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Nucleon Charges from 2+1+1-flavor HISQ and 2+1-flavor clover lattices 7 1 Rajan Gupta∗† 0 2 LosAlamosNationalLaboratory,LosAlamos,NM,87545,U.S.A. E-mail: [email protected] n a J PNDME and NME Collaborations‡ 0 2 Precise estimates of the nucleon charges g , g and g are needed in many phenomenological A S T ] analyses of SM and BSM physics. In this talk, we present results from two sets of calculations t a usingcloverfermionson9ensemblesof2+1+1-flavorHISQand4ensemblesof2+1-flavorclover l - lattices. Weshowthathighstatisticscanbeobtainedcost-effectivelyusingthetruncatedsolver p e method with bias correction and the coherent source sequential propagator technique. By per- h [ formingsimulationsat4–5valuesofthesource-sinkseparationtsep,wedemonstratecontrolover excited-statecontaminationusing2-and3-statefits. Usingthehigh-precision2+1+1-flavordata, 1 v weperformasimultaneousfitina,M andM Ltoobtainourfinalresultsforthecharges. π π 1 5 6 5 0 . 1 0 7 1 : v i X 34thannualInternationalSymposiumonLatticeFieldTheory r a 24-30July2016 UniversityofSouthampton,UK ∗Speaker. †LA-UR-16-29008 ‡Calculationsonthe2+1+1-flavorHISQlatticesarebeingdoneincollaborationwithT.Bhattacharya,V.Cirigliano, Y.C.Jang,H-W.LinandB.Yoon. Calculationsonthe2+1-cloverensemblesarebeingdoneincollaborationwithT. Bhattacharya,V.Cirigliano,J.Green,BálintJoó,Y.C.Jang,H-W.Lin,K.Orginos,D.Richards,S.Syritsen,F.Winter andB.Yoon. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ NucleonIsovectorCharges RajanGupta 1. Introduction In this talk, we highlight three areas in which significant progress has been made to extract matrix elements of quark bilinear operators within nucleon states. (i) cost-effective increase of statisticsusingthetruncatedsolvermethodwithbiascorrectionandthecoherentsourcesequential propagator technique; (ii) inclusion of up to 4-states (3-states) in the analysis of 2-point (3-point) correlationfunctions;and(iii)asimultaneousfitina,M andM Ltodataatdifferenta,M andL π π π togetthephysicalvalue. TheresultspresentedherearebasedonRefs.[10,3,11]. Asummaryof thelatticeensemblesusedandmeasurementsmadeintheclover-on-HISQstudyisgiveninTable1 and in the clover-on-clover study in Table 2. Associated results for the isovector form factors: G (q2),G (q2),G (q2),andGs(Q2)werepresentedbyYong-ChullJangatthisconference[8]. E M A S 2. IncreasingStatisiticsCost-effectively Thevarioussystematicuncertaintiesinthecalculationofmatrixelements(ME)oflocalquark bilinearoperatorswithinnucleonstatesareatthe5%level[3]. Inordertoisolate,understandand address these systematics, one needs data with statistical errors that are significantly smaller. To reduce statistical errors, one needs to make measurements on significant numbers of decorrelated latticesthatadequatelyimportancesamplethephasespaceofthepathintegral. Wehavefoundthe followingthreetechniquestobecost-effectivewaysofreducingthestatisticalerrors. LatticeswithM L≥4canberegardedasconsistingofalargenumberofuncorrelatedregions, π i.e.,measurementsofnucleoncorrelationfunctionsindifferentsub-regionsarestatisticallyuncor- related. Since the generation of lattices with dynamical fermions is expensive, one should assess the dimensions of these sub-regions. For ME within nucleon states we find that