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Monte Carlo studies for parameterization of B-mixing with dilepton events Song, Zhongzhi 2005

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Monte Carlo Studies for Parameterization of B-Mixing with Dilepton Events by Zhongzhi Song B . S c , Anhui University, 1999 M . S c , Peking University, 2002 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in The Faculty of Graduate Studies (Physics) T H E U N I V E R S I T Y O F BRITISH C O L U M B I A July 19, 2005 © Zhongzhi Song, 2005 11 Abstract In this thesis, we report on a Monte Carlo (MC) study of the parameterization of probability distribution functions (PDFs) for neutral B mixing using dilepton events. T(45) decays are generated and those with at least 2 leptons in final state are used in this study. By choosing appropriate functions, a parameteri-zation of the B meson decay time difference At is obtained by studying the B lifetime and the neutral B mixing frequency dependence of the dilepton events using an event re-weighting technique. Our results, combined with Monte Carlo studies of detector resolutions, can be used to obtain more precise measurements of the B mixing frequency and B lifetimes from experimental data. iii Contents Abstrac t ii Contents iii Lis t of Tables v Lis t of F igures vi Acknowledgements viii 1 Introduct ion 1 2 Physics B a c k g r o u n d 9 2.1 Semileptonic B meson decays 9 2.2 B°-B° mixing 12 2.3 Coherent BB states 15 3 BARAR E x p e r i m e n t 18 3.1 The BABAR detector 18 3.2 Event generator software 22 4 N e u t r a l B M i x i n g Frequency Measurement 28 4.1 Brief review of B°-B° mixing measurements 28 4.2 Boost approximation 33 4.3 Point of closest approach 35 5 Ana lys i s Strategy 38 5.1 Event generation 38 Contents iv 5.2 Event classification 41 5.3 Event re-weighting 48 5.4 Model selection and fitting strategy 49 6 Ana lys i s Results 57 6.1 Parameterization fits I - varying B lifetime 57 6.2 Parameterization fits II-varying B mixing frequency 65 6.3 Consistency test of parameterization 67 6.4 Crosschecks of the parameterization approach 70 6.5 Effects of event re-weighting on the P O C A approximation . . . 75 7 Conc lus ion 78 B ib l iography 80 Glossary 84 A M o r e Figures 85 V List of Tables 5.1 B lifetimes and the neutral B mixing frequency. 39 5.2 Monte Carlo event information used in our analysis 40 5.3 Dilepton event categories 41 5.4 Weights for event re-weighting 48 6.1 Charged B lifetime fits with three different methods of calculating Az .\ 76 6.2 Neutral B lifetime fits with three different methods of calculating Az 76 6.3 Neutral B mixing frequency fits with three different methods of calculating Az 77 vi List of Figures 1.1 Feynman diagram for flavour changing charged current 3 1.2 Unitarity triangle in the complex plane 4 2.1 Feynman diagrams for semileptonic b quark decays 11 2.2 Standard Model box diagrams for B^-BQd mixing 13 3.1 BABAR detector longitudinal section 19 4.1 Average of A m ^ measurements from different experiments. . . . 31 4.2 Simultaneous measurements of Am<j and TB° 32 4.3 Two B decay vertices, showing displacement along the boost axis. 34 4.4 Dilepton event topology in the transverse plane 36 5.1 Re-scaled 2-D momentum contour plot of dilepton events 42 5.2 P O C A At distributions for the same B (SB) events 44 5.3 P O C A At distributions for the opposite charged B (OCB) events. 45 5.4 P O C A At distributions for the opposite neutral B (ONB) events. 46 5.5 P O C A At distributions (OS+SS) for the (OCB) events 51 5.6 P O C A At distributions (OS+SS) for the (ONB) events 52 5.7 P O C A At asymmetries (OS-SS)/(OS+SS) for the (ONB) events. 53 6.1 Examples of / ( r ) fits for (OCB-HH) events 59 6.2 Fitted parameters of / ( T ) for (OCB-HH) events 60 6.3 Fitted parameterization function / ( T ) for (OCB) events 61 6.4 Examples of f(r) fits for the (ONB-HH) events 62 6.5 Fitted parameters of / ( T ) for (ONB-HH) events 63 List of Figures vii 6.6 Fitted parameterization function / ( r ) for (ONB) events 64 6.7 Examples of g(Am) fits for (ONB-HH) events 66 6.8 Fitted parameters of g(Am) for (ONB-HH) events 67 6.9 Fitted parameterization function g(Am) for (ONB) events. . . . 68 6.10 Consistency test of the parameterization fits 69 6.11 Lifetime fits for charged B samples 72 6.12 Lifetime fits for neutral B samples 73 6.13 Mixing frequency fits for neutral B samples 74 A . l Examples of f(r) fits for (OCB-HL) events 85 A.2 Fitted parameters of f(r) for (OCB-HL) events 86 A.3 Examples of f(r) fits for ( O C B - L L ) events 87 A.4 Fitted parameters of / ( T ) for ( O C B - L L ) events 88 A.5 Examples of f(r) fits for (ONB-HL) events 89 A.6 Fitted parameters of / ( r ) for (ONB-HL) events 90 A.7 Examples of / ( r ) fits for (ONB-LL) events 91 A.8 Fitted parameters of / ( r ) for (ONB-LL) events 92 A.9 Examples of g(Am) fits for (ONB-HL) events 93 A.10 Fitted parameters of g(Am) for (ONB-HL) events 94 A . 11 Examples of g (Am) fits for (ONB-LL) events 95 A.12 Fitted parameters of g(Am) for (ONB-LL) events 96 V l l l Acknowledgements I would like to thank my supervisor, Dr. Janis McKenna, whose guidance, support, and encouragement make this thesis possible. Special thanks to Dr. Tom Mattison, whose valuable comments and help make this thesis easier. M y special thanks also go to Dave Asgeirsson, whose kind help make my study at U B C very enjoyable. Thanks Dave also for proof reading the first draft of this thesis. I would also like to thank all of the other members of the BABAR group at U B C , Chris, Doug, Neil, Nasim, Bryan, and Patrick. Thanks to all of you for your comments and help during the past two years at U B C . Thank Doug Maas for computing support. 1 Chapter 1 Introduction It is well established in particle physics that there are three generations of quarks and leptons. They are the up-type quarks (u, c, t), down-type quarks (d, s, b), charged leptons (e, / i , r ) , and neutrinos (i/e, Vp, vT). They interact through the exchange of gauge bosons: the weak bosons and Z°, the photon, and the gluons. The Standard Model (SM) of particle physics with 517(3) x 517(2) x 17(1) local gauge invariance provides an elegant theoretical framework and represents our current understanding of these fundamental particles and their interactions. In the strong and electromagnetic interactions, the neutral current only cou-ples with a vector current as described by Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD). One of the most striking features of the charged-current weak interactions, however, is that they do not couple solely to a vector current but to a linear combination of vector and axial vector cur-rents V - A . As a consequence, the electroweak theory is a chiral gauge theory, which means that left- and right-handed fermions transform differently under the electroweak gauge group 517(2) x E7(l)[l]. The right-handed fermions cannot couple to W ± via V - A current, and they are singlets under 517(2): ER = (eR,fj,R,TR); UR = {uR,cR,tR); DR = (dR,sR,bR). (1.1) On the other hand, the left-handed fermions (quarks Q and leptons L) do couple to W±, so they are doublets under 517(2): Chapter 1. Introduction 2 With only gauge fields and fermions, the model is incomplete. In partic-ular, it does not accommodate the observed non-zero masses for the quarks, charged leptons, and weak gauge bosons. To construct gauge invariant inter-actions coupling left- and right-handed fermions, at least one additional scalar field is necessary. It turns out that the massive weak bosons can acquire mass through the spontaneous symmetry breaking of the SU(2) x [7(1) symmetry. This process, called the Higgs mechanism, requires a single Higgs doublet to acquire a vacuum expectation value. The quarks and leptons get mass from the Yukawa couplings to the Higgs doublet. Furthermore, the couplings to the Higgs doublet can also generate flavour mixing and CP violation[2]. The quark sector in the S M of particle physics is characterized by the Cabibbo-Kobayashi-Maskawa ( C K M ) matrix which relates weak eigenstates (d') to mass eigenstates (d) of down-type quarks[3]: 1 ( Vud vus vub \ I d \ I d N Vcd Vcs Vcb s =VCKM s • (1.3) Vtd Vts Vtb J \ b I \ b d' s' b' \ 0 I \ \ 0 I \ 0 J The matrix V C K M is the mixing matrix for three generations of quarks, and it is a 3 x 3 unitary matrix. These elements will show up in the interaction of the flavour-changing charged current between a down-quark of family i and an up-quark of family j as pictured in Fig. 1.1. The amplitude M. can be written as ( Co M2 \ 1'2 ^ = ^ J fij7„(l-76)4-Vy, (1-4) where GF is Fermi constant, My/ is W mass, iij and di are Dirac spinors rep-resenting the quarks, the 7's are Dirac matrices. This means that the coupling strength is proportional to the C K M matrix element Vij. This is an example of charged-current coupling with a V - A current. In the S M , quark flavour mixing arises from the nonzero off-diagonal ele-ments of the C K M matrix and CP violation arises from the single non-zero Chapter 1. Introduction 3 Figure 1.1: Feynman diagrams for flavour changing charged current: the W boson couples to the up-type and down-type quarks. phase which appears in this matrix. Several parameterizations of the C K M matrix have been proposed in the literature. The two widely used ones are the standard parameterization recommended by the Particle Data Group[4] and a generalization of the Wolfenstein parameterization [5, 6]. The absolute values of the elements of the C K M matrix show a hierarchical pattern with the diagonal elements close to unity, the magnitudes \ VUS | and |T^d| of order 0.2 (sine of the Cabibbo angle, sin#c), the elements \Vcb\ and \Vts\ of order 4 x 10~ 2, and the elements \Vub\ and \Vtd\ are of order 5 x 1 0 - 3 . The Wolfenstein parameterization exhibits this hierarchy in a transparent manner. It is an approximate parameterization of the C K M matrix in which each element is expanded as a power series in the sine of the Cabibbo angle 6C: ( VcKM = A A\3(p - irj) \ - A 1 - f ,4A 2 + 0(\4), (1.5) \ AX3(1-p-irj) -AX2 1 where the parameters for P C K M are chosen as A, A, p, and r\. Here A = sin 6C w 0.22, A is of order one, p and r) are constrained by CP violation measurements. Because of the smallness of A and the fact that for each element the expan-sion parameter is actually A 2 , it is sufficient to keep only the first few terms in this expansion. In the case that higher accuracy is needed, the terms of higher order correction have to be included in phenomenological applications. A gener-alization of the Wolfenstein parameterization including 0(X4) and 0 (A 5 ) terms, which satisfies unitarity, can be found in Ref.[6]. The unitarity of the C K M matrix implies various relations between its ele-Chapter 1. Introduction 4 v""Kh / /1 \ 0 P 1 Figure 1.2: Unitarity triangle in the complex plane, ments. In particular, we have the relation vudv:b + vcdv;b + vtdv;b = o. (i.e) Phenomenologically this relation is very interesting as it constrains simulta-neously the elements Vub, Vcb and Vtd which are currently the subject of exten-sive study. The unitarity relation in E q . (1.6) can be represented as a unitarity triangle in the complex (p, 77) plane, where p — p(l — A 2 /2) and 77 = 77(1 — A 2 /2) . Noting that to an excellent accuracy, VcdV*b in the parameterization formula (1.5) is real with \VcdV*b\ = AX3 + 0(X7) and rescaling all terms in E q . (1.6) by AX3 we indeed find that the relation in E q . (1.6) can be represented as a triangle in the complex (p,fj) plane as shown in Fig. 1.2. Since the angles and the sides in this triangle are given by the modulus of the elements of the mix-ing matrix, they are phase-convention independent and physically observable. Consequently they can be measured directly in suitable experiments, such as B meson decays. For example, the quantity sin(2/3), where the /3 is one of the angles in unitarity triangle as shown in Fig. 1.2, has been measured extensively by both BABAR [8] and Belle[9] experiments. One can construct five additional constraints corresponding to other orthogonality relations [7], like the one in E q . (1.6). The areas of all unitarity triangles are equal and are related to the Chapter 1. Introduction 5 measure of CP violation[10]. It was long thought that CP symmetry was exact in nature and only theo-ries that had this exact symmetry property were viable descriptions of the real observed world. The observation of CP violation in the decay K° -> TT+TT~ by Christenson et al. in 1964[11], however, changed that view dramatically. In-terest was heightened by Sakharov's observation in 1967[12] that CP violation is a crucial feature of any theory that attempts to explain the observed asym-metry between matter and anti-matter in a universe which starts from initially symmetric conditions. The principal interest in the studies of B meson decays in the context of the S M lies in that they provide valuable information on the weak mixing matrix-the C K M matrix. In fact, B decays can be used to determine five of the nine C K M matrix elements: Vcb, Vub, Vtd, Vts, and For example, \Vcb\ can be extracted from exclusive B —> D^lv decay processes with the help of Heavy Quark Effective Theory (HQET)[13] or from inclusive semileptonic decays B —>• XEv with the help of Operator Product Expansion theory (OPE)[14]. Like the neutral K mesons, the neutral B mesons are complicated by the fact that different neutral states are relevant to the discussion of different physical processes: there are two flavour eigenstates, which have definite quark content and are most useful to understand particle production and particle decay pro-cesses; and there are eigenstates of the Hamiltonian, namely states of definite mass and lifetime, which propagate through space in a definite fashion. Since the mass eigenstates are not flavour eigenstates, the flavour eigenstates are mixed with one another as they propagate through space. This is why there is the phenomenon of particle-anti-particle mixing in neutral K, D, and B systems. A n excellent testing ground for CP violation is also provided by the B meson system through particle-antiparticle mixing. A particle that is purely B° at time t = 0 will oscillate between 5 ° and B° with a frequency1 A m , where A m is the mass difference between the two neutral B mass eigenstates. If decays to 1We wiD follow the convention c = 1 and h = 1, so that the mass, energy, and momentum are all measured in units of GeV. And we can also convert GeV into ps - 1 , which means that the mass difference Am will have the same unit as frequency. Chapter 1. Introduction 6 a CP eigenstate / are observed, any difference between the rates when starting with a 5 ° or with a B° is a manifestation of CP violation. For example, a state initially produced as a B° (B°) can decay to J/ipK° directly or can oscillate into a B° (B°) and then decay to J/tpK®. With little theoretical uncertainty in the S M , the phase difference between these amplitudes is equal to twice the angle /3 = arg[— VcdV*b/VtdVt*b] of the unitarity triangle as shown in Fig. 1.2. The CP-violating asymmetry measured in such kind of modes can thus provide a crucial test of the SM[15]. Any deviation from unitarity in the C K M matrix would be a clear indication of new physics. The two B mesons produced from T(4S) decays evolve coherently, until one decays. The proper decay time difference At between two B mesons is governed by the following probabilities to observe an unmixed ("+") or a mixed ("—") event: P(At) = -!