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Continuum Results for Light Hadronic Quantities using Domain Wall Fermions with the Iwasaki and DSDR Gauge Actions PDF

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Continuum Results for Light Hadronic Quantities using Domain Wall Fermions with the Iwasaki and 2 1 0 DSDR Gauge Actions 2 n a J 3 ] t ChristopherKelly∗ a l ColumbiaUniversity, - p 905PupinHall, e 116thSt&Broadway, h NewYork,NY10027,USA. [ E-mail: [email protected] 1 v Forthe RBCandUKQCDcollaborations 6 0 7 We present preliminary continuum results for light hadronic quantities obtained by the 0 RBC/UKQCD collaboration using domain wall fermions with both the Iwasaki and the novel . 1 DislocationSuppressingDeterminantRatio(DSDR)gaugeactions. TheDSDRactionallowsus 0 2 tosimulateatnearphysicalquarkmassesonalarger,coarserlattice(a−1=1.4GeV,L=4.6fm) 1 whileretaininggoodchiralsymmetryproperties. We discussourongoingcombinedanalysisof : v thethreeensemblesetsandgiveearlyresultsforthepionandkaondecayconstants,quarkmasses Xi andBK. r a TheXXIXInternationalSymposiumonLatticeFieldTheory-Lattice2011 July10-16,2011 SquawValley,LakeTahoe,California ∗Speaker. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ContinuumResultsforLightHadronicQuantitiesusingDomainWallFermionswiththeIwasakiandDSDRGaugeActions ChristopherKelly Label Size SG b Nens mupni.(MeV) #configs. mPp Q≥(MeV) 32ID 323×64×32 Iwasaki+DSDR 1.75 2 170,250 181,148 140 24I 243×64×16 Iwasaki 2.13 2 330,420 202,178 240 32I 323×64×16 Iwasaki 2.25 3 290,350,400 300,312,252 220 Table1:Asummaryofthepropertiesofthethreeensemblesetsusedinthisanalysis.Here‘S ‘denotesthe G gaugeaction,‘Nens’thenumberofensembles,‘mupni.’theunitarypionmassoneachofthoseensembles,‘# configs’thenumberofgaugeconfigurationsusedinthisanalysis,andmPp Q≥thelightestavailablepartially- quenchedpionmassontheensembleset. 1. Introduction In these proceedings we present continuum limit predictions for the neutral kaon mixing pa- rameter BK, the strange and average up/down quark masses and the pion decay constant fp , ob- tainedviasimultaneouschiral/continuum fitstothethreedomainwallfermionensemblesetsgiven in table 1. Ourmost recently generated ensemble set, referred to bythe label ‘32ID’, uses amod- ified Iwasaki gauge action that includes the Dislocation Supressing Determinant Ratio (DSDR) term, which allows us to simulate with near-physical pion masses on a coarser b =1.75 lattice, whileretaining goodchiralsymmetryandtopology tunneling. Combinedanalysesofthe24Iand32Ilattices,whichwerefertocollectivelyasthe‘2010anal- ysis’,haverecentlybeenpublished[5,6]. Inthesepaperswedevelopedastrategyforthecombined analysis ofmultiple ensemble sets that maximises the use ofthe available data inconstraining the fits. Inthese proceedings wedevelop thisstrategy further toinclude 32ID ensemble set. Notethat thenumberofconfigurations available onthelightest32IDensemblesethasalmostdoubledsince theconference, andtheheavierensemblehasalsoincreased insizeby35%,hencetheconclusions presented heredifferslightly fromthosegivenattheconference. The layout of these proceedings is as follows: We provide a short discussion on the DSDR term,followedbyasummaryofourcombinedfittingstrategy. Wethenpresentresultsforthepion decay constant before brieflytouching onthe non-perturbative renormalisation techniques used in the calculations of B and the physical quark masses. Results for these quantities follow. We K conclude withabriefsummaryandoutlook forthisanalysis. 2. The DSDR Term The2010analysiswasperformedtodataovertherangemp =290−420MeV.Heretheextrap- olations downtothephysical pion massof∼135 MeVprovided thedominant contribution tothe systematic errors on the continuum predictions. For example, the chiral extrapolation systematic on fp was∼4%. Thisprovided astrongmotivationforreaching downtolighterquarkmasses. In order to avoid finite-volume effects, simulations with lighter quark masses require lattices with a larger physical volume. However, at our typical couplings the required increase in the number of lattice sites is beyond the reach of our current computing resources; we are therefore forcedtosimulatewithcoarserlattices. Thishastheunfortunateside-effectofincreasingthesizeof thechiralsymmetrybreaking