UND-HEP-07-BIG01 hep-ph/0701273 Flavour Dynamics & CP Violation in the Standard Model: A Crucial Past – and an Essential Future IkarosI.Bigi 7 PhysicsDept.,UniversityofNotreDameduLac,NotreDame,IN46556, U.S.A. 0 0 emailaddress: [email protected] 2 n Abstract a J Our knowledge of flavour dynamics has undergone a ‘quantum jump’ since 1 just before the turn of the millenium: direct CP violation has been firmly es- 3 tablished in K ππ decays in 1999; the first CP asymmetry outside K L L → decayshasbeendiscoveredin2001inB ψK ,followedbyB π+π−, 1 d S d → → v η′K and B K±π∓, the last one establishing direct CP violation also in S 3 thebeautysec→tor. FurthermoreCKMdynamics allowsadescription ofCPin- 7 sensitive and sensitive B, K and D transitions that is impressively consistent 2 1 also on the quantitative level. Theories of flavour dynamics that could serve 0 as alternatives to CKMhave been ruled out. Yet these novel successes of the 7 0 Standard Model (SM) do not invalidate any of the theoretical arguments for / the incompleteness of the SM. In addition we have also more direct evidence h p forNewPhysics,namelyneutrinooscillations, theobservedbaryonnumberof - theUniverse,darkmatteranddarkenergy. WhiletheNewPhysicsanticipated p e attheTeVscaleisnotlikelytoshedanylightontheSM’smysteriesofflavour, h detailed andcomprehensive studies ofheavy flavourtransitions willbeessen- : v tialindiagnosing salientfeaturesofthatNewPhysics. Strategicprinciples for i X suchstudieswillbeoutlined. r a Contents 1 LectureI:"FlavourDynamicsintheSecondMillenium ( 1999)" . . . . . . . . . . . . . 3 → 1.1 OntheUniqueness oftheSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 BasicsofP,C,T,CP andCPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 TheVerySpecialRoleofCPInvariance anditsViolation . . . . . . . . . . . . . . . . . 10 1.4 FlavourDynamicsandtheCKMAnsatz . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Meson-antimeson Oscillations –onthePowerofQuantum Mysteries . . . . . . . . . . . 15 1.6 CKM–FromaGeneralAnsatztoaSpecificTheory . . . . . . . . . . . . . . . . . . . . 22 1.7 TheSMParadigmofLargeCPViolationinB Decays. . . . . . . . . . . . . . . . . . . 26 1.8 CompletionofaHeroicEra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.9 SummaryofLectureI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2 LectureII:"FlavourDynamics2000-2006"–The‘Expected’ TriumphofaPeculiarTheory 38 2.1 Establishing theCKMAnsatzasaTheory–CPViolationinB Decays . . . . . . . . . . 38 2.2 LoopInduced RareB Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 u,d 2.3 OtherRareDecays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.4 AddingHighAccuracytoHighSensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5 SummaryofLect. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3 LectureIII:ProbingtheFlavourParadigmoftheEmergingNewStandardModel . . . . . . 65 3.1 OntheIncompleteness oftheSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 ∆S = 0–the‘Established Hero’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6 3.3 The‘KingKong’ScenarioforNewPhysicsSearches . . . . . . . . . . . . . . . . . . . 68 3.4 FutureStudiesofB Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 u,d 3.5 B Decays–anIndependent ChapterinNature’sBook . . . . . . . . . . . . . . . . . . 