-e-! A<l/ r (l ± cos(AmAt)), (1.7) 4r where r is the neutral B lifetime. Therefore, we can measure the oscillation frequency as well as the neutral B meson lifetime from these probabilities. In reality, there are three major experimental complications that affect the At distributions. • The tagging algorithms that determine the 6-flavour of a B meson at its time of decay have a finite probability of making a mistake, called the mis-tag probability. As a result, some mixed events will be tagged as unmixed events, and vice versa. • The resolution for At is comparable to the lifetime and must be well understood. • Various background levels and their time evolution properties must be understood in order to extract the lifetime and mixing frequency. Currently, experimental studies of B meson decay are performed at the T(4S) resonance near production threshold, as well as at higher energies in proton-anti-proton collisions. By far the largest samples of B mesons have been Chapter 1. Introduction 7 collected by the e+e~ collider detectors running at T(4S) resonance, which are called "B-Factories". Both BABAR and Belle have accumulated approximately 150 f b _ 1 of data, which make precise measurements possible. The PEP-II B Factory is an asymmetric e+e~~ collider designed to operate at a luminosity of 3 x 10 3 3 c m _ 2 s _ 1 and above, at a center-of-mass energy of 10.58 GeV, the mass of the Y(4S) resonance[16]. The Y(4S) decays almost entirely to B ° B ° or B+B~ pairs with an upper limit on non-BB decays ratio less than 4% of the time[17], and thus provides an ideal laboratory for the study of B mesons. In PEP-II , the electron beam of 9.0 GeV collides head-on with the positron beam of 3.1 GeV , resulting in a Lorentz boost to the Y(4S) of PJ = 0.55. This boost makes it possible to reconstruct the decay vertices of the two B mesons, to determine their relative decay times, and thus to measure the time dependence of neutral B mixing[18]. In this thesis, we study the parameterization of At distributions by using Monte Carlo (MC) generated events at generator level (without full detector simulation) within the BABAR framework. Since in real data we do not have ex-act B momentum (direction), the boost approximation (with an average boost for T(4S)) is used to derive the difference in proper decay times from the dis-tance between two decay vertices along the boost direction. The motivation for this Monte Carlo study is that a more precise probability distribution of exper-imentally measured At values, rather than the distribution of real At (which is not measurable), will help to make more precise measurements of the B meson lifetimes and the neutral B mixing frequency. A brief review of the physics background of B semileptonic decays and the neutral B mixing is given in Chapter 2 of this thesis. In Chapter 3, we provide a brief introduction to the BABAR experiment, the BABAR detector, the simulation tool, and the event generator. In Chapter 4, we will briefly review existing neutral B mixing measurements and explain the two main approximations used in the experimental analysis, namely, the boost approximation and the point of closest approach (POCA) approximation. Strategies for data analysis are given in Chapter 5, where we will describe how the M C events were generated and Chapter 1. Introduction 8 explain the method of event re-weighting and parameterization model selection. The analysis results and independent crosschecks are given in Chapter 6. Finally, we present our conclusions in Chapter 7. 9 Chapter 2 Physics Background B°B° mixing is a result of the oscillations between B° and B°. A n initially produced B° or B° evolves in time into a superposition of B° and B°. The BB state produced from T(45) decays will evolve coherently with time until one of them decays. Semileptonic B meson decays provide a simple way to determine the b quark flavour of the decaying B meson, which is essential for flavour mixing studies. In this chapter, we first review B meson semileptonic decays in Section 2.1 as they are the basic processes for studying the neutral B mixing using dilepton events. Then we provide the quantum mechanical treatment of the two-state system of neutral B mesons in Section 2.2. Finally, in Section 2.3, we apply the knowledge of neutral B mixing from Section 2.2 to the coherent BB states produced from T(4S) decays, which represents the case studied in this thesis. 2.1 Semileptonic B meson decays There are strong motivations for studying semileptonic B decays. First of all, these processes are related to some of the fundamental parameters of the SM. Semileptonic B decays, B ->• Xclv and B Xulv provide an excellent labora-tory to measure moduli of C K M elements, namely \Vcb\ and \Vub\ respectively, because the strong interaction effects are greatly simplified due to the leptons in the final state. Secondly, the semileptonic B decay branching fractions have large effects on other measurements and thus need to be measured precisely. In-deed, the semileptonic decays are an important source of background for many measurements. Finally, the semileptonic B decays are useful to test various Q C D effective theories. Chapter 2. Physics Background 10 B+B~ and B°B° mesons are the lightest of the b-flavoured hadrons, hence they decay via weak interactions. Since the mass of a b-quark is much larger than its partner quark (d or u), B meson decays are mostly described by the b quark decays within the naive spectator model[19]. In the S M , other non-spectator decay processes including both tree level processes such as W-exchange and an-nihilation decays and higher-order loop-induced flavour-changing neutral cur-rent ( F C N C ) 1 decay processes are all suppressed in comparison to the spectator decays. The dominant decay mode of a fe-quark is b —>• cW* where the virtual W* eventually materializes either into a pair of leptons, £v, or into a pair of quarks, qq', which will then hadronize into final states. It is called semileptonic decay if the virtual W* converts into an Itv pair, which is of primary interest for our study. If the virtual W* converts into a quark pair, it is then called hadronic decay, which is more complicated than the semileptonic decay because of strong interactions. A l l other decays that do not occur through the usual b —> c transition are known as rare B decays, which include both semileptonic and hadronic b —> u decays that are suppressed at leading order by the small C K M matrix element Vu&, as well as higher order processes such as electromagnetic and gluonic penguin decays (for details, see review in Ref.[21]). The Cabibbo favoured inclusive decay b -> cl~v is shown in Fig. 2.1(a). We can see that the b quark can only decay directly to an l~. For comparison, a b can only decay directly to an l+ as shown in Fig. 2.1(d). Therefore, the sign of the charge of the direct lepton is the same as the sign of the charge of the b quark in the decaying B meson. That is, a 5 ° or B+ decays directly to a positively charged lepton and a B° or B~ decays directly to a negatively charged lepton. There are also "cascade" background processes, such as b -> c -> s£+v. The decay at quark level is shown in Fig. 2.1(b,e). The virtual W* originating from the b quark will hadronize into an £~V pair or a quark pair (dominated by cs). The leptons produced from c quark decays have the opposite charge compared with the direct leptons (comparing part(a) with part(b); and part(d) 1The suppression is known as the Glashow-Hiopoulus-Maiani (GIM) effect[20]. Chapter 2. Physics Background 11 (a) direct lepton (d) direct lepton (b) cascade lepton-wrong sign (e) cascade lepton-wrong sign (c) cascade lepton-right sign (f) cascade lepton-right sign Figure 2.1: Feynman diagrams for semileptonic b quark decays. with part(e) in Fig. 2.1). In B meson decays, c will mostly hadronize into D mesons and D mesons may then decay and produce the cascade lepton with the opposite charge of a direct lepton. This is called a cascade lepton with wrong sign. Similar cascade processes occur via b —>• ccs and then c —> s£~V as shown in Fig. 2.1(c, f). In these cases the charge of the leptons has the same sign as that of the direct lepton. Therefore these are called cascade leptons with right sign. It should be noted that these cascade leptons with right sign are suppressed because of the smaller fraction for B to two D mesons. These processes are very well described by the O P E , in which the decay rate is expanded in inverse powers of the heavy quark mass m&. For the exclusive Chapter 2. Physics Background 12 decay channels with charmed final states such as B -> D*£v and B —>• . D ^ , these channels will provide information about \Vcb\ and can be studied with H Q E T . For the inclusive charmless semileptonic decays B -> XU£V, these channels only constitute about 1% of the total semileptonic width. For the exclusive B charmless semileptonic decays such as B —> ntP, these channels are well studied within the naive factorization approach 2. In summary, the charges of the final state leptons provide information which can be used to identify the B a s a S 0 (B+) or B° (B~). If a final state can only originate from a B° or B°, then the lepton sign can be used to "tag" the flavour of the B. In a typical experiment, because of the extremely low efficiency in reconstructing both B decays, only one of them is fully reconstructed, and the other B meson is used to tag the pair by determining its b-flavour content. 2.2 B°-B° mixing There are two neutral B°-B° meson systems, Bd-Bd and B°s-B°s, which exhibit particle-antiparticle mixing [23]. The quark content for Bd is Bd = bd, Bd = db. Unless explicitly stated in the rest of this thesis, the Bd system is the only neutral B system that we are interested in, even though the following general discussion in this section might be still applicable to the Bs system with a small modification3. In the S M , transitions B° -»• B° and B° ->• B° are described at lowest order by the weak interaction box diagrams in Fig. 2.2. Such transitions are called | A B | — 2 transitions, because they change the bottom quantum number by two units. In the S M , |AJ5| = 2 amplitudes are small, so measurements of Bd mixing could easily be sensitive to new physics, which can appear in the box diagram in place of the top quark. The amplitudes of box diagrams with internal up or charm quarks are negligible compared to the diagram with two 2 Recently, there has been new progress in the generalization of the naive factorization to exclusive nonleptonic B meson decays, using the so-called "QCD factorization approach" [22]. 3Note that T(4S) is just at the resonance to produce B^B^ and at an energy below the B®B® production threshold. Chapter 2. Physics Background 13 w+ f W W V W y *2 W 52 Figure 2.2: Standard Model Box diagrams inducing Bd-Bd mixing. Similar di-agrams exist in which t quarks are replaced with c or u quarks. internal top quark lines, because the amplitudes of these transitions are roughly proportional to the masses of the internal quarks. We now consider mixing for the neutral meson B° and its antiparticle B° [15]. The two mass eigenstates are BH and Bi, where H and L stand for Heavy and Light, respectively4. These mass eigenstates are a linear combination of the neutral B-meson flavour eigenstates B° and B°, \BL)=p\B°)+q\B°), (2.1) \BH)=p\B°)-q\B0). (2.2) The complex coefficients p and q obey the normalization condition: \P\2 + M 2 = 1. (2-3) The eigenvalue equation is Here M is the mass matrix and T describes the exponential decay of the system. They are both 2 x 2 Hermitian matrices. Since T i 2 M 1 2 , this equation gives \q/p\ ^ 1, which means that the CP violation in mixing is negligible in this study [23]. 4In the neutral K meson system, mixing also has contributions from real intermediate states accessible to both K° and K°. As a result, the two mass eigenstates of the neutral K system have a large decay rate difference Ar but almost identical masses. So it is more convenient to use the long-lived state K\ and short-lived state for the K° system. For the B° system, AF is much smaller than Am<j, therefore, BH and BL are more convenient. Chapter 2. Physics Background 14 Any neutral B state can be written as an admixture of the states BH and BL: \B(t)) = AL{t)\BL) + AH(t)\BH), where the amplitudes of this admixture evolve in time as AH(t) = ^ ( 0 ) e - ( ^ + ' M « ) ' . (2.5) (2.6) (2.7) The proper time evolution of states which at time t = 0 were either a pure B° with AH(0) = AL(0) = l/(2p) or a pure B° with -AH(Q) = AL(0) = l/(2g) is given by \B°(t))=9+(t)\B0) + l9.(t)\B0), P \B°(t)) = ?-g-(t)\B0)+9+(t)\B0), where g+(t) =e-¥-iMicos g- (t) = ie * -iMt sin A m . A r ; ( — ' A m . A I \ ' (2.8) (2.9) (2.10) (2.11) with r = r H ^ " r £ ; M = M h + M l - A m = (MH - ML); AT = TH- TL. (2.12) The time-dependent probabilities for the flavour states remain unchanged (+) or oscillate into the other (—) in the following way: \9±(t)\ 2 _ -rt A r cosh(-^-i) ± cos(Ami) (2.13) In the absence of CP violation, the time-integrated mixing probability for Bd system is given by [4] Xd = !\g-{t)\2dt *d + Vd J\g-(t)\*dt + f\g+(t)\*dt 2(l+xdy where xd = Amd/T, and yd = A r / 2 r . (2.14) Chapter 2. Physics Background 15 In the S M , Amd can be calculated by using a low-energy Hamiltonian and the O P E . It can be shown that[24] Amd oc \VtbVtd\2. (2.15) Therefore, A m j is sensitive to both Vtb and Vtd, and its measurement can provide a constraint on the side of the unitary triangle which has length of \Vt*bVtd\/\Vc*bVcd\ as shown in Fig. 1.2. 2.3 Coherent BB states At a 5 factory, the BB pair produced from the Y(4S) decay is in a coherent state. From Eq . (2.8) and Eq . (2.9), we can easily verify that at any time until one particle decays, there is always exactly one B° and one B° present (for reasons, see below). Once one of the particles decays, however, the other continues to evolve, and thus there are possible events with two B° or two B° decays, whose probabilities are governed by the time difference between the two decays. The two B mesons from an T(4S) decay are identified by the angle 6 that they make with the electron beam (higher energy beam) direction in the T(4S) rest frame. The two-B state5 can be written as[15], S(tf,tb) = [B0phjs(tf,9,4>)B°phys(tb,n-0,4> + n) ~ - ^ h y s ^ ' ^ p h y s ^ ~ e><t> + *)] sin^)' (2-16) where the subscript / refers to the B meson in the forward half-space (6 < TT/2) and the subscript b refers to the backward-moving B meson (6 > TT/2); tf and tb are the proper times for the two B mesons. With the help of E q . (2.8) and E q . (2.9), we can rewrite Eq . (2.16) as S(tf,tb) = - ^ e - ( r / 2 + i M ) ^ + t ^ { c o s [ A m d ( ^ - t b ) / 2 } ( B ° f B ° b - B°fB°b) -ism[Amd(tf - tb)/2](?B°fB°b - -B°fB°b)\ sm(8f). (2.17) q 1 p 1 J 5Note that T(4S) has total spin J = 1 and B mesons have spin S = 0. Momentum conservation tells that the B pair must be an antisymmetric L = 1 P-wave state. Chapter 2. Physics Background 16 Since the mass of the T(4S) state is just above the threshold of BB produc-tion, the B momentum in the T(4S) rest frame is relatively small, ~ 335 MeV. The two B mesons will have equal magnitude, back-to-back momenta in the T(4S) rest frame. To a first approximation, the B rest frame and the T(4S) rest frame are equivalent. Since, in general, the B flight direction in the T(4S) frame is not known, the T(4S) frame is often used in place of the B rest frame. From E q . (2.17), one can easily find that until the time when one or the other of the two -B's decays, there is always one B° and one B° in the system. The decay of one of the particles stops the clock for that particle while the other continues to evolve with time. The probability of getting two B°s or two B°s will depend on the time difference of the two B decays as shown by the sm[Amd(tf — tb)/2] term in Eq . (2.17). From E q . (2.17), one can easily derive the probabilities of getting an event without mixing (i.e., one B° and one B°) and an event with mixing (i.e., two B°s or two B°s). First, we can determine the probability of getting an event by integrating over 6 and 4>, and assuming \q/p\ — 1: P(without mixing^/, tb)) oc e ~ r ( - ^ + t b ) [1 + cos(Amd(tf - tb))} (2.18) P(with mixing^/, tb)) oc e - T ^ ' + i b ) [1 - cos(Amd(tf - tb))] (2.19) Let tshort be the smaller of tf and tb. The probability of an event occurring with the first B decay at tshort is P(tShort) oc e - 2 r t ^ ' (2.20) Therefore, the probabilities of a BB event with and without mixing, which only depend on the time difference of the two B decays, can be obtained by dividing Eq . (2.18) and Eq . (2.19) by E q . (2.20). These results can be written as: e-|A<i/r Bo PIB°B° -> B°B°(At)) = — [1 + cos(Am d Ai)] (2.21) 4TBO _ „ - | A t | / r B o P(B°B° -»• B°B°(At) or B°B°(At)) = — [1 - cos(Am dA*)] (2.22) 4T B O where At = tb — tf, the B lifetime r = 1/T. If A i is measured and the 6-quark flavour of both B mesons at their time of decay is determined, one can extract Chapter 2. Physics Background 17 the B° lifetime Tgo and the mixing frequency Amd. Note that Eq . (2.21) is also valid for B+B~ by taking A m j = 0 and substituting TBO ->• r B + . Therefore, we can extract the charged B lifetime TB+ from experimental data by using E q . (2.21) as well. The mass difference A m ^ between the two mass eigenstates of the (B°B°) system can be measured by constructing the following time-dependent asymme-try: N(B°B°)(At) - [N(B°B°)(At) + N(B°B°)(At)] A(At, Amd) = N(B°B°)(At) + [N(B°B°)(At) + N(B°B°)(At)] cos(AmdAt), (2.23) where At is the time difference of the two B mesons decays. The simplest way to determine the b flavour of the decaying neutral B is to use the charge of the leptons produced from B decays as tagging particles, as described in Section 2.1. By counting the number of "same-sign" events or t~t~ ) and "opposite-sign" events (l+i~), which correspond to "mixed" and "un-mixed" events, respectively, a measurement of Amd can be extracted from the asymmetry: A ( A t > A m d ) - N(in-)-[N(in+) + N(t-i-)} ( 2 - 2 4 ) The fraction of B decays producing direct leptons is around 10 to 15%. Therefore, the direct dilepton events represent 1 to 2% of the T(45) B°B° decays6, which will ensure relatively large statistics for a B mixing analysis. By measuring the asymmetry in Eq . (2.24), we can extract A m ^ from experimental data. 6Here, each lepton is from a direct B decay, so the fraction of dilepton events is square of the fraction of direct leptons from single B decays. If we also include the cascade leptons, we obtain a higher fraction of dilepton events. 