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,...

Fortschr Phys 47 (1999) ± 8, 707±±853 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 presented, including techniques for flavour tagging and data fits The expected performance of an e‡ eÀ and a hadron beam experiment is illustrated with the most promising final states JawKs0 and p‡ pÀ Contents Introduction 709 Particle Anti-Particle Oscillations and CP Violation 2.1 The Unitary CKM Matrix 2.1.1 Unitarity Triangles 2.1.2 Phases and Observables 2.2 Oscillation Phenomenology 2.2.1 Standard Model Predictions 2.2.2 Behaviour of the Four Neutral Meson Anti-Meson Systems 2.2.3 CP Eigenstates Versus Mass Eigenstates 2.2.4 Oscillation at the U(4S) 2.2.5 Determination of the Mixing Parameters of B Mesons 2.2.6 Predictions for xs Y ys and ds 2.3 CP Violation 2.3.1 CP Violation in B Decays 2.3.2 CP Violation in Common Final States of B0 and B"0 2.3.2.1 The Bs aB"s Case 2.3.2.2 CP Violation at the U(4S) 2.3.2.3 Time Integrated Asymmetries 2.3.2.4 Final CP Eigenstates from B0 or Bs Decays 2.3.2.5 The B pp Decay 2.3.2.6 Mixtures of CP Eigenstates 2.3.2.7 Non-Eigenstates 2.3.2.8 The Total Decay Rate 2.3.3 CP Violation in K Decays 710 710 712 715 717 721 724 730 732 735 737 738 739 741 744 745 750 750 754 758 759 759 760 Measurement of CP Violation at B Meson Factories 3.1 B Meson Factories 3.1.1 B Production Cross Sections 3.1.2 B Meson Fractions 3.1.3 B Meson Yields 3.2 Two Typical Detectors 3.2.1 The LHB Detector at a Fixed Target Hadronic 764 765 767 769 770 770 771 B Factory 708 R Waldi, Prospects for the Measurement of the CP Asymmetry 3.2.1.1 Vertex Region 3.2.1.2 Tracking and Particle Identification 3.2.1.3 Trigger 3.2.1.4 Other Experiments 3.2.2 The BABAR Detector at the PEP II e‡ eÀ Storage Ring 3.2.2.1 Vertex Detector 3.2.2.2 Tracking and Particle Identification 3.2.2.3 Electromagnetic Calorimeter 3.2.2.4 Muon and Neutral Hadron Detector 3.2.2.5 Trigger 3.2.2.6 Other Expriments 3.2.3 U (4S) Factories and Hadron Colliders: Two Complementary Concepts 772 773 774 774 774 777 778 780 780 781 781 781 Analysis Techniques and Tools to Estimate Experimental Performance 4.1 Flavour Tagging 4.1.1 Observed Versus True Asymmetry 4.1.2 Statistical Tagging 4.1.3 Specific Tags 4.1.3.1 Lepton Tags 4.1.3.2 Lepton Tags at LHB 4.1.3.3 Kaon Tags 4.1.3.4 Kaon Tags at LHB 4.1.3.5 Charm Tags 4.1.3.6 Other Tags 4.1.4 Combined Tagging Results at the U(4S) 4.1.5 Special Tags at Hadron Machines 4.1.5.1 Vertex Reconstruction and Charged B Tags 4.1.5.2 Tag Jet Charge ‡ 4.1.5.3 The B** B0 p‡ Cascade 4.1.5.4 Same Jet Charge 4.1.6 Combined Tagging for Hadron Machines 4.1.7 Determining I and D from Data 4.2 Fitting CP Asymmetries 4.2.1 Fit to the Time Dependent Asymmetry 4.2.1.1 Fit of L 4.2.1.2 Fit of L and I 4.2.1.3 Fit of Q 4.2.1.4 Fit of Q and I 4.2.1.5 Fit of L and Q 4.2.2 Using Time Integrated Numbers 4.2.2.1 Background 4.2.3 Improved Fit Procedure for Small Data Samples 4.2.3.1 Background 782 783 784 787 791 791 795 798 800 802 804 806 807 808 808 809 810 810 810 813 814 817 818 820 821 822 823 823 824 827 Performance for the Key Final States 5.1 B0 p‡ pÀ 5.2 B0 p‡ pÀ Reconstruction at LHB 5.2.1 Mass Resolution 5.2.2 Trigger Efficiency and Reconstruction 5.2.3 Backgrounds 5.2.3.1 Background from Bs Decays 5.2.3.2 Combinatoric Background 5.2.4 Acceptance 5.3 B0 p‡ pÀ Reconstruction at BABAR 5.3.1 Backgrounds 5.3.2 The Lifetime Measurement 5.4 B0 Jaw Ks0 5.5 B0 Jaw Ks0 Reconstruction at LHB 827 828 829 832 833 834 836 837 839 840 840 841 842 842 Losses 709 Fortschr Phys 47 (1999) 7±±8 5.6 B0 JawKs0 Reconstruction at BABAR 5.7 Present Experimental Information 5.8 Comparison of Future Experiments 5.8.1 B0 JawKs0 and the Determination of b 5.8.2 B0 p‡ pÀ and the Determination of a 845 845 846 846 847 Outlook 848 Acknowledgements 848 References 849 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 À1 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 helicity, or righthanded in the case of anti-neutrinos, thus violating the charge-conjugation symmetry (C) at the same time The product of both discrete symmetries, CP, is almost intact, and seemed to be conserved even in weak interaction processes A small violation has first been observed in 1964 [3] in K 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 violating 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 dis- 710 R Waldi, Prospects for the Measurement of the CP Asymmetry cussed Section presents experimental performance for two examples and in the last section an outlook to the next few years is based on extrapolating these studies to the most promising proposed experiments 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 quantum numbers (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 " D0 aD"0 (c" " d), " and Bs aB"s (bsab" " s) These are K aK"0 (" sdasd), ua" cu), B0 aB"0 (bdab 2.1 The Unitary CKM Matrix The charged current weak interactions responsible for flavour changes are described by the couplings of the W boson to the current H I H I H I H I e d n"e u" À g5 f g € f g À g5 f g f g " Jmcc ˆ d n"m e gm m V Á ‡ …2X1† c g d e dse d e m 2 rY gY b "t n"t t b with a non-trivial transformation matrix V in the quark sector, the Cabibbo±Kobayashi± Maskawa (CKM) Matrix [4, 5]: H I Vud Vus Vub V ˆ d Vcd Vcs Vcb e X Vtd Vts Vtb The quark flavours in (2.1) are defined as the mass eigenstates A completely equivalent picture is to use the states …dH Y sH Y bH † with V  1, and define a non-diagonal mass matrix Since mass generation is accomplished in the Standard Model via couplings to the Higgs field [6], this moves the question of the origin of the CKM matrix elements into the realm of mass generation, which belongs still to the more ``mysteriousº parts [7] of the Standard Model The exploration of the Higgs sector is the main motivation for the LHC storage ring, which is built at CERN and will start operation around 2005 [8] The Higgs-quark couplings alone involve 10 independent parameters of the Standard Model, the quark masses 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 be unitary If there were more than three quark families, this would not hold for the  submatrix, but this possibility is unlikely, given the limit on neutrino flavours from LEP experiments, who find nn ˆ 2X991 Ỉ 0X016 [9] for neutrinos with mass much below the Z mass Thus, if a fourth generation exists, it must incorporate a massive neutrino which is more than a factor 1000 heavier than the tau neutrino, even if we assume the experimental upper limit for the latter From the real parameters of a general unitary matrix, can be absorbed in global phase, relative phases between uY cY t and relative phases between dY sY b which are all subject to convention and in principle unobservable If two quarks within one of these two groups were degenerate in mass, even the sixth phase could be removed by redefining the basis in their two-dimensional subspace 711 Fortschr Phys 47 (1999) 7±±8 Rephasing may be accomplished by applying a phase factor to every row and column: Vjk ei…fj Àfk † Vjk X …2X2† Note that j ˆ uY cY t, k ˆ dY sY b, and the six numbers fu Y fc Y ft Y fd Y fs Y fb represent only five independent phases in the CKM matrix, since different sets of ffj Y fk g yield the same result Any product where each row and column enters once as Vij and once via a complex conjugate Vkl* like Vij Vkl Vil*Vkj* is invariant under the transformation (2.2) This implies that observable phases must always correspond to similar products of CKM matrix elements with equal numbers of V and V * factors and appropriate combination of indices Removing unphysical phases, the CKM matrix is described by real parameters, where only 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 a choice of phases, that leave Vud and Vcb real: I H IH IH c12 s12 0 s13 eÀid13 c13 g f gf gf V ˆ d c23 s23 ed ed Às12 c12 e H Às23 c23 Às13 eid13 c12 c13 f ˆ d Às12 c23 Àc12 s13 s23 eid13 s12 s23 Àc12 s13 c23 e id13 0 c13 s13 eÀid13 s12 c13 g c13 s23 e c12 c23 Às12 s13 s23 eid13 id13 Àc12 s23 Às12 s13 c23 e I …2X3† c13 c23 with cij ˆ cos qij , sij ˆ sin qij , and sij b 0, cij b (0 qij pa2) A convenient substitution1 ) is s12 ˆ l, s23 ˆ Al2 , s13 sin d13 ˆ Al3 h, and s13 cos d13 ˆ Al3 r, which reflects the apparent hierarchy in the size of mixing angles via orders of a parameter l This leads to H l2 l4 À À f f   f f l r ‡ ih À Àl À A f Vˆf f 35 f d l2 Al À …r ‡ ih† À l   l2 A ‡ 1À À l 2   ÀAl2 À Al4 r ‡ ih À I Al …r À ih† g g g g Al g‡ o…l6 † g g g 4e 1À A l …2X4† and agrees to o…l † with the Wolfenstein approximation [12]: H I l2 l Al …r À ih† À f g f g f g Vˆf gX l f g Al Àl À d e Al …1 À r À ih† ÀAl …2X5† Equation (2.4) is more convenient [13] in higher orders than the original proposal of Wolfenstein, or an exact parametrization [14] using the Wolfenstein parameters ) An equivalent choice is l ˆ s12 c13 which leads to the same parametrization to o…l5 † 712 R Waldi, Prospects for the Measurement of the CP Asymmetry Assuming a unitary  matrix, from experimental information these parameters are [9] l ˆ 0X2205 Ỉ 0X0018 Y A ˆ 0X80 Ỉ 0X08 Y p r2 ‡ h2 ˆ 0X36 Ỉ 0X08 while the phase and therefore each individual value of r and h is still very uncertain Inserting these parameters, equation (2.5) shows clearly the dominance of the diagonal matrix elements, indicating that transitions between quarks of different families are suppressed It is the unitarity constraint which makes Vtb ˆ 0X9992 Ỉ 0X0002 the best known matrix element Experimental constraints on the magnitude (90% CL limits [9]) are: H I 0X9745 F F F 0X9757 0X219 F F F 0X224 0X002 F F F 0X005 d 0X218 F F F 0X224 0X9736 F F F 0X9750 0X036 F F F 0X046 e X 0X004 F F F 0X014 0X034 F F F 0X046 0X9989 F F F 0X9993 With already one more family of quarks, we have five additional real parameters, of which two are new non-trivial phases Therefore, the measurement of all CKM matrix elements and their relative phases is an important test of the Standard Model 2.1.1 Unitarity Triangles If nature provides us with just these three families of fermions, unitarity requires the following 12 conditions to be fulfilled: jVud j2 ‡ jVus j2 ‡ jVub j2 ˆ Y …2X6a† jVcd j2 ‡ jVcs j2 ‡ jVcb j2 ˆ Y …2X6b† jVtd j2 ‡ jVts j2 ‡ jVtb j2 ˆ Y …2X6c† jVud j2 ‡ jVcd j2 ‡ jVtd j2 ˆ Y …2X6d† jVus j2 ‡ jVcs j2 ‡ jVts j2 ˆ Y …2X6e† jVub j2 ‡ jVcb j2 ‡ jVtb j2 ˆ Y …2X6f† V *ud Vcd ‡ V *us Vcs ‡ V *ub Vcb ˆ Y …2X6g† V *ud Vtd ‡ V *us Vts ‡ V *ub Vtb ˆ Y …2X6h† V *cd Vtd ‡ V *cs Vts ‡ V *cb Vtb ˆ Y …2X6i† Vud V *us ‡ Vcd V *cs ‡ Vtd V *ts ˆ Y …2X6j† Vud V *ub ‡ Vcd V *cb ‡ Vtd V *tb ˆ Y …2X6k† Vus V *ub ‡ Vcs V *cb ‡ Vts V *tb ˆ X …2X6l† Fortschr Phys 47 (1999) 7±±8 713 Dividing (2.6k) by Al3 % ÀVcd V *cb yields the unitarity triangle2 † as shown in figure 2.1a In the Wolfenstein approximation, it corresponds to …r ‡ ih† À ‡ …1 À r À ih† ˆ X …2X7† A second one from (2.6h) is shown in figure 2.1b Dividing by Al3 % ÀV *us Vts and using the approximation Vud % gives the same triangle (2.7) A closer look, however, reveals slightly different lengths and angles to o…l2 † The angles of the unitarity triangles (2.6k and h) in figure are defined by Vtd Vub V *ud V *tb Y jVtd Vub Vud Vtb j H V *td V *cb Vcd Vtb V *td V *us Vts Vud % eib ˆ À Y eib ˆ À jVtd Vcb Vcd Vtb j jVtd Vus Vts Vud j V *ub V *cd Vcb Vud V *ub V *ts Vus tb H % eig ˆ À X eig ˆ À jVub Vcd Vcb Vud j jVub Vts Vus Vtb j eia ˆ À These are rephasing invariant expressions, hence the angles resemble physical quantities independent of the CKM parametrization It was first emphasized by Jarlskog [17], that CP violation can be described via a rephasing invariant quantity J ˆ Ỉ Im Vij Vkl V *il V *kj % A2 l6 h which is up to a sign independent of iY jY kY l, provided i Tˆ k, j Tˆ l The areas of all six unitarity triangles defined by (2.6g±l) are equal and have the value Ja2 This corresponds to an area % ha2 for the ones in figure 2.1, since their sides have been reduced by the factor Al3 As will be shown below, CP violating observables are typically proportional to the sine of the angles in unitarity triangles, like sin g ˆ Im eig ˆ À Im …V *ub V *cd Vcb Vud † J ˆÀ jVub Vcd Vcb Vud j jVub Vcd Vcb Vud j and vanish for J ˆ 0, i.e if all triangles collapse into lines If the non-trivial phase is or p, the parameter h is and hence J ˆ This would also be the case if two quarks of a given charge had the same mass, since then a rotation between these two flavours could be chosen that removes the phase factors, as can be seen in (2.3) where q13 ˆ would remove all terms with the phase d13 The angles of all six triangles (2.6g±l) can be determined using the standard parametrization (2.3) in a rewritten form H I jVud j jVus j jVub j eeÀi~g jVcb j e V ˆ d ÀjVcd j eif4 jVcs j eeÀif6 …2X8† Àib~ if2 jVtd j ee ÀjVts j ee jVtb j with g~  d13 Here, absolute values and phases are given as separate factors The angles f2 % hl2 , f4 % hA2 l4 , and f6 % hA2 l6 are all positive and very small and their subscript † This geometric interpretation has been pointed out by Bjorken $ 1986; its first documentation in printed form is in ref 15 and more general in ref 16 714 R Waldi, Prospects for the Measurement of the CP Asymmetry Fig 2.1: Unitarity triangles in the complex plane, corresponding to a: (2.6k) and b: (2.6h) Up to 2 corrections of o…l4 † the top points are …rY h† in (b), but …‰1 À l2 Š rY ‰1 À l2 Š h† in (a), and the right2 most points are …1Y 0† in (a), but …1 À l ‰2 À rŠY l h† in (b) The angles are related via g À gH ˆ bH À b % l2 h indicates the order in l of their magnitude The unitarity triangles in figure 2.1 have angles b ˆ b~ ‡ f4 Y H g ˆ g~ À f2 Y bH ˆ b~ ‡ f2 Y a ˆ p À b~ À g~ X g ˆ g~ À f4 Y 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) Àl ‡ l ‡ l À l3 ‡ ‰A2 l5 …r ‡ ih†Š ˆ Y 2 …2X6gH † 715 Fortschr Phys 47 (1999) 7±±8 Al3 …1 À r À ih† À Al3 ‡ Al3 …r ‡ ih† ˆ Y …2X6hH † ‰ÀAl4 …1 À r À ih†Š À Al2 ‡ Al2 ˆ Y …2X6iH † lÀ l À l ‡ l3 À ‰A2 l5 …1 À r À ih†Š ˆ Y 2 …2X6jH † Al3 …r ‡ ih† À Al3 ‡ Al3 …1 À r À ih† ˆ Y …2X6kH † ‰Al4 …r ‡ ih†Š ‡ Al2 À Al2 ˆ …2X6lH † and define three pairs of unitarity triangles, in total: · (2.6hH ) and (2.6kH ) 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 will require very high precision to prove experimentally that they are not degenerate to a line · (2.6iH ) and (2.6lH ) have two sides of length Al2 and one much shorter of order Al4 This limits the small angles, which are f2 ‡ f6 and f2 À f6 , respectively They are close to the differences of angles in the large triangles g À gH ˆ bH À b ˆ f2 À f4 · (2.6gH ) and (2.6jH ) have two sides of length l and one very much shorter of order A2 l5, with a small angle f4 À f6 and f4 ‡ f6 , respectively Both are of order l4 Tiny differences between the two standard unitarity triangles are o…l2 † corrections, Al3 …1 À r À ih† ‡   ‡Al r ‡ ih À ‡ o…l7 † ‡ Al5  Al3 …r ‡ ih† ‡ À ÀAl3 ‡ Al3 …r ‡ ih† ˆ  À r À ih ‡ o…l7 † ÀAl3 ‡ Al5 …r ‡ ih† ‡ o…l7 † ‡ o…l7 † Y Al3 …1 À r À ih† ˆ ‡ o…l7 † ‡ …2X6hHH † …2X6kHH † Al5 …r ‡ ih† ‡ o…l7 † X The angles in these two triangles can be estimated from experimental constraints on a  unitary CKM matrix, leading to 95% CL limits [18] 25 a 125 Y 11 b 35 Y 40 g 145 X All phase angles are only weakly constrained by these limits, and one of the aims of experiments 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 extensions to the Standard Model may imply that the two triangles are dissimilar, or even that they 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 within the 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 that some phases in the CKM matrix are not observables either, and there remains some arbitrariness in the parametrization for this matrix The freedom to choose quark phases may be 716 R Waldi, Prospects for the Measurement of the CP Asymmetry "c Y f "t Y f "d Y f "s Y f "b With the new quark extended to antiquarks, with six more phases f"u Y f states " qHj ˆ eifj qY q"Hj ˆ eifj q"j Y j ˆ uY cY tY dY sY b also the phase induced by the CP operation is changed The transition H CP jqj i ˆ eifCP j j" qj i CP jqHj i ˆ eifCP j j" qHj i requires "j X fHCP j ˆ fCP j ‡ fj À f This equation leaves fHCP j still completely undefined, since all three phases on the righthand side are not observable, and therefore subject to arbitrary changes It becomes meaningful, however, if it is applied to observables, like CP eigenvalues Two CP eigenstates constructed from a meson and anti-meson state with eigenvalues Ỉ1 are related accordingly: à H "  jqj q"k i Ỉ eifCP jk jqk q"j i ˆ i…fj ‡fk † jqHj q"Hk i Ỉ eifCP jk jqHk q"Hj i X H The new states jqHj q"Hk i Æ eifCP jk jqHk q"Hj i have the same eigenvalues, and differ by an overall unobservable phase from the old ones " is The CP operation on a meson, e.g the