DSpace at VNU: Prospects for the Measurement of the CP Asymmetry inBMeson Decay

DSpace at VNU: Prospects for the Measurement of the CP Asymmetry inBMeson Decay tài liệu, giáo án, bài giảng , luận văn,...

Prospects for the Measurement

of the CP Asymmetry in B Meson Decays

Roland Waldi

Institut fuÈr Kern- und Teilchenphysik

Technische UniversitaÈt Dresden

Abstract

The expected effects from CP violation in neutral B decays in the framework of the Standard Model are reviewed Time dependent rates and asymmetries are presented with emphasis on their observability

at recently proposed B factories Detectors and methods to extract CP asymmetry parameters are pre-sented, including techniques for flavour tagging and data fits The expected performance of an e‡eÿ

and a hadron beam experiment is illustrated with the most promising final states J=wK0

s and p‡pÿ

Contents

1 Introduction 709

2 Particle Anti-Particle Oscillations and CP Violation 710

2.1 The Unitary CKM Matrix 710

2.1.1 Unitarity Triangles 712

2.1.2 Phases and Observables 715

2.2 Oscillation Phenomenology 717

2.2.1 Standard Model Predictions 721

2.2.2 Behaviour of the Four Neutral Meson Anti-Meson Systems 724

2.2.3 CP Eigenstates Versus Mass Eigenstates 730

2.2.4 Oscillation at the U(4S) 732

2.2.5 Determination of the Mixing Parameters of B Mesons 735

2.2.6 Predictions for xs; ysand ds 737

2.3 CP Violation 738

2.3.1 CP Violation in B Decays 739

2.3.2 CP Violation in Common Final States of B0and B0 741

2.3.2.1 The Bs= BsCase 744

2.3.2.2 CP Violation at the U(4S) 745

2.3.2.3 Time Integrated Asymmetries 750

2.3.2.4 Final CP Eigenstates from B0or BsDecays 750

2.3.2.5 The B ! pp Decay 754

2.3.2.6 Mixtures of CP Eigenstates 758

2.3.2.7 Non-Eigenstates 759

2.3.2.8 The Total Decay Rate 759

2.3.3 CP Violation in K Decays 760

3 Measurement of CP Violation at B Meson Factories 764

3.1 B Meson Factories 765

3.1.1 B Production Cross Sections 767

3.1.2 B Meson Fractions 769

3.1.3 B Meson Yields 770

3.2 Two Typical Detectors 770

3.2.1 The LHB Detector at a Fixed Target Hadronic B Factory 771 Fortschr Phys 47 (1999) 7 ± 8, 707±±853

3.2.1.1 Vertex Region 772

3.2.1.2 Tracking and Particle Identification 773

3.2.1.3 Trigger 774

3.2.1.4 Other Experiments 774

3.2.2 The BABAR Detector at the PEP II e‡eÿStorage Ring 774

3.2.2.1 Vertex Detector 777

3.2.2.2 Tracking and Particle Identification 778

3.2.2.3 Electromagnetic Calorimeter 780

3.2.2.4 Muon and Neutral Hadron Detector 780

3.2.2.5 Trigger 781

3.2.2.6 Other Expriments 781

3.2.3 U (4S) Factories and Hadron Colliders: Two Complementary Concepts 781

4 Analysis Techniques and Tools to Estimate Experimental Performance 782

4.1 Flavour Tagging 783

4.1.1 Observed Versus True Asymmetry 784

4.1.2 Statistical Tagging 787

4.1.3 Specific Tags 791

4.1.3.1 Lepton Tags 791

4.1.3.2 Lepton Tags at LHB 795

4.1.3.3 Kaon Tags 798

4.1.3.4 Kaon Tags at LHB 800

4.1.3.5 Charm Tags 802

4.1.3.6 Other Tags 804

4.1.4 Combined Tagging Results at the U(4S) 806

4.1.5 Special Tags at Hadron Machines 807

4.1.5.1 Vertex Reconstruction and Charged B Tags 808

4.1.5.2 Tag Jet Charge 808

4.1.5.3 The B**‡! B0p‡Cascade 809

4.1.5.4 Same Jet Charge 810

4.1.6 Combined Tagging for Hadron Machines 810

4.1.7 Determining I and D from Data 810

4.2 Fitting CP Asymmetries 813

4.2.1 Fit to the Time Dependent Asymmetry 814

4.2.1.1 Fit of L 817

4.2.1.2 Fit of L and I 818

4.2.1.3 Fit of Q 820

4.2.1.4 Fit of Q and I 821

4.2.1.5 Fit of L and Q 822

4.2.2 Using Time Integrated Numbers 823

4.2.2.1 Background 823

4.2.3 Improved Fit Procedure for Small Data Samples 824

4.2.3.1 Background 827

5 Performance for the Key Final States 827

5.1 B0! p‡pÿ 828

5.2 B0! p‡pÿ Reconstruction at LHB 829

5.2.1 Mass Resolution 832

5.2.2 Trigger Efficiency and Reconstruction Losses 833

5.2.3 Backgrounds 834

5.2.3.1 Background from BsDecays 836

5.2.3.2 Combinatoric Background 837

5.2.4 Acceptance 839

5.3 B0! p‡pÿ Reconstruction at BABAR 840

5.3.1 Backgrounds 840

5.3.2 The Lifetime Measurement 841

5.4 B0! J=w K0 s 842

5.5 B0! J=w K0 s Reconstruction at LHB 842

5.6 B0! J=wK0

s Reconstruction at BABAR 845

5.7 Present Experimental Information 845

5.8 Comparison of Future Experiments 846

5.8.1 B0! J=wK0 s and the Determination of b 846

5.8.2 B0! p‡pÿand the Determination of a 847

6 Outlook 848

Acknowledgements 848

References 849

1 Introduction

Our understanding of physics in general and particle physics in particular has been mainly put forward by the discovery of symmetries It is remarkable, that most of the symmetries discovered have, however, finally turned out to be only ``almost-symmetriesº, i.e to be more or less broken