18 Chapter 3 BABAR Experiment The primaxy goal of the BABAR experiment is the systematic study of CP asym-metries in the decays of neutral B mesons. In addition to this, a sensitive mea-surement of the C K M matrix elements and measurements of a number of rare B meson decays are possible at BABAR. Studies indicate that the best source of B mesons for such a physics program is an e+e~ collider, operating at the T(45) resonance, but in an asymmetric mode, i.e., with beams of unequal energy. The BB pair from the T(45) decay is almost at rest in the center-of-mass (CM) frame, resulting in the B mesons having significant momenta in the laboratory frame. The PEP-II B Factory is such an asymmetric e+e~ collider with the electron beam of 9.0 GeV colliding head-on with the positron beam of 3.1 GeV [16] . This configuration results in a Lorentz boost to the T(4S) resonance of < /37 >= 0.55 in the lab frame, which enables the proper time of B decays to be inferred from their measurable decay lengths, thus enabling the measurement of the time dependence of their decay rates and neutral B mixing[18]. In this chapter, we first give a brief description of the BABAR detector in Section 3.1. In Section 3.2, we provide some details of running the event gen-erator software program, EvtGen [27], within the BABAR framework, which is used to simulate dilepton events from T(4S) decays for our neutral B mixing study. As we stated in Chapter 1, the crucial test of CP invariance is a comparison of the time-dependent decay rates for B° and B° to a self-conjugate state. For the clean experimental test, in order to maintain acceptable efficiency, we 3.1 The BABAR detector Chapter 3. BABAR Experiment 19 BABAR Coordinate System y Cryogenic Chimney Instrumented Flux Return (IFRJJ Barrel Superconducting Coil ' Cherenkov Detector (DIRCJ Magnetic Shield forDIRC Bucking Coil' Elect ID magnetic Calorimeter (EMC) Drift Chamber ' (DCH) Silicon Vertex Tracker (SVTJ Figure 3.1: BABAR detector longitudinal section. can require events in which one B meson decays to a CP eigenstate that is fully reconstructed and the other B meson is tagged as a B° or a B° by its decay products: a charged lepton, a charged kaon, or other flavour sensitive features such as a low momentum charged pion from a D* decay. The very small branching fractions of B mesons to CP eigenstates, typically of the order 10~ 4, the need for full reconstruction of final states with two or more charged particles and several 7r°s, plus the need to tag the second neutral B, place stringent requirements on the detector. The BABAR detector is a charged and neutral spectrometer with large solid-angle coverage[18]. Fig. 3.1 shows a longitudinal section through the center of the detector. The inner detector consists of a silicon vertex tracker and a drift chamber for charged particle tracking and vertexing, a ring-imaging Cherenkov detector for particle identification, and a Cs l calorimeter for measuring particle energies. These detector systems are surrounded by a super-conducting solenoid Chapter 3. BABAR Experiment 20 that is designed for a field of 1.5 T . Outside of the solenoid is the instrumented flux return which is used for muon and neutral hadron detection. The polar angle coverage extends to 350 mrad in the forward direction and 400 mrad in the backward direction, defined relative to the high energy beam (electron beam). A n overview of the coverage, the segmentation, and performance of the BABAR detector system is presented in Table I of Ref.[18]. Here we only provide a very brief description of each component of the detector (For more details of the detector, see Refs.[15] and [18].) The innermost part of BABAR detector is the silicon vertex tracker (SVT). It has been designed to measure angles and positions of charged particles just outside the beam pipe. It is critical for the separation of the decay vertices of the B mesons and proper reconstruction of short lived particles. The S V T is composed of five layers of double-sided silicon micro-strip detectors. The inner three layers primarily provide position and angle information for the measure-ment of the vertex position. The outer two layers are needed for low PT tracking and providing the coordinate and angle measurements to the drift chamber. The geometrical acceptance of the S V T is 90% of the solid angle in the C M system. Surrounding the S V T is the the drift chamber (DCH), whose principal pur-pose is the momentum measurement for charged particles. The D C H has been designed to measure not only the transverse momenta and positions, but also longitudinal positions of tracks. Longitudinal information is derived from wires placed at small angles to the principal axis. The drift chamber also provides information for lower momentum particles by measuring the ionization energy loss, dE/dx, for particle identification. The D C H is of compact design, with 40 layers of small, approximately hexagonal cells. By choosing low-mass wires and a helium-based gas mixture, multiple scattering inside the D C H is minimized. Outside of the drift chamber is the DIRC, the detector of internally reflected Cherenkov light. It is a novel ring-imaging Cherenkov detector, providing sepa-ration of pions and kaons from about 0.7 GeV to the kinematic limit of 4.5 GeV. The Cherenkov photons are emitted by a particle traveling faster than the speed of light in the surrounding medium. The mass of a charged particle can by deter-Chapter 3. BABAR Experiment 21 mined by measuring the angle of the Cherenkov radiation and the momentum of the particle track. Cherenkov light is produced in the 4.9 m long bars of synthetic fused silica of rectangular cross section, 1.7 cm x 3.5 cm, and trans-ported by total internal reflection, preserving the angle of emission, to an array of photo-multiplier tubes (PMTs). This array forms the backward wall of a toroidal water tank that is located beyond the backward end of the magnet. Images of the Cherenkov rings are reconstructed from the position and time of arrival of the signals in the P M T s . In order to identify the type of particle (e, H, 7T, K, p), an unbinned maximum likelihood method is used to incorporate all the space and time information from the DIRC. The electromagnetic calorimeter (EMC) is designed to detect electromag-netic showers with excellent energy and angular resolution over the energy range from 20 MeV to 4 GeV. This coverage allows the detection of low energy 7r°s and 77°s from B decays as well as from electromagnetic and radiative processes. The E M C is composed of 6,580 thallium-doped cesium iodide crystals. It's divided into two sections, a barrel and a forward end-cap as shown in Fig. 3.1. The crystals absorb electromagnetic showers and produce scintillation light which is read out by two silicon positive-intrinsic-negative (PIN) diodes. To optimize resolution, the amount of material in front of and in-between the crystals is held to a minimum. Low-noise analog circuits and frequent, precise calibration of the electronics and energy response over the full dynamic range are crucial for maintaining the desired performance of the E M C . The instrumented flux return (IFR) for the main magnet also serves to iden-tify muons and to detect neutral hadrons. It was designed to have a high efficiency and good purity over a wide range of momentum down to less than 1 GeV and a high solid angle coverage. The IFR originally consisted of 19 layers of steel and resistive plate chambers (RPCs). It uses the steel of the flux return as a muon filter and hadron absorber. Muons can be identified by measuring how many layers of steel the particle traverses, and comparing that to the ex-pected radiation length for muons. Two additional cylindrical layers of R P C s with four readout planes were placed at a radius just inside the magnet cryostat Chapter 3. BABAR Experiment 22 to detect paxticles exiting the E M C . In 2004, some of the R P C s were replaced with LSTs (Limited Streamer Tubes). For a neutral B mixing analysis, the most important detector capabilities include charged particle tracking, vertex reconstruction, and particle identifica-tion. A l l these requirements can be met by combining the different sub-detectors described above[25]. 3.2 Event generator software A piece of software called "Generators-Framework Interface" (GenFwklnt) is the package used to control event generators in the BABAR framework. One of the GenFwklnt interface modules is Gf iEvtGen which interfaces to the generator called EvtGen [26]. EvtGen is a package for simulating physics processes in decays of B-mesons and other resonances at generator level, i.e., it does not include any material interactions or detector simulation. One of the novel ideas in the design of EvtGen is that decay amplitudes, instead of probabilities, are used for the simulation of decays. The framework uses the amplitude for each node in the decay tree to simulate the entire decay chain, including all angular correlations. EvtGen provides a framework into which decay models may be added. This design is efficient and leads to reusable decay models. The BABAR experiment uses the EvtGen package to simulate generic B decays in its Monte Carlo production. EvtGen implements many detailed models that are important for the physics of B mesons. In particular, it has detailed models for semileptonic decays and CP-violating decays and it uses full angular distributions in sequential decays. EvtGen introduces mixing by generating decays of the T ( 4 £ ) to the proper mix-ture of B°B°, B°B°, and B°B° final states, with the correct distributions of proper decay time difference At. CP asymmetries are generated in modules that modify the generated lifetime distributions of the two B's produced in the decay of the T(4S). One of the particular strengths of EvtGen is for semilep-tonic decays, in which a large number of different models are available. This is Chapter 3. BABAR Experiment 23 important for the neutral B mixing study using dilepton events of this thesis. EvtGen provides decay tables for generic B-meson decays. These tables con-tain detailed models and branching fractions for each B meson decay mode. It provides the default simulation of B-decays in the BABAR simulation. The generator handles only exclusive final states, i.e., it does not perform any frag-mentation. In order to provide a complete table of decays, it also has an interface to the Jetset software package[28] for generation of continuum events at the T(45) and for generic hadronic decays of B mesons that are not implemented in the generator. For the decays of B mesons, about 60% are accounted for by exclusive states, while the other 40% are produced by Jetset7.4. There are 4 basic options that control EvtGen [26]: GEN mode DECAY decay tab le UDECAY ' ' u s e r ' ' decay table PDT p a r t i c l e data table • GEN specifies which decay mode EvtGen will call. There are two options: T(AS) and continuum. • DECAY specifies which decay table to use. The default decay table is called DECAY. DEC, which contains all decay models implemented by EvtGen. • UDECAY specifies the user defined decay table, which will be read after the DEC AY. DEC table. This allows a user to redefine the decays of certain particles after having read the generic decay table. • PDT specifies the particle property table to read by EvtGen. The default particle data table is pdt . tab le . By modifying this file, one can change the properties of particles, such as lifetimes. It's simplest to run the EvtGen within the GeneratorsQA package[29]. This package is setup to run the event generators in standalone mode, i.e., with-out detector simulation or reconstruction. Modification of the event generation procedure will typically involve changing the analysis module or adding a new Chapter 3. BABAR Experiment 24 module. EvtGen is therefore very convenient for conducting a generator-level analysis, or quality assurance studies of the generators and GenFwklnt (for de-tails about GeneratorsQA, see Ref.[29]). The standard way of running EvtGen begins with initializing particle properties according to the pdt. table and pars-ing the full standard decay table DECAY.DEC. After this, a user decay table is parsed in order to redefine the particle decays of interest. In this way, the user decay file will override the standard decay table. The main EvtGen decay table is DECAY.DEC, which contains world average values for the generic decays of B-mesons and other lighter mesons and baryons. For details about the format and syntax of the decay table, see Ref.[27]. Here we provide some simple examples to illustrate the basic format of the decay table. The following is an example for T(4S) —¥ BB decay: Decay Upsilon(4S) 0.50000 B+ B- VSS; 0.49913 B0 anti-BO VSS_BMIX dm; Enddecay A decay entry begins with the key word "Decay" and ends with "Enddecay". Each entry in the table starts with the branching fraction for the decay channel and is followed by a list of the decay products. After each decay product a model name is listed. This model name specifies how to model the decay of the particle. In the example above, "VSS-BMIX" is a model for the decay of a vector meson (the Y(4S)) into a pair of scalar particles (B°B°) with mixing. "VSS" is the same model for T(4S) B+B~, without mixing. Note that 0.087% of T(4S) decays will be non B B final states. These models make proper use of the polarization of the T(4S) such that the B and B are produced with the appropriate decay angles. The variable "dm" is the mixing frequency (Amd) of the neutral B mesons as defined in DECAY. DEC. Final state radiation using the PHOTOS [30] package may be included in each decay mode. This option is invoked by the key word PHOTOS, which is placed after the list of decay daughters, but before the model name. In the following example for B° -¥ D*~t+v , the EvtGen will call the PHOTOS package and use Chapter 3. BABAR Experiment 25 a H Q E T inspired parameterization of the decay spectrum with parameters for p2, Ri, and R2 respectively: Decay BO 0.5000 D*- e+ nu_e PHOTOS HQET3S1 0.92 1.18 0.72; 0.5000 D*- mu+ nu_mu HQET3S1 0.92 1.18 0.72; Enddecay In EvtGen, all particle property information is contained within the particle data table. Here is an example of part of the pdt . table: name i d mass width max. .Dm 3*Q 2*S l i f e t i m e * c Lund-e- 11 0.00051 0 0 -3 1 0 11 e+ -11 0.00051 0 0 3 1 0 0 B0 511 5.2794 0 0 0 0 0.462 107 ant i -B0 -511 5.2794 0 0 0 0 0.462 0 B+ 521 5.2791 0 0 3 0 0.502 108 B - -521 5.2791 0 0 -3 0 0.502 0 The "id" is the particle number according to the standard high energy physics numbering scheme. In columns 3 to 5, the "mass", the "width", the "maximum allowed deviation from the mean mass in the downward direction" all have units of GeV. The sixth column contains 3 times the charge of the particle and the seventh column contains twice the spin. The cr is the lifetime times the speed of light and it is quoted in units of mm. The last column is the L u n d - K C particle identity number[27], which is used for the interface to Jetset . To set the particle data and decay table, and to specify T(45) decays, the following code is placed in the decay Tel script file: module t a l k GfiEvtGen PDT set ~/mypdt.table DECAY set ~/myDECAY.DEC GENERATE set "Upsilon(4S)" ex i t Here "mypdt. table" and "myDECAY. DEC" are the user modified versions of the default p d t . t a b l e and DECAY.DEC, respectively. We can direct the generator to produce only events which satisfy some spe-cific conditions by using a generator-level filter to make the selection. This saves Chapter 3. BABAR Experiment 26 a lot.of computer time which would otherwise be spent simulating events that are of no interest. For example, the following Tel commands will only allow events with e+e~ or fi+fj,~ in the final states of T(4S) decays. module t a l k G e f S e l e c t F i l t e r BooNew P d t l = GefPdtLis t BooNew Pdt2 = GefPdtLis t BooObjects P d t l and e+ e-BooObjects Pdt2 and mu+ mu-BooCompose m y f i l t e r = or P d t l Pdt2 b e f o r e F i l t e r set m y f i l t e r ex i t The filter that we used for our dilepton events generation is very much the same as the example above but included all combinations of dileptons. To generate multiple runs with non-overlapping random numbers, we can change the run number for each run by: module t a l k RacTestlnput run set 1 ex i t where the run number 1 is specified, and it can be changed for the following runs to be 2, 3, 4, etc. After compiling and linking (for details, see Ref.