PoS(Lattice 2010)018 ′ η − η http://pos.sissa.it/ medfor a long the range of processes ce. ), quantities which we are K B roach in lattice simulations (e.g. urbative effects ss a selection of topics in flavour orously if precision flavour physics ogy. Indeed there is no alternative ch a large amount of experimental proved hugely, and the results are mak- , Lattice2010 discovery experiments at the LHC in unravelling the ⊥ P CKM matrix element and decays amplitudes and the spectrum and mixing of us V ive Commons Attribution-NonCommercial-ShareAlike Licen ππ → K quantities for which lattice calculations have been perfor ∗ mature -decays). The improvement in precision and the extension of B [email protected] time (e.g. the determination of the now learning to study (e.g. next level of fundamental physics. which can be studied using lattice QCD hasis to to be continuedvig play a complementary role to large for a wide variety of physical processes. In this talk I discu general method which can be used in the evaluation of nonpert physics, including In recent years the precision ofing lattice calculations a has very im significant impact in particle physics phenomenol Speaker. ) and important phenomenological quantities for whi nonleptonic data is available but which we do not yet understand how to app ∗ Copyright owned by the author(s) under the terms of the Creat c June 14-19, 2010 Villasimius, Italy The XXVIII International Symposium on Lattice Field Theory Christopher Sachrajda School of Physics and Astronomy University of Southampton Southampton SO17 1BJ, UK E-mail: Phenomenology from the Lattice

PoS(Lattice 2010)018 ′ . η − . The s η and the α us V new physics experiments in dis- Christopher Sachrajda ⊥ p and with improving preci- atural question which arises o large lattice calculation of the corre- KM matrix element ressive, due to improved algo- is a large amount of experimen- products of operators. evaluation of other quantities? start with some comments about to be explained by evelopment of our understanding ities for which lattice simulations ificant rôle in the phenomenology. s to evaluate long-distance effects, will discuss two-body nonleptonic within the standard CKM analysis interest to me in recent years, and to relate renormalization schemes , but of course we also need to re- bative QCD effects. In this context ill therefore not attempt to make a l computing resources. It is clearly g theoretical framework and lattice l as many parallel sessions in which es and the coupling constant w elementary particles then precision d on dimensional regularization (such cs beyond the . If, as is be selective and have naturally chosen rospects for the subject more widely. our physics also should not be under- d I will discuss and give examples of have been a number of excellent review ific task of quantifying nonperturbative ng all the exciting new developments and l also not discuss the different formulations tensions decay amplitudes and a recent study of the 2 ππ → K Does the phenomenology require us to strive towards even what next? quantities, i.e. ones which have been studied successfully parameter of neutral kaon mixing, which I will discuss. The n of precision flavour physics is to play a complementary rôle t -decays, a hugely important set of processes for which there K sponding non-perturbative QCD effects. For illustrationB I tal data and yet lattice simulations are not playing any sign Mature system. I will also briefly mention relatively early attempt encoded in the matrix elements of time-ordered (non-local) higher precision or should we set our priorities towards the B for such quantities is sion for many years. These include the determination of the C I will discuss attempts to evaluate can contribute to the quantitative control of the nonpertur Before proceeding to the discussion of specific processes, I Recent progress in lattice phenomenology has been truly imp 1. 3. Quantities for which we don’t yet know how to formulate the 2. Continuing attempts to extend the range of physical quant The examples I use inmember this the talk contributions are which taken lattice fromof QCD Flavour hadronic is Physics structure making and to to the the d determination of mass Even restricting the discussion to flavourto physics, discuss I the have status to ofwhich some I topics believe are which indicative have of been the of exciting progress direct and p the use of continuumwhich perturbation can theory be simulated which in is lattice necessary calculations and ones base rithms and theoretical developments asimpossible well to as do justice more to powerfu theresults title and of I this will talk not by try discussi totalks do on this. specific At topics this conference includingthe there those new in ideas refs. and [1 – resultssystematic 5] compilation have as of been wel the presented latest lattice explicitly. results. I I wil w Lattice Phenomenology 1. Introductory Comments of lattice QCD or effective theories of heavy . Instea will be confirmed and become inconsistencies which will have mission covering and unraveling the next layerexpected, of or fundamental at physi least hoped, theflavour LHC experiments physics discover will ne besimulations necessary will to be determine central to the thishadronic underlyin endeavour, effects. with the The spec discoveryestimated potential however, with of a precision real flav possibility that current PoS(Lattice 2010)018 , K on B MS (the s and α us V decay amplitudes Christopher Sachrajda ππ ently do not know how → . In addition to the techni- K ) MS, while being the scheme i µ i ) are both sufficiently large, is ( distance physics and is calcu- | ative evaluation of Wilson co- µ on-perturbative effects in quan- O MS nd scale dependence cancels be- | , 8] and schemes based on the use O f ix element in such an intermediate ities require renormalization and so m scientific benefit from the results ors or the QCD parameters nderline that the precision of current mmunity to obtain Wilson coefficient I proceed to a discussion of the status nd hence hey can be used in predicting physical f such schemes include the momentum bly such as y. It is therefore preferable to perform are contained in the matrix element(s) ed picture: rectly useful in combining the perturba- × h o calculate the matrix element in the he lattice theory beyond one-loop order, it ing with standard ones such as , computed in the bare lattice theory and with bare 3 i ↑ ↑ is usually calculated in a scheme based on dimen- C i | QCDQCD C ) a ( Perturbative