Effects of a spin-flavour dependent interaction on light-flavoured baryon helicity amplitudes

0.66 MB·English

by  Michael Ronniger

EPJ manuscript No.
(will be inserted by the editor)
Effects of a spin-flavour dependent interaction on light-flavoured
baryon helicity amplitudes
Michael Ronnigera and Bernard Ch. Metsch
Helmholtz Institutefu¨r Strahlen– undKernphysik(Theorie), Universit¨at Bonn, Nußallee 14-16, D-53115 Bonn, Germany
July 12, 2012
2
1 Abstract This paper is a continuation of previous work about the effects of a phenomenological flavour
0
dependentforceinarelativisticallycovariantconstituentquarkmodelbasedontheSalpeterequationonthe
2
structureoflight-flavouredbaryonresonances.Herethelongitudinalandtransversehelicityamplitudesas

l studiedexperimentallyintheelectro-excitation ofnucleon-and∆resonancesarecalculated.Inparticular
u
theamplitudesfortheexcitationofthreeandfourstarresonancesascalculatedinapreviousmodelAare
J
compared to those of the novel model C as well as to existing and partially new experimental data such
1
ase.g. determinedbytheCB-ELSAcollaboration. Abriefdiscussion of someimprovementstomodelC is
1
given after the introduction.

]
h PACS. 11.10.St Boundandunstablestates;Bethe-Salpeterequations–12.39.Ki Relativisticquarkmodel
p – 13.40.Gp Electromagnetic form factors – 13.40.Hq Electromagnetic decays
-
p
e
h 1 Introduction vector coupled Yukawa potential, where the tensor force
[ terms wereneglected. In[1] we founda phenomenological
This paper is a continuation of our previous work [1] on Ansatz whichincludesashortrangedflavour-singletanda
1
v thedescriptionofbaryonresonancesinacovariantBethe- flavour-octet exchange with pseudoscalar-like coupling of
0 Salpeter framework. While in [1] we concentrated on the Gaussian form to be most effective for the improvements
4 description of the mass spectrum in the present paper we mentioned above. This newly introduced interaction ker-
6 discuss the results on electromagnetic transition ampli- nelincreasedthenumberofparametersofseveninthefor-
2 tudesobtainedonthebasisoftheSalpeteramplitudesde- mer model (Model , see [4,5]) to ten in the new version
A
7. terminedpreviously.Thedynamicalingredientsoftherel- (ModelC,see[1]),whichwestillconsidertobeacceptable
0 ativisticallycovariantquarkmodelusedareinstantaneous in view of the multitude of baryon masses described ac-
2 interaction kernels describing confinement, a spin-flavour curately in this manner. For the values of the parameters
1 dependent interactions kernel motivated by instanton ef- we refer to table 1, where we listed an improved set of
v: fects as well as in addition a phenomenologically intro- parametersfortheinteractionkernelsofmodelC together
i ducedflavourdependentinteraction.Thelatterwasfound with the values used in [1] displayedin brackets.Likewise
X to improve the descriptionobtained previously in [3–5] in the parameters for the interactionkernels of model ob-
A
r particular of Roper-like scalar excitations as well as the tained from calculations within a larger model space, see
a positionofsomenegativeparity∆-resonancesslightlybe- also [1] are listed along with the values (in brackets) as
low 2 GeV. With instantaneous interaction kernels the determinedin[4,5]insmallermodelspaces.Notethatthe
Bethe-Salpeter equation reduces to the more tractable two models and , specified in table 1 employ different
A C
Salpeter equation which can be cast in the form of an confinementDirac-structuresfortheconstantpart(offset)
eigenvalue equation for the masses and the Salpeter am- Γ0 andthe linearpart(slope)Γs (see [1,4]formoreinfor-
plitudes. Thesein turndetermine the vertexfunctions for mation). All calculations in the present paper are based
any on-shell momentum of the baryons which then enter on the parameter values of table 1.
the electromagnetic current matrix elements. The details Furthermore we want to point out, that the calcu-
of this procedure can be found in [2]. lation of helicity amplitudes or transition form factors
Flavour dependent effective quark-quark interactions (such as that for the nucleon-∆(1232) magnetic transi-
havealsobeenstudiedpreviouslybytheGrazgroup[9–17] tion) in lowest order as considered here does not intro-
whichobtainedverysatisfactoryresultsforthemassspec- duce any additional parameter in the underlying mod-
tra up to 1.7 GeV as well as for the corresponding nu- els as discussed in [1] before. Since in the new model
C
cleon form factors on the basis of a truncated pseudo- we can account for more baryon excitations accurately
we can now also offer predictions for some ∆-resonances
a e-mail: [email protected] which could not be covered before in model . In partic-
A
2 M. Ronniger et al.:Effects of a spin-flavourdependentinteraction on helicity amplitudes
Table 1. Model parameter values for the novel model C in 2 Improvements to model
C
comparison tothoseofmodelA[4,5].Thebracketednumbers
in the column for model A are the parameters as found in [4, In the course ofthe investigationswithin the novelmodel
5] and the numbers above them are recalculated with higher
[1]a new parameterset wasfound which lead to an im-
numerical accuracy as commented in [1]. The numbers in the C
proved description in particular of the nucleon form fac-
column for model C represent an improved set with respect to
tors.This new set of parametersis listed in table 1 of the
thevaluesquotedin[1]whicharelistedinbrackets.Notethat
introduction. The corresponding baryonmass spectra are
the Dirac structure for the confinement interaction kernel is
verysimilartothosepublishedin[1];onlyforsomehigher
different in models A and C, see also [1].
