Resonances with heavy quarks from lacesasa/talks/prelovsek_sfb16.pdf · Resonances with heavy quarks from lace (Bound states will be discussed by Daniel Mohler) Sasa Prelovsek University

Resonanceswithheavyquarksfromla6ce(BoundstateswillbediscussedbyDanielMohler)

SasaPrelovsekUniversityofLjubljana&JozefStefanInsGtute,Slovenia

SFBmeeGng

UniversityofRegensburg,6and7October2016

1S.Prelovsek,SFBmeeGng2016

incollaboraGonwith:C.B.Lang,D.Mohler,L.Leskovec,U.Skerbis

OutlineLa$ceQCDstudiesconsideringsca+eringoftwohadronsintherestframeq  charmoniumresonancesaboveopen-charmthreshold[Lang,Leskovec,Mohler,S.P.,1503.05363,JHEP2015]

q  la6cesearchforresonanceX(5568)inBsπ+scat.

q  aimedresonancesthatappearinscat.ofhadronswithspinconstrucGonofla6ceoperators[S.P.,Lang,Skerbis,1607.06738]vector-pseudoscalar(e.gJ/ψπforZcresonance)vector-nucleon(e.g.J/ψpforPcresonance)pseudoscalar-nucleon(e.g.πpforRoperresonance)la6cesimulaGonofπNintheRoperchannelwithJP=1/2+[Lang,Leskovec,Padmanath,S.P.,arxiv:1610.01422]

S.Prelovsek,SFBmeeGng2016 2

D0collaboraGon,1602.07588

Bsπ+invariantmass

[Lang,Mohler,S.P.,1607.03185,PRD2016]

H1(p)H2(-p)

AllphysicalstateswithgivenJPCappearasEninprinciple(example:charmoniumwith1++)

•  “single-meson”statesJ/ΨmJ/ψ=E1

•  “two-meson”states


J PC O = qΓq, (qΓ1q) !p1(qΓ2q) !p2

, [qΓ3q ][qΓ4q],...

ψ(3770), 1−− : cc , (cu)(uc) = DD , [cu][cu]

Cij (t) = 0Oi (t)Oj+(0) 0 =

n∑ Zi

n Z jn* e−En t, Zi

n = 0Oi n

Meson(like)systemwithgivenJPCiscreatedbyanumberofinterpolaGngfields

La6cegivesdiscreteenergiesofeigenstates:En

DD,...

forP=0(alerextrapolaGons)

Enrigorouslyrendertwo-hadronscameringmatrix(forexampleDDscameringmatrix)

4

xxx

xx E(L)

Rigoroustreatmentofhadronsnearorabovethresholdscameringoftwospin-lessmesonsinrestframe

δ(E)E(L) energiesfromla6cewithspaGalextentL

scameringphaseshilsatinfinitevolume

S.Prelovsek,SFBmeeGng2016

L

twomesonsinala6cebox

analyGcproposal:Luscher1991 DD̄ ! (3770) ! DD̄example

DD̄

Scameringoftwomesons

5

one-channel(elasGc)scameringwithtotalmomentumP=0:E=Ecm

En(L)δ(E)Luscher’seq.

ScameringmatrixforparGalwavel

Boundstate(B):

cot[δ(EB)]=i,EB<m1+m2

Resonance(R)(ofBreit-Wignertype):

M1(p)

M2(-p)

Re[E]

Im[E]

m1+m2threshold

B

R

LocaGonsofpolesofT(E)forres.andboundst.

exp

S(E) = e2i�(E), S(E) = 1 + 2iT (E), T (E) =

1

cot �(E)� i

T (E) =�E �

E2 �m2R + i E �

, �(E) = g2p2l+1

E2

Twotypesofplotswillbeshown:

•  E(L)energiesoflat.eigenstates

•  mphy=mR,mBextractedfromE(L)

Charmonium

inparGcular

charmoniumresonances



CharmoniawellbelowDDpreciselyknown

Perez-Rubio,Collins,Bali,1503.08440,PRD2015seealso:FNAL/MILC,1412.1057

TheomissionofcharmannihilaGonisthemainremaininguncertainty

m=E(P=0):ag0Lg∞mqgmq

phy

[PDG]