The only unbroken symmetries so far discovered are the U(1) charge-phase symmetry and the SU(3) colour symmetry The consequences are, that the electric and colour charges are exactly conserved in all observed reactions, and that the position in SU(3)-space cannot

be determined, e.g a ``redº and a ``blueº quark cannot be distinguished

Each of the symmetries between leptons and quarks of different flavour is broken by the different masses and electro-weak charges of these particles, and is best approximated in strong interactions as isospin symmetry between the u and d quark due to their almost identical constituent mass

Although physics laws are strictly symmetric under translation or rotation, space-time translational and rotational symmetry is broken through the solutions: The fact that matter

is not distributed homogeneously throughout the universe introduces a locally asymmetric structure of space-time, or asymmetric boundary conditions to any microscopic system The spatial symmetries are best approximated on a macroscopic scale ±± the universe ±± or for microscopic systems isolated from other matter by large distances

Mirror symmetry (parity P) is broken in a more fundamental sense by weak interaction, which makes a maximal distinction between fermions of left and right chirality First ideas

of this unexpected behaviour emerged as a solution of the ``Q t puzzleº, the fact that the neutral kaon decays both to P ˆ ‡1 and P ˆ ÿ1 eigenstates [1], and a direct observation

as left-right-asymmetry in weak beta decays followed soon [2] It is most pronounced in the massless neutrinos, which are produced in weak interactions only with lefthanded heli-city, or righthanded in the case of anti-neutrinos, thus violating the charge-conjugation sym-metry (C) at the same time

The product of both discrete symmetries, CP, is almost intact, and seemed to be con-served even in weak interaction processes A small violation has first been obcon-served in

1964 [3] in K0 decays, which are up to now the only system which does not respect CP symmetry completely The explanation of this violation in the Standard Model will be briefly discussed in the next chapter This is not the only possible description, but the one with no additional assumptions At the same time, the Standard Model predicts CP violat-ing effects in the decay of beauty mesons (B0, Bs, B‡), which should be even large in some rare decay channels

This paper will describe these effects, and discuss techniques and the prospects of their measurement within the next few years After a discussion of meson flavour oscillations and CP violation in the Standard Model, the concept of typical experiments at B meson factories are presented In section 4, analysis techniques and in particular methods for flavour tagging are introduced, and information factors for different fit situations are

cussed Section 5 presents experimental performance for two examples and in the last tion an outlook to the next few years is based on extrapolating these studies to the mostpromising proposed experiments.

sec-2 Particle Anti-Particle Oscillations and CP Violation

Mesons are neither particles nor anti-particles in a strict sense, since they are composed of

a quark and an anti-quark This implies the existence of mesons with vacuum quantumnumbers (e.g f0) More important is the existence of pairs of charge-conjugate mesons,which can be transformed into each other via flavour changing weak interaction transitions.These are K0= K0 (sd=s d), D0= D0 (cu=cu), B0= B0 (bd=b d), and Bs= Bs (bs=bs)

2.1 The Unitary CKM Matrix

The charged current weak interactions responsible for flavour changes are described by thecouplings of the W boson to the current

A gm 1 ÿ g2 5

emt

0B

1C

1C

A gm 1 ÿ g2 5 V

dsb

0B

of mass generation, which belongs still to the more ``mysteriousº parts [7] of the StandardModel The exploration of the Higgs sector is the main motivation for the LHC storagering, which is built at CERN and will start operation around 2005 [8] The Higgs-quarkcouplings alone involve 10 independent parameters of the Standard Model, the quarkmasses and the parameters of the CKM matrix, which are not related within the theory.Local gauge invariance and baryon number conservation requires the CKM matrix to beunitary If there were more than three quark families, this would not hold for the 3  3submatrix, but this possibility is unlikely, given the limit on neutrino flavours from LEPexperiments, who find nnˆ 2:991  0:016 [9] for neutrinos with mass much below the Z0

mass Thus, if a fourth generation exists, it must incorporate a massive neutrino which ismore than a factor 1000 heavier than the tau neutrino, even if we assume the experimentalupper limit for the latter