excitationdeviationsupto30MeVwithrespecttotheval-
parameter model C model A uespresentedin[1]werefound.We thereforerefrainfrom
displayingthemassspectrahere.Thecalculatedmassesof
masses m [MeV] 350.0
n 330.0 those baryons which enter the helicity amplitudes calcu-
[325.0]
latedinthisworkcanbefoundintables2and3.However
m [MeV] 625.0
s 670.0 fornucleonformfactorswhich werealsogivenin[1]some
[600.0]
small but partially significant modifications were found
-370.8 -734.6 and the results will be discussed subsequently.
confinement a [MeV]
[-366.8] [-700.0] InFig.1thecalculatedelectricprotonformfactor(di-
b [MeV/fm] 208.4 453.6 vided by its dipole-shape)
[212.86] [440.0]
instanton gnn [MeV fm3] [334117.5.9] [113360.0.3] GD(Q2)= (1+Q12/M2)2 , (1)
V
260.0 81.8
induced g [MeV fm3]
ns [273.6] [96.0] with M2 = 0.71GeV2 (see [27,28]) in both versions of
interaction λ [fm] 0.4 0.4 V
model is compared to experimental data.
C
oexctcehtange 4gπ82 [MeV fm3] [10101.886.0] – deItscirsibfoeusntdhethdaattawiftohr tmhoemneenwtasettraonfspfearrsamQe2te.rs3mGoedVe2l
C
singlet g02 [MeV fm3] 1715.5 – slightlybetterthanwiththeoldersetof[1],butthemod-
exchange 4π [1897.4] ification is rather small. Fig. 2 shows the electric neu-
λ8 =λ0 [fm] 0.25 –
2.5
MMD[27]
Christy[29]
2 Qattan[30]
∆ula3/r2−th(1es9e40a)rea:n∆d1∆/25+/(21−7(5109)3,0)∆a3/s2+re(p16o0rt0e)d, ∆in1/[21−].(1A9d0d0)i-, 2Q) improvmedodmeoldCel[1]
tionally there now exists new data for photon decay am- (D 1.5 C
plitudes fromAnisovichet al.[6–8]andforhelicityampli- G
/
tudes from Aznauryan et al. [18–21] as well as the MAID 2)
Q 1
analysis[22,23]withinformationalsoonlongitudinalam- (
plitudes which can serve as a test of the present model pGE
beyond the comparison done previously in [2,24,25] on 0.5
the basis of the amplitudes determined in model of [4].
A
For the definition of the helicity amplitudes we use the
conventions as in Capstick and Keister [26] as mentioned 0
0 1 2 3 4 5 6
in Eqs. (7a) to (7c) in the subsequent sec. 3. Q2 [GeV2]
Fig. 1. The electric form factor of the proton divided by
The paper is organised as follows: After a brief re-
the dipole form G (Q2), Eq. (1). MMD-Data are taken from
capitulation on some improvements concerning model D
C Mergell et al. [27], supplemented by data from Christy et
in sec. 2 and the determination of the helicity ampli-
al. [29] and Qattan et al. [30]. The solid line represents the
tudesforelectro-excitationintheSalpetermodel,weshall
resultsfrommodelC withthenewsetofparameters,whilethe
present the results in sec. 3, which contains three subsec-
dashedlinethose from model C [1]. Reddatapointsaretaken
tions: Sec. 3.1.1, covers the helicity amplitudes for the from polarisation experimentsand black ones are obtained by
electro-excitation of nucleon resonances, sec. 3.1.2 con- Rosenbluthseparation.
tains the helicity amplitudes for the electro-excitation of
∆-resonances, while sec. 3.2 summarises the photon de-
cay amplitudes. Sec. 3.3 contains a short discussion of tron form factor where the effect of the new parameter
the magnetic and electric transition form factor of the set yields a significantly improved description, reflecting
∆(1232)resonancebeforeweconcludewithasummaryin the fact that this small quantity is thus very sensitive to
sec. 4. parameterchanges.Although the deviationbetween both
M. Ronniger et al.:Effects of a spin-flavourdependentinteraction on helicity amplitudes 3
curves could be interpreted as an estimate of the uncer- Also for the magnetic form factors displayed in Fig. 3
tainty in the model prediction the new version demon- and4improvementsareobserved:this concernsinpartic-
strates that within the model it is indeed possible to ac- ularthe descriptionofthe magneticprotonformfactorat
countforthemomentumtransferdependenceandthepo- higher momentum transfers Q2 &1GeV2.
sitionof the maximumrather accurately.The new results
2
MMD[27]
0.14 1.8 Anklin[44]
MMD[27] Xu[45]
n
0.12 [31–35] µ 1.6 Kubon[46]
/
0.1 mod[e3l6–4[21]] 2Q) 1.4 AMlaarcdoeny[[4329]]
C (
0.08 improvedmodelC GD 1.2
2(Q) 0.06 2Q)/ 1
nGE 0.04 nG(M 0.8 model [1]
C
0.02 improvedmodel
0.6 C
0
0.4
0 1 2 3 4 5 6
-0.02 Q2 [GeV2]
0 1 2 3 4 5 6
Q2 [GeV2] Fig. 4. The magnetic form factor of the neutron divided by
thedipoleform G (Q2),Eq.(1)andthemagnetic momentof
D
Fig.2. Theelectricformfactoroftheneutron.MMD-Dataare the neutron µ = −1.913µ . MMD-Data are taken from the
n N
takenfromthecompilationofMergelletal.[27].Thesolidline compilationbyMergelletal.[27]andfrommorerecentresults
represents the results from the improved model C; the dashed fromMAMI[44,46].Additionally,polarisationexperimentsare
line is the result from model C [1]. Red data points are taken markedbyreddatapoints.Theblackmarkedonesareobtained
from polarisation experiments and black ones are obtained by byRosenbluth separation.
