PITHA02/02 1 hep-lat/0201011 CP Violation and the CKM Matrix M. Benekea∗ aInstitut fu¨r Theoretische Physik E, Sommerfeldstr. 28, RWTH Aachen, D - 52074 Aachen, Germany ThislectureprovidesageneraloverviewofCPviolation,emphasizingCPviolationinflavour-violatinginteractions, 2 such as dueto theKobayashi-Maskawa mechanism. 0 0 2 n 1. INTRODUCTION the matter-antimatter asymmetric universe that a one observes today. At the electroweak phase J The breaking of CP symmetry (“CP viola- transition the Standard Model satisfies all the 8 tion”),the compositionofparity andchargecon- other necessary criteria for baryogenesis (baryon jugation,isaninterestingphenomenonforseveral number violation, departure from thermal equi- 1 reasons: v librium) [5], but CP violation in the Standard a. CP violation together with CPT symmetry 1 Modelistooweaktoexplaintheobservedbaryon- 1 implies non-invariance of the microscopic equa- to-photon ratio. With the above assumption on 0 tions of motion under motion-reversal. CP vi- the initial condition of the cosmic evolution, our 1 olation rather than C violation implies differ- own existence provides evidence for a source of 0 entphysicalpropertiesofmatter andantimatter. 2 CP violation beyond the Standard Model. These two facts are of fundamental importance 0 Electroweak baryogenesis has the attractive / for our understanding of the laws of Nature, and feature that it couples the required new mech- t a they were perceived as revolutionary, when CP anisms of CP violation to the electroweak scale, l violation was discovered in 1964 [1]. They were - therefore making them testable also in particle p also important in the development of the funda- colliderexperiments. Neverthelessitnowappears e mental theory of particles, since the observation h more likely that the matter-antimatter asymme- of CP violation motivated some early extensions : try is not related to the sources of CP violation v of the Standard Model as it was known at the i thatonemayobserveatcolliders. Twofactshave X time (1973),either by extending the Higgssector contributed to this change of perspective: first, r [2]or by adding a third generationof quarksand the lowerlimitonthe massesofHiggsbosonshas a leptons [3]. Nature has opted for the second pos- been increasing. A heavier Higgs boson implies sibility for certain and the Kobayashi-Maskawa a weaker (first-order) electroweak phase transi- mechanism of CP violation has become part of tion. As a consequence electroweak baryogene- today’s Standard Model. From today’s perspec- sis is already too weak over most of the param- tive motion-reversal non-invariance and the dis- eter space of even the minimal supersymmetric tinction of matter and antimatter, though fun- extensionoftheStandardModel. Second,theob- damental, appear no longer surprising and even servation of small neutrino masses through neu- “natural”. What remains surprising, however, is trino oscillations is explained most naturally by the peculiar way in which CP violation occurs or invoking the seesaw mechanism, which in turn ratherdoesnotoccurinthe StandardModeland is most naturally realized by postulating mas- its possible extensions. siveneutrinos,whicharesingletsundertheStan- b. Assuming that the evolution of the uni- dard Model gauge group. All three necessary verse beganfrom a matter-antimatter symmetric conditions for the generation of lepton number state, CP violation is necessary [4] to generate are naturally realized in the decay of the mas- ∗TalkpresentedatLattice2001,Berlin,August2001. siveneutrino(s). Leptonnumber isthenpartially 2 convertedinto baryonnumber via B+L-violating violation suggest that the Kobayashi-Maskawa (but B–L conserving) sphaleron transitions [6]. mechanismofCPviolationismostlikelythedom- While the leptogenesis scenario is very appeal- inant source of CP violation at the electroweak ing, the new sources of CP violation related to scale. The latest piece of evidence also rules out the Yukawa couplings of the heavy neutrinos oc- that CP symmetry is an approximate symmetry. cur atscalesoforderofthe heavyneutrino mass, A consequence of this is that generic extensions M (1012 1016)GeV (needed to explain the of the Standard Model at the TeV scale, needed R ∼ − small left-handed neutrino masses), and are not to explain the stability of the electroweak scale, directly testable with collider experiments in the suffer from a CP fine-tuning problem since any near future. For this reason, CP violation in the suchextensionimpliesthe existenceofmanynew context of baryogenesiswill not be discussedfur- CP-violating parameters which have no generic ther in this talk. reason to be small. Despite the apparent success c. CP violation