[29]), we are ready to run the executable called "GeneratorsQAApp" from our working directory. We instruct it to generate one million events and then exit as follows: ev begin -nev 1000000 ex i t The output of EvtGen is a set of 4-vector momenta and vertex positions for the decay products. The following is part of an example output for an event generated with EvtGen: Id Px Py Pz E Vx Vy Vz e- -0 1881 -0 0074 8 9953 8 9973 -0 0906 0 2750 0 6745 e+ 0 0664 0 0031 -3 1103 3 1110 -0 0906 0 2750 0 6745 Ups i lon -0 1218 -0 0043 5 8850 12 1083 -0 0906 0 2750 0 6745 ant i -B0 -0 3649 -0 1597 2 9009 6 0371 -0 0906 0 2750 0 6745 B0 0 2431 0 1555 2 9840 6 0712 -0 0906 0 2750 0 6745 D*+ 0 7218 -1 6926 1 8923 3 3176 -0 0934 0 2737 0 6965 mu- 0 0781 1 2371 0 8515 1 5075 -0 0934 0 2737 0 6965 Chapter 3. BABAR Experiment 27 The momenta are in units of GeV and vertex coordinates (Vx, Vy, Vz) are in units of centimeters. These four-vectors are normally used as the input to the down stream detector response simulation, which is outside of the scope of this thesis. A l l of the information of the generated decays will be stored in an NTuple contained within an H B O O K file, which can be converted into a Root file for subsequent analysis. Al l of our data analysis will be based on this Root file, which contains all of the information needed for our Monte Carlo study. 28 Chapter 4 Neutral B Mixing Frequency Measurement Since the neutral B mixing frequency Am<j is sensitive to both of the C K M matrix elements |Vt&| and |Vtd|> its measurement is of fundamental importance in particle physics. The value of A m j has been measured by many experiments around the world since the late 1980's. Recently, more precise measurements of Amd are possible with the help of high luminosities and clean backgrounds at asymmetric B factories. In this chapter we first briefly review the existing neutral B mixing mea-surements in Section 4.1. We will focus on more precise measurements from B factories, for which a neutral B mixing parameterization is studied in this thesis. The boost approximation and the point of closest approach (POCA) method are introduced and explained in Sections 4.2 and 4.3, respectively. These are the two major approximations used in these measurements and in our Monte Carlo analysis as well. 4.1 Brief review of B°-B° mixing measurements Time-integrated measurements of neutral B mixing were published for the first time in 1987 by UA1[31] and A R G U S [32], and since then by many other exper-iments, such as CLEO[33] and L E P experiments. These measurements are typ-ically based on counting same-sign and opposite-sign lepton pairs from semilep-tonic decays of the produced bb pairs. Assuming A r d = 0, the time-integrated Chapter 4. Neutral B Mixing Frequency Measurement 29 mixing probability for Bd system in Eq . (2.14) becomes By using this relation, the measured average Xd = 0.182 ± 0.015 from A R G U S and C L E O converts to[34] Amd = 0.493 ± 0.032 p s _ 1 . (4.2) These time-integrated analyses, however, cannot easily separate the contri-butions from the different 6-hadron species produced from the bb pair. Further-more, A m ^ has to be inferred from the time-integrated probability as shown in Eq . (4.1) (for details, see the review of B°-B° mixing in Ref.[4]). Better sensitivity can be obtained from time-dependent analyses aimed at the direct measurement of the oscillation frequency A m ^ . This frequency can be measured from the proper time distributions of B° candidates identified through their decays in flavor-specific modes, and suitably tagged as mixed or unmixed (see Section 2.1 for details). At high-energy colliders ( L E P and Tevatron), the proper decay time is mea-sured from the distance L between the B production and decay vertices, and from an estimate of the B momentum p, given by = (4-3) h v where TUB is the B meson mass. We will also use this relation to calculate the proper B decay time for event re-weighting in our Monte Carlo study. At asymmetric B factories ( K E K B , PEP-II) with e+e~ -> T(4S) -> B°B°, the proper decay time for a B meson can be measured from its traveled distance z along the beam direction (ignoring the small transverse distance) by using: t x ± (4.4) Since the boosts for the two B mesons are unknown and unequal (due to unequal momenta) in the lab frame, the proper decay time difference between Chapter 4. Neutral B Mixing Frequency Measurement 30 the two B candidates is estimated by an average boost1 as where Az is the spatial separation between the two B decay vertices along the boost direction. Many time-dependent neutral BD oscillation analyses have been published by experiments around the world. A variety of techniques have been used: ALEPH[35], DELPHI[36], L3[37], and OPAL[38] used hadronic Z decays from e +e~ -4- Z -¥ bb; CDF[39] used the Fermilab Tevatron pp Collider operating at y/s = 1.8 TeV; BABAR [40, 41, 42] and Belle[44, 45] used e+e" -> T(45) -> B°B°. A l l individual experiment averages are listed in Fig. 4.1, which are ob-tained by the B oscillations working group within the Heavy Flavour Averaging Group (HFAG) [34]. Combining all published time-dependent measurements and accounting for all identified correlations, yields2: Amd = 0.510 ± 0.003 ± 0.004 ps" 1 . (4.6) Assuming ATd = 0 and no CP violation in mixing, and using the mea-sured BD lifetime, the time-integrated and time-dependent Amd measurements in E q . (4.2) and E q . (4.6) are combined to yield the world average value[34] Amd = 0.509 ± 0 . 0 0 5 ps" 1 . (4.7) It's worth noting that the most recent and precise analyses at the asym-metric B factories measure simultaneously Amd and the neutral B meson life-time T ( B ° ) . The simultaneous measurements of neutral B lifetime and mix-ing frequency by BABAR are based on fully[41] or partially[42] reconstructed B ° ->• D*lv decays. On the other hand, the latest Belle analysis[45], based on fully reconstructed hadronic B ° decays and B ° ->• D*lv decays, extracts si-multaneously Am,i, T(B°), and r ( B + ) . The results of these three simultaneous measurements of Amd and r ( B ° ) are displayed in a two-dimensional plot in 1BABAR and Belle have average boost < 0y > of 0.55 and 0.425, respectively. 2The first quoted error is statistical and the second one is systematic. This applies to all the quoted measurement errors through this thesis. Chapter 4. Neutral B Mixing Frequency Measurement 31 0.44610.02610.019 ps'J 0.519+0.01810.011 ps"1 0.444±0.028±0.028 ps": 0.479±0.018±0.015 ps'; 0.495±0.033±0.027 ps" 0.504±0.025±0.012 ps" 0.456±0.034±0.025 ps" 0.515±0.004±0.006 ps" 0.50910.00410.005 ps" 0.51010.005 ps"1 0.49310.032 ps"1 0.50910.005 ps"1 Figure 4.1: Average of A m ^ measurements from different experiments. This plot is taken from Ref.[34]. Fig. 4.2. Taking into account all statistical and systematic correlations, evalu-ating at the measured charged B lifetime T(B+), the three measurements are combined to yield the average values: Amd = 0.514 ± 0.005 p s - 1 , (4.8) T(B°) = 1.532 ± 0.011 ps, (4.9) with a total correlation of —0.31 [34]. One can see from Fig. 4.1 that measurements from B factories are more precise than others. Since the BAJBAR and Belle experiments are using almost the same technique to measure Amd, we will only provide more details of the measurement analyses made by BABAR, the experiment on which this thesis is ALEPH (3 analyses) DELPHI * (5 analyses) L3 (3 analyses) OPAL (5 analyses) CDF1 * (4 analyses) CDF2 * (2 prel. analyses) DO (1 prel. analysis) BABAR * (1 prel. + 3 analyses) BELLE * (3 analyses) average of above after adjustments ARGUS+CLEO (Xd measurements) world average working group average without adjustments I 1 I 1 I 1 I n h • il H — H —-4 H U n H H 0.4 0.45 0.5 0.55 V A m d (ps"1) Chapter 4. Neutral B Mixing Frequency Measurement 32 O H C 0.54 a <0.52 0.5 0.48 BABAR (prel) D lv part. reco. 88M BB A%2= 1 stat only sfat + syst — \ . / ^ s ^ v Average > t f ^ X \ B E L L E - / - " " " ' \ full hadr. + D lv — \ 1 / \ • \ 1 / 152M BB BABAR D*lv — V *v / / 23MBB I HFAG . Winter 05 i 1 , 1 i 1.45 1.5 1.55 1.6 t(B ) in ps Figure 4.2: Simultaneous measurements of Amd and TBO. The average of the three measurements is indicated by the shaded ellipse. This plot is taken from Ref.[34]. based. BABAR had previously performed several precision measurements of the B ° B ° oscillation frequency Amd [40, 41, 42]. In those analyses, the total decay-rate difference between B ° and B ° , CP violation in mixing and C P T violation were assumed to be negligible. Recently, a generalized B ° B ° mixing analysis was performed which didn't make these assumptions[43]. In all of these analy-ses, the experimental technique consisted of first fully reconstructing a B meson Chapter 4. Neutral B Mixing Frequency Measurement 33 and measuring the vertex position of this "Breco". The vertex of the second B, which is called "Btag", was determined using the remaining charged tracks of the event. The flavour of uBtag" was determined from the charge of leptons and/or other tracks among the remaining tracks, such as kaon(s) and soft pion(s). The distance along the beam direction between the "Brec0" and "Btag" vertices was then used to estimate the lifetime difference At between the two B mesons, using the boost approximation. The reconstructed B pairs were classified in categories which depended on the flavours of "Breco" and "Btag". A fit to the measured At distributions in the different categories was performed to extract the B°B° mixing parameters. In previous measurements, the B°B° pairs are classified as "mixed" or "un-mixed". The parameter Am,* is extracted from the asymmetry in E q . (2.24). In the generalized analysis, the B pairs are classified in 6 different categories: B°B°, B°B°, B°B°, B°B°, B°BCp, B°BCp, where the two B mesons are Btag and Breco, respectively. Unlike the previous mixing analysis, the inclusion of CP eigenstates for Breco in this analysis leads to a more complex fit formula (for details, see Ref.[43]). 4.2 Boost approximation One of the advantages of an asymmetric B factory is that the proper B decay time can be inferred from the corresponding measurable decay length by using the boost approximation. In the following, we will briefly explain how the boost approximation works. Let the 4-momenta of the e~, e+, and T(4S) be (using convention P=(E, Pxi Pyi Pz))'-Pe- = (Pi, 0, 0, pi); Pe+ = (P2, 0, 0, -pi); (4.10) Pr(4S) = (Pi +P2, 0, 0, p i - p 2 ) . Chapter 4. Neutral B Mixing Frequency Measurement 34 T(45) B - - - ' ' ~ - - _ z h Az Figure 4.3: Two B decay vertices, showing displacement along the boost axis, Az. It follows that the boost factor of T(4S) is given by 0 7 = r(4S) I P1-P2 9 .0-3 .1 = 0.558. (4.11) m r (4s ) »^r(4S) 10.58 where the nominal momenta of the e~~ (9.0 GeV) and the e+ (3.1 GeV), and the Y(4S) mass (10.58 GeV) were used. In the real experiment, there are small transverse momenta and small fluc-tuations in the e+e~ 4-momenta. We can randomly select a Monte Carlo event as an example with the following 4-momenta in units of GeV: Pe- = ( 8.9844,-0.1902,-0.0085, 8.9824); Pe+ = ( 3.1099, 0.0666, 0.0025,-3.1092); Pr(4S) = (12.094,-0.1236,-0.0060, 5.8733). (4.12) which results in a boost of 0 7 = - T(4S) I 5.8746 "ir(4S) 10-58 = 0.553. (4.13) Chapter 4. Neutral B Mixing Frequency Measurement 35 In the data analysis, the average boost < 0 7 >= 0.55 was used for the BABAR experiment. We will choose this same value in our Monte Carlo analysis as well. Since the two B mesons are almost at rest in the Y(4S) frame, the proper time difference between their decays is estimated with the help of this boost by, where Az is the z coordinate difference between the two B decay vertices as shown in Fig. 4.3. This approximation neglects the B meson's motion in the Y(4S) rest frame (i.e. a momentum of 335 MeV), and could in principle introduce a systematic Am,! shift which is not negligible at the level of accuracy expected (better than 2% on Amd)- Since it is not possible to determine the exact boost, the effect of this bias was studied with a toy Monte Carlo by comparing the fitted value of Am<2 with the true At and with Az/ < 0 7 >. The relative systematic error related to the correction of the shift is about 0.3% [46]. 4.3 Point of closest approach—POCA The problem of estimating the z coordinate of the B decay vertex with the use of only the lepton track can be solved to a first approximation by taking the z coordinate of the point of closest approach between the track and the beam spot in the x-y plane. This estimator is a fairly good way to determine the z position of the B decay vertices, since the selected direct leptons have rather high momenta, which result in relatively small errors on the track positions. The point of closest approach (POCA) with respect to the decay vertex of the Y(4S) (also the e+e~~ collision point) in the x-y plane is calculated for each leptonic track. The corresponding z coordinates represent the best approxima-tion of the z coordinates of the decay vertices of the two B mesons. More details can be found in Ref.[46]. The topology of the dilepton event in the transverse plane is shown in Fig. 4.4. In experimental analysis, the point on the lepton track closest to Chapter 4. Neutral B Mixing Frequency Measurement 36 Figure 4.4: Dilepton event topology in the transverse plane. the beam spot (average of collision points) is taken as an estimation of the true vertex. That is, in Fig. 4.4, " P O C A 1" and " P O C A 2" are chosen as estima-tions of "True B i vertex" and "True B2 vertex", respectively. Then use x and y coordinates of these P O C A points to find the corresponding z coordinates on the lepton tracks. Therefore, the estimated vertex difference Az is obtained by using this " P O C A " approximation. To simulate real data analysis, we follow this same P O C A approach even though we know the exact B decay vertices in our Monte Carlo simulated events. For example, let the real B decay vertex, the lepton momentum, and the e+e" collision point for one dilepton event be, respectively, X = (x, y, z); P = (Px, Py, Pz); (4.15) X° = (x°, y°, z°). Chapter 4. Neutral B Mixing Frequency Measurement 37 The estimated vertex X' should be on the lepton track. It can be written as, X' = X + s • P, (4.16) where s is a constant. Therefore, the estimated z coordinate of the B decay vertex is z'=z + s-pz. (4.17) The constant s is determined by finding the P O C A within the x-y plane. That is, s is determined by the minimization condition: fl(*i-*i)-(*:-*D=0> . ( 4 _ 1 8 ) OS where the subscript "_L" represents the transverse components of a vector. Solv-ing this equation gives the following value for s: Plugging E q . (4.19) into E q . (4.17) gives our estimate of the z coordinate for the B decay vertex which we will call " P O C A z". The z coordinate difference of the two B vertices, therefore, will be called " P O C A Az". From Eq . (4.20), we see that different values of real Az can end up with the same value of P O C A Az. By using the boost approximation as discussed in Section 4.2, it means that different "real" At will contribute to the same P O C A estimated At. This is usually called the smearing effect of the P O C A approximation. In the real experiment, we cannot know exactly the collision point for each event. Therefore, the nominal beam spot for each run will be used as an esti-mation of the average collision point. Even though we know the exact collision point for each Monte Carlo event, we will follow the same method as that of real experiment data by using the beam spot for each run. The beam spot is fixed for each run in our Monte Carlo generation and is equivalent to the average e+e~~ collision point in that run. 38 Chapter 5 Analysis Strategy To get exact distributions of the measured proper decay time difference At for the two B meson decays, we need to study how these distributions depend on lifetime and mixing frequency. This is achieved by studying At distributions with different values of lifetime and mixing frequency and parameterizing the changes. As a result, the parameterized At distributions can be used to extract values for the B lifetime T B and the neutral B mixing frequency Amd from experimental data. In this chapter, we first explain how our events were generated and classified in Sections 5.1 and 5.2, respectively. The details of event re-weighting are given in Section 5.3. Finally, we explain our choice of parameterization model and fitting strategy in Section 5.4. 