Prospects for B-Physics in the Next Decade

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arXiv:hep-ph/9610305v2 5 Nov 1996 HEPSY 96-01 October 1996 PROSPECTS FOR B-PHYSICS IN THE NEXT DECADE Sheldon Stone Department of Physics Syracuse Univeristy Syracuse, N.Y. 13244-1130 Email: Stone@suhep.phy.syr.edu ABSTRACT In these lectures I review what has been learned from studies of b-quark decays, including semileptonic decays (Vub and Vcb), Bo−B o mixing and rare B decays. Then a discussion on CP violation follows, which leads to a summary of plans for future experiments and what is expected to be learned from them. . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . Presented at NATO Advanced Study Institute on Techniques and Concepts of High Energy Physics, Virgin Islands, July 1996 1 1. INTRODUCTION My assignment is to discuss “Future B Physics Experiments.” But to understand what results we desire, it is necessary to understand past accomplishments and have a ﬁrm theoretical background. In this paper I will give a brief theoretical introduction to the “Standard Model,” and historical introduction to the study of b quark decays. Then I will discuss in some detail the physics already found including: B lifetimes, semileptonic B decays and the CKM couplings Vcb and Vub, Bo −¯ Bo mixing, rare b decays, and CP violation in Ko L decays. Following this is a pedantic discussion on CP violation in B decays, which leads into a discussion of future experiments. 1.1. Theoretical Background The physical states of the “Standard Model” are comprised of left-handed doublets containing leptons and quarks and right handed singlets1 u d ! L c s ! L t b ! L , uR, dR, cR, sR, tR, bR (1) e− νe ! L µ− νµ ! L τ − ντ ! L , e− R, µ− R, τ − R , νeR, νµR, ντ R. (2) The gauge bosons, W ±, γ and Zo couple to mixtures of the physical d, s and b states. This mixing is described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix (see below).2 The Lagrangian for charged current weak decays is Lcc = −g √ 2Jµ ccW † µ + h.c., (3) where Jµ cc = (¯νe, ¯νµ, ¯ντ) γµ    eL µL τL   + (¯uL, ¯cL, ¯tL) γµVCKM    dL sL bL    (4) and VCKM =    Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb   . (5) Multiplying the mass eigenstates (d, s, b) by the CKM matrix leads to the weak eigenstates (d′, s′, b′). There are nine complex CKM elements. These 18 numbers can be reduced to four independent quantities by applying unitarity constraints and the fact that the phases of the quark wave functions are arbitrary. These four remaining numbers are fundamental constants of nature that need to be determined from 2 experiment, like any other fundamental constant such as α or G. In the Wolfenstein approximation∗the matrix is written as3 VCKM =    1 −λ2/2 λ Aλ3(ρ −iη) −λ 1 −λ2/2 Aλ2 Aλ3(1 −ρ −iη) −Aλ2 1    (6) The constants λ and A are determined from charged-current weak decays. To see how this is done, ﬁrst consider muon decay. The muon decays weakly into νµe−¯νe as shown in Fig. 1. The decay width is given by4 Γµ = G2 F 192π3m5 µ × (radiative corrections). (7) µ− ν ν _ e- - W 1 e µ Fig. 1. Diagram for muon decay. The couplings at the vertices are unity for leptons. This process serves to measure the weak interaction decay constant (Fermi constant) GF. − ν _ e- - W s u u u− { } K- πo Vus Fig. 2. Semileptonic K−decay diagram. A charged current decay diagram for strange quark decay is shown in Fig. 2. Here the CKM element Vus is present. The decay rate is given by a formula similar to equation (7), with the muon mass replaced by the s-quark mass and an additional factor of |Vus|2. Two complications arise since we are now measuring a decay process involving hadrons, K−→πoe−¯ν rather than elementary constituents. One is that the ∗In higher order other terms have an imaginary part; in particular the Vcd term becomes −λ − A2λ5(ρ + iη), which is important for CP violation in Ko L decay. 3 s-quark mass is not well deﬁned and the other is that we must make corrections for the probability that the ¯u-spectator-quark indeed forms a πo with the u-quark from the s-quark decay. These considerations will be discussed in greater detail in the semileptonic B decays section. For now5 remember that λ = Vus = 0.2205 ± 0.0018 and, A ≈0.8. Constraints on ρ and η are found from other measurements. These will also be discussed later. 1.2. B Decay Mechanisms Fig. 3 shows sample diagrams for B decays. Semileptonic decays are shown in Fig. 3(a). The name “semileptonic” is given, since there are both hadrons and leptons in the ﬁnal state. The leptons arise from the virtual W −, while the hadrons come from the coupling of the spectator anti-quark with either the c or u quark from the b quark decay. Note that the B is massive enough that all three lepton species can be produced. The simple spectator diagram for hadronic decays (Fig. 3(b)) occurs when the virtual W −materializes as a quark-antiquark pair, rather than a lepton pair. The terminology simple spectator comes from viewing the decay of the b quark, while ignoring the presence of the spectator antiquark. If the colors of the quarks from the virtual W −are the same as the initial b quark, then the color suppressed diagram, Fig. 3(c), can occur. While the amount of color suppression is not well understood, a good ﬁrst order guess is that these modes are suppressed in amplitude by the color factor 1/3 and thus in rate by 1/9, with respect to the non-color suppressed spectator diagram. The annihilation diagram shown in Fig. 3(d) occurs when the b quark and spec- tator anti-quark ﬁnd themselves in the same space-time region and annihilate by coupling to a virtual W −. The probability of such a wave function overlap between the b and ¯u-quarks is proportional to a numerical factor called fB. The decay ampli- tude is also proportional to the coupling Vub. The mixing and penguin diagrams will be discussed later. 2. What is known 2.1. Early history The ﬁrst experimental evidence for b quarks was found at Fermilab by looking at high mass dimuon pairs in 800 GeV proton interactions on nuclear targets.6 Their results are shown in Fig. 4 along with subsequent data from DESY using e+e−anni- hilations which shows narrow peaks at the masses of the Υ and Υ′ resonances.7 The natural width of the peaks is narrower than the energy resolution of either experiment leading to the interpretation that these states are comprised of a bound b¯b quark system. The narrow decay width is similar to the situation in charmonium, 4 b W- q c or u q e, µ, ν b W- q c or u q τ u c d s , a) semileptonic b) hadronic: simple spectator b W- q c or u q u c d s , c) hadronic: color suppressed b W- u , u, c - , d, s ν d) annihilation b d d b W- W- u,c,t u,c,t e) box: mixing b W- s,d γ t,c,u ,g f) Penguin Fig. 3. Various mechanisms for B meson decay. 5 Fig. 4. The data on top is the µ+µ−invariant mass from the Columbia-Fermilab-Stony Brook collaboration and the data shown below is the total e+e−cross-section from the DESY-Heidelberg- Hamburg-Munchen collaboration. i.e. the decay width is proportional to the strong coupling constant α3 s. As the DESY machine was limited in center-of-mass energy at that time, the torch was passed to the CLEO experiment at the CESR e+e−storage ring. An early total cross-section scan is shown in Fig. 5(a). A new narrow state, the Υ′′ (or Υ(3S)), appears along with a state wider than the experimental resolution, the Υ(4S). Fig. 5. Hadronic cross-section scan in the Upsilon region, (a) shows 1S-4S and (b) region above 4S. The mechanism of b quark production in e+e−collisions and the subsequent pro- duction of the ﬁnal states B+B−and Bo ¯Bo from the Υ(4S) are shown in Fig. 6. Subsequent data shown in Fig. 5(b) shows that the cross-section is ≈1 nb and details structures in the total cross-section at higher energies.8 Little data has been taken 6 above the Υ(4S), however. e e b γ - + bu or d u or d B B or B B - + o o Fig. 6. B production mechanism at the Υ(4S). Many properties of B meson decays have been discovered by two e+e−experi- ments operating at the Υ(4S) resonance, CLEO at CESR and ARGUS at DESY (the DESY machine group upgraded the energy so they could do this physics). Fully re- constructed B meson decays were ﬁrst seen by CLEO and the B masses determined.9 Now there are several thousand fully reconstructed decays in many modes allowing for branching ratio determinations. A diﬀerent technique is used to reconstruct B mesons at the Υ(4S) than at other machines. At this resonance we have e+e−→Υ(4S) →B−B+ (8) →BoB o . (9) From energy conservation, the energy of each B is equal to the beam energy, Ebeam (the center-of-mass energy is twice Ebeam). To reconstruct exclusive B meson decays, we ﬁrst require the energy of the decay products be consistent with the beam energy. Suppose the ﬁnal state we are considering is Doπ−. We require that EDo + Eπ−= Ebeam. (10) In practice this means that the diﬀerence between the left-hand side and the right- hand side is less than ≈3 times the error on the measured energy sum. The next step is to compute the invariant mass of the candidate B−using the well known beam energy: mB = q E2 beam −(−→pDo + −→pπ−)2. (11) In practice this technique leads to large background rejections and a B mass resolution of σ ≈2.5 MeV (at CESR) which is due mostly to the energy spread of the beam. A few sample B decay candidate mass plots are shown in Fig. 7 from the CLEO experiment.10 Hadronic production rates for b quarks have been measured at two p¯p colliders, UA1 at the SPS,15 and CDF at the Tevatron.16 E789 has also measured b production using an 800 GeV proton beam hitting nuclear targets.17 CDF has reconstructed B meson decays into modes containing a ψ meson. These are shown14 in Fig. 8. 7 Fig. 7. Beam constrained mass distribution from CLEO for (a) B−→Doπ−, (b) B−→Doρ−, (c) B o →D+π−and (d) B o →D+ρ−. 8 Fig. 8. Invariant mass spectra from CDF for ψK+ and ψK∗o candidates. 9 2.2. Lifetimes Lifetimes are a fundamental property of elementary particles. The b quark lifetime, however, was measured before the individual lifetimes of b ﬂavored hadrons at the higher energy e+e−machines, PEP and PETRA.11 More recent measurements have come from LEP, SLD and CDF.12 The meaning of b quark lifetime is really the average of the B hadron lifetimes over the kinds of B hadrons which happen to be produced in the particular environment. The results are summarized in Table 1.13 Table 1. B lifetime measurements (ps) LEP Avg CDF SLD World Avg b quark 1.54±0.02 1.51±0.03 1.56±0.05 1.53±0.02 B− 1.63±0.06 1.68±0.07 1.65±0.05 Bo 1.52±0.06 1.58±0.09 1.55±0.05 Bo s 1.60±0.10 1.36±0.12 1.50±0.08 Λb 1.21±0.07 1.32±0.17 1.23±0.06 Ξb 1.39+0.34 −0.28 1.39+0.34 −0.28 The meson lifetimes are nearly equal implying the dominance of the spectator diagram. The Λb lifetime appears to be shorter, which implies the existence of other diagrams in baryon decay. This is very diﬀerent from the situation in charm decay where the Do, D+ and Λc have diﬀerent lifetimes. 