Rosenbluth separation.
are very similar to the results from the Bhaduri-Cohler-
3
Nogamiquark model quotedas BCN in [15], whereas the Amaldi[47]
older version from [1] is closer to the result quoted as Brauel[48]
2.5 Bloom[49]
GBE-model in [15].
A D.Guerra[50]
g
)/ 2 Joos[51]
2Q Baker[52]
A(D
G 1.5
MMD[27] /
1.4 ) Miller[53]
Bartel[43] 2
Q 1 Kita.[54,55]
Christy[29] (
/µp 1.2 Qattan[30] 3GA mAolldaesila[5[16]]
2) 0.5 impr.modCel
Q C
(
D 1
G 0
/ model [1] 0 1 2 3 4 5 6
2Q) 0.8 improvedmodCel Q2 [GeV2]
C
(
pM Fig.5.Theaxialformfactorofthenucleondividedbytheax-
G
0.6 ialdipoleform inEq.(2)and theaxialcouplinggA =−1.267.
The solid line is the improved result of model C, the dashed
line the result of model C in [1]. Experimental data are taken
0.4 from thecompilation by Bernard et al. [57].
0 1 2 3 4 5 6
Q2 [GeV2]
Fig. 3. The magnetic form factor of the proton divided by The axial form factor divided by its dipole-shape
tohfethdeipporloetofonrmµpG=D2(.Q7923),µENq..M(1M) Dan-Ddatthaeamreagtankeetnicfmroommtehnet GAD(Q2)= (1+Qg2A/M2)2 , (2)
compilation of Mergell et al. [27]. Additionally, polarisation A
experimentsaremarkedin red.Theblackmarked datapoints with the parameters M = 1.014 0.014GeV and g =
A A
±
are obtained by Rosenbluthseparation. 1.267 taken from Bodek et al. [28] is shown in Fig. 5.
4 M. Ronniger et al.:Effects of a spin-flavourdependentinteraction on helicity amplitudes
Hereaclearpreferenceforeithersetofparameterscannot whereN denotesthegroundstatenucleonN =(p,n)and
be inferred; in general the description is satisfactory in Q2 = (P¯ P¯)2 the momentum transfer. Furthermore
f i
− −
both versions. In subsequent sections all calculations will α := e2 denotes the fine structure-constant. Note, that
use the parameters of the improved model and shall be the ini4tπial baryon mass is denoted by M , the final one
C i
quoted simply as model . by M and that the normalisation of the Salpeter ampli-
C f
tudes entersin the pre-factorsofEqs.(7a),(7b) and(7c).
The amplitudes at the photon point Q2 = 0 correspond
3 Helicity amplitudes and transition form
to the photon decay amplitudes. The longitudinal ampli-
factors from the current matrix elements tude defined in Eq.(7c), as mentionedin [26],canalsobe
redefined as
Followingtheelaborationonthetransitioncurrentmatrix
elements in Merten et al. [2] one finds in lowest order for SN(Q2) =|Pf|CN(Q2), (8)
a initial baryon state with four-momentum P¯i = Mi = 12 Q 12
| |
(M ,0) in its rest frame and a final baryon state with
i
four-momentum P¯f the expression where P2 = (Mi2−Mf2−Q2)2 + Q2 is the three momen-
f 4M2
i
d4p d4p tum of the virtual-photon in the c.m. frame as defined
hP¯f|jµ(0)|Mii=−3 (2π)ξ4 (2π)η4Γ¯PΛ¯f pξ,pη−32q in [26] and [2]. Furthermore hp, P¯f,21|j0E(0)|p, Mi,−12i is
Z Z normalised to +1 at Q2 =0.
S1 1M +p +1p S2 1M p +(cid:0)1p (cid:1)
F 3 i ξ 2 η ⊗ F 3 i− ξ 2 η
⊗S(cid:0)F3 13Mi−pξ−pη(cid:1)+q qγ(cid:0)µSF3 13Mi−pξ−(cid:1)pη 3.1 Helicity amplitudes for electro-excitation
ΓΛ (p ,p ) , (3)
(cid:0) (cid:1)Mi ξ (cid:0)η (cid:1)
b
where the so-called vertex-function ΓΛ (p ,p ) is given Inthelastdecadenewexperimentswereperformedatthe
Mi ξ η Jefferson-Laboratoryinordertostudyhelicityamplitudes
in the rest frame by
upto6GeV2.Thesenewexperimentsweredesignedtode-
ΓΛ (p ,p ):= i dp′ξ dp′η termine the helicity amplitudes for the electro-excitation
Mi ξ η − (2π)4 (2π)4 of the P11(1440), S11(1535) and D13(1520) resonances.
Z Z The results can be found in [18–20] and [22]. In addi-
VΛ(3) pξ,pη;p′ξ,p′η +VΛeff pξ,pη;p′ξ,p′η tion novel data for the longitudinal S1N/2 amplitudes were
h (cid:0) (cid:1)ΦΛ p(cid:0),p , (cid:1)i (4) obtained.