5.1 Event generation The two key physics parameters of interest in generating Monte Carlo events are the B meson lifetime and the neutral B mixing frequency. As explained in Section 3.2, they are defined in the files pdt . tab le and DECAY.DEC, respectively. We can change the values of the B lifetime and the neutral B mixing frequency in these two files for event generation. The default values1 and the ones we used in our Monte Carlo simulation are listed in Table 5.1. We chose the B lifetimes to be 4 times longer than the default values and generated all events without mixing. The reason for using longer lifetimes is that we want to get more statistics for events with larger time difference At, which makes our later event re-weighting more reliable in 1-We quote these default values in the rest of this thesis by using a superscript "0" for each of the symbols: T ^ 0 , , and Am^. Chapter 5. Analysis Strategy 39 Default Value Our Choice World Average (PDG[4]) TflO (PS) rB+ (ps) Amd ( p s - 1 ) 1.541 1.674 0.489 4 x 1.541 4 x 1.674 0 1.536 1.671 0.502 Table 5.1: B lifetimes and the neutral B mixing frequency: Default values versus our choice. The default values are very close to the 2004 world average values with fractional difference less than 3%. that region. Similarly, the reason for choosing zero mixing is to make our event re-weighting reliable along all values of At (more details will follow in Section 5.3.). As explained in Section 3.2, we use a generator-level filter to require at least 2 leptons in the final state of T(4S) decays, where either of the leptons could be an electron or a muon 2 . The two highest-momenta leptons of the final state will be selected as our dilepton signal. The lepton pairs are ordered in their momenta in the lab frame: lepton 1 is always the lepton with the highest momentum and lepton 2 is the lepton with the second highest momentum. Therefore, At (or Az) is a signed quantity, with the value of lepton 1 minus that of lepton 2, i.e., At = t\— t2 and Az = z\ — z2. 80 million events passing our filter were generated by the generator package EvtGen within the BABAR framework. A l l of these events were generated with the parameter values specified in Table 5.1. The At distributions for any value of neutral B mixing frequency can be obtained by re-weighting these generated events. A n additional 35 million dilepton events with default lifetimes and mixing frequency were also independently generated as test events. This second set of 35 million events are used to obtain At distributions without any event re-weighting. We then use these distributions and our parameterization results (from the first 80 million events) to extract the B meson lifetimes and the neutral B mixing frequency. By comparing the fitted values with the expected values 2We do not include tau as our signal lepton because it has a very short lifetime (r ~ 0.29 ps), and is not detected directly. Chapter 5. Analysis Strategy 40 M C Information Particle Used in Part of Analysis ID Leptons, B mesons Event classification Vertex Leptons and T(AS) P O C A and boost Momentum Leptons P O C A approximation Proper Decay Time B mesons Event re-weighting Table 5.2: Monte Carlo event information used in our analysis. for B lifetimes and mixing frequency, we can cross check our parameterization method. A l l of the necessary information that we need for each event in our analysis is listed in Table 5.2. This information is extracted for the generated events from the generator by using a Monte Carlo truth analysis module as described in Section 3.2. It is stored in an NTuple and is ready to be read out by later analysis modules. Here particle identities (ID) for leptons and their B parents are essential for event classification. The vertices of selected leptons and the e+e~~ collision point, which is the same as the T(4S) production vertex, are used in the P O C A and boost approximations as shown in Sections 4.2 and 4.3. Momenta of leptons are needed in the P O C A approximation as well. The proper decay times of the lepton B parents are used to re-weight the generated events (details of event re-weighting will be given in Section 5.3.). A l l the information listed in Table 5.2 is available directly from the MC-truth information for each generated event, except for the proper time of B decays, which are calculated by tproper = L, (5-1) P where L is the distance between the B production and decay vertices, and p is the B meson momentum in the Lab frame 3 . 3 As mentioned in Section 4.1, this technique is also used by other B mixing studies at high-energy colliders, such as Tevatron, where the boost approximation doesn't hold. Chapter 5. Analysis Strategy 41 5.2 Event classification Dilepton events are classified by three combined criteria as shown in Table 5.3. In the following, we explain each of them in detail. Criterion Category Lepton Sign Same Sign (SS) Opposite Sign (OS) Lepton Momentum High-High (HH) High-Low (HL) Low-Low (LL) Lepton B Parent Same B (SB) Opposite Charged B (OCB) Opposite Neutral B (ONB) Table 5.3: Dilepton event categories. A l l events are classified by combining these three criteria. • Lepton Sign of Charge Firstly, the two selected leptons are classified by their sign of charge. A n event is called Same Sign (SS) event if the two leptons have same sign of charge; Otherwise, it is called an Opposite Sign (OS) event. • Lepton Momentum Secondly, the two selected leptons are classified by their momenta with the definition of High (H) and Low (L) momenta as follows, Low: 0.5 GeV < p < 1.2 GeV; High: 1.2 GeV <p< 5.0 GeV. (5-2) To simulate real data analysis, leptons with momenta lower than 0.5 GeV are removed because of their poor resolution and low efficiency in detector, even though there is no detector simulation in our analysis. Momentum of the first and the second leptons are classified into High-High (HH), High-Low (HL), and Low-Low (LL) combinations. Since the two leptons are Chapter 5. Analysis Strategy 42 n n Pi (GeV) Figure 5.1: Re-scaled 2-D momentum contour plot of dilepton events. sorted by their momenta, there is no Low-High category in our selected events. The division of Low and High momentum is obtained with the help of a 2-D contour plot of lepton momentum as shown in Fig. 5.1, where the x-axis is the momentum of lepton 1 and the y-axis is the momentum of lepton 2. The contour plot was obtained by using opposite neutral B events with weight 1 for the correct sign dileptons and with weight -1 for the wrong sign dileptons4. The contours are rescaled by the sum of correct and wrong dilepton events so that they have moduli no larger than 1. In case of equal number of correct and wrong sign dilepton events, we get the zero contour in the plot, which has the same colour as the region without any events (the upper-left part of the plot). We choose 1.2 GeV as our event division as it is very close to the zero contour shown in Fig. 5-1. 4If two leptons have right-right or wrong-wrong signs, they keep the correct sign of their B parents (OS or SS). Therefore, they are called "correct sign" dileptons. On the other hand, if one lepton has the correct sign and the other has the wrong sign, they are called "wrong sign" dileptons. The division of the lepton momentum was chosen as the region with roughly equal right and wrong dilepton events. Chapter 5. Analysis Strategy 43 In this way, the High-High events are mostly correct sign dilepton events, High-Low events are a mixture of correct and wrong sign dileptons, and Low-Low events are mostly wrong sign dileptons as shown by the negative contours. • Lepton B Parent Thirdly, the two selected leptons are also classified by their B parents. If the two leptons of an event come from the same B parent, then it is called a Same B (SB) event. If they come from different B parents, it is then called an Opposite Charged B (OCB) event or an Opposite Neutral B (ONB) event, depending on the charge of the B parents. The sample distributions of the " P O C A At"5 are shown in Fig. 5.2 (SB), Fig. 5.3 (OCB) , and Fig. 5.4 (ONB). A l l of these distributions are obtained from the test events with the default values of B lifetimes and mixing frequency as specified in Table 5.1. In each of the figures, the first column is for the opposite sign events and the second column is for the same sign events. Simi-larly, the three rows correspond to High-High, High-Low, and Low-Low events, respectively. For the same B events shown in Fig. 5.2, most of the dilepton events have opposite sign. This is because the first leptons with highest momentum are mostly direct leptons with the correct sign. The secondary leptons are mostly produced from cascade D decays. As explained in Section 2.1, these cascade leptons mostly have the wrong sign of charge compared to a direct lepton. Therefore, the two leptons produced from the same B decay mostly have oppo-site signs. The reason for having very a small number of same sign events with High-High momenta is that both of the leptons come from B cascade decays, which are very unlikely to both have momentum larger than 1.2 GeV. We get more same sign events in the Low-Low category because the two leptons are more likely to be both cascade leptons with the same sign of charge. Since both 5The At converted from the POCA Az by using the boost approximation is called the "POCA At" in the rest of this thesis. Chapter 5. Analysis Strategy 44 SB-HH-OS M35000 c >30000 25000 20000 15000 10000 5000 Entries 690513 Mean -0.1608 RMS 3.27 (ps) xlO £ 1200 > III 1000 300 600 400 200 SB-HL-OS Entries 2692260 Mean -0.407 RMS 3.044 15 1^0 3 0 (PS) Entries 1396924 Mean -0.144S RMS 3.09 Entries 19782 Mean -0.4949 RMS 2.945 11 0°°l-800 200 -ft SB-LL-SS Entries 24039 Mean -0.08422 RMS 3.072 5 Figure 5.2: P O C A At distributions for the same B (SB) events. In this cate-gory, both of the two leptons come from the same B parent. The two columns correspond to opposite sign (OS) and same sign (SS), respectively. The three rows are High-High (HH), High-Low (HL), and Low-Low (LL) momenta categories. Chapter 5. Analysis Strategy 45 45000 £ 40000 c 9 >35000 30000 25000 20000 15000 10000 5000 n OCB-HH-09 Entries 1472736 Mean -0.01811 RMS 3.146 OCB-HH-SS Entries 239515 Mean 4.5797 RMS 4.11 Entries 1713936 Mean -0.137 RMS 3.149 « 25000 OCB-HL-SS Entries 1098823 Mean -0.5725 RMS 3.523 Entries 575927 Mean -0.1235 RMS 3.428 Figure 5.3: P O C A At distributions for the opposite charged B (OCB) events. In this category, the two leptons come from opposite charged B mesons. That is, one lepton comes from B+ and the other lepton comes from B~. The two columns correspond to opposite sign (OS) and same sign (SS), respectively. The three rows are High-High (HH), High-Low (HL), and Low-Low (LL) momenta categories. Chapter 5. Analysis Strategy 46 ONB-HH-OS HI "35000 C a 30000 25000 20000 15000 10000 5000 Entries 1085836 Mean -0.06328 RMS 2.832 15 -10 -5 ONB-HL-OS Entries1S13S24 Mean -0.1527 RMS 3.352 DNB-HL-SS ONB-LL-OS Entries 792663 Mean 0.08361 RMS 3.616 ONB-LL-SS Entries 501972 Mean -0.5203 RMS 4.593 Entries 1505412 Mean -0.7101 RMS 3.794 Entries 788857 Mean -0.1162 RMS 3.716 Figure 5.4: P O C A At distributions for the opposite neutral B (ONB) events. In this category, the two leptons come from opposite neutral B mesons. Because ofneutral B mixing, the two leptons can come from either of B°B°, B°B°, or B°B°. But they must come from different B parents. The two columns correspond to opposite sign (OS) and same sign (SS), respectively. The three rows are High-High (HH), High-Low (HL), and Low-Low (LL) momenta categories. Chapter 5. Analysis Strategy 47 of the two leptons are coming from the same B parent, the At distributions are independent of the B lifetime and the neutral B mixing frequency. Therefore, no parameterization is needed for these categories. Similar reasoning also holds for the opposite charged B events in Fig. 5.3. Since charge is conserved, we will always have one B+ and one B~~. In cases where both leptons keep the correct sign of their B parents or both are wrong, we get opposite sign events. In case one of the leptons keeps the correct sign and the other keeps the wrong sign of their B parents, we get same sign events. We still have a greater number of opposite sign events than same sign events. Since the two leptons come from opposite charged B mesons, the distributions depend only on the charged B lifetime, but not on the neutral B mixing frequency at all because charged mesons cannot mix. One of the goals of this thesis is to find a parameterization function for these opposite charged B At distributions. When it comes to the opposite neutral B events in Fig. 5.4, there is one additional complexity compared to the opposite charged B events. Due to neutral B mixing, we can have same sign neutral B pairs (i.e., B°B° or B°B°), therefore we still have a chance to get same sign events even when both leptons are direct leptons with correct signs. In this case, the distributions will depend on both the neutral B lifetime and the mixing frequency. The primary goal of this thesis is to find a parameterization function for these distributions, which can be used in precise measurements of the neutral B mixing frequency A m j . For each of the distributions, we fill the events into a histogram with a range of ± 1 5 ps. Each of the histograms is divided into 200 At bins. For each of the 200 At bins, the bin content will change with varying lifetimes and/or mixing frequency. The primary task of the following two sections is to explain how to get distributions with different lifetimes and mixing frequency through event re-weighting, and how this behaviour can be parameterized for each bin of all categories. Chapter 5. Analysis Strategy 48 5.3 Event re-weighting As described in Section 2.3, the probability of getting an event with a neutral B pair produced from T(4S) decays can be written as Pnomix(ti,t2,TBo,Amd) oc - L e - ( * i + t 2 ) / r B 0 [1 + c o s f A m ^ i - t2))}, (5.3) TB° Pmix(ti,t2,TB0,Amd) oc -L e -(*i+*3)/^o [i _ cos(Am d (t i - t2))], (5.4) rB° where tx and t2 are the proper decay times of the two B mesons. Eq . (5.3) and Eq . (5.4) hold for neutral B pairs without mixing (nomix) and with mixing (mix), respectively. Since there is no mixing for charged B mesons, the probability of getting a charged B pair is Pc(tut2,rB+) <x J - e - ( t l + t 2 ) / - B + . (5.5) TB+ A l l the weights that we used in re-weighting the generated events into events with different lifetimes and mixing frequency are listed in Table 5.4. Our choice of lifetimes (4 x r ° ) and mixing frequency (Amd = 0) for generation are used in Generated Category Re-weighted Category Weight O C B (OS/SS) O C B (OS/SS) Pc(t1,t2,TB+) Pc(* l , t 2 ,4x r° + ) O N B (OS) O N B (OS) Pnomix (tl , * 2 , T B 0 ,AlHj) Pnomiz(i l , t2,4XT° 0 ,0) O N B (SS) •Pmiz(il,i2,T B0,Am ( i) P „ o m , x ( t i , t 2 , 4 x r ° ) 0 , 0 ) O N B (SS) O N B (OS) Pmi*(tl,t2,TBo ,Affld) P n o m i » ( t l . * 2 . 