Table Of Content

The CKM matrix and CP violation (in the continuum 6 approximation) 0 0 2 P n a o J 7 S 1 ZoltanLigeti ∗ ( ErnestOrlandoLawrenceBerkeleyNationalLaboratory,UniversityofCalifornia,Berkeley,CA94720 1 L v and 2 CenterforTheoreticalPhysics,MassachusettsInstituteofTechnology,Cambridge,MA02139 A 2 E-mail: [email protected] 0 T 1 0 2 6 Thefirstpartofthistalkreviewsrecentdevelopmentsinflavorphysicsthatcanbemadewithout 0 detailedunderstandingofhadronicphysics,drivenbythedata. Theerrorofsin2b hasshrunkbe- 0 / at low5%,andthemeasurementsofa andg havereachedinterestingprecisions. Forthefirsttime, 0 -l therearesignificantconstraintsonthedeviationsfromthestandardmodelinB Bmixingandin 5 p − e b sandb d transitions. Inthesecondpart,Ireviewsometheoreticaldevelopmentsforex- ) → → h clusivesemileptonicandnonleptonicBdecaysthathavebecomepossibleusingthesoft-collinear : 0 v effectivetheory. Iconcentrateontopicswheretherecentprogresshasmodelindependentimpli- i 1 X cationsforinterpretingthedata. LBNL-59133,MIT-CTP3715 r 2 a XXIIIrdInternationalSymposiumonLatticeFieldTheory 25-30July2005 TrinityCollege,Dublin,Ireland Speaker. ∗ (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ TheCKMmatrixandCPviolation ZoltanLigeti Contents 1. Introduction 3 1.1 Testingtheflavorsector 3 1.2 Constraints fromK andDdecays 4 2. CPviolation inBdecaysandthemeasurementofsin2b 5 2.1 CPviolation indecay 5 2.2 CPVinmixing 5 2.3 CPVintheinterference betweendecaywithandwithoutmixing: B y K 6 S,L → 2.4 OtherCPasymmetries thatareapproximately sin2b intheSM 7 P 3. Measurementsofa andg 8 o 3.1 a fromB pp ,rr andrp 9 → S 3.2 g fromB DK 10 ± ± → ( 4. Implicationsofthea andg measurements 11 L 4.1 NewphysicsinB0 B0 mixing 12 − A 5. Inclusivesemileptonicdecays 14 T 5.1 V andm fromB X ℓn¯ 14 2 cb b c | | → 5.2 B Xuℓn¯,Xsg andXsℓ+ℓ− 15 0 → 0 6. Exclusivesemileptonicheavytolightdecays 16 6.1 Whatweknewafewyearsago 16 5 6.2 A"one-page" introduction toSCET 17 ) 6.3 Thesemileptonic formfactorsand V 18 ub 0 | | 6.3.1 B rg vs.K g 19 → ∗ 1 6.4 Photonpolarization inB K g andX g 19 ∗ s → 2 6.5 CommentsonB tn 20 → 7. Nonleptonicdecays 21 7.1 Factorization inB Dp typedecays 21 → 7.1.1 Colorsuppressed B D( )0M0 decays 22 ∗ → 7.1.2 L b L cp andS (c∗)p decays 22 → 7.2 CharmlessB M M decays 23 1 2 → 7.2.1 B pp amplitudes andCPasymmetries 24 → 8. OutlookandConclusions 26 8.1 Wherecanlatticecontribute themost? 26 8.2 Pastandnearfuturelessons 27 References 28 012/2 TheCKMmatrixandCPviolation ZoltanLigeti 1. Introduction Inthe last fewyears thestudy ofCPviolation and flavor physics has undergone dramatic developments. While for 35 years, until 1999, the only unambiguous measurement ofCP violation (CPV) was e [1], the constraints on the Cabibbo-Kobayashi-Maskawa (CKM) matrix [2, 3] im- K proved tremendously since the B factories turned on. The error of sin2b is now below 5%, and a newsetofmeasurementsstartedtogivethebestconstraints ontheCKMparameters. In the standard model (SM), the masses and mixings of quarks originate from their Yukawa interactions with the Higgs condensate. We do not understand the hierarchy of the quark masses and mixing angles. Moreover, if there is new physics (NP) at the TeV scale, as suggested by the hierarchy problem, then it is not clear why it has not shown up in flavor physics experiments. A