The internal structures of the nucleon resonances $N(1875)$ and $N(2120)$

0.09 MB·English

by  Jun He

The internal structures ofthe nucleon resonances N(1875)and N(2120)
Jun He
∗
Theoretical Physics Division, Institute of Modern Physics,
Chinese Academy of Sciences, Lanzhou 730000, China
ResearchCenterforHadronandCSRPhysics,LanzhouUniversityandInstituteofModernPhysicsofCAS,Lanzhou730000,China and
StateKeyLaboratoryofTheoreticalPhysics,InstituteofTheoreticalPhysics,ChineseAcademyofSciences,Beijing100190,China
A nucleon resonance with spin-parity JP = 3/2 and mass about 2.1 GeV is essential to reproduce the
−
photoproductioncrosssectionsforΛ(1520)releasedbytheLEPSandCLASCollaborations.Itcanbeexplained
asthethirdnucleonresonancestate[3/2 ] intheconstituentquarkmodelsothatthereisnopositiontosettle
− 3
theN(1875) whichislistedinthePDGasthethird N3/2 nucleonresonance. Aninterpretationisproposed
−
thattheN(1875)isfromtheinteractionofadecupletbaryonΣ(1385)andaoctetmesonK,whichisfavoredby
acalculationofbindingenergyanddecaypatterninaBethe-Salpeterapproach.
5
1 PACSnumbers:14.20.Gk,11.10.St,14.20.Pt
0
2
I. INTRODUCTION leased their exclusive photoproduction cross section for the
n
a Λ(1520) for energies from near threshold up to a center of
J In new versions of the Review of Particle physics (PDG) mass energy W of 2.85 GeV with large range of the K pro-

3 aftertheyear2012[1],therearefourN3/2 states, N(1520), duction angle [14]. The reanalyses about the new data in
−
N(1700), N(1875)and N(2120). Thetwo-star state N(2080) Refs. [15, 19] confirmed the previous conclusion that a nu-
h] in previousversionshas been split into a three-star N(1875) cleonresonancenear2.1GeV,N(2120),isessentialtorepro-
t and a two-star N(2120) based on the evidence from BnGa ducetheexperimentaldata[12,13].
-
l analysis[2].
c
UsuallytheN(1520)andtheN(1700)areassignedtostates
u II. ROLEOFTHEN(2120)INTHEΛ(1520)
n with orbital angular momentum L = 1 in quark model, and
PHOTOPRODUCTION
[ mixing effect is very important to explain the decay pattern

of these states [3]. The situation for the internal structures
1 Inthefollowingitwill beshownwhythe N(2120)should
of two N3/2 states with higher mass, the N(1875) and the
v −
beassignedtothethirdstate[N3/2 ] intheconstituentquark
2 N(2120),ismuchlessunclear. In quarkmodel,the N(1875) − 3
2 andthe N(2120)areinthemassregionof N = 3bandstates model in line with the theoretical framework in Ref. [15].
There are five N3/2 states in N = 3 band, of which the
5 ofwhichthemassesanddecaypatternswerepredicted[4,5]. −
0 However,the explicitcorrespondencebetweenpredictedand radiative and Λ(1520)K decay amplitudes were predicted in
0 observed states is unclear. In Large N QCD, the third and Refs.[4,5]aslistedinTable.I.
c
.
1 fourthN3/2−stateshavemasses2101 14and2170 42MeV,
± ±
0 respectively[6]. KlemptandothersclaimedthattheN(1875)
TABLEI:TheN3/2 nucleonresonancesandtheirdecayamplitudes
15 iMsethVemwiitshsinogrbtihtiradngNu(l3a/r2m−)osmtaetnetuinmmLass=re1giaonnd1r8a0d0ia−l 1ex9c0i0- pamrepdliicttueddeisnAtherelaatn−ivdisptaicrtqiaulawrkavmeoddeecla[y4,a5m].plTithuedmesaGss(ℓm)Ra,rheeilnictihtye
: 1/2,3/2
v tation number N = 1 [7], which is also supported by the unitofMeV,10 3/√GeVand √MeV,respectively.
