An Overview of the Hypercentral Constituent Quark Model

10 Pages·0.13 MB·English

by  M. M. Giannini

AN OVERVIEW OF THE HYPERCENTRAL
CONSTITUENT QUARK MODEL
M.M.Giannini,E. SantopintoandA. Vassallo
DipartimentodiFisicadell’Universita`diGenovaandI.N.F.N.,SezionediGenova
February 8, 2008
3
0
0
Abstract
2

n Wereport ontherecentresultsofthehypercentral Constituent QuarkModel
a (hCQM).Themodelcontainsaspinindependentthree-quarkinteractionwhichis
J inspiredbyLatticeQCDcalculationsandreproducestheaverageenergyvaluesof

8 theSU(6) multiplets. The splittingsareobtained withaSU(6)-breaking inter-
action, which can include also an isospin dependent term. The model has been

1 usedforpredictionsconcerningtheelectromagnetictransitionformfactorsgiving
v agooddescriptionofthemediumQ2-behaviour. Inparticularthecalculatedhe-
7
licityamplitudeA1 fortheS11(1535)resonanceagreesverywellwiththerecent
1 2
CLAS data. Furthermore, we have shown for the first time that the decreasing
0
of theratioof theelasticform factorsof theproton isduetorelativisticeffects.
1
0 Finally, theelasticnucleonformfactorshavebeencalculatedusingarelativistic
3 versionofthehCQMandarelativisticquarkcurrent.
0
/
h 1 Introduction
t
-
cl Inrecentyearsmuchattentionhasbeendevotedtothedescriptionoftheinternal
u nucleonstructureintermsofquarkdegreesoffreedom. Besidesthenowclassi-
n calIsgur-Karlmodel[1],theConstituentQuarkModelhasbeenproposedinquite
: differentapproaches: thealgebraicone[2],thehypercentral formulation[3]and
v
thechiralmodel[4,5].InthefollowingthehypercentralConstituentQuarkModel
i
X (hCQM),whichhasbeenusedforasystematiccalculationofvariousbaryonprop-
r erties,willbebrieflyreviewed.
a
2 The hypercentral model
Theexperimental4−and3−starnonstrangeresonancescanbearrangedinSU(6)−multiplets
(seeFig.1).ThismeansthatthequarkdynamicshasadominantSU(6)−invariant
part,whichaccountsfortheaveragemultipletenergies.InthehCQMitisassumed
tobe[3]
τ
V(x)=− + αx, (1)
x
wherexisthehyperradius
x= ρ~2+~λ2 , (2)
q
whereρ~and~λaretheJacobicoordinatesdescribingtheinternalquarkmotion.The
dependenceofthepotentialonthehyperrangleξ=arctg(ρ)hasbeenneglected.
λ
1
(cid:25) = 1 (cid:25) = 1 (cid:25) = (cid:0)1
[GeV]
2
F37
P3100 P3300
F35
1:8 (70;0+) (56;2+)
P1100 P13 D33D130
F15 D15
S110
1:6 P330 (70;1(cid:0)) S31
(56;0+)0 S11
D13
P110
1:4
P33
1:2
(56;0+)
1
P11
0:8
Figure1:Theexperimentalspectrumofthe4-and3-starnonstrangeresonances.Ontheleftthe
standardassignementstoSU(6)multipletsisreported,withthetotalorbitalangularmomentum
andtheparity.
Interactions of the type linear plus Coulomb-like have been used since time
forthemesonsector,e.g. theCornellpotential. Thisformhasbeensupportedby
recentLatticeQCDcalculations[6].Inthecaseofbaryonsasocalledhypercentral
approximationhasbeenintroduced[7,8],whichamountstoaverageanytwo-body
potentialforthethreequarksystemoverthehyperangleξ andworksquitewell,
speciallyforthelowerpartofthespectrum[9]. Inthisrespect, thehypercentral
potentialEq.1canbeconsideredasthehypercentralapproximationoftheLattice
QCDpotential.Ontheotherhand,thehyperradiusxisacollectivecoordinateand
thereforethehypercentralpotentialcontainsalsothree-bodyeffects.