O(100) measure- mentsperlatticeofsizeM L=4arecost-effective[10]. Furthermore,choosingthesourcepoints π randomlywithinthesesub-regionsofalatticeandbetweenlatticesreducescorrelations. Computertimecanbereducedsignificantlybyusingthecoherentsourcesequentialpropagator method[10]. Forexample,ifN measurementsonagivenlatticearedoneinasinglecomputer meas job,thenthenumberofquarkpropagatorsthatneedtobecalculatedusingcoherentsourcesreduces toN +2fromN +2×N inthestandardapproach. meas meas meas Lastly, in a given measurement, one has the freedom to choose the precision with which quark propagators are calculated. The truncated solver method [1] with bias correction [5] significantly reduces the computational time. In Ref. [3], we show that a stopping residue r ≡ LP |residue| /|source|=10−3 reducescostbyafactorofabout17comparedtor =10−10. Possi- LP HP blebiasiscorrectedforusing Cimp= 1 N∑LPC (xLP)+ 1 N∑HP(cid:2)C (xHP)−C (xHP)(cid:3), N LP i N HP i LP i LP i=1 HP i=1 whereC andC are the 2- and 3-point correlation functions calculated in low- (LP) and high- LP HP precision (HP), respectively, and xLP and xHP are the two kinds of source positions. Bias, given i i by the second term, if present, was much smaller than the statistical errors. Speedup is achieved becauseweneed1HPandLPmeasurementforbiascorrectionforevery32LPusedforstatisitcs. 2 NucleonIsovectorCharges RajanGupta EnsembleID a(fm) Msea(MeV) Mval(MeV) L3×T MvalL N NHP NAMA π π π conf meas meas a12m310 0.1207(11) 305.3(4) 310(3) 243×64 4.55 1013 8104 64832 a12m220S 0.1202(12) 218.1(4) 225(2) 243×64 3.29 1000 24000 a12m220 0.1184(10) 216.9(2) 228(2) 323×64 4.38 958 7664 a12m220L 0.1189(9) 217.0(2) 228(2) 403×64 5.49 1010 8080 68680 a09m310 0.0888(8) 312.7(6) 313(3) 323×96 4.51 881 7048 a09m220 0.0872(7) 220.3(2) 226(2) 483×96 4.79 890 7120 a09m130 0.0871(6) 128.2(1) 138(1) 643×96 3.90 883 7064 84768 a06m310 0.0582(4) 319.3(5) 320(2) 483×144 4.52 1000 8000 64000 a06m220 0.0578(4) 229.2(4) 235(2) 643×144 4.41 650 2600 41600 a06m135 0.0568(1) 135.5(2) 136(2) 963×192 3.74 229 1145 36640 Table1: Summaryoftheensemblesusedintheclover-on-HISQstudy. Forthea06m130ensemble, preliminaryresultsonformfactorsonlywerepresentedbyY-PJangatthisconference. EnsembleID a(fm) Msea (MeV) L3×T MvalL N NHP NAMA π π conf meas meas a127m285 0.127(2) 285(3) 323×96 5.85 1000 4020 128480 a094m280 0.094(1) 278(3) 323×64 4.11 1005 3015 96480 a091m170 0.091(1) 166(2) 483×96 3.7 629 2516 80512 a091m170L 0.091(1) 172(6) 643×128 5.08 467 2335 74720 Table2: Summaryoftheensemblesusedintheclover-on-HISQstudy. HereMsea=Mval π π 3. ReducingSystematicUncertainties Theabovetechniquesallowedustomake,oneachensemble,O(105)measurementsonO(1000) lattices. Inthesemeasurementsof2-and3-pointcorrelationfunctions,twosystematicsneedtobe addressedinordertoextractthecharges,formfactors,andotherME:removingexcited-statecon- tamination (ESC) and precise determination of the renormalization factor connecting the lattice operatortosomephenomenologicalschemesuchastheMSschemeatµ =2GeV. ESC arises because typical interpolating operators used to create and annihilate meson and baryon states on the lattice couple to the ground state, all its excitations and multiparticle states with the same quantum numbers. Thus, contributions of all higher states have to be removed to obtain the desired