Mi ′ξ ′η We calculated the corresponding helicity amplitudes
and where the Salpeter-amplitud(cid:0)e ΦΛ (cid:1)is normalised to of these and other states on the basis of the Salpeter am-
√2Mi.Foranarbitraryon-shellmomeMnitumP¯f withP¯f2 = apbliotuvdeewseobatraeinneodwinatbhlee ntoovseollmveodtheel Cei[g1e]n.vAalsume epnrtoibonleemd
M2 the vertex function ΓΛ p ,p 2q is obtained from
f P¯f ξ η−3 with higher numerical accuracy by an expansion into a
ΓΛ (p ,p ) by applying a boost. largerbasiswhichpresentlyincludesallthree-particlehar-
Mf ξ η (cid:0) (cid:1)
The electromagnetic current operator is then defined monic oscillatorstates up to an excitationquantumnum-
as ber Nmax = 18, whereas previously [2,4,5] the results
for baryon masses and amplitudes in model were ob-
jµE(x)= :Ψ¯(x)qˆγµΨ(x): (5) tained with Nmax =12. For comparisonand tAo study the
effects of the newly introduced phenomenologicalflavour-
in terms of the charge operator qˆ and the quark field-
dependentinteractionofmodel wethusalsorecalculated
operator Ψ(x). With the definitions from Capstick and C
the spectrum and the amplitudes for model within the
Keister [26] A
same larger model space. The corresponding changes in
jE(x)= jE(x) ijE(x), (6) the determination of the interaction parameters are indi-
± 1 ± 2 cated in table 1.
the transverse and longitudinal helicity amplitudes AN ,
1/2
AN and CN , respectively, are related to the transition
3/2 1/2 3.1.1 Helicity amplitudes for nucleons
current matrix elements via
πα We will now turn to the discussion of N N helicity
AN(Q2)= N,P¯ ,1 jE(0)B,M , 1 , (7a) →
12 s2Mf(Mi2−Mf2)D f 2(cid:12) + (cid:12) i −2E amplitudes for each angular momentum J and parity π.
(cid:12) (cid:12)
πα (cid:12) (cid:12)
AN(Q2)= N,P¯ ,3 jE(0)B,M ,1 , (7b)
32 s2Mf(Mi2−Mf2)D f 2(cid:12) + (cid:12) i 2E tTrhanesvJer=se 1a/n2d rloensognitaundcineas:l hAeliccoitmypaamrispolintuodfescawlciuthlateexd-
(cid:12) (cid:12)
CN12(Q2)=sMf(Mπi2α−Mf2)DN,P¯f,21(cid:12)j(cid:12)0E(0)(cid:12)B(cid:12),Mi,12E, (7c) prtheeseroivmnaaelnunceteaolifsdtaghtievaetfrnoarinntshvFeeirgeslsee.ca6tmraopn-ledixtuc7id,tearsetsiaoptnetcohtfeivtpehlheyo.StW1o1n(h1pe5ro3eina5st)
(cid:12) (cid:12)
(cid:12) (cid:12)
M. Ronniger et al.:Effects of a spin-flavourdependentinteraction on helicity amplitudes 5
300 50
Ap1/2 Anisovich[8] C1p/2 Aznauryan[18–20]
Ap PDG[58] 40 Cp MAID[22,23]
1/2 1/2
1−3GeV]2 120000 Ap1/2AAAp1z/pn2AaMun1CA/rAy2ap1aIp/PDn2stDi[[[c512Gk982––,[[6522678038]]]]] 1−3GeV]2 123000 CCCC11np11np////2222mmmmooooddddeeeellllAACC
− 1/2 −
0 0 0
1 1
)[ 0 )[ -10
2Q Ap3/2 Fit:Aznauryan[21] 2Q
( Ap model ( -20
NA12-100 An11//22modelAA NC12-30
Ap model
An1/2modelC -40
1/2 C
-200 -50
0 1 2 3 4 5 6 0 1 2 3 4 5
Q2[GeV2] Q2[GeV2]
Fig. 6. Comparison of the S11(1535) transverse helicity am- Fig. 7. Comparison of the S11(1535) longitudinal electro-
plitude AN for electro-excitation from the proton and the excitation helicity amplitude CN of proton and neutron cal-
1/2 1/2
neutron calculated in the model C (solid lines) and model A culated in model C (solid lines) and model A (dashed lines)
(dashedlines)toexperimentaldata[8,18–20,22,23,58–67].The with experimental data [18–20,22,23]. Note that for the data
dottedlineistheresultobtainedbyKeisterandCapstick[68]. pointsoftheMAID-analysisbyTiator et al.[23]noerrorsare
Additionally a fit obtained by Aznauryan et al. [21] has been quoted.Seealso caption to Fig. 6.
displayedinthegreendasheddottedline.Notethattheresults
formodelAwererecalculated withhighernumericalaccuracy
and thusdeviate from the results published previously in [2]. Ap Anisovich[8]
80 1/2
Ap PDG[58]
1/2
An PDG[58]
(Q2 = 0) both for the proton and the neutron are accu- 1−]2 60 Ap1/21/B2urkert[69]
rately reproduced in particular by the new model C, in GeV 40 AAp1/p12/2AMznAaIuDry[a2n2,[1293]]
generalthecalculatedtransverseamplitudesaretoosmall −3 20
byafactoroftwo;incomparisontotheresultsfrommodel 0
1
the amplitudes of model decrease more slowly with )[ 0
iAncreasing momentum transfCer, in better agreement with 2Q
tshtaenceyxpoefritmheenptarlotdoantad.aBtautf,orin0p<artQic2ul<ar1thGeeVne2ariscnoont- N(A12-20 AAn1p1//22mmooddeellAA
reflected by any of the calculated results. For comparison -40 AAnp1/2mmooddeellC
1/2 C
we also plotted the results from the quark model calcu-
-60
lation of the transverse Ap -amplitude by Keister and 0 1 2 3 4 5 6
1/2 Q2[GeV2]
Capstick [68] for Q2 . 3GeV2 and the fit obtained by
Aznauryan et al. [21]. Contrary to this, the momentum Fig. 8. Comparison of the S11(1650) transverse electro-
transfer dependence of the calculated longitudinal helic- excitation helicity amplitude AN1/2 of proton and neutron cal-
ity amplitudes hardly bear any resemblance to what has culatedinmodelC (solid lines)andmodelA(dashedlines)to
been determined experimentally, in particular the mini- experimental data from [19,22,23,58,69]. See also caption to
mumfoundforthe protonatQ2 1.5 GeV2 isnotrepro- Fig. 6.
