Using highly excited baryons to catch the quark mass T.VanCauteren1,a,P.Bicudo2,M.Cardoso2,andFelipeJ.Llanes-Estrada3 1 Dept.PhysicsandAstronomy,GhentUniversity,Proeftuinstraat86,B-9000Gent,Belgium 2 Dpto.F´ısica,InstitutoSuperiorTećnico,Avda.RoviscoPais,1096Lisbon,Portugal 3 Dpto.F´ısicaTeo´ricaI,UniversidadComplutense,Avda.Complutenses/n,28040Madrid,Spain 0 1 Abstract. ChiralsymmetryinQCDcanbesimultaneouslyinWignerandGoldstonemodes,dependingonthe 0 partofthespectrumexamined.Thetransitionregimebetweenboth,exploitingforexampletheonsetofparity 2 doublinginthehighbaryonspectrum,canbeusedtoprobetherunningquarkmassinthemid-IRpower-law regime.Inpassingwealsoarguethatthree-quarkstatesnaturallygroupintosame-flavorquartets,splitintotwo n parity doublets, all splittings decreasing high in the spectrum. We propose that a measurement of masses of a high-partialwave∆ resonancesshouldbesufficienttounambiguouslyestablishtheapproximatedegeneracyand J ∗ seethequarkmassrunning.Wetesttheseconceptswiththefirstcomputationofthespectrumofhigh-Jexcited 4 baryonsinachiral-invariantquarkmodel. ] h p 1 Introduction resonances.Twomainideasareappliedinordertodothis: - insensitivitytochiralsymmetrybreakingleadingtotheap- p e Inthelastthirtyyears,QuantumChromodynamicshasturned pearance of parity doublets in the hadron spectrum [1,2, h out to give an accurate description of many high-energy 3]andlinkingthemasssplittingsbetweenparitypartners [ processes. In the last decennium, a lot of effort has gone withtherunningquarkmass. 2 to solving the QCD equations on a discretized spacetime Ourproposalcanbeconciselyunderstoodwiththehelp v lattice, using an impressive amount of computing power. offigure1. 8 This, together with perturbation theory, has led to very 1 persuasiveevidencethatQCDindeeddescribesthestrong 4 interaction, both at the high-energy end and at the static 2 Chiral quartets 3 end. However, there still remains the question if the the- . 2 ory describes the transition between the perturbative and Chiral symmetry in the strong interactions is the Noether 1 non-perturbativedomain.Asuitablequantityforchecking 9 symmetryrelatedtothechiraltransformation,whichreads this is the quark mass, which should be a function of the 0 ontheclassicalfermionfields quark’s momentum due to the interactions with the QCD : v medium. Asymptotic freedom tells us that at high quark ψ eiαaτaγ5ψ, (1) i momentumk,thenonstrangequarkmassshouldapproach → X thecurrentquarkmassoftheorderof1 5MeVasdeduced r − with αa the parameters describing the rotation in SU(2)- a fromchiralperturbationtheoryorsum-rules,whilehadron flavour space with generators τa. Due to the Dirac matrix phenomenology puts the value of the quark mass at low γ , the chiral transformation is a helicity-dependent rota- momenta to an effective 300 MeV. This means that the 5 ∼ tioninflavourspace. nonstrange quark mass changes by two orders of magni- OnetermintheQCDLagrangianwhichisclearlynot tudewhengoingfromlowmomentumk<<Λ tohigh QCD invariant to the chiral transformation of Eq. 1, is a quark momentumk>>Λ (typicallyΛ 210MeV). QCD QCD massterm ∼ Obviously,adirectcomparisonbetweentheQCDpre- m q¯q. (2) q dictionoftherunningquarkmassandexperimentalobser- vationisnotpossiblesincequarksareconfinedandpropa- Thistermbreakschiralsymmetryexplicitly,andtheamount gate only a distance of order a fermi, too small to detect of breaking is larger when the quark mass increases. As directly. It is a main goal of modern hadron physics to mentionedbefore,thesmallcurrentquarkmassof5MeV glimpse properties of the confined, colored quarks from breakschiralsymmetryonlyslightly.Ontopoftheexplicit colorless hadron properties. We suggest an indirect way breaking of chiral symmetry, the strong interactions also of extracting the