The CKM matrix and the Kobayashi-Maskawa mechanism - Physics

1ContentsA The facilities 11 The B-factories . . . . . . . . . . . . . . . . . . . . . 12 The detectors and collaborations . . . . . . . . . . . 13 Datataking and Monte Carlo production summary . 1B Tools and methods 14 Multivariate methods and analysis optimization . . . 15 Charged particle identification . . . . . . . . . . . . 16 Vertexing . . . . . . . . . . . . . . . . . . . . . . . . 17 B-meson reconstruction . . . . . . . . . . . . . . . .518 B-Flavor tagging . . . . . . . . . . . . . . . . . . . . 19 Background discrimination for B-decays . . . . . . . 110 Mixing and time-dependent analyses . . . . . . . . . 111 Maximum likelihood fitting . . . . . . . . . . . . . . 112 Angular analysis . . . . . . . . . . . . . . . . . . . . 113 Dalitz analysis . . . . . . . . . . . . . . . . . . . . . 114 Blind analysis . . . . . . . . . . . . . . . . . . . . . . 115 Systematic error estimation . . . . . . . . . . . . . . 1C The results and their interpretation 116 The CKM matrix and the Kobayashi-Maskawa mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . 116.1 Historical Background . . . . . . . . . . . . . 1516.2 CP Violation and Baryogenesis . . . . . . . . 216.3 CP Violation in a lagrangian field theory . . . 316.4 The CKM matrix . . . . . . . . . . . . . . . . 316.5 The Unitarity Triangle . . . . . . . . . . . . . 416.6 CP violation phenomenology . . . . . . . . . . 517 B-physics . . . . . . . . . . . . . . . . . . . . . . . . 617.1 V ub and V cb . . . . . . . . . . . . . . . . . . . 617.2 V td and V ts . . . . . . . . . . . . . . . . . . . . 617.3 Hadronic B to charm decays . . . . . . . . . . 617.4 Charmless B decays . . . . . . . . . . . . . . . 617.5 B-meson lifetimes, mixing, EPR correlations,and symmetry violation searches . . . . . . . . 617.6 φ 1, or β . . . . . . . . . . . . . . . . . . . . . 617.7 φ 2, or α . . . . . . . . . . . . . . . . . . . . . 617.8 φ 3, or γ . . . . . . . . . . . . . . . . . . . . . . 617.9 Radiative and electroweak penguin decays . . 63017.10 Leptonic decays, and B → D (∗) τν . . . . . . . 617.11 Rare, exotic, and forbidden decays . . . . . . . 617.12 Baryonic B decays . . . . . . . . . . . . . . . 618 Quarkonium physics . . . . . . . . . . . . . . . . . . 618.1 Conventional charmonium . . . . . . . . . . . 618.2 Exotic charmonium-like states . . . . . . . . . 618.3 Bottomonium . . . . . . . . . . . . . . . . . . 619 Charm physics . . . . . . . . . . . . . . . . . . . . . 619.1 Charmed meson decays . . . . . . . . . . . . . 619.2 D-mixing and CP violation . . . . . . . . . . . 619.3 Charmed meson spectroscopy . . . . . . . . . 619.4 Charmed baryon spectroscopy and decays . . 620 Tau physics . . . . . . . . . . . . . . . . . . . . . . . 621 QED and initial state radiation studies . . . . . . . . 622 Two-photon physics . . . . . . . . . . . . . . . . . . 623 Υ (5S) physics . . . . . . . . . . . . . . . . . . . . . . 624 QCD-related physics . . . . . . . . . . . . . . . . . . 63524.1 Fragmentation . . . . . . . . . . . . . . . . . . 624.2 Pentaquark searches . . . . . . . . . . . . . . . 610202525 Global interpretation . . . . . . . . . . . . . . . . . . 625.1 Global CKM fits . . . . . . . . . . . . . . . . . 625.2 Benchmark “new physics” models . . . . . . . 626 Glossary of terms . . . . . . . . . . . . . . . . . . . . 