≈
duced. Only the non-relativistic calculation of Capstick
and Keister [26] shows a pronounced minimum for the
point of Aznauryan et al. [19] seems to indicate a zero
longitudinal S (1535) amplitude, however this minimum
11
crossingofthis amplitude notreproducedby either ofthe
is predicted at the wrong position.
modelcalculationsoftheCp -amplitudefortheS (1650)-
Also the calculated transverse proton helicity ampli- 1/2 11
tude Ap for the next S (1650) resonance shows a large resonance.
1/2 11 The third and fourth Jπ = 1/2 nucleon resonances
disagreement with experimental data as shown in Fig. 8. −
arepredictedinmodel at1872MeVand1886MeVand
This discrepancy was already found in the previous cal- A
in model at 1839MeV and 1882MeV, respectively. In-
culation of Merten et al. [2] and obviously is not resolved C
deedwithintheBonn-GatchinaAnalysisoftheCB-ELSA
within model . Note, however that the neutron ampli-
tude An calcuClatedatthe photonpointdoes correspond collaborationdata[6,8]evidence foraJπ =1/2− nucleon
1/2 resonance at 1895MeV was found. As can be seen from
to the data from PDG [58], as illustrated in Fig 8. The Fig. 10 the predicted transverse amplitudes for the third
rathersmalllongitudinalS11(1650)amplitude C1N/2 seems resonancebothmodelsareratherlargeandthecalculated
toagreewiththescarcemediumQ2 datafromtheMAID- value at the photon point (Q2 = 0) is much larger than
analysis of [22,23], however for lower Q2 the single data the experimental value quoted in [6,8], but the value of
6 M. Ronniger et al.:Effects of a spin-flavourdependentinteraction on helicity amplitudes
10
C1p/2 Aznauryan[19] 200 Ap1/2 Anisovich[8]
468 C1p/2CCM11npA//22IDmm[oo2dd2ee,ll2AA3] 1−V]2 150 Ap AznAaAun1p1/r/Ay22p1a/PPn2DD[[1G6G89–,[[52578080]]]]
2 Ge 100 1/2Ap MAID[22,23]
y-axis-S - 20 −3)[10 50 1/2AAn1p1//22mmooddeellAA
-4 CC11np//22mmooddeellCC 2N(Q12 0
-6 A
Ap model
1/2 C
-8 -50 An model
1/2 C
-10
0 1 2 3 4 5 6 0 1 2 3 4 5 6
Q2[GeV2] Q2[GeV2]
Fig. 9. Comparison of the S11(1650) longitudinal electro- Fig.11. ComparisonoftheP11(1440)transversehelicityam-
excitation helicity amplitude CN of proton and neutron cal- plitude AN for proton and neutron calculated in model C
1/2 1/2
culated in model C (solid lines) and model A (dashed lines). (solid lines) and model A (dashed lines). See also caption to
See also caption toFig. 6. Fig. 6.
60
Ap1/2 Anisovich[6,8] 60 Cp Aznauryan[18–20]
1−−3GeV]2 2400 AAAAn1p1n1p1////2222mmmmooooddddeeeellllAAAA,,,,4433ttrrhhdd 1−3GeV]2 345000 1/2 MACCICCD11np11np////[222222mmmm,oo2ood3ddd]eeeellCllAACCp
10 0 −0 20
)[ [1
2Q -20 2Q) 10
NA(12-40 AAApn1p1//22mmmooodddeeelllCC,,,433trrhdd N(C12 0
An1/2 modelC,4th -10
1/2 C
-60
-20
0 1 2 3 4 5 6
0 1 2 3 4 5 6
Q2[GeV2] Q2[GeV2]
Fig. 10. Comparison of theS11(1895) transversehelicity am- Fig. 12. Comparison of the P11(1440) longitudinal helicity
p(sloitluiddelinAeN1s/)2anfodrmproodteolnAan(ddasnheeudtrloinnesc)alwcuitlhatethdeisninmgloedpehloC- amplitude C1N/2 for proton and neutron calculated in model C
(solidlines)andmodelA(dashedlines).Notethatforthedata
ton point value from Anisovich et al. [6]. See also caption to pointsoftheMAID-analysisbyTiator et al.[23]noerrorsare
Fig. 11.
quoted.Seealso caption to Fig. 6.
the fourthresonancematchesthe PDGphotondecayam-
plitude before treating the higher excitations P (1875)
plitude. 11
andP (2100).FortheP (1710)resonanceonlythepho-
The transverseandlongitudinalhelicity amplitudes of 11 11
ton decay amplitude is reported [58]. In Figs. 13 and 14
the Roper resonance P (1440) are displayed in Figs. 11
11 wedisplayourpredictionsfortheseamplitudes.Thetrans-
and 12, respectively. It is obvious, that the zero cross-
ing found in the data at Q2 0.5GeV2, see Fig. 11, is verse AN1/2-amplitude of model C matches the PDG data
not reproduced in the calculat≈ed curves, although the Q2 atthephotonpointincontrasttomodel ,whichoveres-
A
timatestheproton-andneutronamplitudesbyafactorof
dependence of the positive values at higher momentum
two. On the other hand this would be in accordance with
transferscanbeaccountedforinbothmodelsafterchang-
the value obtained by Anisovich et al. [8]. The prediction
ing the sign of the old prediction [2]. On the other hand
of the longitudinal CN -amplitudes is given in Fig. 14.