in the Standard Model is es- of the standardtheoryof CP violation,the prob- sentially an electroweak phenomenon originating lem of CP and flavor violation therefore remains from the Yukawa couplings of the quarks to the as mysterious as before. Higgs boson. This implies thatprobes ofCPvio- The three quantities listed above establish CP lationareindirectprobesofthe electroweakscale violation unambiguously, but since the observ- or TeV scale, complementary to direct probes ables depend on decays of mesons at low en- such as the observation of Higgs bosons. This is ergy, interpreting these quantities in terms of probably the most important reason for the cur- CP-violating fundamental parameters of the the- rent interest in CP symmetry breaking: in ad- ory often involves a very difficult strong interac- dition to testing the Kobayashi-Maskawa mech- tionproblem. Chiralperturbationtheoryandthe anism of CP violation in the Standard Model, heavy quark expansion provide analytic tools to experiments directed at CP violation limit the address this problem for kaon and B meson de- constructionofextensionsoftheStandardModel cays, respectively. In addition, lattice QCD can at the TeV scale. There is an analogy between contribute substantially to making the theoreti- CP symmetry and electroweak symmetry break- cal prediction for many (but not all) quantities ing. Both occur at the electroweak scale and for relevant to CP violation more precise. boththeStandardModelprovidesasimplemech- anism. However, neither of the two symmetry 2. CP VIOLATION IN THE STANDARD breaking mechanisms has been sufficiently tested MODEL up to now. Such tests may or may not confirm the Standard Model mechanisms but they may CP violation can occur in the Standard Model also provide answers to questions that concern in three different ways: the structure of the Standard Model in its en- 2.1. The θ term tirety, such as the origin of the electroweak scale The strong interactions could be CP-violating and the origin of flavour and CP violation. [11–13]. Thetopologyofgaugefieldsimpliesthat d. Leaving aside the matter-antimatter asym- the correct vacuum is given by a superposition metryintheuniverseasevidenceforCPviolation θ = einθ n of the degenerate vacua n since this depends on a further assumption, CP | i n | i | i in which pure gauge fields have winding number violation has now been observed in the weak in- P n. Correlation functions in the θ-vacuum can be teractionsofquarksinthreedifferentways: inthe computed by adding to the Lagrangianthe term mixing of the neutral kaon flavoureigenstates (ǫ, 1964)[1];inthedecayamplitudesofneutralkaons g2 (ǫ′/ǫ, 1999)[7,8]; in the mixing of the neutralB =θ s GA G˜A,µν, (1) d Lθ · 32π2 µν meson flavour eigenstates (sin(2β), 2001) [9,10]. It will be seen below that these pieces of data where θ now represents a parameter of the the- together with others not directly related to CP ory. Physical observables can depend on θ only 3 through the combination eiθdet , where is a consequence one may have interesting model- M M the quark mass matrix. A non-zero value of dependent relations between leptogenesis, CP vi- olation in lepton-flavour violating processes and θ˜=θ+argdet (2) neutrino physics, but since the observations are M all indirect through low-energy experiments, one violates CP symmetry. It also implies an elec- may at best hope for accumulating enough ev- tric dipole moment of the neutron of order 10−16θ˜ecm. The non-observation of any such idence to make a particular model particularly electric dipole moment constrains θ˜<10−10 and plausible. Such experiments seem to be possible, butnotinthenearfutureandforthisreasonlep- causes what is known as the strong CP problem, tonic CP violation is not discussed further here. sincetheStandardModelprovidesnomechanism that would require θ˜ to vanish naturally. The Itshouldbenotedthatthereisingeneralnocon- nection between CP violation in the quark and strongCP problem has become more severe with lepton sector except in grand unification models the observationoflargeCP violationin B meson in which the two relevant Yukawa matrices are decayssinceonenowknowswithmoreconfidence related. Even then further assumptions are nec- that the quark mass matrix has no reason to be essary for a quantitative relation. real a priori. There exist mechanisms that render θ˜= 0 ex- 2.3. The CKM matrix actly or very small through renormalization ef- CP violation can appear in the quark sector of fects. None of these mechanisms is convincing theStandardModelatthelevelofrenormalizable enoughto provide a default solution to the prob- interactions [3]. The quark Yukawa interactions lem. What makes the strong CP