2.3. The CKM element Vcb 2.3.1. Theory of semileptonic decays The same type of semileptonic charged current decays used to ﬁnd Vus are used to ﬁnd Vcb and Vub. The basic diagram is shown in Fig. 3(a). We can use either inclusive decays, where we look only at the lepton and ignore the hadronic system at the lower vertex, or exclusive decays where we focus on a particular single hadron. Theory currently can predict either the inclusive decay rate, or the exclusive decay rate when there is only a single hadron in the ﬁnal state. The fraction of semileptonic decays into exclusive ﬁnal states containing either a pseudoscalar or vector meson is given in Table 2. Now let us brieﬂy go through the mathematical formalism of semileptonic decays. Let us start with pseudoscalar to pseudoscalar transitions. The decay amplitude is given by18 A( ¯B →me−¯ν) = GF √ 2VijLµHµ, where (12) Lµ = ¯ueγµ (1 −γ5) vν, and (13) 10 Table 2. Fraction of q →xℓν to 0−or 1−ﬁnal states s quark 100% K →πℓν c quark >90% D →(K + K∗)ℓν ? D →(π + ρ)ℓν b quark ≈66% B →(D + D∗)ℓν ? B →(π + ρ)ℓν t quark 0% t does not form hadrons Hµ = ⟨m|Jµ had(0)|B⟩= f+(q2)(P + p)µ + f−(q2)(P −p)µ, (14) where q2 is the four-momentum transfer squared between the B and the m, and P(p) are four-vectors of the B(m). Hµ is the most general form the hadronic matrix element can have. It is written in terms of the unknown f functions that are called “form-factors.” It turns out that the term multiplying the f−(q2) form-factor is the mass of lepton squared. Thus for electrons and muons (but not τ’s), the decay width is given by dΓsl dq2 = G2 F|Vij|2K3M2 B 24π2 |f+(q2)|2, where (15) K = MB 2 " 1 −m2 −q2 M2 B ! −4m2q2 M4 B #1/2 (16) is the momentum of the particle m (with mass m) in the B rest frame. In principle, dΓsl/dq2 can be measured over all q2. Thus the shape of f+(q2) can be determined experimentally. However, the normalization, f+(0) must be obtained from theory, for Vij to be measured. In other words, ΓSL ∝|Vij|2|f+(0)|2 1 τB Z K3g(q2)dq2, (17) where g(q2) = f+(q2)/f+(0). Measurements of semileptonic B decays give the integral term, while the lifetimes are measured separately, allowing the product |Vij|2|f+(0)|2 to be experimentally determined. For pseudoscalar to vector transistions there are three independent form-factors whose shapes and normalizations must be determined.19 2.3.2. B o →D+ℓ−¯ν CLEO has recently measured the branching ratio and form-factor for the reaction ¯Bo →D+ℓ−¯ν using two diﬀerent techniques.20 In the ﬁrst method the ﬁnal state is reconstructed ﬁnding only lepton and D+ candidates, where the D+ →K−π+π+ decay is used. Then, using the fact that the B′s produced at the Υ(4S) are nearly at rest the missing mass squared (MM2) is calculated as 11 MM2 = E2 ν −−→ Pν 2 (18) = (EB −ED+ −Eℓ)2 −(−→pB −−→pD+ −−→pℓ)2 ≈ (EB −(ED+ + Eℓ))2 −(−→pD+ + −→pℓ)2 ≈ E2 beam + m2 B + m2 D+ + m2 ℓ−2−→pD+ · −→pℓ, where E refers to particle energy, m to mass and −→p to three-momentum. The ap- proximation on the third line results from setting pB to zero. This approximation causes a widening of the MM2 distribution, giving a r.m.s. width of 0.2 GeV2. This analysis is done by ﬁnding the number of D+ events with opposite sign leptons in diﬀerent q2 and MM2 bins. The K−π+π+ mass distributions for the interval 4 > q2 > 2 GeV2 and several MM2 bins are shown in Fig. 9. There is also a large background from ¯Bo →D∗+Xℓ−¯ν decays where the D∗+ → πoD+. These events are reconstructed and their MM2 distribution is directly sub- tracted (after correcting for eﬃciencies) from the candidate signal distribution. We are left with a sample that contains D+ℓ−¯ν decays and also D+Xℓ−¯ν, where X can be a single hadron or hadrons but cannot be the result of ﬁnal state with a D∗+. We ascertain the total number of signal events by ﬁtting the MM2 distribution in the diﬀerent q2 bins as shown in Fig. 10 to a D+ℓ−¯ν signal shape and a background shape for D+Xℓ−¯ν. The second technique reconstructs the neutrino by using missing energy and mo- mentum measurements. Essentially all charged tracks and photons in the event are added up and since the total energy must be equal to the center of mass energy and the total three-momentum must be zero, any diﬀerence is assigned to the neutrino. Events with a second lepton or which do not conserve charge are eliminated. Fur- thermore, the momentum and energy measurements must be consistent. Once the neutrino four-vector is determined, the B can be reconstructed in the “usual” way as shown in Fig. 11. The MM2 technique gives a branching ratio of (1.75 ± 0.25 ± 0.20)%, while the neutrino reconstruction gives (1.89 ± 0.22 ± 0.35)%, giving a combined (preliminary) yield of (1.78 ± 0.20 ± 0.24)%. The statistical errors in both methods are essentially uncorrelated, while the systematic error is almost completely correlated. The q2 distribution from the MM2 method is shown in Fig. 12. The intercept at q2 of zero is proportional to |Vcbf+(0)|2. The curve is a ﬁt to a functional form f+(q2) = f+(0) 1 −q2/M2 V , (19) where MV is left unspeciﬁed but is theorized to be the mass of the vector exchange particle in the t channel, namely the B∗. The results and comparison with diﬀerent models are shown in Table 3. 12 Fig. 9. Invariant K−π+π+ mass spectra from CLEO for events with an opposite sign lepton in the interval 4 > q2 > 2 in four diﬀerent MM2 slices The curves are a ﬁt to a Gaussian signal shape summed with a polynomial background. 13 Fig. 10. Fits to the MM2 distribution for the D+ℓ−¯ν (dashed) and D+Xℓ−¯ν (dotted) compo- nents and the sum (solid) in diﬀerent q2 intervals 14 Fig. 11. Beam constrained mass spectrum for all events passing the cuts. The white area represents the signal events, the hatched area represents the combinatoric background, the crosshatched area represents the D∗+ℓ−¯ν and the shaded area represents all the remaining backgrounds. Fig. 12. The q2 distribution for ¯Bo →D+ℓ−¯ν from the MM2 analysis. 15 Table 3. Results of ¯Bo →D+ℓ−¯ν analysis Model f+(0) prediction |Vcbf+(0)| × 103 |Vcb| × 103 WSB21 0.70 25.7 ± 1.4 ± 1.7 37.3 ± 2.0 ± 2.5 KS22 0.69 25.7 ± 1.4 ± 1.7 36.7 ± 2.0 ± 2.5 Demchuk†23 0.68 24.8 ± 1.1 ± 1.6 36.4 ± 1.6 ± 2.4 Average 36.9 ± 3.7 ± 0.5 † A smaller statistical error is quoted for this model because MV is speciﬁed. For the average value for Vcb, the ﬁrst error is the quadrature of the the systematic and statistical errors in the data, and the fact that the fraction of Bo’s produced in Υ(4S) decay is known only as 0.49±0.05.24 The second error is due only to the model dependence. 2.3.3. Branching Ratio of ¯Bo →D∗+ℓ−¯ν We next turn to measurements of the branching ratio of the pseudoscalar to vector transition ¯Bo →D∗+ℓ−¯ν, shown in Table 4. Table 4. Measurements of B( ¯Bo →D∗+ℓ−¯ν) Experiment B(%) CLEO25 4.1 ± 0.5 ± 0.7 ARGUS26 4.7 ± 0.6 ± 0.6 CLEO II24 4.50 ± 0.44 ± 0.44 ALEPH27 5.18 ± 0.30 ± 0.62 DELPHI28 5.47 ± 0.16 ± 0.67 Average 4.90 ± 0.35 The width predictions of a collection of representative models and the resulting values of Vcb are given in Table 5. Here the ﬁrst error on the average is the from the error on the measured branching ratio (±3.6%) in quadrature with the error on the lifetime (±1.6%) and the second error reﬂects the spread in the models (±5.2%). 2.3.4. Heavy Quark Eﬀective Theory and ¯B →D∗ℓ−¯ν Our next method for ﬁnding Vcb uses “Heavy Quark Eﬀective Theory” (HQET).32 We start with a quick introduction to this theory. It is diﬃcult to solve QCD at long distances, but its possible at short distances. Asymptotic freedom, the fact that the strong coupling constant αs becomes weak in processes with large q2, allows perturbative calculations. Large distances are of the order ∼1/ΛQCD ∼1 fm, since ΛQCD is about 0.2 GeV. Short distances, on the other hand, are of the order of the 16 Table 5. Values of Vcb from B( ¯Bo →D+ℓ−¯ν) Model Predicted Γ(B →D∗ℓν) (ps−1) |Vcb| × 103 ISGW29 25.2|Vcb|2 35.2 ± 1.4 ISGW II30 24.8|Vcb|2 35.5 ± 1.4 KS22 25.7|Vcb|2 34.8 ± 1.4 WBS21 21.9|Vcb|2 37.8 ± 1.5 Jaus131 21.7|Vcb|2 37.9 ± 1.5 Jaus231 21.7|Vcb|2 37.9 ± 1.5 Average 36.5 ± 1.5 ± 1.9 quark Compton wavelength; λQ ∼1/mQ equals 0.04 fm for the b quark and 0.13 fm for the c quark. For hadrons, on the order of 1 fm, the light quarks are sensitive only to the heavy quark’s color electric ﬁeld, not the ﬂavor or spin direction. Thus, as mQ → ∞, hadronic systems which diﬀer only in ﬂavor or heavy quark spin have the same conﬁguration of their light degrees of freedom. The following two predictions follow immediately (the actual experimental values are shown below): mBs −mBd = mDs −mD+ (20) (90 ± 3) MeV (99 ± 1) MeV , and m2 B∗−m2 B = m2 D∗−m2 D. (21) 0.49 GeV2 0.55 GeV2. The agreement is quite good but not exceptional. Since the charmed quark is not that heavy, there is some heavy quark symmetry breaking. This must be accounted for in quantitative predictions, and can probably explain the discrepancies above. The basic idea is that if you replace a b quark with a c quark moving at the same velocity, there should only be small and calculable changes. In lowest order HQET there is only one form-factor function ξ which is a function of the Lorentz invariant four-velocity transfer y, where y = M2 B + M2 D∗−q2 2MBMD∗ . (22) The point y equals one corresponds to the situation where the B decays to a D∗ which is at rest in the B frame. Here the “universal” form-factor function ξ(y) has the value, ξ(1) = 1, in lowest order. This is the point in phase space where the b quark changes to a c quark with zero velocity transfer. The idea is to measure the decay rate at this point, since we know the value of the form-factor, namely unity, and then apply the hopefully small and hopefully well understood corrections. Although this analysis can be applied to ¯B →Dℓ−ν, the vanishing of the decay rate at y equals 1, ( maximum q2, see Fig. 12), makes this inaccurate.20 17 The corrections are of two types: quark mass, characterized as some coeﬃcient times ΛQCD/mQ, and hard gluon, characterized as ηA. The value of the form-factor can then be expressed as33 ξ(1) = ηA 1 + 0 · ΛQCD/mQ + c2 · (ΛQCD/mQ)2 + .... = ηA(1 + δ). (23) The zero coeﬃcient in front of the 1/mQ term reﬂects the fact that the ﬁrst order correction in quark mass vanishes at y equals one. This is called Luke’s Theorem.34 Recent estimates are 0.96±0.007 and −0.55±0.025 for ηA and δ, respectively. The value predicted for ξ(1) then is 0.91±0.03. This is the conclusion of Neubert.33 There has been much controversy surrounding the theoretical prediction of this number.35 To ﬁnd the value of the decay width at y equals one, it is necessary to ﬁt data over a ﬁnite range in y and extrapolate to y of one. HQET does not predict the shape of the form-factor; hence the shape of the dΓ/dy distribution is not speciﬁed. Most experimental groups have done the simplest thing and used a linear ﬁt. The CLEO results with both linear and quadratic ﬁts are shown in Fig. 13. The results from the diﬀerent groups are summarized in Table 6. Also ﬁts of the slope parameter, ρ2, coming from the linear ﬁt are included. 40 30 20 10 0 40 30 20 10 0 1.00 1.10 1.20 1.30 1.40 1.50 y ( a ) Linear Fit ( b ) Quadratic Fit V ξ (y) x 10 cb 3 3330694-013 Fig. 13. Linear and quadratic ﬁts to the CLEO data for the D∗+ℓ−¯ν and D∗oℓ−¯ν. Although the shape of the function is not speciﬁed in HQET general considerations lead to the expectation that the slope is positive: there is a pole in the amplitude 18 Table 6. Values of |Vcb|ξ(1) × 103 Experiment |Vcb|ξ(1) × 103 ρ2 ARGUS36 38.8 ± 4.3 ± 3.5 1.17 ± 0.22 ± 0.06 CLEO II37 35.1 ± 1.9 ± 2.0 0.84 ± 0.12 ± 0.08 ALEPH38 31.4 ± 2.3 ± 2.5 0.39 ± 0.21 ± 0.12 DELPHI39 35.0 ± 1.9 ± 2.3 0.81 ± 0.16 ± 0.10 Average 34.6 ± 1.6 0.82 ± 0.09 as y →−1 and ξ(y) →0 as y increases. Shapes for ξ(y) are suggested by quark models. I have ﬁt the CLEO data to diﬀerent model functions as shown in Fig. 14. The results are shown in Table 7. Fig. 14. Fits to the CLEO data with diﬀerent shapes. The curves are linear (solid), Neubert-Reickert (NR) exponential (dashed), pole (long dash-dot) and exponential (dot-dashed). These shapes give larger values of |Vcb|ξ(1)| than the linear ﬁt by (5±3)%. I call this a model dependent error. The value then obtained for |Vcb|ξ(1)| is (36.3 ± 1.6 ± 1.0) × 10−3, and |Vcb| = 0.0397 ± 0.0021 ± 0.0017 . (24) 19 Table 7. Values of |Vcb|ξ(1) for diﬀerent ﬁt shapes of CLEO II data ξ(y) name ρ |Vcb|ξ(1) × 103 1 −ρ2(y −1) linear 0.90±0.07 0.0351±0.0018±0.0018 2 y+1exp h −(2ρ2 −1) y−1 y+1 i NR exp 0.90±0.12 0.0366±0.0024±0.0018 2 y+1 2ρ2 pole 1.07±0.11 0.0364±0.0023±0.0018 exp [−ρ2(y −1)] exp 1.01±0.10 0.0360±0.0022±0.0018 2.3.5. |Vcb| using inclusive semileptonic decays The inclusive semileptonic branching ratio can also be used to measure Vcb. While B(B →Xe−¯ν) this has traditionally been done by measuring the inclusive lepton momentum spectrum using only single lepton data, recently dilepton data have been used. The inclusive lepton spectrum from the latest CLEO II data40 is shown in Fig. 15. Both electrons and muons are shown. Leptons which arise from the contin- 0.5 1.0 1.5 2.0 2.5 3.0 0.00 0.05 0.10 0.15 0.20 Lepton Momentum (GeV/c) 1/N(ϒ(4S)) x (dN/dp) (GeV/c)-1 e± µ± fit b→clν b→ulν b→c→xlν Fig. 15. Fit to the CLEO inclusive lepton spectrum with the ACM model. uum have been statistically subtracted using the below resonance sample. The peak at low momentum is due to the decay chain ¯B →DX, D →Y ℓ+ν. The data are ﬁt to two shapes whose normalizations are allowed to ﬂoat. The ﬁrst shape is taken from models of B decay while the second comes from the measured shape of leptons from D mesons produced nearly at rest at the ψ′′, which is then smeared using the measured 20 momentum distribution of D′s produced in B decay. CLEO ﬁnds Bsl of 10.5±0.2% and 11.1±0.3% in the ACM43 and ISGW∗models, respectively.40 The ACM model will be described below. The ISGW∗model is a variant of the ISGW29 model. The ISGW model includes all the exclusive single hadron modes, D, D∗, and D∗∗which contains several components. CLEO lets the normalization of the D∗∗components ﬂoat in the ﬁt, and calls this model ISGW∗. Next, I discuss how to use dilepton events to eliminate the secondary leptons at low momentum. Consider the sign of the lepton charges for the four leptons in the following decay sequence: Υ(4S) →B−B+; B−→Dℓ− 1 ¯ν, B+ →¯Dℓ+ 3 ν; D → Y ℓ+ 2 ν, ¯D →Y ′ℓ− 4 ¯ν. If a high momentum negative lepton (ℓ− 1 ) is found, then if the second lepton is also negative it must come from the cascade decay of the B+ (i.e. it must be ℓ− 4 ). On the other hand the second lepton being positive shows that it must be either the primary lepton from the opposite B+, (ℓ+ 3 ), or the cascade from the same B−, (ℓ+ 2 ). However the cascades from the same B−can be greatly reduced by insisting that the cosine of the opening angle between the two leptons be greater than zero as they tend to be aligned. The same arguments are applicable to Υ(4S) →Bo ¯Bo, except that an additional correction must be made to account for B ¯B mixing. The CLEO II data are shown in Fig. 16. The data ﬁt nicely to either the ACM or ISGW∗model. They ﬁnd that the semileptonic branching ratio, Bsl, equals (10.36 ± 0.17 ± 0.40)% with a negligible dependence on the model.41 This result conﬁrms that the B model shapes are appropriate down to lepton momenta of 0.6 GeV/c. ARGUS42 did the ﬁrst analysis using this technique and found Bsl = (9.6 ± 0.5 ± 0.4)%. 