effectsinthedomainwallfermionformulation, duetothefollowing mechanism. The size of the chiral symmetry breaking is parameterised by an additive mass-shift known as the ‘residual mass’ m . This quantity is governed by the eigenvalue density r of the four- res dimensional Hamiltonian H =2tanh−1 HW describing quark propagation through the fifth T 2+DW dimension. Here D is the Wilson Dirac(cid:16)operato(cid:17)r and H =g 5D isthe hermitian Wilson Dirac W W W operator. The equation above implies a relationship between the eigenmodes of H and H . The T W modesofthelatteraredividedintotworegionsbyamobilityedgel : thoseabovel areextended c c 2 ContinuumResultsforLightHadronicQuantitiesusingDomainWallFermionswiththeIwasakiandDSDRGaugeActions ChristopherKelly Figure1: AnillustrationofthemoleculardynamicsforceimpartedbytheDSDRtermatvariousvaluesof e ande ,reproducedfromref.[3] f b over the lattice whereas those below are localized - in fact this property is essential in ensuring that the four-dimensional effective theory of the fields on the boundary is local. The structure of eigenmodes ofH impliesthatm hasthefollowingdependence onL [7]: W res s e−lcLs 1 m ∼R4r (l ) +R4r (0) , (2.1) res e c L l L s s wherethefirsttermcontains contributions fromtheextended modesandthesecondtermfromthe localized near-zero modes. In modern simulations, L is typically large enough that m is dominated by the near-zero s res mode contribution. These modes are associated with small tears or ‘dislocations’ in the gauge field, which occur more often as we approach the disordered strong-coupling region. In order to retain goodchiralsymmetryatstronger coupling wemusttherefore seektosuppress thesemodes. However we must retain enough of the very-near-zero modes that allow topological tunneling to occur. This can be achieved by introducing a weighting term, the DSDR term, into the gauge action, givenby[1,2,3] W(M;e ;e )= det DW(−M+iebg 5)†DW(−M+iebg 5) =(cid:213) l i2+e 2f , (2.2) f b det[D (−M+ie g 5)†D (−M+ie g 5)] l 2+e 2 (cid:2) W f W f (cid:3) i i b where l are eigenvalues of H and e and e are tunable parameters. This introduces a force in i W f b themolecular dynamicsevolution oftheform d l 2+e 2 F(e ,e )= −log i f , (2.3) i f b dl l 2+e 2 i i b! which can betuned topeak in thenear-zero region without further suppressing thevery-near-zero modes. In figure 1 wereproduce aplot from ref. [3] which shows the force as a function of l for severalcombinations ofe ande onatestsimulation, demonstrating thissuppression. f b 3. Simultaneous FittingProcedure Weobtainourfitformsviaadualexpansiontonext-to-leading order(NLO)ina2andthelight andheavyquarkmassesm andm ,adoptingapowercountingthatdiscardstermslikeO(a2m)and l h O(m2)andhigheratthisorder. Expandingaroundanon-zeromasspoint(m ,m )andabsorbing l0 h0 constant termscontaining m intotheleadingcoefficient, weobtaintheanalyticfitfunction l0 (m˜ +m˜ ) f =Cfp 1+C a2 +Cfp x y +Cfp m˜ +Cfp (m˜ −m ) (3.1) ll 0 f 1 2 2 l 3 h h0 (cid:0) (cid:1) 3 ContinuumResultsforLightHadronicQuantitiesusingDomainWallFermionswiththeIwasakiandDSDRGaugeActions ChristopherKelly 2010analysis Thisanalysis(alldata) Thisanalysis(mp ≤350MeV) fp (MeV) 124(2)(5)(2) 125(2)(2)(2) 127(3)(0.5)(3) mMS(2GeV)(MeV) 3.59(13)(12)(6)(8) 3.48(6)(7)(3)(8) 3.39(9)(4)(2)(7) u/d mMS(2GeV)(MeV) 96.2(1.5)(0.2)(0.1)(2.1) 94.9(1.2)(1.4)(0.2)(2.1) 94.2(1.9)(1.0)(0.4)(2.1) s Bˆ 0.749(7)(21)(3)(15) 0.748(6)(15)(4)(15) 0.751(11)(8)(4)(14) K Table2: Resultsfor fp ,theaverageup/downquarkmass,thestrangequarkmassandBK. Thefirstcolumn containsthe 2010analysisresult[5, 6], thesecondtheresultobtainedin thisanalysisbyfitting tothe full rangeofavailabledata,andthethirdbyfittingonlytodatawithmp ≤350MeV.Theerrorsarestatistical, chiral,finite-volumeandNPRerrors(whereappropriate)respectively. for the pion decay constant and similar forms for other quantities. Here m˜ =m+m . We also res consider expanding about the SU(2) chiral limit (with and without finite-volume corrections), in which case we obtain the usual NLO ChPT fit forms with an extra a2 and (m −m ) term. We h h0 refertotheseastheChPTandChPTFVfitforms. Usingthesethreefitansätze, wesimultaneously fit the 32I, 24I and 32ID ensemble sets, with the coefficients shared between all three data sets. Forensemble