Thehypercoulomb term1/xhasimportantfeatures[3,10]: itcanbesolved
analytically and the resulting form factors have a power-law behaviour, at vari-
ancewiththewidelyusedharmonicoscillator;moreover,thenegativeparitystates
are exactly degenerate with the first positive parity excitation, providing a good
startingpointforthedescriptionofthespectrum.
Thesplittingswithinthemultipletsareproducedbyaperturbativetermbreak-
ingSU(6),whichasafirstapproximationcanbeassumedtobethestandardhy-
perfineinteractionH [1].ThethreequarkhamiltonianforthehCQMisthen:
hyp
p2 p2 τ
H = λ + ρ − + αx+H , (3)
2m 2m x hyp
wheremisthequarkmass(takenequalto1/3ofthenucleonmass).Thestrength
2
ofthehyperfineinteractionisdeterminedinordertoreproducethe∆−N mass
difference,theremainingtwofreeparametersarefittedtothespectrum,reported
inFig.2,leadingtothefollowingvalues:
−2
α=1.16fm , τ =4.59. (4)
N D
M
@MeVD
1900
1700
1500
1300
1100
900
1+ 1- 3+ 3- 5+ 5- 1+ 1- 3+ 3- 5+ 7+
€€€€ €€€€ €€€€ €€€€ €€€€ €€€€ €€€€ €€€€ €€€€ €€€€ €€€€ €€€€
2 2 2 2 2 2 2 2 2 2 2 2
Figure2: ThespectrumobtainedwiththehypercentralmodelEq. (3)andtheparametersEq.
(4)(fulllines)),comparedwiththeexperimentaldataofPDG[11](greyboxes).
Keepingtheseparametersfixed, themodelhasbeenappliedtocalculatevar-
iousphysicalquantitiesofinterest: thephotocouplings [12],theelectromagnetic
transitionamplitudes[13],theelasticnucleonformfactors[14]andtheratiobe-
tween the electricand magnetic proton form factors [15]. Some results of such
parameterfreecalculationsarepresentedinthenextsection.
3 The results
Theelectromagnetictransitionamplitudes,A1/2andA3/2,aredefinedasthema-
trix elements of the transverse electromagnetic interaction, Ht , between the
e.m.
nucleon,N,andtheresonance,B,states:
A1/2 = hB,J′,Jz′ = 21 |Hetm|N,J = 12,Jz =−12 i
(5)
A3/2 = hB,J′,Jz′ = 32 |Hetm|N,J = 12,Jz = 12 i
Thetransitionoperatorisassumedtobe
3
e
Ht = − j (p~ ·A~ + A~ ·p~) + 2µ s~ ·(∇~ ×A~ ) , (6)
em Xj=1 (cid:20)2mj j j j j j j j (cid:21)
wherespin-orbit and higher order correctionsareneglected[16,17]. InEq. (6)
m , e , s~ , p~ and µ = gej denote the mass, the electric charge, the
j j j j j 2mj
3
spin,themomentumandthemagneticmomentofthej-thquark,respectively,and
A~ = A~ (r~)isthephotonfield;quarksareassumedtobepointlike.
j j j
TheprotonphotocouplingsofthehCQM[12](Eq.(5)calculatedatthephoton
point), in comparison with other calculations [2, 17, 21], have the same overall
behaviour,havingthesameSU(6)structureincommon,butinmanycasestheyall
showalackofstrength.
Figure 3: The helicity amplitudes for the D13(1520) resonance, calculated with the
hCQMofEqs. (3)and(4)andtheelectromagnetictransitionoperatorofEq. (6)(full
curve, [13]). The dashed curve is obtained with the analytical version of the hCQM
([10]), where the behaviour of the quark wave function is determined mainly by the
hypercoulombpotential.Thedataarefromthecompilationofref.[19]
Taking into account the Q2−behaviour of the transition matrix elements of
Eq.(5),onecancalculatethehCQMhelicityamplitudesintheBreitframe[13].