6Chapter 16The CKM matrix and theKobayashi-Maskawa mechanismEditors:Adrian Bevan and Soeren Prell (BABAR)Boštjan Golob and Bruce Yabsley (Belle)Thomas Mannel (theory)16.1 Historical BackgroundThe beginning of modern flavour physics was marked bythe discovery of strange particles. Their decays into nonstrangeparticles had lifetimes too long to be classified asstrong decays: this led to the introduction of the strangenessquantum number, which is conserved in strong decays butmay change in a weak decay. In turn, the small couplingstrength of strangeness-changing processes, compared tostrangeness-conserving processes, eventually led to the parameterizationof quark mixing by Cabibbo (1963). Inmodern language this implies that the up quark u couplesto a rotated combination d cos θ C + s sin θ C of the downquark d and the strange quark s. The value θ C ∼ 13 ◦for the Cabibbo angle explained the observed pattern ofbranching ratios in baryon decays.Experiments at that time only probed the three lightestquarks, and there was no known reason for the branchingfraction of the flavour changing neutral current (FCNC)decay K + → π + l + l − to be extremely suppressed with respectto the charged current decay K + → π 0 l + ν. Theresolution of this puzzle was found by Glashow, Iliopoulos,and Maiani (1970): one includes the charm quark, withthe same quantum number as the up quark, but couplingto the orthogonal combination −d sin θ C + s cos θ C .FCNC processes are suppressed by this “GIM mechanism”.In fact, FCNC’s in the kaon system involve a transitionof an s quark into a d quark. This can be achievedby two successive charged current processes involving (inthe two-family picture) either the up or the charm quarkas an intermediate state. Taking Cabibbo mixing into account,these amplitudes areA(s → d) = A(s → u → d) + A(s → c → d)= sin θ C cos θ C [f(m u ) − f(m c )]. (16.1.1)Hence, if the up and charm quark masses were degenerate,K 0 − K 0 mixing and other kaon FCNC processes wouldnot occur.However, the up and charm masses are not degenerateand thus K 0 − K 0 mixing can occur. Neglecting the small

2up-quark mass, the mixing amplitude turns out to beA(K → K) ∝ sin 2 θ C cos 2 θ C . m2 cMW2 . (16.1.2)suppression factors of the formCKM Factor × 116π 2 m 2 t − m 2 uM 2 W(16.1.3)40455055606570758085This implies that a mass difference ∆m K appears in theneutral kaon system. From this mass difference (an expressionanalogous to Eq. (??)) Gaillard and Lee (1974) couldextract the prediction that the charm-quark mass shouldbe about m c ∼ 1.5 GeV, and it was one of the great triumphsof particle physics when narrow resonances withmasses of about 3 GeV were discovered a few monthslater (Aubert et al., 1974; Augustin et al., 1974): thesewere identified as cc bound states. This discovery completedthe second particle family and introduced a 2 × 290quark mixing matrix into the phenomenology of weak interactions.Almost ten years before the discovery of charm, CPviolation was observed in the study of rare kaon decaysby Christenson, Cronin, Fitch, and Turlay (1964). Thiseffect is difficult to accommodate for two families, but anextension to three families yields a natural way to accountfor CP violation. The “six-quark model” was proposed byKobayashi and Maskawa (1973), extending Cabibbo’s 2×2quark mixing matrix into the 3 × 3 Cabibbo-Kobayashi-Maskawa (CKM) matrix. The GIM mechanism for the sixquark model is implemented by the unitarity of the CKMmatrix.While the observation of decays K 0 L→ 2π meant thatCP was violated, the data at that time only required CPviolation in mixing (see Sec. 16.6 for the classification ofCP -violating effects). The observed strength of CP violationin mixing, ε K ≃ 2.3 × 10 −3 , was consistent with the 100Kobayashi-Maskawa (KM) mechanism. However, this didnot constitute a proof that the KM mechanism was reallythe origin of the observed CP violation; the measurementof the single parameter ε K could not be used to test theKM mechanism. Such a test was performed for the first 105time by the measurement of sin 2φ 1 (sin 2β) at the B Factories.Without this, alternative explanations of the originof ε K were still possible.