four-quark operator (sd¯)2/L 2 with O(1) coefficient would give a contribution exceeding the P NP measuredvalueofe unlessL >104TeV. Similarly,(db¯)2/L 2 yieldsD m aboveitsmeasured o K NP NP Bd valueunlessL >103TeV. Flav∼orphysicsprovidessignificantconstraintsonNPmodelbuilding; S NP ∼ for example generic SUSYmodels have 43 newCP violating phases [4, 5], and wealready know ( thatmanyofthemhavetobesuppressed nottocontradict theexperimental data. L Flavor andCP violation were excellent probes of new physics in the past: (i) the absence of A K m +m predicted thecharm quark; (ii)e predicted thethirdgeneration; (iii)D m predicted L − K K → the charm mass; (iv) D m predicted the heavy top mass. From these measurements we knew T B already before the B factories turned on that if there is NP at the TeV scale, it must have a very 2 special flavorandCPstructure tosatisfy theseconstraints. Sowhatdoesthenewdatatellus? 0 Sections2–4summarizethestatusofCPviolationmeasurementsandtheirimplicationswithin 0 andbeyondtheSM,concentrating onmeasurementswherethedatacanbeinterpreted withoutde- 5 tailed understanding of the hadronic physics. Sections 5–7 deal with some recent model indepen- ) denttheoretical developments andtheirimplications. 0 1.1 Testingtheflavorsector 1 2 Theonly interaction that distinguishes between the fermion generations is their Yukawa cou- plingstotheHiggscondensate. ThissectoroftheSMcontains10physicalquarkflavorparameters, the 6 quark masses and the 4parameters in the CKMmatrix: 3 mixing angles and 1CP violating phase (for reviews, see, e.g., [5, 6]). Therefore, the SM predicts intricate correlations between dozens of different decays of s, c, b, andt quarks, and in particular betweenCP violating observables. Possibledeviations fromCKMparadigm mayupsetsomepredictions: Subtle (or not so subtle) changes in correlations, e.g., constraints from B and K decays in- • consistent, orCPasymmetries notequalinB y K andB f K ,etc.; S S → → Flavor-changing neutral currents at an unexpected level, e.g., B mixing incompatible with s • SM,enhanced B ℓ+ℓ ,etc.; (s) − → Enhanced (orsuppressed)CPviolation, e.g.,inB K g orB yf . ∗ s • → → ThegoaloftheprogramisnotjusttodetermineSMparametersaspreciselyaspossible,butto test by many overconstraining measurements whether all observable flavor-changing interactions can be explained by the SM, i.e., by integrating out virtual W and Z bosons and quarks. It is 012/3 TheCKMmatrixandCPviolation ZoltanLigeti (r,h) a V V* V V* ud ub td tb V V* V V* cd cb cd cb g b (0,0) (1,0) Figure1: Sketchoftheunitaritytriangle. convenient tousetheWolfenstein parameterization1 oftheCKMmatrix, V V V 1 1l 2 l Al 3(r¯ ih¯) P ud us ub −2 − VCKM =Vcd Vcs Vcb= −l 1−12l 2 Al 2 +..., (1.2) o V V V Al 3(1 r¯ ih¯) Al 2 1  td ts tb  − − −  S which exhibits its hierarchical structure by expanding in l 0.23, and is valid to order l 4. The ( ≃ unitarity of the CKM matrix implies (cid:229) V V = d and (cid:229) V V = d , and the six vanishing L i ij i∗k jk j ij k∗j ik combinations can be represented by triangles in a complex plane. The ones obtained by taking A scalarproducts ofneighboring rowsorcolumnsarenearlydegenerate, sooneusuallyconsiders T V V +V V +V V =0. (1.3) 2 ud u∗b cd c∗b td t∗b 0 Agraphicalrepresentationofthisistheunitaritytriangle,obtainedbyrescalingthebest-knownside 