ThehCQMresultsfortheD13(1520)andtheS11(1535)resonances[13]aregiven
inFig. 3and4,respectively. TheagreementinthecaseoftheS11 isremarkable,
themoresosincethehCQMcurvehasbeenpublishedthreeyearsinadvancewith
respecttotherecentTJNAFdata[20].
Ingeneral theQ2-behaviour isreproduced, except fordiscrepancies atsmall
Q2, especially in the Ap3/2 amplitude of the transition to the D13(1520) state.
Thesediscrepancies,astheonesobservedinthephotocouplings,canbeascribed
eithertothenon-relativisticcharacterofthemodelortothelackofexplicitquark-
antiquark configurations, which may be important at low Q2 . The kinematical
relativisticcorrectionsatthelevelofboostingthenucleonandtheresonancestates
toacommonframearenotresponsibleforthesediscrepancies,aswehavedemon-
4
160 Capstick and Keister (Rel)
Old data
Armstrong et. al.
140
CLAS published
New CLAS data
120
Aiello, Giannini, Santopinto
1/2) 100
-V
e
G
3
-0 80
1
(2
1/
A
60
40
20
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Q2 (GeV2)
Figure 4: ThehelicityamplitudefortheS11(1535) resonance, calculatedwiththehCQMof
Eqs. (3)and(4)andtheelectromagnetictransitionoperatorofEq. (6)(dashedcurve,[13])and
themodelofref.[21].Thedataaretakenfromthecompilationofref.[22]
stratedinRef.[23]. Similarresultsareobtainedfortheothernegativeparityreso-
nances[13].Itshouldbementionedthatther.m.s.radiusoftheprotoncorrespond-
ingtotheparametersofEq.(4)is0.48fm,whichisjustthevalueobtainedin[16]
inordertoreproducetheD13photocoupling.Thereforethemissingstrengthatlow
Q2canbeascribedtothelackofquark-antiquarkeffects,probablymoreimportant
intheouterregionofthenucleon.
4 The isospin dependence
ThewellknownGuersey-Radicatimassformula[24]containsaflavourdependent
term,whichisessentialforthedescriptionofthestrangebaryonspectrum. Inthe
chiral Constituent QuarkModel [4,5], thenon confining part of the potential is
providedbytheinteractionwiththeGoldstonebosons, givingrisetoaspin-and
flavour-dependentpart,whichiscrucialinthisapproachforthedescriptionofthe
lowerpartofthespectrum.Moregenerally,onecanexpectthatthequark-antiquark
pair production can lead to an effective residual quark interaction containing an
isospin(flavour)dependentterm.
Therefore,wehaveintroducedisospindependenttermsinthehCQMhamilto-
nian.Thecompleteinteractionusedisgivenby[25]
Hint = V(x)+HS+HI+HSI, (7)
5
N D
M
@MeVD
1900
1700
1500
1300
1100
900
1+ 1- 3+ 3- 5+ 5- 1+ 1- 3+ 3- 5+ 7+ States
€€€€ €€€€ €€€€ €€€€ €€€€ €€€€ €€€€ €€€€ €€€€ €€€€ €€€€ €€€€
2 2 2 2 2 2 2 2 2 2 2 2
Figure 5: The spectrum obtained with the hypercentral model containing isospin dependent
termsEq.(7)[25](fulllines)),comparedwiththeexperimentaldataofPDG[11](greyboxes).
whereV(x)isthelinear plushypercoulomb SU(6)-invariant potential of Eq. 1,
whiletheremainingtermsaretheresidualSU(6)-breakinginteraction,responsible
forthesplittingswithinthemultiplets. HS isasmearedstandardhyperfineterm,
HIisisospindependentandHSIspin-isospindependent. Theresultingspectrum
for the 3*- and 4*- resonances is shown inFig. 5 [25]. The contribution of the
hyperfineinteractiontotheN −∆massdifferenceisonlyabout35%,whilethe
remainingsplittingcomesfromthespin-isospinterm,(50%),andfromtheisospin
one, (15%). It should be noted that the position of the Roper and the negative
paritystatesiswellreproduced.