75 s 3.6 Instead ofaSummary: OntheFutureHEPLandscape –aCalltoWell-Reasoned Action 77 In my lecture series I will sketch the past evolution of central concepts of the Standard Model (SM), which are of particular importance for its flavour dynamics. The reason is not primarily of a historical nature. I hope these sketches will illuminate the main message I want to convey, namely that we find ourselves in the midst of a great intellectual adventure: even with the recent novel successes of the SM the case for New Physics at the TeV (and at higher scales) is as strong as ever. While there is a crowd favourite for the TeV scale New Physics, namely some implementation of Supersymmetry (SUSY) – an expectation I happen to share – we better allow for many diverse scenarios. To deduce which one is realized in nature we will need all the experimental information we can get, including the impactoftheNewPhysicsonflavourdynamics. YetbasedonthepresentsuccessesoftheSM,wecannot countonthatimpactbeingnumericallymassive. Iwillemphasizegeneralprinciplesfordesigningsearch strategies forNewPhysicsoverspecificanddetailedexamples. Forataschool likethiswewanttohelp youprepareyourself forafutureleadership role;thatrequiresthatyoudoyourownthinkingratherthan ‘out-source’ it. Theoutline ofmythreelectures isasfollows: – LectureI:"FlavourDynamicsintheSecondMillenium( 1999)"–Basicsofflavourdynam- → icsandCPviolation, CKMtheory, K0 andB0 oscillations, theSM‘Paradigm oflarge CPviola- tioninB decays’. – Lecture II: "Flavour Dynamics 2000 - 2006" – Verifying the SM ‘Paradigm of large CP vio- lation in B decays’, praising EPRcorrelations & hadronization, Heavy Quark Theory, extracting CKMparameters andCKMtriangle fits. 2 – LectureIII:"ProbingtheFlavourParadigmoftheEmergingNewStandardModel–Indirect searches forNewPhysics,‘KingKong’scenarios (EDM’s,charm,τ leptons)vs. precision probes (beauty),thecaseforaSuper-FlavourFactoryandanewgenerationofkaonexperimentsinHEP’s futurelandscape. ToalargedegreeIwillfollowthehistoricaldevelopment, becauseitdemonstrates best,whyitisadvan- tageoustolistentopredictions fromtheory–butalsogoagainstitattimes! 1 Lecture I:"FlavourDynamicsinthe Second Millenium (→ 1999)" Memento∆S = 0dynamics: 6 – The‘θ τ puzzle’–theobservation thattwoparticlesdecayingintofinalstatesofoppositeparity − (θ 2π, τ 3π) exhibited the same mass and lifetime – lead to the realization that parity was → → violated inweakinteractions, andactuallytoamaximaldegreeinchargedcurrents. – The observation that the production rate of strange hadrons exceeded their decay rates by many ordersofmagnitude–afeaturethatgaverisetotheterm‘strangeness’–wasattributedto‘associate production’ meaning the strong and electromagnetic forces conserve this new quantum number ‘strangeness’, while weak dynamics do not. Subsequently it gave rise to the notion of quark families. – ThegreatsuppressionofflavourchangingneutralcurrentsasevidencedbythetinyratesforK L → µ+µ−, γγ and the minute size for ∆M , lead some daring spirits to postulate the existence of a K newquantum numberforquarks, namelycharm. – The observation of K π+π− established that CP invariance was not fully implemented in L → nature and induced two other daring spirits to postulate the existence of yet another, the third, quarkfamily,withthetopquark, aswelearntlater, beingtwohundredtimesheavier thankaons. Allthesefeatures, whicharepillarsoftheSMnow,represented ‘NewPhysics’atthattime! 1.1 OntheUniquenessoftheSM A famous American Football coach once declared:"Winning is not the greatest thing – it is the only thing!" This quote provides some useful criteria for sketching the status of the different components of theStandardModel(SM).Itcanbecharacterized bythecarriersofitsstrongandelectroweakforcesthat aredescribed bygaugedynamics andthemassmatricesforitsquarksandleptons asfollows: SM∗ = SU(3) SU(2) ‘CKM′( ‘PMNS′) (1) C L × ⊕ ⊕ I have attached the asteriks to ‘SM’ to emphasize the SM contains a very peculiar pattern of fermion mass parameters that is not illuminated at all by its gauge structure. Next I will address the status of thesecomponents. 