0 to unit length (see Fig. 1). Its sides and angles can be determined in many "redundant" ways, by measuringCP violating and conserving observables. Comparing constraints on r¯ and h¯ provides 5 aconvenient language tocomparetheoverconstraining measurements. ) 0 1.2 ConstraintsfromK andDdecays 1 Weknewfrom themeasurement ofe thatCPVintheK system isatalevelcompatible with K 2 theSM,ase canbeaccommodated withanO(1)valueoftheKMphase[3]. Theotherobserved K CP violating quantity in kaon decay, e , is notoriously hard to interpret, because for the large top K′ quark mass the electromagnetic and gluonic penguin contributions tend to cancel [10], thereby significantlyamplifyingthehadronicuncertainties. Atpresent,wecannotevenruleoutthatalarge partofthemeasuredvalueofe isduetoNP,andsowecannotuseittoteststheKMmechanism. K′ In the kaon sector precise tests will come from the study of K pn n¯ decays. The K p 0n n¯ L → → decayisCPviolating,andthereforetheoreticallyveryclean,andthereisprogressinunderstanding thelargest uncertainties inK p n n¯ duetocharm andlight quarkloops[11,12]. Inthismode ± ± → threeeventshavebeenobservedsofar,yielding[13] B(K+ p +n n¯)= 1.5+1.3 10 10. (1.4) → 0.9 × − − (cid:0) (cid:1) 1Weusethefollowingdefinitions[7,8,9],sothattheapexoftheunitaritytriangleinFig.1isexactlyr¯,h¯: l = p|Vud|V|2u+s||Vus|2, A= l1(cid:12)(cid:12)(cid:12)VVcubs(cid:12)(cid:12)(cid:12), Vu∗b=Al 3(r +ih )= √A1l−3l(r2¯[+1−ih¯A)√2l14−(r¯A+2lih4¯)]. (1.1) (cid:12) (cid:12) 012/4 TheCKMmatrixandCPviolation ZoltanLigeti Thisisconsistent withtheSMwithinthelargeuncertainties, butmuchmorestatistics isneededto makedefinitivetests. The D meson system is complementary to K and B mesons, because flavor andCP violation are suppressed both by the GIM mechanism and by the Cabibbo angle. Therefore, CPV in D decays, rare D decays, and D D mixing are predicted to be small in the SM and have not been − observed. This is the only neutral meson system in which mixing generated by down-type quarks in the SM (or up-type squarks in SUSY). The strongest hint for D0 D0 mixing is the lifetime − difference betweentheCP-evenand-oddstates[14] G (CPeven) G (CPodd) y = − =(0.9 0.4)%. (1.5) CP G (CPeven)+G (CPodd) ± Unfortunately, due to hadronic uncertainties, this central value alone could not be interpreted as a P sign of new physics [15]. At the present level of sensitivity, CPV or enhanced rare decays would o betheonlycleansignalofNPintheDsector. S ( 2. CP violationinB decays and the measurement ofsin2b L 2.1 CPviolation indecay A Thisisthesimplest form ofCPviolation, whichcan beobserved in both charged and neutral T mesonaswellasinbaryon decays. Ifatleasttwoamplitudes withnonzero relativeweak(f )and k 2 strong(d )phases contribute toadecay, k 0 Af =hf|H |Bi=(cid:229) Akeidkeifk, Af =hf|H |Bi=(cid:229) Akeidke−ifk, (2.1) 0 k k 5 thenitispossible that A /A =1,andthusCPisviolated. | f f|6 ) ThistypeofCPviolationisunambiguously observedinthekaonsectorbye =0,andnowit K′ 6 0 isalsoestablished inBdecays [16,17], 1 G (B0 K p +) G (B0 K+p ) − − AK−p + ≡ G (B0→K p +)−+G (B0→K+p ) =−0.115±0.018. (2.2) 2 − − → → Thisissimplyacountingexperiment: thereare 20%moreB0 K+p thanB0 K p +decays. − − ∼ → → Thismeasurement implies thatafter the"K-superweak" model[18], nowalso"B-superweak" models are excluded. I.e., models