(One such explanation was offered by the superweakmodel of Wolfenstein (1964), where CP violation was dueto a new, very weak four-fermion interaction that changedstrangeness by 2 units (∆S = 2). However, this possibilitywas ruled out by the observation of direct CP violationin K L → ππ decays, Re(ε ′ K /ε K) = (1.65 ± 0.26) × 10 −3(Alavi-Harati et al., 1999; Burkhardt et al., 1988; Fantiet al., 1999).)After the discovery of the bottom quark and the τlepton the third generation remained incomplete, since thetop quark turned out to be quite heavy. The first hintof the large top-quark mass was the discovery of B 0 −oscillations by ARGUS (Albrecht et al., 1987). Themeasured ∆m d implied a heavy top with a mass m t above50−70 GeV, if the standard six quark model was assumed 115(Bigi and Sanda, 1987; Ellis, Hagelin, and Rudaz, 1987).B 0In fact, if the top mass had been significantly smaller,ARGUS could not have observed B 0 −B 0 oscillations. TheGIM mechanism for down-type quarks leads generally to95110and hence the GIM suppression for the bottom quark ismuch weaker than in the up-quark sector, where the correspondingfactor isCKM Factor × 1 m 2 b − m2 d16π 2 . (16.1.4)M 2 WHence FCNC decays of B-mesons have branching ratiosin the measurable region, while FCNC processes for D-mesons are heavily suppressed.16.2 CP Violation and BaryogenesisParticle physics experiments of the past thirty years haveconfirmed the standard model (SM) even at the quantumlevel, including quark mixing and CP violation. However,the observed matter-antimatter asymmetry of the universeindicates that there must be additional sources ofCP violation, since the amount of CP violation impliedby the CKM mechanism is insufficient to create the observedmatter-antimatter asymmetry.In fact, the excess of baryons over antibaryons in theuniverse∆ = n B − n B(16.2.1)is presumably non-zero, but still small compared to thenumber of photons: the ratio is measured to be ∆/n γ ∼10 −10 . Although it is conceivable that there might be regionsin the universe consisting of antimatter, just as ourneighborhood consists of matter, no mechanism is knownwhich could, from the Big Bang, produce regions of matter(or antimatter) as large as we observe today.The conditions under which a non-vanishing ∆ canemerge dynamically from the symmetric situation ∆ =0 have been discussed by Sakharov (1967). He identifiedthree ingredients1. There must be baryon number violating interactionsH eff (∆B ≠ 0) ≠ 0.2. There must be CP violating interactions. If CP wereunbroken, then we would have for every process i → fmediated by H eff (∆B ≠ 0) the CP conjugate one withthe same probabilityΓ (i → f) = Γ (i → f) (16.2.2)which would erase any matter-antimatter asymmetry.3. The universe must have been out of thermal equilibrium.Under the assumption of locality, causality, andLorentz invariance, CP T is conserved. Since in an equilibriumstate time becomes irrelevant on the globalscale, CP T reduces to CP , and the argument of point2 applies.

