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PoS(KAON09)053 http://pos.sissa.it/ physics which have been reported e Commons Attribution-NonCommercial-ShareAlike Licence. ∗ [email protected] Speaker. at this conference. I summarize the precision tests of the in ∗ Copyright owned by the author(s) under the terms of the Creativ c ° 2009 KAON International Conference KAON09, June 09 - 12 2009 Tsukuba, Japan Gilberto Colangelo Albert Einstein Center for Fundamental Physics Institute for theoretical physics University of Bern Sidlerstrasse 5 3012 Bern, Switzerland E-mail: Precise Standard Model tests PoS(KAON09)053 (1.1) ally light ence half in breaking in three pieces ed in combina- ospin limit only dominated by the Gilberto Colangelo anizers for having CD is a fully first- ibutions and topics e if one admits the ticity and unitarity, enomenology. The ed as either tests of resented by a series hiro Okada who had d. On the other hand CD phenomena: the enomenology is now ing QCD mass term , ments analyticity and what is known and to the small and the tiny nference, it is useful to . . Having made clear that D becomes a parameter- tinction between the two the consequences of symme- xed, for example to correctly . Z weak and L W + d m – the heavy masses, which are also − iso — u s m m L L + 2 ) + and d = ˆ u m ( QCD QED = L L d = = m = SM iso — u L L m In the last few years lattice calculations with dynamical and with realistic At energies of the order of the kaon mass and below, the SM Lagrangian is Before discussing what precision SM tests have been reported at this co At present there are three main approaches to tackle nonperturbative Q physics experiments of the last three decades can always be view quark masses have become available, so that a direct comparison with the ph QCD term. This partthe of masses the of Lagrangian the has light only , two free parameters (in the is define an ordering principle, which allowstouched us to without navigate losing among the the many overviewwhich, contr of at this the field. energy scale Ione: of will the split kaon, the can SM be Lagrangian described as the dominant, present, are fixed at their value).reproduce Once these the two lowest free part parameters are offree fi the theory spectrum and (the we and mustnon-perturbative nature kaon face of mass), the the QC challenge problem makes to this describe a with highly nontrivial it task alattice, very the rich effective ph field theoryprinciple and approach; the in the dispersion-relation effective method. field theorytry Lattice one in efficiently Q a derives quantum field theoryalbeit framework perturbatively); (and so in automatically the respectingunitarity. analy dispersion The theory three approach methods one aretion complementary exactly in and imple order indeed to they obtain are a sometimes prediction us which can be compared to experiments. The dominant term is the QCD Lagrangianpart, in which the isospin can limit, itself the be small split oneand into is the the the tiny isosp QED piece Lagrangian is andof due the nonrenormalizable isospin-break to terms weak generated interactions, by which the at exchange of this energy is best rep 2. Strong interactions the standard model (SM)points or of as view searches is for somewhatpossibility artificial, new that because physics it testing may (NP). a be The wrong, theory andthe dis only so search makes that sens new for phenomena new may phenomena bewhat discovere only level of makes precision, sense which if means,this that one separation one can is has clearly somewhat tested define artificial, thedecided “old I theory” to can make also this express splitting, mythat which gratitude difficult. made to I the the am impossible org sure task thatto of this summarize gratitude summarizing the is a part also of confer shared the by conference my related colleague to Yasu the search for NP [1]. Standard Model 1. Introduction PoS(KAON09)053 2 ℓ π allow 6 merical and requires p 2 ible since ℓ ud . Moreover eem to lead K 4 V t an estimate p / rovide unique ecay constants, us ts, necessary for V . Improving this . A purely chiral to order Gilberto Colangelo 4 the determination and logy. Such an ini- us pends on unknown p 2 t known from other V p scia [6]. Moreover, if net European network the CKM matrix. The vel of precision which 3)-breaking correction ues and setups show a hed today if one wants ummarize lattice results moment concentrating on which reflects the fact that timates which any theorist , see below). masses. The description of us V one obtains two independent , whereas those on ) or of the ratio ν us ℓ us V ) V 0 → ( + π f . Symmetry arguments (the Ademollo- ( are nowadays possible and have already π Γ F π from 1 is quadratic in the SU(3)-breaking / / F ) ) PT is quite satisfactory and has led to a deter- / K ν 0 χ F K ℓ ( F + PT Lagrangian both at order f 3 → from superallowed Fermi decay of nuclei together χ and 2 scattering length have been calculated by different and K ) ud ( depends on LECs already at order ) 0 = V Γ ( 0 π . The extraction of I ( + ) F f + π / f [7, 8]. This nontrivial result