in which CP violation only occurs in mixing are no longer viable. Thismeasurementalsoestablishesthattherearesizablestrongphasesbetweenthetree(T) andpenguin (P)amplitudes incharmless Bdecays, since T/P isestimatedtobenotmuchlarger | | than AK p + . Such information on strong phases will have broader implications for charmless | − | nonleptonic decays andforunderstanding theB Kp andpp ratesdiscussed inSec.7.2.1. → Thebottomlineisthat,similartoe ,ourtheoreticalunderstanding atpresentisinsufficientto K′ eitherproveorruleoutthattheCPasymmetryinEq.(2.2)isduetoNP. 2.2 CPVinmixing ThetwoBmesonmasseigenstates arerelatedtotheflavoreigenstates via B = pB0 qB0 . (2.3) L,H | i | i± | i 012/5 TheCKMmatrixandCPviolation ZoltanLigeti CPisviolatedifthemasseigenstatesarenotequaltotheCPeigenstates. Thishappensif q/p =1, | |6 i.e., if the physical states are not orthogonal, B B = 0, showing that this is an intrinsically H L h | i 6 quantum mechanical phenomenon. The simplest example of this type of CP violation is the semileptonic decay asymmetry to "wrongsign"leptons. Themeasurementsgive[19] G (B0(t) ℓ+X) G (B0(t) ℓ X) 1 q/p4 ASL= G (B0(t)→ℓ+X)−+G (B0(t)→ℓ−X) = 1−+|q/p|4 =−(3.0±7.8)×10−3, (2.4) → → − | | implying q/p =1.0015 0.0039, wheretheaverageisdominatedbyarecentBELLEresult[20]. | | ± In semileptonic kaon decays the similar asymmetry was measured [21], in agreement with the expectation thatitisequalto4Ree . The calculation of A is possible from first principles only in the m L limit, using P SL b QCD ≫ anoperator product expansion toevaluate therelevant nonleptonic rates. LastyeartheNLOQCD o calculation was completed [22, 23], predicting A = (5.5 1.3) 10 4, where I averaged the SL − S − ± × central values and quoted the larger of the two theory error estimates. (The similar asymmetry ( in the B sector is expected to be l 2 smaller.) Although the experimental error in Eq. (2.4) is s L an order of magnitude larger than the SM expectation, this measurement already constraints new physics[24],asthem2/m2 suppression ofA intheSMcanbeavoided byNP. A c b SL T 2.3 CPVintheinterference betweendecaywithandwithoutmixing: B y K S,L → 2 It is possible to obtain theoretically clean information on weak phases in B decays to certain 0 CP eigenstate final states. Theinterference phenomena between B0 f and B0 B0 f is CP CP → → → 0 described by q A q A 5 l = fCP =h fCP , (2.5) fCP p AfCP fCP p AfCP ) where h fCP =±1 is theCP eigenvalue of fCP. Experimentally one can study the time dependent 0 CPasymmetry, 1 a = G [B0(t)→ f]−G [B0(t)→ f] =S sin(D mt) C cos(D mt), (2.6) 2 fCP G [B0(t) f]+G [B0(t) f] fCP − fCP → → where 2Iml 1 l 2 f f S = , C (= A )= −| | . (2.7) f 1+ l 2 f − f 1+ l 2 f f | | | | Ifamplitudeswithoneweakphasedominateadecaythena measuresaphaseintheLagrangian fCP theoretically cleanly. In this caseC =0, and S =Iml =sin(argl ), where argl is the f fCP fCP fCP fCP phasedifference betweentheB0 f andB0 B0 f decaypaths. → → → The theoretically cleanest example of this type of CP violation is B y K0. While there → are tree and penguin contributions to the decay with different weak phases, the dominant part of the penguin amplitudes have the same weakphase as the tree amplitude. Therefore, contributions with the tree amplitude’s weak phase dominate, to an accuracy better than 1%. In