power-law behaviour of the quark mass showspontaneouschiralsymmetrybreakingduetotheap- function from the experimentally obtainable masses of ∆ pearance of the quark condensate < q¯q > (a constituent quark mass of 300 MeV arises and breaks it to a much a e-mail:timvancauteren@gmail.com largerdegree). EPJWebofConferences Running quark mass (matched to lattice) 0.4 0.3 V) Ge0.2 M( 0.1 0 0 0.5 1 1.5 2 2.5 3 p (GeV) MeV 400 300 200 100 J 3/2 5/2 7/2 9/2 11/2 13/2 15/2 Fig.1. Therunningofthequarkmass(top),fallingwithmomen- tumasapowerlaw,isaccessiblefromthehadronspectruminthe regionm(k)< k (continuousredline).Alloneneedstodoisfit h i apowerlawtothedecreasingparitysplittingsforexcitedbaryon states (bottom, where we show a variational-Montecarlo model calculationthereofasboxesandexperimentaldatafromRef.[4] as circles). Then the exponent of the quark mass power-law in Eq.(10)isobtainedfromEq.(9). ThisistheQCDexplanationfortheabsenceofparity doubling in the low spectrum, as the symmetry is spontaneously broken. The community now believes that the highlyexcitedstateshoweverareinsensitivetothissponta- neousbreaking(seefig.2foraclassicalanalogy).Indeed, sincethequarkmassislargeatlowmomentaandsmallat high momenta, one may expect that the chiral symmetry breaking is less important in systems where the average quark momentum < k > is high, than in systems where Fig.2. Aclassicalanalogyofthreesymmetryrealizationsina < k > is large. This leads to the idea of insensitivity to quantum theory. Top: the two buildings have equal mass as re- chiralsymmetrybreakinghighinthehadronspectrum[1, quired by Reflection symmetry in Wigner mode. Middle: if in- 2,3,5,6] : in highly excited meson or baryon states, the mersedinafluid(non-trivialgroundstate)withadensitygradient averagequarkmomentumcanbecomelargerthanΛQCD fromlefttoright,thetwobuildings,filledwithit,havenowdif- andtheexplicitchiral-symmetrybreakingquarkmassterm ferentmass(thesymmetryisinGoldstonemode).Bottom:even is small. This is illustrated in Fig. 3, where typical quark ifthefluidbreakstheReflectionsymmetry,tallenoughbuildings momentumdistributionsareshownforthelowest-lying∆- haveclosetoequalmasses(insensitivitytosymmetrybreaking). resonancefordifferentspins,togetherwiththelatticeQCD calculationoftherunningquarkmassfromRef.[7]. showsupbothinthequarkspinorsandintheQCDHamil- 2.1 Seriesexpansioninm(k)/k tonian. Since we want to find out what happens at small quarkmassandhighmomentum,aseriesexpansionofthe In order to assess the effect of a nonzero quark mass to a spinors and Hamiltonian in the parameter m(k)/k seems hadron mass, it is important to note that the quark mass appropriate. 19th InternationalIUPAPConferenceonFew-BodyProblemsinPhysics theexplicitexpressionforthespinors,thechiralchargecan bewrittenas d3k τa k Qa = (6) m(k) (GeV) q 000...453 m∆∆∆∆357q///(222k) m×(kh)(σ5·kˆ)Zλλ′((cid:16)2Bπ†kλ)3fcλBλX′kfλ′f′fc′c +2D!f†−fk′λ′pf′ckD2−+kλmfc2(cid:17)(+k) Lattice 0.2 ∆∆91/12/2 k (iσ2)λλ′ B†kλfcD†−kλ′f′c+Bkλ′f′cD−kλfc # . 13/2 (cid:16) (cid:17) 0.1 The first term between the square brackets represents a quark and antiquark number operators flipping spin and 0 2 4 6 k (GeV) parity.Whenm(k)< c M 2 ± signofthelowercomponentofthechiralinvariantterm. c2 √j → → WhenexpandingtheQCDHamiltonian[8]intheweak √α 3. Thechirallyinvariantterm()cancelsout sense(thatis,notoftheHamiltonianitself,butarestriction χ | | in∆M: thereof to the Hilbert space of highly excited resonances, M+ M << M where< k >islarge),wenotethattheleadingterminthe − ± | − | m(k)/k expansion is chiral invariant, while the first order and term may involve nonchiral, spin-dependent potentials in m(k) m(k) thequark-quarkinteraction: |M+−M−|→< k HχQCD′ >→ c3 k < HχQCD′ > m(k) hn|HQCD|n′i≃hn|HχQCD|n′i+hn| k HχQCD′|n′i+... . (4) 