Table Of ContentNuclear
Physics A
NuclearPhysicsA00(2013)1–4
Heavy-quark spin symmetry for charmed and strange baryon
resonances
3
1 Olena Romanetsa, Laura Tolosb,c, Carmen Garc´ıa-Reciod, Juan Nievese,
0 Lorenzo Luis Salcedod, Rob Timmermansa
2
n aTheoryGroup,KVI,UniversityofGroningen,Zernikelaan25,9747AAGroningen,TheNetherlands
a bInstitutodeCienciasdelEspacio(IEEC/CSIC),CampusUniversitatAuto`nomadeBarcelona,FacultatdeCie`ncies,TorreC5,E-08193
J Bellaterra(Barcelona),Spain
8 cFrankfurtInstituteforAdvancedStudies,JohannWolfgangGoetheUniversity,Ruth-Moufang-Str.1,60438FrankfurtamMain,Germany
1 dDepartamentodeF´ısicaAto´mica,MolecularyNuclear,andInstitutoCarlosIdeF´ısicaTeo´ricayComputacional,UniversidaddeGranada,
E-18071Granada,Spain
] eInstitutodeF´ısicaCorpuscular(centromixtoCSIC-UV),InstitutosdeInvestigacio´ndePaterna,Aptdo.22085,46071,Valencia,Spain
h
p
-
p
e
h Abstract
[
Westudycharmedandstrangeodd-paritybaryonresonancesthataregenerateddynamicallybyaunitarybaryon-mesoncoupled-
2 channelsmodelwhichincorporatesheavy-quarkspinsymmetry.ThisisaccomplishedbyextendingtheSU(3)Weinberg-Tomozawa
v chiralLagrangiantoSU(8)spin-flavorsymmetryplusasuitablesymmetrybreaking. Themodelgeneratesresonanceswithnega-
3 tiveparityfromthes-waveinteractionofpseudoscalarandvectormesonswith1/2+and3/2+baryonsinalltheisospin,spin,and
4
strangesectorswithone, two, andthreecharmunits. Someof ourresultscanbeidentifiedwithexperimentaldatafromseveral
9
facilities,suchastheCLEO,Belle,orBaBarCollaborations,aswellaswithothertheoreticalmodels,whereasothersdonothave
3
astraightforwardidentificationandrequirethecompilationofmoredataandalsoarefinementofthemodel.
.
2
1 c 2012PublishedbyElsevierLtd.
2 (cid:13)
1 Keywords: Charm,heavy-quarkspinsymmetry,dynamicallygeneratedbaryonresonances
:
v
i
X 1. Introduction
r
a The propertiesof heavy-flavoredhadronicresonanceshave attracted a lot of attentionlately. The study of such
states can help in the interpretation of the nature of particles found in past and ongoing experiments (e.g. CLEO,
BaBar, Belle, LHCb) [1], as well as in understanding states which will be discovered in future experiments (e.g.
PANDA at GSI [2]). It is importantto understandwhetherbaryon(meson)resonancescan be describedas excited
three-quark(quark-antiquark)states or rather as hadronmolecules; also a combined interpretationof such states is
possible.
Atpresentthereisalackofarobustschemetosystematicallyconstructaneffectivefieldtheoryapproachtostudy
fourflavorphysics. Some stepsin thatdirectionhave beentakenby recentstudiesusingcoupled-channelsmodels.
Among them one can find unitarized coupled-channels models [3, 4, 5, 6], the Ju¨lich meson-exchange model [7]
andschemesbasedonhiddengaugeformalism[8]. Thesemodelsarenotfullyconsistentwiththeheavy-quarkspin
symmetry (HQSS) [9], which is a propersymmetry of Quantum Chromodynamics(QCD) in the limit of infinitely
heavyquarkmasses. Therehavealsobeensomeattemptstobuildaschemebasedonchiralperturbationtheoryfor
hadronswhichcontainheavyquarks[10]. Moreover,aneffectivetheorywhichincorporatesheavy-quark,chiraland
1
OlenaRomanetsetal./NuclearPhysicsA00(2013)1–4 2
hiddenlocalgaugesymmetrieswasdevelopedforstudyingbaryon-baryoninteractions[11]. Besides,anSU(8)spin-
flavor symmetric unitarized coupled-channelsmodel has been recently developedand used for variousnumbers of
charmandstrangeness[12,13].