pseudoscalar B0 meson jbdi, CP jB0 i ˆ eifCP B jB"0 i Y …2X9† where the phase factor eifCP B ˆ hB"0 j CP jB0 i depends on the parity of the bound-state wave function, and the chosen quark and antiquark phase convention It is thus an unobservable, arbitrary phase Quark phase changes can in principle be compensated by phase changes of the CKM matrix elements according to (2.2), leaving terms like hqj j Vjk jqk i invariant However, this is not a physical requirement, and in fact the CP transformed qj j V *jk j" qk i Y eifCP kj h" …2X10† 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 be made independent of the choice of quark phases The appearance of an additional phase factor in (2.10) can be avoided by the restriction "j ˆ Àfj for quark phase changes, and an appropriate phase convention which makes f terms related by a CPT transformation relatively real If a choice of phases is possible where all CKM matrix elements can be made real, also charged current weak interactions would not violate CP symmetry Phase conventions will also enter into relations among decay amplitudes An amplitude for a weak decay B0 X via a single well defined process can be written as A ˆ hXj H jB0 i ˆ hXj OV jB0 i Y …2X11† 840 R Waldi, Prospects for the Measurement of the CP Asymmetry 5.3 B0 p +p ± Reconstruction at BABAR The reconstruction of B0 p‡ pÀ at the U (4S) resonance has been exercised by the CLEO collaboration, who have first observed evidence for this and/or the K ‡ pÀ final state [160] They had an acceptance of 38% in their first analysis, which included the effect of hard cuts to suppress continuum background With an increased sample, they could raise this number to 44% [164] This gives a lower limit on the achievable acceptance at BABAR that is safer than the estimates at hadron experiments which are based solely on Monte Carlo simulations with large systematic uncertainties The data sample of BABAR after one nominal year will be 10 times larger than the data sample used in [164], which allows a worse signal to background ratio, and hence much softer cuts with negligible signal loss With a vertex resolution of sz % 40 mm [132, 174], the continuum background will decrease rapidly at those time differences between the apparent signal and tag, which contribute to the amplitude measurement of the sin xT % sin 370Dzmm asymmetry This will result in an efficiency of 60 to 70%: Besides the geometrical acceptance for the two tracks, this contains about 5% losses in the tails of the vertex fit probability (significance level cut), the particle identification quality, the energy and the mass distribution In contrast to a hadron collision experiment, effectively no losses are imposed by the trigger 5.3.1 Backgrounds The mass resolution is 28 MeV (see figure 5.9b), so background from three pion final states are well separated in mass Similarly, the energy in the U (4S) rest frame must be half the U (4S) mass, and provides a powerful tool to separate backgrounds with missing particles Fitting the mass with this energy constraint gives a better signal enhancement over combinatoric background The only harmful background from B decays is the penguin channel K ‡ pÀ , which has to be suppressed via the particle identification systems The ionization measurements alone provide Kp separation at about the 2s level (see figure 5.10) Most of the pions from B0 p‡ pÀ fall in the barrel region, in which the DIRC is providing the hadron identification The number of photoelectrons observed by the DIRC for pions from this source is $ 40 at the backward end, reducing to about $ 20 at cos qlab ˆ 0X0, and increasing again to Fig 5.10: The Kap separation for pions from B0 p‡ pÀ as a function of cos qCMS , from the DIRC and from dEadx Fortschr Phys 47 (1999) 7±±8 841 $ 50 at the forward end of the DIRC The momentum of the forward pions is larger, which makes the Kap separation more difficult Figure 5.10 shows the number of standard deviations of Kap separation for pions from this decay mode as a function of cos qCMS It drops from complete separation for backward pions to a separation of about 4.5 standard deviations at cos qCMS ˆ 0X0 It is then constant for forward angles up to cos qCMS ˆ 0X7, which corresponds to the forward edge of the DIRC The Kap separation from dEadx is also shown for reference, and is the only tool for background suppression in the endcap For B0 p‡ pÀ events with both pions in the acceptance of the tracking system, the fraction of pions which miss the DIRC is about 11% in total, 4% due to cracks between the quartz bars in azimuth and about 7% in the forward endcap region These tracks will be identified by dEadx alone, leaving some background from the Kp channel All of the B0 p‡ pÀ events with a pion in the backward endcap region have their other pions striking the B1 magnet, so there is no acceptance lost by not having high momentum hadron identification there Another background is from random combinations of pions, both from BB" and from q" q continuum events Here, compared to the CLEO analysis, the reconstruction of the B0 vertex will help to reduce this combinatoric background The amount is further reduced by the presence of a high quality flavour tag, e.g a fast lepton, thereby reducing its actual influence on the CP asymmetry fit The behaviour of the background can be studied by a careful investigation of the sidebands in the B mass distribution, taking into account the satellite peaks from multibody B decays in this region 5.3.2 The Lifetime Measurement Since the B production vertex …x0 Y y0 Y z0 † is not known, we