Lattice latt calculated in the same scheme or equivalently to translate O and the matrix element, as long as the ultraviolet cut-off | provided that they are calculated in the same renormalizati C f i (NPR), by choosing a renormalization condition which can be C i h | MS scheme. Unfortunately however, we are not able to perform -decay amplitudes). O | B f h Physics = and C , to the corresponding renormalized operators ) a is the lattice spacing) and the renormalization scale ( ( a latt masses and mixing and finally to some quantities which we curr ′ O (perhaps a Wilson coefficient function) contains the short- η C − Lattice simulations are used to compute the long-distance n One possibility for matching , where η 1 MS) which cannot, but which are generally used in the perturb − a operators to use perturbation theory to relate to evaluate at all (non-leptonic 1.1 Continuum Perturbation Theory tities such as the matrix elements of local composite operat through quantities which we are beginning to evaluate relia strong coupling constant) and thehave quark to masses. be combined These with quant perturbativeobservables. calculations This before is t sketched in the following oversimplifi where lated in perturbation theoryof and one the long-distance or effects moretween local the operators. two factors The renormalization scheme a Lattice Phenomenology as Non-Perturbative Renormalization imposed directly in lattice computations [6]schemes and RI-Mom examples o [6] and itsof recent the generalization RI-SMom Schrödinger [7 Functionalscheme, [9]. we Having need to obtained combine the it matr with lattice simulations in a non-integer number of dimensions a is found however, that the series frequently converge poorl and of choice in continuum perturbative calculations,tive is and not lattice di results.scheme In entirely particular, non-perturbatively. it is not possible t cal difficulty of performing perturbative calculations in t efficient functions. The main pointlattice of computations this is discussion such is that,we to need in u to order work to with the getfunctions higher-order the in perturbation theory schemes maximu co we can simulate.of the After evaluation this of digression a variety of physical quantities, start ( sional regularization, such as the scheme. Because of its practical advantages PoS(Lattice 2010)018 (2.1) (2.2) 1 (red + 2 = semileptonic 2 flavours of 2 flavours of . The current f π ud = = N al extrapolation. V f f / → N N us Christopher Sachrajda K V with ) LO perturbation theory 0 n ( 1 and 1 and + , where the argument 0 in f ) + + m FLAG [14], in which are 0 2 the ratio 2 atus of the results for the form ( 0 f = = and ) and so it is the difference from s f f ic et al. [15], it has also become π MS needs to be performed beyond orem implies that these corrections oments of correlators on the lattice nd the Karlsruhe perturbation theo- f essional N m N tion with the QCD perturbation the- N between the kaon and pion satisfies / example [10]), there have been recent )= to . q nificance of our lattice results. = K 0 MS schemes at two-loop order [11, 12]. f . are by now standard and from the ratio ( emarkable when compared to what was ations. In addition to John Gracey, who d + with with K ons being performed; this enhances con- 008 m f f . 006 ) π 0 . f MS. The precision of lattice calculations is = 0 0 ( ± / u and + K ± m f f π f 4 956 1 flavours of sea quarks is [14] . 193 0 . + 1 2 = )= = 0 f π ( K , the calculational techniques are such that we would obtain f f ) + N 0 f ( + f and π f precisely by combining the experimental results for / K us f 2 (dashed blue oval). V , and the main uncertainty for this quantity is due to the chir = ) f 0 ( N + f from Lattice simulations , combined with the experimental leptonic widths, we obtain sea-quark (shown separately); oval) and sea-quark (shown separately); 0 [16 – 18]. The FLAG collaboration summarises the current st π us The principal lesson of this section is that close collabora Lattice calculations of the decay constants The recent results are nicely summarized in figure 1, also fro f V = (i) the allowed regions from calculations of / (ii) the allowed regions from calculations of 2 (iii) the allowed regions combining the calculations of K f status obtained from simulations with In the last 6possible years to determine or so, following the suggestion of Becirev Lattice Phenomenology the matrix element from the intermediate scheme to the parentheses indicates that theq four-momentum transfer For both the quantities decays with lattice determinations of the form factor Note also the joint work byrists the on HPQCD the lattice evaluation of collaboration the a and charm-quark in mass the by continuum matching [13]. m ory community is increasingly necessary to optimise the sig 2. now such that this translation from the intermediate scheme one-loop order and hence is left most effectively to the prof shown: possible just a few years ago. The precision of the results in eqs. (2.1) and (2.2) is truly r factor as [14] siderably the phenomenological reach ofhas the been lattice performing calcul such calculations for some time (see for specialists. It is therefore pleasing to see such calculati calculations relating the quark masses in the RI-SMom and one which we are actually computing.are The small Ademollo-Gatto for the precisely 1 in the SU(3) flavour symmetry limit ( PoS(Lattice 2010)018 h and (2.6) (2.5) (2.3) (2.4) 2 | ud ]. V | 1, representing are with Stan- = ) , 0 2 | ( ) us + . Christopher Sachrajda V 59 f | ) ( -decays and an arc of + r than the uncertainties 47 β 2 ( | and 1927 ud . π f V 1 | [19]: / y colleagues in the FLAG [14] 21661 = in (2.1) and (2.2). ) K . f 0 ) 0 ee that everything is remarkably π K ( f 0 f + ( = row of the CKM matrix. Although f + | ons for four unknowns. As the fourth e unitarity constraint: f ) , decays of kaons and pions give us two 0 . ) ( her in the next subsection. ults for ) , obtained on the lattice with the experimental and + | 46 1 and f π ( 22 us f ( π = V f us / | 2 V K / | | f 9608 K . us f , and 0 V 5 | 97425 | us . V ud 0 from super-allowed nuclear + and , V )= | 2 = | 0 ud ud | ( V is very small compared to the uncertainties in ) V ud + ud 2 V f | | from the recent analysis in ref. [20] based on 20 different 59 V ( | ub in the Standard Model without lattice calculations and whic ud V , | that follows if the three-flavour CKM-matrix is unitary. [14 V ) us | V us 95 27599 . ( V | 0 transitions. The dotted arc is part of the circle = and 1. Since