4 X T » 0 , 0 ) O N B (SS) P.omii (tl , * 2 , T B 0 ,AtHj) P n o m i » ( t l , < 2 , 4 X T ° 3 0 , 0 ) Table 5.4: Weights for event re-weighting. The probability functions are given in Eqs. (5.3)-(5.5). Re-weighting can change the lepton sign for oppo-site neutral B events (ONB), but not for opposite charged B events (OCB) nor the same B events (SB). Chapter 5. Analysis Strategy 49 the denominators of the weights. The values of lifetimes and mixing frequency in the numerators of the weights can be any value of interest. In our analysis, we vary these values by ± 1 0 % around the default values shown in Table 5.1. For charged B events, the weight only depends on the charged B lifetime. Event re-weighting won't change the event category. For neutral B events, the event category can change when we re-weight the generated events with mixing. In this case, event re-weighting changes the sign type of some of the leptons (OS i—> SS). This is how we recover the neutral B mixing from the generated un-mixed events. It is the right time now to explain the physics parameters we chose in our event generation. Firstly, we can see from Table 5.4 that all the weights are ratios of two probabilities. From the probability function in Eq . (5.3) for un-mixed events, we can see that there is zero probability for some values of At if Amd is non-zero. This would make our event re-weighting unreliable since E q . (5.3) appears in the denominators of the weights for opposite neutral B events. Therefore, Amd = 0 is a very good choice and makes our event re-weighting reliable for all values of At. Secondly, the probability of getting an event with longer B decay times of ti and t2 becomes larger with longer lifetime. Therefore longer lifetime in the generation makes our event re-weighting more reliable in the larger At region (the tails in the At distributions). 5.4 Model selection and fitting strategy With the help of the event categories and event re-weighting described above, we are now ready to introduce our model selection for the parameterization of At distributions. As mentioned in Section 2.3, the probability of getting a direct dilepton event can be written as P(At,rB0,Amd) = e - | A t | / r B o [1 ± cos(AmdAt)] (5.6) P(At,rB+) = 4TBO [ 1 ± 1 ] (5.7) 4r B + Chapter 5. Analysis Strategy 50 where the "+" sign is for OS events and the "—" sign is for SS events. Eq . (5.6) and E q . (5.7) describe opposite neutral B events and opposite charged B events, respectively. As explained in Chapter 4, we have to use the P O C A and boost approxi-mations to estimate At from Az. Therefore, we need to study the distributions for this P O C A At even though we know the distributions for the unmeasurable "real" At as shown in Eqs. (5.6) and (5.7). Furthermore, the probabilities listed in the equations above only apply to direct dileptons ( where both leptons come from B meson decays directly). We want to know the distributions not only for these direct dileptons but also for other events where one or both leptons comes from B cascade decays. We need to choose our own parameterization functions for the P O C A At in our Monte Carlo analysis. For charged B events, the distributions only depend on the charged B life-time. As an example, the At distributions for the sum of same sign and the op-posite sign (OS+SS) events with varying B lifetime are shown in Fig. 5.5, where the lifetime is changed by ± 1 0 % around the default value (TB+ = 1.674ps). A l l these distributions are obtained from event re-weighting, as explained in Section 5.3. Therefore, we need only one function of B lifetime, which can be written as P(At,r) = f(At,r), (5.8) where f(At,r) is a function of At and T . Similarly, distributions for neutral B events depend on both the neutral B lifetime and the mixing frequency. For example, the At distributions with varying B lifetime and mixing frequency are shown in Fig. 5.6 (OS+SS) and Fig. 5.7 ((OS-SS)/(OS+SS)), where the lifetime and the mixing frequency are both changed by ± 1 0 % around the central values (TB0 = 1.541 ps and Amd = 0.489ps - 1 ) . As explained in Section 5.2, the High-High category has mostly op-posite sign dileptons while the High-Low and Low-Low categories have almost equal number of the same sign and opposite sign dileptons. This explains why the asymmetries change with momentum in Fig. 5.7. A l l of these distributions Chapter 5. Analysis Strategy 51 0.2 0.13 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 •1 OCB-HH-(OS+SS) E_ lifetime (in T°(B )) E_ 90% 100% 110% 3 OCB-HL-(OS+SS) 0 . 1 8 § _ lifetime ( inx°(B )) 90% 100% 110% 0.18 (z_ lifetime (in x°(B )) 0.16 OCB-LL-(OS+SS) Entries 3900620 Mean -0.09421 RMS 3.137 Entries 6392177 Mean -0.3056 RMS 3.147 Entries 2789672 Mean -0.06388 RMS 3.168 Figure 5.5: P O C A At distributions of the sum of the same sign and opposite sign (OS+SS) for opposite charged B (OCB) events. For each cate-gory, we changed the charged B lifetime by ± 1 0 % around the default value (T° + ). Chapter 5. Analysis Strategy 52 3 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 ONB-HH-(OS+SS) §_ lifetime (in i ° ( B )) E_ 90% 100% 110% ONB-HL-(OS+SS) 5 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 ONB-LL-(OS+SS) — lifetime (in x°(B )) E_ 90% = 100% 110% Entries 3635155 Mean -0.2134 RMS 3.403 Entries 6908140 Mean -0.429 RMS 3.465 15 (PS) Entries 3621096 Mean -0.03076 RMS 3.551 Figure 5.6: P O C A At distributions of the sum of the same sign and opposite sign (OS+SS) for opposite neutral B (ONB) events. For each cate-gory, we changed the neutral B lifetime by ± 1 0 % around the default value (TJJO)-Chapter 5. Analysis Strategy 53 a = 0.6 E | 0.4 0.2 0 42 , lQNB-HL-(OS-SS)/(OS+SS) ONB-HH-(OS-SS)/(OS+SS) Entries 3635155 - mixing frequency (in Am°) Mean -1.431 ; 90% ~ 100% ~ 110% i5 ' ' ' ' -i'o ' ' ' ' -5 1 RMS 7.28 ' ' ' o' ' ' ' ' i 1 ' 10 15 At (PS) Figure 5.7: P O C A A t asymmetries between the opposite sign and the same sign events (OS-SS)/(OS+SS) for opposite neutral B (ONB) categories. For each category, the neutral B lifetime is fixed at default value (TBQ) and the mixing frequency varied by ±10% around the default value (Amd). Chapter 5. Analysis Strategy 54 are obtained from event re-weighting and the distributions in Fig. 5.7 are eval-uated with the default lifetime. Therefore, we need one function of lifetime, similar to that for charged B events and another function of Am<j as follows: P(At,r,Am) = f(At,r) \ ± \a(&t, Am) (5.9) where g(At, Am) is a function of A i and A m , and " ± " stands for opposite sign and same sign events, respectively. The primary goal of this study is to determine this g(At, Am). The lifetime part, / ( A t , T) , only depends on the B lifetime. The reason for the choice of the mixing part comes from the fact that the sum of OS and SS events will not depend the value of the mixing frequency A m . Therefore the mixing parts must cancel out in the sum of the opposite sign and same sign events. With some simple algebra, one can show that the constant 1/2 in E q . (5.9) will not cause any problem and is chosen for convenience. If we choose any other constant, it will not change any of our results in an important way. To make our fitting more reliable (with a smaller number of parameters6) and to test our choice of function forms, we fit the lifetime and mixing functions step by step. First, we fit the distributions of OS+SS to obtain / ( r ) by Pos + Pss = f(r) = Tle0°+P^+M^f+03(^)\ (5.10) T where r ° is the default B lifetime, and the /3's are parameters. This functional form is motivated by the fact that it should be an exponential decay term with a normalization factor as shown in E q . (5.7). We then use the asymmetry between OS and SS to obtain the function g (Am) by Pos - Pss Pos + Pss = g(Am) = 7o + 7 i (Am) 2 + 72(Am) 4 + 7s(Am) 6 , (5.11) 6If there are too many parameters to fit, the minimization and error analysis package Hinuit [47] is very poor at parameter estimation. But with a reasonable number of parameters, it works very well for all the fits in our analysis. Our experience is that Hinuit is not good at estimating the initial value of parameters. This can be overcome by using Fumili [48], another minimization package (not as good as Minuit in error analysis), to estimate reasonable initial values of parameters and then pass them to Hinuit. This solved all the problems we had during our analysis. Chapter 5. Analysis Strategy 55 where the 7's are parameters. Here we choose only even power of the mixing frequency through the cos(AmAt) term^which is even in A m . Note that there is no At dependence in Eqs. (5.10) and (5.11). This is because we use the same functional form in the fitting for each of the bins with a different value of A t in the distributions. Therefore all A t information will be kept in the parameters: /3(At)'s and 7(At)'s. Therefore, the primary goal of the next step fitting is to extract all these parameters for each bin in the A t distributions. We also tried other functional forms which have clearer physics interpretation than that listed above but the fitted results were not as good (as determined by residuals) as the one we chose in our final analysis. For example, we also proposed, These functional forms have very nice physics interpretation. For each bin with a fixed small region of At , there are contributions from events with real A t that is different from the P O C A A t 7 . Therefore, we add extra terms (/3 in E q . (5.12) and 7 in E q . (5.13)) to A t to account for the average effect of this smearing. In the special case of no smearing effect, i.e., /3 = 7 = 0, Eqs. (5.12) and (5.13) are equivalent with Eqs. (5.6) and Eq . (5.7) for real A t . It turns out that they work very well for direct dileptons but not for the cascade leptons, which have more complicated smearing that cannot be accounted for linear shift in A t . Hence the functions in Eqs. (5.10) and (5.11) are used in the final analysis. In our analysis, the normalized A t distributions with different lifetime and mixing frequency for opposite sign and same sign events will be substituted into Eqs. (5.10) and (5.11) for Pos and Pss, respectively. We fit the parameteri-zation function f(r) first to extract /3(At)'s for each A t bin. This applies to 7This is the smearing effect from the POCA approximation as discussed in Section 4.3 frequency by noting the fact that neutral B events only depend on mixing e-\At+0\/r Pos + Pss = / ( r , A t ) = ; 4r u O S l n S S = p ( A m , A t ) = c o s [ A m ( A t + 7)]. (5.12) (5.13) Chapter 5. Analysis Strategy 56 both charged and neutral B events. For neutral B events, we then calculate the asymmetry in E q . (5.11) by using the fitted (Pos + Pss) in the denominator from the first step fits, and a perform parameterization fit for g(Am) to extract 7(A£)'s. By doing this calculation, we are actually simultaneously fitting the lifetime and mixing frequency but in two steps (see reasons explained above). In summary, we perform parameterization fits of / ( r ) for all 200 At bins in each distribution (charged or neutral B events of different momentum cate-gories), and have parameterization fits of g(Am) for all 200 At bins in each of the neutral B event distributions. After all of these fits are done, we then plug all the /3's and 7's back in Eqs. (5.10) and (5.11). With all these parameters known, the functions depend only on lifetimes and mixing frequency. Finally, we get f(rB+) for charged B events, / ( T B O ) and g(Am<i) for neutral B events. Al l of these results are presented in the next chapter. Chapter 6 Analysis Results 57 We parameterize the P O C A At distributions in terms of their dependence on B lifetime and mixing frequency. As a result, these parameterized distributions are written in the form of functions of the B lifetime and the neutral B mixing frequency. Therefore, measurements of the P O C A At distributions can be used to extract these two physics parameters from experimental data. In this chapter, we first present our results of parameterization fits from varying the B lifetime and the mixing frequency in Sections 6.1 and 6.2, respec-tively. The consistency test of these fits is given in Section 6.3. To test our parameterization method, we provide cross-checks in Section 6.4. Effects of the event re-weighting on the P O C A approximation, especially the effect on using the beam spot as an estimation of the collision point, are discussed in Section 6.5. 6.1 Parameterization fits I— varying B lifetime As mentioned in Chapter 5, the sum of the same sign and opposite sign events (OS+SS) depends only on the B lifetime. Therefore, we vary the B lifetime through event re-weighting to study this dependence. Our goal is to parame-terize the (OS+SS) distributions as functions of the B lifetime. In our analysis, we vary lifetimes for both neutral and charged B mesons by ± 1 0 % around the default values by choosing 11 different sets of lifetimes, i.e., changing the lifetimes by 2% for each set1. As a result, we get 11 sets of distributions with different B lifetimes for each of the event categories through xTo test the applicability of the parameterization functions for larger ranges of lifetime and mixing frequency, we vary them by ±50%. After that test, we vary them only ±10% to get better interpolation results in this smaller region around the default values as shown in Table 5.1. Chapter 6. Analysis Results 58 event re-weighting as discussed in Section 5.3. Each of the 200 At bins of each distribution will yield 11 probability values, corresponding to 11 different values of B lifetime. These probability values therefore can be parameterized as an analytic function of B lifetime. As explained in Section 5.4, we choose the parameterization function as Pos + Pss = f(r) = Tle^+^+^f+^)\ (6.!) T where r ° is the default B lifetime ( T ^ + for charged B and r^0 for neutral B), and the /3's are fit parameters. These parameters can be extracted from a fit for the 11 probability values with varying lifetime T . Examples of the parameterization fits with varying B lifetime are shown in Fig. 6.1 for the opposite charged B events. Here we only show 8 out of 200 fits from the High-High momentum event category. For examples of High-Low and Low-Low momentum event categories, see Fig. .(A.1-A.4) in Appendix A . In each of the graphs in Fig. 6.1, the 11 points are the probabilities for that At bin with 11 different values of lifetime (r). By fitting2 these points using the function / ( r ) in Eq . (6.1), we obtain the corresponding parameter values for the /3(At)'s, which are shown in Fig. 6.2 for all 200 At bins (200 points in the graphs). We can see that 03(At) is very close to zero in a large range ( ± 1 0 p s ) of At . This indicates that higher powers of ( r 0 / r ) are not necessary for / ( r ) in Eq . (6.1). Since we do not need the errors of these parameters in our analysis, we ignore them at this moment. We show a meaningful way to test the consistency of these fits in Section 6.3. We then plug these fitted parameter values into E q . (6.1), so that it is now a function of charged B lifetime only. The results of the fitted parameterization function / ( T ) for opposite charged B categories are shown in Fig. 6.3. In the figure, we provide examples of / ( r ) with three different values of r: 0.9r^ + , T £ + , l . l r ^ + (where r £ + = 1.674ps). This covers the range we chose in varying the charged B lifetime in the parameterization fits. Following the same procedure as described above for charged B categories, 2We use a x2 minimization method for all fits in this thesis. Chapter 6. Analysis Results 59 OCB-HH-bin30 •_l , , , 1 1 , , , !_ AB')h 2 umL I £ OCB-HH-bin70 r°(B>i 0CB-HH-bin11d 0.109 '— oioe — T»(B> 0.0O7B 0.007 OCB-HH-bin150 0.0O9 OCB-HH-bin50 :"(B>i OCB-HH-bin90 _j—,—,—,—,—i—,—,—,—_ T°(B> S S 0.03 OCB-HH-bin130 j -J o.a o.B6 1 x'(B> OCB-HH-bin170 7 OB 0.95 1 1.0S 1.1 t«(B> Figure 6.1: Examples of f(r) fits for the opposite charged B High-High mo-mentum category events (OCB-HH) . Here we only show 8 out of 200 fits for the 200 A r bins. The x axis is the ratio between the default lifetime and the varying lifetime: TB+/T. The y axis is the sum of the same sign and opposite sign (OS+SS) probabilities. Chapter 6. Analysis Results 60 — 1.5 o.s 0 -1.5 OCB-HH y • * . * - \ / • ' ' 1 • ' • • 1 -15 -10 -5 0 5 10 15 At ( PS) 5 10 15 At (ps) OCB-HH Q -0.5 -1 f - * * / • -1.5 -2 -2.5 -3 r ^ * -3.5 -i i i i i i i i * i i > 1 — 1 — 1 — 1 — 1.. -15 -10 -5 0 5 10 At 15 (ps) Figure 6.2: Fitted parameters (/3(At)'s) of / ( r ) for the opposite charged B High-High momentum category events (OCB-HH) . examples of the / ( r ) fits for opposite neutral B High-High momentum category are shown in Fig. 6.4. The corresponding fitted parameter values as functions of At are shown in Fig. 6.5. For examples of fits and the fitted parameter values for High-Low and Low-Low categories, see Figs. (A.5-A.8) in Appendix A . By plugging the fitted parameter values into Eq . (6.1), we obtain the fitted / ( T ) for opposite neutral B categories as shown in Fig. 6.6. In the figure, we plot Chapter 6. Analysis Results 61 0.2 0.18 0.16 D.14 0.12 0.1 0.08 0.06 0.04 0.02 0 f(T>OCB-HH A . lifetime (in t°(B*)) = i^k — 9 0 % z Jf \ 100% E-J \ 110% IIIIIIIIIIIII ' ' --I—1—~-—•—•— 15 at (ps) f(T)-OCB-HL gs lifetime (in z°{B*)) i- fp& 90% §- / \ 100% iliiiliiiliul / \ —110% Milniliii < .