0 1 2 3 0.00 0.05 0.10 0.15 pe (GeV/c) dB/dp (0.1 GeV/c)-1 Fig. 16. The lepton momentum spectrum in dilepton events from CLEO. The solid points are for opposite sign leptons, while the open circles indicate like sign lepton pairs. The ﬁt is to the ACM model. The next topic is to measure Vcb using the inclusive lepton spectrum. Consider 21 Γsl ≡Γ(B →Xe−¯ν) in the simplest parton model: Γsl = G2 Fm5 b 192π3 pc|Vcb|2 + pu|Vub|2 ηQCD, (25) where the p’s are phase space factors, and the QCD correction, ηQCD = 1 −2αs/3π. Since |Vub| << |Vcb|, we ignore the 2nd term. To use the semileptonic width to extract |Vcb| using this expression requires a knowledge of m5 b, which is poorly understood. A way around this dilemma was found by Altarelli et al.43 They make two important corrections to the simple parton model. First they treat the spectator quark in the B meson as a quasi-free particle with a Gaussian spectrum of Fermi-momentum, p: f(p) = 4p2 √πp3 f exp(−p2/p2 f). (26) The average value, pf, is a free parameter in the model. Secondly, they include the eﬀects of gluon radiation from the quarks, which lowers the spectrum at high lepton momentum. The semileptonic width is given explicitly as: dΓ(B →DXℓ−¯ν) dx = m5 bG2 FV 2 cb 96π3 · [Φ(x, ǫ) −G(x, ǫ)] , (27) where x = 2Eℓ/mb, Eℓbeing the lepton energy, ǫ = mc/mb, G(x, ǫ) is a complicated gluon radiation function and Φ(x, ǫ) = x2(1 −ǫ2 −x)2 (1 −x)3 h (1 −x)(3 −2x) + (3 −x)ǫ2i . (28) Each value of the Fermi-momentum, p, leads to a diﬀerent value of mb and hence a diﬀerent distribution for dΓ dx which must be convoluted with Eq. (27) to ﬁnd the total theoretical lepton momentum spectrum. The relationship between mb and p is just given by kinematics m2 b = m2 B + m2 sp −2mB q (p2 + m2sp). (29) Here mB is the known value of the B meson mass of 5.280 GeV and msp is the spectator quark mass. A ﬁt to the shape of the lepton energy spectrum then is needed to determine the free parameters pf, ǫ and msp. In turns out that one can ﬁx msp and any latent dependence is absorbed by the other two. So a ﬁt to the data will determine Bsl, pf and ǫ. In this way Altarelli et al. remove the explicit dependence of the m5 b term in the total decay rate. The ISGW and ISGW∗models are also used. The resulting values are given in Table 8. The representative value of |Vcb| found from this analysis alone is |Vcb| = 0.039 ± 0.001 ± 0.004 . (30) 22 Table 8. Vcb Values from Inclusive leptons Model Experiment Vcb ACM CLEO I 0.042±0.002±0.004 ACM ARGUS 0.039±0.001±0.003 ACM CLEO II 0.040±0.001±0.004 ISGW CLEO I 0.039±0.002±0.004 ISGW ARGUS 0.039±0.001±0.005 ISGW CLEO II 0.040±0.001±0.004 ISGW∗ CLEO I 0.037±0.002±0.004 ISGW∗ CLEO II 0.040±0.002±0.004 There are determinations of the inclusive B semileptonic branching ratio from LEP. These measurements average over more B species that at the Υ(4S). Since the lifetimes of some of these, especially the Λb appears to be shorter than for the ground state mesons, the semileptonic branching ratio measured at LEP should be lower than that measured on the Υ(4S), yet it is somewhat higher.44 Since the measurement at LEP is far more complicated, I have chosen to leave out these results. The results of using all four methods to ﬁnd Vcb are shown in Fig. 17. It is remarkable that all four separate methods give such consistent results. Advocates for any particular method can choose among these results. I have chosen to average them. The errors are handled by adding the statistical and systematic errors on each method and then adding the diﬀerent methods in quadrature. This should give a generous estimate of the ﬁnal error. The average value of Vcb is 0.0381±0.0021, which gives a value for the CKM parameter A = 0.784 ± 0.043 . (31) 2.4. The CKM element Vub The ﬁrst evidence of a non-zero value of Vub was obtained by CLEO I who saw a non-zero excess beyond the endpoint allowed for B →Dℓν transitions.45 This result was quickly conﬁrmed by ARGUS.46 The latest evidence from CLEO II47 is shown in Fig. 18. R2 is the second Fox-Wolfram event shape variable,48 which tends to zero for spherical events, such as Υ(4S) decays and to one for jet-like events. Pmiss is the missing momentum in the event. The branching ratios are small. CLEO ﬁnds that the rate in the lepton momentum interval 2.6 > pℓ> 2.4 GeV/c, Bu(p), is (1.5 ± 0.2 ± 0.2) × 10−4. To extract Vub from this measurement we need to use theoretical models. It is convenient to deﬁne: Γ(b →uℓν) = γu|Vub|2, and Γ(b →cℓν) = γc|Vcb|2. In addition, fu(p) is the fraction of the spectrum predicted in the end point region by diﬀerent models, and Bsl is the 23 30 35 40 45 |V |x10 cb 3 HQET (1) AVERAGE 36.5±1.5±1.9 39.7±2.1±1.7 39±1±4 38.1±2.1 36.9±3.7±0.5 B(B X → ν) ξ B(B D → ν) B(B D → ν ) * Fig. 17. Results of four diﬀerent methods used to evaluate Vcb, and the resulting average. The horizontal lines show the values, the statistical errors out to the thin vertical lines, and the systematic errors added on linearly out to the thick vertical lines. semileptonic branching ratio. Then: |Vub|2 |Vcb|2 = Bu(p) Bsl · γc fu(p)γu . (32) These models disagree as to which ﬁnal states populate the endpoint region. Most models agree roughly on values of γc. However, models diﬀer greatly in the value of the product γu · fu(p). There are two important reasons for these diﬀerences. First of all, diﬀerent authors disagree as to the importance of the speciﬁc exclusive ﬁnal states such as πℓν, ρℓν in the lepton endpoint region. For example, the Altarelli et al. model doesn’t consider individual ﬁnal states and thus can be seriously misleading if the endpoint region is dominated by only one or two ﬁnal states. In fact, several inventors of exclusive models have claimed that the endpoint is dominated by only a few ﬁnal states.29,21 Secondly, even among the exclusive form-factor models there are large diﬀerences in the absolute decay rate predictions. This is illustrated in Fig. 19. The diﬀerences in the exclusive models are much larger in b →u transitions than in b →c transitions because the q2 range is much larger. Artuso has explicitly shown that the q2 distributions were very diﬀerent in the ACM and original ISGW model.49 However, the new ISGW II model agrees much better with ACM (see Fig. 20).50 Measurement of exclusive charmless semileptonic decays can put constraints on 24 Lepton Momentum (GeV/c) 0 40 80 120 Number of Events / 50 MeV/c 2.00 2.25 2.50 2.75 3.00 0 250 a) b) Fig. 18. Lepton yield versus momentum from CLEO II for the “strict” cut sample, R2 < 0.2, Pmiss > 1 GeV/c and the lepton and missing momentum direction point into opposite hemispheres, (a) and the R2 < 0.3 sample (b). The ﬁlled points are from data taken on the peak of the Υ(4S), while the open points are continuum data scaled appropriately. The dashed curves are ﬁts to the continuum data, while the solid histograms are predictions of the sum of b →cℓν and continuum lepton production. Fig. 19. Lepton momentum spectra, for B →ρℓν in the KS and the original ISGW model. the models and therefore restrict the model dependence. In principle, the ratio of rates for πℓν and ρℓν can be measured as well as the q2 dependence of the form-factors. However, measurement of these rates is diﬃcult. CLEO has recently succeeded in 25 Fig. 20. q2 distribution, for charmless semileptonic b decays in the model of Altarelli et al.(ACCMM) and the orginal ISGW model shown on top, and the new ISGW model shown on the bottom. The areas reﬂect the predicted widths, but the vertical scale is arbitrary. The high q2 tails on the ISGW models arise from the πℓν ﬁnal state. measuring the branching ratios.51 A neutrino reconstruction technique is used. The neutrino energy and momentum is determined by evaluating the missing momentum and energy in the entire event: Emiss = 2Ebeam − X i Ei (33) −→pmiss = X i −→pi . (34) Criteria are imposed to guard against events with false large missing energies. First, the net charge is required to be zero. Secondly, events with two identiﬁed leptons (implying two neutrinos) are rejected. Leptons are required to have momenta greater than 1.5 GeV/c in the case of πℓν and greater than 2.0 GeV/c in the case of ρℓν. In addition, the candidate neutrino mass is calculated as M2 ν = E2 miss −−→p 2 miss . (35) Candidate events containing a neutrino are kept if M2 ν /2Emiss < 300 MeV. Then the semileptonic B decay candidates (πo, π+, ρo, ωo, ρ+)ℓν are reconstructed using the neutrino four-vector found from the missing energy measurement.53 The beam constrained invariant mass, Mcand is deﬁned as M2 cand = E2 beam − −→pν + −→pℓ+ −→p(π or ρ) 2 , (36) 26 and with the use of the neutrino four-vector is essentially the same as any other full B reconstruction analysis done at the Υ(4S). The Mcand distributions are shown in Fig. 21. 1850696-001 35 30 25 20 15 10 0 50 40 30 20 10 0 40 20 0 40 20 0 1.5 2.0 2.5 3.0 1.5 2.0 2.5 3.0 / 7.5 MeV (0.2 GeV/c) Events 5.12 5.14 5.16 5.18 5.20 5.22 5.24 5.26 5.28 5.30 Mcand (GeV) 5 Events / 7.5 MeV Events (0.2 GeV/c) Events plepton (GeV / c) plepton (GeV / c) Fig. 21. The B candidate mass distributions, Mcand, for the sum of the scalar π+ℓν and πoℓν (top) and the vector modes (ρ and ω) (bottom). The points are the data after continuum and fake background subtractions. The unshaded histogram is the signal, while the dark shaded shows the b →cX background estimate, the cross-hatched, estimated b →uℓν feedown. For the π (vector) modes, the light-shaded and hatched histograms are π →π (vector→vector) and vector→π (π →vector) crossfeed, respectively. The insets show the lepton momentum spectra for the events in the B mass peak (the arrows indicate the momentum cuts). It is often diﬃcult to prove that a ππ system indeed is dominantly resonant ρ. CLEO attempts to show ρ dominance by plotting the π+π−and π+πo summed mass spectrum in Fig. 22. They also show a test case of πoπoℓν, which cannot be ρ, since ρo cannot decay to πoπo. There is an enhancement in the π+π−plus π+πo sum, while the πoπo shows a relatively ﬂat spectrum that is explained by background. The 3π spectrum shows little evidence of resonant ω, however. More data is needed to settle this issue. CLEO proceeds by assuming they are seeing purely resonant decays in the vector channel. 27 1850696-002 Events / 190 MeV 50 40 30 20 10 0 15 10 5 0 Events / 40 MeV Events / 190 MeV 15 10 5 0 0.7 0.8 0.9 1.0 0.5 1.0 1.5 0.5 1.0 1.5 ππ π 3π Mass (GeV) o oπ Mass (GeV) Mass (GeV) Fig. 22. Mass distributions for π+π−plus π+πo (left), 3π (upper right) and πoπo (lower right), for events which are candidates B →xℓν decays which satisfy all the other B candidate cuts including a cut on the B mass.The shading is the same as on the previous ﬁgure. The arrows indicate the mass range used in the analysis. The measured branching ratio is model dependent due to diﬀerent form-factor de- pendences on q2 and lepton momentum. Therefore, CLEO reports diﬀerent branching ratios for a selection of models. The ratio of ρℓν/πℓν is also given, see Table 9, and compared to model predictions; the errors are non-Gaussian, but the KS model has only a 0.5% likelihood of being consistent with the data. The values of Vub obtained from both the exclusive and the inclusive analyses are summarized in Fig. 23. For the inclusive analysis, results from CLEO I and ARGUS have been included in the average.52 Since the KS model predicts the wrong pseudoscalar/vector ratio, it is excluded from the average. The ISGW model has been dropped in favor of the ISGW II model. The range of model predictions is now narrowed compared to former analyses. However, the model variations still dominate the error. A conservative estimate gives Vub Vcb = 0.080 ± 0.015 , (37) which provides a constraint 1 λ2 Vub Vcb 2 = ρ2 + η2 = (0.36 ± 0.07)2 . (38) 28 Table 9. Results from exclusive semileptonic b →u transistions Model B(B →πℓν) B(B →ρℓν) Γ(ρ)/Γ(π) Γ(ρ)/Γ(π) ×104 ×104 predicted ISGW II 2.0 ± 0.5 ± 0.3 2.2 ± 0.4+0.4 −0.6 1.1+0.5+0.2 −0.3−0.3 1.47 WSB 1.8 ± 0.5 ± 0.3 2.8 ± 0.5+0.5 −0.8 1.6+0.7+0.3 −0.5−0.4 3.51 KS 1.9 ± 0.5 ± 0.3 1.9 ± 0.3+0.4 −0.5 1.0+0.5+0.2 −0.3−0.3 4.55 Melikhov† 1.8 ± 0.4 ± 0.3 ± 0.2 2.8 ± 0.5+0.5 −0.8 ± 0.4 1.6+0.7+0.3 −0.5−0.4 ± 0.11 1.53±0.15 † The 3rd error arises from uncertainties in the estimated form-factors ISGW II WSB KS Melnikov from and * ρ ν ν from lepton end- point ISGW II ACM WSB KS RDB Best estimate 0.089 0.076±0.010 0.058 0.105 0.080±0.006 0.082±0.006 0.081±0.006 0.062±0.005 0.074±0.007 0.080±0.015 +0.006 +0.010 -0.011 -0.008 +0.018 -0.019 0.04 0.06 0.08 0.10 0.12 V V ub cb *assumes V = 0.0381 cb π Fig. 23. Values of Vub/Vcb obtained from the exclusive πℓν and ρℓν analyses combined and taking Vcb = 0.0381, and results from the inclusive endpoint analysis. The best estimate combining all models except KS is also given. 