β | 22544 π . K = f f ud -transitions: 0 2 V | | β us ud = us V V |

V | us V | + 2 | has been dropped from the left-hand side because it is smalle ud 2 | V | The plot compares the information for ub within the Standard Model V | us , this circle represents the unitarity condition on the first I’d now like to share a simple analysis, which I learned from m 2 V | us V result extracted from nuclear in excellent agreement with the lattice results for Figure 1: Lattice Phenomenology the circle the correlation between dard Model expectations. Experimental studiesprecise of constraints leptonic for the four quantities, Combining equations (2.3), (2.4) and (2.5) we obtain: superallowed nuclear | Within the standard model, a third equation is provided by th Also shown on this plot is the result for it would have been moreconsistent exciting with the to standard find model, an a point inconsistency, I we underline s furt 2.1 in the remaining terms. Now eqs.equation (2.3) and we (2.4) can give 3 take equati the value of where collaboration, which determines underlines again how remarkably consistent the lattice res PoS(Lattice 2010)018 and is the π q decays) → 3 ℓ B K (where n table 1, where 2 q start with the eval- decays ( Christopher Sachrajda ) 0 π ( 0 . It is a little puzzling 0 f → ) = d K 2 max m )= q ( 0 ( 0 = esented as functions of the pion f f the LECs, the above discussion + decreases, they are only doing so u f ptonic ation theory on the other hand, at m π for the difference but they have the alue is given by one of the very few ytic terms have the correct sign and r chiral logarithms) should account ch the initial kaon and final pion are d that the one-loop chiral logarithms e entries in table 1 appear to a long [22, 23]. Estimating the SU(2) LECs m theory to investigate whether the very in table 1 and eq. (2.7). Of course the π y the ETM collaboration with 2 flavours of f mit, re the same features hold for 26 (2.7) man relation, which I present here in the . / ons of 1 π K f m ≃ decreases is expected to accelerate towards the π K 575 MeV470 MeV 1.00016(6) 435 1.00272(34) MeV375 1.00416(43) MeV 1.00961(123) 300 MeV 1.01923(121) 260 MeV 1.03097(224) f f π 6 m is the largest physical value of obtained by the RBC-UKQCD collaboration with 2+1 ) are proportional to unknown low-energy constants 2 )= π π ) ) π m m 2 max m q 2 max ( q − 0 ( f K 0 f m in table 1 are increasing as = ( ) 2 max 2 max as a function of q q π ( ) observed in table 1 as –– – – 0 m ) f can be calculated with excellent precision as illustrated i 2 max q 555 MeV415 1.00192(34) MeV330 1.00887(89) MeV 1.02143(132) 670 MeV 1.00029(6) ) 2 max ( 0 q f ( , where 2 max 0 ) q f ( 0 2 max f q ( 0 f Values of decays and SU(2) chiral perturbation theory semileptonic decays. 3 ℓ π I end this section with some observations relating to semile It should be stressed here that since, apart from the values o K → Lattice Phenomenology D have a large coefficient andwrong are sign, of the implying correct that sizefor the to approximately analytic account twice terms the (or differenceanalytic higher between terms orde the (both linear results and quadratic in one-loop order the LECs can be expressed in terms of relied only on SU(2) chiral perturbation theory and therefo value in eq. (2.7) for smaller values of the pion mass. We foun (LECs) and hence are not calculable. In SU(3) chiral perturb very slowly. In ref. [21], weslow increase used in SU(2) chiral perturbation by converting results from SU(3)(large) ChPT magnitude suggests to that account the for anal the difference. Table 1: flavours of Domain Wall Fermions (left two columns) [16] and b 2.2 twisted mass fermions (right two columns) [17]. where all the quantities are evaluated in the SU(2) chiral li using SU(2) chiral perturbation theory. Lattice calculati uation of momentum transfer) and corresponds to theboth kinematics at in whi rest. that while the values of the results from the RBC-UKQCD andmass. ETM collaborations The are reason pr thatway I below the present value these expected results inanalytic the here, non-perturbative SU(2) results is chiral in that limit. QCD, th This theform v Callan-Trei PoS(Lattice 2010)018 has (2.9) (2.8) ld be K B 2 oper- kaon is he chiral = S ∆ denotes the number on theory. In spite f , Christopher Sachrajda N .   , the bag parameter of K denotes the left-handed 2 2 π π K ˆ B m m L B orm-factor at the standard + − c c + + hris Dawson, the rapporteur at K , but the results in table 2 were nomenology for which the pre-   ˆ B approach has been generalized to 2 2 π π 2 2 p invariant µ µ matrix element of the m m uate the chiral logarithms, since they s has been d be extended to two loops in chiral ely due to the fact that there were in- stration here I tabulate the key results ding significantly to our confidence in   ed review of recent results for state (where 0 log log K atistical and the second is systematic. f 2 2 f f N 2 2 π 2 π 2 π π m m 16 16 7 semileptonic decays [25]. 4 4 3 3 and final π 0 − − ¯ 1 1 K →   0. Since the chiral extrapolation of the lattice results is D − + F F = loops, finding [21] and 2 q π ) = ) = any information about the chiral behaviour is useful (it wou Publication 0 0 ( ( ) → 0 − + ETMC 2010 [32] 2 0.729(30) ( JLQCD 2008 [31] 2 0.758(6)(71) B f f internal Bae et al. 2010 [29] 2+1 0.724(12)(43) 0 hard-pion chiral perturbation theory f Aubin et al. 2009 [28] 2+1 0.724(8)(29) RBC-UKQCD 2007[27] 2+1 0.720(13)(37) RBC-UKQCD 2010 [30] 2+1 0.749(7)(26) between an initial )= 0 0. At this point, the energy of the pion in the rest frame of the ) ( L 0 s 2 and so cannot be considered soft in SU(2) chiral perturbati = f are unknown LECs. In this way we have some information about t µ / 2 γ K ± q L ¯ c m d )( L and s µ decays [24] and to A comparison of recent results for the renormalization grou ± γ F L ¯ d ππ K One of the very important quantities in particle physics phe Of course, we really want to know the chiral behaviour of the f ( B → been presented at this conferencein by table Jack Laiho 2. [5], Not forobtained illu only with are a variety the of errors fermionthe reduced actions evaluation and by of techniques a the ad systematic factor uncertainties. of 3 or so where component of the spinorthis field). conference, was As reporting recently resultssufficient as with unquenched five a results years 12% at error ago, light larg C masses [26]. A detail Lattice Phenomenology behaviour of the form factors at 3. Table 2: cision of lattice results has improved hugely in recent year neutral kaon mixing. It is defined as the suitably normalized of dynamical quark flavours. In each case, the first error is st reference point very useful indeed ifperturbation the theory). results This in eqs.(2.8) and (2.9) coul the largest uncertainty in approximately of the hard external pion,can we be found calculated that from it soft is possible to eval ator K PoS(Lattice 2010)018 es ∆Γ lue hed vacuum and vacuum tensions K =1/2 rule → I → ty triangle. M ∆ K ∆ K 4 ( and / decay amplitudes and π π π Christopher Sachrajda ππ → → 4 4 2 amplitude by a factor K → e experimental value of − − / K 4 1 K 10 − 10 = × 10 × I nity is the reliable calculation ) ) ε × ∆ 3 8 ese calculations, the authors did . In the past lattice calculations will continue towards 1% preci- . / . ′ decays, we should include terms 2 1 -2) K ε B ± ± utations of ÷ mation from different physical pro- that in the theoretical expression for ansion and I start with a brief review evel (see for example [33 – 37]), and ππ 0 obtained from mesons) [38 – 40]. Necessary theoret- 2 . s (see sec. 6). cussion of the direct evaluation of the ays in general and on the . y non-trivial. ffects, for which we need to determine S 4 ε egin considering corrections which had (-7 is not precisely equal ( → / K 17 ′ e for new physics, a number of ) ( vacuum matrix elements al expansion. − ε K amplitude with the two pions in a state with ∆Γ and → 2 / L 8 K A K ππ . K 1 M 8 → 12 ∆ /Re ± and 2 2 rule and 0 ÷ K ( 3 22.2 / . 9 A π 1 25 = → Re I is the K ∆ 2 transitions) in particular. The non-zero experimental va 0 problem, i.e. the numerical subtraction of power divergenc / A 3 2 rule (the enhancement of the / = chiral expansion one can determine the 1 vacuum matrix elements. In 2001, two collaborations publis I ) 3 = ∆ → ( I (where K are an important ingredient in global studies of the unitari ∆ 0 SU ultraviolet A K RBC [41] B Experiments and CP-PACS [42] Collaboration(s) /Re π 0 A contribute to these tensions. Calculations of → K K B decay amplitudes from decays decay amplitudes in general and attempts to reproduce the th Quenched 2001 results on the ππ in particular (see table 3). In spite of the limitations of th ππ was historically the first evidence for direct CP-violation ππ ε ε → / → ′ / → and to understand the ′ An important challenge for the lattice phenomenology commu At lowest order in the Lattice results for K ε ε K ε K , the parameter monitoring indirect CP-violation in / ′ K have arisen in recent yearsthe at results the for 1.5 – 3 standard deviation l sion, but already with current precisionpreviously it been is neglected. necessary For to example b it has been stressed Although in general thecesses remarkable significantly consistency restricts of the the possible infor parameter spac for ical improvements include the evaluation ofmatrix long elements distance of e the time-ordered product of two operator 4. of about 22 relative to that for are the differences of the masses and widths of the and the renormalization of the weak operators. This is highl of ε ε Table 3: Lattice Phenomenology have tried to estimate the matrixmatrix elements by elements combining with comp the lowest orderof terms the in status the of chiral such exp matrix calculations elements before with proceeding two-pion to final a states. dis 4.1 achieve the control of the by calculating isospin 0) and recognise that the phase arctan proportional to Im matrix elements using the lowest order term in the SU(3) chir sone interesting quenched results on non-leptonic kaon dec and PoS(Lattice 2010)018 (4.1) (4.2) prob- prob- matrix ππ is an integer. . For illustra- → infrared n ) 2 ultraviolet K ∗ q matrix elements di- + Christopher Sachrajda π 2 ππ m ( 4 . are the masses of the sea and → ) and the value of this curve L z s = ) =0.001 =0.005 =0.01 in the pion-mass range 240- K x x x =0.001 =0.005 =0.01 m x x x m , the lowest order LEC for the 2 ¯ ud E 27 = ( sured values of the energies one and α =0.005 m =0.005 m =0.005 m =0.01 m =0.01 m =0.01 m L l l l l l l d major study. The have also heard about a proposed ntly reliable and m m m m m m SU(3) extrapolation ) x by idean volume is also understood as m ates, the quantization condition for ich diverge as inverse powers of the ¯ m e form [46 – 48] su ing that the one-loop corrections are E oach. This is illustrated in fig.2 where , = l ntre-of-mass frame, but they have also g NLO SU(3) ChPT, but the corrections on of u ering into states other than two pions). m + ( ones limited to the two pions at rest [45]. π cant statistical errors. The m

