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Computing the long-distance contribution to the 2 1 e kaon mixing parameter 0 K 2 n a J 0 1 ] NormanChrist∗† t a DepartmentofPhysics l ColumbiaUniversity,USA - p E-mail: [email protected] e h RBC andUKQCDcollaborations [ 1 v The largest contribution to the CP violating K K mixing parameter e comes from second L S K 5 − order weak interactions at short distances and can be accurately determined by a combination 6 0 of electroweak perturbation theory and the calculation of the parameter B from lattice QCD. K 2 However,thereisanadditionallongdistancecontributiontoe whichisestimatedtobeoforder . K 1 5%. Here recentlyintroducedlattice techniquesfor computingthe long-distancecomponentof 0 2 theK K massdifferencearegeneralizedtothislong-distancecontributiontoe . L S K 1 − : v i X r a XXIXInternationalSymposiumonLatticeFieldTheory July10U˝ 162011 SquawValley,LakeTahoe,California Speaker. ∗ †ThisworkwassupportedinpartbyU.S.DOEgrantDE-FG02-92ER40699 (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ Long-distancecontributiontoe NormanChrist K 1. Introduction Some of the most stringent tests of the standard model and most promising windows into possible new physics come from process which occur at second order in the Fermi constant G . F These include the standard model prediction for indirect CP violation in the K0 K0 system and − the bound on interactions which change strangeness by two units coming from the small K K L S − mass difference. Such O(G2) amplitudes involve the exchange of twointermediate bosons which F requires that eachpairofvertices joined bysuch aW exchange areseparated byashort distance ± ontheorderof1/m . However,thespatialseparationbetweenthesetwopairsofverticesneednot W besmall,and“longdistance” separations oftheorderof1/L arepossible. QCD ThisiswellillustratedbythefamiliarWigner-Weisskopfformuladescribingthetimeevolution oftheK0 K0 system: − d K0 M M i G G K0 i = 00 00 00 00 (1.1) dt K0 M M −2 G G K0 ! ( 00 00! 00 00!) ! wherethe2 2matricesM andG aregivenby: × ¥ G = 2p (cid:229) dE iH a (E) a (E)H j d (E m ) (1.2) ij W W K a Z2mp h| | ih | | i − M = (cid:229) P ¥ dEhi|HW|a (E)iha (E)|HW|ji. (1.3) ij m E a Z2mp K− We are using the subscripts 0 and 0 to represent the K0 and K0 states. The generalized sum over a and integral over the energy represents the sum over a complete set of energy eigenstates nor- malized as a ′(E′)a (E) = d (E′ E)d a a . These two matrices determine the two important h | i − ′ quantities, m m ande : KS− KL K m m = 2Re M (1.4) KS− KL { 00} i ImM iImG ImA e = 00−2 00 +i 0 (1.5) K 2(ReM00−2iReG 00) ReA0 The absorptive part, given by the energy conserving matrix G , is formed from products of ij first-order weak amplitudes which are now the targets of increasingly precise lattice calculations, see for example Refs. [1]and [2]. Thedispersive part, M is intrinsically second order in G and ij F containsbothashort-distance partwerebothexchangedW’sareseparatedbydistancescalesmuch smallerthan1/L andthelong-distance partdiscussed above. QCD The computation of the short distance contributions to e or m m is now a highly suc- K KS− KL cessfulapplicationoflatticeQCD.AtthescaleofQCD,phenomenatakingplaceatadistancescale of1/m canbeaccuratelyrepresentedbyaneffectivefour-Fermioperatorwhosecoefficientcanbe W computed in electro-weak perturbation theory and whose matrix element between kaon states can becomputedusinglatticeQCD.Fore theseshort-distancesdominateandthelong-distancepartis K estimated tobeafewpercentcorrection [3]. However,currentresults forthisshortcontribution to e arenowaccurateonthe5%level,givingtheunknownlong-distancepartincreasingimportance. K 