problem so dif- read ficult to solve is that one does not have a clue at what energy scale the solution should be sought. = ydQ¯′Hd′ yuQ¯′ǫH∗u′ +h.c., (4) LY − ij i j − ij j Strong CP violation is not discussed further in with Q′ the left-handed quark SU(2)-doublets, this talk (see the discussion in [14]). u′ and d′ the right-handed SU(2) singlets and 2.2. The neutrino mass matrix i = 1,2,3 the generation index. The complex The Standard Model is an effective theory de- mass matrices that arise after electroweak sym- finedbyitsgaugesymmetriesanditsparticlecon- metry breaking are diagonalizedby separateuni- tent. CP violationappears in the lepton sector if tary transformations Uu,d of the left- and right- L,R neutrinos are massive. The leading operator in handed up- and down-type fields. Only the com- the effective Lagrangianis [15] bination fΛij ·[(LTǫ)iiσ2H][HTiσ2Lj]. (3) VCKM =ULu†ULd = VVucdd VVucss VVucbb , (5) V V V td ts tb After electroweak symmetry breaking this gener-   atesaMajorananeutrinomassmatrixwiththree referred to as the CKM matrix, is observable, CP-violating phases. One of these phases could since the charged current interactions now read beobservedinneutrinooscillations,theothertwo e phases only in observables sensitive to the Majo- − √2sinθ u¯iγµ[VCKM]ijdjWµ++h.c.. (6) rana nature of neutrinos. W Unless the f are extremely small, the scale Flavour and CP violation in the quark sector ij Λ must be large to account for small neutrinos can occur in the Standard Model only through masses, which suggests that leptonic CP viola- charged current interactions (assuming θ˜ = 0). tion is related to very large scales. For exam- With three generations of quarks, the CKM ma- ple, the standard see-saw mechanism makes the trix contains one physical CP-violating phase. f dependent on the CP-violating phases in the Any CP-violating observable in flavour-violating ij heavy gauge-singlet neutrino mass matrix. As processes must be related to this single phase. 4 Model. More precisely, CP-violating observables ((cid:26)(cid:22);(cid:17)(cid:22)) are either small numbers, or else they are con- structed out of small numbers such as small (cid:11) VudVu(cid:3)b VtdVt(cid:3)b branching fractions of rare decays. The hierar- V dV (cid:3)b V dV (cid:3)b chyofquarkmassesandmixinganglesrepresents a puzzle, sometimes called the flavour problem, (cid:13) (cid:12) which will also not be discussed further in this (0;0) (1;0) talk. Typical attempts to solve the flavour prob- lem focus on broken generation symmetries. Figure 1. The unitarity triangle. 3. CONSTRAINTS ON THE UNITAR- ITY TRIANGLE The verification or, perhaps rather, falsification of this highly constrainedscenariois the primary InthefollowingIreviewthecurrentconstraints goal of many current B- and K-physics exper- on (ρ¯,η¯), the apex of the unitarity triangle, and iments. This type of CP violation is therefore the prospects for improving these constraints. discussed in some detail in later sections. (The definitions ρ¯/ρ = η¯/η = 1 λ2/2 render − For reasons not understood the CKM matrix the location ofthe apex accurateto orderλ5 [17] hasahierarchicalstructureasregardstransitions and will be used in the following.) It is not the between generations. It is therefore often repre- purpose of this talk to go into the details of the sented in the approximate form [16] theoretical calculations that contribute to these constraints. Recentsummariesoftherelevantlat- 1 λ2/2 λ Aλ3(ρ iη) − − tice calculations can be found in [18–20].  λ 1 λ2/2 Aλ2 , (7) − − 3.1. CP-conserving observables Aλ3(1 ρ iη) Aλ2 1  The lengths of the three sides of the triangle  − − −    are determined from CP-conserving observables. where λ 0.224andA, ρ,η arecountedasorder unity an≈d corrections are of order λ4. It is the Semileptonic decays. |Vcb|=0.041±0.002 sets the scale of the sides of the triangle and is deter- great achievement of heavy quark theory of the mined from exclusive [21] and inclusive semilep- 1990s to have determined V , i.e. A, to the cb | | tonic B decays [22,23]. Both methods rely on accuracyofafewpercent,whereasdeterminingρ the heavy quarkexpansion. The currenterroron and η with this accuracy remains a challenge for V is not a limiting factor in the determination this decade. The unitarity of the CKM matrix | cb| of (ρ¯,η¯), but it may become important for rare leads to a number of relations between rows and kaondecays,whichdependonAtoahighpower. columns of the matrix. The one which is most The inclusive method has probably reached its useful for B-physics is obtained by multiplying intrinsic limits [24]. Further improvement then the first column by the complex conjugate of the depends on how well