n L s respectively. ) ππ ¯ )= dd ∗ → ( q is a known kinematic function and ( K 9 − φ φ L ) res )+ ¯ uu + m matrix element. z ∗ ( m q π  ( L δ ) → Decay Amplitudes ¯ K sd phase shift, ππ = ( ) ππ 0 0.01 0.02 0.03 0.04 0.05 → 1 , 2 / K 27 1 is related to the two-pion energy ( 3

matrix elements with

0.8

1.6 1.4 1.2 0.6

K K

π π

f f m m / > K | O |

< < ∗

π O + (27,1)(3/2) + π q → K The mass dependence of the is the physical s-wave 2 (27,1) operator: / δ 3 From the above discussion we conclude that we must calculate The RBC-UKQCD collaboration have repeated the calculation = I 4.2 Direct Calculations of lem of extracting the spectrumlong and as amplitudes in we a neglect finiteAssuming the Eucl that inelastic the contribution rescattering (i.e. istwo-pion states dominated rescatt in by a the finite volume s-wave derived st by Lüscher takes th Satisfactory fits for the mass dependence were obtainedwere usin found to be very large, castingthe serious dashed doubt curve on represents the the appr results in the SU(3) limit ( Figure 2: Lattice Phenomenology elements have to be computedmethod directly. to combine At this conference we lem of the subtraction oflattice power spacing) divergences remains (i.e. tractable of at terms the expense wh of signifi where rectly and the RBC-UKQCD collaboration is now undertaking a ∆ tion, all the formulae inbeen this generalised section to are moving presented frames in [49 the – ce 51]. Thus from the mea The relative momentum in the chiral limit isvery much large. below the Thus data points, the demonstrat use of soft-pion theorems is not sufficie 420 MeV [43, 44]. For illustration consider the determinati valence light quark and the valence "heavy" (strange) quark PoS(Lattice 2010)018 ) ) 8 , ¯ ud 8 (4.4) (4.5) (4.3) (4.6) (4.7) ( )( chiral ¯ in (4.4) R 2 decay sd ) ( / ) 3 1 1 ( , = -quark say, so SU 27 I ( d ∆ × e L is simply , ) i 2 2 3 Christopher Sachrajda / / ( + , 3 3 R L R K O -component of isospin. ) ) ) | SU j j z i 2 2 / / d d d j j j 3 3 in eq. (4.4) – (4.6). ¯ ¯ ¯ , u u u O ( ( ( | 2 2 2 matrix elements to unphysical | / / L L ) L 0 1 3 ) ) . The subscript 2 ) dy with Domain Wall Fermions i j h isospin 2 is relatively straight- M i 2 lson fermions was completed in π p | O ed results from dynamical simu- u elements of these operators is the u u q i ( isconnected diagrams to evaluate. i i + lume effects at non-zero total mo- ) ¯