−
Xi Asodns/bQeCtwDee[n8]p.reTdhiecitredcoanncdluosbiosenrvisedonmlyasbsaesse.dAosneanlcigohmtepnaerid- State mR A1p/2 A3p/2 G(ℓ1) G(ℓ2)
ar by Isgur, “in a complex system like the baryon resonances, [N3/2−]3 1960 36 -43 −2.6 −0.2
predictingthespectrumofstatesisnotaverystringenttestof [N3/2−]4 2055 16 0 −0.5 0.0
amodel”[9]. Decaypatternprovidesmoreinformationabout [N3/2−]5 2095 -9 -14 0.4 0.0
hadroninternalstructure. [N3/2−]6 2165 −− −− 0.4 0.0
Many analyses suggested that a N3/2− state with mass [N3/2−]7 2180 −− −− 1.1 0.1
about2.1GeVisessentialtoexplainexperimentalresults[10–
13]. Before the year 2012, it is related to the only N3/2 The predicted radiative and strong decay amplitudes sug-
−
state listed in the PDG with mass higher than 1.8 GeV, the gesttheimportanceofthenucleonresonance[N3/2−]3inthe
N(2080),andexplainedasthethirdstate[N3/2 ] predicted Λ(1520) photoproduction, which is the first state in N = 3
− 3
in theconstituentquarkmodel. Forexample,the N(2080)is bandstatesandthethirdstateinallnucleonresonanceswith
foundtoplaythemostimportantroleinthephotoproduction JP =3/2−predictedintheconstituentquarkmodel.
of Λ(1520) off proton target [12, 13]. Recently, the CLAS InRef.[15],basedonthehighprecisionexperimentaldata
Collaboration at Jefferson National Accelerator Facility re- releasedbytheCLASandLEPSCollaborationsrecently,the
interactionmechanismofthephotoproductionofΛ(1520)off
aprotontargetisinvestigatedwithinaRegge-plus-resonance
approach. Theinclusionofthe N(2120)asstate[N3/2 ] in
− 3
Electronicaddress:[email protected] theconstituentquarkmodelreducedtheχ2 obviously. Inthat
∗
2
work, mass and width are fixed at 2.12 GeV and 0.33 GeV, III. THEN(1875)ASAΣ(1385)K BOUNDSTATE
respectively. Here, a mass scan is made for the N(2120) by
fittingthedatafromtheCLASandLEPSCollaborations.Ex-
Nowthatthe N(1875)cannotbeexplainedintheconven-
ceptmass,thewidthΓ ,whichwasfixedat0.33GeVinpre-
R tional quark model, it may be a exotic hadron. In meson
viouswork[15],isalsosetasafreeparameter. Thebehavior
sector, some of particles which can not be explained in the
ofχ2ispresentedinFig.1.
quark model framework, such as XYZ particles observed in
recent years, have been suggested to be hadronic molecular
states. The lightscalarsa (980), f (980)and f (500)areof-
0 0 0
300
tenconsideredasmeson-mesonresonances.Inbaryonsector,
N(2120) as [N3/2−] in CQM someauthorsproposedthattheΛ(1405)maybeexplainedas
250 3
N(2120) as [N3/2−] in CQM a NK¯ boundstate[16–18]. ThemassoftheN(1875)isclose
4
200 to the Σ(1385)K threshold, which encouragesus to interpret
N(1875)asaboundstateofΣ andK(hereandhereafterIde-
∗
150 noteΣ(1385)asΣ ). Assaidabove,theinternalstructureofa
2 ∗
hadroncannotbejudgedonlythroughitsmass. Inthiswork,
dc 100 both mass and decay pattern of N(1875)as a bound state of
Σ andKwillbecalculatedwithamethoddevelopedbasedon
50 ∗
thecovariantspectatorformalismoftheBethe-Salpeterequa-
tion [20–25], which has been used to study the BB¯ and the
0 ∗
DD¯ systems.
∗
2.05 2.1 2.15 2.2 2.25 AnalogoustoRef.[24,25],withhelpofonshellnessofthe
heavy constituent 1, Σ , the numerator of propagator Pµν is
∗ 1
MN3/2− (GeV) rewritten as λuµ1λu¯ν1λ with uµ1λ being the Rarita-Schwinger
spinorwithhelicityλ. Theequationforvertexisinaform
P
FIG.1: (Color online) Thechange of χ2 inmassscan. Thesolid
and dashed lines are for the results with assuming the N(2120) as Γ = G Γ , (1)
λ λλ 0 λ
| i V ′ | ′i
state[N3/2−]3andwithassumingtheN(2120)asstate[N3/2−]4,re- Xλ′
spectively.