5 Relativity
Therelativisticeffectsthatonecanintroducestartingfromanonrelativisticquark
model are: a) the relativistic kinetic energy; b) the boosts from the rest frames
oftheinitialandfinalbaryontoacommon(saytheBreit)frame;c)arelativistic
quarkcurrent.Allthesefeaturesarenotequivalenttoafullyrelativisticdynamics,
whichisstillbeyondthepresentcapabilitiesofthevariousmodels.
The potential of Eq.1 has been refitted using the correct relativistic kinetic
energy
τ
H = p2 +m2− + αx+H . (8)
rel ij x hyp
Xi<j q
Theresultingspectrumisnotmuchdifferentfromthenonrelativisticoneandthe
parametersofthepotentialareonlyslightlymodified.
Theboostsandarelativisticquarkcurrentexpandeduptolowestorderinthe
quark momenta has been used both for the elastic form factors of the nucleon
[14] and the helicity amplitudes [23]. In the latter case, as already mentioned,
the relativistic effects are quite small and do not alter the agreement with data
discussedpreviously. Fortheelasticformfactors,therelativisticeffectsarequite
6
1
0.8
0.6
2Q)
p(E 0.4
G
0.2
0
-0.2
0 1 2 3 4 5 6
Q2 (GeV2)
Figure6: Theelectricprotonformfactor,calculatedwiththerelativistichCQMofEq. (8)and
arelativisticquarkcurrent[27].
strongandbringthetheoreticalcurvesmuchclosertothedata;inanycasetheyare
responsibleforthedecreaseoftheratiobetweentheelectricandmagneticproton
formfactors, as ithas been shown for thefirsttimeinRef. [15], inqualitative
agreementwiththerecentJlabdata[26].
Arelativisticquarkcurrent,withnoexpansioninthequarkmomenta,andthe
booststotheBreitframehavebeenappliedtothecalculationoftheelasticform
factorsintherelativisticversionof thehCQMEq. (8)[27]. Theresultingtheo-
retical form factors of the proton, calculated, it should be stressed, without free
parameters and assuming pointlike quarks, are good (see Figs. 6 and 7), with
somediscrepanciesatlowQ2,which,asdiscussedpreviously,canbeattributedto
thelackingofthequark-antiquark paireffects. Thecorresponding ratiobetween
theelectricandmagneticprotonformfactorsisgiveninFig.8:thedeviationfrom
unityreachesalmostthe50%level,notfarfromthenewTJNAFdata[28].
6 Conclusions
The hCQM is a generalization to the baryon sector of the widely used quark-
antiquarkpotentialcontainingacoulombplusalinearconfiningterm. Thethree
freeparametershavebeenadjustedtofitthespectrum[3]andthenthemodelhas
beenusedforasystematiccalculationofvariousphysicalquantities:thephotocou-
plings[12],thehelicityamplitudesfortheelectromagneticexcitationofnegative
paritybaryonresonances[13,23],theelasticformfactorsofthenucleon[14,27]
andtheratiobetweentheelectricandmagneticprotonformfactors[15,27]. The
agreement with data is quite good, specially for the helicity amplitudes, which
arereproducedinthemedium-highQ2 behaviour, leavingsomediscrepanciesat
low (or zero) Q2, where the lacking quark-antiquark contributions are expected
7
2.2
2
1.8
1.6
1.4
2Q) 1.2
p(M
G 1
0.8
0.6
0.4
0.2
0
0 1 2 3 4 5 6
Q2 (GeV2)
Figure 7: Themagneticprotonformfactor, calculatedwiththerelativistichCQMofEq. (8)
andarelativisticquarkcurrent[27].
tobeeffective. Itshouldbenotedthatthehypercoulombterminthepotentialis
themainresponsibleofsuchanagreement[10],whileforthespectrumafurther
fundamentalaspectisprovidedbytheisospindependentinteractions[25].
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10