¯ ¯ s s s + 5 π ) 56]. Before discussing the results I sured Euclidean matrix elements and γ ∗ π st, it is not now necessary to isolate an lties when evaluating , so that the evaluation of the matrix ) q + ( + ( + ( → d 1 ( ∓ ′ the use of periodic boundary conditions,

p

+ m 1 P R ( L R ( ) ) K φ ) + j j µ i = π γ d d d u j j j h , the change in the 1 representation of the ¯ ¯ )+ ¯ z ¯ q d d 2 3 d m I ∗ ) ( ( ( q ∆ 1 = ( = , 10 ′ − − − i R δ L R R , , where the arguments represent the momenta in lat- 27 ) ) ) 1 effective Hamiltonian and transform as the + L  i ( i j j ) ) u K 2 2 u u 3 K j L = | j j ∗ . (4.3) has been generalized to moving frames in [50, 51]. q ¯ 2 2 / ¯ ¯ u ∗ m q u u 1 S are equivalent, but there are a number of advantages in us- / / ( ¯ ( ( q π q 1 3 structure of the operators 2 ∆ 2 2  (   / / − O 2 components of the electroweak penguin operators labelled V L 1 3 | L L ( ¯ / ) ud π ) ) ) j i i . O + 3 ) 8 2 R d d d π p i i i ¯ ) su ) = ¯ ¯ ¯ ( = s s s 3 0 L I and ( 2 states. The flavour structure of the operators | / π 2 2 i ) 2 operators whose matrix elements need to be evaluated: / / A π = ( = ( = ( ) + ( | SU 1 3 3 + ( / ) p 2 2 ¯ 3 π + O 1 dd / / × ( , 7 3 8 3 2 + π L + | / are the = 27 − ) π O O ( 3 π | I 2 3 8 h , / O ¯ ( ∆ Decays uu -state we can impose antiperiodic boundary conditions on th 2 7 3 2 + = )( O I SU = π I ¯ sd ) + ( are colour indices and π ones [57]: j ππ + ( denotes differentiation w.r.t. ′ π → + and i π K in the standard notation for the There are three With the The calculation of decay amplitudes into two pion states wit A significant simplification in the calculation of the matrix → 8 , + 7 tice units (up to finite-volume corrections). In contrast to for which the ground state corresponds to the two-pions at re amplitudes. An exploratory2004 quenched [54], study but with at improved thatmentum. Wi time Results we from had RBC-UKQCD’s not exploratory understoodwere quenched the presented stu finite-vo in 2009lations [55] with and almost this physical yearmention pions, M.Lightman some but present theoretical points. on a course lattice [ ing the fully extended forward; there are noWe power will divergences see to in subtract section 4.2.2 nor that any these d are significant difficu rather than the elements of the operators Eq. (4.7) is an exact relation in the isospin limit symmetry group. Lattice Phenomenology can determine the phase-shift. The relation between the mea the physical amplitudes is given by [52, 53] where the 4.2.1 denotes that the operator transforms as the where K that the ground state is where the subscript on the operators indicates representation of O use of the Wigner-Eckart theorem to relate the physical PoS(Lattice 2010)018 ); = = × × 2 the ) I π A 4 ∆ syst . m ) 2 25 ± ass ( 04 stat . ) 0 07 ± ( 6 . 56 9 . Christopher Sachrajda ( 1 − = almost physical pions to oration presented at this 2 and the second pion with decays in the RBC-UKQCD )= A GeV . The non-perturbative L 2 MeV) and choosing the mo- / ππ 8 f the Wigner-Eckart theorem A ) ( θ − 2 → ematic error will decrease from ( 7 MeV). The properties of these 10 m . the lattice spacing. It appears that K BC-UKQCD programme. y [63]. × 2 519 e Im edure was tested in an exploratory / For the physical decay, the minimum 493 atio) gauge action on a course lattice evaluated are illustrated in fig.3. The tude is Re 3 + + operators which contribute to Im ( = 50 ach of the three operators are presented . π π = = K ng the measured matrix elements to the wisted boundary conditions [59 – 62] on tor requires knowledge of the derivative I K m ∆ m 180 MeV) and the kaon mass is 519(2) MeV, ≃ MeV, π s-wave phase-shift can then be obtained by the ) m 11 5 2 2 6 MeV, / / ( . 3 3 ππ 6 . O s 32 lattice using a Domain Wall Fermion action for the 139 which allows for the derivative of the phase-shift to be × 145 = θ = 64 π K π m × (it is sufficient to perform the twist in a single direction), m 3 , the energy of the two-pion system is found to be 516(9) MeV so θ L 2 matrix elements is very considerably harder than that for / / π 1 2 amplitudes were obtained with a partially quenched pion of m = √ I . The corresponding ∆ ππ L / Decays ) → π 0 K 2 = I ) − θ ππ ( ( 4 GeV). The motivation for such a coarse lattice was to enable Schematic diagram illustrating the contractions for . → 1 MeV (the corresponding unitary pion has GeV [56], but after the renormalization is complete the syst K ) GeV, to be compared to the experimental number of 1 -quark with twisting angle ≃ The evaluation of With the masses given above ( The final theoretical point I wish to make here is that the use o 6 I now briefly summarise the results from the RBC-UKQCD Collab d 13 8 ( 1 − − − a ( quarks and the DSDR (dislocation suppressing determinant r ensembles were discussed at this conference by Bob Mawhinne be simulated, the Figure 3: Lattice Phenomenology momentum excited state in order to have a decay into two-moving pions. calculation [55]. size of the lattice is halved from about 6 fm to 3also fm. allows us evaluate the Lellouch-Lüscher factor relati the two-pion ground state now corresponds to a pion with momentu Lüscher formula (4.2) as a function of conference [56]. The correlation functionssimulations which were need performed to on be 32 145 physical amplitudes directly [58].of In the particular, phase-shift (see this eq. fac (4.3)). By imposing partially-t evaluated directly at thesimulation masses [58] but being has simulated. not yet been This implemented in proc the main R mentum of the pions to be guesstimating the renormalization factors RBC-UKQCD quot that the kinematics is almost matched. Samplein plateaus fig. for 4. e The preliminary result for the real part of the ampli both close to their physical values ( about 25% to 15%, dominated bythese the calculations uncertainty are in possible the with value good of precision. 4.2.2 10 renormalization has not been completed for the electroweak 10 PoS(Lattice 2010)018 , . ) R t d vy ) 5 i d a γ 25 d ππ (4.8) (4.9) s j ¯ E u 0 two- ( − L ( = 20 ) j I . d i ¯ ider the four s 15 ( , 73 GeV and an t 4 . GeV 3 1 or. 10 13 GeV − = Christopher Sachrajda 11 and (c) 1 correlation function is 10 − 5 R − V ) ) a 10 hown in fig. 7 and a total −15 ¯ 08 ) ud ππ . ( x 10 2 0 exploratory study by RBC- 4 3 2 1