the B D(∗) form factors third: → can be computed with lattice QCD. V V∗ +V V∗ +V V∗ =0. (8) The determination of V uses semileptonic ud ub cd cb td tb ub | | b u decays and gives Ifη =0(whichimpliesCPviolation)thisrelation → 6 can be represented as a triangle in the complex 1 λ2/2 V plane, called the unitarity triangle. See Figure 1, ρ¯2+η¯2 = − ub . (9) λ V which also introduces some notation for the an- p (cid:12)(cid:12) cb(cid:12)(cid:12) gles of the triangle that will be referred to later However, V /V (cid:12)0.08(cid:12)5 is currently known ub cb (cid:12) (cid:12) | | ≈ on. only within an error of about 20%. V can ub ± | | The hierarchyofthe CKMmatrix implies that alsobedeterminedfrominclusiveorexclusivede- CP violation is a small effect in the Standard cays. Theinclusivetreatmentwouldparallelthat 5 of V if not the background from b c transi- Since V is already determined by the unitar- cb ts | | → tions had to be suppressed. Distributions in var- ity of the CKM matrix, ∆M alone does not Bs ious kinematic variables (lepton energy,hadronic constrain the unitarity triangle further. How- invariantmass)havebeenconsideredattheprice ever,theratio(∆M /∆M )1/2alsodetermines Bd Bs of a more complicated and uncertain theory. A ((1 ρ¯)2 + η¯2)1/2 and involves only the ratio − cut on the leptonic and hadronic invariant mass ξ = f B1/2/f B1/2, which is believed to be avoids the kinematic region, where the heavy knownBfsromBs latBtidceBQdCD with an error ( 6%) quark expansion (in local operators) is invalid, smaller than the error on each of the had±ronic but the scale of the expansion is now around parameters individually. The current lower limit 2GeV rather than mb [25–27]. While the ulti- on∆MBs thenprovidesanimportantupperlimit mate accuracy of this method is not known, it on the length of the relevantside of the unitarity shouldbepossibletohalvetheerroron Vub . The triangle,which turns into the mostimportantre- | | exclusive determination of Vub from B Mlν strictiononthe upper limitfor the angleγ in the | | → must rely on lattice QCD for the B M form combined (ρ¯,η¯) fit. → factor. V is extracted by comparing the lep- ub ton inva|rian|t mass spectrum at largeq2 with the 3.2. CP violation in kaon decays form factor in this region, where it is computed CP violation in mixing (indirect, 1964). Due most reliably at present. The exclusive method to CP violation in KK¯ mixing, the neutral kaon is not yet competitive, but it will certainly play mass eigenstates are superpositions of CP-even an important role in an accurate determination and CP-odd components. The long-lived kaon ofT|Vhueb|Vin t-hceonfusttruarien.ton(ρ¯,η¯)isparticularlyim- sbtuatted,eKcaLys≈inKto2+twǫ¯oKp1,ioisnsprtehdroomugihnaintstlysmCaPll-oCdPd-, ub portant|, si|nce it is the only constraint that is even component K1. The decay KL ππ con- → based on a tree decay and therefore arguably in- stituted thefirstobservationofCPviolationever sensitive to non-Standard Model interactions. It [1]. The quantity ǫ =2.27 10−3 (equal to ǫ¯ in | | · | | constrains the parameters of the CKM matrix the standard phase convention to very good ac- even in the presence of new physics and helps to curacy)hasnowbeenmeasuredinmanydifferent define the Standard Model reference point. The ways. error on V is also a major component of the KK¯ mixing is dominated by top and charm ub errorin th|e in|direct determination of sin2β from box diagrams. The long-distance contributions the globalfit to (ρ¯,η¯) discussedbelow. Its reduc- are encoded in the matrix element of a four- tionwouldthereforesharpenthe consistencytest fermion operator similar to BB¯ mixing, conven- with the direct measurement of sin2β. tionally parameterized by BK and computed in BB¯ mixing. In the Standard Model the BB¯ lattice QCD.Thereiscurrentlyanuncertaintyof mass difference is dominated by the top quark at least 15% on this parameter. ǫ determines ± | | box diagram, proportional to V V∗ 2 (q =d,s). η¯(1.3 0.05 ρ¯). This is the fourth and last This determines the length of|thteq rtebm| aining side constra±int tha−t enters the “standard” unitarity of unitarity triangle. The large mass of the top triangle fit. quarkimpliesthatthemassdifferencecanbecal- CP violation in decay (direct, 1999). CP- culated up to the matrix element of the local op- violating effects can also be seen in the interfer- erator (q¯b) (q¯b) , conventionally parame- enceoftwodecayamplitudeswithdifferentCKM V−A V−A terized by f2 B . For ∆M one obtains phases. The double ratio Bq Bq Bd (1 ρ¯)2+η¯2 =(0.83 0.03) fBdBB1/d2. (10) ΓΓ((KKL →ππ00ππ00))ΓΓ((KKS →ππ++ππ−−)) S L − ± × 230MeV → → ǫ′ Tphe use of this result is limited by an error of 1 6Re , (11) about ±15% on