confirms that the systematic F K necessary for the extraction of F ud us ) V 0 V ( ( / + ) f K F decays measure the product us from the ratio 3 V ℓ ( us K V correction was essential in order to have a precise extraction of to extract in terms of masses and decay constants only [3] and give an unambiguos nu ) 6 π 4 PT and dispersion relations a direct comparison to the phenomenology is poss p F p χ ( / LECs. Estimating these with resonance saturation or other methods does not s K O . Experiments on F ) There have been several talks about lattice results at this conference, Several lattice groups have studied pion physics in detail: the pion mass and d For 6 us p V ( prediction. Unfortunately this isto not test enough the at unitarity the ofof level of the the precision CKM reac matrix, and in fact it was realized early on [4] tha to the necessary precision. The ratio with – moreover the effective Lagrangian methodof allows order one to express the SU( one to identify unambiguously theO unitarity contribution, but the final result de prediction for these is impossible, becausesources. the relevant Lattice LEC calculations combination for is no both nowadays lattice QCD provides invaluable information onthe hadronic SM matrix elemen analysis of kaonof phenomenology. Probably the best example today is estimate is difficult. Calculations of the chiral expansion of the form factor up decays provide the ratio mination of several low energy constants of the reached the necessary precision tocurrent make status a of sensitive these test has ofone the been uses unitarity reviewed the of in independent determination the of talks of Boyle [5] and Me Gatto theorem [2]) imply that the deviation of much longer and what wehas have been seen reached. at this conference is the remarkable le 2.1 Lattice results input on the hadronic matrix elements Standard Model becoming possible. In some testsinformation of (like the in SM the this case is of essential, as lattice calculations p results obtained with different actionsgood and level of different agreement. computational In techniq viewin of a this unified it way appears and to whereor be possible useful experimentalists to to could offer try then averages and directly or s usetiative lattice-based in es is his/her being analysis carried of forward the(the by phenomeno acronym FLAG, stands a for working FLAVIAnet group Lattice of Averaging the Group), FLAVIA for the uncertainties of lattice calculations are under good control. and compatible determinations of the quark mass dependence of these quantities with groups with different actions and at different values of the light quark the scalar and vector form factors and the PoS(KAON09)053 , il per phys s s m m ). Since ≤ borations s phys s m acing and is not overlook m calculation of elow the kaon analysis of the lattice, the one iders only very matrix elements = dence on : either by doing such calculations s Gilberto Colangelo boration performs lculations relevant lower pion masses m ges come at a very ππ al value of the kaon close to the physical s ave reported a failure pansion successfully. one can describe data → m s well. K y this collaboration show , I do not doubt that some imulations at each the kaon-mass region 1], Mescia [6] and Noaki [9]. PT is quite dramatic. Of course, χ , it is clear that the chiral expansion matrix elements discussed by Norman phys s π m 1) dynamical quarks with the overlap for- − = + K s m by considering the as heavy and 4 matrix elements with three light dynamical flavour i.e. π − based on a Taylor expansion around K PT , these data can barely improve the precision of the LEC i.e. χ PT , which is nowadays done at NNLO. χ mainly and ) – at this conference I have presented the 400 MeV (cf. [5]). This is not that surprising, after all, in view of ∼ i.e. 0, an approach pioneered by Roessl [10]), or by doing polynomial fits π M = m – it will then be interesting to understand why the chiral expansion seems to fa such that m easy 0 is not needed to reach the physical point. On the other hand, one should and beyond with NNLO = dependence of lattice data ( PT in describing the quark-mass dependence of the lattice data for K s s χ M m m PT at NLO describes well the pion mass dependence only for pion masses b This issue is even more urgent for the four-quark A different picture comes out of the lattice data on kaons, even if one cons At this conference there has been one talk dedicated to pion physics on the The reader interested in learning more about the current status of lattice ca χ , and since it is possible to calculate these directly, this is the route that lattice colla ε . This does require to invest considerable efforts in doing expensive s / ′ value, but for ˆ the possibility to learn something interesting about QCD and investigate the depen Christ in his talk [11], becausethe for main motivation those for quantities carrying the out such failure difficultwhich of calculations is are to get relevant the in the CP-violating decays andinterested in play these problems an will take important (astime announced role in in [11]). in the Nonetheless future the lattice calculations of mass (similar conclusions have also beenit reached is by other necessary groups). to include To r NNLOaround terms. One must add, however, that even if by J.