wpriothpa|gΓaλtior=Gu¯µλf|oΓrµµp′airutµRi′claen1daVndλλ2′ w=ithu¯µλmVaµsνs′umνλ′′.anTdhemrewstriot-f
0 1 2
tendowninthecenterofmassframewhereP=(W,0)is
Here the results with assuming the N(2120) as state
[N3/2−]3andwithassumingtheN(2120)as[N3/2−]4arepro- δ+(k2 m2) δ+(k0 E (k))
vided.IftheN(2120)isassumedtobethethirdstate[N3/2 ] G =2πi 1− 1 =2πi 1− 1 ,(2)
intheconstituentquarkmodel,thechangeofχ2willdecrea−se3 0 k22−m22 2E1(k)[(W−E1(k)2−E22(k)]
andreachminimumat2.13GeVwiththeincreaseofmass. If
assumedtobethefourthstate[N3/2−]4,thechangeofχ2keep where k1 = (k10,k) = (E1(k),k), k2 = (k20,−k) = (W −
stable around 150, which means that the experimental data E (k), k)withE (k)= m2 + k2.
can not be well reproduced. Obviously, the N(2120) should 1 − 1,2 q 1,2 | |
Theintegralequationcanbewrittenexplicitlyas
beassignedasstate[N3/2 ] insteadofstate[N3/2 ] inthe
− 3 − 4
constituent quark model. Since the N(1875) is much lower
(W E (k) E (k))φ (k)
than the N(2120), it is unnaturalto assign it to the fourthor − 1 − 2 λ
dk
higher states. The first and second states in the constituent = ′ V (k,k,W)φ (k), (3)
quarkmodelhavebeenassignedtothefour-starN(1520)and Xλ Z (2π)3 λλ′ ′ λ′ ′
′
thethree-starN(1700)inthePDG,whichhasbeenconfirmed
by manyexperimentaland theoreticalevidences[1]. Hence, with
there is no position to settle the N(1875) in the constituent
quarkmodel. Vλλ(k,k′,W)= iV¯λλ′(k,k′,W) , (4)
′ 2E (k)2E (k)2E (k)2E (k)
Thestate[N3/2−]3predictedintheconstituentquarkmodel 1 2 1′ ′ 2′ ′
is much lower than the N(2120) even if model uncertainty, p
about100MeV,isconsidered. Itiswell-knownthatloopef- where the reduced potential kernel ¯λλ = F(k) λλF(k′)
V ′ V ′
fectwillleadtomassshift. Thestate[N3/2−]3hasalargede- witha factoras F(k) = √2E2(k)/(W−E1(k)+E2(k)). The
caywidthinΛ(1520)Kchannelaspredictedintheconstituent normalized wave function can be related to vertex as φ =
quarkmodel[4,5].Moreover,theΛ(1520)Kthresholdisnear Nψ = NF−1G0 Γλ with the normalization factor N(|ki) =
| i | i
baremass ofstate [N3/2 ] . Hence, the largedifferencebe- 2E (k)E (k)/(2π)5W.
− 3 1 2
tweenbaremassandobservedmasscanbeexplainedbyboth pSince K is a pseudoscalar particle, it is forbidden to ex-
uncertaintyoftheconstituentquarkmodelandmassshiftaris- change pseudoscalar meson between K and Σ . The vector
∗
ingfromtheΛ(1520)Kloopeffect. mesonexchanges,ρ, ωandφ,isdominantintheinteraction.
3
Thepotentialkernel canbeobtainedfromtheeffectiveLa- a) σ b) ρ
V K K
grangiansdescribingtheinteractionsforvectormesonsVwith
K andΣ , N∗ N∗
∗ K K
= ig K Vµ∂ K, (5)
LKKV KKV † µ κ Σ∗ N Σ∗ N
= g Σ [γν Σ∗Σ∗Vσµρ∂ ]VνΣ . (6)
LΣ∗Σ∗V Σ∗Σ∗V ∗µ† − 2mΣ ρ µ c) ω d) K
K K
Inthisworktheisospinstructuresarefollowingthestandard
form in Ref. [26] and omitted in the Lagrangians. The cou- N∗ K N∗ ρ
plingconstantsforvectormesonsρ, ωandφ interactedwith
iKnaqnudarΣk∗mcaondbeel[o2b7ta]ianneddfrreolamtiognρπsπg=6.1=99gand/f2ρ∆=∆ =g−4.3=0 Σ∗ N Σ∗ Λ,Σ
KKρ ρππ KKω
√2g = g /2andg = g = g /√2 = g
undeKrKSφU(3)ρπsπymmetry.Σ∗ΣH∗ρere −diffΣe∗Σre∗ωnt defiΣn∗Σi∗tφions betwe∆e∆nρ e) K π f) Σ∗ π
Ref. [27] and this work have been considered. Since the N∗ N∗
constituent 2 is off shell, a monopole form factor is intro- K∗ Σ,Λ
ducedat the vertexfor each off-shellkaon meson with mass
mthKe eaxschh(akn2g)ed=mΛe4s/o[n(mw2Kit−hkm2)a2s+s Λm4].isTchheosfeonrmasfafc(tqo2r)fo=r Σ∗ N K N
V
(Λ2 m2)/(Λ2 q2). EmpiricallythecutoffΛshouldbenot FIG.2: (Color online) Thedecays of theΣ K bound state. a) Nσ
− V − ∗
farfrom1GeV. channelwithK exchange. b) NρchannelwithK exchange. c) Nω
The 3-dimensional equation can be reduced to a one- channel with K exchange. d) KΛ/Σ channel with ρ exchange. e)
dimensionalequationwithpartialwaveexpansion. Thewave and f) are for Nπ decay channel with K∗ exchange and with Λ/Σ
functionhasanangulardependentas exchange,respectively.