the quantity fBdBB1/d2. Only lat- ≈ − (cid:18)ǫ(cid:19) tice QCD can possibly improve upon this error. ifdifferentfromunity,impliessuchaneffect,since 6 both ratios would equal ǫ, if CP violation oc- It has also been demonstrated that, in princi- | | curred only in mixing. The existence of this ef- ple,latticeQCDcansettlethematrixelementis- fect has been conclusively demonstrated by two sue definitively, since K ππ matrix elements → experiments in 1999,following the first hints of a computed in a lattice of finite volume, can be non-vanishing ǫ′/ǫ in 1992 [28]. The new results matchedtocontinuum,infinite-volumematrixel- of2001havefurtherclarifiedthe situation,which ements including all information on rescattering is now summarized by [29,30] [34,35]. Due to the potential cancellations the matrix elements are needed with high precision. ǫ′ (15.3 2.6) 10−4 NA48 (97-99) = ± · (12) They are also needed soon, since further CP- ǫ (20.7 2.8) 10−4 KTeV (96/97). (cid:26) ± · violatingobservablesthatwillbemeasuredinthe The theory of ǫ′/ǫ is more complicated than future will diminish the importance of ǫ′/ǫ. Rare kaon decays (future). There exist sev- that of any other quantity discussed so far. The eral proposals to measure the very rare decays short-distance contributions are many-fold, but K+ π+νν¯ and K π0νν¯ with expected have been worked out to next-to-leading order L → → branchingfractionsofabout7 10−11and3 10−11, [31,32]. The following, approximate representa- · · respectively. The first of these decays is CP- tion of the result, conserving and constrains (ρ¯,η¯) to lie on a cer- ǫ′ ImV∗V 110MeV 2 tain ellipse in the (ρ¯,η¯)-plane. The second decay =16 10−4 ts td (13) ǫ · 1.2 10−4 m (2GeV) is CP-violating and determines η¯. The branch- (cid:20) · (cid:21)(cid:18) s (cid:19) ingfractionsarepredictedtheoreticallywithhigh m¯ 2.5 B(1/2)(1 Ω ) 0.4 t B(3/2) , precision,sothatthesetwokaonmodesalonecan ×( 6 − IB − 165GeV 8 ) in principle fix the shape of the unitarity trian- (cid:16) (cid:17) gle, or uncover inconsistencies with other con- QCD penguin EW penguin | {z } straints [36]. Two K+ π+νν¯ events have in | {z } → illustratesthe difficulty thatarisesfromacancel- fact been observed [37] resulting in a branching lation between strong and electroweak penguin fraction somewhat larger than expected but con- contributions and the need to know the hadronic sistentwithexpectationswithintheexperimental matrix elements B ππ O K , which involve error. i i ∝ h | | i a two-pion final state, accurately. Before 1999 it 3.3. Summing up was commonly, though not universally, assumed The four quantities V /V , ∆M and ǫ thatB6,8 1neartheirvacuumsaturationvalue, | ub cb| Bd,s | | ≈ are usually combined into a global fit of (ρ¯,η¯). and with the isospin breaking factor Ω 0.25, IB this gives only about 6 10−4. The experi≈mental Differentgroupsusedifferentstatisticalmethods, · but since the dominant errors of all input quan- resulthastriggeredalargetheoreticalactivitydi- tities are theoretical, no sophisticated procedure rectedtowardsunderstandingbetterthehadronic can conceal the fact that there is a difficulty in matrix elements. Different approaches continue quantifyingsucherrorsobjectively. Currentlythe to disagree by large factors, but it appears now various procedures appear to give similar results certain that serious matrix element calculations when the same inputs are used. Figure 2 shows mustinonewayoranotheraccountforfinalstate the result of one such global fit [38]. interactionsofthetwopions. Chiralperturbation The four quantities are in remarkable agree- theorycombinedwithalarge-N matchingofthe c ment. This results in an indirect determination non-leptonic operators can probably go furthest of the angles of the unitarity triangle,in particu- towards this goal with analytic methods. The calculationreportedin[33]findsB(1/2) enhanced lar 6 by a factor 1.55 through rescattering and this, sin(2β)=0.68 0.21, (14) together with a reevaluation of isospin-breaking, γ =(58 24)◦.± (15) may account for the experimental result within ± theoretical uncertainties. As discussed above the precision of the indirect 7 2Imλ 1 λ2 = sin(∆M t) −| | cos(∆M t). 