1.1.1 QCD–the‘Only’Thing 1.1.1.1 ‘Derivation’ ofQCD While it is important to subject QCD again and again to quantitative tests as the theory for the strong interactions, one should note that these serve more as tests of our computational control over QCD dynamicsthanofQCDitself. Foritsfeaturescanbeinferred fromafewgeneralrequirements andbasic observations. Asimplifiedlistreadsasfollows: 3 – Our understanding of chiral symmetry as a spontaneously realized one – which allows treating pionsasGoldstonebosonsimplyingvarioussoftpiontheorems–requiresvectorcouplingsforthe gluons. – The ratio R = σ(e+e− had.)/σ(e+e− µ+µ−) and the branching ratios for π0 γγ, → → → τ− e−ν¯ ν andB lνX pointtotheneedforthreecolours. e τ c → → – Colourhastobeimplementedasanunbrokensymmetry. Localgaugetheoriesaretheonlyknown way to couple massless spin-one fields in a Lorentz invariant way. The basic challenge is easily stated: 4 = 2;i.e.,whileLorentzcovariance requires fourcomponent todescribe aspin-one field, 6 the latter contains only two physical degrees of freedom for massless fields. (For massive vector fieldsonecangototheirrestframetoreduceandprojectoutonecomponentinaLorentzinvariant waytoarriveatthethreephysicaldegreesoffreedom.) – Combiningconfinementwithasymptoticfreedom requires anon-abelian gaugetheory. Insummary: fordescribing thestronginteractions QCDistheuniquechoiceamonglocalquantumfield theories. A true failure of QCD would thus create a genuine paradigm shift, for one had to adopt an intrinsically non-local description. It should be remembered that string theory was first put forward for describing thestronginteractions. 1.1.1.2 ‘Fly-in-the-Ointment’: theStrongCPProblemofQCD A theoretical problem arises for QCD from an unexpected quarter that is very relevant for our context here: QCDdoesnotautomatically conserve P,T andCP.Toreflectthenontrivial topological structure of QCD’s ground state one employs an effective Lagrangian containing an additional term to the usual QCDLagrangian[1]: g2 i = +θ S G G˜µν , G˜ = ǫ Gρσ (2) Leff LQCD 32π2 µν µν 2 µνρσ Since G G˜µν is agauge invariant operator, its appearance in general cannot beforbidden, and what is µν notforbiddenhastobeconsideredallowedinaquantumfieldtheory. Itrepresentsatotaldivergence,yet inQCD–unlikeinQED–itcannotbeignored duetothetopological structure ofthegroundstate. SinceunderparityPandtimereversalTonehas G G˜µν =P,T G G˜µν , (3) µν µν ⇒ − thelastterminEq.(2)violatesPaswellasT. SinceG G˜µν isflavour-diagonal, itgeneratesanelectric µν dipolemoment(EDM)fortheneutron. Fromtheupperboundonthelatter d < 0.63 10−25 ecm (4) N · oneinfers[1] θ < 10−9 . (5) Being the coefficient of a dimension-four operator θ can be renormalized to any value, even zero. Yet the modern view of renomalization is more demanding: requiring the renormalized value to be smaller than its ‘natural’ one by orders of magnitude is frowned upon, since it requires finetuning between the loop corrections and the counterterms. This is what happens here. For purely within QCD the only intrinsically ‘natural’ scale for θ is unity. If θ 0.1 or even 0.01 were found, one would not be overly ∼ concerned. Yet the bound of Eq.(5) is viewed with great alarm as very unnatural – unless a symmetry canbecalledupon. Ifanyquark weremassless –mostlikely theuquark–chiral rotations representing symmetrytransformations inthatcasecouldbeemployed toremoveθ contributions. Yetaconsiderable phenomenological bodyrulesagainstsuchascenario. 