120125130135140145150155In order to illustrate the first two Saharov conditions,we employ a very simplistic example. Assume that in theearly universe, there was a particle X that could decay toonly two final states |f 1 〉 and |f 2 〉, with baryon numbersN (1)Band N(2)Brespectively, and decay ratesΓ (X → f 1 ) = Γ 0 r and Γ (X → f 2 ) = Γ 0 (1 − r) ,(16.2.3)where Γ 0 is the total width of X. Taking the CP conjugate,the particle X decays to the state f 1 with baryon number−N (1)B and f 2 with baryon number −N (2)B; the rates areΓ (X → f 1 ) = Γ 0 r and Γ (X → f 2 ) = Γ 0 (1 − r),160(16.2.4)where Γ 0 is the same as for X due to CP T invariance.The overall change ∆N B in baryon number inducedby the decay of an equal number of X and X particles is∆N B = rN (1)(2) (1)B+ (1 − r)NB− rNB()= (r − r) N (1)B − N (2)B− (1 − r)N(2)B(16.2.5)Thus ∆N B ≠ 0 means that we have to have CP violation(r ≠ r) and a violation of baryon number (N (1)B ≠ N (2)B ),illustrating the first two conditions.165Sakharov’s paper remained mostly unnoticed until thefirst formulation of Grand Unified Theories (GUTs). Inthese theories, for the first time, all the necessary ingredientswere present. In particular, baryon number violationappears naturally since quarks and leptons appear in the 170same multiplets of the GUT symmetry group. Furthermore,there are additional sources of CP violation, and aphase transition takes place at the scale M GUT , which hasto be quite high to prevent proton decay.One may also consider electroweak baryogenesis. Theelectroweak interaction provides CP violation through theCKM mechanism, and the electroweak phase transitionhas been thoroughly studied. The first ingredient is alsopresent, as the current corresponding to baryon numberis conserved only at the classical level: electroweak quantumeffects violate baryon number, but still conserve thedifference B − L of baryon and lepton number. However,although all the ingredients are present, this cannot explain∆. In particular, the CKM CP violation is too smallby several orders of magnitude.Given the firm evidence for non-vanishing neutrinomasses, there could be new sources of CP violation in thelepton sector, and even (although there is no evidence forthis as yet) lepton-number violation. This could lead toviolation of baryon number via leptogenesis, with the surplusof leptons transferred to the baryonic sector through(B − L)-conserving interactions.In any case, an additional source(s) of CP violation isneeded, beyond the phase of the CKM matrix, in order toexplain the matter-antimatter asymmetry of the universe.The search for this new interaction is one of the mainmotivations for flavor-physics experiments.16.3 CP Violation in a lagrangian field theoryThe SM is formulated as a quantum field theory based ona Lagrangian derived from symmetry principles. To thisend, the (hermitian) Lagrangian of the SM is given interms of scalar operators O i with couplings a iL(x) = ∑ )(a i O i (x) + a ∗ i O † i (x) , (16.3.1)iwhere the O i are composed of the SM quark, lepton, andgauge fields. It is straightforward to verify that CP conservationimplies that all couplings a i can be made realby suitable phase redefinitions of the fields composing theO i . In turn, CP is violated in a lagrangian field theory ifthere is no choice of phases that renders all a i real.In the SM there are in principle two sources of CPviolation. The so-called “strong CP violation” originatesfrom special features of the QCD vacuum, resulting in acontribution of the formL strong CP = θ α s8π Gµν,a ˜Ga µν (16.3.2)where G µν ( ˜G µν ) is the (dual) field strength of the gluonfield. This term is P and CP violating due to its pseudoscalarnature. However, experimentally it turns out thatthe coupling parameter θ is extremely small; the reasonfor this has not yet been discovered. This is known as the“strong CP problem” (see for example Cheng, 1988; Kimand Carosi, 2010); we shall ignore this in what follows bysetting θ = 0.The second source of CP violation is the CKM matrix.It turns out that all terms in the SM Lagrangian are CPinvariant except for the charged current interaction termH cc =g √2(uL c L t L)VCKM γ µ ⎛3⎝ d LsL ⎠ W µ + . (16.3.3)bL⎞Under a CP transformation we have⎛( )uL c L t L VCKM γ µ ⎝ d LsL ⎠ W µ + (16.3.4)⎞bL⎛CP−→ ( )d L s L b L VTCKM γ µ ⎝ u ⎞LcLtL⎠ W − µ (16.3.5)and hence the combination H cc +H † cc appearing in the SMlagrangian is CP invariant, ifV T CKM = V † CKM or V CKM = V ∗ CKM. (16.3.6)This statement refers to a specific phase convention forthe quark fields; in general terms it means that in theCP -invariant case, the CKM matrix can be made real byan appropriate phase redefinition of the quark fields.