In this paper we study dynamically-generatedbaryon resonances, using the SU(8) spin-flavormodel. We have
payed special attention to analyze the underlying symmetry of the interaction. In particular, we have studied the
originalgroupmultipletsfromwhereeachofthefoundbaryonresonancesoriginatesandobtainedthedifferentHQSS
multiplets. Ourstudiescoveredstates with charmand strangeness, and in the followingsectionswe will show and
discussourresultsforbaryonresonanceswithcharmC =1,namelyΛ (strangenessS =0,isospinI =0),Σ (S =0,
c c
I =1),Ξ (S = 1,I =1/2)andΩ (S = 2,I =0).
c c
− −
2. Theoreticalframework
We use the SU(8) spin-flavor model of [12, 13]. The interaction potential is an extension of the SU(3) chiral
Weinberg-Tomozawa potential to the SU(8) symmetry. In this model vector mesons are treated on equal footing
withpseudoscalarmesonsandbothspin-1/2andspin-3/2baryonsare takenintoaccount. We onlyconsiders-wave
interaction,whichisappropriatecloseto themeson-baryonthresholds. InthisSU(8)schemethemesonsfallin the
63-plet,andthebaryonsareplacedinthe120-plet. Consequently,inthes-channel,thebaryon-mesonspacereduces
intofourSU(8)irreps,threeofwhich(120,168and4752)areattractive. Wefindthatthemultiplets120and168are
themostattractiveones,andthereforewehaveconcentratedourstudyonthestateswhichbelongtothesetwoirreps
intheSU(8)symmetriclimit.
WeconsiderthereductionoftheSU(8)symmetrySU(8) SU(6) SU (2) U (1),whereSU(6)isthespin-flavor
C C
⊃ × ×
groupforthreelightflavors,SU (2)isthecharmquarkrotationgroup,andU (1)isthegroupgeneratedbythecharm
C C
quantumnumberC. TheSU(6)multipletscanbereducedunderSU(3) SU(2),whereSU(2)referstothespinof
l l
×
thelightquarks. WefurtherreduceSU(2) SU (2) SU(2)whereSU(2)referstothetotalspin J;inthiswaywe
l C
× ⊃
maketheconnectionwiththelabeling(C,S,I,J).
Thecontacttree-levelmeson-baryoninteractionoftheextendedSU(8)symmetricWeinberg-Tomozawapotential
reads
2√s M M E +M E +M
V (s)=D − i− j i i j j . (1)
ij ij 4 fifj r 2Mi s 2Mj
Here,iand jaretheoutgoingandincomingbaryon-mesonchannels; M, E,and f stand,respectively,forthemass
i i i
andthecenter-of-massenergyofthebaryonandthemesondecayconstantintheichannel;D arethematrixelements
ij
forthevariousCSIJ sectorsconsideredinthiswork,whichcanbefoundintheAppendicesof[12,14].
Thescatteringamplitudesarecalculatedbysolvingtheon-shellBethe-Salpeterequationinthecoupledchannels:
1
T(s)= V(s). (2)
1 V(s)G(s)
−
HereG(s)isadiagonalmatrixcontainingthebaryon-mesonpropagatorforeachchannel,andD,T andVarematrices
incoupled-channelsspace.
ThebareloopfunctionG0(s)islogarithmicallyultravioletdivergentandneedstoberenormalized.Wehaverenor-
ii
malizedouramplitudesbyusingasubtractionpointregularization,withasubtractionpoint √s = µ = m2 +M2,
i th th
wheremth andMth are,respectively,themassesofthemesonandthebaryonofthechannelwiththeloweqstthreshold
inthegivenCSI sector,
G (s)=G0(s) G0(µ2). (3)
ii ii − ii i
Inordertogetabetterfine-tuningwiththeexperimentaldata,onecandefinethesubtractionpointasµ = α(m2 +M2),
i th th
withαbeingslightlydifferentfromone. q
Thebaryonresonancesareobtainedaspolesofthescatteringamplitudeonthecomplex-energyplane. Themass
m andthewidthΓ ofthestatecanbeobtainedfromthecoordinate √s ofthecorrespondingpoleonthecomplex
R R R
energy √splane, √s = m iΓ ,andthecouplingstothemeson-baryonchannelscanbefoundfromtheresidues
R R− 2 R
oftheT-matrixaroundthepole.