2 Long-distancecontributiontoe NormanChrist K FortheK K massdifference thesizeofthelong-distance contribution islesscertain andcould L S − provide asmuchas50%ofthefullresult. Itisthelatticecalculationoftheselongdistancepartswhichisthesubjectofthistalk. Wewill briefly review theapproach presented inRef. [4]forthe calculation of thelong-distance contribution to the K K mass difference and then describe its generalization to the more complex case L S − ofe . K 2. Strategy forlatticecalculationK K massdifference. L S − Aspresented inRef.[4]aEuclidean-space, finitevolumecalculation oftheK K massdif- L S − ferenceD m containsthreeimportantingredients. Thefirstisarelationbetweenthesecond-order, K infinite-volume, principalpartintegralinEq.1.3givingD m =2Re(M )andafinite-volume,dis- K 00 crete perturbation theory sum. This relation is obtained through a generalization of the first-order formula of Lellouch and Luscher for the K pp decay width [5]. The second ingredient is a → computational strategy for evaluating that perturbation theory sum using lattice QCD.While with sufficient statistics the resulting lattice calculation will correctly determine the long-distance part of D m , the short distance contribution to this result will be incorrect, reflecting the details of the K latticeregulator. Thus,thethirdingredientisamethodtoreplacethiserroneousshort-distancecon- tribution with the correct short distance part of the continuum theory. We will now briefly review eachoftheseinturn. The relation between D m and a finite volume perturbation theory sum follows from the K Luscher finite-volume quantization condition [6] relating the allowed finite volume energies, E , n foratwo-pionsystem andthep p scattingphaseshift: − f (k L/2p )+d (E )+d (E )=np (2.1) n 0 n W n wherekn= En2/4 mp2 andtheknownfunctionf (y)isdefinedinRef.[6]. Herewehavedivided − thep p scatting phaseshiftintothestrongs-wavephaseshiftd andthatcausedbytheresonant − p 0 scattering through the weakly coupled K state. Since the location of the kaon pole in d (E ) is S W n shifted by D m , Eq. 2.1 can be used to relate D m to the finite-volume, second-order perturba- K K tion theory sum which determines E . It is this argument which we will generalize to determine n Im(M ). 00 Since the resulting finite volume sum that must be evaluated to determine D m is closely K related to perturbation theory, the entire sum can be obtained from an integrated Green’s function constructed from four operators: two interpolating operators which create an initial K0 state at a time t and destroy a final K0 state at t and two D S = 1 weak operators H (t ) and H (t ) i f W 1 W 2 evaluated atintermediate timesandintegrated overaranget t t t t fork=1,2. The f b k a i ≫ ≥ ≥ ≫ term proportional to length ofthe integration interval, t t gives the desired perturbation theory b a − sumwhichdeterminesD m . Whiletheevaluationofthistime-integrated four-pointfunctionposes K substantial computational challenges, ourfirstexploratory studygivesencouraging results[7]. Since D m receives both short- and long-distance contributions, the integrated amplitude de- K scribed in the previous paragraph will receive important contributions from the region in which the two weak vertices coincide. If a charm quark is included to exploit GIM cancelation a substantial, incorrect short-distance contribution will remain, of size ln(m a). Fortunately, the same W 3 Long-distancecontributiontoe NormanChrist K techniques whichallowthiscontinuum short-distance parttobeprecisely definedandevaluatedin a lattice calculation can be employed here to isolate and replace this lattice artifact by the correct continuum piece. If the four quarks making up the two kaon interpolating operators in the above four-point function are instead treated separately and the resulting Green’s function evaluated in Landau gauge and given large, off-shell momentum the short distance contribution can be evaluated numerically in a fashion that is free of infrared singularity allowing a simple substitution by the correct continuum short distance part. Again, first numerical experiments [7] suggest that this canbedonewithoutdifficulty. 