2.5. Bo d −¯Bo d Mixing Neutral B mesons can transform to their anti-particles before they decay. The diagrams for this process are shown in Fig. 24. Although u, c and t quark exchanges 29 are all shown, the t quark plays a dominant role mainly due to its mass, as the amplitude of this process is proportional to the mass of the exchanged fermion. (We will discuss the phenomenon of mixing in more detail in section 3.2). b d t,c,u t,c,u W- b d b d t,c,u t,c,u W- b d Fig. 24. The two diagrams for Bd mixing. The probability of mixing is given by54 x ≡∆m Γ = G2 F 6π2BBf 2 BmBτB|V ∗ tbVtd|2m2 tF m2 t M2 W ! ηQCD, (39) where BB is a parameter related to the probability of the d and ¯b quarks forming a hadron and must be estimated theoretically, F is a known function which increases approximately as m2 t, and ηQCD is a QCD correction, with value about 0.8. By far the largest uncertainty arises from the unknown decay constant, fB. Bd mixing was ﬁrst discovered by the ARGUS experiment.55 (There was a previous measurement by UA1 indicating mixing for a mixture of Bo d and Bo s.56 At the time it was quite a surprise, since mt was thought to be in the 30 GeV range. Since |V ∗ tbVtd|2 ∝|(1 −ρ −iη)|2 = (ρ −1)2 + η2, (40) measuring mixing gives a circle centered at (1,0) in the ρ −η plane. The best recent mixing measurements have been done at LEP, where the time- dependent oscillations have been measured. The OPAL data57 is shown in Fig. 25. Averaging over all LEP experiments x=0.728±0.025.58 2.6. Rare B Decays The term “rare B decays” is loosely deﬁned. The spectator process shown in Fig. 26(a) is included since b →u doesn’t occur very often (≈1%), and the mixing process which occurs often(≈17%) is included since it involves two gauge bosons (the so called box diagram Fig. 26(b)). Other loop or box diagrams are shown in Fig. 26(d-f). CLEO found the ﬁrst unambiguous loop process, the one shown in Fig. 26(c).59 These decays involving a loop diagram are sometimes called “penguins,” an indefen- sible if amusing term that was injected into the literature as a result of a bet. For the Standard Model to be correct these decays must exist. In fact, penguins are expected 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2 0 2 4 6 8 10 t (ps) t (ps) t (ps) t (ps) R OPAL Fig. 25. The ratio, R, of like-sign to total events as a function of proper decay time, for selected B →D∗+Xℓ−¯ν events. The jet charge in the opposite hemisphere is used to determine the sign correlation. The curve is the result of a ﬁt to the mixing parameter. to play an important role in kaon decay, but there are no unique penguin ﬁnal states in kaon decay. Since penguins are expected to be quite small in charm decay, it is only in B decay that penguins can clearly be discerned. CLEO ﬁrst found the exclusive ﬁnal state B →K∗γ. An updated value for the branching ratio is60 B(B →K∗γ) = (4.2 ± 0.8 ± 0.6) × 10−5 . (41) This analysis uses the standard B reconstruction technique, summarized in equa- tion (11) used at the Υ(4S), combined with some additional background suppression cuts. These are separated into trying to insure that one is dealing with a real K∗and trying to supress background leading to hard photons. The latter comes from initial state radiation (ISR), where one of the beams radiates a photon and then subsequently annihilates and from continuum quark-antiquark production (Q ¯Q). Suppression of ISR and Q ¯Q is accomplished by combining event shape variables into a Fischer dis- criminant. A Fischer discriminant61 is a linear combination of several variables which individually may have poor separation between signal and background, but when taken together yield acceptable background rejection, the correlations between the variables helping. The Fischer output distribution for Monte Carlo simulations of 31 b W- (a) b d t,c,u t,c,u W- b d b W- s,d γ t,c,u (c) b W- s,d g t,c,u (d) (b) b W- s,d t,c,u γ,Z + (e) b t,c,u W- s,d W- + - ν (f) c,u Fig. 26. A compendium of rare b decay diagrams. (a) The spectator diagram, rare when b →u; (b) one of the mixing diagrams; (c) a radiative penguin diagram; (d) a gluonic penguin diagram; (e) and (f) are dilepton penguin diagrams. 32 signal, ISR and Q ¯Q backgrounds are shown in Fig. 27. 0.00 0.25 0.50 0.75 1.00 Fisher Discriminant Output Number of Scaled Events/bin Signal QQ ISR Fig. 27. The distribution of the Fischer discriminant output for Monte Carlo samples of Bo → K∗oγ(K∗o →K+π−) signal, Q ¯Q and ISR backgrounds. The histograms have equal area and the x axis has been rescaled to make the Fischer discriminant output lie between 0 and 1. The branching ratio is extracted by making a maximum likelihood ﬁt to four distributions, MB, ∆E, the Kπ invariant mass m(Kπ), and the Fischer discriminant. To illustrate what the signal shapes look like, projection plots are made by applying restrictive selection criteria on three of the four likelihood variables and projecting the remaining events onto the axis of the fourth variable. This is shown for the K∗o →K−π+ mode in Fig. 28. The extraction of the inclusive rate for b →sγ is more diﬃcult. There are two separate CLEO analyses.62 The ﬁrst one measures the inclusive photon spectrum from B decay near the maximum momentum end, similar to what is done to extract an inclusive b →Xℓν signal, but with the additional problem that the expected branching ratio is much lower. The main problem is to reduce the ISR and Q ¯Q backgrounds. Here instead of using a Fischer discriminant, a set of event shape variables and energies formed in a series of cones parallel and antiparallel to the candidate photon direction are fed into a neural net trained on Monte Carlo. The result is shown in Fig. 29(leftside). The second technique constructs the inclusive rate by summing up the possible exclusive ﬁnal states. Since the photons are expected to be at high momentum, and therefore take away up to half the B’s rest energy, the number of hadrons in the ﬁnal state is quite limited. The analysis looks for the ﬁnal states B →K nπγ where n is allowed to be a maximum of 4, but only one can be a πo. Only one entry per event 33 0.00 0.50 1.00 Fisher Output 0 4 8 (a) 5.200 5.250 5.300 MB 0 4 8 (b) 0.746 0.896 1.046 MKπ 0 4 8 12 (c) -0.35 -0.10 0.15 ∆Eππ 0 4 8 (d) Number of Events / Bin Width B0→K*0 γ (K*0→K+ π-) Fig. 28. Projections of Bo →K∗oγ(K∗o →K+π−) data events (histograms) and maximum likelihood ﬁts (curves) onto the four ﬁt variables: (a) Fischer discriminant output, (b) MB, (c)MKπ and (d) ∆Eππ, which is the diﬀerence between the candidate B energy and the beam energy assuming both charged tracks are pions. 1851194-002 3000 2000 1000 0 150 50 0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 ( a ) ( b ) Eγ (GeV) Weighted Events / 0.1 GeV 1851194-003 ( a ) ( b ) 200 150 100 50 0 25 0 -25 1.8 2.0 2.2 2.4 2.6 2.8 Eγ (GeV) Events / 0.1 GeV Fig. 29. Photon energy spectra from the neural net analysis shown on the left side, and from the B reconstruction analysis, shown on the right side. In (a) the on resonance date are the solid lines, the scaled oﬀresonance data are the dashed lines, and the sum of backgrounds from oﬀresonance data and b →c Monte Carlo are shown as the square points with error bars. In (b) the backgrounds have been subtracted to show the net signal for b →sγ; the solid lines are ﬁts of the signal using a spectator model prediction. 34 is allowed. Here background reduction is accomplished by using the full power of the exclusive B reconstruction analysis. The resulting γ energy spectrum is shown on the right side of Fig. 29. The branching ratios found are (1.88 ± 0.74) × 10−4 and (2.75 ± 0.67) × 10−4 for the neural net and B reconstruction analyses, respectively. The average of the two results, taking into account the correlations between the two techniques is B(b →sγ) = (2.3 ± 0.5 ± 0.4) × 10−4 . (42) The theoretical prediction for the branching ratio is given by63 Γ(b →sγ) Γ(b →cℓν) = V ∗ tsVtb Vcb α 6πg(mc/mb)|Ceff 7 (µ)|2, (43) where g(mc/mb) is a known function. While C7 is calculated perturbatively at µ equal to the W mass, the evolution to b mass scale causes ≈25% uncertainty in the prediction, since the proper point could be mb/2 or 2mb. In the leading log approximation the theoretical prediction is B(b →sγ) = (2.8 ± 0.8) × 10−4,63 while an incomplete next to leading order calculation, gives ∼1.9 × 10−4.64 A recently completed next to leading order calculation gives 3.3 × 10−4.65 In all cases the data are consistent with the prediction. The second analysis also produces the mass spectrum of the K nπ system, shown in Fig. 30. A clear K∗(890) component is observed. The best way to measure the fraction of K∗(890) is to divide the exclusive result by the average inclusive result. This number can test theoretical models, but mostly we are testing the prediction of the exclusive rate which is the far more diﬃcult calculation than the inclusive rate. 1851194-004 20 15 10 5 0 -5 -10 0.6 0.8 1.0 1.2 1.4 1.6 1.8 M (Xs) (GeV) Events / 0.1 GeV Fig. 30. The apparent K nπ mass distribution for the B reconstruction analysis. The points are the background subtracted data, not eﬃciency corrected, the solid histogram is ﬁt to the data using several K∗resonance as input to a Monte Carlo simulation, while the dotted histogram shows all the ﬁt components but the K∗(890). 35 The CLEO result is60 Γ(B →K∗γ) Γ(b →sγ) = 0.181 ± 0.068 . (44) Model predictions vary between 4 and 40%.68 Rare hadronic ﬁnal states have also been measured. CLEO reported a signal in the sum of K±π∓and π+π−ﬁnal states.66 The particle identiﬁcation could not uniquely separate high momentum kaons and pions. While the Kπ mode results from a penguin diagram the ππ mode results mainly from a b →u spectator diagram. The reconstructed B mass plot is shown in Fig. 31, along with the results of several other searches from an updated analysis,67 based on 2.4 fb−1 of integrated luminosity on the Υ(4S). Here a best guess is made as to which ﬁnal state is present. The resulting rate is B(Bo →K±π∓+ π+π−) = (1.8+0.6+0.2 −0.5−0.3 ± 0.2) × 10−5 . (45) Events / 2 MeV MB (GeV) K+ ( a ) K+ π+ π π ( b ) o / o K+ π+ π- / π- ( c ) πo πo ( d ) K oπo ( e ) K o π+ K- / 2 0 2 0 2 0 4 0 4 0 8 5.22 5.24 5.26 5.28 5.30 0990695-013 Fig. 31. MB plots for (a) Bo →π+π−(unshaded), Bo →K+π−, and Bo →K+K−(black) (b) B+ →π+πo (unshaded) and B+ →K+πo (grey), (c) Bo →πoπo, (d) Bo →Koπo, and (e) B+ →Koπ+. The projection of the total likelihood ﬁt (solid curve) and the continuum background component (dotted curve) are overlaid. An attempt to separate the kaon and pion components using the small diﬀerence in reconstructed energy and whatever particle identiﬁcation power exists leads to the 36 dipion fraction shown in Fig. 32. The best current guess is that approximately half of the rate is due to π+π−. Nsum Nππ / Nsum 1σ 2 3 4σ σ σ 30 25 20 15 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0990695 007 5 Fig. 32. The central value (+) of the likelihood ﬁt to Nsum ≡Nππ + NKπ and the fraction Nππ/Nsum. The solid curves are the nσ contours and the dotted curve is the 1.28σ contour. CLEO also has found a signal in the sum of ωπ+ and ωK+ decays.69 The B mass plot is shown in Fig. 33. The signal is 10 events observed on a background of 2±0.3 events. The branching ratio is B(B+ →ωπ+ + ωK+) = (2.8 ± 1.0 ± 0.5) × 10−5 . (46) Fig. 33. The MB projection for a) B+ →ηh+ and b) B+ →ωh+ after all other cuts, including the ∆E cut. The arrows indicate the signal region. DELPHI also reports a signal of 11 “rare” events over a background of 1 event. The invariant mass plot is shown in Fig. 34.70 One of these events appears to be 37 DELPHI B 0 → π +π -, K +π -, K +K - B - → ρ 0π -,K *0π -,K -ρ 0 B 0 → K +a1 - Invariant Mass (GeV/c 2) Entries / 0.1 GeV/c2 Fig. 34. Invariant mass distribution for two-body charmless hadronic B decays. The points with error bars represent the real data and the histograms the mass distributions expected in the absence of such decays as obtained from simulation. The curve represents the shape expected for the signal events normalized to the number of candidates selected in real data in the signal mass region. uniquely identiﬁed as a K∗oπ−ﬁnal state and this then is an unambiguous hadronic penguin decay. The evidence is shown in Fig 35. Fig. 35. The candidate B−→K∗oπ−decay: a magniﬁed view of the extrapolated tracks at the vertex is displayed above. The primary and secondary vertices are indicated by error ellipses corresponding to 3σ regions. The plot below summarizes the hadron identiﬁcation properties. The lines represent the expected response to pions (upper), kaons (middle) and protons (lower), and the points with error bars the measured values for the reconstructed B decay products. 