can not determine the signal B lifetime ts ˆ zbs Àzg0 and the tag-B lifetime tt ˆ zbt Àzg0 separately, in order to calculate the relezs s zt t vant time difference Dt ˆ ts À tt The easiest approximation is to use bzt % bzs % b, the boost of the U (4S), and Dt ˆ zs À zt X bg This infers an rms error on the lifetime difference of 0X22 ps, taking into account the angular distribution of the pseudoscalar B mesons shown in figure 5.9a (a flat distribution would increase the width to 0X30 ps) On the other hand the signal B is fully reconstructed, hence we know its full momentum vector This can be exploited, assuming the first decaying B to have lived its average lifetime, which is ta2, to estimate the location of z0 This yields the following recipe where vs XXˆ bzs gs ˆ pz amB of the p‡ pÀ combination, v Xˆ bg, and rm XXˆ m(U (4S))am…B0 † % 2X00: V zs À zt rm v À 2vs b b ‡t zs ! zt Y ` vs 2vs Dt ˆ zs À zt rm v À 2vs b b zs zt X ‡t X 2v À vs 2…rm v À vs † The resolution is improved considerably, with an rms of 0X09 ps and essentially no bias The vertex resolution for zs is around 35 mm, with non-Gaussian tails of the distribution that make the rms value as big as 55 mm The main uncertainty in Dt, however, comes from the resolution on the tag vertex coordinate zt A detailed Monte Carlo study [174] uses a simple strategy: all tracks are fit to a common vertex, which can be improved if 842 R Waldi, Prospects for the Measurement of the CP Asymmetry exclusive final states from charmed mesons can be reconstructed The common vertex has a correctable bias of a few mm due to the cascade charmed particle decay, and a resolution ranging from 100 mm to 200 mm This corresponds to an error in the scaled lifetime difference s T ˆ 0X3 F F F 0X5 which is still substantially smaller than half an oscillation period of pax % 4X5 Fortunately, the better resolutions are achieved in events which also have a good flavour tag, hence they contribute more to the overall fit than the events with less precise vertex information Therefore, the effective dilution factor Dr from the time measurement is expected to be around 0X95 5.4 B0 Jaw K 0S The product branching fraction for the second gold-plated channel is f…B0 Jaw KS0 † Á f…Jaw l‡ lÀ † ˆ …2X9 Ỉ 0X2† Á 10À5 from the average of ARGUS and CLEO measurements of charged and neutral B decays to JawK (see table 5.9) A factor two applies if one uses as leptons l ˆ m and e, adding up to …5X8 Ỉ 0X4† Á 10À5 Reconstructing only KS0 p‡ pÀ results in a fraction of …2X0 Ỉ 0X2† Á 10À5 for one type of lepton Amazingly, this is the same order of magnitude as the p‡ pÀ branching fraction, since secondary decays are required for the Jaw reconstruction If the assumption of equal decay fractions for both B mesons is dropped, the number of reconstructable events has to be estimated from the B0 results alone, which yield f…B0 Jaw KS0 † Á f…Jaw l‡ lÀ † ˆ …2X5 Ỉ 0X4† Á 10À5 X A bias due to unequal production rates will cancel in the predicted rate at the U (4S), but not for other B production mechanisms Table 5.9 Product f…B0 Jaw KS0 † Á f…Jaw l‡ lÀ †, where l = e or m, assuming f0 X f‡ ˆ 0X50 X 0X50 and f…KS0 p‡ pÀ † ˆ 68X67 experiment n…l‡ lÀ p‡ pÀ † E nU …4S† a103 f Á f Á 105 ARGUS [175] CLEO [176] CLEO II [158] 45X5 Ỉ 7X3 6X6 192 Ỉ 10 240 3360 2.9 Ỉ 2.0 2.1 Ỉ 1.2 2X53 Ỉ 0X41 0X37 average 0.26 0.44 0.39 2.50 Ỉ 0X37 ‡ ‡ avg from B Jaw K avg assuming isospin symmetry 2.98 Ỉ 0.22 2.86 Æ 0.19 The knowledge on the branching fraction f…Jaw l‡ lÀ † is irrelevant, since the experimental results are already the product branching fractions If one takes f…Jaw l‡ lÀ † ˆ …6X02 Ỉ 0X19†% [9], the best estimate corresponds to f…B0 Jaw K † ˆ …9X9 Ỉ 0X7† Á 10À4 5.5 B0 Ja K0S Reconstruction at LHB The Jaw KS0 decay can be observed in two final states: m‡ mÀ p‡ pÀ and e‡ eÀ p‡ pÀ The first one is reconstructed by 843 Fortschr Phys 47 (1999) 7±±8 · requiring a m‡ mÀ pair identified in the muon chambers from a reconstructed vertex (the B vertex) with the Ja w mass, · a KS0 p‡ pÀ from an isolated secondary vertex with the K mass, · and subsequently calculating the Jaw KS0 invariant mass using the momenta from mass constraint fits to the Jaw and KS0 The high mass of the B0 meson is exploited in transverse momentum cuts max fpc …l‡ †Y pc …lÀ †g b 1X2 GeVac Y pc …KS0 † b GeVac X A reconstructed, isolated vertex is required for both the KS0 and the Jaw Within the vertex detector we use a distance cut of 12 mm to define a common vertex, i.e both tracks must match in space within this distance, and no other track may come as close The vertex must be within the range mm ` Dvz …Jaw† ` 150 mm Y vz …KS0 † ` 17 m X The cut on the B0 decay vertex implies a cut on the minimum lifetime and therefore increases the sensitivity to L (4.17) to S0 ˆ 0X68 All cuts are summarized in table 5.10, which gives also an estimate on the error on sin 2b after one year of data taking Table 5.10 Number of events, acceptance factors and measurement errors for the determination of b using B0 Jaw KS0 , after one year of data taking N bb" B0 ‡ B"0 f…B0 Jw KS0 m‡ mÀ p‡ pÀ † reduction/step Á 1010 Á 109 160 000 Á 0X4 Á 10À5 l ‡ K tag 28 800 …l† 57 600 …K† 0X002 ` ql ` 0X075 jpl j b 10 GeVac max pc …l† b 1X2 GeVac 20 700 41 500 72% 500 17 000 41% B vertex found mm ` Dvz …B† ` 150 mm Jaw mass cut …Ỉ120MeV† d0B ` 80 mm pB ` TeVac B mass cut …Ỉ80 MeV† 200 10 400 61% track reconstruction factorized trigger eff 600 600 KS0 reconstruction 18% …l† +36% …K† a Dt (tagging dilution) Dm ˆ À 2c s…sin 2b† ˆ 1aDtot S0 s…sin 2b† a 90% 0.70 p N tracks match within 10 mm 0.045 0.034 