the usual ∼ phaseconventionSy KS,L =∓sin[(B-mixing=−2b )+(decay=0)+(K-mixing=0)],soweexpect Sy KS,L =±sin2b andCy KS,L =0toasimilaraccuracy. Thecurrentworldaverageis sin2b =0.687 0.032, (2.8) ± 012/6 TheCKMmatrixandCPviolation ZoltanLigeti 1.5 excluded area has CL > 0.95 D m d 1 ssiinn 22bb D ms & D md 0.5 e a K g b 0 h |V /V | ub cb -0.5 e K -1 CKM f i t t e r LP 2005 P -1.5 -1 -0.5 0 0.5 1 1.5 2 o r S Figure2: ThepresentCKMfitusingthemeasurementsofe , V /V ,∆m ,andsin2b . K ub cb d,s | | ( L whichisnowa5%measurement. Inthelasttwoyearsthe2b vs.p 2b discreteambiguityhasalso − A been resolved by ingenious studies of the time dependent angular analysis of B y K 0 and the ∗ timedependentDalitzplotanalysisofB0 D0h0 withD0 K p +p ,pioneered→byBABAR[25] T S − → → andBELLE[26],respectively. Asaresult, thenegativecos2b solutions areexcluded, eliminating 2 twoofthefourdiscrete ambiguities. 0 To summarize, Sy K was the first observation of CP violation outside the kaon sector, and 0 the first observation of an O(1)CP violating effect. It implies that models with approximate CP 5 symmetry (in the sense that all CPVphases are small) areexcluded. Theconstraints on the CKM ) matrix from the measurements of Sy K, Vub/Vcb , eK, B and Bs mixing are shown in Fig. 2 using | | 0 the CKMfitterpackage [27,9]. Theresults throughout this paper are based on the latest averages, exceptfor V ,forwhichthepre-Lepton-Photon2005valueisused, V =(4.05 0.13 0.50) 1 ub ub | | | | ± ± × 10−3, as explained in Sec. 5.2. The overall consistency between these measurements wasthe first 2 precisetestoftheCKMpicture. ItalsoimpliesthatitisunlikelythatO(1)deviationsfromtheSM canoccur, andoneshouldlookforcorrections ratherthanalternatives oftheCKMpicture. 2.4 OtherCPasymmetries thatareapproximately sin2b intheSM Theb stransitions,suchasB0 f K,h K,K+K K ,etc.,aredominatedbyone-loop(pen- ′ − S → → guin)diagramsintheSM,andthereforenewphysicscouldcompetewiththeSMcontributions[28]. UsingCKMunitaritywecanwritethecontributionstosuchdecaysasatermproportionaltoV V cb c∗s and another proportional toV V . Since their ratio is about 0.02, we expect amplitudes with the ub u∗s V V weakphasetodominatethesedecaysaswell. Thus,intheSM,themeasurementsof h S cb c∗s f f − should agreewithSy K (andCf shouldvanish)toanaccuracy oforderl 2 0.05. ∼ IftheSMandNPcontributions arebothsignificant, theCPasymmetriesdependontheirrela- tive size and phase, which depend on hadronic matrix elements. Sincethese are mode-dependent, the asymmetries will, in general, bedifferent between the various modes, and different from Sy K. OnemayalsofindC substantially different from0. f 012/7 TheCKMmatrixandCPviolation ZoltanLigeti Dominant SMallowedrangeof f ∗ h S C process CP |−h fCPSfCP−sin2b | − f f f b cc¯s y K0 <0.01 +0.687 0.032 +0.016 0.046 → ± ± b cc¯d yp 0 0.2 +0.69 0.25 0.11 0.20 → ∼ ± − ± D +D 0.2 +0.67 0.25 +0.09 0.12 ∗ ∗− ∼ ± ± D+D 0.2 +0.29 0.63 +0.11 0.36 − ∼ ± ± b sq¯q f K0 <0.05 +0.47 0.19 0.09 0.14 → ± − ± h K0 <0.05 +0.48 0.09 0.08 0.07 ′ ± − ± p 0K 0.15 +0.31 0.26 0.02 0.13 S ∼ ± − ± K+K K 0.15 +0.51 0.17 +0.15 0.09 − S ∼ ± ± K K K 0.15 +0.61 0.23 0.31 0.17 S S S ∼ ± − ± P f0K 0.25 +0.75 0.24 +0.06 0.21 S ∼ ± ± ωK 0.25 +0.63 0.30 0.44 0.23 o S ∼ ± − ± S Table1:CPasymmetriesforwhichtheSMpredicts h S sin2b .The3rdcolumncontainsmyestimates