38 3. Importance of Further Study of B Decays 3.1. Tests of the Standard Model via the CKM triangle The unitarity of the CKM matrix†allows us to construct six relationships. The most useful turns out to be VudV ∗ td + VusV ∗ ts + VubV ∗ tb = 0 . (47) To a good approximation Vud ≈V ∗ tb ≈1 and V ∗ ts ≈−Vcb, (48) then Vub Vcb + V ∗ td Vcb −Vus = 0 . (49) Since Vus = λ, we can deﬁne a triangle with sides 1 (50) Vtd Aλ3 = 1 λ q (ρ −1)2 + η2 = 1 λ Vtd Vts (51) Vub Aλ3 = 1 λ q ρ2 + η2 = 1 λ Vub Vcb . (52) The CKM triangle is depicted in Fig. 36. We know two sides already: the base is η ρ 0 1 β α γ Fig. 36. The CKM triangle shown in the ρ −η plane. The left side is determined by |Vub/Vcb| and the right side can be determined using mixing in the neutral B system. The angles can be found by making measurements of CP violation in B decays. †Unitarity implies that any pair of rows or columns are orthogonal. 39 deﬁned as unity and the left side is determined by the measurements of |Vub/Vcb|. The right side can be determined using mixing measurements in the neutral B system. We will see, however, that there is a large error due to the uncertainty in fB. Later we will discuss other measurements that can access this side. The ﬁgure also shows the angles as α, β, and γ. These angles can be determined by measuring CP violation in the B system. First we discuss CP violation in the Ko L system which also provides constraints on ρ and η. To test the Standard Model we can measure all three sides and all three angles. If we see consistency between all of these measurements we have deﬁned the parameters of the Standard Model. If we see inconsistency, the breakdown can point us beyond the Standard Model. 3.2. CP Violation The fact that the CKM matrix is complex allows CP violation. This is not only true for three generations of quark doublets, but for any number greater than two. Now let us explain what we mean by CP violation. C is a quantum mechanical operator that changes particle to antiparticle, while P switches left to right, i.e. x → −x. Thus under a P operation, −→p →−−→p since t is unaﬀected. Examples of CP violation have been found in the Ko system. Let us examine one such measurement. Consider the Ko to be composed of long lived and short lived components having equal weight, so the wave function is |Ko⟩= 1 √ 2 (|KS⟩+ |KL⟩) . (53) In the case of neutral kaons there is a large diﬀerence in lifetimes between the short lived and long lived components. The lifetimes are 9 × 10−11 sec and 5 × 10−8 sec. Suppose we set up a detector far away from the Ko production target. Then after the KS decay away we have only a KL beam. We ﬁnd both KL →e+νeπ−and KL →e−¯νeπ+ (54) are present. Now the initial state was a Ko, which contains an ¯s quark and can only decay semileptonically into the e+νeπ−ﬁnal state as shown in Fig. 37. Thus we have found evidence that both Ko and Ko are present. This phenomenon, Ko ⇔Ko is called mixing and can be depicted by the diagram shown in Fig. 38, much like the diagram for BoBo mixing. However, here the c-quark loop has the largest amplitude, unlike the B case, where the t-quark is dominant. (This is because the CKM couplings are so much larger, i.e. Vcs and Vcd ≫Vts and Vtd and this compensates for the decrease due to (mc/mt)2.) There are also hadronic intermediate states which contribute to the real part of the mixing amplitude, such as Ko →ππ →¯Ko. An example of CP violation is the measured rate asymmetry in our Ko L detector5 δ = 2Re(ǫ) = #(KL →e+νeπ−) −#(KL →e−¯νeπ+) #(KL →e+νeπ−) + #(KL →e−¯νeπ+) = 3.3 × 10−3 . (55) 40 − ν _ e- - W s d u { } K o π+ Vus −d Fig. 37. Semileptonic decay of a s quark contained in a Ko meson. s d t,c,u t,c,u W- d s s d t,c,u t,c,u W- d s Fig. 38. Ko −¯Ko mixing diagrams. Let us look at why this violates CP. In Fig. 39 the momentum and spin vectors for the two ﬁnal states are shown. The CP operation transforms the e+νeπ−to the e−¯νeπ+ ﬁnal state and vice-versa. Thus CP invariance would imply equal rates for the two processes, contrary to what is observed. CP → → → → → → → → → →↔ σ σ ν ν e e Z Z p p e e- + - - + π π Fig. 39. The momentum and spin orientations of the two ﬁnal states in semileptonic Ko L decay, showing that they are mapped into one another by a CP transformation. 41 CP violation thus far has only been seen in the neutral kaon system.‡ If we can ﬁnd CP violation in the B system we could see if the CKM model works or perhaps go beyond the model. Speculation has it that CP violation is responsible for the baryon-antibaryon asymmetry in our section of the Universe. If so, to understand the mechanism of CP violation is critical in our conjectures of why we exist.71 There is a constraint on ρ and η given by the Ko L CP violation measurement (ǫ), given by72 η h (1 −ρ)A2(1.4 ± 0.2) + 0.35 i A2 BK 0.75 = (0.30 ± 0.06), (56) where the errors arise from uncertainties on mt and mc. The constraints on ρ versus η from the Vub/Vcb measurement, ǫ and B mixing are shown in Fig. 40. The width of the B mixing band is caused mainly by the uncertainty on fB, taken here as 240 > fB > 160 MeV. The width of the ǫ band is caused by errors in A, mt, mc and BK. The size of these error sources is shown in Fig. 41. The largest error still comes from the measurement of Vcb, with the theoretical estimate of BK being a close second. The errors on mt and mc are less important. Excluded by B mixing s from ε from B mixing d Allowed -0.6 -0.4 0.2 0 0.2 0.4 0.6 ρ 0.8 0.6 0.4 0.2 η m = 170±15 GeV t γ β from V V ub cb α Fig. 40. The regions in ρ −η space (shaded) consistent with measurements of CP violation in Ko L decay (ǫ), Vub/Vcb in semileptonic B decay, Bo d mixing, and the excluded region from limits on Bo s mixing. The allowed region is deﬁned by the overlap of the 3 permitted areas, and is where the apex of the CKM triangle sits. ‡The other observed example of CP violation is the decay Ko L →ππ. 42 0.15 0.10 0.05 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 ρ Error sources for ε(ρ,η) δη Vcb k B mt mc Fig. 41. Error sources in units of δη on the value of η as a function of ρ provided by the CP violation constraint in Ko L decay. 3.3. Ways of Measuring CP violation in B Decays 3.3.1. CP Violation in Charged B Decays The theoretical basis of the study of CP violation in B decays was given in series of papers by Carter and Sanda and Bigi and Sanda.73 We start with charged B decays. Consider the ﬁnal states f ± which can be reached by two distinct weak processes A and B. Then the strong (s) and weak (w) parts are A = aseiθsaweiθw, B = bseiδsbweiδw . (57) Under the CP operation the strong phases remain constant but the weak phases change sign, so A = aseiθsawe−iθw, B = bseiδsbwe−iδw . (58) The rate diﬀerence is Γ −Γ = |A + B|2 −|A + B|2 (59) = 2asawbsbw sin(δs −θs) sin(δw −θw) . (60) A weak phase diﬀerence is guaranteed in the appropriate decay mode (diﬀerent CKM phases), but the strong phase diﬀerence is not; it is very diﬃcult to predict the magnitude of strong phase diﬀerences. As an example consider the possibility of observing CP violation by measuring a rate diﬀerence between B−→K−πo and B+ →K+πo. The K−πo ﬁnal state can be 43 b W- (a) b W- s g t (c) u s}K u u u} πo u u u u}K }πo b W- (b) u s} K u u u} πo b W- s g t (d) u d ud}K }π o Fig. 42. Diagrams for B−→K−πo, (a) and (b) are tree level diagrams where (b) is color suppressed; (c) is a penguin diagram. (d) shows B−→Koπ−, which cannot be produced via a tree diagram. reached either by tree or penguin diagrams as shown in Fig. 42. The tree diagram has an imaginary part coming from the Vub coupling, while the penguin term does not, thus insuring a weak phase diﬀerence. This type of CP violation is called “direct.” Note also that the process B−→Koπ−can only be produced by the penguin diagram in Fig. 42(d). Therefore, we do not expect a rate diﬀerence between B−→Koπ− and B+ →Koπ+. 3.3.2. Formalism in neutral B decays Consider the operations of C and P: C|B(−→p )⟩= |B(−→p )⟩, C|B(−→p )⟩= |B(−→p )⟩ (61) P|B(−→p )⟩= −|B(−−→p )⟩, P|B(−→p )⟩= −|B(−−→p )⟩ (62) CP|B(−→p )⟩= −|B(−−→p )⟩, CP|B(−→p )⟩= −|B(−−→p )⟩. (63) For neutral mesons we can construct the CP eigenstates |Bo 1⟩ = 1 √ 2 |Bo⟩−|B o⟩ , (64) |Bo 2⟩ = 1 √ 2 |Bo⟩+ |B o⟩ , (65) where CP|Bo 1⟩ = |Bo 1⟩, (66) 44 CP|Bo 2⟩ = −|Bo 2⟩. (67) Since Bo and B o can mix, the mass eigenstates are a superposition of a|Bo⟩+ b|B o⟩ which obey the Schrodinger equation i d dt a b ! = H a b ! = M −i 2Γ a b ! . (68) If CP is not conserved then the eigenvectors, the mass eigenstates |BL⟩and |BH⟩, are not the CP eigenstates but are |BL⟩= p|Bo⟩+ q|B o⟩, |BH⟩= p|Bo⟩−q|B o⟩, (69) where p = 1 √ 2 1 + ǫB q 1 + |ǫB|2, q = 1 √ 2 1 −ǫB q 1 + |ǫB|2. (70) CP is violated if ǫB̸ = 0, which occurs if |q/p|̸ = 1. The time dependence of the mass eigenstates is |BL(t)⟩ = e−ΓLt/2eimLt/2|BL(0)⟩ (71) |BH(t)⟩ = e−ΓHt/2eimHt/2|BH(0)⟩, (72) leading to the time evolution of the ﬂavor eigenstates as |Bo(t)⟩ = e−(im+ Γ 2)t cos ∆mt 2 |Bo(0)⟩+ iq p sin ∆mt 2 |B o(0)⟩ ! (73) |B o(t)⟩ = e−(im+ Γ 2)t ip q sin ∆mt 2 |Bo(0)⟩+ cos ∆mt 2 |B o(0)⟩ ! , (74) where m = (mL + mH)/2, ∆m = mH −mL and Γ = ΓL ≈ΓH. Note, that the probability of a Bo decay as a function of t is given by ⟨Bo(t)|Bo(t)⟩∗, and is a pure exponential, e−Γt/2, in the absence of CP violation. 3.3.3. Indirect CP violation in the neutral B system As in the example described earlier for KL decay, we can look for the rate asym- metry asl = Γ B o(t) →Xℓ+ν −Γ (Bo(t) →Xℓ−ν) Γ B o(t) →Xℓ+ν + Γ (Bo(t) →Xℓ−¯ν) (75) = 1 − q p 4 1 + q p 4 ≈O 10−2 . (76) 45 These ﬁnal states occur only through mixing as the direct decay occurs only as Bo →Xℓ+ν. To generate CP violation we need an interference between two dia- grams. In this case the two diagrams are the mixing diagram with the t-quark and the mixing diagram with the c-quark quark. This is identical to what happens in the Ko L case. This type of CP violation is called “indirect.” The small size of the ex- pected asymmetry is caused by the oﬀdiagonal elements of the Γ matrix in equation (68) being very small compared to the oﬀdiagonal elements of the mass matrix, i.e. |Γ12/M12| << 1. This results from the nearly equal widths of the Bo L and Bo H.74 3.3.4. CP violation for B via interference of mixing and decays Here we choose a ﬁnal state f which is accessible to both Bo and B o decays. The second amplitude necessary for interference is provided by mixing. Fig. 43 shows the decay into f either directly or indirectly via mixing. It is necessary only that f be Bo Bo f Fig. 43. Two interfering ways for a Bo to decay into a ﬁnal state f. accessible directly from either state, however if f is a CP eigenstate the situation is far simpler. For CP eigenstates CP|fCP⟩= ±|fCP⟩. (77) It is useful to deﬁne the amplitudes A = ⟨fCP|H|Bo⟩, ¯A = ⟨fCP|H|B o⟩. (78) If ¯ A A ̸ = 1, then we have “direct” CP violation in the decay amplitude, which we will discuss in detail later. Here CP can be violated by having λ = q p · ¯A A̸ = 1, (79) which requires only that λ acquire a non-zero phase, i.e. |λ| could be unity and CP violation can occur. A comment on neutral B production at e+e−colliders is in order. At the Υ(4S) resonance there is coherent production of Bo ¯Bo pairs. This puts the B’s in a C = −1 state. In hadron colliders, or at e+e−machines operating at the Zo, the B’s are 46 produced incoherently. For the rest of this article I will assume incoherent production except where explicitly noted. The asymmetry, in this case, is deﬁned as afCP = Γ (Bo(t) →fCP) −Γ B o(t) →fCP