0 . 2

L 4.5 3.5 2.5 1.5 Quotient of Correlators of Quotient 0 t 2 ) 1 2 he statistical precision, quark ± = ¯ sd ± 4 in fig.7 are needed to subtract tions, at this conference Qi Liu ( I 79 9 4 3 . . 7 2 e vacuum subtraction. At least the ( ( e four components of the in fig. 6 [64], from which we see that mes. , (b) 25 ents of the pseudoscalar density ¯ , albeit at unphysical kinematics with − − L are colour labels. 2 ) j A ution with sufficient precision. = = , ¯ ud i 20 2 0 ( tical cancelations to obtain the exp L A A and ) 0 ¯ , and 15 sd Im Im ) 1 ( A j t d 12 ) 0 has vacuum quantum numbers and the vacuum 4 3 2 10 5 , , γ = ∓ I 5 1 GeV GeV ( µ 7 8 −16 γ − − j 0 correlation function is proportional to 2D+C-6R+3V. The x 10 0 ¯ 6 4 2 8

u

14 12 10 Quotient of Correlators of Quotient 10 10 = )( ) ) appears to be tractable for this simplified kinematics at hea i I 9 we will require more statistics to confirm that there is indee 1 2 d . 0 ) 0 0 5 A 045 . γ A 4 3 ± 0 t t 0 t 25 − 0 16 lattice with an inverse lattice spacing of . ± 1 3 ( × The four diagrams which contribute to the two-pion propagat µ is the energy of the two-pion state. To illustrate this, cons 20 γ i 394 = ( ¯ DCR . 32 s 5 0 ππ × A = ( E 15 3 = ( R t 0 0 , Re 2 1 2 L (a) (b) (c) Figure 5: ) A 10 ¯ ud ( Re L Sample plateau for the matrix elements (a) 5 ) ¯ sd −17 420 MeV and with the pions at rest [64]. There are 6 diagrams, s ( x 10

To demonstrate the above, I now present some results from the In spite of the difficulty of subtracting the vacuum contribu 0

6 5 4 3 2 1 7 Quotient of Correlators of Quotient ≃ 2 operators. The two-pion state with π / pion masses, whereas for Im The results for this unphysical kinematics are [64] The precision is limited bycalculation the of disconnected the diagrams real and part th of of 48 Wick contractions. Thethe diagrams power labeled divergences which by are Mix3 proportional and to Mix matrix elem where 3 Lattice Phenomenology diagrams in fig 5 whichproportional make to up D-C the whereas two-pion the propagator. The Figure 4: behaviour, where m UKQCD on a 16 major practical difficulty is to subtract the vacuum contrib contributions have to be subtracted requiring large statis unphysically heavy pion withpropagators mass are generated 420 MeV from each [64]. of the 32 To time increase slices. t Th pion correlation function 2D+C-6R+3V are shownthe separately error on the 3V component grows significantly at larger ti presented the results of a complete calculation of PoS(Lattice 2010)018 (5.1) π π π π Christopher Sachrajda s , l Mix4 Type3 s s 16 , s K K cent study on a lattice of spatial disconnected diagrams and this is 5 γ 14 C ¯ s decays. The blue circles represent the 3V 6R 2D ere I mention one other longstanding 0 two-pion correlation function. = orming the corresponding calculations 12 ππ s = I O → π ltonian and the squares that of the pseudoscalar π π π 10 K t 8 and 13 6 d mesons. To determine the propagators we need to 5 Mix3 Type2 γ ′ ¯ s s d 4 η 2 + √ u 2 73 GeV [65]. Using interpolating operators of the form and 5 K K . γ 1 ¯ u η 0 = = 1 l − 16 15 14 18 17

O a

10 10 10 10 10 Corr(t) π π π π s , l Contribution of the four components to the Type1 Type4 s s Mesons ′ K K η and lattice spacing Types of correlation function corresponding to 3 Figure 6: and In the previous section we saw the importance of controlling η Figure 7: the case for many important phenomenologicalissue, quantities. the H spectrum and mixing of Lattice Phenomenology signal. The greater challenge is now to proceed towards perf 5. insertion of a four quark operator appearingindensity. the weak Hami at physical kinematics. extent 16 evaluate the diagrams shown in fig. 8 and I report here on our re PoS(Lattice 2010)018 ′ . ′ η η ted (5.2) (5.3) MeV, and and the ) s η ss O C 14 142 ( represent the s and ’

s 12 η η 947 ll est mass used in C and = 10 l ′ ls s Christopher Sachrajda η D , 8 l m t s (b) = l 6 O β system. , ′ α 4 η l re denoted by - nerated with sources at each of O MeV and η 2 ) ss s s eature that the uncertainties on the 6 C ( s n the connected ones. The effective 0 where O 573 and a short, but clear, plateau for the 0 1

i

0.4 0.2 0.8 0.6 eff = M η s he simulations; see [65] for details. ) . (b) The effective mass plots for the , ′ s t η t O ( m ! β ss ls O ss D ) D 14 ′ D 2 t − √ + ss l t s − C ( O 16 α ll sl (not shown). O D s are shown in fig. 9(b), again at the lightest masses used in the D 14 h sl 2 2 O ) 0 D t ss ll ls ss ll − ( = √ ll 31 C D D D C ′ 12 ∑ l l t ll , from which we see that the usual expectation that disconnec X C − t and C l 1 10 32 ls

O D t , = 8 (a) ss l ) = l t D O ( 6 , ll αβ D . In fig. 9(a) we present the values of the diagrams at the light ll 4 X ′ t D 2 l ) and strange quarks respectively. The connected diagrams a 0 d 5 9 8 7 6 Diagrams contributing to the correlation functions for the