sets i other than the primary set, chosen as the 32I set, wemustinclude appropriate factors of the ratio of lattice spacings R =a32I/ai and quark masses R =m˜32I/m˜i in the fit a l/h l/h l/h forms such that the coefficients C can be assumed equal. In this analysis we adopt the so-called ‘generic scaling’ approach[5]inwhichtheseratiosaredeterminedasfreeparametersinthefit. Before taking the continuum limit, wemust fixthe primary lattice spacing a32I and thequark masses m32I andm32I in32I normalisation. Thisisachieved byvarying these parameters until the u/d s continuum values of the pion, kaon and Omega baryon masses agree with their physical values. These quantities define the ‘scaling trajectory’ along which the continuum limit is defined. Note that this makes the a2 coefficients of these quantities zero by definition. As the above conditions areapplied inthecontinuum limit,theprocedure isnecessarily iterative. The inclusion of the Iwasaki+DSDR data complicates the situation slightly over the 2010 analysis, in that the coefficients of the a2 terms must now be allowed to differ between the two differentgaugeactions. Althoughthisintroduces oneextradegreeoffreedomperquantityintothe fits,anyalgorithmicinstabilitythatthismaycauseisoffsetbytheincreasednumberofdatapoints. Recall that eqn. 3.1 contains a term in the heavy sea-quark mass m . Wevary this parameter h using reweighting [4,5], whereby the weight of agiven gauge configuration isre-evaluated inthe path integral atseveral strange quark masses shifted by up to20% from the simulated value. This allows us to explore the strange sea quark dependence with minimal cost and to ultimately quote predictions atthephysical strangequarkmassatthepenalty ofanincreaseinstatistical error. 4. Preliminary Results For fp fp is obtained from the h0|A0|p i matrix elements in the usual way, the only difference being that for domain wall fermions one must correctly renormalise the 4d axial current to match the continuum current. Following ref. [5], the renormalisation factor is obtained from the ratio of the 5dDWFconservedvector currenttothe4dvectorcurrent: thiswasshowntobemoreprecisethan the ratio of axial currents due to the unknown renormalisation coefficient between the 5d DWF PCACcurrentandthecontinuum current. Figure2showsthechiralextrapolation of fp downtothephysical up/downquarkmassusing theanalytic andChPTFVansätze. Followingthe2010analysis weestimate theerroronthechiral extrapolationthroughthedifferenceoftheChPTFVandanalyticpredictions,andthefinite-volume error from the difference of the ChPTFV and ChPT predictions. Weobtain the value given in the second column of table 2. Thisresult issome 4% (1.5s ) below the physical value of 130.7 MeV. Asimilardiscrepancy wasnotedinthe2010analysis(firstcolumnofthetable),andwasattributed 4 ContinuumResultsforLightHadronicQuantitiesusingDomainWallFermionswiththeIwasakiandDSDRGaugeActions ChristopherKelly Figure2:Comparisonsoftheunitary fp datacorrectedtothecontinuumlimitandfitusingtheChPTFVand analyticansätze,wherethefitsareperformedtothefulldataset(left)andonlytothedatawithmp ≤350 MeV(right). Thedataincludedinthefitaremarkedwithcircles,andthoseexcludedwithdiamonds. The continuumlimitsaremarkedwithsquares.Pointsinpastelcoloursarecorrectedtothecontinuumusingthe ChPTFVansatz,andinboldcoloursbytheanalyticansatz. to the systematic error on the chiral extrapolation. There, the strategy of estimating the chiral errorfromthedifferenceoftheChPTFVandanalyticresultsproducedasystematicerrorsufficient to explain the discrepancy, but for this analysis it appears to be an underestimate. However, the introduction of the light DSDR data allows us to perform stable fits even after removing some of the heavier data from the 32I and 24I ensemble sets. We can therefore restrict our fits to a region of lighter mass in which we would expect the fit ansätze to perform better. Cutting the heaviest twoensembles (cf. table 1)suchthattheheaviest pionhasamassof350MeV,weobtain the result given in the third column of table 2. Here, as a result of the cut, the central value has increased suchthattheresultisnowconsistent withthephysicalvalue,evenwithoutincorporating thechiralsystematic. Noticealsothatouradhocchiralsystematichasdecreasedsubstantially, and the c 2/d.o.f. alsodecreased, whichsuggests thatthe fitsdoindeed perform better whenrestricted tothislightermassrange. Although thisanalysisisstillpreliminary, theseresultsarepromising. 