1+ λ2 B − 1+ λ2 B | | | | 1 InthespecialcasethatAisdominatedbyasingle ∆m ∆m/∆m d s d weak phase, A = AeiδW (so that λ = 1), the | | | | asymmetryisproportionalto sin2(β+δ ),the |ε| ∆md ± W K sign depending on the CP eigenvalue of f. sin 2β ThefinalstateJ/ψKS (andrelatedones)satis- WA η 0 fiesthisspecialconditiontotheaccuracyofaper- |Vub/Vcb| cent. FurthermoreδW ≈0forb→cc¯s. Hencethe mixing-inducedCPasymmetryinB J/ψKde- |εK| cay determines the BB¯ mixing phase→(relative to b cc¯s), or sin(2β) in the StandardModel, with -1 lit→tle theoretical uncertainty [39,40]. It deter- minestheBB¯ mixingphasealsobeyondtheStan- CKM f i t t e r dard Model, since it is unlikely that the CKM- -1 0 1 2 favoured b cc¯s transition acquires a large CP- ρ → violating phase from new flavour-changing inter- actions. Figure 2. Summary of unitarity triangle con- The asymmetry is now precisely measured by straints (excluding the direct measurement of the two B factories. The central values reported sin(2β) which is overlaid)[38]. by both experiments have been increasing over the past year as the statistics of the experiments improved and now reads [9,10] fitrelies onthe accuracyto whicha few hadronic 0.59 0.15 BaBar matrix elements are known. It is therefore clear sin(2β)= ± (17) 0.99 0.15 Belle, that the future of the standard unitarity triangle (cid:26) ± fit(basedonthefourquantitiesabove)isnowen- yielding the world average sin(2β)=0.79 0.10. tirelyinthehandsoflatticeQCD(upto,perhaps, ± The fact that this asymmetry is large and in V ). | ub| agreementwith the indirect determination of the angleβ leadstotwoimportantconclusionsonthe 3.4. sin(2β) nature of CP violation: In2001CP violationhasbeen observedalsoin B meson decays, more precisely in the interfer- CP is not an approximate symmetry of na- ence of mixing and decay. Assume that both, B0 • ture (as could have been if CP violation and B¯0, can decay into a CP eigenstate f, call in kaon decays were caused by some non- the amplitude of the former decay A, the latter standard interactions). A¯ and define λ=e−2iβA¯/A with 2β the phase of the BB¯ mixing amplitude (standard phase con- the Kobayashi-Maskawa mechanism of CP • vention). A B meson identified as B0 at time violationismostlikelythedominantsource t = 0 can decay into f at a later time t either of CP violation at the electroweak scale. directly or indirectly through its B¯0 component Newflavour-changinginteractionscertainlycould acquired by mixing. If there is CP violation, the haveaffectedBB¯ mixingandcouldhavebeenre- amplitude for the CP conjugate process will be vealedfirstbythedirectsin(2β)measurement. If different, resulting in a time-dependent asymme- the CKM matrix were the only source of flavour- try changing processes also in an extension of the Γ(B¯0(t) f) Γ(B0(t) f) Standard Model, then the new interactions that ACP(t)= Γ(B¯0(t)→f)+−Γ(B0(t)→f) (16) modify BB¯ mixing also affect KK¯ mixing. One → → 8 then finds (if one adds an additional assumption with different weak phases, thattherearenonewoperatorsinthelow-energy A(B f)=A eiδS1eiδW1 +A eiδS2eiδW2. (18) effectiveweakHamiltonian)thatonlysmallmod- 1 2 → ifications of the BB¯ mixing phase (still related If the stronginteractionphasesarealsodifferent, to the phase of V in such models of “minimal td the partial width of the decay differs from that flavour violation”) could have been possible, in of its CP-conjugate, Γ(B f) = Γ(B¯ f¯). particular sin(2β) > 0.42 with a preferred range → 6 → Many rare B decays are expected to exhibit CP from0.5to0.8[41–44]. Viceversatheobservation violationindecayduetotheinterferenceofatree of a very small or very large sin(2β) would have and a sizeable or even dominant penguin ampli- implied a new mechanism of flavour violation tude. The weak phase difference can only be de- with (probably) new CP phases. As discussed termined, however, if the strong interaction am- below in a more generalcontext, this would have plitudes are known. This is also necessary for led to a CP problem. mixing-inducedCPasymmetries,ifthedecayam- plitude is not dominated by a single term. 4. MORE CP VIOLATION IN B MESON There exist two complementary approaches to DECAYS obtain the strong interaction amplitudes. The first employs a general parameterization of the The search for CP violation will continue in decay amplitudes of a set of related decays, im- many ways (B decays, D decays, K decays, plementing SU(2)-isospin relations. The remain- electric dipole moments), but the Kobayashi- ing strong interactionparameters are then deter- Maskawa mechanism predicts large effects only mined from data (often also using SU(3) flavour in B decays and very rare K decays. The pri- symmetryand“little”furtherassumptionsonthe mary focus of the coming years will be to verify magnitudes of some amplitudes). Very often this relations between different observables predicted needs difficult measurements. The second ap- intheKobayashi-Maskawascenarioandtosearch proach attempts to calculate the strong interac- for small deviations. tion amplitudes directly from QCD with factor- AnexampleofthistypeisB¯d φK duetothe ization methods also used in high-energy strong → penguin b ss¯s transition at the quark level. interaction