From the 9 real parameters of a general unitary matrix, 5 can be absorbed in 1 globalphase, 2 relative phases between u; c; t and 2 relative phases between d; s; b which are allsubject to convention and in principle unobservable If two quarks within one of these twogroups were degenerate in mass, even the sixth phase could be removed by redefining thebasis in their two-dimensional subspace

Rephasing may be accomplished by applying a phase factor to every row and column:

Note that j ˆ u; c; t, k ˆ d; s; b, and the six numbers fu; fc; ft; fd; fs; fb represent onlyfive independent phases in the CKM matrix, since different sets of ffj; fkg yield the sameresult Any product where each row and column enters once as Vijand once via a complexconjugate Vkl* like VijVklVil*Vkj* is invariant under the transformation (2.2) This implies thatobservable phases must always correspond to similar products of CKM matrix elementswith equal numbers of V and V* factors and appropriate combination of indices

Removing unphysical phases, the CKM matrix is described by 4 real parameters, whereonly one is a phase parameter, while the other three are rotation angles in flavour space.The standard parametrization [9] (first proposed in [10], notation follows [11]) uses achoice of phases, that leave Vud and Vcbreal:

1C

with cijˆ cos qij, sijˆ sin qij, and sij> 0, cij> 0 (0  qij p=2)

A convenient substitution1) is s12ˆ l, s23ˆ Al2, s13sin d13ˆ Al3h, and

s13cos d13 ˆ Al3r, which reflects the apparent hierarchy in the size of mixing angles viaorders of a parameter l This leads to

‡ o…l6†

…2:4†and agrees to o…l3† with the Wolfenstein approximation [12]:

Assuming a unitary 3  3 matrix, from experimental information these parameters are [9]

Dividing (2.6k) by Al3 ÿVcdV*cb yields the unitarity triangle2† as shown in figure 2.1a.

In the Wolfenstein approximation, it corresponds to

A second one from (2.6h) is shown in figure 2.1b Dividing by Al3 ÿV*usVts and usingthe approximation Vud  1 gives the same triangle (2.7) A closer look, however, revealsslightly different lengths and angles to o…l2†

The angles of the unitarity triangles (2.6k and h) in figure are defined by

sin g ˆ Im eigˆ ÿIm …V*jV ubV*cdVcbVud†

ubVcdVcbVudj ˆ ÿ

J

jVubVcdVcbVudjand vanish for J ˆ 0, i.e if all triangles collapse into lines If the non-trivial phase is 0 or

p, the parameter h is 0 and hence J ˆ 0 This would also be the case if two quarks of agiven charge had the same mass, since then a rotation between these two flavours could bechosen that removes the phase factors, as can be seen in (2.3) where q13ˆ 0 would removeall terms with the phase d13

The angles of all six triangles (2.6g±l) can be determined using the standard tion (2.3) in a rewritten form

with ~g  d13 Here, absolute values and phases are given as separate factors The angles

f2 hl2, f4 hA2l4, and f6 hA2l6 are all positive and very small and their subscript

2† This geometric interpretation has been pointed out by Bjorken  1986; its first documentation inprinted form is in ref 15 and more general in ref 16

indicates the order in l of their magnitude The unitarity triangles in figure 2.1 have angles

b ˆ ~b ‡ f4; b0ˆ ~b ‡ f2; g ˆ ~g ÿ f4;

g0ˆ ~g ÿ f2; a ˆ p ÿ ~b ÿ ~g :

In the Wolfenstein approximation, the unitarity relations read (all terms given to order l3

or, if this is still 0, [in brackets] to leading order)

and define three pairs of unitarity triangles, 6 in total:

· (2.6h0) and (2.6k0) are the ones shown in figure 2.1 with three sides of similar length, all

of order Al3 This is ``the unitarity triangleº The other ones are quite flat, and it willrequire very high precision to prove experimentally that they are not degenerate to a line

· (2.6i0) and (2.6l0) have two sides of length Al2 and one much shorter of order Al4 Thislimits the small angles, which are f2‡ f6 and f2ÿ f6, respectively They are close tothe differences of angles in the large triangles g ÿ g0ˆ b0ÿ b ˆ f2ÿ f4

· (2.6g0) and (2.6j0) have two sides of length l and one very much shorter of order A2l5,with a small angle f4ÿ f6 and f4‡ f6, respectively Both are of order l4

Tiny differences between the two standard unitarity triangles are o…l2† corrections,

ÿ12 Al5…r ‡ ih† ‡ o…l7† ‡ o…l7† ‡ 12 Al5…r ‡ ih† ‡ o…l7† :The angles in these two triangles can be estimated from experimental constraints on a 3  3unitary CKM matrix, leading to 95% CL limits [18]

25  a  125; 11  b  35; 40 g  145:

All phase angles are only weakly constrained by these limits, and one of the aims ofexperiments designed to observe CP violation in B meson decays is a first measurement,and ultimately a precise determination of their values However, deviations from or exten-sions to the Standard Model may imply that the two triangles are dissimilar, or even thatthey are no (closed) triangles at all Therefore, it is important to distinguish measurements

of different parameters, even if they are expected to have identical or close values withinthe three family Standard Model

2.1.2 Phases and Observables

The fact that phases of quark fields are unobservable numbers has been used to show thatsome phases in the CKM matrix are not observables either, and there remains some arbi-trariness in the parametrization for this matrix The freedom to choose quark phases may be

extended to antiquarks, with six more phases fu; fc; ft; fd; fs; fb With the new quarkstates

q0

jˆ eif jq; q0

jˆ ei  f jqj; j ˆ u; c; t; d; s; balso the phase induced by the CP operation is changed The transition

CP jqji ˆ eifCP jjqji ! CP jq0

ji ˆ eif 0

CP jjq0

jirequires

f0

CP jˆ fCP j‡ fjÿ fj:

This equation leaves f0

CP j still completely undefined, since all three phases on the hand side are not observable, and therefore subject to arbitrary changes It becomes mean-ingful, however, if it is applied to observables, like CP eigenvalues Two CP eigenstatesconstructed from a meson and anti-meson state with eigenvalues 1 are related accord-ingly:

Quark phase changes can in principle be compensated by phase changes of the CKMmatrix elements according to (2.2), leaving terms like

hqjj Vjkjqki

invariant However, this is not a physical requirement, and in fact the CP transformed

has a phase which changes with the quark phases Since none of the two terms corresponds

to an observable, the actual choice of phases in the CKM matrix parametrization can bemade independent of the choice of quark phases

The appearance of an additional phase factor in (2.10) can be avoided by the restriction



fjˆ ÿfj for quark phase changes, and an appropriate phase convention which makesterms related by a CPT transformation relatively real If a choice of phases is possiblewhere all CKM matrix elements can be made real, also charged current weak interactionswould not violate CP symmetry

Phase conventions will also enter into relations among decay amplitudes An amplitudefor a weak decay B0! X via a single well defined process can be written as

where V is a product of the appropriate CKM matrix elements and O is an operator describingthe rest of the weak and possibly also subsequent strong interaction processes involved in thetransition Since strong interaction and also weak interaction ±± except for nontrivial phases

in V ±± are CP invariant, the charge conjugate mirror process B0! X has an amplitude

relates the two amplitudes, and the ratio A=A flips sign with the CP eigenvalue

All physical observables must be independent of the choice of phases This is the case ifonly absolute values of amplitudes are involved, but for interference terms the phase con-vention cancels often in a more subtle way Some examples will be shown in the followingchapters On the other hand, expressions where the arbitrary phases are still present cannot

0B

@

1C

where M and G are hermitian, but H is not [19] If the B0= B0 system is taken as a sentative to illustrate the behaviour of oscillating meson pairs, the indices 1 and 2 corre-spond to base vectors jB0i and j B0i, respectively

repre-CPT invariance requires m11ˆ m22:ˆ m and G11 ˆ G22:ˆ G, reducing the number ofreal parameters of the Hamiltonian to six

0B

@

1C

CPT invariance is one of the indispensable premises of any relativistic field theory within

or beyond the Standard Model [20] The generalized phenomenology including CPT tion will therefore not be considered here, but can be found in textbooks [21]

viola-Solving the eigenvalue problem det …H ÿ a
1† ˆ …H ÿ a†2ÿ H12H21ˆ 0, one obtainstwo eigenstates with eigenvalues a ˆ H pH12H21, explicitly

where x is a non-negative real number, and y may only assume values between ÿ1 and 1

It is an asymmetry parameter in the widths or, equivalently, in the lifetimes tL; tH

The eigenvectors jBL; Hi ˆ qp

are found by inserting (2.16) into H jBL; Hi ˆ

aL;HjBL; Hi, giving the ratio

and single particle eigenstates are described by one complex parameter hm This meter3† is defined only up to an arbitrary phase, and only jhmj is a measurable quantity.The value of the phase depends on conventions, one of them is the definition of thephase fCP Bˆ arg h B0j CP jB0i This makes also E (sometimes also denoted E, e.g in [23])

para-an arbitrary qupara-antity The stpara-andard choice of the CKM matrix (2.3) para-and fCP K ˆ 0 makejEj small in the K0= K0 system, but a consistent convention fCP Bˆ 0 leaves it at

o…0:1 1† in the B0= B0 system A different definition of E for the kaon system given in[22] is independent of arbitrary phases In general, convention independent parameterscan be defined if decays are involved They can usually be expressed via the unitarityangles (see fig 2.1) and will be given for the B and K systems at the appropriate placesbelow