417518018519019520016.4 The CKM matrixThe CKM matrix V CKM appearing in (16.3.3) is explicitlywritten as⎛V CKM = ⎝ V ⎞ud V us V ubV cd V cs V cb⎠ . (16.4.1)V td V ts V tbV CKM =Here the V ij are the couplings of quark mixing transitionsfrom an up-type quark i = u, c, t to a down-type quarkj = d, s, b.205In the SM the CKM matrix is unitary by construction.Using the freedom of phase redefinitions for the quarkfields, the CKM matrix has (n − 1) 2 physical parametersfor the case of n families. Out of these, n(n − 1)/2 are(real) rotation angles, and ((n − 3)n + 2)/2 are phases,which induce CP violation. For n = 2, no CP violationis possible, while for n = 3 a single phase appears. Thisis the unique source of CP violation in the SM, once thepossibility of strong CP violation is ignored.The CKM matrix for 3 families may be represented bythree rotations and a matrix generating the phase⎡U 12 = ⎣ c ⎤12 s 12 0−s 12 c 12 0 ⎦ ,0 0 1210⎡U 13 = ⎣ c ⎤13 0 s 130 1 0 ⎦ ,−s 13 0 c 13⎡U 23 = ⎣ 1 0 0⎤0 c 23 s 23⎦ ,0 −s 23 c 23⎡U δ = ⎣ 1 0 0⎤0 1 0 ⎦ , (16.4.2)0 0 e −iδ13where c ij = cos θ ij , s ij = sin θ ij , and δ is the complexphase responsible for CP violation; by convention the mixingangles θ ij are chosen to lie in the first quadrant so thatthe s ij and c ij are positive. Then (Chau and Keung, 1984)V CKM = U 23 U † δ U 13U δ U 12⎛⎞c 12c 13 s 12c 13 s 13e −iδ= ⎝ −s 12c 23 − c 12s 23s 13e iδ c 12c 23 − s 12s 23s 13e iδ s 23c 13⎠ ;s 12s 23 − c 12c 23s 13e iδ −c 12s 23 − s 12c 23s 13e iδ c 23c 13(16.4.3)this is the representation used by the PDG (Nakamura,2010).The elements of the CKM matrix exhibit a pronouncedhierarchy. While the diagonal elements are close to unity,the off-diagonal elements are small, such that e.g. V ud ≫V us ≫ V ub . In terms of the angles θ ij we have θ 12 ≫θ 23 ≫ θ 13 . This fact is usually expressed in terms of theWolfenstein parameterization (Wolfenstein, 1983), whichcan be understood as an expansion in λ = |V us |. It readsup to order λ 3⎛⎝1 − λ 2 /2 λ Aλ 3 (ρ − iη)−λ 1 − λ 2 /2 Aλ 2Aλ 3 (1 − ρ − iη) −Aλ 2 1⎞⎠ + O(λ 4 ).(16.4.4)The parameters A, ρ and η are assumed to be of orderone. When using this parameterization, one has to keepin mind that unitarity is satisfied only up to order λ 4 . Asit turns out that both ρ and η are also of order λ, theextension to higher orders becomes non-trivial, and onehas to consider redefining the parameters accordingly; thishas been studied by Ahn, Cheng, and Oh (2011).One can obtain an exact parameterization of the CKMmatrix in terms of A, λ, ρ, and η, for example, by followingthe convention of Buras, Lautenbacher, and Ostermaier(1994), whereλ = s 12 , (16.4.5)A = s 23 /λ 2 , (16.4.6)Aλ 3 (ρ − iη) = s 13 e −iδ , (16.4.7)and substituting Eqs (16.4.5) through (16.4.7) into Eq. (16.4.3),and noting that sin 2 θ = 1 − cos 2 θ. Such a parametrizationis described in Sec. ?? to illustrate CP violation inthe charm sector.Sometimes a slightly different convention for the Wolfensteinparameters is used, with parameters denoted ρ and η.These parameters are defined (Buras, Lautenbacher, andOstermaier, 1994; Charles et al., 2005) such thatρ + iη = − V udVub∗V cd Vcb∗ . (16.4.8)The difference with the parameterization defined aboveappears only at higher orders in the Wolfenstein expansion;the relation between the two schemes is√1 − A2 λρ + iη = (ρ + iη) √ 41 − λ2 [1 − A 2 λ 4 (ρ + iη)] . (16.4.9)16.5 The Unitarity TriangleThe unitarity relations V CKM · V † CKM = 1 and V † CKM ·V CKM = 1 yield six independent relations correspondingto the off-diagonal zeros