4. InHχQCD,thespin-orbitLi·Sitermiscrucialtocorrect the angular momentum in the centrifugal barrier term from L2 to the chirally invariant L2 +2L S = J2 i i i · i i − 3.Duetothesigndifferenceinthehelicity-dependent 2.2 Chiralchargeandthree-quarkstates 4 term σ kînthespinor,thespin-orbitterminHQCD′ χ ∼− · Ifthequarksweremassless,andtherewouldbenochiral adds to the mass difference ∆M, instead of cancelling noninvariantmassterminthestronginteractions,thechiral out as it does for HχQCD. Since the centrifugal barrier charge[9] scales like M± for high j, the spin-orbit term scales τa withonepowerof jless: Qa = dxψ (x)γ ψ(x) (5) 5 † 5 2 Z c 1 wchoiuraldl scyommmmeuttreywwitohutlhdesQtilClDbeHsapmoinlttoanneiaonu.sNlyevberrotkheenlesbsy, < HχQCD′ >→c5M±j−1 → √5αs j . (7) thegroundstate, Qa0 , 0,leadingtoalargequarkmass 5| i Combiningthesefourarguments,weobtain inthequarkpropagator,pseudo-Goldstonebosonsandthe loss of parity-degeneracy in ground-state baryons. Using c c M+ M 3 5 m()j 1 . (8) Bogoliubov-rotatedquark/antiquarkoperatorsBandDand | − −|→ c √α − 2 EPJWebofConferences An experimental extraction can be done by fitting the ex- Hamiltonianusingasetofbasisfunctionsforthree-quark ponent iof jinthesplitting states. − Constructinganorthonormalbasisofthree-quarkstates M+ M j i. (9) | − −|∝ − is not an easy task. The basis functions need to form an antisymmetrized and orthonormal set of states of definite Then,thepower-lawbehaviouroftherunningquarkmass spin and parity. We have built this basis starting from ba- isgivenby sis states which are a combination of harmonic-oscillator m(k) k 2i+2. (10) ∝ − (HO)stateswithparameterαanda2 2 2=8-component × × The experimentally known masses of lowest-lying ∆ res- spinor onances for j = 1/2, ,15/2 are shown in Fig. 4. From ··· this, it is clear that the present state of our experimental α (p ,p ,S )=ϕα (p )ϕα (p )χ (S ), knowledgeofthe∆-spectrumisnotsufficienttoderivethe BN,l,mρ,mλ,Ri ρ λ i nρN,l,lρN,l,mρ ρ nλN,l,lλN,l,mλ λ Ri i exponenti.Knowledgeofthemassesoftheparitydoubler (12) forspins j>9/2wouldgreatlyenhancethis. wherepρ andpλ aretherelativeJacobimomenta,andthe indices are needed to denote the quantum numbers of the HOwavefunctionsandthespinor. 4 PDG, natural parity Thefollowingsubsequentstepsaretakeninconstruct- PDG, parity doubler ingthebasis 3.5 3 V) – A unitary transformation is applied to the basis func- Ge2.5 tions of Eq. 12, using Clebsch-Gordan coefficients to M ( ensureafixedspinofthebasisfunction.Theparityof 2 the state is automatically fixed for the HO basis func- 1.5 tions:HOshellswitheven (odd)shellnumber N give risetopositive(negative)paritystates. 11 2 3 4 5 6 7 8 9 – The resulting set of basis functions are symmetrised J (thecolourpartofthewavefunctionistakentobean- Fig.4.Experimentallyknownmassesforthelowest-lying∆res- tisymmetric and is left out in this discussion) by sim- onanceforspinsupto15/2andeachparity.Thedegeneracyseen plysummingoverthesixpossiblequarkpermutations. at j=9/2canbeaccidentalandmaybeconfirmedifthemasses Thisgivesrisetoanoverdeterminedsetofnon-orthogonal oftheparitydoublersat j>9/2aremeasured. symmetrizedbasisfunctionsoffixedspinandparity. – Anorthonormalsetofbasisfunctionsiscreatedbyper- formingaGram-Schmidtprocedureonthenon-orthogonal set.Forthis,onehastocomputetheoverlapsbetween thenon-orthogonalwavefunctionsusingthevanBeveren- 4 A basis for ∆ states Ribeiro-Moschinskycoefficients[14]. The idea of extracting the running quark mass from the hadron spectrum can be investigated using a chirally in- Wehaveperformedthisorthonormalizationprocedure variant quark model. The Hamiltonian we have used in anddiagonalizedthemodelHamiltonianforspin1/2and Ref.