dominated腾讯qq好友恢复

Exploringpossiblyexistingqqbbtetraquarkstateswithqq=ud,ss,ccAntjePetersa,PedroBicudob,KrzysztofCichya,c,BjrnWagenbacha,MarcWagneraaGoethe-UniversittFrankfurtamMain,InstitutfürTheoretischePhysik,Max-von-Laue-Strae1,D-60438FrankfurtamMain,GermanybCFTP,DepartamentodeFísica,InstitutoSuperiorTécnico,UniversidadedeLisboa,AvenidaRoviscoPais,1049-001Lisboa,PortugalcAdamMickiewiczUniversity,FacultyofPhysics,Umultowska85,61-614Poznan,PolandE-mail:peters@th.
physik.
uni-frankfurt.
de,bicudo@tecnico.
ulisboa.
pt,kcichy@th.
physik.
uni-frankfurt.
de,wagenbach@th.
physik.
uni-frankfurt.
de,mwagner@th.
physik.
uni-frankfurt.
deWecomputepotentialsoftwostaticantiquarksinthepresenceoftwoquarksqqofnitemassusinglatticeQCD.
InasecondstepwesolvetheSchrdingerequation,todetermine,whethertheresultingpotentialsaresufcientlyattractivetohostaboundstate,whichwouldindicatetheexistenceofastableqqbbtetraquark.
Wendaboundstateforqq=(uddu)/√2withcorrespondingquantumnumbersI(JP)=0(1+)andevidenceagainsttheexistenceofboundstateswithisospinI=1orqq∈{cc,ss}.
The33rdInternationalSymposiumonLatticeFieldTheory14-18July2015KobeInternationalConferenceCenter,Kobe,JapanSpeaker.
cCopyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence.
http://pos.
sissa.
it/arXiv:1508.
00343v1[hep-lat]3Aug2015Exploringpossiblyexistingqqbbtetraquarkstateswithqq=ud,ss,ccAntjePeters1.
MotivationAnumberofmesonsobservedinexperimentslikeLHCborBellearenotwellunderstood.
Thosemesonshavemassesandquantumnumbers,whicharenottypicalforstandardquark-antiquarkstates,butindicateanexoticfour-quarkstructure.
Prominentexamplesarethechargedcharmonium-likeandbottomonium-likestatesZc±andZb±(cf.
e.
g.
[1]).
Theirmassesanddecayproductssuggestthepresenceofaccorbbpair,respectively.
Ontheotherhandtheirelectricchargeindicatesadditionallyalightquark-antiquarkpairudordu.
Thosefour-quarksystems,inthefol-lowingalsoreferredtoastetraquarks,areexpectedtobestudiedinmoredetailinthenearfuturebyexperimentalcollaborations.
Therefore,asoundtheoreticalunderstandingofthosesystemsiscrucialandofgreatinterest.
Herewesummarizethemainresultsofourrecentlypublishedwork[2],wherewehavestudiedfour-quarksystemswithtwoheavyantiquarksbbandtwolighterquarksqqusinglatticeQCDandtheBorn-Oppenheimerapproximation.
Firstbbpotentialsinthepresenceoflighterquarksqqarecomputed.
ThentheSchrdingerequationissolvedusingthesepotentials,wherepossiblyexistingboundstatesindicatestabletetraquarks.
Otherpapersstudyingthesamesystemswithsimilarmethodsinclude[3,4,5,6,7,8,9,10,11,12,13,14].
2.
QualitativediscussionofqqbbsystemsAtsmallbbseparationsthebbinteractionisdominatedby1-gluonexchange.
Foraboundstatethebbpairmust,therefore,beinanattractivecolortriplet.
DuetothePauliprincipleandbecauseweassumeaspatiallysymmetrics-wave,bbhastoformanantisymmetriccolor-spin-avorcombinationand,hence,asymmetricspincombination,i.
e.
bbspinjb=1.
Sincethecompletefour-quarksystemiscolorneutral,thelightquarksqqmustbeinanantisymmetriccolorantitriplet.
AgainduetothePauliprincipleqqhastoformanantisymmetriccolor-spin-avorcombinationand,hence,asymmetricspin-avorcombination.
Candidatesfortetraquarksare,therefore,the(spin)scalarisosinglet(i.
e.
aqqspinsingletj=0withantisymmetricavor,e.
g.
qq∈{(uddu)/√2,(s(1)s(2)s(2)s(1))/√2,(c(1)c(2)c(2)c(1))/√2}1)andthe(spin)vectorisotriplet(i.
e.
aqqspintripletj=1withsymmetricavor,e.
g.
qq∈{uu,(ud+du)/√2,dd,ss,cc}).
TheoverallquantumnumbersofaboundqqbbsystemareI(JP)=0(1+)forthescalarisosingletchannelandI(JP)∈{1(0+),1(1+),1(2+)}forthevectorisotripletchannel.