4 Amuchmoreattractivesolutionwouldbeprovidedbytransformingθfromafixedparameterinto the manifestation of a dynamical field – as is done for gauge and fermion masses through the Higgs- Kibble mechanism, see below – and imposing a Peccei-Quinn symmetry would lead naturally to θ ≪ (10−9). Alas – this attractive solution does not come ‘for free’: it requires the existence of axions. O Thosehavenotbeenobserved despitegreateffortstofindthem. Thisisapurelytheoreticalproblem. YetIconsiderthefactthatitremainsunresolvedasignificant chink intheSM∗’sarmour. Istillhave notgivenuphopethat‘victory canbesnatched from thejawsof defeat’: establishing aPeccei-Quinn-type solution wouldbeamajortriumphfortheory. 1.1.1.3 TheoreticalTechnologies forQCD True theorists tend to think that by writing down, say, a Lagrangian one has defined a theory. Yet to makecontactwithexperiment oneneedstheoretical technologies toinferobservable quantities fromthe Lagrangian. Thatisthetaskthatengineers andplumbers likemehavesetforthemselves. Examples for suchtechnologies are: – perturbation theory; – chiralperturbation; – QCDsumrules; – heavyquarkexpansions (whichwillbedescribed insomedetailinLectureII). Exceptforthefirstonetheyincorporate thetreatmentofnonperturbative effects. None of these can claim universal validity; i.e., they are all ‘protestant’ in nature. There is only one‘catholic’ technology, namelylatticegaugetheory1: – Itcanbeappliedtononperturbativedynamicsinalldomains(withthepossiblepracticallimitation concerning strongfinalstateinteractions). – Itstheoretical uncertainties canbereduced inasystematic way. Chiralperturbation theoryisQCDatlowenergies describing thedynamicsofsoftpionsandkaons. The heavyquarkexpansionstreatingthenonperturbativeeffectsinheavyflavourdecaysthroughanexpansion ininversepowersoftheheavyquarkmassaretailormadefordescribingB decays;towhichdegreetheir application can be extended down to the charm scale is a more iffy question. Different formulations of lattice QCDcan approach the nonperturbative dynamics at the charm scale from below as well as from above. The degree to which they yield the same results for charm provides an essential cross check on theirnumericalreliability. Inthatsensethestudyofcharmdecaysservesasanimportantbridgebetween ourunderstanding ofnonperturbative effectsinheavyandlightflavours. 1.1.2 SU(2) ×U(1)–noteventheGreatestThing L 1.1.2.1 Prehistory It was recognized from early on that the four-fermion-coupling of Fermi’s theory for the weak forces yields an effective description only that cannot be extended to very high energies. The lowest order contribution violates unitarity around 250 GeV. Higher order contributions cannot be called upon to remedy thesituation, sinceduetothetheory being non-renormalizable those comewithmoreandmore untamable infinities. Introducing massivecharged vectorbosonssoftens theproblem,yetdoesnotsolve it. Considerthepropagator foramassivespin-one bosoncarrying momentumk: g + kµkν − µν MW2 (6) k2 M2 − W 1Ihastentoaddthatlatticegaugetheory–whilecatholicinsubstance–exhibitsadifferentsociology: ithasnotdeveloped aninquisitionanddealswithhereticsinarathergentleway. 5 Thesecondterminthenumeratorhasgreatpotentialtocausetrouble. Foritcanactlikeacouplingterm withdimension 1/(mass)2;this isquite analogous totheoriginal ansatz ofFermi’stheory andamounts toanon-renormalizable coupling. Itisactuallythelongitudinal componentofthevectorbosonthatisat thebottomofthisproblem. Thispotentialproblemisneutralized,ifthesemassivevectorbosonscoupletoconservedcurrents. To guarantee the latter property, one needs a non-abelian gauge theory, which implies the existence of neutralweakcurrents. 