(a) Values of each of diagrams as a function of 10 10 10 10 10 l Corr(t) and O u disconnected ones by light ( After the chiral extrapolation we obtain the values Figure 8: Lattice Phenomenology mesons. In each case the results are for the lightest mass in t we calculate the correlation functions Figure 9: where the diagrams are definedthe 32 in time fig. 8. slices The propagators are ge the simulations as a function of simulations, from which we see a good plateau for the diagrams are small does notdisconnected apply here diagrams and are also significantly themass expected larger plots f for than the those eigenvalues of o PoS(Lattice 2010)018 .A ◦ (6.1) 10 fore of − (statistical to ◦ is a weak or ◦ ) 8 . J 20 2 -products of the ( − T 1 . Christopher Sachrajda 14 d s i − ltonian, K = | θ )] y ( µ s power like). As lattice results J ) s to physical processes, generally ciple to evaluate the long distance s x 2 local operator has been discussed d alue of e mixing angle. Once SU(3) flavour ates that QCD can explain the large o consider only the symmetric octet ( nce contributions effectively and this h the SU(3) octet state and provides a ion is the dominant term, but long- experimental values of 548 MeV and states are simply linear combinations element of a local operator. A good ) = ng the reach and precision of lattice Q it is a standard phenomenological as- to be done now to reduce the statistical es lying in the range ′ [ direction include ref. [66] for rare kaon µ S η T diagrams play an major role. ple if the GIM suppression is logarithmic / ∆ | es. Progress in the efficient determination t is possible to check the orthogonality of W π h and M y ( . η iq C − 15 y e 4 = x d 4 decays. This requires the evaluation of d d s ¯ ν Z parameter in neutral-kaon mixing illustrated by V πν K N is a volume factor. The generic lattice calculation is there B → V )= N K 2 q ( W W J . , and Q T s − d ℓ + ℓ π → K is one of the four-quark operators appearing in the weak Hami Q As explained in ref. [65], it is also possible to determine th We are used to calculating the short-distance contribution In spite of the limited precision, our calculation demonstr In ref. [66], the authors conclude that it is possible in prin of the octet and singletsumption that flavour they combinations; are. nevertheless Based on this assumption we obtain a v error only) in agreement with the phenomenological estimat Lattice Phenomenology where only the statistical errors are shown, compared to the 958 MeV respectively. symmetry is broken it is not clear that the physical The determination of the matrix element of the resulting form electromagnetic current and of all-to-all propagators will be very important in improvi particularly important feature of the calculation is that i mass of the ninth pseudoscalar andfirst its benchmark small for mixing future wit calculations. There remainserrors much and to quantify in detail the systematic uncertainti calculations of physical quantities in which disconnected formulating the calculation asexample the is evaluation the of evaluation of the the matrix the mixing matrix confirming that it is a good approximation t 6. Long-Distance Contributions to Physical Quantities and singlet states in order to understand the mixing. effects for where in the section 3.distance contributions In are many not always cases negligible, for theor exam if short-distance there contribut is abecome CKM more enhancement precise, (even we should if try the torepresents GIM compute a the suppression new long-dista type i of calculation.decays Early and thoughts ref.[67] in for this neutral kaon mixing. 6.1 Rare Kaon Decays PoS(Lattice 2010)018 → (6.2) K mesons respec- atrix elements is K , channels, has been i Christopher Sachrajda 0 | -mesons into two light and ) B k ~ π , K t K ( ε † K ng and short distance physics, on to restrict consideration of φ -meson means that very many g ideas how the long-distance B )] lating a lattice approach, namely esses in flavour physics for which, 0 nto a practicable method, the main l arguments by one-loop perturba- ( etries and the GIM mechanism, the finite-volume corrections to second c i e contribution; retical framework for treating this in x element in a way that is consistent e matrix elements of local operators, l calculations have been performed. Q sive decays of ably the CP-asymmetry in the golden- are light mesons. In contrast to e matrix elements of the product of two of the field of interesting physical applications er divergences and the consequences of 2 − CD effects sufficiently precisely. , been taken). The main issue discussed in ) 1 0 M ( u i Q might be evaluated [67]. This requires ) [ K x ε 16 ( mass difference and µ X S J K ) - p L ,~ K π t 1 operator and ( π = φ are the interpolating operators for the | B 0 K ∆ h φ x · iq − is a and mass difference and i S π x e O φ 4 K d − 0. -Decays L Z i B < K − ), our ability to deduce fundamental information about CKM m K , where t S i -decays. A huge amount of precise information, from over 100 K B B | ψ ) 1, four-quark weak operators between kaon states; J 0 ( 0 and = i → S O > | B with the original explicit evaluation of the short-distanc order in the . ∆ 2 π t I end this talk with a discussion of an important class of proc At this conference Norman Christ presented some interestin The operator product expansion can be used to separate the lo decays discussed in section 4, where it is a good approximati M 1 (i) the calculation of the long-distance contribution to th (ii) the subtraction of the short distance part of this matri (iii) a generalization of the Lellouch-Lüscher approach to M up to now at least,non-leptonic little or no progress has been made in formu Although much work remains to be donetheoretical to steps develop have these now ideas been i taken. 7. Non-leptonic ππ mesons. Unfortunately, with just a fewmode exceptions (most not Lattice Phenomenology correlation functions of the form final-state rescattering to two-pion states, the large mass contribution to the obtained about decay rates and CP-asymmetries for the exclu with power divergences can be removed andtive calculations. they They check believe their that their genera studyfor opens the a new lattice community, although to date no such numerica 6.2 Long-distance contribution to the contact terms. The authors conclude that, as a result of symm tively (after the three-dimensional Fourier transformref. has [66] is that of renormalization, the subtraction of pow limited by our inability to quantify the non-perturbative Q so that the non-perturbative QCDh effects are contained in th intermediate states contribute. We dosimulations not in yet(?) Euclidean space. have a theo PoS(Lattice 2010)018 ) π b f are and m v (7.2) (7.1) / 1 M QCD and Λ → , the long- u ( ∞ B O corrections). . ) is not so large, i ) is shown as the b → 2 d b meson transition b A ¯ m represented by the B M m / − | m A V → A − Christopher Sachrajda ) ↔ QCD V − . This is illustrated in B ) ¯ V Λ 1 du ) ( ¯ ( du M ( A O ¯ decays. On the other hand ub urbation theory. − ( π | V ) and and + . The local four-quark operator ised that as ) + ( → − A ¯ ) π − ub u s + , π − ( terms are important. B π ( are light-cone distribution ampli- he right-hand side are known or α -decays were based on naïve fac- ) V π 2 i h + ) he significance of the factorization ) ( u B y either reflect universal properties b M Φ π 0 ( O ¯ | ts of matrix elements of 2 ub ities which appear on the right-hand m ator Φ o the fact that since ( A ) M / → antities [68 – 70] nal (naïve) factorization is recovered as u − ptonic puted systematically and the results are, on-scale dependence of the two sides do Φ at all strong interaction phases are gener- d ( rtional to the leptonic decay constant udes) or refer to a ting V tion formulae are valid up to ¯ I ) B ) i j QCD v ( ¯ Λ du 1 u ( ty. ( M duT | O 1 Φ − 0 ) Z π 17 ξ ) transitions, the h ¯ ( 2 2 d 1 B ? = m M ( Φ i 1 ) b d M v ¯ → B , → | u B B j , A F ξ − d ( V j ¯ B ) II ∑ i ¯ du = ( i A ¨ ve factorization for the decay − B dudvT | V i ) ξ O d ¯ | ub 1 ( 2 0 | Z M − , + 1 π is shown as a product of the bilinear operators + M are short distance contributions and are calculable in pert and/or vice-versa, depending on the flavour quantum numbers A are the form factors for π h − 2 1 II h , V Illustration of na I M ) M T → ¯ du B j ( → F A − In 1999, together with Beneke, Buchalla and Neubert, we real In the past phenomenological approaches to nonleptonic V ) ¯ ub matrix element of a local currenta (form rigorous factors). prediction Conventio in the infinite mass limit (i.e. neglec CKM and chiral enhancements to non-factorizable tudes and and the second is proportional to the form-factors of semile Lattice Phenomenology where the limitations of such a modelnot are match clear; and the rescattering renormalizati effects are not included. the momentum fractions carried by theformula quarks stems in from the mesons. the T side fact are that much the simpler non-perturbative than quant of the a original single meson matrix state elements. (the light-cone The distribution amplit ( Perturbative corrections to naïve factorization canin general, be process com dependent. It is aated remarkable perturbatively in feature the th heavy quark limit. The factoriza Figure 10: corrections and the main limitation of the framework is due t black circles. The two black circles are separated for clari torization in which the matrix element was reduced to produc distance QCD effects do factorise into simpler universal qu The motivation for assumingcalculable (7.1) (in is principle at that least). the The two first factors factor on is propo t product of the matrix elements of the two currents: vacuum fig. 10, where the matrix element of the local four-quark oper PoS(Lattice 2010)018 , ) + + ˜ k ˜ k ( (7.4) (7.3) of the B φ and we a / decays and re the parton 2 M decays directly, 1 2 M M 1 Christopher Sachrajda , → 0 M B = ⊥ → z , sical pions. This improve- + ons, at least up to now, we B z