The original Hamiltonian can be rewritten using the parameter hm as

@

1CA

…2:21†and the mass and flavour eigenstates are related by the equations

In contrast to E the real number d is an observable The deviation of jhmj from one (called

jw…t†i ˆ bHeÿi …m H ÿiG H =2† tjBHi ‡ bLeÿi …m L ÿiG L =2† tjBLi

ˆ eÿimtÿT=2 ei …xÿiy† T=2‡ eÿi …xÿiy† T=2

Starting with pure B0mesons at t ˆ 0 corresponds to a ˆ 0 and

jw…t†i ˆ a eÿimtÿT=2 cos …x ÿ iy† T2 jB0i ‡ ihmsin …x ÿ iy† T2 j B0i

Starting with pure B0 mesons at t ˆ 0 is described by replacing hm$ 1=hm The numbers

of B0 and B0 at time T for N0 pure B0 mesons at T ˆ 0 are

NB0…t† ˆ N0jhB0j w…t; a ˆ 0; a ˆ 1†ij2ˆ N0 eÿT

2 …cosh yT ‡ cos xT† ;

NB 0…t† ˆ N0jh B0j w…t; a ˆ 0; a ˆ 1†ij2ˆ N0jhmj2 eÿT2 …cosh yT ÿ cos xT† : …2:26†

These numbers, however, can not be observed What is accessible by experiment is onlythe rate of decays to flavour specific final states X and X at a given time T These decaymodes are often called tagging modes, since they serve as a ``tagº to indicate the flavour

of the mother particle at decay time The rates can be obtained from (2.25) by multiplyingwith hXj H or hXj H, respectively, to obtain the amplitudes They are converted into rates

_NB 0 !X…t† ˆ N0

…dPS jhXj H jw…t; a ˆ 0†ij2ˆ 12 N0eÿTGX…cosh yT ‡ cos xT† ;

_NB0 !X…t† ˆ N0

…dPS jhXj H jw…t; aˆ0†ij2ˆ 12 N0jhmj2eÿTGX…cosh yT ÿcos xT† ;

…2:27†where

GX ˆ„ dPS jhXj H jB0ij2ˆ„ dPS jhXj H j B0ij2

is the partial width for a non-oscillating meson It agrees in value for the two CP conjugateprocesses if the amplitudes differ only by phases Integrating over all times the total num-ber of decays are

NB0 !Xˆ

1 0

The corresponding numbers for initial B0 mesons are obtained with the replacement

hm ! 1=hm If we ignore CP violating effects in the oscillation, i.e for jhmj ˆ 1, we candefine a meaningful branching fraction as

B…B0! X† ˆN1

0

1 0

2.2.1 Standard Model Predictions

The Hamiltonian (2.15) can be obtained using

which has the stable flavour eigenstates B0 and B0, and Hw is the weak interaction tion The Wigner-Weisskopf approximation for small Hw leads to [24]

mjk ˆ 12 …Hjk‡ H*kj† ˆ E0djk‡ hjj Hwjki ‡P

X P

…dPShjj HwEjXi hXj Hwjki

The off-diagonal elements H12; 21 have non-zero contributions in the sum from states Xwhich can be reached in weak decays of both B0 and B0 In contrast to the neutral kaonsystem, for B0= B0 these are only a small fraction of all B decays, and they contribute withalternating signs Therefore H12; 21 are dominated by the leading term hB0j Hwj B0i whichcorresponds to the box diagrams

They give approximately [25]

t=m2

W Anevaluation of the product S…m2

which can be used to determine jVtdj from experimental results on B0= B0 mixing Theeigenstates are determined by

hmˆ ÿjmm*12

12jˆ eifCP B

V*2

tdV2 td

jV2

tbV2

with ÿ~b ˆ arg V*tbVtd This phase depends on the CKM parametrization and is ±± like the

CP phase ±± not an observable The arbitrariness cancels only in physical observables,which include decay amplitudes with further CKM elements and a CP phase The corre-sponding

E ˆ ÿi sin arg hm

m2 b

m2

This ratio applies to both the B0and Bssystems

To leading order in G12=m12 equation (2.20) yields

with jdj  1 where b is the CKM unitarity angle in figure 2.1a Since this result is based

on a leading order quark diagram, the number should be taken only as an order of tude In particular, at this level of precision it can not be used to measure b

2.2.2 Behaviour of the Four Neutral Meson Anti-Meson Systems

All four meson pairs K0= K0, D0= D0, B0= B0, and Bs= Bs show a different oscillation iour, since they have all different relations of G, DG, and Dm The same symbols will beused for all four systems Only when two specific systems shall be compared, their param-eters will be distinguished by the subscripts K, D, d, and s, respectively The dimensionlessparameters x and y give the ratios of time constants involved: t ˆ 1=G is the harmonicaverage of the lifetimes, toscˆ 2p=Dm ˆ 2pt=x is the period of the oscillation, and

behav-trelˆ 2=DG ˆ t=y is the lifetime of the oscillation amplitude, i.e the damping time stant of a relaxation process Numerical values are summarized in table 2.1

con-While the parameters of the K0= K0 system are well measured [9], theoretical tions enter into the B meson columns Many precise lifetime measurements for neutral Bmesons have become available last year All lifetime measurements are summarized intable 2.2, and average to td ˆ …1:54  0:03† ps