—.—.— 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 15 at (ps) f(x)-OCB-LL 0.16 A lifetime (in z°(B*)) 0.16 J ^ . 90% 0.14 f \ —100% 0.12 / \ 110% 0.1 0.08 0.06 0.04 0.02 a 15 at (ps) Figure 6.3: Fitted parameterization function / ( r ) for the opposite charged B category events (OCB) . The three rows are the High-High (HH), High-Low (HL), and Low-Low (LL) momenta categories. Chapter 6. Analysis Results 62 ONB-HH-bin30 T«(B");t ONB-HH-bin50 0.0 OM ONB-HH-bin70 z'(B°)h - 0.124 "8 0-«3 ONB-HH-bin90 :°(B> ONB-HH-bin110 0.0 0.96 i T«(B0)ft ONB-HH-bin150 ONB-HH-bin130 :"(B> ONB-HH-bin170 :"(BV Figure 6.4: Examples of / ( r ) fits for the opposite neutral B High-High momen-tum category events (ONB-HH). Here we only show 8 out of 200 fits for the 200 At bins. The x axis is the ratio between the default lifetime and the varying lifetime: TB0/T. The y axis is the sum of the same sign and opposite sign (OS+SS) probabilities. Chapter 6. Analysis Results 63 -15 -10 -5 0 5 10 15 At (PS) ca. o -2 -14 ONB-HH i / / \ .* \ ; t _ y \ 1? r 4 ; : , > 1 > , > > 1 > , 1 1 1 ' 1 1 1 -II... -15 -10 -5 5 10 15 At (ps) -15 -10 -5 0 5 10 At 15 (ps) ONB-HH m to. ; 0 -0.5 * / : / f > > --1 -1.5 r jr -2 \ -2.5 -3 " , , I , , , , I , , . 11 • •. 11. 1 1 I 1 1 -15 -10 -5 0 5 10 15 At (PS) Figure 6.5: Fitted parameters (/3(At)'s) of / ( r ) for opposite neutral B High-High category events (ONB-HH). fitted / ( r ) for opposite neutral B category events, again, with three different values of r: 0 .9T£ 0 , rB0, 1.1TB0 (where T £ 0 = 1.541 ps). This covers the range we chose in varying the neutral B lifetime in the parameterization fits. Chapter 6. Analysis Results 64 f(T)-ONB-HH at (ps) f(x)-ONB-HL 0.18 0.16 0.14 — A lifetime (in T°(B 0)) 0.12 =- / \ 100% 0.1 iiln / \ 110% 0.08 E-0.06 0.04 0.02 0 , . , i . . . . i . . . . i . ^ - i — , — . — . — _ ofe- , •- 7 , , , 1 , , , , I , , , , I , , -1- ; —, . . . -15 -10 -5 0 S 10 15 a (ps) f(x)-ONB-LL ai (ps) Figure 6.6: Fitted parameterization function / ( r ) for opposite neutral B cate-gory events (ONB). The three rows are the High-High (HH), High-Low (HL), and Low-Low (LL) momenta categories. Chapter 6. Analysis Results 65 6.2 Parameterization fits II-varying B mixing frequency To examine the dependence on neutral B mixing, we use the asymmetry between the opposite sign and the same sign probabilities. The parameterization function g(Am) is chosen in Section 5.4 as P O S ~ = fl(Am) = 70 + 7 i (Am) 2 + 72(Am) 4 + 73(Am) 6, (6.2) ^os + ^ss where the 7's are parameters which can be extracted from bin by bin fits. As explained in Section 5.4, we use the fitted value for the sum of the opposite sign and the same sign probabilities (OS+SS) in the denominator of Eq . (6.2), so that the P O C A A r distributions are simultaneously parameterized in the B lifetime and the mixing frequency. In our case, the asymmetries are obtained from Eq . (6.2) by using the fitted values in Fig. 6.6. Examples of g (Am) fits are shown in Fig. 6.7 for opposite neutral B High-High momentum category events. The points in the figure correspond to the asymmetry between the opposite and the same sign probabilities for that bin with 11 different values of mixing frequency A m . These fits determine the values of the parameters in Eq . (6.2) (7(Ar)'s), which are shown in Fig. 6.8. The corresponding results for High-Low and Low-Low momentum categories are shown in Fig. .(A.9-A.12) in Appendix A . Since the parameters are estimated from these fits, we then plug them into E q . (6.2). As a result, g(Am) depends only on the mixing frequency A m . The fitted parameterization function g(Am) for opposite neutral B categories is shown in Fig. 6.9. Similarly, we provide in the figure results of fitted g (Am) with three different values of A m : 0.9Am^, A m ^ , l . l A m ° (where A m ^ = 0.489ps - 1 ) . This covers the range we chose in varying the neutral B mixing frequency in the parameterization fits. Chapter 6. Analysis Results 66 t -0.14 H -0.16 E ONB-HH-bin30 i • • • i • • • Am (ps1) i 3 -OOB ONB-HH-bin70 r * L_, 1—, , i 1 ,_ o.*4 o.«a Am (ps*) ONB-HH-bin110 L,—i—.—.—.—i u , i . , 0 44 0.46 Am (ps") ONB-HH-bin150 0.4S 0*8 0.5 0.S3 E -0.22 | 0.S3 ONB-HH-bin50 0.44 0*6 0.40 O.S 0.52 0.M Am (ps'1) ONB-HH-bin90 0.44 0.46 Am (ps") ONB-HH-bin130 L * *• „ CU6 0.82 O.M Am (ps") ONB-HH-bin170 • • 0.44 Q.46 D.4B 0.6! UH (Pi") Figure 6.7: Examples of a(Am) fits for opposite neutral B High-High category (ONB-HH). Here we only show 8 out of 200 fits for 200 At bins. The x axis is the varying B mixing frequency A m . The y axis is the asymmetry between the opposite and the same sign probabilities. Chapter 6. Analysis Results 67 ONB-HH ONB-HH 0.6 0.4 : t * / \ f •v * 4 2 - * V 0.2 -0 f r r E f 0 » A i * -0.2 -0.4 — -2 : i : / -* » i \ •'_ -0.6 -4 \ f V -0.8 " , i , , , , i , • . i — i i — *. . i . . . . i . . . 1 . . . . i . i . . i . . . . i , , , I , , -15 -10 -5 0 5 10 15 At ( PS) -15 -10 -5 0 5 10 At 15 (ps) ONB-HH ONB-HH *~ 20 -15 J*' A ** 15 -10 / X 10 -5 5 0 / 0 -5 t • / J 1 7 v*. w \ '* v v -5 -10 -15 \ v' f • s \ V g / -10 •:, , i , . . . i . . . i . . . . i . . . . i . . . . i -20 -25 n . I . . . . I . . . i 1 i i 1 I . I 1 1 I i i i I • i -15 -10 -5 0 5 10 15 At ( PS) -15 -10 -5 0 5 10 At 15 (ps) Figure 6.8: Fitted parameters (7(Ar)'s) of g(Am) for opposite neutral B High-High momentum category events (ONB-HH). 6.3 Consistency test of parameterization In this section, we test the consistency of our parameterization fits presented in Sections 6.1 and 6.2 in getting functions / ( r ) and g(Am). The idea is as follows: we fit re-weighted distributions, from which the pa-rameterization functions (/(r) and g(Am)) are obtained, to extract the associ-Chapter 6. Analysis Results 68 g(Am)-ONB-HH g(Am)-ONB-HL I I < -0 -0.1 -0.2 -0.3 -0.4 g(Am)-ONB-LL mixing frequency (inAm°) 90% 100% 110% 15 at (ps) mixing frequency (in Am£) 90% 100% 110% 15 at (ps) Figure 6.9: Fitted parameterization function g(Am) for opposite neutral B cat-egories events (ONB). The three rows are the High-High (HH), High-Low (HL), and Low-Low (LL) momenta categories. Chapter 6. Analysis Results 69 x10'5 OCB-HH 2 -• • 1 : ' • 1i . ' * 2-i . . . . i . . . . i . . . • 0.8 0.95 1 1.05 1.1 t'(B*)/T x10's OCB-HL 3 -2- . . 1 -°r 1-• • 2- * . i . . . . i . . 0.8 0.95 1 1.09 1.1 x10'5 OCB-LL • > ' • • t'(B> 0.2 -0.1 -0---0.2-OCB-Combined 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 ONB-HH 0.85 1 1.05 1.1 0.051-•o'r -0.05 i -0.1 j--0.15 I--0.2 L ONB-HL xlO'* ONB-LL 0.14 0.12 • 0.1 0.08 0.06 • 0.04 • • 0.02 0 -o n? i . . . . i 0 y 0.85 1.05 1.1 t°(8°)ft X10'S ONB-Combined 0.06 : • 0.06 "• 0.04 J- # 0.02 • • 0 r • • -0.02 • •OM 4.06 -0.08 -0.1 0.8 0.85 x10's ONB-HH 0.2 • 0.15 • 0.1 -0.05 -0 • 0.05 f • • -0.1 4.15 r . ' X10'5 ONB-HL 0.4 • 0.3 0.2 0.1 • 0 • -0.1 • 0.2 0.15 0.1 0.05 -0 -0.05 -0.1 X10'! ONB-LL am/Am" 0.2 0.15 0.1 0.05-•Or -0.05--0.1 --0.15-ONB-Combined Figure 6.10: Consistency test of the parameterization fits. The points are the fractional difference between the fitted values and the "true" values of the lifetime. The first two columns are / ( T ) tests for O C B and O N B categories. The last column is the g(Am) tests for O N B categories. The first three rows are H H , H L , and L L momenta categories. The fourth row is the simultaneous fits for the three momenta categories. Chapter 6. Analysis Results 70 ated B lifetimes and the neutral B mixing frequency. We then compare these fitted values with the expected values as used in the event re-weighting. If our parameterization model and the fits are good enough, there should no sizable discrepancy between the original and fitted values of the B lifetimes and the neutral B mixing frequency. In Fig. 6.10, we show the test of consistency of the parameterization fits for all categories. It is measured by the difference between the fitted values and the expected values of.the B lifetimes and the neutral B mixing frequency. For easy comparison, all these values are rescaled by the default values as specified in Table 5.1. The first two columns are / ( r ) tests for opposite charged B (OCB) and opposite neutral B (ONB) categories. The third column is the g(Am) tests for O N B categories. The first three rows are the High-High, High-Low, and Low-Low momenta categories. The fourth row is the simultaneous fits for all three momenta categories. From the figure, we can see that the fitted values (both lifetimes and mixing frequency) are very close to the expected values, with deviations of less than 5 parts per million. This shows that our functional form and the parameterization fits provide excellent results. That is, the fitted parameterization functions describe the distributions for all re-weighted events with extremely high accuracy. 6.4 Crosschecks of the parameterization approach With the parameterization results presented in Sections 6.1 and 6.2, whose consistency is tested in Section 6.3, we are now ready to use them to extract B lifetimes and mixing frequency from independently generated Monte Carlo test events (still without any detector simulation). This serves as a cross check of our parameterization method. These test events are generated with default lifetimes (TB+ and TB0) and mixing frequency (Amd) as described in Section 5.1. Following the same classification as discussed in Section 5.2, the same sign Chapter 6. Analysis Results 71 and opposite sign events are summed together (OS+SS) to extract B lifetimes. We also calculate the asymmetry between the opposite sign and same sign ((0S-SS)/(OS+SS)) for opposite neutral B events to extract the B mixing frequency. The results of fitting the (OS+SS) P O C A At distributions with f(f) are shown in Fig. 6.11 and Fig. 6.12 for charged and neutral B events, respectively. In the figures, the first three rows are fits for the High-High, High-Low, and Low-Low momenta events. These three momenta events are chained together in the fourth row. That is, the first peak of the fourth row is for High-High momentum events, the middle peak is for High-Low events, and the third peak is for Low-Low momentum events. Therefore, the fourth row is a simultaneous lifetime fit for these three momenta events. The ratios between the default and fitted lifetimes are also printed on the graphs. The expected value of the ratio is 1. Our fitted ratios are very close to this expected value with less than 0.2% deviation. A l l these fitted ratios are consistent with the expected values without biases if statistical errors are included. For the mixing frequency fits, we first calculate the asymmetry between the opposite sign and the same sign events by using the fitted results in Fig. 6.12 for (OS+SS) (for reasons, see Section 5.4). We then fit these asymmetries by using the parameterization function g(Am). The fitted mixing frequencies are shown in Fig. 6.13. In the figures, the first three rows are fits for the High-High, High-Low, and Low-Low momenta events. Similar to lifetime fits, a simultaneous mixing frequency fit for these three momenta events is also shown in the figure. The ratios between A m and A m ^ are very close to the expected value of 1 with less than 1% deviation. Note the exception of the Low-Low category, which is insensitive to the mixing frequency (see the third row in Fig. 6.9) because the mixing signals are highly diluted. Taking into account statistical errors, all of these fitted ratios are consistent with the expected values without biases. A l l of these tests prove that our analyses, especially the event re-weighting and the parameterization function fits, are consistent with less than 1% unbiased deviations from the expected values of the B lifetimes and the neutral B mixing frequency. Chapter 6. Analysis Results 72 T?(B*)fr 0.99810.001229 lifetime fit -f(x) — MC data x°(B+)/x 1± 0.001518 lifetime fit - f (x) - M C data -OCB-combined x°(B*)/x 0.9989 + 0.000682 0.18 0.16 =• A ifetime fit A 0.12 i- / \ - m \ 0.1 0.08 | - / \ — MCdata / \ 0.04 %• / \ 1 V / V 0X2 0 Figure 6.11: Lifetime fits for charged B samples. The fitted lifetimes are quoted as ratios to the default value, rB+ fr. The first three rows are H H , H L , and L L momenta categories. The fourth row is a simultaneous fit of all three momenta categories shown in the first three rows. Chapter 6. Analysis Results 73 T°(B°)/X 0.998710.001493 lifetime fit — f(t) — MC data at (ps) ONB-HL 0.16 0.14 0.12 O06 O04 O02 016 0.14 012 O06 OOt 0.02 0.18 016 0.1-1 012 0.0b 0.04 0.02 x°(B )/T 1± 0.001158 lifetime fit -%%) — M C data ——it at (ps) ONB-LL T°(BV 1-002+0.001714 mill lifetime fit = v — f (x) 5 10 —1 1 1 . l _ -5 0 \ — M C data at (ps) ONB-combined i°(B )/T 1 ± 0.0008072 lifetime fit — m Figure 6.12: Lifetime fits for neutral B samples. The fitted lifetimes are quoted as ratios to the default value, TB<S/T. The first three rows are H H , H L , and L L momenta categories. The fourth row is a simultaneous fit of all three momenta categories shown in the first three rows. Chapter 6. Analysis Results 74 Figure 6.13: Mixing frequency fits for neutral B samples. The fitted B mixing frequencies are quoted as ratios to the default value, A m / A m 0 . The first three rows are the H H , H L , and L L momenta categories. The fourth row is a simultaneous fit of all three momenta categories shown in the first three rows. Chapter 6. Analysis Results 75 6.5 Effects of event re-weighting on the P O C A approximation In this thesis, we use the event re-weighting technique in getting the parameter-ization for At distributions of B mixing. As mentioned before, we use the beam spot rather than the precise e+e~ collision point for each event in the P O C A Az calculation 3. In our Monte Carlo generated events, however, we know both the real Az and the precise collision point for each event. Therefore, we can study effects of event re-weighting on the P O C A approximation, especially the effect on averaging the collision points by using the beam spot in the P O C A Az calculation. The procedure is exactly the same as described in Sections 6.1, 6.2, and 6.4. We repeat the crosschecks in Section 6.4 by using different methods for calculating Az for both the parameterization and the test events. The three methods to calculate Az are: • Using the MC-Truth Az directly; • Using the precise collision point for each event in the P O C A Az calcula-tion; • Using the beam spot for each run as an estimate of the collision point in the P O C A Az calculation. This method is the same one used in the main body of this thesis. In the following, we present results of crosschecks for these three methods together for easier comparison. For details, see Section 6.4 as an example. Results of the fitted charged B lifetimes with different methods of calculating Az are listed in Table 6.1. The results are quoted in the same way as in Section 6.4. The quoted errors are obtained from x2 fits. We can see that results from using real Az and the P O C A Az with precise collision point are very close to each other. They are also very close to the results from the P O C A Az using beam spot. We conclude that our event re-weighting 3The boost approximation is always used to convert Az into At. Chapter 6. Analysis Results 76 Category \ Method Real A z Collision point Beam spot High-High 1.0000 ± 0 . 0 0 0 8 1.0000 ± 0.0009 0.9980 ± 0.0012 High-Low 0.9997 ± 0.0007 0.9993 ± 0.0008 0.9988 ± 0.0010 Low-Low 1.0030 ± 0 . 0 0 1 1 1.0010 ± 0.0013 1.0000 ± 0.0015 Combined 1.0000 ± 0.0005 1.0000 ± 0.0005 0.9989 ± 0.0007 Table 6.1: Charged B lifetime fits with three different methods of calculating Az: real A z ; P O C A Az with precise collision point; and P O C A Az using beam spot for each run. The quoted results are the ratio between the default lifetime and the fitted lifetime: TB+/T. Category \ Method Real Az Collision point Beam spot High-High 1.0020 ± 0 . 