in the unit matrix. They representtriangles in the complex plane; each triangle has thesame area, reflecting the fact that (with three families)there is only one irreducible phase. A non-trivial triangle— one with angles other than 0 or π — indicates CP violation,proportional to the triangles’ common area. Bigiand Sanda (2000) provide a detailed discussion of the varioustriangles, their interpretation, and the possibilities toprobe them. Only two triangles have sides of comparablelength, which means that they are of the same order in

5215220225230235240245the Wolfenstein parameter λ. The corresponding relationsareV ud Vub ∗ + V cd Vcb ∗ + V td Vtb ∗ = 0 (16.5.1)V ud Vtd ∗ + V us Vts ∗ + V ub Vtb ∗ = 0. (16.5.2)Inserting the Wolfenstein parameterization, both relationsturn out to be identical, up to terms of order λ 5 ; the apexof the unitarity triangle is given by the coordinate (ρ, η).The three sides of this triangle (Fig. 16.5.1) — usually referredto as “the” CKM triangle — control semi-leptonicand non-leptonic B d transitions, including B d − B d oscillations.Due to the sizeable angles, one expects large250CP asymmetries in B decays in the SM; this was actuallyrealized before the discovery of “long” B lifetimes.__(,)V td V*V ud Vub2= V VV cd V* cdcb(0,0) = 3*tb*cb = Fig. 16.5.1. The Unitarity Triangle.The angles of the unitarity triangle are defined as1(1,0)φ 1 = β ≡ arg [−V cd Vcb/V ∗td Vtb] ∗ , (16.5.3)φ 2 = α ≡ arg [−V td Vtb/V ∗ud Vub] ∗ , (16.5.4)φ 3 = γ ≡ arg [−V ud Vub/V ∗cd Vcb] ∗255, (16.5.5)where this definition is independent of the specific phasechoice expressed in (16.4.3). Different notation conventionshave been used in the literature for these angles.260In particular the BABAR experiment has used α, β, andγ, whereas the Belle experiment has reported results interms of φ 1 , φ 2 , and φ 3 . We use the latter for brevitywhen discussing results in later sections.The SM allows us to construct “the” CKM triangleby its angles or its sides or any combinations of them.Any discrepancy between the observed and predicted valuesindicates a manifestation of dynamics beyond the SM.Clearly this requires a good control of experimental andtheoretical uncertainties, both in their CP sensitive andinsensitive rates.Measurements of the magnitudes of CKM matrix elementsV ub and V cb can be found in Section 17.1, andmeasurements of V td and V ts in Section 17.2. Measurementsof the angles φ 1 , φ 2 , and φ 3 are discussed in Sections17.6, 17.7, and 17.8 respectively. It is possible toperform global fits, using data from many decay processesand theoretical input from Lattice QCD calculations, toover-constrain our knowledge of the CKM mechanism in 265the determination of the apex of the triangle. These globalfits are discussed in Chapter 25, both in the context of theSM (Section 25.1) and allowing for physics beyond the SM(Section 25.2).16.6 CP violation phenomenologySince CP violation is due to irreducible phases of couplingconstants, it becomes observable through interference effects.The simplest example is an amplitude consisting oftwo distinct contributionsA f = λ 1 〈f|O 1 |B〉 + λ 2 〈f|O 2 |B〉 (16.6.1)where λ 1,2 are (complex) coupling constants (in our casecombinations of CKM matrix elements) and 〈f|O 1,2 |B〉are matrix elements of interaction operators between theinitial and final state.The CP conjugate is the process B → f, yieldingA f= λ ∗ 1〈f|O † 1 |B〉 + λ∗ 2〈f|O † 2 |B〉. (16.6.2)The matrix elements of O (†)1,2 involve only strong interactions,which we assume to be CP -invariant. Hence we have〈f|O † 1 |B〉 = 〈f|O 1|B〉 and 〈f|O † 2 |B〉 = 〈f|O 2|B〉.