processes. This approach makes es- → In the Standard Model the time-dependent CP sential use of the fact that the b quark mass is asymmetry of this decay is also proportional to large. There is currently no theoretical frame- sin(2β) to reasonable (though not as good) pre- work that also covers 1/m -corrections system- b cision. However,new interactions are more likely atically, so there is an intrinsic limitation to the toaffecttheloop-inducedpenguintransitionthan accuracythatonecanexpectfromthisapproach. thetreedecayb cc¯sandmayberevealedifthe Nonetheless, the additional information on the time-dependent→asymmetry in B¯d φK turns dynamics of the decay providedby this approach out to be different from that in B¯→d J/ψK. is important as long as data is sparse and will → However, if the difference is small, its interpre- continue to be useful later on. tation requires that one controls the strong in- teractioneffects connectedwith the presenceofa 4.2. The angle γ small up-quark penguin amplitude with a differ- In the Standard Model the angle β is obtained entweakphase. Thisdifficultyisofaverygeneral accuratelyfromthetime-dependentCPasymme- nature in B decays. try in Bd J/ψK. It remains to determine di- → rectly the angle γ, the phase of V∗. ub 4.1. CP violation in decay The preferred methods rely on decays with in- The need to control strong interaction effects terference of b cu¯D (no phase) and b uc¯D → → is closely related to the possibility of observing (phase γ) transitions and their conjugates (D = CP violation in the decay amplitude. The decay d,s). These decays receive no penguin contribu- amplitude has to have at least two components tionsandarearguablyinsensitivetonewflavour- 9 changing interactions. γ can be extracted from 0.6 either of the following decay classes, B (t) d → 0.5 D±π∓ [45], B (t) DK [46], B± K±D d S CP [47],B (t) D±K→∓[48],sinceeveryo→neofthem 0.4 providess su→fficiesntly many observables to elimi- 0.3 (cid:22)(cid:17) nate all strong interaction parameters, thus pro- 0.2 vidingnice illustrationsofthe firstofthe twoap- 0.1 proaches mentioned above. None of these strate- 0 gies is simple to carry out experimentally, how- -0.6 -0.4 -0.2 0 0.2 0.4 0.6 PSfrag repla ements ever,since they involveeither smallCP asymme- tries, or small branching fractions, or disparate (cid:26)(cid:22) amplitudes, or rapid B oscillations. s Figure 3. 95% (solid), 90% (dashed) and 68% Thepossibilitytodetermineγfromdecayswith (short-dashed) confidence level contours in the interference of b uu¯D (tree, phase γ) and → (ρ¯,η¯) plane obtained from a global fit to the CP b Dqq¯(penguin, phase 0 (D = s), β (D =d)) → averaged B πK,ππ branching fractions, using transitionshasthereforebeenthoroughlyinvesti- → the scanning method [38]. The darker dot shows gatedrecently,inparticularthe decaysB πK. → the overall best fit, whereas the light dot indi- Thebranchingfractionsforthesemodesareofor- cates the best fit for the default parameter set. der10−5andhavealreadymeasuredwithanerror The light-shaded region indicates the region pre- of (10 20)%, including first measurements of ± − ferred by the standard global fit, excluding the directCPasymmetries(allcompatiblewithzero). direct measurement of sin(2β). The drawback of these and related modes is that the amplitudes contain more strong interaction parameters than there are observables. SU(3) symmetry andthe structure ofthe weakeffective turbativehard-scatteringkernels,whichalsocon- Hamiltonian allow one to construct a number of tainthestrongrescatteringphases. Thisresultis interesting bounds on γ [49,50], but a full under- valid up to 1/m corrections, some of which can b standing of these modes requires a calculation of be large. The extent to which the QCD factor- the penguin-to-tree amplitude ratio including its ization formalism can be of quantitative use is strong rescattering phase. In the following I de- not yetfully known. The approachhas been suc- scribe very briefly one such method. cessful in explaining the universality of strong- interactioneffectsinclass-IB D+lightmeson 4.3. QCD factorization → decaysandunderstandingthenon-universalityin Intheheavyquarklimitthebquarkdecaysinto the corresponding class-II decays [52]. It also very energetic quarks (and gluons), which must appears to account naturally for the magnitude recombine to form two mesons. Methods from of the πK branching fractions, sometimes con- the heavyquarkexpansionandsoft-collinearfac- sidered as unexpectedly large, but there is cur- torization(“colour-transparency”)can the be ar- rently no test that would allow one to conclude gued to imply a factorizedform of the amplitude thatthecomputationofstronginteractionphases ofadecayintotwolightmesons[51,52]. Schemat- whichareeither oforderα or1/m isreliablein s b ically, the case of penguin-dominated final states [53]. 