Excitedcharmoniawithinsingle-hadronapproach


justccinterpolatorsused,strongdecayofresonancesaboveDDnottakenintoaccount

Had.Spec.Coll,1610.01073

Resonanceψ(3770)andboundst.ψ(2S)fromDDscameringinp-wave

Lang,Leskovec,Mohler,S.P.,1503.05363,JHEP2015S.Prelovsek,SFBmeeGng2016 9

cquarku,dquarks

DD̄ ! (3770) ! DD̄First(exploratory)la6cesimulaGonofacharmoniumresonanceaboveopen-charmthresholdtakingintoaccountitsstrongdecay

WickcontracGons

evaluatedusing

-disGllaGon[Peardonet.al.,2009]

-stochasGcdisGllaGon

[Morningstaretal.,2011]

Eigen-energiesEninfinitevolume

Lang,Leskovec,Mohler,S.P.,1503.05363,JHEP2015S.Prelovsek,SFBmeeGng2016 10

En ! �(En)

mπ=266MeVL=2fm

PACS-CSmπ=156MeV

L=2.9fm


Lang,Leskovec,Mohler,S.P.,1503.05363,JHEP2015


En ! �(En) ! p3 cot �(p)/ps

ThisquanGtypresentedasitisapprox.linearnearsimpleBWresonance:•  BWfit(i):

•  fit(ii):capturesalsoboundst.:R:mR:zero,ΓR:slopenearzero

B:cotδ=i

DDthreshold

mπ=266MeVmπ=156MeV

p = i|p| ! p

3cot� = (i|p|)3i = |p|3 > 0

p3 cot �ps

(2S)

DDscat.inp-waveissimulatedT-matrixisdeterminedfromEnFitofT-matrixgives:BWresonanceψ(3770):mR(magentadiamonds):δ=π/2Γ(givenbelow):fromslopenearδ=π/2Boundstateψ(2S)frompoleinT:mB(magetnatriangles):cotδ=i

ψ(3770),fit(ii) Mass[MeV] g(nounit)

Lat(mπ=266MeV) 3774±6±10 19.7±1.4

Lat(mπ=156MeV) 3789±68±10 28±21

Exp. 3773.15±0.33 18.7±1.4


Lang,Leskovec,Mohler,S.P.,1503.05363,JHEP2015]

O : c c, DD, J PC =1−−

extractedmassesofhadrons


� =g2

6⇡

p3

s

SearchforresonanceX(5568)inBsπ+scamering


D0coll.foundresonanceX(5568)inBsπ+scamering


D0collaboraGonfebruary20161602.07588•  Γ=22MeV•  JPnotmeasured•  Theonlyhadronwith4differentflavors?!

Bsπ+invariantmass

SearchforX(5568)inBsπ+scamering


•  soonalerD0result,severalphenomenologicalstudiesappeared•  thosewhocouldfindsupport,suggestedithasJP=0+•  ifX(5568)isJP=0+:-theonlystrongdecaychannelisBsπ+

-thenextthresholdisBKanditis210MeVaboveX(5568)!-exoGcresonanceinelasGcchannel!?Thisissomethingla6ceQCDcando!

•  note:allotherexoGccandidates(Zc,Zb,....)areresonancesbutlie-nexttoahigherthreshold-aboveseveralthresholds

thisismuchmorechallengingthatiswhyitisdifficulttoestablishthoseexoGcstatesonla6ce



AnalyGcexpectaGonforEnifX(5568)exists

basedonmXandΓXasmeasuredbyD0exp

Employedoperators:Bsπ+andBK

BKthreshold210MeVhigher;theinterpolatorsareemployedjustforcompleteness


ResultsofEnfromactualla6cesimulaGon


AnalyGcexpectaGonforEnifX(5568)exists

basedonmXandΓXasmeasuredbyD0expLa6ceAnalyGc(ifXexists)

L=2.9fm

La6ceresult(obtainedbeforemarch22nd2016–seenextslide):•  noindicaGonofX(5568)•  interacGonsinBsπ+andBKsystemaresmall[Lang,Mohler,S.P.,1607.03185,PRD2016]


LHCbdatapublishedin1608.00435:PRL2016

Bsπ+invariantmass Bsπ+invariantmass

La6ceoperatorsforscameringofparGcleswithspin


Previoustwoexamplesconcernedscameringoftwospinlesshadrons:pseudoscalar-pseudoscalar(DDandBsπ+)

[S.P.,Lang,Skerbis,1607.06738]

MoGvaGon

La6ceoperatorsforscameringofparGcleswithspin 20

•  MainlyPPscameringwassimulatedonla6ceuptonowscameringphaseshilextracted(Phasnospin)

•  H(1)H(2):whereoneorbothHcarryspinwasexploredmostlyonlyforL=0

manyinteresGngchannelssGllunexplored,parGcularlyforL>0

IwillconsiderconstrucGonofH(1)H(2)interpolators

whereHisoneofP,V,Nhadrons,whichis(almost)stablewithrespecttostrongdecay:

P=psuedoscalar(JP=0^-)=π,K,D,B,ηc,...