ME within the ground state. The behavior of the 2- and 3-point correlation functions, given by the spectral decomposition, has contributions from a tower of intermediate statesintermsofunknownamplitudesA,energiesM andME(cid:104)f|O |i(cid:105)thatareextractedfromfits. i i Γ Formally,thegroundstateME(cid:104)0|O |0(cid:105)canbeobtainedbycalculatingthe3-pointfunctionswith Γ very large source-sink separationt . For baryons, the signal in 2- and 3-point function degrades sep exponentially,andforvaluesoft accessiblewithO(105)measurements,theESCisfoundtobe sep significant. Thus,methodstomiminizeESCincalculationswith1< t <1.5fmareneeded. ∼ sep∼ We demonstrate control over ESC by (i) constructing the interpolating operators using tuned 3 2 1.5 1 NucleonIsovectorCharges RajanGupta 0.5 0 →∞ t t =10 t =12 t =14 t =16 t =18 sep sep sep sep sep sep 1.40 1.40 1.35 1.35 d 1.30 d 1.30 u-A u-A g g 1.25 1.25 1.20 1.20 χ2/d.o.f = 0.71 χ2/d.o.f. = 0.84 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 τ - t /2 τ - t /2 sep sep Figure1: Unrenormalizedgu−d onthea081m3152+1-flavorcloverensemblewithσ =5smearing. A Theleft(right)panelshowsthedata,2-state(3-state)fit,andt →∞estimate(greyband). sep smearedsourcesinthegenerationofthequarkpropagators;(ii)performingthecalculationatmulti- plevaluesoft ;(iii)insertingtheoperatoratallintermediatetimeslicesτ betweenthesourceand sep sink; (iv) analyzing the 2- and 3-point correlators by including increasing number of intermediate states. Forexample,the2-statetruncationofthezero-momentumcorrelationfunctionsis C2pt(t ,t)=|A |2e−M0(tf−ti)+|A |2e−M1(tf−ti), (3.1) f i 0 1 C3pt(t ,τ,t)=|A |2(cid:104)0|O |0(cid:105)e−M0(tf−ti)+|A |2(cid:104)1|O |1(cid:105)e−M1(tf−ti)+ Γ f i 0 Γ 1 Γ A A∗(cid:104)0|O |1(cid:105)e−M0(τ−ti)e−M1(tf−τ)+A∗A (cid:104)1|O |0(cid:105)e−M1(τ−ti)e−M0(tf−τ), (3.2) 0 1 Γ 0 1 Γ where τ is the operator insertion time and t −t =t in the 3-point function calculation. The f i sep states |0(cid:105) and |1(cid:105) represent the ground and “first excited” nucleon states, respectively. In 2-state analysis, the four parameters, M , M , A and A are estimated first from fits to the 2-point data 0 1 0 1 and then used as input in fits to 3-point functions to obtain the three ME (cid:104)0|O |0(cid:105), (cid:104)0|O |1(cid:105) and Γ Γ (cid:104)1|O |1(cid:105). Theestimateofthechargeg =(cid:104)0|O |0(cid:105)improveswithnumberoft ,theprecisionof Γ Γ Γ sep thedata,andthenumberofstatesincludedinthefits. WefindthatwithO(105)measurements,fits with 4 states (3 states) to the 2-point (3 point) functions with full covariance matrix can be made. Stableandconsistentestimatesofthechargesinthet →∞limitareobtainedusingdatawith4–5 sep valuesoft intherange1–1.5fm. Acomparisonofthe2-and3-statefits,andtheconsistencyof sep thet →∞valueobtainedfortheisovectoraxialchargegu−d isillustratedinFig.1. sep A Results for the various ME are then renormalized by multiplicative factors Z calculated us- Γ ing the RI-sMOM scheme as discussed in Ref. [3]. Errors in the ME and Z are combined in Γ quadratures. Thisgivesusasetofrenomalizedlatticeestimatesasfunctionsofa,M andM L. π π 4. Simultaneousfitina,M andM L π π Withtherenormalizedestimates,calculatedasfunctionsofa,M andM L,inhand,resultsin π π the limit a→0, M =135 MeV and M L→∞, are obtained using a simultaneous