -5 0 5 0 10 20 30 40 50 60 / ndf 2 χ 75.13 / 57 Constant 1.93± 44.03 Mean 0.03487± -0.02669 Sigma 0.0291± 0.9977 pullf -5 0 5 0 10 20 30 40 50 / ndf 2 χ 42.84 / 47 Constant 1.81± 44.91 Mean 0.03394± -0.03604 Sigma 0.026± 1.019 pullδ -5 0 5 0 10 20 30 40 50 60 / ndf 2 χ 39.79 / 47 Constant 1.97± 48.54 Mean 0.0315± -0.1018 Sigma 0.0235± 0.9436 pull 2 | - s |A -5 0 5 0 10 20 30 40 50 60 / ndf 2 χ 52.63 / 48 Constant 1.87± 45.89 Mean 0.0334± -0.2175 Sigma 0.0250± 0.9866 pull 2 | + s |A -5 0 5 0 10 20 30 40 50 / ndf 2 χ 67.71 / 49 Constant 1.94± 44.11 Mean 0.03725± 0.03007 Sigma 0.031± 1.014 pull 2 | ss |A -5 0 5 0 10 20 30 40 50 60 / ndf 2 χ 104.9 / 54 Constant 2.13± 44.29 Mean 0.04091± -0.02769 Sigma 0.0346± 0.9652 pull - s δ -5 0 5 0 10 20 30 40 50 60 / ndf 2 χ 66.82 / 45 Constant 2.01± 46.88 Mean 0.0341± -0.1166 Sigma 0.0268± 0.9501 pull + s δ - δ -5 0 5 0 10 20 30 40 50 / ndf 2 χ 38.72 / 52 Constant 1.84± 44.52 Mean 0.03402± 0.05267 Sigma 0.027± 1.032 pull ss δ -5 0 5 0 10 20 30 40 50 60 / ndf 2 χ 42.21 / 45 Constant 1.99± 48.12 Mean 0.0318± -0.1609 Sigma 0.0258± 0.9528 Figure 6.32: Pull distributions of the parameters in the fit obtained from toy experiments simulating data used for the final result including the effect of the mass resolution. ) 2 ) (MeV/cπM(K 750 800 850 900 950 1000 ) 2 Entries/(20 MeV/c 0 20 40 60 80 100 120 140 ) 2 ) (MeV/cπM(K 750 800 850 900 950 1000 Swave 0 1 2 3 4 5 6 7 8 9 10 LASS (1430); (A) 0 + K*κ α(1430); (B) 0 + K*κ α(1430); (C) 0 + K*κ α(1430) 0 K* LASS (1430); (A) 0 + K*κ α(1430); (B) 0 + K*κ α(1430); (C) 0 + K*κ α(1430) 0 K* Figure 6.33: Different models for the S–wave mass propagator. 121 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 Table 6.22: Fitted values of the parameters in the invariant mass model for B0→ K+K−K±π∓. The parameters quoted without error were fixed to the indicated value in the fit. µdis the mean of the B0peak, fCB is the fraction of the Crystal Ball (1 −fCB is the fraction of the Gaussian), σCB is the width of the Crystal Ball, aCB and nCB are Crystal Ball parameters and σGauss is the width of the signal Gaussian distribution. cbkg is the slope of the exponential distribution describing the combinatorial background. kP hysBkg and mP hy sBkg are the parameters of the ARGUS shape describing partially reconstructed events, that represent a fraction fP hysBkg of the total background. Parameter Value NB01049 ±33 NB0 s27.1±6.4 NBkg 234 ±18 µd( MeV/c2) 5283.98 ±0.50 σCB ( MeV/c2) 14.65 ±0.41 aCB 2.56 nCB 1.1 fCB 0.915 σGauss ( MeV/c2) 24.5 cbkg (( MeV/c2)−1) (−0.8±1.3) ×10−3 fP hysBkg 0.613 ±0.091 kP hysBkg (1.40 ±0.41) ×10−2 mP hysBkg 5153.4±9.3 128 6.6 Branching ratio of B0 s→K∗0K∗0 6.6.2.1 Generator efficiency Generator level efficiencies are found by generating simulated signal events using the MC11a release of Gauss (v41r4) and then counting how many events have all their final state particles falling within the LHCb geometric acceptance. The generator level efficiencies for signal and control channel are given in Table 6.23. Additionally, EvtGen will not generate resonances with masses further m0±15 ×Γ0, where m0and Γ0are the mass and the width of the resonance. This cutoff in the mass of the resonance applied at the generator level should also be taken into account in the global efficiency. We compute the global efficiency as ε=N Ngen (6.21) where Nis the number of simulated events we have after reconstruction, trigger and selection processes, and Ngen is the number of generated events. To properly calculate the global efficiency we should replace Ngen by Ngen/(1−η) (or εby ε×(1−η)), where η is the fraction of events with |m−m0|>15Γ0according to the lineshape description used in generation [98]. In the case of B0 s→K∗0K∗0, this number is calculated by integrating the 2D distribution in m(K+π−) and m(K−π+), η=RR∞ mK∗0+15ΓK∗0P DF (m1, m2)dm1dm2 