V=vector(JP=1^-)=D*,B*,J/ψ,ϒb,Bc*,...(butnotdirectlyapplicabletoρasisunstable...)

N=nucleon(JP=1/2^+)=p,n,Λ,Λc,Σ,...(butnotdirectlyapplicabletoN-(1535)asisunstable...)

Iwillconsiderinterpolatorsforchannels:

PV:mesonresonancesandQQ-likeexoGcs(e.g.πJ/ψ,DD*..)

PN:baryonresonances(e.g.πN,KN...)andpentaquarks

NV:baryonresonancesandpentaquarksNN:nucleon-nucleonanddeuterium,baryon-baryon

SasaPrelovsek

MoGvaGon


•  MainlyPPscameringwassimulatedonla6ceuptonowscameringphaseshilextracted(Phasnospin)

•  H(1)H(2):whereoneorbothHcarryspinwasexploredmostlyonlyforL=0

manyinteresGngchannelssGllunexplored,parGcularlyforL>0

IwillconsiderconstrucGonofH(1)H(2)interpolators

whereHisoneofP,V,Nhadrons,whichis(almost)stablewithrespecttostrongdecay:

P=psuedoscalar(JP=0^-)=π,K,D,B,ηc,...

V=vector(JP=1^-)=D*,B*,J/ψ,ϒb,Bc*,...(butnotdirectlyapplicabletoρasisunstable...)

N=nucleon(JP=1/2^+)=p,n,Λ,Λc,Σ,...(butnotdirectlyapplicabletoN-(1535)asisunstable...)

Iwillconsiderinterpolatorsforchannels:

PV:mesonresonancesandQQ-likeexoGcs(e.g.πJ/ψ,DD*..)

PN:baryonresonances(e.g.πN,KN...)andpentaquarks

NV:baryonresonancesandpentaquarksNN:nucleon-nucleonanddeuterium,baryon-baryon

SasaPrelovsek

•  O=HHneededtocreate/annihilateHHsystem

•  EnrelatedtophaseshilsforHHscamering-twospinlessparGclesLuscher(1991):

-twoparGcleswitharbitraryspin

Briceno,PRD89,074507(2014)

(otherauthors:somespecificcases)

Somepreviousrelatedworkonla6ceHHoperatorsforhadronswithspinandL≠0

SasaPrelovsek La6ceoperatorsforscameringofparGcleswithspin 3

ParGal-wavemethodforHH:Berkowitz,Kurth,Nicolson,Joo,Rinaldi,Strother,Walker-Loud,1508.00886Wallace,Phys.Rev.D92,034520(2015),[arXiv:1506.05492]ProjecGonmethodforHH:M.Göckeleretal.,Phys.Rev.D86,094513(2012),[arXiv:1206.4141].Helicityoperatorsforsingle-H:Thomas,EdwardsandDudek,Phys.Rev.D85,014507(2012),[arXiv:1107.1930]SomeaspectsofhelicityoperatorsforHH:Wallace,Phys.Rev.D92,034520(2015),[arXiv:1506.05492].Dudek,EdwardsandThomas,Phys.Rev.D86,034031(2012),[arXiv:1203.6041].WhichCGofH1andH2toH1H2irrepsarenonzero;valuesofCGnotpublished:MooreandFleming,Phys.Rev.D74,054504(2006),[arXiv:hep-lat/0607004].etc...However:forala6cepracGGonerwhowasinterestedinacertainchannel,forexample(PVscameringinL=2orVNscameringwithλV=1andλN=1/2)thereweresGlllotsofpuzzlestobeatbeforeconstrucGngareliableinterpolator..