-I. Noaki [9] oncalculations with behalf two of (and the more JLQCD/TWQCD recently even collaboration. with 2 This colla determination – one should rather aimand at determine having the more LECs and only in more the precise region data where at the chiral series converge mulation. This formulation respectstherefore chiral particularly symmetry well exactly suited even at tohigh finite approach computer lattice the cost, sp chiral unfortunately, limit.can which be makes performed These it and advanta produce even competitive morethat results. The remarkable results that obtained b se but by doing this one mayphenomenology reliably based determine on the SU(3) SU(3) LECs and shed light on the ε Standard Model low-energy ( mass is already at theFrom the border practical of point where of onean view can SU(2) this hope chiral difficulty to analysis has apply been of the overcome data chiral in on ex two kaons ways ( the fact that in this case the kaon mass is around 600 MeV and that the physic status of this project [7]. The first paper of FLAG will appear soon. around expanding only around ˆ it is no problem for lattice calculations to work at of the will become here and at what values of the strange quark mass it breaks down. simple quantities like masses and decayof constants. NLO Different collaborations h for kaon decays is referred to the contributions by Boyle [5], Christ [1 PoS(KAON09)053 0 π 3 (2.2) (2.1) → r of the digozhin L K measurement. syst 4 e seventies, when K t the few-percent Gilberto Colangelo scattering lengths d result of the two 0016 . 0 ± lations. The calculation ntum): . cusp effect in nks to a dramatic increase stat by NA48/2 and band for the 2 0 a 0010 0044 . . 0 and 0 Cusp ChPT 0 0 decays of NA48/2 and of the cusp ± ± a ing the cusp and the 4 e (NA48/2 [17, 18]). At this conference K DIRAC 0444 0 0429 . . π 0 0 0 − − π decays [13], but only in the last few years a ± 4 = = e π 2 0 2 0 K a 5 a 4 → ± 10 · K syst 005 in a different channel has been discussed, on behalf of the . 2 0 0 a ± 0015 . − 0 0 0 NA48/2 combined Ke4 + Cusp NA48/2 combined Ke4 (stat. + syst.) errors 68% CL contour Ke4 a 0.2 0.21 0.22 0.23 0.24 0.25 0.26 220 ± UB . decays (E865 [14] and NA48/2 [15]), from the lifetime of pionium 0 stat 4 e 0 = K very close to the theoretical one and a remarkably good agreement. 0 0 0.01 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 a 0 0 0047 a . 0 ± -wave scattering lengths represent one of the best examples of the powe S 2210 . decays have been presented by Brigitte Bloch-Devaux [19] and Dmitry Ma 0 Ellipses representing the experimental measurements of π ππ = 3 scattering lengths -wave scattering lengths based on this approach [12] yields a precision a 0 0 S a → A first attempt to measure The ππ was extracted from a measurement of 3 K 0 0 (DIRAC [16]) and from the cusp effect in KTeV collaboration, by Ed Blucher [21], who presented evidence for a The first experimental measurementsa of the scattering lengths date back to the effective Lagrangian method, especially when combined withof dispersion the re with an uncertainty in DIRAC measurement. The small (red) one is obtained by combin level (the superscript indicates the isospin, the subscript the angular mome precision close to the theoretical one has been reached. This not only tha 2.2 Figure courtesy of Brigitte Bloch-Devaux. in the event statistics, butcan also now thanks be to measured the in use of different methods: the two [20]. The most interesting outcome ofmeasurements, these as preliminary illustrated analyses in is Fig. the 1, combine which gives [19] Figure 1: Standard Model preliminary results of analyses of the full data sample of in PoS(KAON09)053 0 0 es a PT , 031, (2.3) . χ 0 ) one can ± high values ir own cusp plays a role 215 g lengths, as anslate high- . introduced in tible with the 2 0 0 a parameter which = ble extrapolation Gilberto Colangelo adronic processes and an additional s which have been 2 0 surement (2.2). a . Let me emphasize ) 4 ) ates large corrections − p 1 0 0 scattering lengths from ( ) enters as a subtraction a + 2 0 O lts by the NA48/2 collab- reshold are unfortunately the Dalitz-plot parameter. a e is something to be better 2 -wave scattering length is as heavy and apply SU(2) ππ 5 S , 24] only and does not take = − ) f 0 0 2 N a ( 2 state, and that = , f PT NLO formula). The results are world première = χ N I ( decays, on the other hand, is mainly , π 3 NPLQCD ETM → ) K is very interesting also because this quantity can at the measured NA48/2 value they can also ob- 6 38 2 0 2 0 a a ) )( − 42 28 ( ( 0 0 phase shifts with solutions of the Roy equations (since a , see (2.2), is not yet comparable to the theoretical one, 2 0 a scattering. In particular one may rely on a soft-pion theorem ππ 04330 04385 decays are never in an . . N 0 0 4 . Neither of the two experiments provides a reasonably precise π e 2 0 − − 300 MeV and the use of the K (more precisely the combination 2 a ( 2 0 < − a -wave. The cusp in π 0 0 S a )= . 