0.052 77% 0.78 83% 0.45 S0 ˆ 0X68 844 R Waldi, Prospects for the Measurement of the CP Asymmetry Fig 5.11: Invariant mass of reconstructed B0 Jaw KS0 m‡ mÀ p‡ pÀ The narrow peak with s ˆ 30 MeVac2 is obtained after mass constraint fits of the Jaw and KS0 momenta, while the thin line gives the measurement result without mass constraints A detector resolution of s…1apz † ˆ Á 10À4 a GeV, s t—n qx ˆ st—n qy ˆ Á 10À4 is used Multiple scattering is assumed to contribute with sp ap ˆ 0X3% and s t—n qx ˆ s t—n qy ˆ Á 10À3 ap The mass resolution obtained in our simulation is s…mJaw † ˆ 58 MeVac2 and s…mKS0 † ˆ 12 MeVac2 After cutting at Ỉ2s, we apply a mass constraint fit to improve the measurement of their momenta, resulting in a significant improvement in B mass resolution to s ˆ 30 MeVac2 (see figure 5.11) Background to the Jaw KS0 signal from three sources have been investigated [172]: · Other B or Bs decays They can be separated kinematically, as is shown 0in figure 5.12 for the most important channels A small contribution from Bs Jaw K * l‡ lÀ p‡ pÀ p0 remains in the peak region · Random combinations of Jaw and KS0 If they originate from the process gg gJaw…X†, which has a cross section s…gg g…c" c† Jaw X† F 0X4 mb, most are rejected by the cut Dvz b mm, and the requirement of a tag lepton or kaon The only remaining back- Fig 5.12: Invariant mass of reconstructed B0 Jaw KS0 m‡ mÀ p‡ pÀ compared to background from B0 Jaw K * m‡ mÀ p‡ pÀ p0 , where the p0 is ignored, taking the product branching fraction of 5X8 Á 10À5 from the present average of ARGUS and CLEO measurements (a) The small signal from Bs Jaw KS0 in (a) is hardly visible, while Bs Jaw K * m‡ mÀ p‡ pÀ p0 , where the p0 is ignored, leaves some background under the peak (b) Fortschr Phys 47 (1999) 7±±8 845 ground is from interactions in the silicon detector, if they have not been identified as such More serious background comes from Jaw mesons, which originate from bb" events This background is expected to be less than 4% of the signal in the 160 MeV mass interval around the B0 peak · Random combinations of real and fake leptons, forming accidental vertices If they fall inside our Jaw mass window, and are combined with random KS0 mesons in the event, they yield a flat distribution in the B mass region Their contribution is estimated to be negligible The second channel is often taken as doubling the statistics This is, however, not true due to the photon radiation by the electron and positron, which leaves a significant tail on the left side of the Jaw mass peak, and has therefore a reduced efficiency for the mass cut The exact losses depend on the ability of the reconstruction program to detect and correct those losses A rough estimate gives a reduction by less than 20% 5.6 B0 Jaw K 0S Reconstruction at BABAR The Jaw KS0 decay can be observed in four final states: m‡ mÀ p‡ pÀ , m‡ mÀ p0 p0 , e‡ eÀ p‡ pÀ , and e‡ eÀ p0 p0 For the reconstruction of m‡ mÀ p‡ pÀ , the angular limits set by the position of the closest beam-separation magnets of PEP II correspond to a geometric acceptance of about 65% for observing all four charged tracks of m‡ mÀ p‡ pÀ The cut around the Jaw mass can safely be made wide enough to lose at most 1% of the muon pairs Due to the good mass resolution of 13 MeVac2 only a weak lepton identification is required, with 98% efficiency [177] KS0 p‡ pÀ reconstruction can proceed similarly, combining two oppositely charged tracks Due to the high resolution vertex detector, a soft cut on an isolated secondary vertex rejecting less than 5% of the real events is expected to reduce combinatoric background to a negligible level The signal vertex is determined from the lepton pair The resolution is around 42 mm, with tails making the effective rms 65 mm [174] This is below the tag-vertex uncertainty Since events close to Dt ˆ not contribute to the asymmetry, a modest cut on jDtj can help to further suppress background The corresponding reduction in efficiency is compensated by the increase in the statistical sensitivity S0 (solid line in figure 4.26), therefore this cut is not included in the acceptance calculation The estimate from the Technical Design Report [132] for the total efficiency is 59% For a more conservative estimate, the comparison with the CLEO experiment achieving 40% efficiency for muon and electron pairs [158] leads to a value around 50% for the muon pair channel at BABAR The only background is from wrong combinations, e.g with a Jaw from one B and a KS0 from the other, or from continuum events Monte Carlo studies show, that these backgrounds not exceed 6% of the signal This fact is supported by the background-free small event samples observed by ARGUS and CLEO Again, the electron channels will suffer from the radiative tail in the Jaw candidate mass distribution Here, some radiation losses may be recovered by detecting the photon in the calorimeter, and associate it with the radiating electron The channels with KS0 p0 p0 4g require harder cuts on the four photons, making use of the two pion masses, the KS0 mass, and the geometry, assuming a common origin and taking into account the KS0 lifetime [177] The overall efficiency is about 60% of that for the charged KS0 decay Combinatoric background has been found to be again below 6% 5.7 Present Experimental Information Although already samples from ten to about 200 tagged B0 Jaw KS0 events have been reconstructed [178, 179], the available information on their CP asymmetry or sin 2b is still 846 R Waldi, Prospects for the Measurement of the CP Asymmetry poor A first hint