Using highly excited baryons to catch the quark mass

0.13 MB·English

by  T. Van Cauteren

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,InstitutoSuperiorTe´cnico,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:[email protected] 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 spon-
taneously 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)<<k,itdominatesthesecondtermrep-
resentingthecreationorannihilationofapion(andtheRe-
Fig. 3. Typical momentum distributions of increasingly excited
forerealizeschiralsymmetrynonlinearly).
∆ , ,∆ resonancesoverlaplessandlesswiththedynam-
3/2 ··· 13/2 AsisargumentedinRefs.[10,11,12],repeatedaction
ically generated infrared quark mass. The momentum distribu-
ofthechiralchargeonathree-quarkstateleadstoaquartet
tions are computed using variational wave functions (not nor-
malized for visibility) for a linear potential with string tension ofstates,twoofeachparity,whichdynamicallybreaksinto
σ=0.135GeV2. twodoubletsofparitypartners.Thesepartnersbecomede-
generatewhenm(k)vanishes.Moreover,themasssplitting
betweenpartnersisadirectmeasureofm(k).
Thespinorsareexpandedas
U = 1 √E(k)+m(k)χλ (3) 3 The running quark mass and the ∆
kλ 2E(k) √E(k) m(k)σ kˆχλ spectrum
" − · #
1 χ 1 m(k) χ
−k−−→−−→∞ √2"σ·kˆλχλ#+ 2√2 k "−σ·λkˆχλ# , iItnyodroduebrlteotltionkthteherumnnasinsgspqluitatrinkgm|MasPs=,+w−ewMiPll=l−o|oinkaatptahre-
lowest-lying∆paritydoubletsforincreasingspin jandwe
with E(k) = k2+m(k)2. We have kept the leading chi-
usethefollowingfourarguments
rallyinvarianttermandtheleadingchiral-symmetrybreak-
p
ingtermwhichisnecessarilyoforderm(k)/k.Notethatthe 1. Reggetrajectories: j=α +αM 2 j→∞ αM 2
0 ± ±
lower component of the first-order term has the opposite −→
2. Relativistic virial theorem [13]: < k > c M
2 ±
signofthelowercomponentofthechiralinvariantterm. c2 √j → →
WhenexpandingtheQCDHamiltonian[8]intheweak √α
3. Thechirallyinvariantterm(<nHQCDn>)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 kˆinthespinor,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(<k>)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.
University. FLE acknowledges useful conversations with
Craig Roberts, Eric Swanson and Christoph Hanhart dur-
5 Conclusions ingtherecentBad-Honnefmeetings.
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