2J+1
φ (k)= DJ (φ,θ,0)φ (k), (7)
λ r 4π λR∗,λ λ,λR | | constituents as shown in Fig. 2 are taken as the main decay
channelsoftheΣ K boundstate.
whereDJ (φ,θ,0)istherotationmatrixwithλ beingthehe- ∗
λR∗,λ R Thedecayamplitudescanbewrittenas
licityofboundstatewithangularmomentumJ. Thepotential
afterpartialwaveexpansionis
= A G Γ = A FN 1φ
λ 0 λ λ − λ
M | i | i
Xλ Xλ
VλJλ′(|k|,|k|′)=2πZ dcosθk,k′dλJ,λ′(θk,k′)Vλλ′(k,k′), (8) ≡ Z d4kδ((2kπ10)−4 E1)FN((|kk|)φλ(k)Aλ,λ′1λ′2(k,k′),(10)
where θ is angle between k and k. The one-dimensional Xλ | |
k,k ′
′
integralequationreads
where λ are helicities for two final particles and k and k
′1,2 ′
(W E (k) E (k))φJ(k) arethemomentaforΣ∗ andfinalmesoninthecenterofmass
− 1 | | − 2 | | λ | | frame. A istheamplitudesfortwoconstituentsΣ andK
k 2d k λ,λ′λ′2 ∗
= Xλ Z | (′2|π)|3′|VλJλ′(|k′|,|k′|)φλJ′(|k′|). (9) wtoatvweofufinncatliopnarφticalneds,GN0πh,aNvσe baenednsuoseodn.inTthheeddeefirinviatitoionnsooff
′ Eq. (10). The normalizationof wave functionφ insures that
To study the decay property of a bound state, the infor- thereisnofreetotalfactorinourcalculationofamplitude.
mation about coupling of a bound state to its constituents is Besides the Lagrangians in Eq. (4), the following La-
essential. In literatures it is often achieved with the method grangiansareusedtocalculatetheamplitudesA ,
proposedbyWeinberg[28,29]. Inthiswork,thevertexwave λ,λ′λ′2
function, which contains the information about coupling of = g 2m ∂ K ∂µKσ, (11)
KKσ KKσ π µ †
boundstatetoitsconstituents,isobtainedduringsolvingthe L
= ig K µ(π∂µK ∂µπK), (12)
binding energy. It make a study of the decay pattern of the LK∗Kπ K∗Kπ ∗ −
f
Σ∗KSinbcoeuntwdost-abtoedpyosdseibclaey. of a molecular state occurs only LKNY = mNK+NYmYN¯γµγ5Y∂µK+H.c., (13)
through hadron loop mechanism, it is suggested that three- = fPBΣ∗∂ KΣ¯ µN+H.c., (14)
body decay may be larger than two-body decay [30]. As LPBΣ∗ m µ ∗
P
shown in Refs. [30–32], it was found that three-body decay f
= i VBΣ∗Σ¯ µγνγ [∂µρ ∂ ρµ]B+H.c., (15)
haspositivecorrelationtothedecaywidthoftheconstituents. VBΣ ∗ 5 ν ν
L ∗ − m −
V
Hence, the three-bodydecay of the bound state Σ K is sup-
∗
presseddueto thesmalldecaywidthofΣ . Inthisworkthe where PB means KN, πΛ or πΣ, VB means ρΛ, ρΣ or K N
∗ ∗
two-bodydecaysthroughexchanginga particlebetweentwo andY meansΣ or Λ. Thecouplingconstantsare adoptedas
4
g2 /4π = 0.25 [33], g = 3.23, f = 13.24 and changeispronetohappenintheboundstate. Comparedwith
KKσ K∗Kπ − KNΛ
f =3.58[34]. ThecouplingconstantsaboutΣ canbeob- thePDGandBnGavaluesaboutmassandtotalwidth,thebest
KNΣ ∗
tainedthrough f = 1.27, f = 3.22[34]and f = cut off Λ 1.80 GeV is reasonable and consentient to the
ΣΛπ KNΣ ∆ρN
∗ ∗ − ≈