RR∞ mπ+mKP DF (m1, m2)dm1dm2 (6.22) since (mπ+mK)>(mK∗0−15ΓK∗0). A equivalent expression can be written for B0→ φK∗0. The values obtained for ηin both channels is shown in Table 6.23. Table 6.23: Generator efficiencies for B0 s→K∗0K∗0and B0→φK∗0decays and correction from the generator level mass cutoff. Channel εgen (%) η B0→φK∗017.47 ±0.04 0.09093 ±0.00001 B0 s→K∗0K∗016.02 ±0.04 0.02659 ±0.00001 6.6.2.2 Selection efficiency The reconstruction and selection efficiencies, englobed in εsel , were calculated from B0 s→ K∗0K∗0and B0→φK∗0simulated data by applying the selection in table Table 6.2 and Table 6.21 respectively, with the exception of the PID cuts. Table 6.24 contains the selection efficiencies for both signal and normalization channel. The uncertainties in the efficiencies are calculated using the binomial formula, σ(ε) = rε(1 −ε) N(6.23) where N is the total number of events before the selection is applied. 129 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 Table 6.24: Reconstruction and selection efficiencies for B0 s→K∗0K∗0and B0→φK∗0 decays. Channel εsel (%) B0→φK∗05.600 ±0.023 B0 s→K∗0K∗04.605 ±0.015 ratio 1.216 ±0.006 6.6.2.3 PID efficiency The efficiency associated with the particle identification cuts in the signal and normalisation channel selections is estimated in this section. As already mentioned, the discrepancies between data and MC in the PID variables, leading to discrepancies in the efficiencies, need to be taken into account. In order to do so, the same approach explained in Sect. 6.4.3 has been followed.Calibration samples have been used to determine the efficiency of selecting a known ID track by applying a certain PID cutas a function of the track properties. In this case, the momentum and pseudorapidity of the track were used. Fig. 6.13 shows the PID performance histograms for pions and kaons selected by imposing the DLLa−b(a, b =K, π, p) requirementes in Table 6.2. This information is then used to reweight the B0 s→K∗0K∗0MC sample and the average PID efficiency is computed. The same procedure is followed to reweight the B0→φK∗0MC according to the PID efficiencies corresponding to the requirements in Table 6.21. The resulting PID efficiencies for signal and normalization channel are summarized in Table 6.25. Table 6.25: PID efficiencies for B0 s→K∗0K∗0and B0→φK∗0decays. The efficiencies are calculated separately for different magnet polarities (Magnet Up and Magnet Down) and the average is computed. Channel εsel MU (%) εsel MD(%) εsel (%) B0→φK∗048.70 ±0.14 48.44 ±0.15 48.57 ±0.10 B0 s→K∗0K∗045.00 ±0.11 45.30 ±0.11 48.57 ±0.08 ratio - - 1.076 ±0.003 6.6.2.4 Trigger efficiency The trigger efficiencies for the signal and normalization channels have been computed in two steps: the level-0 trigger (L0) efficiency and the High level trigger (HLT) efficiency, εtrig =εL0×εHLT .(6.24) The L0 is responsible for most of the trigger acceptance effect in B0 s→K∗0K∗0and B0→φK∗0, and its efficiency has been calculated directly from data as explained below. Simulated events have been used to determine the efficiency corresponding to the High level trigger. The efficiency of any choice of trigger lines can be measured on real data using 130 6.6 Branching ratio of B0 s→K∗0K∗0 Table 6.26: L0 efficiencies for signal and normalization channels calculated from data using the TISTOS technique. Channel εL0(%) B0→φK∗046.5±3.9 B0 s→K∗0K∗047.8±5.0 ratio 0.97 ±0.13 TISTOS technique [122] as follows εtrig =εT IS Ntrig NT IS (6.25) where TIS or “Triggered Independent of Signal” designate candidates triggered by tracks that do not belong to the final state of interest (for instance, tracks coming from decays of the accompanying b-quark) and Ntrig is the total number of triggered events. The TIS efficiency, εT IS, can also be determined from data as εT IS i=NT IS&T OS NT OS i (6.26) ibeing a small enough