5. RenormalisationforB andthe quark masses K Before giving results for the quark masses and B , we present a short summary of the non- K perturbative renormalisation techniques thatareusedtoconvert thesequantities intothecanonical MSschemewithunprecendented precision. Direct conversion into the MS scheme on the lattice is not possible as this scheme is regu- larised in non-integer dimensions. Instead we first convert to a convenient intermediate scheme, which is then run to high energy and matched to MS using perturbation theory. We use variants of the Rome-Southampton Regularisation-Invariant Momentum (RI-MOM) scheme, in which the renormalisation coefficientsaredefinedfromtheratiooftheamputatedGreen’sfunction ofanop- erator toitstree-level value intheLandau gaugeataparticular momentum scale. IntheRI-MOM scheme, the Green’s functions are formed using massless propagators of equal momentum p, and thescaleisdefinedasm 2= p2. Howeverthisisaso-called‘exceptional’ momentumconfiguration in whichthe hardexternal momenta canbe routed insuch awaythat parts ofthediagram contain only soft momenta -this enhances the effect of the spontaneous chiral symmetry breaking at high momenta. In order to avoid this problem we follow the 2010 analysis in adopting ‘symmetric’ momentum conditions (SMOM), for which the external momenta are not equal but rather obey m 2 = p2 = p2 =(p −p )2. Note that this applies both for the mass renormalisation, which we 1 1 2 obtain from the scalar bilinear vertex, and the four-pointVV +AAweak operator contained inthe definition of B ; for the latter the Wick contractions comprise traces of two bilinear vertices -the K momenta p and p areassigned tothetwopropagators formingeachofthosebilinears. 1 2 5 ContinuumResultsforLightHadronicQuantitiesusingDomainWallFermionswiththeIwasakiandDSDRGaugeActions ChristopherKelly Asinthe 2010 analysis, weuse volume source propagators inthese calculations, which offer a significant improvement over the traditional point sources, giving statistical errors of the order 0.1%evenwithonlyO(20)gaugeconfigurations. One of the dominant systematic errors on the NPR in our earlier analyses arose through the use ofmomenta whose unit vectors were not equivalent under hypercubic rotations, and hence do not have the same discretisation errors; this induces a scatter in the RI-SMOM renormalisation coefficients as a function of momentum. In ref. [6] we described how this can be corrected using twistedboundary conditions inthevalence sector, whichallowustosmoothly varythemagnitude ofthemomentumwhilekeepingthedirectionfixed. Wealsodiscussedhowanotherofthedominant systematic errors, that associated with the truncation of the perturbative series, can be reduced by afactoroftwobyperformingthematchingtotheMSschemeat3GeVratherthanthecanonical 2 GeV. For this analysis these improvements have been applied in the case of B but have yet to be K applied tothequarkmassrenormalisation. 6. Results fortheQuark Masses The continuum physical quark masses are obtained in the normalisation of the 32I ensemble set, and hence must be renormalised into the MS scheme to remove any cutoff dependence. The renormalisation coefficientsaredeterminedbyperformingthecontinuumextrapolation ofZ /R m l/h overthetwoIwasakilattices,whereR arethequarkmassratiosdefinedinsection3andR32I≡1 l/h l/h by definition. Notice that the coefficient on the 32ID lattice is not needed for this procedure. As mentioned above, the MS(3 GeV) lattice coefficients with twisted boundary conditions have yet to be determined, hence for this analysis we reuse those given in ref. [5], although the continuum extrapolation is performed anew with the lattice spacings and R obtained here. Following the l/h 2010analysis procedure, wechooseourbestNPRschemeforthecentralvaluetakethetruncation errorontherenormalisation fromthesizeofthetwo-loopcontribution totheMSmatching. From a fit to the full data set, we obtain the values given in the second column of table 2. Forcomparison, wegivethe 2010 analysis result inthe firstcolumn. Wefindresults that are very consistent, and