r i appear.) Although at large B owever, there are quantities for ) | ynamical quarks with masses of can and are being computed [71] an error which was not possible elped to organise which was held ) s to particle physics phenomenol- + ical ideas for lattice simulations to rediction that it will take about 30 to fulfill the prediction, but we are 2 ˜ 0 k , utations beyond the standard quan- ( ( is convoluted with the perturbative 1 ey diverge as powers of 1 and the positive moments of the wealth of experimental measure- ics, which has been the focus of this results presented at the annual lattice et for which we do not yet know how α . M B le to lattice studies. b ) torization formula (7.2)? The moments ] Φ + 0 ces with sufficient precision. ns g log ˜ , k z ve QCD effects in ( B )[ z Φ ( β + ¯ + ˜ u k discovery experiments at the LHC in unravelling represents the part-ordered exponential of gauge ˜ k | d 18 ] 0 ⊥ 0 h ∞ p , 0 − z z Z [ + ˜ k i = e 2 b − λ √ dz Z or its moments and I end this section with a brief explanation and the relevant quantity is )= B II + i ˜ k Φ T ( αβ B is defined by Φ B form factors. What we do not know how to compute at this stage a and 0. In evaluating matrix elements, Φ z , the convolution in eq.(7.4) is finite. In lattice calculati M + ˜ k → / 1 B denote light-cone coordinate and MeV or so until today we arrive at simulations with almost phy ∼ ± ) ) Although we do not know how to evaluate the matrix elements fo This discussion underlines the fact that we need new theoret At this conference we have seen many beautiful contribution At the previous lattice conference which Guido Martinelli h + ˜ k 500 ( ( B we can ask what can lattice simulationsof contribute the to light-cone the distribution fac functions for the light meso φ Lattice Phenomenology as are the know how to calculate the matrix elements of local operators (In higher orders of perturbation theory factors containin can indeed be written in terms of local operators. However th distribution amplitudes of where fields between hard-scattering amplitude reasons for this. still need to develop techniques to subtract these divergen start contributing to the evaluation of the non-perturbati hence to enable fundamental information toments be of obtained the from rates and CP-asymmetries. 8. Conclusions ogy, both in improved precision andtities. in We the readily extensions forget that of it comp symposium was were only a largely few in yearsto ago the quantify that quenched reliably. approximation, We with O then moved on to a brief period with d ment has to be continued vigorouslytalk, is if to precision play flavour phys a complementary role to large in 1989 in Capri, Ken Wilsonyears made to the have seemingly precision pessimistic Lattice p QCD.now We well only on have our 9 way. years left the next level of fundamental physics. As was noted in sec. 7 h which a large amount of experimentalto data begin is formulating available, the and calculations y to make them accessib PoS(Lattice 2010)018 et [hep-ph]. Christopher Sachrajda ith whom many of the , 64-80 (2009). B812 , 81-108 (1995). [hep-lat/9411010]. r his longstanding encouragement T/G000557/1 and from the EU con- btained and my colleagues from the (1992) 168-228. [hep-lat/9207009]. critical discussions placing lattice re- B445 (2008) 054513 [arXiv:0805.2999 [hep-lat]]. , H. Pedroso de Lima, C. T. Sachrajda, ons about the evaluation of long-distance (2008) 054510. [arXiv:0712.1061 [hep-lat]]. Collaboration [ETM Collaboration], Phys. yler, V. Lubicz, S. Necco, C. T. Sachrajda B384 (2009) 014501. [arXiv:0901.2599 [hep-ph]]. 78 escia, S. Simula, C. Tarantino, G. Villadoro, D78 D80 19 (2009) 055502 [arXiv:0812.1202 [nucl-ex]]. , Nucl. Phys. 79 (2010) 054017. [arXiv:1004.4613 [hep-ph]]. , Nucl. Phys. et al. , Phys. Rev. (2008) 141601 [arXiv:0710.5136 [hep-lat]]. (2010) 159-167. [arXiv:1004.0886 [hep-lat]]. (2010) 001. [arXiv:1011.3660 [hep-lat]]. , Phys. Rev. D82 et al. (2010) 011. [arXiv:1102.0410 [hep-lat]]. (2010) 010. [arXiv:1103.1523 [hep-lat]]. 100 et al. C69 (2003) 247-278. [hep-ph/0304113]. et al. B662 LATTICE2010 [FlaviaNet Working Group on Kaon Decays], arXiv:0801.1817 (2005) 339-362. [hep-ph/0403217]. LATTICE2010 , Eur. Phys. J. LATTICE2010 [ RBC and UKQCD Collaborations ], Nucl. Phys. , Phys. Rev. Lett. [HPQCD Collaboration], Phys. Rev. D et al. et al. B705 et al. et al. (2009) 111502 [arXiv:0906.4728 [hep-lat]]. et al. 80 , [arXiv:1011.4408 [hep-lat]]. al. Nucl. Phys. [arXiv:0809.1229 [hep-ph]]. Rev. D J. M. Zanotti I thank my colleagues from the RBC and UKQCD collaborations w [9] M. Luscher, R. Narayanan, P. Weisz [7] Y. Aoki, P. A. Boyle, N. H. Christ [4] C. Hoelbling, PoS [8] C. Sturm, Y. Aoki, N. H. Christ [1] C. Alexandrou, PoS [5] J.Laiho, these proceedings. [6] G. Martinelli, C. 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