rotaGonsRinversionI

Werestricttototalmomentumzero


H(1)(p)H(2)(-p),Ptot=0AdvantageofPtot=0:•  parityPisagoodnumber•  channelswithevenandoddLdonotmixinthesameirrep

nottrueforPtot≠0

SasaPrelovsek

BuildingblocksH:requiredtransformaGonproperGesofH

annihilaGonfield

msisagoodquantumnumberatp=0:msisnotgoodquantumnumberingeneralforp≠0:inthiscaseitdenotesmsofcorrespondingHms(p=0)

toprovecorrecttransformaGonproperGesofHH

WignerDmatrix

Non-pracGcalchoiceofH:canonicalfieldsH(c)


L(p)isboostfrom0top;drawback:H(c)(p)dependonm,E,..

withcorrecttransformaGonproperGesunderRandI

Non-pracGcalchoiceofH:canonicalfieldsH(c)


L(p)isboostfrom0top;drawback:H(c)(p)dependonm,E,..

withcorrecttransformaGonproperGesunderRandI

PracGcalchoiceofHwithcorrecttransformaGonproperGesunderRandI

TheseHareemployedasbuildingblockinourHHoperatorssimpleexamples

relevantrotaGons::Owith24el.forJ=integer;O2with48elementsforJ=half-integerThegroupincludinginversionI:Ohwith48el.forJ=integer;O2

hwith96elementsforJ=half-integer

TherepresentaGonOJreducibleunderO(2).IrreduciblerepresentaGons(irreps)aredenotedbyΓandrowsr

RequiredtransformaGonproperGesofO=HH


T(R)givenforallirrepsinBernard,Lage,Meißner,Rusetsky,JHEP2008,0806.4495WeusesameconvenGonsforrows.

goodparitysincePtot=0!

conGnuumR

discreteR

“seed”:EachHacanhaveanypolarizaGonmsanddirecGonpwithgiven|p|.DifferentchoicesleadtodifferentlinearlyindependentOn

MethodI:ProjecGonoperators

8

Someexamplesfor|p|=1:PVinT1+,nmax=2:

PNinH+,nmax=1:

VNinH-,nmax=3:

Disadvantage:

notinformaGvewhichconGnuumnumbers(parGalwaveLorhelicity)eachOncorresponds

Thisisremediedinnexttwomethod.s

T(R)givenforallirrepsinBernard,Lage,Meißner,Rusetsky,JHEP2008,0806.4495

SasaPrelovsek La6ceoperatorsforscameringofparGcleswithspin

Proof(inourpaperandbackupslides):thecorrecttransformaGonproperGes

followfromtransformaGonsofH(slide4)andproperGesofC,YlmandD.

ExampleofPVoperatorsSubducGontoirrepsdiscussedlateron.

MethodII:ParGal-waveoperators


ProposedforNNin[Berkowitz,Kurth,Nicolson,Joo,Rinaldi,Strother,Walker-Loud,CALLAT,1508.00886]ThereYlm*appearswherewehaveYlm

Clebsch-GordansSphericalHarmonics

buildingblocksH

menGonedonslide6belowStarGngannihilaGonoperator(beforesubducGontoirreps)

MethodIII:helicityoperators

SasaPrelovsek 10

•  buildingblocksinparGal-waveoperatorsareHms(p)andmsisnotgoodforp≠0:•  HelicityλisprojecGonofStop.ItisgoodalsoforparGclesinflight

•  DefiniGonofsingle-hadronhelicityoperator•  HelicityisnotmodifiedunderR(pandStransformthesameway)

•  Two-hadronO:•  Proof:

[HHinconGnuum:Jacob,Wick(1959)][forsingleHonla6ce:HSC,Thomasetal.(2012)][notwidelyusedforHHonla6ceyet]

pisarbitrarymomentumingivenshell|p|;Rdoesnotmodifyλ1,2,soH1,2havechosenλ1,2inallterms

rotaGonfrompztop

goodms

R’=RaR

denotedbysuperscripth

MethodIII:helicityoperators(conGnued)


UsingdefiniGonsofandparityprojecGon

•  Harebuildingblocksfromslide6below:acGonsofRandIonHms(p)aregiveninslide4•  ThereareseveralchoicesofR0pwhichrotatefrompztop:

-theseleadtodifferentphasesindefiniGonofHλh:inconvenience

-buttheyleadtothesameOabove(moduloirrelevantoverallfactor):sonoproblemforsuchconstrucGon

•  Simplechoiceformomentumshell|p|=1:p=pzandR0p=IdenGty

•  paperprovidesdetailshowtousefuncGonsfromMathemaGcaforconstrucGon,alsosinceMathemaGcausesnon-convenGonaldefniGonofD

,choiceofsigninourpaper

SubducGonofOJtoirreduciblerepresentaGons


ParGal-waveoperatorsOJ,mJ,L,S

HelicityoperatorsOJ,mJ,λ1,λ2

TherepresentaGonOJisirreducibleunderconGnuumR.ButitisreducibleunderRindiscretegroupla6ceO(2).