2
OlenaRomanetsetal./NuclearPhysicsA00(2013)1–4 3
ThematrixelementsD displayexactSU(8)invariance,butthissymmetryisseverelybrokeninnature.Therefore
ij
weimplementsymmetry-breakingmechanisms.Itshouldbementionedherethatwehaveremovedthechannelswith
thecc¯pairstobeconsistentwiththeHQSS.Inthepresentworkweusephysicalvaluesforthemassesofthehadrons
andforthedecayconstantsofthemesons. ThesymmetryisbrokenfollowingthechainSU(8) SU(6) SU(3)
⊃ ⊃ ⊃
SU(2),wherethelastgroupSU(2)referstoisospin. Thissymmetrybreakingwasperformedbyadiabaticchangeof
thehadronmassesandmesonweakdecayconstants.Inthiswaywecanlabeleachbaryonresonancewiththeoriginal
groupmultipletanddefinetheHQSSmultiplets.
3. Dynamicallygeneratedbaryonresonances
LetusbeginwiththeΛ states. OurmodelgeneratesfourΛ baryonresonances,threewithspin J =1/2andone
c c
with J = 3/2. By comparingthedominantchannelswiththe decaychannelsofthe experimentalstates, twoofour
Λ ’shavebeenidentifiedwithexperimentallyknownstates. We identifytheexperimentalΛ (2595)resonancewith
c c
thestatethatwefoundaround2618.8 i0.6MeV. TheexperimentalvalueofthewidthofΛ (2595)3.6+2.0MeVisnot
− c 1.3
reproduced,duetothefactthatwe havenotincludedthe three-bodydecaychannelΛ ππ,whichalre−adyrepresents
c
almostonethirdofthedecayevents[15].OurresultforΛ (2595)agreeswiththeresultsfromt channelvector-meson
c
−
exchange(TVME)models[4,5],but, asitwasfirstpointedoutinRef.[12], weclaimadominantND component
∗
initsstructure,whereasintheTVMEmodelthe Λ (2595)isgeneratedmostlyasa NDboundstate. Wealsoobtain
c
a broadresonance with a mass veryclose to the Λ (2595), namely at 2617.3MeV. It couplesstrongly to the open
c
channel Σ π. The other pole with J = 1/2 that we find around 2828 i0.4MeV has not been identified with any
c
−
knownexperimentalstate.
In the J = 1/2 sector there are 16 coupled channels, which can generate Λ resonances. Every found baryon
c
resonance couples strongly only to some of the coupled channels, see [14]. Therefore, we study how the features
(masses, widths and couplings) of the Λ resonances change, when we consider only the dominant meson-baryon
c
coupledchannels. It turnsoutthat the masses and widths, as well as couplingsdo notchangedrastically when we
onlyconsidertherestrictedcoupled-channelsspace. The widthofthe Λ (2617.3)resonanceincreasesfrom89.8to
c
97.3MeV,whereasthemassandthecouplingtotheΣ πstayunchanged.TheΛ (2618.8)resonanceslightlyincreases
C c
itsmassby2.6MeV,andthewidthdecreasesfrom1.2to1.1MeV,whilethecouplingtoND channelremainsalmost
∗
thesame. Finally,themassoftheΛ (2828.4)stateraisesby8.6MeV,anditswidthisnow1.0MeV;thecouplingsto
c
thedominantΛ ηandΣ ρchannelsslightlyvary.
c ∗c
Further,wefindoneΛ resonancewith J =3/2locatedat(2666.6 i26.7MeV). Weidentifythisresonancewith
c
−
the experimentalΛ (2625) [15]. The experimental Λ (2625) has a very narrow width, Γ < 0.97MeV, and decays
c c
mostlytoΛ ππ. Bychangingthesubtractionpoint,suchthatthemassoftheresonanceisclosertothevalueofthe
c
experimental one, the phase space would be reduced. A similar resonance was found at 2660MeV in the TVME
modelofRef.[6]. However,inourcalculationweobtainanon-negligiblecontributionfromthebaryon-vectormeson
channelstothegenerationofthisresonance,asalreadyobservedinRef.[12].Whenrestrictingthenumberofcoupled
channels to the four ones, to which Λ (2666.6) couples the most, namely Σ π, ND , Σ ρ and Σ ρ, the resonance
∗c ∗c ∗ c ∗c
featuresarechangedasfollows. Themasssomewhatincreasesby1.2MeV,whilethewidthgrowsby8.2MeV,and
couplingsremainalmostunchanged.