Γ (Bo(t) →fCP) + Γ B o(t) →fCP , (80) which for |q/p| = 1 gives afCP = (1 −|λ|2) cos (∆mt) −2Imλ sin(∆mt) 1 + |λ|2 . (81) For the cases where there is only one decay amplitude A, |λ| equals 1, and we have afCP = −Imλ sin(∆mt). (82) Only the amplitude, −Imλ contains information about the level of CP violation, the sine term is determined only by Bd mixing. In fact, the time integrated asymmetry is given by afCP = − x 1 + x2Imλ = −0.48Imλ . (83) This is quite lucky as the maximum size of the coeﬃcient is −0.5. Let us now ﬁnd out how Imλ relates to the CKM parameters. Recall λ = q p · ¯ A A. The ﬁrst term is the part that comes from mixing: q p = (V ∗ tbVtd)2 |VtbVtd|2 = (1 −ρ −iη)2 (1 −ρ + iη) (1 −ρ −iη) = e−2iβ and (84) Imq p = − 2(1 −ρ)η (1 −ρ)2 + η2 = sin(2β). (85) To evaluate the decay part we need to consider speciﬁc ﬁnal states. For example, consider f ≡π+π−. The simple spectator decay diagram is shown in Fig. 44. For the moment we will assume that this is the only diagram which contributes. Later I will show why this is not true. For this b →u¯ud process we have ¯A A = (V ∗ udVub)2 |VudVub|2 = (ρ −iη)2 (ρ −iη)(ρ + iη) = e−2iγ, (86) and Im(λ) = Im(e−2iβe−2iγ) = Im(e2iα) = sin(2α). (87) For our next example let’s consider the ﬁnal state ψKS. The decay diagram is shown in Fig. 45. In this case we do not get a phase from the decay part because ¯A A = (VcbV ∗ cs)2 |VcbVcs|2 (88) 47 b W- u d}π d u} π + d Fig. 44. Decay diagram at the tree level for Bo →π+π−. b W- c } ψ K s } d d s c Fig. 45. Decay diagram at the tree level for Bo →ψKS. is real. In this case the ﬁnal state is a state of negative CP, i.e. CP|ψKS⟩= −|ψKS⟩. This introduces an additional minus sign in the result for Imλ. Before ﬁnishing discussion of this ﬁnal state we need to consider in more detail the presence of the KS in the ﬁnal state. Since neutral kaons can mix, we pick up another mixing phase (see Fig. 38). This term creates a phase given by q p ! K = (V ∗ cdVcs)2 |VcdVcs|2 , (89) which is real. It necessary to include this term, however, since there are other for- mulations of the CKM matrix than Wolfenstein, which have the phase in a diﬀerent location. It is important that the physics predictions not depend on the CKM con- vention.§ In summary, for the case of f = ψKS, Imλ = −sin(2β). 3.3.5. Comment on Penguin Amplitude In principle all processes can have penguin components. One such diagram is shown in Fig. 46. The π+π−ﬁnal state is expected to have a rather large penguin §Here we don’t include CP violation in the neutral kaon since it is much smaller than what is expected in the B decay. 48 amplitude ∼10% of the tree amplitude. Then |λ|̸ = 1 and aππ(t) develops a cos(∆mt) term. It turns out (see Gronau75), that sin(2α) can be extracted using isospin consid- erations and measurements of the branching ratios for B+ →π+πo and Bo →πoπo. b W- d g t u u} } d d + π- π Fig. 46. Penguin diagram for Bo →π+π−. In the ψKS case, the penguin amplitude is expected to be small since a c¯c pair must be “popped” from the vacuum. Even if the penguin decay amplitude were of signiﬁcant size, the decay phase is the same as the tree level process, namely zero. 3.3.6. What actually has to be measured? In charged B decays we only have to measure a branching ratio diﬀerence between B+ and B−to see CP violation. For neutral B decays we must ﬁnd the ﬂavor of the other b-quark produced in the event (this is called tagging), since we do not have any Bo beams. We then measure a rate asymmetry aasy = #(f, ℓ+) −#(f, ℓ−) #(f, ℓ+) + #(f, ℓ−), (90) where ℓ± indicates the charge of the lepton from the “other” b and thus provides a ﬂavor tag. In Fig. 47(a) the time dependence for the Bo and ¯Bo are shown as a function of t in the B rest frame for 500 experiments of an average of 2000 events each with an input asymmetry of 0.3. In Fig. 47(b) the ﬁtted asymmetry is shown for 500 diﬀerent “experiments.” 3.4. Better Measurements of the sides of the CKM triangle One side of the triangle is determined by |Vub/Vcb|. It appears that the best way to improve the values now is to measure the form-factors in the reactions B →πℓν and B →ρℓν. This will decrease the model dependence error, still the dominant errors, in the Vub determination. Lattice gauge model calculations are appearing and should be quite useful. 49 Fig. 47. (a) Time dependence of Bo and B o decaying into a CP eigenstate, for an asymmetry of 0.3 for a total of 1 million events. The x-axis is proper time. In (b) the ﬁtted asymmetry results are shown for 500 “experiments” of average of 2000 events each. The other side of the triangle can determined by measuring Bs mixing, using the ratio xs xd = Bs B fBs fB !2 τBs τB Vtd Vts 2 , (91) where Vtd Vts 2 = λ2 h (ρ −1)2 + η2i . (92) The large uncertainty in using the Bd mixing measurement to constrain ρ and η is largely removed as the ratio of the ﬁrst three factors in equation (91) is already known to 10%. As an alternative to measuring xs, we can measure the ratio of the penguin decay rates B(B →ργ) B(B →K∗γ) = ξ Vtd Vts 2 , (93) where ξ is a model dependent correction due to diﬀerent form-factors. Soni76 has claimed that “long distance” eﬀects, basically other diagrams spoil this simple rela- tionship. This is unlikely for ρoγ but possible for ρ+γ.¶If this occurs, however, then ¶One example is the B−decay which proceeds via b →uW −, where the W −→¯ud →ρ−and the u combines with the spectator ¯u to form a photon. 50 it is possible to ﬁnd CP violation by looking for a diﬀerence in rate between ρ+γ and ρ−γ. The CLEO II data are already background limited. The limit quoted is60 B(B →ργ) B(B →K∗γ) < 0.19 (94) at 90% conﬁdence level. 3.5. Rare decays as Probes beyond the Standard Model Rare decays have loops in the decay diagrams so they are sensitive to high mass gauge bosons and fermions. However, it must be kept in mind that any new eﬀect must be consistent with already measured phenomena such as Bo d mixing and b →sγ. Let us now consider searches for other rare b decay processes. The process b → sℓ+ℓ−can result from the diagrams in Fig. 26(e or f). When searching for such decays, care must be taken to eliminate the mass region in the vicinity of the ψ or ψ′ resonances, lest these more proliﬁc processes, which are not rare decays, contaminate the sample. The result of searches are shown in Table 10. Table 10. Searches for b →sℓ+ℓ−decays b decay mode 90% c.l. upper limit Group Ali et al. Prediction77 sµ+µ− 50 × 10−6 UA178 K∗oµ+µ− 25 × 10−6 CDF80 2.9 × 10−6 23 × 10−6 UA178 31 × 10−6 CLEO79 K∗oe+e− 16 × 10−6 CLEO79 5.6 × 10−6 K−µ+µ− 9 × 10−6 CLEO79 0.6 × 10−6 10 × 10−6 CDF80 K−e+e− 12 × 10−6 CLEO79 0.6 × 10−6 B’s can also decay into dilepton ﬁnal states. The Standard Model diagrams are shown in Fig. 48. In (a) the decay rate is proportional to |VubfB|2. The diagram in (b) is much larger for Bs than Bd, again the factor of |Vts/Vtd|2. Results of searches are given in Table 11. 4. Future Experiments 4.1. e+e−machines operating at the Υ(4S) Recall that only B meson pairs are produced at the Υ(4S) as shown in Fig. 6. Since each B has about 30 MeV of kinetic energy, it moves on the average only 30 51 b W- u - ν b d t,c,u W- ν - + (b) (a) Fig. 48. Decay diagrams resulting in dilepton ﬁnal states. (a) is an annihilation diagram, and (b) is a box diagram. Table 11. Upper limits on b →dilepton decays (@90% c.l.) B(Bo →ℓ+ℓ−) B(Bs →ℓ+ℓ−) B(B−→ℓ−¯ν) e+e− µ+µ− µ+µ− e−¯ν µ−¯ν τ −¯ν SM† 2 × 10−15 8 × 10−11 2 × 10−9 10−15 10−8 10−5 UA178 8.3 × 10−6 CLEO81 5.9 × 10−6 5.9 × 10−6 1.5 × 10−5 2.1 × 10−5 2.2 × 10−3 CDF80 1.6 × 10−6 8.4 × 10−6 ALEPH82 1.8 × 10−3 †SM is the Standard Model prediction.83 µm before it decays. Another important consequence is that the decay products mix together and do not appear in distinct jets. To measure the important time diﬀerence required in CP violation experiments via mixing, it is necessary to to give the B’s a Lorentz boost which can be accomplished by using asymmetric beam energies.84 Let me amplify on this last statement. The asymmetry I presented afCP = −Imλ sin(∆mt), (95) is calculated for incoherent production of the Bo and another b quark (t is the time from production of the Bo until it decays). In e+e−production the B’s can be produced in a coherent state. At the Υ(4S) C = −1, while at higher energies, where B∗¯B (B∗→Bγ) is produced, C = +1. For coherent production equation (95) gets modiﬁed to afCP C=± = −Imλ sin (∆m(t ± t′)) , (96) where t refers to the decay time of fCP and t′ the decay time of the tagging B. In principle, afCP can be measured by taking a time integral. For incoherent production this works ﬁne (see equation (83)). Here, however, the integral over the C = −1 case gives exactly zero, necessitating the time dependent measurement. The integral over the C = +1 case, does not give zero, but the measured cross-section for B∗¯B is about 1/7 that of the Υ(4S).85,86 The one serious disadvantage of the Υ(4S) machines is that the cross-section is only 1 nb, so at a peak luminosity of 3 × 1033, we expect only 60 million Bo’s/year. For example, for a rare process with a branching ratio of 5 × 10−6 and a “typical” eﬃciency of 20%, we get only 60 events/year. 52 It is also important to note that there will not be much more B physics from LEP. The data sample has been collected and there are no current plans to get another large sample of Zo decays to add to the brilliant b physics already done. The CESR machine will be upgraded to produce a luminosity in excess of 2 × 1033cm−2s−1, albeit with symmetric energy beams. Both the KEK laboratory in Japan and SLAC in Stanford, Cal. will construct asymmetric energy machines with planned luminosities in excess of 3 × 1033cm−2s−1. The advantages of such machines are that the b cross-section is 1/4 of the total, and the relatively clean enviornment and low interaction rates allow for superb pho- ton detection using CsI crystal calorimeters87 and for planned particle identiﬁcation systems which should provided excellent π/K separation.88 4.2. Hadron machines Let us ﬁrst discuss the characteristics of hadronic b production. Hadronic b pro- duction mechanisms are shown in Fig. 49.89 The relative contribution of the terms a) q q Q Q Q Q g g Q Q g g g g Q Q b) q q Q Q g Q Q g g Q Q g g g Q Q g g g Fig. 49. Feynman diagrams for heavy quark production in hadronic collisions (a) of order α2 s, and (b) some diagrams of order α3 s. proportional to α2 s and those proportional to α3 s is not well known. This is an im- portant issue since the correlations in rapidity, η and in azimuthal angle between the b-quark and the ¯b-quark depends on the production mechanism. It is generally thought that |ηb −η¯b| < 2. In Fig. 50 I show the azimuthal opening angle distribu- tion between a muon from a b quark decay and the ¯b jet as measured by CDF90 and compare with the MNR predictions.91 The model does a good job in representing the shape which shows a strong back-to-back correlation. The normalization is about a factor of two higher in the data than the theory, which is generally true of CDF b 53 cross-section measurements.92 In hadron colliders all B species are produced at the same time. 10 10 2 0 0.5 1 1.5 2 2.5 3 Fig. 50. The diﬀerential δφ cross-sections for pµ T > 9 GeV/c, |ηµ| <0.6, E¯b T >10 GeV, η¯b < 1.5 compared with theoretical predictions. The data points have a common systematic uncertainty of ±9.5%. The uncertainty in the theory curve arises from the error on the muonic branching ratio and the uncertainty in the fragmentation model. The B meson transverse momentum distribution is severely limited and peaks near the B meson mass. The distribution in η, however is spread widely. In Fig. 51 I show the predicted (Pythia) distribution at the Tevatron collider. It should be realized Fig. 51. The predicted distribution of B’s versus η for 1.8 TeV p¯p collisions. 54 that this distribution in η reﬂects into a sharply peaked distribution in spatial angle (cos(θ)). The laboratory angular distributions of the B and B mesons expected at the LHC are shown in Fig. 52. Most of the events are far forward with the B and B being strongly correlated.93 0 0.5 1 1.5 2 2.5 3 0 1 2 3 0 200 400 600 800 1000 Fig. 52. Production anglesof B versus production angle of the B in the laboratory (in radians) for the LHC collider calculated using PYTHIA. Let us review some properties of current and proposed hadron b collider experiments.94 • The CDF and D0 detectors already exist at the Fermilab collider. The b cross- section is ∼50 µb, with the ratio σ(b)/σ(total) = 10−3. The luminosity is now close to 1031 and will increase with the advent of the main injector to 1032. However, the restrictive trigger limits the b sample. • The HERA-b experiment at DESY collides the HERA proton beam with ﬁxed wire targets. The b cross-section is only ∼6 nb with σ(b)/σ(total) = 10−6. In order to produce enough b’s they plan on four interactions per crossing. The goal is to measure CP violation in the ψKS decay mode and possibly investigate other modes that are accessible by triggering on dileptons. The experiment is now under construction. • The LHC-B experiment is being planned. At the LHC the b cross-section is ∼300 µb, with the ratio σ(b)/σ(total) = 3 × 10−3. The experiment can run at a luminosity of 1032, ≈240 Billion Bo/year are produced. • Also at the LHC, the Atlas and CMS experiments will have some B capabilities. • There is now a proposal for a dedicated B collider experiment at Fermilab called BTEV. Here ≈60 Billion Bo/year are produced. 