OperatorsthattransformaccordingtoirrepΓandrowrobtainedviasubducGon.

SubducGonmatricesS[Dudeketal.,PRD82,034508(2010)

Edwardsetal,PRD84,074508(2011)]

conGnumR discreteRindiscretegroupO(2)subducGon

Single-hadronoperatorsH:experiencebyHadronSpectrumcollaboraGonPhys.Rev.D82,034508(2010)•  subducedoperatorsO[J]

ΓcarrymemoryofconGnuumspinanddominantlycoupletostateswiththisJ

ExpectaGonforparGal-waveandhelicityoperatorsHHobtainedbysubducGon:

•  woulddominantlycoupletoeigen-stateswithconGnuum(J,L,S)

•  woulddominantlycoupletoeigen-stateswithconGnuum(J,λ1,λ2)

valuableforsimulaGons

givephysicsintuiGononquant.num.

onelaststepbeforereachingtheresults...

P(1)V(-1)operators,T1+,row=r=1


JP=1+,S=1,L=0

JP=1+,S=1,L=2JP=1+,λV=0JP=1+,λV=1

projecGonop.parGal-waveop.helicityop.

ParGal-waveandhelicityoperatorsexpressedintermsofprojecGonoperatorsthroughout.

N(1)P(-1)operators,G1+,row=r=1,p-wave


JP=1/2+,S=1/2,L=1

JP=1/2+,λN=1/2

projecGonop.parGal-waveop.helicityop.

employedinla6cesimulaGonofπNsca+eringinp-wavetostudytheRoperchannelwithJP=1/2+[Lang,Leskovec,Padmanath,S.P.,arxiv:1610.01422]

PV,PN,VN,NNallirreps,|p|=0,1consistentresultsfoundinthreemethods


givenin[S.Prelovsek,U.Skerbis,C.B.Lang,arXiv:1607:06738]

ExplicitexpressionsforallH(1)(p)H(2)(-p)

PV:mesonresonancesandQQ-likeexoGcs(e.g.πJ/ψ,DD*forZc)PN:baryonresonances(e.g.πNforN*,KN...)andpentaquarksNV:baryonresonancesandpentaquarks(e.g.pJ/ψforPc)NN:nucleon-nucleonanddeuterium,baryon-baryon

Operatorsvaluabletostudyresonanceswithheavyquarks

Conclusions


Ihavediscussed•  firststudyofcharmoniumresonancethattakesintoaccountitsstrongdecay•  searchforanexoGcstateX(5568)thatcontainsfourdifferentflavorsinelasGcscamering:wedidnotfindevidenceforitfromla6ce,inagreementwithrecentLHCbresult

•  construcGonofla6ceoperatorsforsimulaGngscameringoftwo-hadronswithspin:neccessarytoextractinformaGononresonancesthatappearinPV,PNandVN,NNscamering

Backup


La6cesetup


•  Wilon-cloverquarks

•  Fermilabmethodforcandb:[ElKhadra,Kronfeldetal,1997]ResthadronenergieshavesizablediscreGzaGonerrorsbuttheselargelycancelinspli6ngs.

Onlyspli6ngswithrespecttoachosenreferencemassarecomparedtoexperiment.

•  evaluaGngWickcontracGonstosimulatescameringonthela6ceischallengingandcomputaGonallyintensive–thatispartofthereasonwhyasmallnumberofstudieshavebeenmade.Weapplytwomethods

-disGllaGon(Ensemble1)[Peardonet.al.,HSC,2009]-stochasGcdisGllaGon(Ensemble2)[Morningstaretal.,2011]

PACS-CS

Resonances with heavy quarks from lacesasa/talks/prelovsek_sfb16.pdf · Resonances with heavy quarks from lace (Bound states will be discussed by Daniel Mohler) Sasa Prelovsek University

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