Weobtainthreespin-1/2Σ resonances,withmasses2571.5,2622.7and2643.4MeV.Thesestatesarepredictions
c
ofourmodel,sincethereisnoexperimentaldatainthisenergyregion.IntheSU(4)modelofRef.[5]twoΣ spin-1/2
c
resonancesarepredicted.Inthisreference,thefirst Σ resonancehasamass2551MeVandawidthof0.15MeV,and
c
itcanbeassociatedwiththeΣ (2572)stateofourmodelwhichwegeneratewiththewidthΓ=0.8MeV. However,in
c
ourmodelthisresonancecouplesmoststronglytothechannelswhichincorporatevectormesons,whereasinRef.[5]
it is notthe case. The otherresonance predictedin Ref. [5] cannotbe comparedto any of ourresults. Further, we
obtaintwospin-3/2Σ resonances.Thefirstone,aboundstateat2568.4MeV,liesbelowthethresholdofanypossible
c
decaychannelandisthoughttobethecharmedcounterpartofthehyperonicΣ(1670)resonance. Thesecondstateat
2692.9 i33.5 MeVhasnodirectcomparisonwiththeavailableexperimentaldata.
−
OurmodelgeneratessixΞ stateswithJ =1/2andthreeoneswith J =3/2.Inthissectortherearetwonegative-
c
parityexperimentallyknownresonancesthatcanbeidentifiedwithsomeofourdynamically-generatedstates,namely
experimentalΞ (2790)JP =1/2 andΞ (2815)JP =3/2 [15]. ThestateΞ (2790)hasawidthofΓ<12 15MeV
c − c − c
−
anditdecaystoΞ π,withΞ Ξ γ. Weassignittothe2804.8 i13.5MeVstatefoundinourmodelbecauseofthe
′c ′c → c −
3
OlenaRomanetsetal./NuclearPhysicsA00(2013)1–4 4
largeΞ πcoupling.Aslightmodificationofthesubtractionpointcanlowerthepositionofourresonanceto2790MeV
′c
andmostprobablyreduceitswidthasitwillgetclosertothe Ξ πchannel. Itcouldbealsopossibletoidentifyour
′c
poleat2733MeVwiththeexperimentalΞ (2790)state. Inthatcase,onewouldexpectthatiftheresonanceposition
c
getscloserto the physicalmass of2790MeV, its width will increase andit will easily reach valuesof the orderof
10MeV. The full width of the experimentalΞ resonancewith JP = 3/2 is expectedto be less than 3.5MeVfor
c −
Ξ+(2815)andlessthan6.5MeVforΞ0(2815),andthedecaymodesareΞ+π+π ,Ξ0π+π . Weobtaintworesonances
c c c − c −
at2819.7 i16.2MeVand2845.2 i22.0MeV,respectively,thatcouplestronglytoΞ π,withΞ Ξ π. Allowing
− − ∗c ∗c → c
forthis possible indirectthree-bodydecaychannel, we mightidentifyone of themto the experimentalresult. This
assignmentis possible forthe state at 2845.2MeVif we slightly changethe subtractionpoint, which will lowerits
positionandreduceitswidthasitgetsclosertothethresholdoftheopenΞ πchannel.
∗c
WeobtainthreeΩ boundstateswithmasses2810.9,2884.5and2941.6MeV.Thereisnoexperimentalinforma-
c
tiononthoseexcitedstates. However,ourpredictionscanbecomparedtorecentcalculationsofRef.[5]. Inthiswork
three Ω resonancesare predicted,with masses higherthanthe onesofourresonancesbyapproximately100MeV.
c
Further,weobtaintwospin-3/2boundstatesΩ withmasses2814.3and2980.0MeV,whichmainlycoupletoΞD
c ∗
andΞ D ,andtoΞ K¯,respectively.Asinthe J =1/2sector,noexperimentalinformationisavailablehere.