fit in the three π π variables. Withthecurrent9clover-on-HISQdatapoints,fitsaresensitivetoonlythelowestorder 4 NucleonIsovectorCharges RajanGupta �� - - - �� (��) �π�(��) �π� �� - - - �� (��) �π�(��) �π� �� - - - �� (��) �π�(��) �π� Figure 2: The 9-point fit using Eqs. (4.1) and (4.2) to the data for the renormalized isovector charges,gu−d,gu−d andgu−d,intheMSschemeat2GeV. Theresultofthesimultaneousextrapo- A S T lationtothephysicalpointdefinedbya→0,M →Mphys =135MeVandL→∞aremarkedby π π0 aredstar. Theerrorbandsineachpanelshowthesimultaneousfitasafunctionofagivenvariable holdingtheothertwoattheirphysicalvalue. Thedataareshownprojectedontoeachofthethree planes. Theoverlayinthemiddlefigureswiththedashedlinewithinthegreyband,isthefittothe dataversusM2 neglectingdependenceontheothertwovariables. π correctionterms [3]: g (a,M ,L)=c +c a+c M2+c M2e−MπL, (4.1) A,T π 1 2 3 π 4 π g (a,M ,L)=c +c a+c(cid:48)M +c(cid:48)M e−MπL. (4.2) S π 1 2 3 π 4 π Adding next order terms such as chiral logs did not improve the fits (based on the Akaike Infor- mationCriteria)andtheircoefficientswerepoorlydetermined. Variationinestimatesonincluding chirallogswere, nevertheless, usedtoobtainfirstestimatesofthepossiblesystematicuncertainty duetousingthelowestorderfitansatz. OurfinalfitsusingEqs.(4.1)and(4.2)areshowninFig.2. TheClover-on-cloveresimateson4ensemblesareconsistentwiththosefromclover-on-HISQ atsimilarvaluesofthelatticeparameters. Toperformanalogousfitstoobtainresultsata→0and M =135MeV,clover-on-clovercalculationsarebeingextendedtoadditionalvaluesofaandM . π π 5. Results: NucleonChargestoquarkEDM (I) Our results for the isovector nucleon charges, using the simultaneous fit ansatz defined in 5 NucleonIsovectorCharges RajanGupta Eqs.(4.1)and(4.2)tothe9clover-on-HISQdatapoints,areshowninFig.2andgive[3] gu−d =1.195(33)(20); gu−d =0.97(12)(6); gu−d =0.987(51)(20). (5.1) A S A (II)Usingtheconservedvectorcurrentrelation∂ (dγ u)=(m −m )du,latticeestimatesofm − µ µ d u d m =2.67(35)givenbyFLAG[9],andourresultforgu−d/gu−d weobtain u S V (M −M )QCD=gu−d(m −m )/gu−d =2.59(49)MeV. (5.2) N P S d u V (III)Constraintsonnovelscalarandtensorcouplings,ε andε ,attheTeVscaleusinglow-energy S T experimentsandourgu−d andgu−d arederivedandcomparedwiththosefromtheLHCinFig.3. S T (IV) The leading opertors in a low-energy effective theory that contribute to the neutron electric dipole moment (nEDM) are the Θ-term, the quark EDM operator and the quark chromo EDM operators. The ME of the quark EDM operator, same as the flavor diagonal tensor charges gu,d,s, T aredeterminedtobe[2] gu =0.792(42); gd =−0.194(14); gs =0.007(8). (5.3) T T T In these estimates, the disconnected contributions to gu and gd have been neglected as they were T T O(1%)(smallerthanthequotederrors)andpoorlydetermined. Usingtheseresultsandtheexper- imentalboundontheneutronEDM,weperformedafirstanalysisofconstraintsonpossiblequark EDMcouplingsgeneratedattheTeVscaleandimplicationsforasplitSUSYmodelinRef.[2,4]. 