region of the signal B-meson phasespace (i.e. a B pTbin) so the signal and underlying event properties are uncorrelated. This procedure has been followed to calculate the level-0 trigger efficiency for B0 s→ K∗0K∗0and B0→φK∗0. The bias induced in the calculation of εT IS when no binning in the Bphasespace is considered is correlated with the topology of the decay under study. Given the similar topology of B0 s→K∗0K∗0and B0→φK∗0, the cancelation of this bias in the ratio of the trigger efficiencies is assumed to be a good approximation. This approximation was tested in simulated data as it is explaind in Sect. 6.6.5.3. The number of L0T IS,L0T OS and L0T IS&L0T OS events for signal and normalization channels have been determined from a fit to the four body invariant mass spectrum of the events in each category, see Appendix E. The resulting L0 efficiencies are shown in Table 6.26. The HLT efficiency, that includes the effect of HLT1 and HLT2 steps, has been determined using MC simulated B0 s→K∗0K∗0and B0→φK∗0events. Table 6.27 shows the HLT trigger efficiencies estimated for both channels. Table 6.27: HLT trigger efficiencies for B0 s→K∗0K∗0and B0→φK∗0decays calculated from MC simulated data. Channel εHLT (%) B0→φK∗086.07 ±0.29 B0 s→K∗0K∗085.97 ±0.22 ratio 1.001 ±0.004 6.6.3 Purity The number of B0 scandidates in the invariant mass fit correspond rigorously to the number of B0 s→(K+π−)(K−π+) selected and triggered, with a Kπ mass in a 150 MeV/c2 131 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 window around the nominal K∗0mass. This includes K∗0, but also an S–wave component and the interference between the S–wave and P–wave components. Here, everything that is not S–wave or P–wave and S–wave interference is considered as K∗0since the contamination from higher resonances is expected to be negligible, see Sect. 6.5.3.1. Therefore, the fraction of B0 s→K∗0K∗0present in the sample is given by fB0 s→K∗0K∗0=ZP DF (X;|A+ s|=|A− s|=|Ass|= 0) ε(X) dX ZP DF (X)ε(X) dX ,(6.27) where X={Ω, m1, m2},P DF (X) is the probability density function given by (4.18) and ε(X) is the acceptance function (the efficiency as a function of X). Introducing the parameter values obtained from the amplitude analysis, see Table 6.6, this expression yields: fB0 s→K∗0K∗0= 0.405 ±0.036.(6.28) where we have propagated the statistical uncertainties in the magnitude and phase of the different amplitudes. A equivalent approach is needed to determine the fraction of B0→φK∗0within the B0 s→K+π−K+K−data. Using the results in [117], the value fB0→φK∗0= 0.760 ±0.018 (6.29) is obtained. 6.6.4 Overall angular acceptance The last effect we need to take into account is the dependence of the overall (recontruction, selection and trigger) efficiency, ε, with the angular distribution of the particles in the final state of both signal and normalization channels. This efficiency will be proportional to the integral of the observed angular distribution, divided by the integral of the physical distribution. For the B0 s→K∗0K∗0signal (no S–wave) this means ε∝RP DF (X;|A+ s|=|A− s|=|Ass|= 0) ε(X) dX RP DF (X;|A+ s|=|A− s|=|Ass|= 0) dX(6.30) The calculation of the efficiencies in the previous sections rely on MC simulated data which was generated with a certain choice of amplitudes that might differ from those measured in data. Thus the overall efficiency must be corrected by the factor λfL=εdata εMC =RXP DF (X;|A+ s|=|A− s|=|Ass|= 0) ε(X) dX (1 −|A+ s|2−|A− s|2−|Ass |2)RP DFMC(X)ε(X) dX(6.31) where P DFMC(X) represents the probability density function used in the generation of the MC sample. The numerator of (6.31) appears also in the expression used to compute the fraction of B0 s→K∗0K∗0, (6.27). Therefore, it is convenient to evaluate together 132 6.6 Branching ratio of B0 s→K∗0K∗0 Table 6.28: Values of the κfactor calculated for B0 s→K∗0K∗0and