on its value comes from CDF [179], who find an asymmetry amplitude sin 2b ˆ 1X8 Ỉ 1X1 Ỉ 0X3 in about 200 reconstructed B0 JawKS0 The unphysical value of an asymmetry parameter reflects the systematic uncertainty on the tagging dilution, and demonstrates the importance of this number for a quantitative result on the CKM phase angles However, the sign of this number is unaffected by this scale uncertainty and indicates agreement with the sign predicted by the Standard Model 5.8 Comparison of Future Experiments The calculation of performance figures requires knowledge on the number of B mesons produced at a given site This is a number with a pronounced uncertainty for any proposed machine, mostly due to unproven machine parameters for e‡ eÀ colliders, and due to poorly known cross sections for hadron machines The estimates of the precision on the five experiments BABAR, HERA-B, LHB, CDF, and LHCb are based on the luminosities and cross sections given in table 3.3, with all their uncertainty discussed already there 5.8.1 B0 Jaw K 0S and the Determination of b The performance on the Jaw KS0 m‡ mÀ p‡ pÀ final state after the first year of operation of these experiments is estimated in table 5.11 The product branching fraction for this state is …2X0 Ỉ 0X2† Á 10À5 (table 5.9) Since the efficiencies for trigger, signal selection and tagging are not independent, usually a total efficiency e including all these effects is given in the table In this case, et is taken as in the effective tagging dilution The efficiencies claimed by different experiments vary widely in the detail of their detector simulation, and in the systematic uncertainty involved in predicting detector responses The numbers can therefore only be taken as a guideline to compare orders of magnitude of what can be achieved Systematic variations are also to be expected from the Monte Carlo models used for B decays and background event generation The e‡ eÀ B factory is represented by the more conservative PEP-II design with head-on collisions Within expected improvements in machine design, by the start of the second generation CP experiments LHCb and/or BTEV, a factor increase of luminosity will probably be reached, and the accumulated data per experiment should correspond roughly to 10 times the given first-year number BELLE/KEKB claims a factor higher luminosity already at startup time Most experiments use the channel Jaw e‡ eÀ to increase the number of events by a varying amount of 10±±80% corresponding to the lower acceptance, both from lost electrons due to bremsstrahlung in the detector material, and ±± at hadron beam experiments ±± due to different trigger efficiency If kinks can be detected close to the vertex, or if photons can be associated with the radiating electron to correct its momentum, a large efficiency is obtained Also, low combinatoric background allows wider cuts in the Jaw mass On the other hand, if much material is transversed before the spectrometer part of the detector, many electrons will be registered with only a fraction of their original energy, and the reconstructed e‡ eÀ mass will be too far away from the Jaw mass to be accepted At the e‡ eÀ colliders, further gain in precision comes from KS0 p0 p0 and the investigation of channels from a Jaw KL0 (with opposite sign in the asymmetry and higher background) These channels are also included in table 5.11 Systematic errors have to be added to the statistical ones given in table 5.11 Besides detector specific uncertainties, there is the error from the determination of the dilution factors Since always many more events are available from control channels that may be used for the dilution determination, this error is proportional to sin 2b and always much smaller 847 Fortschr Phys 47 (1999) 7±±8 Table 5.11 Number of Jaw KS0 m‡ mÀ p‡ pÀ events in one year and estimated precision on sin 2b for some of the experiments being presently constructed The systematic error from the determination of the total dilution D and the intrinsic asymmetry I is not included e‡ eÀ U (4S) BABAR p targ 820 GeV HERA-B [112] p targ TeV LHB p" p TeV CDF [113] pp 14 TeV LHCb [111] B0 ‡ B"0 Á 107 Âf…B0 m‡ mÀ p‡ pÀ † 600 2X8 Á 108 5600 Á 109 1X6 Á 105 Á 1010 1X6 Á 106 Á 1011 1X2 Á 107 e trigX ‡ recoX …‡ tag† 0.50 300 0.17 950 0.065 10 000 0.004 6500 0.011 130 000 et Á …Dt Á Dm †2 D2r (time resolution) Dc (background) 0.35 0.90 1.00 0.10 1.00 1.00 0.19 1.00 1.00 0.07 1.00 0.7 0.065 1.00 0.77 S0 s…sin 2b† 0.58 0.18 0.66 0.16 0.68 0.034 0.58 0.10 0.60 0.020 ‡Jaw e‡ eÀ ‡KS0 p0 p0 ‡250 ‡330 ‡500 ‡8000 0 ‡7000 ‡KL0 Dc (background) ‡300 0.80 0 0 s…sin 2b† 0.09 0.11 0.025 0.10 0.019 than the statistical error Similar arguments are true for the intrinsic asymmetry which occurs only at hadron beam experiments Any detector at an e‡ eÀ collider on the U (4S) resonance will trigger every BB" event, " Obvious candiand allows to investigate more channels with the same quark content c" cd d ‡ ‡ À dates are D‡ DÀ , or states with mixtures of amplitudes as D* DÀ or D* D* The angle H b which is close to b in the Standard Model can be investigated with KS0 p0 or KS0 hH, provided these decays proceed predominantly via b s loop diagrams All these additional channels have different systematics, and will therefore be valuable tools not only to improve the statistical accuracy on sin 2b, but also as consistency checks for the Standard Model description of CP violation 5.8.2 B0 p +p ± and the Determination of a The B0 p‡ pÀ events can be triggered and reconstructed by all experiments Background contributes mainly at Dt ˆ at