f f − ≈ oflimitsonthedeviationsfromsin2b intheSM(strictboundsareworse),andthelasttwocolumnsshow ( theworldaverages[19]. (TheCP-evenfractionsinK+K−KS andD∗+D∗− aredeterminedexperimentally.) L A TheaveragesofthelatestBABARandBELLEresultsareshowninTable1. Thetwodatasets T arefairlyconsistent bynow. Thelargest hintofadeviation fromtheSMisnowintheηK mode, ′ 2 Sy K Sh K =0.21 0.10, Sy K Sf K =0.22 0.19, (2.9) 0 − ′ ± − ± 0 whichis2σ. TheaverageCPasymmetryinallb smodes,whichalsoequalsSy KintheSM,hasa bit moresignificant deviation, Sy K ηfSf(b →s) =0.18 0.07. Thisiscurrently a2.6σeffect, 5 however, this average is not too m−eahn−ingful, b→eciause som±e of the modes included may deviate ) significantly from Sy K in the SM. The third column in Table 1 shows my estimates of limits on 0 thedeviationsfromSy K intheSM.ThehadronicmatrixelementsmultiplyingthegenericO(0.05) 1 suppression of the "SM pollution" are hard to bound model independently [29], so strict bounds 2 areweaker, whilemodelcalculations obtainbetterlimits. To understand the significance of these measurements, note that a very conservative bound usingSU(3)flavorsymmetryusingthecurrentexperimentallimitsonrelatedmodesgives[29,30] Sy K Sh K <0.2 in the SM. Estimates based on factorization [31] obtain deviations at the 0.02 | − ′ | level. Thus, Sy K Sh K 0.2wouldbeasignofNPifestablishedatthe5σlevel. (Thedeviation | − ′ |≈ of Sf K from Sy K is now statistically insignificant, but the present central value with a smaller errorcouldstillestablish NP.)Suchadiscovery wouldexclude inaddition totheSM,modelswith minimalflavorviolation, anduniversal SUSYmodels,suchasgaugemediatedSUSYbreaking. 3. Measurements ofa and g Toclarify terminology, I’ll call ameasurement of γthe determination ofthe phase difference between b u and b c transitions, while α( π β γ) will refer to the measurements of γ → → ≡ − − in the presence of B B mixing. The main difference between the measurements of γand those − of the other two angles is that γis measured in entirely tree-level processes, so it is unlikely that 012/8 TheCKMmatrixandCPviolation ZoltanLigeti B → pp (S/C from BABAR) CKM +– 1.2 Morfi oi nt dt e0 5r B → pp (S/A+– from Belle) Combined no C/A 00 1 0.8 L C – 1 0.6 0.4 CKM fit 0.2 no a meas. in fit 0 0 20 40 60 80 100 120 140 160 180 a (deg) P o Figure3: Constraintsona fromB pp .Someoftheeightmirrorsolutionsmayoverlap. → S new physics could modify it. It is therefore very important in searching for and constraining new ( physics. Interestingly, thebestmethodsformeasuring bothαandγarenewsince2003. L 3.1 αfromB ππ,ρρandρπ A → T IncontrasttoB ψK,whichisdominatedbyamplitudeswithoneweakphase,inB π+π − → → there are two comparable contributions with different weak phases. Therefore, to determine α 2 model independently, it is necessary to carry out an isospin analysis [32] (for other possibilities, 0 seeSec.7.2.1). Thehardestingredients arethemeasurementoftheπ0π0 rate, 0 B(B π0π0)=(1.45 0.29) 10 6, (3.1) 5 − → ± × ) andtheCP-asymmetry, 0 G (B π0π0) G (B π0π0) → − → =0.28 0.39. (3.2) 1 G (B π0π0)+G (B π0π0) ± → → 2 If these measurements were precise, one could pin down from the isospin analysis the penguin pollution, D α≡α−αeff (2αeff =argλp +p − =arcsin[Sp +p −/(1−Cp2+p −)1/2]). In Fig. 3, the dark shaded region shows the confidence