0 decay [26] – this issue clearly calls for further investigations. corrections. The analysis in [31], however, has shown that with the valu = latt i.e. M + ) ( away from the value obtained by the NA48/2 experiment. The discrepancy I 2 2 0 π K alone, but since they are sensitive to two slightly different combinations of a σ M 2 0 5 ( . a O , despite their algebraic suppression, which is surprising. Alternatively, ) scattering [29]. The amplitude has been calculated at two loops [30]) in SU(3 6 p K ( π scattering with dispersion relations – the Roy-Steiner equations – [32] and tr O , the combined analysis yields a rather precise measurement of both scatterin K 2 0 π e.g. The experimental uncertainty in The NPLQCD collaboration has also calculated scattering lengths for other h a PT [10], just like one calculates χ made with dynamical quarks andto small the enough physical quark point masses ( to allow for a relia be calculated on the lattice. At the moment there are two independent calculation illustrated in Fig. 1. This first measurement of and constant also in the subject only to [27, 28] although one may doubt whether thisof theory center is of still mass in energy its applicability even. domain One at such can, however, consider the kaon energy data into constraints on theto scattering lengths. the This soft-pion analysis theorem. also indic current algebra The prediction. lattice This short calculation summary [29],understood clearly indicates here. on that ther the Direct other measurements ofnot hand available this is – scattering compa the process same near methodology th used by DIRAC to measure the this way). In this framework these only embody analyticity, unitarity and crossing symmetry, no theory bias is which is about 1 which states that the current algebra prediction for the isospin-odd measurement of Standard Model [22]. Making a two-parameter fit to the Dalitz-plot distribution they obtain but the result is nonetheless most remarkable as it represents a is not particularly worrying in viewis of unknown the otherwise. strong Indeed correlation if with they the fix Dalitz-plot tain a good fit (albeitMoreover, slightly the less analysis so), is but get basedinto a on account very the the different rediative Cabibbo-Isidori value corrections approach for calculated [23 oration in on [25]. the Preliminary resu same measurementmeasurement also in point the in the direction of some tension with the and nicely agree both with the chiral calculation (2.1) and the experimentallike, mea of the LECs estimated in [30], these corrections are of about 10% at sensitive to the difference treat 10% at that pions in the final state of here only because one describes the PoS(KAON09)053 (3.1) decay is effects): 0 ulation the 2 π e 3 ments of the l calculations ttice approach → f the theoretical een calculated to L Gilberto Colangelo e been performed too early to draw f isospin breaking ffects start to play ibutions to K s one of the future teractions or by the into account. Most simulations in a cost sis is based on [35], cts in those measure- ve corrections to their ange quarks in the sea. After having discussed imation. More recently latter is even consistent or the fact that neutral nference Taku Izubuchi who whenever possible or the cusp effect, on the scattering length in QCD reaking effects and obtain analysis the isospin break- 4 e sical phenomenon – without ππ scattering lengths, and indeed e formation of such has K K π MeV [41] ) MeV [42] ) 55 1396 GeV). While such a quantity cannot ( . 10 ( 0 443 20 = . . 1 1 π 7 M ( = ) 2 e PT calculations are for the ( ] χ 0 K M − decays on [38]. In particular for the + tuned such that K 4 e d M K [ m = on [23, 24, 36, 37] (in fact also the calculation for the u π m 3 → is split into a part due to the quark mass difference and the part due to K 0 K M − + K Isospin breaking effects, whether they are induced by electromagnetic in At the level of precision reached by lattice calculations, isospin breaking e M available, see [25]), and for These results indicate rather small violationswith of no Dashen’s violation theorem at (and all), the whichperformed is in in contrast the to what framework is of typically models obtained in [43, analytica 44, 45, 46]. While it is probably up and masstry difference, to ignore are them mostly – a unlessin nuisance the detail for precision the reached theorists, forces high themcalculations, level not it of to is do precision appropriate so. to of commentments. the on the scattering Indeed role both length of the measurement isospin lattice breaking and and effe o the an important role.and The if QCD the spectrum precision isand reaches one charged the of members of percent the an fundamental level,mass. isospin tests then multiplet In for one receive addition, different the has strong QEDlattice la to calculations radiati isospin-breaking so account contributions far f have must been also performedis in be well the known isospin taken limit, in but the thealready community issue more o and than ten first