3. Extension to e K We now turn to the central topic of this talk, the extension of the method reviewed above to the case of Im(M ) and e . Since there are no apparent added difficulties associated with the 00 K numericalevaluationoftheresultingfinite-volumeexpressionorthecorrectionofitsshort-distance part these topics will not be discussed further. However, on first-sight using a Lellouch-Luscher style argument toobtain afinitevolume formulaforIm(M )mayseemunlikely tosucceed. First 00 e describes the mixing of the K and K states and the K state with its predominate decay into K S L L three pions would appear inaccessible to Luscher’s formalism which applies only to two-particle scattering. Further the energies that arethe basis oftheLellouch-Luscher method areintrinsically CPconserving suggesting nosensitivity totheCPviolating phaseofM . 00 Fortunatelybothoftheseproblemscanbeavoidedbyaddingafictitious,D S=2“superweak” term to the Hamiltonian whose effects can be combined with (but will not alter at leading order) thoseofthestandard weakinteractions: HSW=(w +iw ) s(1+g 5)g m d s(1+g 5)g m d +hermitianconjugate. (3.1) r i Thequantities w andw arethe(cid:8)nchosensothat (cid:9) r i w K0 O K0 +ReM = 0 (3.2) rh | LL| i 00 w K0 O K0 G , (3.3) i LL ij h | | i≫| | foralliand j. HereO istheoperator withinthecurlybrackets inEq.3.1. Withtheoff-diagonal, LL real part ofM canceled andthe off-diagonal term proportional tow dominating theoff-diagonal ij i parts of G , the eigenstates of the time development operator in Eq. 1.1 become the unfamiliar ij combinations K =(K0 iK0 )/√2— both states which decay predominantly into twopions, consistent withL±usch|er’sifi±ni|teviolumeformalism. Inaddition, thelargecontribution ofw insures i that the two p p resonances associated with the states K are non-overlapping and will each − ± contribute independent resonant behavior to that scattering. The eigenvalues of the 2 2 time × development matrixontheright-hand sideofEq.1.1forthestates K canbewritten: | ±i 1 l = (M +M ) w K0 O K0 +ImM ± 2 00 00 ∓ ih | LL| i 00 (cid:16) 1 (cid:17) +i (G +G iG iG ) (3.4) 4 00 00∓ 00± 00 1 = M w K0 O K0 +ImM +i (G ImG ) (3.5) 00∓ ih | LL| i 00 2 00∓ 00 (cid:16) (cid:17) 4 Long-distancecontributiontoe NormanChrist K We first examine the finite volume problem. Here the box size L is adjusted so that we have threenearlydegeneratestates: thep p state n withenergyE aswellasthetwosingle-particle states, K0 and K0 . Using second−order, d|eg0einerate perturbant0ion theory, we can determine the | i | i finitevolumeenergies ofinterest astheeigenvalues ofthe3 3matrix: × mK+n=(cid:229) n0|hnm|HKW|KEn0i|2 n=(cid:229) n0hK0|HWm|nKihnE|nHW|K0i+d m hK0|HW|n0i  n=(cid:229) n0hK0|HWm6|nKihnE|nHW|K−0i+d m∗ 6 mK+n=(cid:229) n−0|hnm|HKW|KEn0i|2 hK0|HW|n0i , (3.6)  6 hn0|H−W|K0i hn06|HW|K0−i En0+n6=K(cid:229) 0,K0|hnE|Hn0W−|nE0ni|2    with the rows and columns corresponding to K0 , K0 and n in increasing order. Here the 0 | i | i | i complexquantity d misgivenby: d m=(w +iw ) K0 O K0 . (3.7) r i LL h | | i UsingtheCPandtime-reversal operations, the3 3matrixinEq.3.6canbewrittenas: × Ed1 M e−id0A  M∗ Ed1 e−id0A∗. (3.8) eid0A∗ eid0A Ed2     The eigenvalues of this matrix can be determined by diagonalizing the large part of the matrix proportional to the first-order amplitude A and then applying perturbation theory to diagonalize theremainder toorderH2: W Re M(A )2 ∗ E = E (3.9) 0 d1− A 2 (cid:8) | | (cid:9) 1 Re M(A )2 E = (E +E ) √2 A + ∗ . (3.10) ± 2 d1 d2 ± | | 2A 2 (cid:8) | | (cid:9) We now require that the Luscher finite volume condition, Eq. 2.1, be obeyed when the total p p phase shift, written as a combination of the usual strong s-wave phase shift, the resonant − scattering through the K states and astandard second-order-weak Bornterm is evaluated atthe | ±i finite-volume energy eigenvalues E giveninEq.3.10: ± k L G (E )/2 G (E )/2 np = f (cid:18) 2±p (cid:19)+d 0(E±)+arctan(cid:18) M++−±E± (cid:19)+arctan(cid:18) M−−−±E± (cid:19) (3.11) p (cid:229) |hb |HW|pp (E)i|2. − E Eb b =K0,K0 − 6 HereM andG canbedeterminedfromtherealandimaginarypartsoftheinfinite-volumeeigen- ± ± valuesgiveninEq.3.5: M = m +M w K0 O K0 +ImM (3.12) ± K 00∓ ih | LL| i 00 G = G ImG (cid:16). (cid:17) (3.13) ± 00∓ 00 If the finite volume is adjusted so that the zero-order p p energy E =m , then to zeroth − n0 K order in H , Eq. 3.11 reduces to the original Luscher quantization condition relating the strong W 5 Long-distancecontributiontoe NormanChrist K s-wave phase shift d (E ) and the finite-volume p p energy E . If, for the same value of E , 0 ± − n0 n0 Eq. 3.11 is expanded to first order in H then the Lellouch-Luscher relation between the finite W volumeK pp matrixelementA andtheK pp widthG isobtained. 00 − → ConsiderablymorealgebraisneededtoobtainarelationbetweenthefinitevolumesuminM and the infinite volume amplitude M . As in the earlier study of D m , we first evaluate Eq. 3.11 00 K for a volume which makes the p p energy E close to m on the scale of QCDbut sufficiently − n0 K different that a perturbative expansion in H /(E m ) is possible. Equation. 3.11 can then be W n0− K evaluated in a power series in H . The resulting formula contains a pole at E =m . Requiring W n0 K thatthispoletermvanishgivesthestandardLellouch-Luscherrelation. Requiringthatthenon-pole piece vanishes at E =m then relates the infinite-volume, second-order Born term in Eq. 3.11 n0 K andthefinite-volume, second-order suminthelowerleftcornerofthematrixinEq.3.6. Inthenextstep,Eq.3.11isevaluatedforavolumemakingthetwo-pionenergyE =m and n0 K the result expanded to second order in H , including the resonant denominators. The expression W thatresultscontainsdenominators oforderH2 andnumerators oforderH4 andcanbewritten: W W ¶ (f +d 0) (cid:229) nHW K0 2 0 = |h | | i| M ¶ E m E − 00 (n=n0 K− n 6 + A1 2Re (A∗)2 (cid:229) hK0|HWm|nihnE|HW|K0i−M00 | | nn6=n0 K− n o!) ¶ 2(f +d ) ¶ ¶ (f +d ) + 0 A 2 2 0 n H K0 2 , (3.14) ¶ E2 | | − ¶ E ¶ E |h 0| W| i| n0 (cid:26) (cid:27) givingafinitevolumeexpressionforacombinationoftheinfinitevolumeamplitudesM andM . 00 00 TodetermineM alone,asecondequationisneeded. AsinthesimplercaseexaminedRef.[4], 00 a second relation follows is we require that the expectation value of the matrix M for the state ij K =(A K0 A K0 )/(2A 2) agree with the corresponding finite volume expression up to ∗ | i | i− | i | | termsexponentiallysmallinthesizeofthatvolume. Herethestate K ischosensothatitdoesnot | i ceouple tothetwo-pion, on-shell state, justifying theneglect offinitevolumeeffects. Theresulting equation canbecombinedwithEq.3.14toeliminateM givingtheerelation: 00 0 = Re (A∗)2 (cid:229) hK0|HWm|nihnE|HW|K0i−M00 (3.15) ( n=n0 K− n 6 1 1¶ 2(f +d ) ¶ ¶ (f +d ) + 0 A2 0 K0 H n n H K0 . ¶ (f¶+Ed0)"2 ¶ E2 −¶ En0 (cid:26) ¶ E h | W| 0ih 0| W| i(cid:27)#!) Since the K pp amplitude A is non-zero with a phase that is independent of the phase of the → amplitude in curved brackets, (...), we can conclude that (...)