1.1.2.2 StrongPoints Therequirementsofunitarity,whichisnonnegotiable, andofrenormalazibility, whichistosomedegree, severely restrict possible theories of the electroweak interactions. It makes the generation of mass a highlynontrivial one,assketched below. Thereareotherstrongpointsaswell: SincethereisasingleSU(2) group,thereisasinglesetofgaugebosons. Theirself-couplingcontrols L ⊕ also, how they couple to the fermion fields. As explain later in more detail, this implies the property of ‘weakuniversality’. The SM truly predicted the existence of neutral currents characterized by one parameter, the weak ⊕ angleθ ,andthemassesoftheW andZ bosons. W : Most remarkably the SU(2) U(1) gauge theory combines QED with a pure parity conserving L ⊕ × vector coupling to a massless neutral force field with the weak interactions, where the charged currents exhibitmaximalparityviolationduetotheirV AcouplingandaveryshortrangeduetoM > M Z W − ≫ m . π 1.1.2.3 GeneratingMass Amassivespin-one fieldwithmomentumk andspins hasfour(Lorentz)components. Goingintoits µ µ restframeonerealizesthattheLorentzinvariantconstraintk s = 0canbeimposed,whichleavesthree · independent components, asithastobe. Amasslessspin-onefieldisstilldescribed byfourcomponents, yethasonlytwophysicaldegrees offreedom. Itneedsanotherphysicaldegreeoffreedomtotransmogrifyitselfintoamassivefield. Thisis achievedbyhavingthegaugesymmetryrealizedspontaneously. Forthecaseathandthisisimplemented through an ansatz that should be – although rarely is – referred to as Higgs-Brout-Englert-Guralnik- Hagen-Kibble mechanism (HBEGHK).Sufficeittoconsider thesimplest caseofacomplexscalar field φwithapotential invariant underφ(x) eiα(x)φ(x),sincethismechanism hasbeendescribed ingreat → detailinPich’slectures [2]: m2 V(φ) = λ φ4 φ2 (7) | | − 2 | | Its minimum is obviously not at φ = 0, but at m2/4λ. Thus rather than having a unique ground | | state with φ = 0 one has an infinity of different, yet equivalent ground states with φ = m2/4λ. | | p | | Tounderstand the physical content ofsuch a scenario, one considers oscillations ofthe fieldaround the p minimum of the potential: oscillations in the radial direction of the field φ represent a scalar particle with mass; in the polar direction (i.e. the phase of φ) the potential is at its minimum, i.e. flat, and the corresponding fieldcomponent constitutes amasslessfield. It turns out that this massless scalar field can be combined with the two transverse components of a M = 0 spin-one gauge field to take on the role of the latter’s longitudinal component leading to the emergence of a massive spin-one field. Its mass is thus controlled by the nonperturbative quantity 0φ0 . h | | i Applying this generic construction to the SM one finds that a priori both SU(2) doublet and L tripletHiggsfieldscouldgenerate massesfortheweakvectorbosons. TheratioobservedfortheW and 6 Z massesisfullyconsistentwithonlydoubletscontributing. Intriguinglyenoughsuchdoubletfieldscan eoipsogeneratefermionmassesaswell. In the SMone adds a single complex scalar doublet fieldto the mix ofvector boson and fermion fields. Threeofitsfourcomponentsslipintotheroleofthelongitudinal componentsofW± andZ0;the fourth one emerges asan independent physical field– ‘the’ Higgs field. Fermion masses are then given by the product of the single vacuum expectation value (VEV) 0φ0 and their Yukawa couplings – a h | | i pointwewillreturnto. 1.1.2.4 TriangleorABJAnomaly The diagram with an internal loop of only fermion lines, to which three external axial vector (or one axial vector andtwovector) lines areattached, generates a‘quantum anomaly’ 2: itremoves aclassical symmetry as expressed through the existence of a conserved current. In this specific case it affects the conservation oftheaxialvector currentJ5. Classicallywehave∂µJ5 =0formasslessfermions;yetthe µ µ triangleanomalyleadsto g2 ∂µJ5 = S G G˜ =0 (8) µ 16π2 · 6 even for massless fermions; G and G˜ denote the gluonic field strength tensor and its dual, respectively, asintroduced inEq.