the U (4S) or t ˆ at hadron beam experiments, and dies out more rapidly than the signal Since almost no information on the CP asymmetry comes from the region near T ˆ 0, its effect on the error in L0 is more realistically estimated when a cut on jTj is employed The p‡ pÀ asymmetry can be used to determine the angle a However, as discussed in section 2.3.2.5, additional input is required due to a sizeable penguin amplitude in this decay [86] This is either theoretical information on the ratio of the penguin and tree amplitudes, or only experimental information on the branching fractions to pỈ p0 and flavourtagged p0 p0 [90], which will be provided by BABAR, BELLE and CLEO on a similar time scale For the latter rates, which are expected to be low in the Standard Model, even upper limits on their sum help to considerably reduce the error on a [180] The first option 848 R Waldi, Prospects for the Measurement of the CP Asymmetry is the one accessible to hadron beam experiments without further input from e‡ eÀ colliders Both approaches require a two-parameter fit to the time dependent asymmetry (2.57) leading to results on the asymmetry amplitudes L0 and Q0 The additional systematic error from the theoretical input required at hadron beam experiments can be reduced in a combined analysis of pp and Kp final states [181] An estimate of the best expected performance are the precisions of a one parameter fit, which are dL0 % 0X1 from the study of the two, three and four pion final states after one year of BABAR operation It will improve to dL0 % 0X04 for the analysis of one year's p‡ pÀ data of LHCb using two fit parameters Accuracies beyond these orders of magnitude cannot be predicted The final precision that will be reached in a combined analysis of many final states depends not only on experimental precision and their mostly unknown branching fractions, but also on the values of these yet unknown parameters Outlook The measurement of CP violation in the B meson system is a powerful test of the Standard Model which could open windows to many extensions Of utmost importance on the way to this aim is the redundant information on the unitarity triangles which becomes available through new measurements of phase angles and absolute values of CKM matrix elements Thereby, the triangles will become overconstrained, and inconsistencies may show up There are many possible origins that can lead to inconsistent results within the Standard Model description: · Four families of quarks lead to violation of the  CKM matrix unitarity conditions · Additional Higgs boson multiplets are likely to induce CP violation by themselves, which are probably not compatible with the Standard model expectations · Couplings to supersymmetric particles or new heavy flavour changing bosons may add new parameters, including new mixings which cannot be absorbed in the Standard Model parametrization These are just three of the most obvious extensions to the Standard Model If the unextended present Standard Model is sufficient to explain CP violation, we will see consistency checks and first significant numbers on the phase angles in the unitarity triangle within the next few years, almost certainly at e‡ eÀ collider B factories, but also from hadron beam experiments at Tevatron and from HERA-B These experiments will reach a combined precision in the few 0X01 range for the unitarity angle b, some larger error on a and probably first rough estimates of g Improved data on the triangle sides will result from better understanding of the theoretical uncertainties, and from extended acceptance ranges in high statistics experiments due to more powerful background separation Further constraints will come from the rare decays K ‡ p‡ n" n, KL0 p0 n" n [182], but also from different measurements at B factories: the measurement of f…B‡ t‡ n† gives a value of fB Vub , and Bs aB"s oscillation determines Vtd aVts These results, if consistent with the Standard Model or not, will unavoidably call for higher precision on the CP violation parameters This will only be available from second generation experiments, which can exploit the high B meson rate at hadron beams with optimized detectors BTEV at Tevatron could be one candidate, but a desirable continuation of the B factory program will be a dedicated detector LHCb at the LHC collider, which will provide accuracy below the 0X01 level after several years of running Acknowledgements My understanding of heavy flavour physics and BaB" mixing has evolved during my long stay with the ARGUS collaboration, and I owe thanks to many colleagues there for having improved my knowledge Fortschr Phys 47 (1999) 7±±8 849 in theory, data analysis and experimental techniques During my work for LHB and BABAR, I have profited greatly from the common work with hundreds of physicists in these collaborations all around the world I especially enjoyed a few months stay at Pisa with Giovanni Carboni, where most of the LHB related work was done Many ideas written in this paper have been evolved or clarified in very fruitful discussions with other colleagues, among them Giovanni Carboni, Nando Ferroni, Robert Fleischer, Joachim Graf, Frank Krauss, Christof Kreuter, Francois Le Diberder, Ralph MuÈller, Dominic PoÈtschke, Helen Quinn, Klaus R Schubert, Bernhard Spaan, JoÈrg Urban, and Stefan Werner Last I want to express special thanks to my mentor Klaus Schubert, who is a recognized expert in the field and has contributed to this work with many ideas, stimulating questions and continuous 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