0 0 1 0 1.0010 ± 0 . 0 0 1 1 0.9987 ± 0 . 0 0 1 5 High-Low 1.0010 ± 0 . 0 0 0 8 1.0020 ± 0.0009 1.0000 ± 0.0012 Low-Low 0.9984 ± 0.0012 0.9992 ± 0.0014 1.0020 ± 0.0017 Combined 1.0010 ± 0 . 0 0 0 5 1.0010 ± 0.0006 1.0000 ± 0.0008 Table 6.2: Neutral B lifetime fits with three different methods of calculating Az: real Az; P O C A Az with precise collision point; and P O C A Az using beam spot for each run. The quoted results are the ratio between the default lifetime and the fitted lifetime: rB0/r. does not change the smearing effect of P O C A approximation (including using the beam spot) in a significant way for charged B events. In Table 6.2, we list the results for the fitted neutral B lifetime with different methods of calculating A z . Similar to the charged B lifetime fitting, all three methods give almost identical results and all are unbiased with deviation less than 1%. Results for the fitted neutral B mixing frequency with different methods of calculating the A z are listed in Table 6.3. We see that results from using real A z and the the P O C A A z with precise collision point are still very close to each other. They give better results 4 than the method of using beam spot for each 4 As we discussed before, the Low-Low momentum events are insensitive to the B mixing frequency. Therefore, we compare only results of the simultaneous fits in the last row in Table 6.3. Chapter 6. Analysis Results 77 . Category \ Method Real Az Collision point Beam spot High-High 0.9994 ± 0.0017 0.9984 ± 0.0018 0.9921 ± 0.0022 High-Low 1.0070 ± 0.0056 1.0010 ± 0.0052 1.0080 ± 0.0065 Low-Low 1.0700 ± 0.0235 1.0860 ± 0.0226 1.0420 ± 0.0272 Combined 1.0000 ± 0.0017 0.9992 ± 0.0017 0.9941 ± 0.0021 Table 6.3: Neutral B mixing frequency fits with three different methods of calculating Az: real Az; P O C A Az with precise collision point; and P O C A A z using beam spot for each run. The quoted results are the ratio between the fitted mixing frequency and the default value: Am/Amd. run as an estimation of the collision point. In summary, the effect of the event re-weighting on the P O C A approximation (comparing results from using the real Az and the P O C A Az with collision point in the tables) is insignificant compared to the statistical uncertainties in the fitting procedures. The effect of event re-weighting on using the beam spot for each run in the P O C A Az calculation (comparing results from using the P O C A Az using collision point with the P O C A Az using the beam spot) is also very small except in the neutral B mixing frequency fitting. This might be explained by the fact that the event re-weighting does not change any smearing effect of the P O C A approximation (as explained in Section 4.3) in a sizable way. Since the lifetime fitting is based on the sum of the same sign and the opposite sign events, it should be insensible to any small smearing shift. For the mixing frequency fitting, however, we use the asymmetry between the same sign and the opposite sign, which might be more sensitive to even a small smearing shift due to the event re-weighting. Chapter 7 Conclusion 78 The time evolution of B mesons is governed by the overall decay rate TB = 1/TB and by the mass difference Amd of the two neutral mass eigenstates. A precision measurement of the B°B° mixing frequency is of fundamental importance since it is sensitive to the C K M matrix element | V i 0 | which has not previously been measured with great precision and sensitive to new physics beyond the S M . At the same time, a precise determination of the B lifetimes reduces the systematic error on measurements of the the elements of the C K M matrix, such as \VCb\ and \VUb\- Therefore, simultaneous and precise measurements of the neutral B mixing frequency and B lifetimes, provide stringent constraints on the C K M matrix. In this thesis, we studied the parameterization of the neutral B mixing prob-ability distribution functions of the proper decay time difference A r of the B pairs resulting from T(4S) decays. Dilepton events from T(4S) decays were generated by the EvtGen generator package within the BABAR framework. Following the same method applied in the real experimental data analysis, the boost approximation and the point of closest approach (POCA) were used to estimate the proper decay time difference A r in our Monte Carlo analysis. By choosing appropriate functions, the parameterization was obtained from bin by bin fits of At distributions with different B lifetimes and neutral B mixing frequencies. These distributions were obtained from re-weighting the Monte Carlo generated events, which have a fixed lifetime and mixing frequency. For each of the distributions, events are divided into 200 At bins within ± 15 ps. For each of the At bins, we perform parameterization fits by varying lifetimes and mixing frequency. As the charged B events are independent of Chapter 7. Conclusion 79 neutral B mixing, we obtained the parameterization results as a function of only the charged B lifetime. For the neutral B events, the parameterization depends on both the neutral B lifetime and the neutral B mixing frequency. Therefore, we obtained parameterization results for neutral B events as functions of the neutral B mixing frequency and the neutral B lifetime. The parameterization approach was tested with another independently gen-erated set of test dilepton events. The deviations in extracting the B lifetimes and the neutral B mixing frequency are less than 1% from the expected val-ues, without biases. We also examined the effects of event re-weighting on the P O C A approximation, especially the effect on using the beam spot for each run as an estimate of precise e +e~ collision point. A l l of these effects turn out to be insignificant. Our results presented in this thesis, combined with Monte Carlo studies of detector resolutions, provide a tested technique to perform precise measurements of the neutral B mixing frequency and B lifetimes from experimental data. 80 Bibliography [1] S. L . Glashow, Nucl. Phys. 22, 579 (1961); S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); A . Salam, "in Proceedings of the 8th Nobel symposium", ed. N . Swartholm, et al (1968). [2] K . Anikeev, et al, Fermi Lab-Pub-01/197, hep-ph/0201071, (2002). [3] N . Cabibbo, Phys. Rev. Lett. 10, 531 (1963); M . Kobayashi and T . Maskawa, Prog. T h . Phys. 49, 652 (1973). [4] Particle Data Group, S. Eidelman et al, Phys. Lett. B 592 1 (2004). [5] L . Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983). [6] A . J . Buras, M . E . Lautenbacher and G . Ostermaier, Phys. Rev. D 50, 3433 (1994). [7] R. Aleksan, B. Kayser and D . London, Phys. Rev. Lett. 73, 18 (1994); J.P. Silva and L . Wolfenstein, Phys. Rev. D 55, 5331 (1997); I.I. Bigi and A.I. 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Meth A 462, 152 (2001); A . Ryd, et al, BABAR Analysis Document 522, (2003); U R L : http://www.lns.cornell.edu/~ryd/ Bibliography 82 [28] T . Sjostrand, Comp. Phys. Commun. 82, 74 (1994). [29] http://www.slac.stanford.edu/BFROOT/www/Physics/Tools/generators [30] E . Richter-Was, Phys. Lett. B 303, 163 (1993). [31] UA1 Collaboration, C . Albajar et al, Phys. Lett. B 186, 247 (1987). [32] A R G U S Collaboration, H . Albrecht et al, Phys. Lett. B 192, 245 (1987); Z. Phys. C 55, 357 (1992); Phys. Lett. B 324, 249 (1994). [33] C L E O Collaboration, B . H . Behrens et al, Phys. Rev. Lett. 71,1680 (1993); Phys. Lett. B 490, 36 (2000). [34] H F A G , hep-ex/0505100; http://www.slac.stanford.edu/xorg/hfag/osc [35] A L E P H Collaboration, D. Buskulic et al, Z. Phys. C 75, 397 (1997). [36] D E L P H I Collaboration, J . Abdallah et al, Eur. Phys. Jour. C 28, 155 (233). [37] L3 Collaboration, M . Acciarri et al, Eur. Phys. Jour. C 5, 195 (1998). [38] O P A L Collaboration, G . Abbiendi et al, Phys. Lett. B 493, 266 (2000). [39] C D F Collaboration, T . Affolder et al, Phys. Rev. D 60, 112004 (1999). [40] BABAR Collaboration, B. Aubert et al, Phys. Rev. Lett. 88, 221802 (2002); Phys. Rev. D 66, 032003 (2002); Phys. Rev. Lett. 88 221803 (2002); [41] BABAR Collaboration, B. Aubert et al, Phys. Rev. D 67, 072002 (2003). [42] BABAR Collaboration, B . Aubert et al, hep-ex/0408039. [43] BABAR Collaboration, B. Aubert et al, Phys. Rev. Lett. 92, 181801 (2004); Phys. Rev. D 70, 012007 (2004). [44] Belle Collaboration, K . Abe et al, Phys. Rev. Lett. 86, 3228 (2001); K . Hara et al, Phys. Rev. Lett. 89, 251803 (2002); T . Tomura et al, Phys. Lett. B 542, 207 (2002); N . C . Hastings et al, Phys. Rev. D 67, 052004 (2002); Y . Zheng et al, Phys. Rev. D 67, 092004 (2003). Bibliography 83 [45] Belle Collaboration, K . Abe et al, Phys. Rev. D 71, 072003 (2005). [46] BABAR Collaboration, C . Bozzi et al, B A D Note 28 (2000). [47] http://wwwasdoc.web.cern.ch/wwwasdoc/minuit/ [48] http://root.cern.ch/root/html/TFumili.html 84 Glossary C K M Cabibbo-Kobayashi-Maskawa C M Center-of-mass D C H Drift Chamber DIRC Detector of Internally Reflected Cherenkov light E M C Electromagnetic Calorimeter F C N C Flavour-changing Neutral Current G I M Glashow-Hiopoulus-Maiani H H High-High H L High-Low H Q E T Heavy Quark Effective Theory IFR Instrumented Flux Return L L Low-Low M C Monte Carlo O C B Opposite Charged B O N B Opposite Neutral B O P E Operator Product Expansion OS Opposite Sign P D F Probability Distribution Function P D G Particle Data Group P M T Photo-multiplier Tubes P O C A Point of Closest Approach Q C D Quantum Chromodynamics Q E D Quantum Electrodynamics R P C Resistive Plate Chamber SB Same B S M Standard Model SS Same Sign S V T Silicon Vertex Tracker Tel Tool Command Language Appendix A More Figures 85 OCB-HL-bin30 t°(B*)ft OCB-HL-bin70 OCB-HL-bin110 -•(B> £, O.ODB S 0.0075 i tan OCB-HL-bin150 0.011 0.0105 D0OK ti.aoar. OCB-HL-bin50 OCB-HL-bin90 OB 0M 1.09 1.1 T°(B> OCB-HL-bin130-\ OCB-HL-bin170 T°(B*)/T T°(B-)rt Figure A . l : Examples of / ( r ) fits for the opposite charged i? High-Low momen-tum category events ( O C B - H L ) . Here we only show 8 out of 200 fits for the 200 A t bins. The x axis is the ratio between the default lifetime and the varying lifetime: TB+/T. The y axis is the sum of the same sign and opposite sign (OS+SS) probabilities. Appendix A. More Figures 86 I0CB-HLI 1.5-1-0 5 : o --0.5 -i --1.5 • • 1 1 1 i 1 1 1 1 i -15 -10 5 10 15 At ( PS) -10 -20; -2 -2.5 5 10 15 At ( PS) OCB-HL - / / r \ * < \ \ < V - , I , , , . I , , . I . . . . I . K ill 1 .,-II. -15 -10 -5 0 5 10 15 At ( PS) OCB-HL : l d \ i ,\ 1 •K • \ V t. ...i..- ......... in r . \ i • I • i i * I * i i > I i i -15 -10 -5 0 5 10 15 At ( PS) Figure A.2: Fitted parameters (/?(A£)'s) of / ( r ) for the opposite charged B High-Low momentum category events ( O C B - H L ) . Appendix A. More Figures 87 = 0.3034 O 0.0032 0 003 0.0026 0.0024 D.0022 OCB-LL-bin30 i i ' ' 1 'Bis t°(B*)/x T°(B*)ft OCB-LL-birt110 0.114 0.113 am am an o in D.rOB 0.107 0.106 T°(8*)/X OCB-LL-bin150 fev s 0.009 o.oou:, — aooa L 0.007S : 0.007 1 0.0061, T°(B*)/T A 0.01 0.00*5 0.009 OCB-LL-bin50 oi ' ' ' ' lis l'(B )/T OCB-LL-bin90 i i ' ' ' 'tis' ' ' ' I ' ' ' 't.k' T°(B )/T n 0.032 3 £ 0.031 OCB-LL-bin130 ».» ' ' ' ' lis' i • • • 'iii • T°(B*)ft OCB-LL-bin170 0.9 • ' ' 'iii' Figure A.3: Examples of / ( r ) fits for the opposite charged B Low-Low category momentum events ( O C B - L L ) . Here we only show 8 out of 200 fits for the 200 At bins. The x axis is the ratio between the default lifetime and the varying lifetime: Tb+ Jr. The y axis is the sum of the same sign and opposite sign (OS+SS) probabilities. Appendix A. More Figures 88 OCB-LL \ • 1 1 ' ' • • 1 ' ' 1 1 1 1 1 , 1 1 1 1 , i . , 1 1 1 , 1 1 , 1 1 1 -15 -10 10 At 15 (ps) -5 -15 OCB-LL : \ f \ \ \ u \ V _L I I 1 1 I 1 I 1 1 — i , , , , i — i , , -15 -10 -5 10 15 At ( PS) OCB-LL -15 -10 -5 5 10 15 At (ps) °E -0.5; -1 I -1.5] •2--2.5': -3 ; -3.5; -4C. -15 -10 • ' ' ' ' ' • -5 0 5 10 15 At (ps) Figure A.4: Fitted parameters (/3(A£)'s) of / ( r ) for the opposite charged B Low-Low momentum category events ( O C B - L L ) . Appendix A. More Figures 89 S 0.008 s •§ OJXW 0.00*6 ONB-HL-bin30 ONB-HL-bin70 •_i—,—,—,—,—i—,—,—,—,_ i"(B> £ 0.0*7 ONB-HL-bin110 ONB-HL-bin150 2 o.oi x ONB-HL-bin50 t"(B'yt ONB-HL-bin90 Hi—_—,—,—i—,—,—,—,—1_ ONB-HL-bin130 ONB-HL-bin170 1*. ;'(B')ft .•(BV Figure A.5: Examples of / ( r ) fits for the opposite neutral B High-Low momen-tum category events (ONB-HL) . Here we only show 8 out of 200 fits for the 200 At bins. The x axis is the ratio between the default lifetime and the varying lifetime: TB0/T. The y axis is the sum of the same sign and opposite sign (OS+SS) probabilities. Appendix A. More Figures 90 ONB-HL a ca -0 -0.2 -0.4 -0.6 -0.8 : . V - ' ~- •: *'* -_ *. c \ .. -1 -1.2 % \ i -1.4 -1.6 • -1.8 -, , I . . , , I , i l l . . • l>< -15 -10 -5 0 5 10 15 At (ps) 2h ONB-HL 1 • \ \ - % X % \ \ \ \ . , 1 . . . . 1 , , • , 1 , ?TTT i r , i , , , , i , , , , i , , co. 0 -2 +• •6| -8 -10; -12 -14 ONB-HL I / i i i \ \ \ i \ * • - \ i > : . / ft \ ., .** •* -111111 11 , i — — i — i — i . , -0.5 -1.5 -15 -10 -5 0 5 10 15 At ( P S) ONB-HL 1 / i V '•l s % L y r • '• • i . . . . i . . • 1 i i , i 1 , , , 11,, , i , , , , i , . -15 -10 5 10 15 At (ps) -15 -10 5 10 15 At ( P S) Figure A.6: Fitted parameters (/3(A£)'s) of /fir) for the opposite neutral B High-Low momentum category events (ONB-HL) . Appendix A. More Figures 91 ONB-LL-bin30 >. 0.038 % % 0.037 ONB-LL-bin70 = 0.108| z 2 0.1071 a 0.1061 MOB 0.104 0.103 0.102 ONB-LL-bin110 ill ' ' ' ' nit11 1 1 I 'i.»5 ' ' ' ' )'i ' ONB-LL-bin50 0.S &S6 :°(BV f 0.122 5 0.12 ONB-LL-bin90 si ' ' ' 'tis ' ONB-LL-bin130 T ~ i • • . . . . . . l_L t"(B> ONB-LL-bin150 ONB-LL-bin170 t»(BV i0(B"yT Figure A.7: Examples of / ( r ) fits for the opposite neutral B Low-Low momen-tum category events (ONB-LL) . Here we only show 8 out of 200 fits for the 200 A t bins. The x axis is the ratio between the default lifetime and the varying lifetime: TB0/T. The y axis is the sum of the same sign and opposite sign (OS+SS) probabilities. Appendix: A. More Figures 92 ONB-LL -0.2 -0.4 -0.6 - i -0.8 -1 -1.2 : ' c y I. ' :_ ' V J ? -1.4 -1.6 -1.8 - , i , , , , I , , , , I , . i , , . . i . . . . i , . , , i , , -15 -10 -5 0 5 10 15 At (ps) 5 10 15 At (ps) 5 10 15 At (ps) 5 10 15 At (ps) Figure A.8: Fitted parameters (/?(At)'s) of / ( r ) for the opposite neutral B Low-Low momentum category events (ONB-LL) . Appendix A. More Figures 93 ONB-HL-bin30 £ °" | .... £> -0.0S I I 0.14 0.46 U.4U Am (ps-1) ONB-HL-bin70 r _ — * Am (ps"') ONB-HL-bin110 Am (ps1) ONB-HL-bin150 L-.—i—,—i—,—i—,—, , | , 074  0M g -0.18 s ONB-HL-bin50 OA 0.S2 0.S4 Am (ps1) ONB-HL-bin90 -ONB-HL-bin130 Am (ps ) ONB-HL-bin170 L» Am (ps") Figure A.9: Examples of g(Am) fits for the opposite neutral B High-Low mo-mentum category events (ONB-HL) . Here we only show 8 out of 200 fits for the 200 At bins. The x axis is the varying B mixing frequency A m . The y axis is the asymmetry between the opposite and the same sign probabilities. Appendix A. More Figures 94 -0.2 -0.4 ONB-HL J / » / >-•. . i . , . . i . . . i . , , . i , , . , i . . . , i , . , , i . , -15 -10 -5 0 5 10 15 At (ps) ONB-HL 3 2.5 2 1.5 1 0.5 0 -0.5 -1 ONB-HL v\'.V 1.5h • I i . • r I • • i i I i i • i I i i i i I . i i i I • . 10 -5 k -10 -15 -10 -5 0 5 10 15 At (ps) ONB-HL V r I , , . . I , . I 1 I , 1 1 . I 1 , 1 1 I , , -15 -10 -5 0 5 10 15 At (ps) Figure A.10: Fitted parameters (7(Ar)'s) of g(Am) for the opposite neutral B High-Low momentum category events (ONB-HL) . Appendix A. More Figures 95 ONB-LL-bin30 | -0.03 f. -O.04 -0.05 -OJW -0.07 4JM •0.08 ONB-LL-bin50 'lia' -0.076 1 -OJM •0.088 •0.09 -0.005 ONB-LL-bin70 im (ps"1) s 0-066 1 0.0B4 ym < 0.D5 0JM6 0.046 ONB-LL-bin110 Am (ps'1) >. 4.006 | -0.01 f. 4.012 < 4.014 •0-011 •0.018 403 •0.022 -0JJ24 -0.026 •0.028 ONB-LL-bin90 ""Trie"* 0.^ 4 Am (ps'1; Am (ps ) Am (ps'1) Figure A . l l : Examples of g(Am) fits for the opposite neutral B Low-Low mo-mentum category events (ONB-LL) . Here we only show 8 out of 200 fits for the 200 At bins. The x axis is the varying B mixing frequency A m . The y axis is the asymmetry between the opposite and the same sign probabilities. Appendix A. More Figures 96 0.3 T 0.2 R 0.1 j-" r -0.1 j--0.2 |--0.3 j--0.4^ -0.5 L , ONB-LL ONB-LL I •.•V ' S ^ ' -15 -10 -5 0 5 10 15 At ( PS) 2 -1 -o ' 7 -0.5 h -1 --1 .5 -5 10 15 At (ps) 5 10 15 At (ps) 5 10 15 At (ps) Figure A.12: Fitted parameters (7(At)'s) of g(Am) for the opposite neutral B Low-Low momentum category events (ONB-LL) . 

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