(2). While byitself ityields afiniteresult onthe right hand sideof Eq.(8), itdestroys therenormaliz- abilityofthetheory. Itcannotbe‘renormalized away’(sinceinfourdimensionsitcannotberegularized in a gauge invariant way). Instead it has to be neutralized by requiring that adding up this contribution fromalltypesoffermionsinthetheoryyieldsavanishing result. For the SM this requirement can be expressed very concisely that all electric charges of the fermions of a given family have to add up to zero. This imposes a connection between the charges ofquarks andleptons, yetdoesnotexplainit. 1.1.2.5 TheoreticalDeficiencies Withalltheimpressive,evenamazingsuccessesoftheSM,itisnaturaltoaskwhyisthecommunitynot happywithit. Thereareseveraldrawbacks: SincethegaugegroupisSU(2) U(1),onlypartialunificationhasbeenachieved. L ⊖ × TheHBEGHKmechanismisviewedasprovidingmerelyan‘engineering’solution,inparticularsince ⊖ the physical Higgs fieldhas not been observed yet. Evenif orwhen it is, theorists in particular willnot feel relieved, since scalar dynamics induce quadratic mass renormalization and are viewed as highly ‘unnatural’,asexemplifiedthroughthegaugehierarchyproblem. Thisconcernhasleadtotheconjecture ofNewPhysicsentering aroundtheTeVscale, whichhasprovidedthejustification fortheLHCandthe motivationfortheILC. maximal violation of parity is implemented for the charged weak currents ‘par ordre du mufti’ 3, i.e. ⊖ basedonthedatawithnodeeperunderstanding. Likewiseneutrino masseshadbeensettozero‘parordredumufti’. ⊖ Theobservedquantization ofelectricchargeiseasilyimplementedandisinstrumental inneutralizing ⊖ thetriangleanomaly–yetthereisnounderstanding ofit. Onemightsaythesedeficienciesaremerely‘warts’thathardlydetractfromthebeautyoftheSM. Alas–thereisthewholeissueoffamilyreplication! 2Itisreferredtoas‘triangle’anomalyduetotheformoftheunderlyingdiagramorA(dler)B(ell)J(ackiw)anomalydueto theauthorsthatidentifiedit[3]. 3A French saying describing a situation, where a decision is imposed on someone with no explanation and no right of appeal. 7 1.1.3 TheFamilyMystery Thetwelveknownquarks and leptons arearranged into three families. Thosefamilies possess identical gauge couplings and are distinguished only by their mass terms, i.e. their Yukawa couplings. We do not understand this family replication or why there are three families. It is not even clear whether the numberoffamiliesrepresentsafundamentalquantityorisduetothemoreorlessaccidentalinterplayof complexforcesasoneencounterswhenanalyzingthestructureofnuclei. Theonlyhopeforatheoretical understanding we can spot on the horizon is superstring or M theory – which is merely a euphemistic wayofsayingwehavenoclue. Yet the circumstantial evidence that we miss completely a central element of Nature’s ‘Grand Design’isevenstrongerinviewofthestronglyhierarchical patterninthemassesforup-anddown-type quarks, charged leptonsandneutrinos andtheCKMparametersasdiscussed later. 1.2 BasicsofP,C,T,CP andCPT 1.2.1 Definitions Paritytransformations flipthesignofpositionvectors~rwhileleaving thetimecoordinate tunchanged: P (~r,t) ( ~r,t) (9) −→ − Momentachangetheirsignsaswell,yetorbitalandotherangular momentadonot: p~ P p~ vs. ~l ~r p~ P ~l (10) −→ − ≡ × −→ Parityoddvectors–~r,p~–andparityevenones–~l–arereferredtoaspolarandaxialvectors,respectively. P P Likewise one talks about scalars S and pseudoscalars P with S S and P P. Examples are S = ~p ~p , ~l ~l and P = ~l p~ . Parity transformations are−→equivalent to−m→irr−or transformations 1 2 1 2 1 2 · · · followedbyarotation. Theyaredescribed byalinearoperatorP. Chargeconjugation exchangesparticlesandantiparticlesandthusflipsthesignofallchargeslike electriccharge, hyper-charge