years efforts ago in [39],a at calculating method the these time has effects still been in hav proposed the to quenchedeffective approx evaluate electromagnetic way effects [40]. in dynamical masses In with [41] two a dynamical lattice Domaincoupling Wall of calculation Fermions the of has field the been to thehas performed electromagnetic sea presented – quarks contr an has in update been this of neglected. this calc For At calculation this example [42] co the which now electromagnetic includes contributionbe also to ( str the kaon mass difference has b in the isospin limit ( at this conference Ewaprojects Rondio at [33] CERN. Just has before discussed thebeen conference announced this the by exciting first the evidence DIRAC possibility of collaboration a th [34]. 3. Isospin breaking Standard Model the pionium lifetime could however also be used to measure the ing corrections make a spectacular effect.other For hand, the they decay of are pionium notthem and these corrections, f measurements but would are not be responsible at for all the possible. phy the quantities in thescattering isospin lengths limit. discussed This above: hasfor for in the pionic cusp fact atoms in been the done experimental for analy all three measure be directly measured, one can correct the experimental data for isospin b PoS(KAON09)053 and ′ Z teractions s are partic- ]. t difference is el at low energy e heavy degrees gh to observe the gauge couplings. es of rare decays, the subject and a Gilberto Colangelo ysics of an energy the lattice calcula- eak interactions in articles exchanged rkable, as illustrated n reached nowadays ns of these effects is er lattice calculations cessary precision one rrections (at NLO) to ecays of kaons, which would decouple or not , as illustrated in (1.1) g talk by Professor Lim ients expressed in terms on). They break various gy the weak interactions and in the corresponding se of the size of the strong and the flavour symmetry, tly like new physics, since nrenormalizable operators. en into account at the level us tors which are so generated. V CP , the P 8 penguin contributions Z bosons, or the , or their supersymmetric partners or a Z and W Weak interactions are the last tiny bit of the Lagrangian of the Standard Mod In the long step from the electroweak scale down to the kaon mass the strong in Isospin breaking effects play also an essential role in the study of weak d Among the various nonrenormalizable interactions those which involve bosons which are maybe a factor two or three heavier. A technically importan ′ tests of unitarity of the CKMradiative matrix, corrections or in and tests strong of isospin leptonof breaking universality, or effects precision in have analys of to today’s be experiments. tak summarized in The the contributions status of of Emilie the Passemar [47] theoretical and calculatio Christopher Smith [48 4. Weak interactions is the subject of the next section. Whether in the determination of of course that forof the freedom SM and derive weak the interactionsDoing exact this form one one of obtains can the the explicit explicitly nonrenormalizable form ofof opera integrate the CKM corresponding out Wilson matrix coeffic th elements,These masses steps of have been the illustrated heavy with[49], a degrees emphasizing historical of perspective in in particular freedom the the and openin question– of of the whether the nondecoupling the effects heavy top are quark verythe intimately SM. related to the structure of the w Standard Model definitive conclusions on this issue, ittions is is clear possible, that whereas a on systematic theof improvement analytical these of side effects, this see is the rather unlikely. literature For cited oth in [42]. and play a role in thedecay analysis of of otherwise the stable phenomenology onlysymmetries (like if the which one pions, are waits the otherwise long kaons conserved enou and in the QCD, neutr like the and make kaon physics especially interesting asscale they several allow us orders to of glimpse magnitude at the higherare ph than represented the by kaon mass. a At– series this from of ener a nonrenormalizable low-energy terms perspectiveat the in weak the the interactions kaon Lagrangian are mass treatedin it exac loops does are not the make much of a difference whether the heavy p play again a crucial role as theyThis are is responsible an for effect the which can running be ofcoupling dealt the constant with no in and perturbation the theory, but largehas becau ratio to of go the beyond scales thein involved, leading to evaluating log QCD reach approximation. the corrections, Indeed ne knownthe the at nonrenormalizable level part NNLO, of of precisio and the even SMin electroweak Lagrangian the co at contribution low by energy Martin iscomprehensive Gorbahn review, quite see rema [50] [51]). (for a very detailed introduction4.1 to Semileptonic decays, photon and W PoS(KAON09)053 (4.1) (4.2) (4.3) present with no be tested against st, and with it an Gilberto Colangelo no running between hich the hadronic part trix elements of quark r of those due to real as it is proportional to 5