B0→φK∗0decays. This factor contains the purity and angular acceptance corrections to the efficiencies entering the B(B0 s→K∗0K∗0) calculation. Channel κ B0→φK∗01.382 ±0.035 B0 s→K∗0K∗03.123 ±0.257 ratio 0.442 ±0.038 λfL/fB0 s→K∗0K∗0to take better into account the correlations when propagating the statistical uncertainties. The factor that enters the branching ratio expression, κ, is defined as κB0 s→K∗0K∗0≡λfL(B0 s→K∗0K∗0)×1 fB0 s→K∗0K∗0 =RP DF (X)ε(X) dX RP DFMC(X)ε(X) dX×1 1−|A+ s|2−|A− s|2−|Ass |2(6.32) This correction, associated to B0 s→K∗0K∗0, has been evaluated from the results in Table 6.6. The measurement in [117] allow an equivalent calculation for the κB0→φK∗0. Both results are show in Table 6.28. 6.6.5 B(B0 s→K∗0K∗0) systematic uncertainties Three main systematic sources were considered in the calculation of the B(B0 s→K∗0K∗0): the selection efficiency, the trigger efficiency and the mass fit bias and systematic uncertainties. The systematic uncertainties associated to the purity and angular correction are calculated by propagating the uncertainties on the results from the amplitude analysis assessed at Sect. 6.5.4. 6.6.5.1 Invariant mass fit The systematic uncertainties induced by the model used in the mass fit are studied in this section. As explained before the B0 ssignal (as well as the B0signal) is described using a double CB, where the different tails account for radiative processes and low resolution events respectively. An alternative model, consisting of a combination of a Crystal Ball and a Gaussian was tried and the variation in the number of B0 ssignal events was taken as a systematic uncertainty. Regarding the background description, the contribution coming from B0→ρK∗0 decays is the most likely to cause a bias in the B0 syield, since this kind of events accumulate between the B0and B0 speaks. However, the size of this contamination after the selection in Table 6.2 is expected to be small. As explained in Sect. 6.3.3, if the fraction of B0→ρK∗0events is let free to float during the minimization, the lower limit of the parameter allowed interval is hit. Thus, in the nominal fit the contribution from this decay is fixed to zero. In order to estimate the impact of this assumption, the nominal result is compared with the one obtained when the number of B0→ρK∗0events is fixed to the one estimated from simulation (NB0→ρK∗0= 8.9±3.5). 133 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 Table 6.29: Variation in the number of signal events, when the parameterization of the different components of the fit is changed. The sum in quadrature of all the components is taken as systematic uncertainty. B0→ρK∗0background Decay Signal Data model MC model Syst. Total channel free fixed free fixed Systematic B0 s→K∗0K∗08.2 0.12 6.54 0.04 4.15 6.54 10.5 (1.5%) B0→φK∗07.2 - 7.2 (0.7%) Moreover, two different shapes were used to describe the shape of the B0→ρK∗0 contribution. The first one is described in Sect. 6.3.3, where a tightly selected sample of B0→ρK∗0decays is used to fix the parameters of the model, a single Crystal Ball distribution. For the second one, a sample of B0→ρK∗0MC events was selected using the B0 s→K∗0K∗0stripping and offline selection (with the exception of the offline PID cuts) and fitted with the same shape to determine another set of parameters. Using these models for the B0→ρK∗0background the mass fit was repeated and the largest variation in the number of B0 scandidates used as systematic uncertainty. In the case of the normalization channel, the fit was repeated with a different parameterization of the signal shape, a combination of two Crystal Ball distributions. Table 6.29 summarizes the systematic uncertainties associated to the model used in the invariant mass fit for both channels. Aditionally, biases in the invariant mass fit have been searched for, by generating and