Atlargebbseparationsthebbinteractionisscreenedbythelightquarksqq,i.
e.
thefourquarksformasystemoftwoheavy-lightmesons.
Oneexpectsstrongerscreeningforincreasingquarkmassmq,becausethewavefunctionsofthecorrespondingmesonsqbarethenmorecompact.
3.
LatticeQCDcomputationofstaticantiquark-antiquarkpotentialsWeextractpotentialsoftwostaticantiquarksQQ(approximatingthetwobquarksoftheqqbbsystem)inthepresenceoftwolightquarksqqfromcorrelationfunctionsC(t,r)=|O(t)O(0)|∝t→∞exp(V(r)t).
(3.
1)1Tobeabletostudyavorantisymmetricqqcombinationswithq=s,weconsidertwohypotheticaldegenerateavorswiththemassofthesquark,s(1)ands(2),andsimilarlyforq=c,c(1)andc(2).
2Exploringpossiblyexistingqqbbtetraquarkstateswithqq=ud,ss,ccAntjePetersOdenotesafour-quarkcreationoperator,O=(CΓ)AB(CΓ)CDQC(r1)q(1)A(r1)QD(r2)q(2)B(r2),r=|r1r2|,(3.
2)whereΓisanappropriatecombinationofγmatricesaccountingfordenedquantumnumberslightquarkspin|jz|,parityPandPx(cf.
[8]fordetails).
Γ∈{(1γ0)γ5,(1γ0)γj}doesnotaffecttheresultingpotentialV(r),sincethestaticquarkspinisirrelevant.
C=γ0γ2denotesthechargeconjugationmatrix.
Notethatoperatorslike(3.
2)generateoverlapnotonlytomesonicmoleculestructures,butalsotodiquark-antidiquarkstructures[15,16].
Theasymptoticvalueofapotentialandwhetheritisattractiveorrepulsivedependsonthequantumnumbers(|jz|,P,Px)and,hence,onΓ.
Inthefollowingweareexclusivelyinterestedinat-tractivepotentialsbetweentwogroundstatestatic-lightmesons:thescalarisosingletcorrespondingtoΓ=(1+γ0)γ5andthevectorisotripletcorrespondingtoΓ=(1+γ0)γj.
Computationshavebeenperformedusingtwoensemblesofgaugelinkcongurationsgener-atedbytheEuropeanTwistedMassCollaboration(ETMC)withdynamicalu/dquarks.
Informa-tionontheseensemblescanbefoundinTable1and[17,18].
βlatticesizelainfmmπinMeV#congurations3.
90243*480.
004000.
0793404804.
35323*640.
001750.
042352100Table1:Ensemblesofgaugelinkcongurations(β:inversegaugecoupling;l:bareu/dquarkmassinlatticeunits;a:latticespacing;mπ:pionmass).
4.
qqbbtetraquarksintheBorn-OppenheimerapproximationTodetermineananalyticalexpressionfortheQQpotentialorequivalentlybbpotential,wettheansatzV(r)=αrexprd2+V0(4.
1)withrespecttoα,dandV0tothelatticeQCDresultsobtainedintheprevioussection.
TheconstantV0accountsfortwicethemassofthegroundstatestatic-lightmeson.
Weinsertthetheanalyticalexpression(4.
1)intheSchrdingerequationfortheradialcoordi-nateofthetwobquarks(whichweassumetobeinans-wave),12d2dr2+U(r)R(r)=EBR(r)(4.
2)withU(r)=V(r)|V0=0and=mb/2anddeterminethelowesteigenvalueEB.
IfEB0,thereisnobinding,i.
e.
thefour-quarksystemwillalwaysbeasystemoftwounboundBmesons.
Noticethatthisso-calledBorn-Oppenheimerapproximationisvalidformqmb,whichiscertainlythecaseforq∈{u,d,s}andatleastcrudelyfullledforq=c.
Toquantifythesystematicerrorsofdifferentchannels(scalarisosingletandvectorisotriplet,differentlightavorsq∈{u,d,s,c}),weperformalargenumberoftsvaryingtherangeof3Exploringpossiblyexistingqqbbtetraquarkstateswithqq=ud,ss,ccAntjePeterstemporalseparationstmin≤t≤tmaxofthecorrelationfunctionC(t,r)(cf.
eq.
(3.
1)),atwhichthelatticepotentialisreadoff,aswellastherangeofspatialbbseparationsrmin≤r≤rmaxconsideredintheχ2minimizingtofeq.
(4.
1)tothelatticepotential.
Detailsonthisparametervariationcanbefoundin[2].
Foreachsetofinputparameters(tmin,tmax,rmin,rmax)wedetermineα,dandEB.
Thenwegeneratehistogramsforα,dandEBweightedaccordingtothecorrespondingχ2/dof.