we do find a satisfactory description of the longitudinal 1/2
Cp -amplitude displayed in Fig. 12 in particular in the Finally we present the results for the fourth and fifth
1/2
new model . Jπ = 1+-nucleon state in Fig. 15, where we show the
C 2
Helicity amplitudes of higher lying resonances in the transverse helicity amplitudes only. The corresponding
P channelareonlypoorlystudiedinexperiments.Never- massespredictedbymodel are1905MeVforthe fourth
11
A
theless we shall discuss briefly the P (1710) helicity am- and1953MeVforthefifthstate;formodel thepredicted
11
C
M. Ronniger et al.:Effects of a spin-flavourdependentinteraction on helicity amplitudes 7
70 80
Ap Anisovich[8] AnisovichN+ (1875)[6]
60 1/2Ap PDG[58] AnisovichN1+/2(1875)[8]
1/2 1/2
1−−310GeV]2 1234500000 AAAAAn1/n1p1n1p1//2//2222PmmmmDooooGddddeeee[ll5llA8ACC] 1−−310GeV]2 4600 0022 AAAAAAn1p1n1p1n1p1//////222222mmmmmmooooooddddddeeeeeellllllAAAACC,,,,,,554444tttttthhhhhh
)[ )[ 20
2N(Q12-1 00 2N(Q12 01 AAn1p1//22 mmooddeellCC,,55tthh
A -20 A 0
-30 01
-40 -20
0 1 2 3 4 5 6 0 1 2 3 4 5 6
Q2[GeV2] Q2[GeV2]
Fig. 13. ComparisonoftheP11(1710)transversehelicityam- Fig. 15. Prediction of theP11 transverse helicity amplitudes
plitude AN for proton and neutron calculated in model C AN for thethird andfourth excitation of theproton and the
1/2 1/2
(solid lines) and model A (dashed lines). See also caption to neutron calculated within model C (solid lines) and model A
′′ ′′
Fig. 6. (dashedlines).Thedataatthephotonpointmarked 01 and
′′02′′ were reported in [6,8] as alternatives for the N+ (1875)
1/2
resonance. See also caption to Fig. 11.
50
Cp model
40 C11n//22modelAA
1−]2 30 CC11np//22mmooddeellCC 114600 Ap1/2AApniPsoDviGch[5[88]]
GeV 20 1]2 120 Ap AAzp11n//22auPrDyaGn[[5189]]
−30 10 −V 100 A1/p12/2 MAID[22,23]
2Q)[1 -1 00 −30Ge 6800 AAAn1p1p//22mmmooodddeeelllAA
NC(12-20 2)[1 40 An11//22modelCC
Q
-30 ( 20
N12
A 0
-40
0 1 2 3 4 5 6
-20
Q2[GeV2]
-40
Fig.14. PredictionoftheP11(1710)longitudinalhelicityam- 0 1 2 3 4 5 6
plitude CN for proton and neutron calculated in model C Q2[GeV2]
1/2
(solid lines) and model A (dashed lines). See also caption to
Fig.16. ComparisonoftheP13(1720)transversehelicityam-
Fig. 6. plitudeAN ofprotonandneutroncalculatedinmodelC(solid
1/2
lines)andmodelA(dashedlines).Notethatforthedatapoints
oftheMAID-analysisbyTiatoretal.[23]noerrorsarequoted.
masses are 1872MeV and 1968MeV, respectively. The
Seealso caption to Fig. 6.
two dataat the photonpoint marked 01 and 02 were
′′ ′′ ′′ ′′
obtainedbytheCB-ELSAcollaborationwithintheBonn-
Gatchina Analysis as reported in [6,8] for the N1+/2(1875) analysis [22,23] indicate a sign change for the Ap ampli-
resonance. They correspond to two different partial wave 1/2
tude at Q2 2GeV2 not reproduced by either model. In
solutions in order to extract the corresponding baryon
spite of not≈being able to account at all for the large Ap
massandhelicityamplitudes.Thepredictionforthefourth 3/2
state lies between these values, the values found for the amplitude found experimentally, the longitudinal helicity
fifth state are much smaller. This also applies for higher amplitude as reported in the MAID analysis with excep-
Jπ = 1+ excitations not displayed here. tion of the value at Q2 1GeV2 is reproduced by both
2 models rather well, as sh≈own in Fig. 18.
ForthetransversehelicityamplitudeAp (seeFig.19)
1/2
The J = 3/2 resonances: In Figs. 16 and 17 the trans- of the D (1520)-resonance we find a reasonable quanti-
13
verse helicity amplitudes of the P (1720) resonance are tative agreement with experimental data for low momen-
13
displayed. Althoughareasonableagreementwiththedata tum transfers, while apart from the fact that in model
C
ofAznauryanet al.[19]andwiththephotondecayampli- the amplitude is too small by about a factor of two the
tude is found for both models, the data from the MAID Q2 dependence is reproduced up to Q2 6GeV2. The
≈
8 M. Ronniger et al.:Effects of a spin-flavourdependentinteraction on helicity amplitudes
200
Ap Anisovich[8] Ap Anisovich[8]
1/2 1/2
Ap PDG[58] 0
1]2 150 Ap AAzp33n//2a2uPryDaGn[[5189]] 1]2 Ap PDG[58]
−V 100 A3/p32/2MAID[22,23] −V -50 Ap11//22PDG[58]
−3Ge 50 AAn3p3//22mmooddeellAA −3Ge -100 AAp1p1//22BAuhrrkeenrst[[7619]]
0 0 Ap Aznauryan[18–20]
)[1 0 )[1 1/2Ap1/2 MAID[22,23]
2Q 2Q -150 Ap3/2 Fit:Aznauryan[21]
NA(32-1-5000 AAn3p3//22mmooddeellCC N(A12-200 AAAAn1p1np1///222mmmmooooddddeeeellllAAC
1/2 C
-150 -250
0 1 2 3 4 5 6 0 1 2 3 4 5 6
Q2[GeV2] Q2[GeV2]
Fig. 17. ComparisonoftheP13(1720)transversehelicityam- Fig.19. ComparisonoftheD13(1520)transversehelicityam-
plitudeAN ofprotonandneutroncalculatedinmodelC(solid plitudeAN ofprotonandneutroncalculatedinmodelC(solid
3/2 1/2
lines)andmodelA(dashedlines).Notethatforthedatapoints lines) and model A (dashed lines). Seealso caption toFig. 6.
oftheMAID-analysisbyTiatoretal.[23]noerrorsarequoted.