esented at this conference

− ) ud

)

) 2

+ µ l3 V

0 (K (K →

NA62 – prelim. KLOE ud us + 5 V

5 /V −

(0 −

. us ’s and the universality assumption (on

10

ud V · 10 V fit with unitarity W )) · , as discussed in detail by Palutan [8] and )) unitarity ) γ ) γ ( ud ( PT . The calculation of the matrix elements V ν 001 syst χ / . e µν 0 us V 019 → ± 9 → . fit 5 as performed by the Flavianet Kaon Working group [8]. 0 − K K ( ( ud ± 477 10 Γ . Γ V · 2 stat ) ≡ and = ( K Kaon WG 016 025 us R . . lavi V SM K 0 0 net R ± ± and of the ratio 0.970.97 0.975 0.975 us 500 493

V

. .

0.220.22 0.230.23

us 2 2 have been discussed in Sect. 2.1. These allow a precise determination 0.2250.225 V ( ( π ( F / = K F K R Summary of the analysis of and the ratio ) Even safer against hadronic uncertainties are ratios of decay rates in w 0 ( + f cancels out completely, like in the square of the ratioexperiments of requires the an analysis masses.photon of emission. A This radiative has precise corrections been SM provided and prediction recently in in that [52] can particula and reads In this ratio one tests the coupling of the to the which the SM is based). In the SM this ratio is tiny, of the order of 10 illustrated in Figure 2: as seen there, the CKM passes perfectly the unitarity te essential building block of the SM is experimentally confirmed. A precise experimental measurement has been recentlyboth performed by and KLOE pr [53] and NA62 [54], and their results read of the CKM matrix elements Figure courtesy of Matteo Palutan. ularly easy to deal with:the electroweak the and quark the currents low-energy scalebilinears do to among not worry kaon renormalize about. and and Moreover pion theremore the states is than ma can two be hadrons calculated in reliably the on external the states) lattice or (at in Figure 2: Standard Model thus beautifully confirming the SM. PoS(KAON09)053 K → B ± K ot only rately on the uark operators, oefficients have hanging neutral channel, because -violating contri- repancy with the equire discussing Gilberto Colangelo PT to obtain from ibutions, including . NA48 and KTeV ¯ der to fully exploit ν χ 2 amplitudes will be CP ] leads to an accurate KM matrix elements alculate, in particular / πν 3 ct 56]. These remarkable has started a new major are several other modes they will attack it from a ately. On the theory side is of the different modes → = ould bring up the question I nic matrix elements, on the [50] have been evaluated to K ombination of these lattice ∆ Norman Christ [11], because , as the successful calculations of has been presented for the first time at matrix elements, following the method ε / ′ ππ per se ε → K 10 vacuum matrix elements and use → 2 amplitudes are more difficult and will take longer but / K 1 physics, which is the topic of the parallel summary talk by = and K I ∆ π . These measurements are not new, but an update of the analysis → K ππ and photon penguins. Among these there are the “golden modes” → Z , for which the standard model prediction is dominated by the uncertainty in , with a theoretical error (excluding the parametric one coming from the K ¯ ν K ν ε matrix elements has shown to lead to results with uncontrolled uncertainties. 0 π ππ → L → K K and -physics observables which provide an alternative measurement of the C ¯ For nonleptonic kaon decays the situation is unfortunately less satisfactory There are several reasons that make these decay amplitudes difficult to c In the broad class of semileptonic decays belong also those due to flavour c ν B ν ± (see, [60, 61, 62],calculations with reviewed the precise at perturbative analysis this of theSM conference weak Hamiltonian prediction in [50 for [5]) show. The c have measured these accurately andbutions in in particular both the the decay direct and indire the fact that thereon are the two other light hand, is in not the anymore a final major state. obstacle Dealing with four-q proposed by Lellouch