1:64  0:08  0:08 SLD 97 [36]1:58  0:09  0:02 CDF 96 [37]1:474  0:0390:052 CDF 98 [38]

Figures 2.2±±2.5 show the number of mesons and anti-mesons as a function of the ing lifetime variable T ˆ t=t and the asymmetry

scal-a…T† ˆ _N…X ! X† ÿ _N…X ! X†

_N…X ! X† ‡ _N…X ! X†

T

ˆ…1 ÿ jhmj2† cosh yT ‡ …1 ‡ jhmj2† cos xT

…1 ‡ jhmj2† cosh yT ‡ …1 ÿ jhmj2† cos xT

…2:36†for a meson produced at T ˆ 0 as a flavour eigenstate X, and decaying to a flavour-specificfinal state as X or X at a later time T Expressed via the small real parameter d instead of

flavour-a…T† ˆ _N…X ! X† ÿ _N…X ! X†

_N…X ! X† ‡ _N…X ! X†

Tˆ ÿcos xT ÿ d cosh yTcosh yT ÿ d cos xT :The approximation jhmj ˆ 1 corresponding to d ˆ 0 leads to a simpler expression

where x is clearly seen as the oscillation parameter, and y as the damping parameter.The kaon has both x  1 and y  ÿ1, i.e the long-living state is the heavier mass eigen-state With these parameters one half of a sample of kaons of either flavour decays rapidly,mainly into two pions with CP ˆ ‡1, and the other half transforms to a sample of thelong-living K0

L states, which decay (aside from the small CP violation) to CP ˆ ÿ1 states and to flavour-specific states The ratio of lifetimes of the two states (table 2.1) isapproximately 580 The time evolution of an initially pure K0 flavour eigenstate is shown

eigen-in figure 2.2 The upper diagram shows the number of remaeigen-ineigen-ing K0 and K0 after a scaledtime T ˆ Gt, where G  GS=2 ˆ 1=…2tS† is the average width of the short- and long-livingstate The decay rate into flavour-specific final states is proportional to these numbers,while the dominant decays to CP eigenstates follow different evolution functions due to CPviolation, and will be discussed below

The D0 meson decays mainly to flavour specific states with well defined strangeness,with only a few decays to CP ˆ ‡1 eigenstates, as pp, K K, K0

Lp0 and CP ˆ ÿ1 states, as

K0p0 or K0w This leads to equal lifetimes for the two eigenstates, i.e y  0 The sponding box graph has a b quark as the heaviest particle in the loop, which is accompanied bythe small CKM elements Vcband Vub The mass difference induced that way by the StandardModel is very small, corresponding to x 0.002 Therefore, almost no asymmetry is visible infigure 2.3, although the number x ˆ 0:02 used for the plot is a factor 10 higher The value

corre-x ˆ 0:002 corresponds to a total micorre-xed fraction of initially pure D0states given by

c ˆ ND0 !X

as c  2
10ÿ6

The parameters of the B0= B0 system have been introduced above A good approximation

is y ˆ 0 and d ˆ 0, which leads for N0 pure B0 at t ˆ 0 to

This asymmetry can be observed using a flavour-tagging decay, like B0! Dÿl‡n The rate

of mesons decaying at time T into the channel X are given by (2.27) where y ˆ 1 makes

Fig 2.2: K0= K0 mixing is determined bythe parameters x ˆ 0:95, y ˆ 0:996, and

jhmj2ˆ 0:994 T ˆ t=t is the lifetime inunits of t  2tS, the inverse of the averagewidth of K0 and K0 The upper diagramshows the number of K0 (solid) and K0

(dotted) as a function of T for a samplestarting with 100% K0 mesons The lower

a ˆ …NKÿ NK†=…NK‡ NK† The relaxationprocess soon dominates, leaving only K0

after not much more than one oscillation

cosh yT ˆ 1 leading to the same asymmetry function a…T† ˆ cos xT Integrating over alltimes, the observed numbers are

NB0 !Xˆ

…_NB 0 !X…T† dt ˆ 12 N0 GGX 2 ‡ x1 ‡ x22 ;

NB 0 !Xˆ

…_NB 0 !X…T† dt ˆ 12 N0 GX

G

x2

1 ‡ x2 :Their asymmetry becomes

and the mixing probability is as in (2.39)

c ˆ NB0 !X

It was this net effect which gave the first proof for a sizeable mixing parameter x  0:7 inthe B0 meson system in 1987 [39] The time-dependent particle anti-particle oscillations ofthe neutral B meson have been first seen six years later by experiments at LEP [40] With

x  0:7, about one period is visible before most of the mesons are decayed

If we assume the Standard Model predictions to be true, the Bs meson is a very ing case There will be a small y and a very large x Figure 2.5 is plotted with xsˆ 15,which is close to the lower limit of the theoretical range The time-integrated mixing prob-

interest-Fig 2.4: B0= B0 evolution is dominated bythe oscillating part, with the parameters

x ˆ 0:70, y ˆ 0, and jhmj ˆ 1 The ratio ofthe areas under the dotted and solid curve inthe upper plot is the mixing probability c.The zero transition in the asymmetry, whichmarks the crossover point in the upper plot,

is at T ˆp

2x

ability is for jhmj ˆ 1

c ˆ2…1 ‡ xx2‡ y22†:

For large xs 1, this approaches its maximum value of 0:5, where a measurement of thisquantity has no sensitivity on x any more To observe the rapid oscillations, a very goodlifetime resolution will be required Experimentally, a lower limit xs> 18 has been found atLEP (see below)

In the general case jhmj 6ˆ 1, the integrated mixing probability depends on the initialflavour It is

c ˆ jhmj2…x2‡ y2†

2 ‡ x2…1 ‡ jhmj2† ÿ y2…1 ÿ jhmj2†ˆ

…1 ÿ d† …x2‡ y2†2‰1 ‡ x2‡ d…1 ÿ y2†Š …2:43a†

Fig 2.5: Bs= Bs is expected to be the mostrapidly oscillating system, with a longer re-laxation time This plot assumes xsˆ 15,

ysˆ 0:10, and jhmj ˆ 1

for an initial B and

2 jhmj2‡ x2…1 ‡ jhmj2† ‡ y2…1 ÿ jhmj2†ˆ

…1 ‡ d† …x2‡ y2†2‰1 ‡ x2ÿ d…1 ÿ y2†Š …2:43b†for an initial B (which is c with jhmj replaced by 1=jhmj or d by ÿd) This exhibits already

CP violation, since the probabilities P…X ! X† and P…X ! X† are different It is also Tviolation, since the transition X ! X is the time reversed process X ! X

2.2.3 CP Eigenstates Versus Mass Eigenstates

The following discussion will again use B0= B0 as an example, but is applicable to each ofthe four systems accordingly

The standard phase convention requires all JPCˆ 0ÿ‡ mesons to have CP jXi ˆ ÿjXi,fixing fCP Bˆ p Independent of any convention, two orthogonal CP eigenstates

The mass eigenstates of the B0= B0 system are not CP eigenstates Using

CP jB0i ˆ eif CP Bj B0i, they are transformed by a CP operation as

A meaningful question is which of BH or BL decays more often into CP ˆ 1 states In contrast to the neutral kaon system, most final states from B decays are flavour-specific, and both mass eigenstates decay into them via either their B0 or their B0 compo-nent The small fraction of states that can be reached both by B0 and B0 includes thecontribution from CP eigenstates which appear mainly through three processes On the treelevel, there are two main decay channels that can produce CP eigenstates: b ! ccd with theccd d final state, and b ! uud with the quark content uud d A state of the first kind willhave decay amplitudes

eigen-A ˆ hXccj H jB0i ˆ V*cbVcdA0; A ˆ hXccj H j B0i ˆ hXeÿifCP BVcbV*cdA0;

where hX ˆ 1 is the CP eigenvalue of the state The corresponding decay amplitudes of

1 ÿ hXcos 2b

which is for b <p

4 less than 1 for hXˆ ‡1 and vice versa In this case, the heavier state

BH will decay more often into states with negative CP eigenvalue, hXˆ ÿ1

Accordingly, for the uud d states

depends on the angle a, which is likely to be larger than p

4 This would give the oppositeanswer, i.e the heavier state BH will decay more often into states with positive CP eigen-value, hX ˆ ‡1

Some decays with an intermediate state ccds or ccds proceed into K0 or K0, whichfinally result in ccd d via a K0

For decays via W exchange, like b d ! cc or b d ! uu, the same CKM elements areinvolved, and the same arguments lead to the same answers as above Also, the favouredpenguin-type transition b ! s with subsequent hadronization into a K0

L or K0

S has a netphase close to b0 leading to the ratio (2.45)

CP eigenstates with quark content d d can be reached via CKM-suppressed penguin-typeloops Due to the top quark dominance the amplitudes are

All these results receive corrections from non-leading terms, like c quark loops in the lastcase, or b ! d penguin corrections to the b ! u transition final states The general case for

The situation would be different for a purely real CKM matrix (up to phases that can beremoved by quark phase changes) In this case, all unitarity triangles would be degenerate

to lines, and their angles would be 0 or p Therefore, cos 2a ˆ cos 2b ˆ 1, and the heavierstate would be the only to decay to CP ˆ ÿ1, while CP ˆ ‡1 final states would bereached exclusively via decays of BL For decay products which are CP eigenstates thissituation would correspond to a perfectly predictable CP eigenvalue corresponding to themass eigenstate BL or BH A natural choice of phases in this case would force all terms

of the weak interaction Hamiltonian to be real, corresponding to hm ˆ eif CP B Then

CP jBLi ˆ ‡jBLi ˆ jB‡i and CP jBHi ˆ ÿjBHi ˆ ÿjBÿi, and CP is conserved in decayswhere this quantum number is meaningful