[10]comesfromafieldtheoryupgradeoftheCornell 3/2statesofeachparity.Theresultingmassesareplotted model and we use a linear quark-quark potential V (r) = inFig.5.Here,wecanalreadyseeatendencyofdecreas- L σr with string tension σ = 0.135 GeV2. This leads to an ingmassdifferencebetweenstatesofdifferentparitywhen analytic expression for the mass splitting between parity lookingathigher-excitedstates. partnersof However, it is at this point not clear which states are paritypartnerssincethedensityofstatesissohigh.Inor- M M =3 d3k1 d3k2 2 d3q (11) dertodisentanglethepairsinmodelterms,onejustneeds +− − (2π)3(2π)3 3 (2π)3 tocheckouttheoverlapsofthechiralchargebetweenpos- ×Vˆ(q)12 m(k|k1|) +Zm(k|k1++qq|) F∗ λ1λ!2λZ3(k1,k2) isttiavteesanardencelogsaetivtoebpeairnitgyscuacnhddideagteense.rIaftheBp−a|rQtna5e|Brs+.iH∼ow1,evtheer | 1| | 1 | ! thisisdifficulttoimplementwithexperimentaldataalone. I σkˆ σk +q Fµ1λ2λ3(k +q,k q). 1 1 1 2 Thusweadvocatereachinghighexcitationnotby“ra- × − λ1µ1 − dial”excitations(successiveresonancesofincreasingmass (cid:16) (cid:17) HereVˆ istheFouriedrtransformofthelinearpotentialwhich butequalquantumnumbers),butinsteadbyangularexcita- fallslikeq 4,andthe F isthewavefunctionofoneofthe tions(thegroundstateineachJ-channel).Itisstraightfor- − parity partners. This wavefunction can be obtained vari- wardtomatchthegroundstatesinanexperimentalanaly- ationally as done in Ref. [10]. However, it is only possi- sis,aslongasgoodpartialwaveexpansionsareachieved. bletodothisfortheloweststateforeachspinandparity. WelookforwardforJLABandELSAprovidingtheseex- Higher-lyingstatescanbecomputedbydiagonalizingthe cited∆ spectrum. J 19th InternationalIUPAPConferenceonFew-BodyProblemsinPhysics Parity doubling in the ∆ spectrum? quarkmodels,thesecondtheNambu-Jona-Lasiniomodel). 4 Thus one needs to resort to global-color models that are non-local, and we employ the field-theory upgrade of the Cornell quark model. In this contribution we have shown 3 resultsforhighlyexcited∆baryonstomorethan20eigen- V) values (out of reach for conceivable lattice calculations), M (Ge2 11//22-+ wraintgheaomfbuacshislvaregcetorrvsa.rOiautriocnoamlpbuatsaitsioinnst,hfeorhbuontdhrespdi-nm1o/r2e and3/2,clearlyshowtheparitydoubling. 1 However,thisavenueisnotpromisingforexperimental extraction,asitisthendifficulttomatchthepartnerssince 00 2 4 6 8 10 12 14 16 18 20 thedensityofstatesislarge,andtheyoverlapastheirwidth Eigenvalue number growshighinthespectrum.Weadvocateanexperimental Parity doubling in the ∆ spectrum? measurementofthedoublingfortheground-statesofeach 4 angularmomentumchannelatJLABandELSA. Withthedatainhand,onecanthenuseourm(k)/kfirst- orderexpansiontoobtaintherunningofthequarkmassin 3 theinfrared,ahithertounaccessedquantityinQCD.Other V) effortsareunderway toaccessthisinterestingpropertyof M (Ge2 33//22-+ apacroisnofinnsehdooubldjepctrofvroeminhteardersotinngst.ructuredata1 andacom- 1 Acknowledgments 0 0 2 4 6 8 10 12 14 16 18 20 Eigenvalue number WorksupportedbygrantnumbersUCM-BSCHGR58/08 Fig.5.Amassivevariational-Montecarlomodelcalculation[15] 910309,FPA2007-29115-E,FPA2008-00592,FIS2008-01323, allows to check that parity doubling occurs high in the high CERN/FP/83582/2008,POCI/FP/81933/2007,/81913/2007, baryonspectrum,here∆baryonswithspin1/2and3/2.Thecom- PDCT/FP /63907/2005 and /63923/2005, Spain-Portugal putationofhighlyexcitedstatesiscurrentlyimpossibleinlattice bilateralgrantHP2006-0018/E-56/07,aswellastheFlan- gaugetheory.Eveninoursimplifiedchiralfield-theorymodelof QCD,thepresentcomputationisformidableandrequired 105 dersResearchFoundation(FWO).Computationalresources ∼ and services used in this work were provided by Ghent CPU-hoursat2-3GHz. 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Using highly excited baryons to catch the quark mass

Similar Using highly excited baryons to catch the quark mass