etc. Itisalsodescribed byalinearoperator C. Timereversalisoperationally definedasareversalofmotion (p~,~l) T (p~,~l), (11) −→ − which follows from (~r,t) T (~r, t). While the Euclidean scalar~l p~ is invariant under the time 1 2 −→ − · reversaloperatorT,thetriplecorrelations of(angular) momentaarenot: ~v (~v ~v ) T ~v (~v ~v ) with ~v = p~,~l . (12) 1 2 3 1 2 3 · × −→ − · × Theexpectation valueofsuchtriplecorrelations accordingly arereferredtoasToddmoments. IncontrasttoP orC theToperator isantilinear: T(αa +β b ) = α∗T a +β∗T b (13) | i | i | i | i Thisproperty ofTisenforced bythecommutation relation[X,P] = i~,since T−1[X,P]T = [X,P] (14) − T−1i~T = i~ (15) − Theanti-linearity ofTimpliesthreeimportantproperties: 8 – Tviolationmanifestsitselfthroughcomplexphases. CPTinvariancethenimpliesthatalsoCPvi- olationentersthrough complexphasesintherelevantcouplings. ForTorCPviolation tobecome observable in a decay transition one thus needs the contribution from two different, yet coherent amplitudes. – While a non-vanishing P odd moment establishes unequivocally P violation, this is not necessar- ily so for T odd moments; i.e., even T invariant dynamics can generate a non-vanishing T odd moment. T being antilinear comes into play when the transition amplitude is described through second (or even higher) order in the effective interaction, i.e. when final state interactions are included denoted symbolically by i i i T−1( ∆t+ ( ∆t)2+...)T = ∆t ( ∆t)2+... = ∆t+ ( ∆t)2+... eff eff eff eff eff eff L 2 L L −2 L 6 L 2 L (16) evenfor[T, ] = 0. eff L – ‘Kramer’s degeneracy’ [4]: With T being anti-unitary, the Hilbert space – for T invariance – can be decomposed into two disjoint sectors, one with T2 = 1 and the other with T2 = 1, and the − latteroneisatleastdoublydegenerate inenergy. Itturnsoutthatforbosonic states onehasT2 = 1andforfermionic onesT2 = 1. Theamazing − thing is that the necessary anti-unitarity of the T operator already anticipates the existence of fermions and bosons – without any reference to spin. Maybe a better way of expressing it is as follows. While nature seems to be fond of realizing mathematical structures, it does so in a very efficientway: itcanhavebosons–statessymmetricunderpermutationofidenticalparticles–and fermions, which are antisymmetric; it can contain states with half integer and integer spin, and finallyitallowsforstateswithT2 = 1. Itimplementsallthesestructuresanddoessointhemost ± efficientway,namelybybosons[fermions]carrying [half]integerspinandT2 = +[ ]1. − Kramer’s degeneracy has practical applications as well, for example in solid state physics: con- siderelectrons insideanexternal electrostatic field. Suchafieldbreaks rotational invariance; thus angularmomentumisnolongerconserved. yetnomatterhowcomplicatedthisfieldis,foranodd numberofelectrons therealwayshastobeatleasttwo-folddegeneracy. 1.2.2 Macroscopic TViolationor‘ArrowofTime’ Letusconsiderasimpleexamplefromclassicalmechanics: themotionofbilliardball(s)acrossabilliard tableinthreedifferent scenarios. (i)Watchingamovieshowingasingleballrolearoundandbounceoffthewallsofthetableonecouldnot decidewhetheronewasseeingtheeventsintheactualtimesequenceorinthereverseorder,i.e. whether onewasseeingthemovierunningbackwards. Forbothsequences arepossible andequallylikely. (ii)Seeingoneballmoveinandhitanotherballatrestleadingtobothballsmovingoffindifferentdirec- tions is a possible and ordinary sequence. The reverse – two balls moving in from different directions, hitting each other with one ball coming to a complete rest and the other one moving off in a different direction –isstill apossible sequence yet arather unlikely one since itrequires finetuning between the