Exactly this situation is almost true for the K0= K0 system The light K0

L decays to about99:9% into the CP ˆ ‡1 eigenstates p‡pÿand p0p0, while the K0 decays to one third into

a CP ˆ ÿ1 eigenstate with 3 pions, the rest being mainly flavour specific semileptonicdecays, and only 0:3% are to the CP ˆ ‡1 two pion state [9] Therefore, a parametrization

is chosen where K0 K‡ and K0

L Kÿ If we have a K0 as decay product of the B, weare used to assign it a CP ˆ ‡1 eigenvalue contribution to the whole final state To beprecise, this is only correct if the K0 decays into a CP ˆ ‡1 final state In this case also a

K0

L! pp will be assigned the same CP ˆ ‡1 eigenvalue, i.e the ``K0º denotes its finalstate rather than the undecayed particle, and a K0! pln as a flavour specific state is notincluded in this use of the label K0

2.2.4 Oscillation at the U (4S)

The B B system from strong interaction U (4S) decay is in an odd C and P eigenstate withangular momentum L ˆ 1, retaining the quantum numbers JPCˆ 1ÿÿ of the mother parti-cle This system has to be treated as a coherent quantum state The time evolution of a statewith odd symmetry is different from that of one with even symmetry This is due to thefact, that only one anti-symmetric XX state,

and their relative amplitudes may change with time The quantum numbers characterizing thetwo different mesons, which are represented by (1) and (2) here, can be thought of as the spatialwave functions w…x† and w…ÿx† or alternatively the states in momentum space j pi and jÿ pi.Explicitly, for initial B B states of well defined symmetry,

wÿ…t† ˆ eÿ2imteÿT‰jB0…1† B0…2†i ÿ jB0…2† B0…1†iŠ …2:48a†for ‡:

a flavour eigenstate Only when one decays into a state revealing its flavour (not rily the first one that decays) the state ``collapsesº and the second one continues as a one-particle state evolving in time according to (2.24)

necessa-The second case (2.48b) of an even wave function leads to a probability oscillating with twicethe single-B frequency between a like-sign (BB or B B) and opposite-sign (B B) flavour state.For different times T1 and T2 of B meson (1) and (2), e.g the times of decay of the two

B mesons, we have for the anti-symmetric state

Again it is seen that for T1ˆ T2 only the anti-symmetric state is present, and mixed states,i.e two final states indicating the same beauty flavour, will show up only at T16ˆ T2 Themixing probability (B and B denote the flavour at decay time)

For incoherent B0B0 pair production, e.g in b b jet fragmentation, the integrated rate is determined by two independent mixing probabilities

mixed-N…BB ‡ B B†

N…BB ‡ B B ‡ BB ‡ B B†ˆ 2c…1 ÿ c† :

Equation (2.49b) is an expansion in the two mass eigenstates The anti-symmetric wavefunction is always composed of two different states, there will be never BHBH or BLBL,even at different decay times

The question of CP eigenstates can only be answered after both B mesons have decayed.This involves the phases in decay amplitudes, and includes all effects of CP violationwhich will be discussed in detail below

For the symmetric state, the wave function is

‡ i sin …x ÿ iy†T1‡ T2 2 …jBL…1† BL…2†i ‡ jBH…1† BH…2†i†

:…2:51†This is very similar to the function of an anti-symmetric state, but the oscillation is in thesum of the two lifetimes instead of the lifetime difference

In the approximation jhmj ˆ 1 and y ˆ 0, the integrated mixed-rate is

The expansion in mass eigenstates shows, that the symmetric wave functions consistsalways of two eigenstates with the same mass, i.e BHBH or BLBL.

2.2.5 Determination of the Mixing Parameters of B Mesons

Only a small fraction of B meson decays has been fully reconstructed However, the flavour

of a B meson can be identified by various ``tagsº The first observation of a then pected large B0B0 mixing by ARGUS in 1987 [39] used the best flavour tags available: Inmultihadron events on the U (4S) resonance, 25 like-sign lepton pairs were observed whichcould not be attributed to other sources but semileptonic B decays The charge of the leptonfrom b ! l‡nc in these decays is identical to the beauty (or bottomness) quantum number

unex-of the meson, and these events had to be attributed to B0B0 and B0B0 final states from

U (4S) decays The mixing probability c can be calculated from the number of like- andopposite-sign dilepton events as

While only the integrated effect can be observed on the U (4S) at presently existing metric colliders, an observation of the oscillating behaviour was possible at the Z0, where

the lifetime can be measured This yields directly the frequency Dm from the asymmetry

a…t† ˆ _N…B† ÿ _N… B†

_N…B† ‡ _N… B†

... observe the difference of matter and anti-matter atfar regions of the universe, the absence of regions of matter anti-matter annihilation bound-aries suggests that the whole universe is made of matter,... transforms to a sample of thelong-living K0

L states, which decay (aside from the small CP violation) to CP ˆ ÿ1 states and to flavour-specific states The ratio... heavierstate would be the only to decay to CP ˆ ÿ1, while CP ˆ ‡1 final states would bereached exclusively via decays of BL For decay products which are CP eigenstates thissituation would