fitting 1000 toy MC experiments using the nominal invariant mass model and the parameter values extracted from data. The number of events generated for each toy experiment corresponds to the total number of events seen in data. The pull distribution for the signal yield is shown in Fig. 6.35. The central value and width of this distribution can be found in Table 6.30. Since the observed bias is compatible with zero, no correction to the number of B0 scandidates is applied. The uncertainty in the central value of the pull distribution is taken as a systematic uncertainty. The same procedure was applied to the reference channel mass fit. Similarly, no Entries 1000 Mean -0.0119 RMS 0.995 / ndf 2 χ 50.29 / 54 Constant 1.55± 38.15 Mean 0.0335802± 0.0005939 Sigma 0.0256± 0.9982 pull s B N -4 -2 0 2 4 0 5 10 15 20 25 30 35 40 45 Entries 1000 Mean -0.0119 RMS 0.995 / ndf 2 χ 50.29 / 54 Constant 1.55± 38.15 Mean 0.0335802± 0.0005939 Sigma 0.0256± 0.9982 Entries 1000 Mean 0.00131 RMS 1 / ndf 2 χ 46.54 / 54 Constant 1.61± 38.87 Mean 0.032949± 0.009939 Sigma 0.0269± 0.9846 pull d B N -4 -2 0 2 4 0 10 20 30 40 50 Entries 1000 Mean 0.00131 RMS 1 / ndf 2 χ 46.54 / 54 Constant 1.61± 38.87 Mean 0.032949± 0.009939 Sigma 0.0269± 0.9846 Figure 6.35: B0 s→(K+π−)(K−π+) (left) and B0→K+K−K±π∓(right) yield pulls obtained from 1000 toy MC experiments generated and fitted with the corresponding mass model. 134 6.6 Branching ratio of B0 s→K∗0K∗0 Table 6.30: Mean and width of the B0 s→(K+π−)(K−π+) and B0→K+K−K±π∓yield pulls distribution obtained from 1000 toy MC experiments. Yield Pull mean Pull width Bias Systematic NB0 s0.001 ±0.034 0.998 ±0.026 0.02 1.3 NB00.010 ±0.033 0.985 ±0.027 0.33 1.1 significant bias is observed, therefore no correction is applied and a systematic uncertainty is derived from the uncertainty in the pull mean. 6.6.5.2 Selection efficiencies In order to assess the systematic uncertainty associated to the determination of the selection efficiency, the distribution of the main variables involved in the selection have been compared between B0 s→K∗0K∗0data and MC simulation. In Appendix D.3 this comparison is shown. The most important differencies appear for the B0 smeson pTand IP significance, as well as for the secondary vertex χ2. The effect of these discrepancies is expected to be small in the ratio of selection efficiencies εsel B0→φK∗0/εsel B0 s→K∗0K∗0. Nevertheless, the size of the effect was estimated by correcting the MC distributions mentioned above to match those seen in data. The ratio of efficiencies was then recalculated, and the difference with respect to the nominal one was found to be 0.0089 (0.74%). This value is assigned as the systematic uncertainty. 6.6.5.3 Trigger efficiencies In Sect. 6.6.2.4 the L0 trigger efficiency was determined directly from data using the TISTOS technique, considering no binning in the signal B-meson phase space. The bias induced in the ratio of efficiencies by that approximation can be determined using MC simulated data, where the L0 trigger efficiency can be computed directly as εL0 MC =NL0 Nsel (6.33) where Nsel is the number of MC events that survive the offline selection and Ntrig is the number of those which also were selected by the L0 trigger decision. The L0 efficiency calculated above can be compared with the one obtained by applying the TISTOS technique to the MC sample. This comparison is shown in Table 6.31. The variation in the ratio of efficiencies estimated with the two different methods is considered as a systematic uncertainty in the branching ratio calculation. Table 6.31: L0 trigger efficiencies for B0 s→K∗0K∗0and B0→φK∗0calculated from MC simulated data with two different methods. Channel εL0 MC (%) εL0 MC;T IST OS (%) B0→φK∗037.54 ±0.25 42.06 ±0.20 B0 s→K∗0K∗050.38 ±1.47 53.33 ±1.14 ratio 0.893 ±0.007 0.945 ±0.034 135 Chapter 6. Time-integrated angular