Thewidthsofthesehistogramsaretakenassystematicerrorsofα,dandEB[19],whilethestatisticalerrorsareobtainedviaajackknifeanalysis.
InFigure1examplehistogramsforthescalarisosingletforqq=udareshown.
Figure1:Histogramsforthescalarisosingletforqq=ud.
Thered/green/bluebarsindicatethestatistical/systematic/combinederrors.
Theresultingpotentialstsfordifferentchannels,i.
e.
eq.
(4.
1)withcorrespondingvaluesforαandd,arecollectedinFigure2.
Theerrorbandsrepresentthecombinedsystematicandstatisticalerrors.
Onecanobservethatthepotentialsarewideranddeeperforlighterqqquarkmasses.
Moreover,thescalarchannelsaremoreattractivethantherespectivevectorchannels.
Correspondingly,itturnsoutthatthereisaboundstateonlyforthescalarisosingletwithqq=udwithbindingenergyEB=93+4743MeV,i.
e.
aboundstatewitharound2σcondencelevel.
InFigure3wepresentourresultsinanalternativegraphicalway.
Thethreeplotscorrespondtou/d,sandclightquarksqq,respectively.
Eachtofeq.
(4.
1)tolatticepotentialresultsisrepresentedbyadot(red:scalarchannels;green:vectorchannels;crosses:rmin=2a;boxes:rmin=3a).
Theextensionsofthepointcloudsrepresentthesystematicuncertaintieswithrespecttoαandd.
IfapointcloudislocalizedaboveorleftoftheisolinewithEB=0.
1MeV(essentiallythebindingthreshold),thecorrespondingfourquarksqqbbcannotformaboundstate.
Alocalizationbeloworrightofthatisolineisastrongindicationfortheexistenceofatetraquark.
Againthereisclearevidenceforatetraquarkstateinthescalaru/dchannel.
Thescalarschannelisslightlyabove,butratherclosetothebindingthreshold.
Thescalarcandallvectorchannelsclearlydonothostboundfour-quarkstates.
5.
SummaryandoutlookWehavefoundaudbbtetraquarkwithquantumnumbersI(JP)=0(1+)(i.
e.
inthescalarisosingletchannelwithqq=ud)withacondencelevelofaround2σ.
Thereseemtoexistno4Exploringpossiblyexistingqqbbtetraquarkstateswithqq=ud,ss,ccAntjePetersFigure2:Potentialstsfordifferentchannels(upperline:scalarisosinglet;lowerline:vectorisotriplet).
Thecurveswithoutanerrorbandarecopiedfromtherespectiveotherplotsinthesamelineforeasycomparison.
Verticallinesindicatetheavailablelatticebbseparations.
tetraquarksfortheotherchannels.
InthisworklatticeQCDcomputationshavebeenperformedforlightu/dquarkscorrespond-ingtomπ≈340MeV.
Weplantorepeattheanalysisforatleastanotherpionmassandthenextrapolatetothephysicalpoint.
Itwillthenbemostinterestingtocheck,whetheraboundstatewillalsoappearinthevectorisotripletchannelwithqq=ud.
AnotheraspectistoinvestigatethestructureofthefoundI(JP)=0(1+)tetraquark,i.
e.
toexplore,whetheritisratheramesonicmoleculeoradiquark-antidiquarkpair.
Wealsoplantoincludecorrectionsduetotheheavyquarkspins(forrstpreliminaryresultscf.
[14]).
Finally,oneshouldstudytheexperimentallymoreaccessible,buttheoreticallymorechallengingcaseofqqbbsystems.
AcknowledgmentsP.
B.
thanksIFTforhospitalityandCFTP,grantFCTUID/FIS/00777/2013,forsupport.
M.
W.
andA.
P.
acknowledgesupportbytheEmmyNoetherProgrammeoftheDFG(GermanResearchFoundation),grantWA3000/1-1.
ThisworkwassupportedinpartbytheHelmholtzInternationalCenterforFAIRwithintheframeworkoftheLOEWEprogramlaunchedbytheStateofHesse.
5Exploringpossiblyexistingqqbbtetraquarkstateswithqq=ud,ss,ccAntjePetersFigure3:BindingenergyisolinesEB=constantintheα-d-planeforu/d,sandclightquarksqqtogetherwiththetresultsofeq.
(4.
1)tolatticepotentials.
6Exploringpossiblyexistingqqbbtetraquarkstateswithqq=ud,ss,ccAntjePetersCalculationsontheLOEWE-CSChigh-performancecomputerofJohannWolfgangGoethe-UniversityFrankfurtamMainwereconductedforthisresearch.
WewouldliketothankHPC-Hessen,fundedbytheStateMinistryofHigherEducation,ResearchandtheArts,forprogrammingadvice.
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