observe a factor of two reduction in the statistical and chiral systematic errors on the up/down quark mass over the 2010 analysis as a result of including the lighter data. We also see a reduction in the statistical error on the strange quark mass, but also a large increase in the chiral error. This is likely a result of allowing the mass ratios Z and Z to differ between the fit l h ansätze, wherebefore theywerefixedtovalues obtained bymatching thelattices atanunphysical massscale(thefixedtrajectory method). Weintendtoinvestigate thisfurther. Wealso investigate the effect of cutting out the heaviest twoensembles on the quark masses. We obtain the result given in the third column of the table. Here as with fp , we see significant improvements intheestimatedchiralerror,attheexpense ofanincreaseinstatistical error. 7. Results forB K We obtain results for B through an independent simultaneous fit over our three ensemble K sets, using the lattice spacings, quark masses, etc., obtained from the fits above. In particular, we constrain the ChPT fits by including the lowest-order ChPT parameters f and B from the main analysis. The fits are performed to MS-renormalised data, where we use the coefficients obtained bymatchingtoourlatticeschemeat3GeVasdiscussedabove,afterwhichweconverttotheRGI- scheme for the convenience of the reader. The NPR error is obtained by taking the difference of ourtwobestschemes,asdiscussedinref.[6]. Fromthefitstothefulldataset,weobtaintheresult giveninthesecondcolumnoftable2. Thisresultisveryconsistentwiththe2010result,andshows a30%improvementinthechiralerror. Fittingonlytothedatawithmp ≤350MeV,weobtainthe result given in the third column of the table. As before wesee a substantial decrease in the chiral 6 ContinuumResultsforLightHadronicQuantitiesusingDomainWallFermionswiththeIwasakiandDSDRGaugeActions ChristopherKelly Figure3: ComparisonsoftheunitaryBK dataintheSMOM(/q,/q)schemecorrectedtothecontinuumlimit and fit using the ChPTFV and analytic ansätze, where the fits are performedto the full data set (left) and onlytothedatawithmp ≤350MeV(right). Thesymbolsandtheircoloursaredescribedinthecaptionof figure2. error,butheretheincreaseinthestatisticalerrorislargerthanbefore. Thisislikelyduetothelack ofstatistical resolution onthelightest32IDdatapoints. 8. Conclusions andOutlook Using the Iwasaki+DSDR gauge action, we have been able to simulate with near physical pions while retaining good topological tunneling properties and small finite-volume corrections. Including these data in simultaneous fits with our Iwasaki lattices, we have been able to substan- tially improve our continuum predictions over the 2010 analysis, especially after cutting out the heaviest twoensembles suchthattheheaviest pionnowhasamassofonly350MeV-performing this cut we observed factor of two reductions in our estimated chiral systematic. In addition, our preliminary prediction for fp ,127(4) MeV,isnowconsistent withthephysical value. Weobserve that NLOchiral perturbation theory extrapolations overthe mass range of140−350 MeV should be expected to fail at the 5% level, hence weexpect this value to rise further towards the physical valueaswefurtherrestrictthefitrangeinfutureanalyses. Itisourintentiontopublishananalysisofthesedatashortly,afterwhich,alongsidecontinuing togenerate moredataonthe32ID ensemble set, weintend tocommencethegeneration offurther domainwallfermionensembleswithnearphysicalpions,takingadvantageofthelargeincreasein computing power provided by our upcoming IBM Blue Gene/Q resources. This will allow us to further refineourcontinuum predictions. References [1] P.M.Vranas,[hep-lat/0001006]. [2] P.M.Vranas,Phys.Rev.D74 (2006)034512.[hep-lat/0606014]. [3] D.Renfrew,T.Blum,N.Christ,R.Mawhinney,P.Vranas,PoSLATTICE2008 (2008)048. [arXiv:0902.2587[hep-lat]]. [4] C.Jung,PoSLAT2009 (2009)002.[arXiv:1001.0941[hep-lat]]. [5] Y.Aokietal.[RBCandUKQCDCollaborations],Phys.Rev.D83 (2011)074508. [arXiv:1011.0892[hep-lat]]. [6] Y.Aoki,R.Arthur,T.Blum,P.A.Boyle,D.Brommel,N.H.Christ,C.Dawson,T.Izubuchietal., Phys.Rev.D84 (2011)014503.[arXiv:1012.4178[hep-lat]]. [7] D.J.Antonioetal.[RBCandUKQCDCollaborations],Phys.Rev.D77 (2008)014509. [arXiv:0705.2340[hep-lat]]. 7

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