∗ ∗ ∗c
4. Summary
Charmedbaryonresonances,inparticularΛ ,Σ ,Ξ andΩ odd-paritystateshavebeenstudiedwithinacoupled-
c c c c
channelsunitaryapproachthatimplementsHQSS.ForthispurposetheSU(8)spin-flavorsymmetricmodelofRef.[12]
hasbeenused. We haveobtainedfourΛ baryonresonances,two ofwhichcan beidentifiedwith the experimental
c
Λ (2595)andΛ (2625)states. Whenthenumberofcoupledchannelsisreducedtothedominantones,thefeatures
c c
(mass, width, couplingconstants) of the correspondingresonance do not change significantly. Further, five Σ and
c
nine Ξ resonances are obtained. Some of our resonances can be identified with experimentallyknown Σ and Ξ
c c c
states,whileothersrequirethecompilationofmoredataandarefinementofthemodel.
5. Acknowledgments
ThisresearchwassupportedbyDGIandFEDERfunds,undercontractsFIS2011-28853-C02-02,FIS2011-24149,
FPA2010-16963and the Spanish Consolider-Ingenio2010 Programme CPAN (CSD2007-00042), by Junta de An-
daluc´ıagrantFQM-225,by GeneralitatValencianaundercontractPROMETEO/2009/0090and by the EU Hadron-
Physics2project,grantagreementn. 227431. O.R.wishesto acknowledgesupportfromtheRosalindFranklinFel-
lowship. L.T.acknowledgessupportfromRamonyCajalResearchProgramme,andfromFP7-PEOPLE-2011-CIG
undercontractPCIG09-GA-2011-291679.
References
[1] http://www.lepp.cornell.edu/Research/EPP/CLEO/;http://belle.kek.jp/;http://www.slac.stanford.edu/BF/;http://lhcb.web.cern.ch/lhcb/.
[2] http://www-panda.gsi.de/.
[3] M.F.M.LutzandE.E.Kolomeitsev,Nucl.Phys.A730,110(2004);M.F.M.LutzandE.E.Kolomeitsev,Nucl.Phys.A755,29(2005);
[4] L.Tolos,J.Schaffner-BielichandA.Mishra,Phys.Rev.C70,025203(2004);T.MizutaniandA.Ramos,Phys.Rev.C74,065201(2006).
[5] C.E.Jimenez-Tejero,A.RamosandI.Vidana,Phys.Rev.C80,055206(2009).
[6] J.HofmannandM.F.M.Lutz,Nucl.Phys.A763,90(2005);J.HofmannandM.F.M.Lutz,Nucl.Phys.A776,17(2006).
[7] J.Haidenbauer, G.Krein,U.G.MeissnerandA.Sibirtsev,Eur.Phys.J.A33,107(2007);J.Haidenbauer, G.Krein,U.G.Meissnerand
A.Sibirtsev,Eur.Phys.J.A37,55(2008);J.Haidenbauer,G.Krein,U.G.MeissnerandL.Tolos,Eur.Phys.J.A47,18(2011).
[8] J.-J.Wu,R.Molina,E.OsetandB.S.Zou,Phys.Rev.Lett.105,232001(2010);K.P.Khemchandanietal.,Phys.Rev.D83,114041(2011);
K.P.Khemchandanietal.,Phys.Rev.D84,094018(2011);K.P.Khemchandanietal.,Phys.Rev.D85,114020(2012).
[9] N.Isgur,M.B.Wise,Phys.Lett.B232,113(1989);M.Neubert,Phys.Rept.245,259-396(1994);A.V.ManoharandM.B.Wise,Camb.
Monogr.Part.Phys.Nucl.Phys.Cosmol.10,1(2000).
[10] M.B.Wise,Phys.Rev.D45,(1992)R2188;Casalbuonietal.,Phys.Lett.B292,(1992)371;Yanetal.,Phys.Rev.D46,(1992)1148.
[11] Y.R.Liu,M.Oka,Phys.Rev.D85,014015(2012);W.Meguro,Y.R.LiuandM.Oka,Phys.Lett.B704,547-550(2011).
[12] C.Garcia-Recio,V.K.Magas,T.Mizutani,J.Nieves,A.Ramos,L.L.SalcedoandL.Tolos,Phys.Rev.D79,054004(2009).
[13] D.Gamermann,C.Garcia-Recio,J.Nieves,L.L.SalcedoandL.Tolos,Phys.Rev.D81,094016(2010).
[14] O.Romanets,L.Tolos,C.Garcia-Recio,J.Nieves,L.L.SalcedoandR.G.E.Timmermans,Phys.Rev.D85,114032(2012).
[15] J.Beringeretal.(ParticleDataGroup),Phys. Rev. D86,010001(2012).
4