6.08 [27] with the SU(3) symmetry relations f /m = valueintheliterature[35]. ThebranchratiosofN(1875)are
πΣΣ π
− ∗
−√13 fπmΛπΣ∗, fρmΛρΣ∗ = √12f∆ρN/mρ, fρmΣΣρ∗ = −√16f∆ρN/mρ and smtaobslteimcopmorptaanretddewciatyhcbhinadninnegl,eanbeorugty5.5T%he,wNhσichchiasncnoenlsiisstethnet
fKm∗TKN∗Σh∗e=Σ−∗ c√1a6rfr∆ieρNs/smpiρn.-parity JP = 3/2+ and isospin I = 1. twhiethBtnhGeaPDanGalsyusgisg.eTstheedmvaaliunesde2c4a±y2ch4a%nnaenldo6f0th±e1[2N%3/f2ro−m]3
A system composed of Σ∗ and kaon carries I = 1/2 or 3/2. predictedintheconstituentquarkmodelisNρwhichismuch
In this work, all states with J 3/2 are considered and the largerthanotherdecaychannels.Itconflictwithboththeval-
≤
rangesof the cutoffs in form factors are chosen as 1 < Λ < ues suggested by the PDG and these obtained by the BnGa
5 GeV. The bound state solutions with the binding energies analysis. Hence, the decay pattern of N(1875) disfavors the
E =m1+m2−W arelistedinTableIIandcomparedwiththe assignmentas[N3/2−]3.
valuesfromthePDGandtheBnGagroups[1,2].
TABLEII:ThebindingenergiesEforΣ Ksystemwithdifferentcut IV. SUMMARY
∗
offΛThecutoffΛ,bindingenergyandbranchratioareintheunits
ofGeV,MeVand%,respectively.
In this work,the internal structures of the (1875) and the
N(2120)areinvestigated. Theexperimentaldataforthepho-
Λ E Γ Nσ Nρ Nω Nπ ΛK ΣK toproductionofΛ(1520)offprotonreleasedbytheCLASand
1.68 3 41 55.9 4.7 14.1 22.4 2.3 0.6 LEPSCollaborationssuggesttheexplanationofthe N(2120)
1.72 8 73 55.8 4.7 14.0 22.6 2.3 0.6 as the third state with JP = 3/2 in the constituent quark
−
1.76 16 111 55.7 4.7 14.0 22.7 2.2 0.6 model. The N(1875) is explained as a bound state from the
1.80 28 155 55.6 4.8 14.2 22.8 2.1 0.5 interactionofΣ∗ andkaon,whichissupportedbythenumer-
1.84 44 204 55.3 4.9 14.6 22.7 2.0 0.5 ical results of both binding energy and decay pattern of the
bound state of Σ K system with isospin I = 1/2 and spin-
1.88 67 257 54.9 5.1 14.9 22.9 1.8 0.4 ∗
parityJP =3/2 .
1.92 100 312 53.6 5.1 14.7 24.8 1.5 0.3 −
PDG[1] 30+25 24+24 6+6 20+4 7+6 0.7+0.4
25 24 6 4 6 0.4
BnGa[2] 0+−20 200+20 60−+12 − − 3+−2 4+2 15−+8
20 20 12 2 2 8
[N(23−)]3 -−85 3−24 − 57.1 12.3 20−.8 9−.7 −0 Acknowledgments
This project is partially supported by the Major State
OnlyoneboundstatesolutionwithI =1/2andJP = 3/2 Basic Research Development Program in China (No.
−
isfoundfromtheinteractionofΣ andK.Thedecaywidthbe- 2014CB845405),theNationalNaturalScienceFoundationof
∗
comeslargerwithincreaseofthebindingenergy. Itisunder- China(GrantsNo.11275235,No.11035006)andtheChinese
standablebecausethelargebindingenergymeansthatthedis- AcademyofSciences(theKnowledgeInnovationProjectun-
tancebetweentwoconstituentsissmallersothatthequarkex- derGrantNo. KJCX2-EW-N01).
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