momentaofthetwoincomingbilliardballs. (iii) Onebilliard ball hitting a triangle of ten billiard balls atrest and scattering them in all directions is amostordinary sequence foranybody butthe mostinept billiard player. Thereverse sequence –eleven billiard balls coming in from all different directions, hitting each other in such a way that ten come to restinaneatlyarrangedtrianglewhiletheeleventhonemovesoff–isapracticallyimpossibleone,since itrequires amostdelicate finetuningoftheinitialconditions. Therearecountless otherexamplesofonetimesequencebeingordinarywhilethereversedoneis (practically) impossible – take β decay n pe−ν¯, the scattering of a plane wave off an object leading → to an outgoing spherical wave in addition to the continuing plane wave or the challenge of parking a 9 car in a tight spot compared with the relative ease to get out of it. These daily experiences do not tell us anything about T violation in the underlying dynamics; they reflect asymmetries in the macroscopic initialconditions, whichareofastatistical nature. Yetacentral message ofmylectures isthatmicroscopic Tviolation hasbeenobserved, i.e. Tvi- olation that resides in the basic dynamics of the SM. It is conceivable though that in a more complete theoryitreflectsanasymmetryintheinitialconditions insomehighersense. 1.3 TheVerySpecialRoleofCPInvarianceanditsViolation WhilethediscoveryofPviolationintheweakdynamicsin1957causedawelldocumentedshockinthe community, even the theorists quickly recovered. Why then was the discovery of CP violation in 1964 not viewed as a ‘deja vue all over again’ in the language of Yogi Berra? There are several reasons for thatasillustrated bythefollowingstatements: – Letmestartwithananalogyfrompolitics. Inmydaysasastudent–atatimelongagoandaplace faraway–politicswashotlydebated. Oneofthesubjectsdrawingoutthegreatestpassionswasthe questions ofwhatdistinguished the‘left’from the‘right’. Ifyou listened toit, youquickly found out that people almost universally defined ‘left’ and ‘right’ in terms of ‘positive’ and ‘negative’. Theonlyproblemwastheycouldnotquiteagreewhothegoodguysandthebadguysare. Therearisesasimilarconundrum whenconsidering decayslikeπ eν. Whensayingthatapion → decay produces a left handed charged lepton one had π− e−ν¯ in mind. However π+ e+ν → L → R yields aright handed charged lepton. ‘Left’isthusdefinedintermsof‘negative’. Nomatterhow muchPisviolated,CPinvarianceimposesequalratesfortheseπ±modes,anditisuntruetoclaim that nature makes an absolute distinction between ‘left’ and ‘right’. The situation isanalogous to thesayingthat‘thethumbisleftontherighthand’–acorrect,yetuselessstatement,sincecircular. CP violation is required to define ‘matter’ vs. ‘antimatter’, ‘left’ vs. ‘right’, ‘positive’ vs. ‘nega- tive’inaconvention independent way. – Duetothealmostunavoidable CPTsymmetryviolation ofCPimpliesoneofT. – Itisthesmallestobserved violation ofasymmetryasexpressed through ImMK ImMK 1.1 10−8 eV 12 2.2 10−17 (17) 12 ≃ · ↔ M ≃ · K – It is one of the key ingredients in the Sakharov conditions for baryogenesis [5]: to obtain the observed baryonnumber ofourUniverseasadynamically generated quantity rather thananarbi- traryinitialconditiononeneedsbaryonnumberviolating transitionswithCPviolationtooccurin aperiod, whereourUniversehadbeenoutofthermalequilibrium. 1.4 FlavourDynamicsandtheCKMAnsatz 1.4.1 TheGIMMechanism Astrikingfeatureof(semi)leptonickaondecaysarethehugesuppressionofstrangenesschangingneutral currentmodes: Γ(K+ π+e+e−) Γ(K µ+µ−) → 6 10−6 , L → 3 10−9 (18) Γ(K+ π0e+ν) ∼ · Γ(K+ µ+ν) ∼ · → → Embeddingweakcharged currents withtheirCabibbocouplings J(+) = cosθ d¯ γ u +sinθ s¯ γ u µ C L µ L C L µ L J(−) = cosθ u¯ γ d +sinθ u¯ γ s (19) µ C L µ L C L µ L 10