1 Such tests will be possible soon and the non- A(B¯ M M )=FB→M1(0) duTI(u)Φ (u) → 1 2 M2 observation of direct CP violation at the current Z0 level of sensitivity alreadysupports the idea that + dξdudvTII(ξ,u,v)ΦB(ξ)ΦM1(v)ΦM2(u),(19) strong rescattering effects are suppressed. Z Figure 3 shows the result of a global fit of where FB→M1 is a form factor, Φ denote light- (ρ¯,η¯) to CP-averaged B πK,ππ branching X → cone distribution amplitudes and TI,II are per- fractions with a QCD factorization computation 10 used as an input [53]. The result is consistent thattheStandardModelhasnotfinallygivenway with the standardfitbasedonmesonmixingand to a more fundamental theory. However, return- V ,butshowsapreferenceforlargerγ orsmaller ing to the perspective of the year 1973,when the ub V . Iftheestimateofthetheoryuncertainty(in- mechanismwasconceived,onecanhardlyfeelthis ub cludedinthecurvesintheFigure)iscorrect,non- way. After all, the Kobayashi-Maskawa mecha- leptonic decays together with V fromsemilep- nism predicted a new generation of particles on ub | | tonicdecaysalreadyimply theexistenceofaCP- the basis of the tiny and obscure effect of CP vi- violating phase of V . olation in KK¯ mixing. It then predicted rela- ub tions between CP-violating quantities in K, D, 4.4. Resum´e B-physics which a priori might be very different. The B factories are already providing data on The fact that it has taken nearly 30 years to as- dozens ofrareB decaymodes. QCDcalculations semble the experimental tools to test this frame- – thoughprobably not veryprecise – will be nec- work does not diminish the spectacular fact that essary to interpret these data beyond “simple” once again Nature has realized a structure that quantitieslikethemixing-inducedCPasymmetry was concepted from pure reasoning. in Bd J/ψK. The immediate future shouldbe Nevertheless several arguments make it plau- → very interesting since the measurements of direct sible that the Kobayashi-Maskawa mechanism is CP asymmetries at the few percent level and the not the final word on CP violation. The strong mixing-induced CP asymmetry in Bd π+π− andcosmologicalCPproblem(baryogenesis)con- → decay provide tests of the theoretical framework tinue to call for an explanation, probably related and further information on CP violation. Subse- to high energy scales. There may be an aesthetic quent second generation B physics experiments appeal to realizing the full Poincar´e group as a will probably supply enough data to rely more symmetry of the Lagrangian, in which case CP and more on measurements and symmetries. Al- and P symmetry breaking must be spontaneous. togetherthe Kobayashi-Maskawamechanismwill One of the strongest arguments is, however,that be decisively and precisely tested, but on the the electroweak hierarchy problem seems to re- wayonecanexpectmanydiscussionsonhadronic quire an extension of the Standard Model at the physics,imaginedand,perhaps,truenewphysics TeV scale. Generic extensions have more sources signals. of CP violation than the CKM matrix. These Latticecalculationswillcontinuetoplayanim- have not (yet) been seen, suggesting that there portantrolebymakingmoreprecisethestandard is some unknown principle that singles out the unitarity triangle fit. They could also provide CKM matrix as the dominant source of flavour some of the non-perturbative quantities (form andCPviolation. Inthe followingI givearather factors, light-cone-distribution amplitudes) that colloquialoverviewofCPviolationingenericTeV are needed in factorization-based calculations of scale extensions of the Standard Model. This is non-leptonic decay amplitudes. It will be much perhaps an academic catalogue, but it illustrates harder for lattice QCD to make an impact on how restrictive the Kobayashi-Maskawa frame- non-leptonic, exclusive decays directly, since in- work is. elastic rescattering dominates final-state interac- tionsandthereiscurrentlynomethodthatwould 5.1. Extended Higgs sector allow one to compute this on the lattice. Extending the Higgs sector by just a second doublet opens many new possibilities. The Higgs potential may now contain complex couplings, 5. CPVIOLATION INEXTENSIONSOF leading to Higgs bosons without definite CP par- THE STANDARD MODEL ity,toCPviolationinchargedHiggsinteractions, The emerging success of the Kobayashi- flavour-changing neutral currents, and CP viola- MaskawamechanismofCPviolationissometimes tioninflavour-conservinginteractionssuchastt¯H accompanied by a sentiment of disappointment and electric dipole moments. The Lagrangian