See also caption toFig. 6.
250
Ap Anisovich[8]
1/2
200 Ap3/2 PDG[58]
Ap PDG[58]
2300 1−V]2 150 AAp3p3//223/BA2uhrrkeenrst[[7619]]
1−V]2 10 −310Ge 1 5000 AAp3/p3/22FAAip3tz/:n2AaMuznrAyaaIuDnry[[a12n82–,[222031]]]
3Ge 0 2Q)[ 0
−2NC(Q)[1012---321000 C1p/2CCM1npA/2IDmm[oo2dd2ee,ll2A3] N(A32--11-550000 AAAAn3p3n3p3////2222mmmmooooddddeeeellllAACC
1/2 A 0 1 2 3 4 5 6
-40 CC1np/2mmooddeellC Q2[GeV2]
1/2 C
-50 0 1 2 3 4 5 6 Fig.20. ComparisonoftheD13(1520)transversehelicityam-
Q2[GeV2] plitude AN3/2 for proton and neutron calculated in model C
(solid lines) and model A (dashed lines). See also caption to
Fig. 18. Comparison of the P13(1720) longitudinal electro- Fig. 6.
excitation helicity amplitude CN of proton and neutron cal-
1/2
culated in model C (solid lines) and model A (dashed lines).
Note that for thedata points of the MAID-analysis by Tiator The transverse amplitudes for the next 3/2− nucleon
et al. [23] no errors are quoted.Seealso caption to Fig. 6. resonance,i.e.D (1700),aredisplayedinFigs.22and23.
13
In contrast to the situation for the D (1520)-resonance
13
describedabove,herebothmodelsareinaccordancewith
the PDG-data [58] as well as with the data from Az-
minimum at Q2 1GeV2 is clearly visible for model
≈ A nauryan et al. [19]. The prediction for the longitudinal
whereas this feature is not so pronounced in model .
The Ap -amplitudes are displayed in Fig. 20; here boCth D13(1700) amplitudes is given is Fig. 24. The calculated
3/2 amplitudes turn out to be rather small.
model underestimates the data by more than a factor of
three. Likewise the calculated neutron An - and An -
1/2 3/2
amplitudesatthephotonpointaretoosmall.Inparticular The J = 5/2 resonances: Although the transverse
for the An -amplitude the predicted value close to zero D (1675)helicity amplitudes atthe photonpoint repro-
1/2 15
is in contradiction to the experimental value 59 9 duce the experimental data from MAID [22,23] and the
− ± ×
10−3GeV−1/2 fromPDG[58].Thesituationismuchmore PDG[58]ratherwell,asdisplayedinFigs.25and26,both
favorablefor the longitudinalamplitude Sp ,see Fig.21. calculations cannot account for the apparent zero of the
1/2 experimental Ap -amplitude atQ2 1.5GeV2. Further-
Heremodel fitsthedataofAznauryanetal.[18–20]and 1/2 ≈
MAID [22] vCery well. Only for lower transition momenta more the Ap -amplitude, displayed in Fig. 26 is severely
3/2
it underestimates the experimental values slightly. underestimated in magnitude by both models and model
M. Ronniger et al.:Effects of a spin-flavourdependentinteraction on helicity amplitudes 9
100 50
Cp Aznauryan[18–20] Ap Anisovich[8]
1/2 1/2
80 Cp MAID[22,23] 40 Ap PDG[58]
1/2 3/2
1−]2 60 CC11np//22mmooddeellAA 1−]2 30 Ap3/2 AAzn3n/2auPrDyaGn[[5189]]
GeV 40 CC11np//22 mmooddeellCC GeV 1200 AAn3p3//22mmooddeellAA
3 20 3
− −
0 0 0
1 0 1
)[ )[ -10
2(Q -20 2(Q -20 AAnp3/2mmooddeellC
N12 N32 3/2 C
C -40 A -30
-60 -40
-80 -50
0 1 2 3 4 5 6 0 1 2 3 4 5 6
Q2[GeV2] Q2[GeV2]
Fig. 21. Comparison of the D13(1520) longitudinal helicity Fig.23. ComparisonoftheD13(1700)transversehelicityam-
amplitude CN for proton and neutron calculated in model C plitude AN for proton and neutron calculated in model C
1/2 3/2
(solid lines) and model A (dashed lines). See also caption to (solid lines) and model A (dashed lines). See also caption to
Fig. 6. Fig. 6.
60
1−GeV]2 2400 Ap1A/2p1/A2AAAAzn1p1An/n1/p1//2na222uiPPsrmmoyDDvoaoGiGnddchee[[[ll515[A889A8]]]] 1−eV]2 112 5050 CCCC11np11np////2222mmmmooooddddeeeellllAACC
3 G
−0 0 −3 0
2Q)[1 -20 AAn1p1//22mmooddeellCC 2)[10 -5
( Q -10
N12 (
A N12
-40 C -15
-20
-60
0 1 2 3 4 5 6 -25
Q2[GeV2] 0 1 2 3 4 5 6
Q2[GeV2]
Fig.22. ComparisonoftheD13(1700)transversehelicityam-
plitude AN for proton and neutron calculated in model C Fig. 24. Prediction of the D13(1700) longitudinal helicity
1/2 amplitude CN for proton and neutron calculated in model C
(solid lines) and model A (dashed lines). See also caption to 1/2
(solid lines) and model A (dashed lines). See also caption to
Fig. 6.