55 4.3. Detector Considerations For an experiment to do frontier B physics the following components appear to be necessary: • Silicon vertex detector • Charged particle tracking with magnetic analysis • Cherenkov identiﬁcation of charged hadrons • Electromagnetic shower detection • Muon detection with iron A precision vertex detector is necessary to use the long B lifetime to reject back- ground. Silicon is the current technology of choice; it can be realized as strips or as pixels. Charged particle tracking with magnetic analysis is important for momentum measurement as it is in most experiments. In order to pick out speciﬁc B decay modes, such as K+π−from π+π−or ργ from K∗γ, it is crucial to have kaon and pion identiﬁ- cation. Currently this is best provided using Cherenkov radiation.88 Electromagnetic shower detection and muon identiﬁcation are required to study semileptonic decays and provide ﬂavor tags. The BELLE experiment, shown in Fig. 53 is an example of a detector that has all of these elements. There are important constraints on all of these detection elements. Radiation damage implies various limits and certain technologies. The number of interactions per second implies a rate limit on detector elements. It appears that the maximum rate on any detector element is about 107/sec. The total detector readout rate is limited to about 10-100 MB/sec. (The smaller number is given by current technology and the larger number is based on expected improvement.) For an event size of 100 KB, this gives a maximum readout rate of 1000 events/sec. Next, I will discuss the trigger. e+e−experiments have a distinct advantage here, since they merely trigger on everything. Experiments at hadron collider must trigger very selectively, or the data transmission rate will be swamped by background. There are several trigger strategies which have been developed. The one with the highest background rejection is B →ψX, ψ →ℓ+ℓ−. Unfortunately the branching ratio for the former is only 1.1% and the latter 12%, giving a maximum triggerable B event rate of only 2.6 × 10−3. This must be reduced by eﬃciency of the apparatus and kinematic cuts. Another strategy is to trigger on semileptonic decays, where the 10% branching ratio to both muons or electrons is attractive. Furthermore, for CP violation mea- surements through mixing, this trigger also provides a tag. It has been traditionally easier to trigger on muons because electrons can easily be faked by photon conversions near the vertex or Dalitz decays of the πo. 56 Belle Detector For the KEK B factory BELLE SVD CDC PID (Aerogel) TOF CsI KL/µ Superconducting Solenoid Fig. 53. Diagram of the Belle detector. The most progressive strategy is to trigger on detached vertices. Recent simula- tions for BTEV have shown that it is possible to achieve a good eﬃciency > 70% on B decay events with a rejection on light quark background in excess of 100:1. To achieve this it is necessary to use a forward geometry with the silicon vertex detector inside the beam pipe.95 A test of this concept was done at CERN by experiment P238.96 A sketch of the silicon detector arrangement is shown in Fig. 54. It is also possible to consider triggering on speciﬁc low multiplicity ﬁnal states such as Bo →π+π−by using hadrons with pt >1 GeV/c. 57 p beam p beam Fig. 54. Side view of the P238 silicon detector assembly and Roman pots. The 6 silicon planes are the vertical lines just above and below the beam line. The bellows (zig-zag lines) allow movements in the vertical direction of the pots, which are the thin vertical lines close to the bellows (they have 2 mm wall thickness). The edges of the 200 µm-thick aluminum RF shields closest to the beam (shown as the thin curved lines near the silicon detectors) normally ran at a distance of 1.5 mm from the circulating beams. The black horizontal pieces at top and bottom are the vacuum bulkheads bolted to the Roman pots. The crucial issue in all of the trigger strategies is what the background rates are for a high signal eﬃciency. Does this give enough signal events with simultaneously rejecting background at the 100:1 level? 4.4. Hadron Geometries There is a choice between two basic geometrical conﬁgurations that can be used for collider hadron B experiments. One is a central detector. An example is given by the planned upgraded CDF detector, shown in Fig. 55. Here the detector elements are arranged in an almost cylindrical manner about the beam pipe, so that the detector is very good near η equals zero. Notice that there are no detector elements for particle identiﬁcation, though some information may be available from dE/dx measurements in the tracking chamber. An example of a forward detector is the proposed LHC-B experiment shown in Fig. 56. Here the vertex detector is inside a ﬂared beam pipe. There are three diﬀerent radiators for the RICH detectors. In hadron colliders the most important rejection of non-B background is accom- plished by seeing a detached decay vertex. In Fig. 57 I show the normalized decay length expressed in terms of L/σ where L is the decay length and σ is the error on L for the Bo →π+π−decay.97 This study was done for the Fermilab Tevatron. 58 Scintillator Counter Silicon Tracker Fiber Tracker Electromagnetic Calorimeter Hadronic Calorimeter Drift Chamber Solenoid Coil Toroid Steel Shielding Key: Low b Quad h = 1.0 h = 2.0 h = 3.0 Low b Quad h = 1.5 Fig. 55. A schematic diagram of the CDF upgrade. The symbol ‘h’ refers to rapidity. Note that the ﬁber tracker may change to a diﬀerent technology. The forward detector clearly has a much more favorable L/σ distribution. In Fig. 58 we show the time resolution in picoseconds for the forward and central detectors for the reaction Bs →ψKs, which has been suggested as a possible way to measure Bs mixing.98 Remarkably the time resolution is a factor of 10 smaller for the forward detector. A comparison of diﬀerent B experiments is shown in Table 12. 5. Conclusions B decay physics started in the 1980’s and the ﬁrst generation of experiments at CESR, DORIS, PEP, PETRA, LEP and CDF have made great contributions including the ﬁrst fully reconstructed B’s and precise measurement of the B meson masses, measurement of the B lifetimes, discovery of Bo−¯ Bo mixing, the measurement of the CKM parameters Vcb and Vub and the sighting of the ﬁrst rare decays. Many mysteries, however, remain to be untangled. Measuring independently all sides and angles of the CKM triangle may point us beyond the Standard Model if the data are inconsistent. This will require measuring all three CP violating angles, 59 LHC-B 1 2 5 4 3 6 7 8 9 10 p → ← p 0 5 10 15 [m] [m] 2 4 1) Vertex detector 2) Aerogel and Gas RICH's 3) Magnet yoke 4) Coils 5) Magnetic field shielding plates 6) Tracking chambers 7) Gas RICH 8) Electromagnetic calorimeter 9) Hadron calorimeter 10) Muon system 300 mrad 400 mrad Fig. 56. A schematic diagram of the proposed LHC-B detector. Table 12. Comparison of B decay detectors Experiment Particle Vertex Photon σ(b) σ(b) I. D. detection detection σ(T) Babar Excellent Good Excellent 1 nb 0.25 Belle Good Good Excellent 1 nb 0.25 CLEO Excellent Mediocre† Excellent 1 nb 0.25 CDF Poor Good Poor 50 µb 10−3 D0 Poor Good Poor 50 µb 10−3 HERA-B Excellent Excellent Poor 6 nb 10−6 LHC-B Excellent Excellent Poor 300 µb 3 × 10−3 † detector is excellent but low B velocity compromises vertex detection measuring Bs mixing and precisely determining Vub/Vcb. Furthermore, observation of rare B decays may also point us beyond the Standard Model.99 e+e−threshold machines are great for future B physics. They will surely produce precision measurements of Vub and Vcb and the important measurement of sin(2β) using the ψKS decay mode. Posssibly sin(γ) can be measured using charged B decays and there are some who think these machines can measure sin(2α), but I ﬁnd that unlikely. However, these experiments are limited by the total number of B mesons. 60 L/σ 0 100 200 300 400 1 101 102 103 104 105 Central Forward Fig. 57. Comparison of the L/σ distributions for the decay Bo →π+π−in central and forward detectors produced at a hadron collider with a center of mass energy of 1.8 TeV. beta * gamma time resolution - ps Forward Detector 0 0.01 0.02 0.03 4 6 8 10 12 14 beta * gamma time resolution - ps Central Detector 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 Fig. 58. The time resolution plotted as a function of βγ for a forward detector (2.0 < η < 4.5) and a central detector (|η| < 1.5) for the decay Bs →ψK ∗produced at a hadron collider with a center of mass energy of 1.8 TeV. Even if these machines reach luminosities of 1034cm−2s−1, there are not enough B’s to probe most rare phenomena. The prospects for Bs mixing, Λb and Bc studies are dim. 61 There is a fantastic potential for studying CP violation phenomena and rare B studies in hadronic machines but it’s not easy. Let us consider the calculation of the error on an asymmetry measurement: σ(aasy) = 1 D q Neff · ǫ · B , (97) where Neff = N Signal Signal + Background, (98) B is the branching ratio of the ﬁnal state of interest, ǫ is the overall eﬃciency including the tagging eﬃciency. D is the dilution factor caused by anything which causes a wrong-sign tag to be found, such as away side mixing, lepton misidentiﬁcation etc.. A sample calculation is shown in Table 13. Table 13. Sensitivity Calculation for Observing a CP asymmetry in ψKS CM energy 2 TeV Cross-section 50 µb Luminosity 1032cm−2s−1 NBo/‘Snowmass’ year 3.75 × 1010 B(Bo →ψKS) 5.5 × 10−4 B(Bo →ψ(µ+µ−)KS(π+π−)) 2.2 × 10−5 N(Bo →µ+µ−π+π−)/year 8.2 × 105 Semi-leptonic decay of away side tag 0.10 Tagged N(Bo →µ+µ−π+π−)/year 8.2 × 104 Triggering eﬃciency 0.8 Reconstruction eﬃciency of µµππ 0.25 Reconstruction eﬃciency µ tag 0.25 Vertex ﬁnding eﬃciency 0.9 Cleanup & analysis cuts 0.7 Dilution factors: Shape dependence Dt−int 0.47 mixing of muon tag 0.75 muon tag misidentiﬁcation 0.9 Time resolution and cuts 0.95 Background 0.95 Total sensitivity 0.07 This calculation shows an error in the asymmetry of 7%. To see if that is in the range of interest, I show in Fig. 59 the expectations for the three CP violating angles and xs. These plots merely reﬂect the “allowed” region shown in Fig. 40. It should 62 -0.4 - 0.2 0 0.2 0.4 ρ 0.8 0.6 0.4 0.2 β sin(2 ) -0.4 -0.2 0 0.2 0.4 ρ α sin(2 ) 0.5 0 -0.5 -0.4 -0.2 0 0.2 0.4 ρ xs 80 60 40 20 0.8 0.6 0.4 0.2 -0.4 - 0.2 0 0.2 0.4 ρ γ sin( ) Fig. 59. The allowed values of three CP violating angles and the Bs mixing parameter xs as a function of ρ, taken from the allowed region in Fig. 40. be emphasized that this is not the result of a sophisticated analysis, which is diﬃcult to do because of the non-Gaussian nature of the theoretical errors. The decay modes which will probably be used to measure the CP violating angles are given in Table 14, with their branching ratios. Finally, I list in Table 15 the CP violation and Bs mixing measurements of prime importance and my guess on which experiments, should they be built, are likely to perform these measurements and which could possibly perform them. The B system challenges us with the possibility of very diverse and important measurements. Hopefully this physics will be done by the machines and experiments in the next and future decades. 