(16.6.3)Thus for the CP asymmetry we findΓ (B → f) − Γ (B → f)A CP (B → f) ≡Γ (B → f) + Γ (B → f)(16.6.4)∝ 2 Im[λ 1 λ ∗ 2] Im[〈f|O 1 |B〉〈f|O 2 |B〉 ∗ ].Consequently, in order to create CP violation, there hasto be — aside from the “weak phase” due to the complexphases of the CKM matrix — also a “strong phase”, i.e.a phase difference between the matrix elements 〈f|O 1 |B〉and 〈f|O 2 |B〉. In the SM these two contributions correspondto different diagram topologies. In many cases, onecan identify tree-level contributions which carry differentCKM factors compared to penguin contributions. CP violationthen emerges from the interference of “trees” and“penguins”.For a quantum-coherent pair of neutral B-mesons (likethe color-singlet B 0 B 0 pair from Υ (4S) decay) the timeevolution generates a phase difference ∆m ∆t, which actslike the strong phase difference between the amplitudesfor B → f and for B → B → f. Hence we make use ofthe time-dependent CP asymmetryA B→fCP (∆t) ≡ Γ (B0 (∆t) → f) − Γ (B 0 (∆t) → f)Γ (B 0 (∆t) → f) − Γ (B 0 (∆t) → f)(16.6.5)= S B→f sin (∆m d ∆t) − C B→f cos (∆m d ∆t) .(16.6.6)The derivation (see the discussion in Chapter 10 leadingto Eq. (??)) neglects the small lifetime difference ∆Γ inthe B d system; the expressions for S and C can be foundin Eqs (??) and (??).

6270275280We may distinguish three different types of CP violationaccording to the various sources from which itemerges. CP violation in decays stems from different ratesfor a process and for its CP conjugate: hence we have|A f /A f | ≠ 1. This contribution leads to C B→f ≠ 0: it isalready present at ∆t = 0, and remains in time-integratedmeasurements. Indirect CP violation, sometimes referredto as direct CP violation, emerges from mixing in caseswhere we have |p/q| ̸= 1; 1 one observable related to thisis the semileptonic decay asymmetry a SL . Finally, mixinginducedCP violation occurs for Imλ ≠ 0, in which caseinterference of the amplitudes B → f and B → B → fleads to CP violation.In the B d system we have to a very good approximationqp = exp(−2iφ 1) . (16.6.7)This follows from Eq. (??) and by inspection of the boxdiagram contributing to the B d mixing (Fig. ??), fromwhich it can be seen that the CKM matrix elements appearingin the amplitude yield φ M12 = 2φ 1 . Hence in allcases where A = A, we find |λ| = 1 and Imλ = − sin(2φ 1 ),leading toC B→f = 0 and S B→f = − sin(2φ 1 ). (16.6.8)This holds for the gold-plated mode B → J/ψK s wherethere is no relative weak phase between A and A. However,if there appears a relative weak phase in the decayamplitudes, then we may still have |A| = |A| and hence|λ| = 1, and thus no direct CP violation. For example,the tree amplitude in B → ππ carries a weak phase e −iφ3which (neglecting penguin contributions) would lead toλ = exp(−2i(φ 1 + φ 3 )) = exp(+2iφ 2 ). (16.6.9)However, the penguin contribution in B → ππ cannot beneglected; in particular it leads to |λ| ̸= 1 and to directCP violation in these decays.In general we have the “unitarity relation” betweenthe quantities S B→f and C B→f ,(CB→f ) 2+(SB→f ) 2= 1 −(DB→f ) 2≤ 1 (16.6.10)285whereD B→f =2 Reλ1 + |λ| 2 . (16.6.11)However, in the limit of vanishing lifetime difference thetime-dependent CP asymmetry does not depend on D B→f ,and hence a direct measurement of this quantity in the B dsystem is difficult.1 For a definition of the quantities p, q, and λ, we refer toChapter 10, where time evolution is considered.

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