Fig. 6.
even yields the wrong sign. The transverse amplitudes
C
for the neutronarepredicted to be negative,here the cal- Incontrastto this the Ap -amplitudes are againseverely
3/2
culatedvalue atthe photonpointfor model is closerto
A underestimated in magnitude, see Fig.29. In contrast,for
athmepelxitpuedreims aenretaclavlacululaettehdantofobremvoerdyelsCm.aTllhfeorlobnogtihtumdionda-l thelongitudinal C1p/2-amplitude weobservearathergood
agreementwiththedataasdisplayedinFig.30,thevalues
els. There exists only experimental data from the MAID-
obtained in model being too small at lower momentum
analysis [23], indicating that the experimental values are C
transfers.
consistent with zero.
There also exist data for the helicity amplitudes of
the F (1680)-resonance.The comparison with the calcu-
15
lated values is given in Figs. 28 and 29. In particular in
model areasonabledescriptionoftheAp amplitudesis The J = 7/2 resonances: Concerning the J = 7/2 res-
C 1/2
foundforthenewerdatafromAznauryanetal.[18,19]and onances there exists only one negative parity resonance
MAID[22,23]bothatthephotonpointandforthevalues withmorethanatleastathreestarrating,theG (2190).
17
at higher momentum transfers. The calculated values in The correspondingpredictions fortransverseandlongitu-
model are in better accordancewith the the older data dinal helicity amplitudes are shown in Figs. 31 and 32.
A
from Burkert et al. [69] which are larger in magnitude.
10 M. Ronniger et al.:Effects of a spin-flavourdependentinteraction on helicity amplitudes
30 4
Ap Anisovich[8] Cp MAID[23]
1/2 1/2
20 Ap PDG[58] 3.5 Cp model
An1/2PDG[58] C1n/2modelA
1]2 10 1/2 1]2 3 C1p/2 modelA
−V 0 −V 2.5 C11n//22modelCC
Ge Ge 2
3 -10 3
− −
0 Ap MAID[23] 0 1.5
2(Q)[1 --3200 1AAA/2n1p1p//22mmmooodddeeelllAA 2(Q)[1 0 .15
NA12-40 An11//22modelCC NC12 0
-50 -0.5
-60 -1
0 1 2 3 4 5 6 0 1 2 3 4 5 6
Q2[GeV2] Q2[GeV2]
Fig.25. ComparisonoftheD15(1675)transversehelicityam- Fig. 27. Comparison of the D15(1675) longitudinal helicity
plitude AN for proton and neutron calculated in model C amplitude CN for proton and neutron calculated in model C
1/2 1/2
(solid lines) and model A (dashed lines). See also caption to (solid lines) and model A (dashed lines). See also caption to
Fig. 6. Fig. 6.
40
100 Ap1/2 Anisovich[8]
Ap PDG[58]
1/2
20 An PDG[58]
1]2 1]2 Ap 1/B2urkert[69]
− − 1/2
V 0 V 50 Ap1/2 Aznauryan[18,19]
Ge Ge Ap1/2 MAID[22,23]
−3 Ap Anisovich[8] −3
)[10 -20 1/2AAnp3/2PPDDGG[[5588]] )[10 0
2Q -40 Ap3/2MAID[23] 2Q
NA(32-60 3AAAA/2n3p3n3p3////2222mmmmooooddddeeeellllAACC N(A12-50 AAAAn1p1n1p1////2222mmmmooooddddeeeellllAACC
-80
0 1 2 3 4 5 6 0 1 2 3 4 5 6
Q2[GeV2] Q2[GeV2]
Fig. 26. Comparison of the D15(1675) transverse electro- Fig.28. ComparisonoftheF15(1680)transversehelicityam-
excitation helicity amplitudeAN forproton andneutroncal- plitude AN for proton and neutron calculated in model C
3/2 1/2
culated in model C (solid lines) and model A (dashed lines). (solid lines) and model A (dashed lines). See also caption to
See also caption toFig. 6. Fig. 6.
The J = 9/2 resonances: The transverse and longitu- tudes for the Jπ =1/2 I (2600)-resonances.So far no
− 111
dinal helicity amplitudes of the Jπ = 9/2+ resonance data available.
G (2250) are predicted to be very small as shown in
19
Figs. 33 and 34 and coincide with the estimate by Aniso-
vichetal.[8]forthetransverseamplitudes.Obviously,the 3.1.2 Nucleon ∆ helicity amplitudes
Ap amplitudeofmodel andthelongitudinalamplitudes →
3/2 C
of model are effectively zero. We now turn to a discussion of the results for N ∆
AlthouAgh the resonance with Jπ = 9/2 , H (2220) electro-excitation. →
− 19
hasafourstarratingbythePDG,onlytheprotonphoton
decayamplitudehasbeenestimatedin[8].Thecalculated
valuesaredisplayedinFig.35andFig.36;theamplitudes The J = 1/2 resonances: We start the discussion with
turnouttobe muchsmallerinmodel thaninmodel . thepositiveparityS (1620)-resonance.FortheS (1620)
C A 31 31
transverse and longitudinal helicity amplitudes, depicted
in Fig. 39, a wide variety of experimental data at and
nearthephotonpointexists.Thecalculatedvaluesliewell
The J = 11/2 resonances: Figs. 37 and 38 shows pre- withintheregionofexperimentaldataobtainedduetothe
dictions of the transverse and longitudinal helicity ampli- spread in partially contradictory experimental data but