analysis of B0 s→K∗0K∗0 Table 6.32: Summary of the relevant quantities in the B(B0 s→K∗0K∗0) calculation. The first uncertainty is statistical, the second systematic (if just one is quoted, it represents the combination of statistical and systematic). Parameter Value NB0 s697 ±31 ±11 NB01049 ±33 ±7 κB0→φK∗0/κB0 s→K∗0K∗00.442 ±0.036 ±0.024 εB0→φK∗0/εB0 s→K∗0K∗01.30 ±0.17 ±0.07 6.6.6 B(B0 s→K∗0K∗0) result The results obtained in the previous sections were joined together in B(B0 s→K∗0K∗0) B(B0→φK∗0)=fd fs×εB0→φK∗0 B0 s→K∗0K∗0×κB0→φK∗0 κB0 s→K∗0K∗0×NB0 s NB0×B(φ→K+K−) B(K∗0→K+π−)(6.34) to determine the B(B0 s→K∗0K∗0) relative to B(B0→φK∗0). The values and uncertainties of the parameters in the previous expression are summarized in Table 6.32. The ratio between the hadronization fractions has been taken from [123] and is fs fd= 0.259 ±0.015. The value of B(φ→K+K−) = (0.489 ±0.005) is taken from [16] and B(K∗0→K+π−)=2/3. Using these numbers, (6.34) leads to B(B0 s→K∗0K∗0) B(B0→φK∗0)= 1.080 ±0.182(stat.)±0.081(syst.)±0.063(fd/fs) Considering the value B(B0→φK∗0) = (9.8±0.6) ×10−6from [16], B(B0 s→K∗0K∗0) = (10.6±1.8(stat.)±1.0(syst)±0.6(fd/fs)) ×10−6 136 6.7 Triple products and direct CP asymmetries 6.7 Triple products and direct CP asymmetries The “true” TP asymmetries and direct CP asymmetries have been calculated for B0 s→ K∗0K∗0using (4.36 -4.39) and (4.41 -4.41), respectively. These quantities have been independently determined for TOS and non-TOS events. The angular distributions of the background have been parameterized using events from the high B0 smass sideband, and subtracted according to the fraction present in each sample, see Sect. 6.5.1.1. After correcting for the angular acceptance, the data distribution in the relevant angular functions are shown in Fig. 6.36 (TOS) and Fig. 6.37 (non-TOS). Table 6.33 contains the TP and CP asymmetries measured for each sample. A weighted average between TOS and non-TOS is taken as the final result. One of the main sources of systematic error in these measurements is the effect of the angular acceptance. The angular acceptance correction is more relevant in the case of the four CP asymmetries with respect to the triple product asymmetries, as can be seen in Fig. 6.38. Two sources of systematics related to the angular acceptance have been considered, following the same approach as in the case of the amplitude analysis: the difference in the pTspectra between data and MC (Sect. 6.5.4.3) and the statistical error in the acceptance description (Sect. 6.5.4.2). The lifetime dependence of the different amplitudes are modified by the lifetime biasing cuts in the selection. This could induce a bias in the measured TPA and CP asymmetries. To determine the size of this effect, a set of toy MC samples were generated using a time-dependent PDF that includes the change in efficiency as a function of the lifetime of the B0 s. The lifetime acceptance was parameterized using full generated B0 s→K∗0K∗0 MC, see Appendix D.5. From the comparison of the values measured for the different asymmetries with the ones that had been generated, the systematic uncertainty coming from the lifetime acceptance was estimated. Finally, the background fraction has been changed in ±1σin order to estimate the systematic effect in the measured asymmetries. The results of the systematic studies are summarized in Table 6.34. Table 6.33: TP asymmetries and CP asymmetries measured with B0 s→K∗0K∗0TOS and non-TOS events. Asymmetry TOS non-TOS Average A1 T0.028 ±0.058 -0.023 ±0.059 0.003 ±0.041 A2 T-0.034 ±0.058 0.052 ±0.059 0.009 ±0.041 A3 T-0.014 ±0.058 0.051 ±0.059 0.019 ±0.041 A4 T-0.011 ±0.058 -0.069 ±0.059 -0.040 ±0.041 A1 D-0.004 ±0.058 -0.117 ±0.059 -0.061 ±0.041 A2 D0.087 ±0.058 0.075 ±0.059 0.081 ±0.041 A3 D-0.140 ±0.057 -0.018 ±0.059 -0.079 ±0.041 A4 D-0.118 ±0.057 -0.044 ±0.059 -0.081 ±0.041 137