and Lüscher [59] — a 10-20% accuracy for the by the KTeV experiment andthis their conference final [55]. number The for summaryexperimental of results the do NA48 not analysis provide can athe be significant calculation found test of in of the [ the effective SM, weakbeen performed unfortun Hamiltonian to and NLO the accuracy running [57,other 58]. of hand, the The cannot calculation Wilson yet of c be the performed hadro reliably,the as strategy discussed to in calculate detail the by of much larger long-distance contributions.and a For detailed a discussion comprehensive of analys howtheir essential potential it in is testing to the SM, measure see all [48]. these modes4.2 in or Nonleptonic decays reached in about two years.are also The “within reach” [11]. the relevant CKM matrix elements. For these modes both the long-distance contr isospin-breaking ones [48], as well assuch the an short-distance accuracy contributions that thewhere total the theory electroweak penguins error contribute, amounts but to none about is 4%. as clean There as the involved, which is outside the scope ofof this possible conference. new Moreover, this physics w signals in π Standard Model uncertainties in the CKMto matrix provide elements) an of about examplelattice, 6%. of but a also I difficult because mention matrixSM this this may element is quantity start which becoming one here visible can of n [63,the the be 64]. A few calculated detailed quantities accu analysis where of this a issue possible would r disc Okada [1] — better for me to stop here. different side: it has been announcedproject that with the the RBC/UKQCD collaboration goal of calculating directly the This does not mean that they have given up the calculation, but rather that these the currents originated by PoS(KAON09)053 Devaux and size that at low with small cor- is very interest- elements. Gilberto Colangelo ays allow us to test interactions, which –CT–2006–035482 s have played an im- ter, they still provide Innovations- und Ko- Albert Einstein Center ically one which would itive to yet smaller non- vel of precision. No evi- tromagnetic origin. With mmarizing a conference is e the culprit could be either enz SUK/CRUS”. This work (2001) 125 [arXiv:hep-ph/0103088]. 603 9. (1964) 264. (1985) 517. 11 13 250 (1984) 91. 25 (1999) 507 [arXiv:hep-ph/9904230]. 555 ). net I gratefully acknowledge useful comments on the manuscript by Brigitte Bloch- Since the very discovery of kaons the experimental studies of their decay [1] Yasuhiro Okada, these proceedings, PoS(KAON09)054. [2] M. Ademollo and R. Gatto, Phys. Rev. Lett. [3] J. Gasser and H. Leutwyler, Nucl. Phys. B [4] H. Leutwyler and M. Roos, Z. Phys. C [5] P.A. Boyle, these proceedings, PoS(KAON09)002. [6] F. Mescia, these proceedings, PoS(KAON09)028. [7] Gilberto Colangelo, these proceedings, PoS(KAON09)02 [8] Matteo Palutan, these proceedings, PoS(KAON09)008. [9] Jun-Ichi Noaki, these proceedings, PoS(KAON09)030. portant role in shaping thestringent Standard tests Model. of this Today, theory, more to thanan an impossible sixty ever task, years increasing even level la if of it precision.energy, is the Su split SM in is two – torections the a approach given good I by approximation have isospin–breaking dominated taken contributions is byenough either to the precision of empha strong in strong interactions, the or data elec be or zero if because of one a looks symmetryrenormalizable at of terms the the in proper strong the Lagrangian interactions) observable — can (typ become the then low-energy fully be remnant dinamical sens of only the at weak muchall higher these energies. aspects Experiments of in the kaon SM,dent dec even discrepancies its have high-energy emerged scales, to so a farsome remarkable other experimental le than issue or in our a difficulty few in cases calculating wher strong-interaction matrix Acknowledgments Standard Model 5. 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