level using Eq. (3.2), while the light shaded region is the constraint without it. One finds D α <37 at 90% CL, a small improvement over the 39 bound ◦ ◦ | | without Eq. (3.2). This indicates that it will take a lot more data to determine α precisely. In addition, theBABAR[33]andBELLE[34]resultsarestillnotquiteconsistent; seeTable2. The B ρρmode is more complicated than B ππ, because a vector-vector (VV) final → → state is a mixture ofCP-even (L=0 and 2) and -odd (L=1) components. The B ππisospin → B π+π− Sp +p Cp +p → − − BABAR 0.30 0.17 0.09 0.15 − ± − ± BELLE 0.67 0.17 0.56 0.13 − ± − ± average 0.50 0.12[0.18] 0.37 0.10[0.23] − ± − ± Table2:CPasymmetriesinB p +p . ThebracketsshowtheerrorsinflatedusingthePDGperscription. − → 012/9 TheCKMmatrixandCPviolation ZoltanLigeti analysis applies foreachLinB ρρ(orforeachtransversity, and,therefore, forthelongitudinal → polarization component as well). The situation is simplified dramatically by the experimental observation thatintheρ+ρ andρ+ρ0 modesthelongitudinal polarization fraction isnearunity, so − theCP-evenfractiondominates. Thus,onecansimplyboundD αfrom[35] B(B ρ0ρ0)<1.1 10 6 (90%CL). (3.3) − → × The smallness of this rate implies that D α in B ρρ is much smaller than in B ππ. To → → appreciate the difference, note that B(B π0π0)/B(B π+π0)=0.26 0.06, while B(B → → ± → ρ0ρ0)/B(B ρ+ρ0)<0.04 (90%CL). FromSr +r andtheisospinboundonD αoneobtains → − α=96 10 4 11 (α α ). (3.4) ◦ eff ± ± ± − P UltimatelytheisospinanalysisismorecomplicatedinB ρρthaninππ,becausethefinitewidth → o of the ρallows for the final state to be in an isospin-1 state [36]. This only affects the results at theO(G r2/m2r )level, whichissmallerthanothererrors atpresent. Withhigherstatistics, itwillbe S possible toconstrain thiseffectusingthedata[36]. ( Finally, in B ρπit ispossible in principle to use a Dalitz plot analysis [37]of the interfer- L → enceregions oftheπ+π π0 finalstatetoobtainamodelindependent determination ofα,without − A discrete ambiguities. Thefirstsuchanalysisgives[38] T α=(113+2177±6)◦. (3.5) 2 − ViewingB ρπastwo-bodydecays,isospinsymmetrygivestwopentagonrelations[39]. Solving 0 → them would require measurements of the rates andCP asymmetries in all the B ρ+π , ρ π+, 0 − − → andρ0π0modes,whicharenotavailable. BABARandBELLEagreeonthedirectCPasymmetries, 5 andtheiraverage ) Ap −r + =−0.47+00..1143, Ap +r − =−0.15±0.09, (3.6) 0 − is a 3.6σ signal of direct CP violation, i.e., (Ap r +, Ap +r ) = (0,0). Translating the available 1 − − 6 measurementstoavalueofαinvolvesassumptionsaboutfactorizationandSU(3)flavorsymmetry, 2 andaretheoryerrordominated. Combiningtheρρandππisospin analyses withtheρπDalitzplotanalysis yields[9] α=(99+12) , (3.7) 9 ◦ − whichisshowninFig.4. Thisdirectdetermination ofαisalreadymoreprecisethanitisfromthe CKMfit(withoutusingαandγ),whichgivesα=(98 16) . ◦ ± 3.2 γfromB DK ± ± → Heretheideaistomeasuretheinterference ofB D0K (b cu¯s)andB D0K (b − − − − → → → → uc¯s) transitions, which can be studied in final states accessible in both D0 and D0 decays. The key is to extract the B and D decay amplitudes, the relative strong phases, and the weak phase γ fromthedata. Apractical complication isthattheprecision depends sensitively ontheratioofthe interfering amplitudes, A(B D0K ) − − r → , (3.8) B≡ A(B D0K ) − − → 012/10