6. ConclusionsandOutlook Our goal is to calculate the charges and the form factors with O(1%) uncertainty on each ensembleandobtainresultsinthea→0,M =135MeVlimitwithatotalerrorof2%. Thiswill π requiresimulationswithO(106)measurementsat4–5valuesofthelatticespacingandonmultiple values of the light quark masses close to the physical pion mass. To achieve this goal over the next 5–10 years will require further improvements in algorithms for generating lattices, physics analysis,andthecalculationofrenormalizationfactors. Worktowardsthesethreegoalsisongoing. Acknowledgments We thank the MILC Collaboration for providing the 2+1+1-flavor HISQ ensembles and the JLab/W&M collaboration for the 2+1 clover lattices. Simulations were carried out on computer facilities of (i) Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, whichissupportedbytheOfficeofScienceoftheU.S.DepartmentofEnergyunderContractNo. DE-AC05-00OR22725;(ii)theUSQCDCollaboration,whicharefundedbytheOfficeofScience oftheU.S.DepartmentofEnergy;(iii)theNationalEnergyResearchScientificComputingCenter, aDOEOfficeofScienceUserFacilitysupportedbytheOfficeofScienceoftheU.S.Department of Energy under Contract No. DE-AC02-05CH11231; and (iv) Institutional Computing at Los Alamos National Laboratory; and (v) the Extreme Science and Engineering Discovery Environ- ment (XSEDE), which is supported by the NSF Grant No. ACI-1053575. The calculations used the Chroma software suite [6]. Work supported by the U.S. Department of Energy, NSF and the LANLLDRDprogram. 6 NucleonIsovectorCharges RajanGupta CURRENT CONSTRAINTS PROSPECTIVE CONSTRAINTS Low-energy: Low-energy: Low-energy: Low-energy: gS,T from quark model gS,T from lattice gS,T from quark model gS,T from lattice LHC: LHC: √s = 8 TeV √s = 14 TeV L = 20, fb-1 L = 10, 300 fb-1 Figure 3: Left panel: current 90% C.L. constraints on ε and ε from beta decays (π → eνγ S T √ and 0+ →0+) and the LHC (pp→eν+X) at s=8 TeV. Right panel: prospective 90% C.L. √ constraintsonε andε frombetadecaysandtheLHC(pp→eν+X)at s=14TeV.Thelow- S T energyconstraintscorrespondto10−3 measurementsofB,binneutrondecayandbin6Hedecay. Bothpanelspresentlow-energyconstraintsundertwodifferentscenariosforthescalarandtensor charges g : quark model [7] (large dashed contour) and lattice QCD results given in Eq. (5.1) S,T (smallsolidcontour). LHCconstraintrsareshownasdottedcontours. References [1] GunnarS.Bali,SaraCollins,andAndreasSchafer. Effectivenoisereductiontechniquesfor disconnectedloopsinLatticeQCD. Comput.Phys.Commun.,181:1570–1583,2010. [2] T.Bhattacharya,V.Cirigliano,S.Cohen,R.Gupta,A.Joseph,H-W.Lin,andB.Yoon. Iso-vectorand Iso-scalarTensorChargesoftheNucleonfromLatticeQCD. Phys.Rev.,D92(9):094511,2015. [3] T.Bhattacharya,V.Cirigliano,S.Cohen,R.Gupta,H-W.Lin,andB.Yoon. Axial,ScalarandTensor ChargesoftheNucleonfrom2+1+1-flavorLatticeQCD. Phys.Rev.,D94(5):054508,2016. [4] T.Bhattacharya,V.Cirigliano,R.Gupta,H-W.Lin,andB.Yoon. NeutronElectricDipoleMoment andTensorChargesfromLatticeQCD. Phys.Rev.Lett.,115(21):212002,2015. [5] ThomasBlum,TakuIzubuchi,andEigoShintani. Newclassofvariance-reductiontechniquesusing latticesymmetries. Phys.Rev.,D88(9):094503,2013. [6] RobertG.EdwardsandBalintJoo. TheChromasoftwaresystemforlatticeQCD. Nucl.Phys.Proc.Suppl.,140:832,2005. [7] P.Herczeg. Betadecaybeyondthestandardmodel. Prog.Part.Nucl.Phys.,46:413–457,2001. [8] Y.C.Jangetal. ibid,2016. [9] TheFlavorLatticeAveragingGroup(FLAG),2016. [10] B.Yoonetal. ControllingExcited-StateContaminationinNucleonMatrixElements. Phys.Rev., D93(11):114506,2016. [11] B.Yoonetal. Isovectorchargesofthenucleonfrom2+1-flavorQCDwithcloverfermions,2016. 7