=0, our desired finite-volume ex- pressionforM . TherealpartofthisequationgivestheearlierresultforRe M . Theimaginary 00 { 00} partgivesthenewresultforIm M : { 00} Im M = Im (cid:229) Im hK0|HW|nihn|HW|K0i + 1 1¶ 2(f +d 0) K0 H n n H K0 { 00} (n=n0 n mK−En o ¶ (f¶+Ed0)"2 ¶ E2 h | W| 0ih 0| W i 6 ¶ ¶ (f +d ) 0 K0 H n n H K0 . (3.16) −¶ E ¶ E h | W| 0ih 0| W i n0 (cid:26) (cid:27)#) 6 Long-distancecontributiontoe NormanChrist K Thus, by exploiting the interference between the complex, CP-violating amplitude A and the second-order, Wigner-Weisskopf mass matrix, we have been able to determine the CP-violating partofM fromafinite-volume energy whichisintrinsically CP-even. 00 Heretheleft hand side ofEq.3.16isthe infinitevolume quantity which determines thelong- and short-distance dispersive parts of e . The right-hand side contains only finite volume matrix K elements which can be evaluated in a lattice QCD calculation. The similarity of the results for Re M ,Im M andM (notdisplayed)suggeststhatamoredirectderivationmaybepossible. { 00} { 00} 00 4. Conclusion Giventhecontrol overthesingularity associated withtheprincipal partandrelated finitevol- ume effects implied by these results and those in Ref. [4], both the K K mass difference D m L S K − and the complete, dispersive part of e can in principle be computed using lattice methods. In K fact, the first exploratory results of Jianglei Yu, presented in Ref. [7], suggest that such a lattice calculation maybepossible withthenextgeneration ofhighperformance computers. However, many obstacles must be overcome: (a) This initial work suggests that the short- distance partsofD m ande canbeproperly treatedbyasinglesubtraction, definedusingRome- K K Southampton techniques. Such a single subtraction is adequate only if GIM cancelation has been realizedinthelatticecalculation, requiringtheinclusionofthecharmquarkwhichinturnrequires asmall lattice spacing. (b)Manymoreoperators and contractions mustbe included inacomplete calculation than have been attempted in Ref. [7]. (c) These initial calculations use a relatively massive, 410 MeVpion. Thesuccessful subtraction of anexponentially growing single-pion contribution demonstrated in Ref. [7] will become more difficult as the pion mass is reduced. Even greater difficulty will result from the vacuum subtraction needed when disconnected graphs are included. We believe that these difficulties can be overcome with improved numerical methods and thesustained, 100Tflopscapability ofthenextgeneration machines nowbeing installed. The authoracknowledges theimportantcontributions ofhisRBC/UKQCDcollaborators tothiswork. References [1] T.Blum,P.Boyle,N.Christ,N.Garron,E.Goodeetal.,K topp DecayamplitudesfromLatticeQCD, arXiv:1106.2714 [hep-lat]. [2] T.Blum,P.Boyle,N.Christ,N.Garron,E.Goodeetal.,TheK (pp ) DecayAmplitudefrom I=2 → LatticeQCD,arXiv:arXiv:1111.1699 [hep-lat]. [3] A.J.Buras,D.GuadagnoliandG.Isidori,One beyondlowestorderintheOperatorProduct K Expansion,Phys.Lett.B688(2010)309–313[arXiv:1002.3612 [hep-ph]]. [4] RBCandUKQCDCollaboration,N.H.Christ,Computingthelong-distancecontributiontosecond orderweakamplitudes,PoSLATTICE2010(2010)300[arXiv:1012.6034 [hep-lat]]. [5] L.LellouchandM.Luscher,Weaktransitionmatrixelementsfromfinite-volumecorrelationfunctions, Commun.Math.Phys.219(2001)31–44[hep-lat/0003023]. [6] M.Luscher,Twoparticlestatesonatorusandtheirrelationtothescatteringmatrix,Nucl.Phys.B354 (1991)531–578. [7] J.Yu,LongdistancecontributiontoK -K massdifference,arXiv:1111.6953 [hep-lat]. L S 7