63 Table 14. Branching ratios for decay modes used in measuring CP violation CKM angle Modes B Product B β ψKS 0.4 × 10−3 3.7 × 10−5 α π+π− 0.9 × 10−5 0.9 × 10−5 γ100 DoK− 3.3 × 10−4 4.0 × 10−5 D oK− 4.1 × 10−6 4.9 × 10−7 Do CPK− 2.2 × 10−5 2.6 × 10−6 γ101 K±πo, π±πo ≈10−5 ≈10−5 Koπ±, K±η(′) each each Table 15. Prospects for CP violation and Bs mixing measurements Quantity Modes Possible Likely sin(2α) π+π− Babar, Belle LHC-B, BTEV sin(2β) ψKS HERA-B, CDF, CLEO Babar, Belle, LHC-B, BTEV sin(2γ) Kπ Babar, Belle, CLEO sin(2γ) DoK− Babar, Belle, CLEO LHC-B, BTEV xs ψK∗ LHC-B, BTEV 6. Acknowledgements I have beneﬁted greatly by physics discussion with many of my colleagues, most recently with M. Artuso, K. Berkelman, T. Skwarnicki, M. Witherell, J. Rosner and M. Gronau, M. Neubert, C. Sachrajda, A. Ali and A. Buras. I have also learned a lot from people associated with new B eﬀorts including, J. Butler, C. Bebek, M. Procario, P. Mcbride, T. Ypsilantis and P. Schlein. I also thank Thomas Ferbel and Barbara Ferbel for arranging such an interesting school. I learned a lot and so did Julia. My special thanks to J. Rosner, K. Berkelman, M. Witherell and T. Skwarnicki for reading through the manuscript and making many useful comments. 64 7. REFERENCES 1. S. L. Glashow, Nucl. Phys. 22, 579 (1961); S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); A. Salam in Elementary Particle Theory, ed. N. Svartholm, Almqvist and Wisksell, Stockholm (1968). 2. N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963); M. Kobayashi and K. Maskawa Prog. Theor. Phys. 49, 652 (1973). 3. L. Wolfenstein Phys. Rev. Lett. 51, 1945 (1983). 4. W. J. Marciano and A. Sirlin, Phys. Rev. D22, 2695 (1980). 5. R. M. Barnett et al., Phys. Rev. D54, 1 (1996). 6. S. W. Herb et al., (CFS Collaboration) Phys. Rev. Lett. 39, 252 (1977). 7. J. K. Bienlein et al., Phys. Lett. B78, 360 (1978). See also C. W. Darden et al., Phys. Lett. B78, 364 (1978). 8. D. Besson et al., Phys. Rev. Lett. 54, 381 (1985). 9. S. Behrends et al., Phys. Rev. Lett. 50, 881 (1983). 10. M. S. Alam, et al., Phys. Rev. D50, 43 (1994). 11. For an excellent description of the techniques used at PEP and PETRA see W. B. Atwood and J. A. Jaros, “Lifetimes,” in B Decays 2nd edition revised, ed. S. Stone, World Scientiﬁc, Singapore (1994). 12. For an excellent description of the techniques used at LEP and CDF see V. A. Sharma and F. V. Weber “Recent Measurements of Lifetimes of b Hadrons,” in B Decays 2nd edition revised, ed. S. Stone, World Scientiﬁc, Singapore (1994). 13. CDF results from T. Huﬀman, “CDF B Results and Upgrade,” and LEP re- sults from R. Hawkings, “B-Lifetimes,” at “BEAUTY ’96,” 4th International Workshop on B-Physics at Hadron Machines June 17-21, 1996 - Roma, Italy to appear in proceedings. SLD results from K. Abe et al.Phys. Rev. Lett. 75, 3623 (1995). 14. CDF Collaboration, “Branching Fractions of B+ →ψ(2S)K+ and Bo → ψ(2S)K∗o Decays at CDF,” ICHEP96/pa01-86c, FERMILAB-Conf-96/160-E 15. C. Albajar et al., Phys. Lett. B186, 237 (1987); B213, 405 (1988); B256, 121 (1991). 16. K. Abe et al., Phys. Rev. Lett. 75, 1451 (1995). 17. D. M. Jansen et al., Phys. Rev. Lett. 74, 3118 (1995). 18. B. Grinstein, N. Isgur and M. B. Wise, Phys. Rev. Lett. 56, 258 (1986); F. J. Gilman and R. Singleton, Phys. Rev. D41, 142 (1990); K. Hagiwara, A. D. Martin and M. F. Wade, Nucl. Phys. B327, 569 (1989). 19. J. D. Richman and P. R. Burchat, Rev .Mod. Phys. 67, 893 (1995), and references contained therein. 20. T. Bergfeld et al., “Measurement of B( ¯Bo →D+ℓ−¯ν) and Extraction of |Vcb|. CLEO-CONF 96-3, ICHEP-96 PA05-78 (1996). 65 21. M. Wirbel, B. Stech and M. Bauer Z. Phys. C29, 637 (1985); M. Bauer and M. Wirbel,Z. Phys. C42, 671(1989). 22. J. G. Korner and G. A. Schuler Z. Phys. C38, 511(1988); ibid, (erratum) C41 690 (1989). 23. N. B. Demchuk, I. L. Grach, I. M. Narodetski, S. Simula, “ Heavy-to-heavy and heavy-to-light weak decay form factors in the light-front approach: the exclusive 0−to 0−case,” INFN-ISS 95/18, hep-ph/9601369 (1996). 24. B. Barish et al., Phys. Rev. D51, 1014 (1995). 25. D. Bortoletto et al., Phys. Rev. Lett. 16, 1667 (1989). 26. H. Albrecht et al., Z. Phys. C57, 533(1993). 27. D. Buskulic et al., Phys. Lett. B359, 236 (1995). 28. P. Abreu et al., CERN-PPE/96-11 (1996). 29. B. Grinstein, N. Isgur and M. B. Wise, Phys. Rev. Lett. 56, 258 (1986). 30. N. Isgur, D. Scora, B. Grinstein, and M. B. Wise, Phys. Rev. D39, 799 (1989). 31. W. Jaus, Phys. Rev D41, 3394 (1990). 32. N. Isgur and M. B. Wise, “Heavy Quark Symmetry,” in B Decays Revised 2nd Edition, World Scientiﬁc, Singapore (1994) p231; N.Isgur and M.B. Wise, Phys. Rev. D42 2388 (1990); N. Isgur and M. B .Wise, Phys. Lett. B232 113 (1989); ibidem B237, 527 (1990); M. B. Voloshin and M. A. Shifman,Sov. J. Nucl.Phys. 45 292 (1987); ibidem 47, 511 (1988); H. D. Politzer and M. B. Wise, Phys. Lett. 206B 681 (1988) 681; ibidem B208, 504 (1988); E. Eichten and B. Hill, Phys. Lett. B234, 511 (1990); H.Georgi, Phys .Lett. B240B 447 (1990); B.Grinstein, Nucl. Phys. B339, 253 (1990); A.F.Falk, H.Georgi, B.Grinstein and M. B .Wise, Nucl. Phys. B343, 1 (1990). 33. M. Neubert, Int. J. Mod. Phys. A11, 4173 (1996), hep-ph/9604412. 34. M. E. Luke, Phys. Lett. B252, 447 (1990). 35. A. Czarnecki, Phys. Rev. Lett. 76, 4121 (1996); M. Shifman, N.G. Uraltsev and A. Vainshtein, Phys. Rev. D51, 2217 (1995); M. Shifman and N. G. Uraltsev, Int. J. Mod. Phys. A10, 4705 (1995). 36. H. Albrecht et al., Z. Phys. C57, 533 (1993). 37. B. Barish et al., Phys. Rev. D51, 1041 (1995). 38. D. Buskulic et al., Phys. Lett. B359, 236 (1995). 39. P. Abreu et al., CERN-PPE/96-11 (1996), submitted to Z. Phys. C. 40. J. Bartelt et al., “Inclusive Measurements of B-meson Semileptonic Branching Fractions,” submitted to Lepton Photon conf., Cornell (1993), CLEO-CONF 93-19. 41. B. Barish et al., Phys. Rev. Lett. 76, 1570 (1996). 42. H. Albrecht et al., (ARGUS) Phys. Lett B318, 397 (1993). 43. G. Altarelli, N. Cabibbo, G. Corbo and L. Maiani Nuclear Phys. B208, 365 (1982). 44. T. Skwarnicki, “Decays of b Quark,” in Proc. of Lepton Photon Conf., Beijing, China August 1995, hep-ph/9512395. 66 45. R, Fulton et al., Phys. Rev. Lett. 64, 16 (1990). 46. H. Albrecht, et al., Phys. Lett. B234, 409 (1990). 47. J. Bartelt et al., Phys. Rev. Lett. 71, 4111 (1993). 48. G. Fox and S. Wolfram, Phys. Rev. Lett. 41, 1581 (1978). 49. M. Artuso, Phys. Lett. B311, 307 (1993). 50. M. Artuso, “CLEO Results,” presented at Beauty ’96, Rome, Italy, June 1996, to appear in proceedings. 51. J. Alexander et al., preprint CLNS 96/1419, CLEO 96-9 (1996). 52. R. Fulton et al. (CLEO), Phys. Rev. Lett. 64, 16 (1990); H. Albrecht et al. (ARGUS), Phys. Lett. B234, 409 (1990); a table of results and the average is given in S. Stone, “Semileptonic B Decays,” in B Decays revised 2nd edition, ed. S. Stone, World Scientiﬁc, Singapore (1994) p349. 53. Since the resolution on is about 2.5 worse than the resolution on |pmiss| (≈110 MeV), the neutrino four-vector is deﬁned as (Eν, −→p ν) = (|−→p miss|, −→p miss). 54. M. Gaillard and B. Lee, Phys. Rev. D10, 897, (1974); J. Hagelin, Phys. Rev. D20, 2893, (1979); A. Ali and A. Aydin, Nucl. Phys. B148, 165 (1979); T. Brown and S. Pakvasa, Phys. Rev. D31, 1661, (1985); S. Pakvasa, Phys. Rev. D28, 2915, (1985); I. Bigi and A. Sanda, Phys. Rev. D29, 1393, (1984). 55. H. Albrecht et al., Phys. Lett. B192, 245 (1983). 56. Using the ratio of like-sign to opposite sign dilepton events, UA1 published a mixing signal at the 3σ level that resulted from a combination of Bs and Bd, see C. Albajar et al., Phys. Lett B186, 245 (1987). 57. R. Akers et al., A Measurement of the B0 d Oscillation Frequency using Letponts and D∗+ℓ−mesons, CERN-PPE/96-74 (1996); R. Akers et al., Z. Phys. C66 555 (1995). 58. C. Zeitnitz, “Oscillations and Mixing,” presented at BEAUTY 96. 59. R. Ammar et al., Phys. Rev. Lett. 71, 674 (1993). 60. R. Ammar et al., “Radiative Penguin Decays of the B Meson,” CLEO-CONF 96-6 (1996). 61. R. A. Fischer, “The Use of Multiple Measurements in Taxonomic Problems,” Annals of Eugenics 7, 179 (1936); M. G. Kendall and A. Stuart, “The Advanced Theory of Statistics,” Volume III, Hafner Publishing, NY 2nd edition (1968). 62. M. S. Alam et al., Phys. Rev. Lett. 74, 2885 (1995). 63. S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); R. Grigjanis et al., Phys. Lett. B 213, 35 (1988); B. Grinstein, R. Springer and M. Wise Nucl. Phys. B339, 269 (1990); see also N. G. Desphande, “Theory of Penguins in B Decay,” in B Decays Revised 2nd Edition, ed. S. Stone, World Scientiﬁc, Singapore (1994). 64. M. Ciuchini et al., Phys. Lett. B334, 137 (1994). 65. K. Chetyrkin and M. Misiak, ‘b →sγ Decay Beyond Leading Logarithms,” pa08-005, presented at ICHEP ’96, Warsaw, Poland, August (1996). 66. M. Battle et al., Phys. Rev. Lett. 71, 3922 (1993). 67 67. D. Asner et al., Phys. Rev. D53, 1039 (1996). 68. S. Playfer and S. Stone, “Rare B Decays,” Int. Journal of Mod. Phys. A10, No 29. 4107, (1995). 69. B. Barish et al., “Observation of the Decay B →ωπ+,” CONF 96-23, ICHEP- 96 PA05-95 (1996). 70. W. Adam et al., “Study of Rare B Decays with the DELPHI Detector at LEP,” CERN PPE 96-67 (1996). 71. P. Langacker, “CP Violation and Cosmology,” in CP Violation, ed. C. Jarl- skog, World Scientiﬁc, Singapore p 552 (1989). 72. A. J. Buras, “Theoretical Review of B-physics,” in BEAUTY ’95 ed. N. Harnew and P. E. Schlein, Nucl. Instr. and Meth. A368, 1 (1995). 73. A. Carter and A. I. Sanda, Phys. Rev. Lett. 45, 952 (1980); Phys. Rev. D23, 1567 (1981); I. I. Bigi and A. I. Sanda, Nucl. Phys. B193, 85 (1981); ibid B281, 41 (1987). 74. I. Bigi, V. Khoze, N. Uraltsev, in CP Violation, ed. C. Jarlskog, World Scientiﬁc, Singapore 175 (1989). 75. M. Gronau, Phys. Rev. Lett. 63, 1451 (1989); M. Gronau and D. London, Phys. Rev. Lett. 65, 3381 (1990). 76. D. Atwood, B. Blok and A. Soni, Int. J. Mod. Phys. A11, 3743 (1996); see also N. Deshpande, X. He & J. Trampetic, Preprint OITS-564-REV (1994); see also J. M. Soares, Phys. Rev. D53, 241 (1996). G. Eilam, A. Ioannissian R. R. Mendel and P. Singer, Phys. Rev. D53, 3629 (1996). 77. A. Ali, C. Greub and T. Mannel, “Rare B Decays in the Standard Model, ” in Hamburg 1992, Proceedings, ECFA Workshop on a European B-meson Factory, Eds. R. Aleksan and A. Ali, p155 (1993). 78. C. Albajar et al., Phys. Lett. B262, 163 (1991). 79. R. Balest et al., “Search for B →Kℓ+ℓ−and B →K∗ℓ+ℓ−decays,” CLEO- CONF 94-4 (1994). 80. F. Abe et al., Phys. Rev. Lett. 76, 2015 (1996). 81. R. Ammar et al., Phys. Rev. D49, 5701 (1994); M. Artuso, et al., Phys. Rev. Lett. 75, 785 (1995). 82. D. Buskulic et al., Phys. Lett. B343, 444 (1995). 83. A. Ali and T. Mannel, Phys. Lett. B264, 447 (1991). Erratum, ibid, B274, 526 (1992). 84. The use of asymmetric beam energies to study CP violation in e+e−machines operating the Υ(4S) was ﬁrst suggested by Pierre Oddone. 85. D. S. Akerib et al., Phys. Rev. Lett. 67, 1692 (1991). 86. S. Stone, Mod. Phys. Lett. A3, 541 (1988). 87. E. Blucher et al., Nucl. Instr. and Meth. A249, 201 (1986); Y. Kubota et al., Nucl. Instr. and Meth. A320, 66 (1992). 88. M. Artuso, Nucl. Instr. and Meth. A371, 324 (1996); B. N. Ratcliﬀ, Nucl. Instr. and Meth. A371, 309 (1996) 68 89. M. Gluck, J. F. Owens and E. Reya, Phys. Rev. D17, 2324 (1978); B. L Cambridge, Nucl. Phys. B151, 429 (1979); P. Nason et al., Nucl. Phys. B327, 49 (1989); W. Beenakker et al., Nucl. Phys. B351, 507 (1991), Phys. Rev. D40, 54 (1989). 90. F. Abe et al., Phys .Rev. D53, 1051 (1996). See also, M. Artuso, “Experimental Facilities for b-Quark Physics, ” in B Decays revised 2nd Edition, Ed. S. Stone, World Scientiﬁc, Sinagapore (1994). 91. M. Mangano, P. Nason and G. Ridolﬁ, Nucl. Phys. B373, 295 (1992). 92. F. Abe et al., Phys. Rev. Lett. 75, 1451 (1995). Previous UA1 measurements agreed with the theoretical predictions, see C. Albajar et al., Phys. Lett. B256, 121 (1991). Recent D0 measurements agree with both the CDF measurements and the high side of the theoretically allowed range. See S. Abachi et al., Phys. Rev. Lett. 74 3548 (1995). 93. S. Erhan, Nucl. Instr. and Meth. A368, 133 (1995). 94. For more information, the reader is referred to proceedings of conferences called “Beauty 93” through “Beauty 96,” the proceedings of which are published by Nucl. Instru. and Methods volumes A333 (1993), A351 (1994), A368 (1995). 95. J. N. Butler, “The BTEV Experiment at Fermilab,” presented at 2nd Int. Conf. on Hyperons, Charm and Beauty Hadrons, Montreal, Quebec, Canada, Aug. 1996. 96. J. Ellet et al., Nucl. Instr. and Meth. A317 28, (1992). 97. M. Procario, “B Physics Prospects Beyond the year 2000,” presented at the 10th Topical Workshop on Proton-Antiproton Collider Physics, to appear in proceedings, Fermilab-BONF-95/166. 98. P. McBride and S. Stone, Nucl. Instr. and Meth. A368, 38 (1995). 99. A. G. Cohen et al., “B-Factory Physics from Eﬀective Supersymmetry,” UW- PT-95-22, hep-ph/9610252 (1996); J. L. Hewett and J. D. Wells, “Searching for supersymmetry in rare B decays,” SLAC-PUB-7290, hep-ph/9610323 (1996); Y. Nir and H. R. Quinn, “Theory of CP Violation in B Decays,” in B Decays ed. S. Stone, World Scientiﬁc, Singapore p520 (1994). 100. M. Gronau and D. Wyler, Phys. Lett. B265, 172 (1991). 101. M. Gronau, J. L. Rosner and D. London, Phys. Rev. Lett. 73, 21 (1994). 69