Roper Electroproduction Amplitudes in a Chiral Confinement Model

10 Pages·0.22 MB·English

by  M. Fiolhais

ROPER ELECTROPRODUCTION AMPLITUDES IN A
CHIRAL CONFINEMENT MODEL
M. FIOLHAIS,P. ALBERTO AND J. MARQUES
Departamento de F´ısica and Centro de F´ısica Computacional, Universidade de
Coimbra, P-3004-516 Coimbra, Portugal
E-mail: tmanuel@teor.fis.uc.pt
2
0
B. GOLLI
0
2 Faculty of Education, University of Ljubljana and J. Stefan Institute, Ljubljana,
Slovenia
n
a
AdescriptionoftheRoperusingthechiralchromodielectricmodelispresentedand
J
thetransverseA1/2andthescalarS1/2helicityamplitudesfortheelectromagnetic
4 Nucleon–Roper transition are obtained for small and moderate Q2. The sign of

theamplitudesiscorrectbutthemodelpredictionsunderestimatethedataatthe
1
photon point. Our results do not indicate a change of sign inany amplitudes up
v to Q2 ∼ 1 GeV2. The contribution of the scalar meson excitations to the Roper
0 electroproductionistakenintoaccountbutitturnsouttobesmallincomparison
2 with the quark contribution. However, it is argued that mesonic excitations may
0 playamoreprominentroleinhigherexcited states.
1
0
2 1 Introduction
0
/
h Several properties of the nucleon and its excited states can be successfully
p explained in the framework of the constituent quark model (CQM), either in
-
p itsnon-relativisticorrelativisticversion. Thereare,however,processeswhere
e thedescriptionintermsofonlyvalencequarksisnotadequatesuggestingthat
h other degrees of freedom may be important in the description of baryons, in
:
v particularthechiralmesons. Typicalexamples–apartofdecayprocesses–are
i electromagnetic and weak production amplitudes of the nucleon resonances.
X
Alreadytheproductionamplitudesforthelowestexcitedstate,the∆,indicate
r
a the important role of the pion cloud in the baryons. The other well known
example is the Roper resonance, N(1440), which has been a challenge to any
effective modelof QCDat low orintermediate energies. Due to the relatively
low excitation energy, a simple picture in which one quark populates the
2s level does not work. It has been suggested that the inclusion of explicit
excitations of gluons and/orglueballs,or explicit excitations of chiralmesons
may be necessary to explain its properties.
The other problem relatedto the CQM is the difficulty to introduce con-
sistently the electromagnetic and the axial currents as well as the interaction
emi2001: submitted to World Scientific on February 1, 2008 1
with pions which is necessary to describe the leading decay modes of reso-
nances. Such problems do not exist in relativistic quark models based on
effectiveLagrangianswhichincorporateproperlythechiralsymmetry. Unfor-
tunately, severalchiralmodels for baryons,suchas the linear sigma model or
various versions of the Nambu–Jona-Las´ınio model, though able to describe
properly the ∆ resonance, are simply not suited to describe higher excited
states since they do not confine: for the nucleon, the three valence quarks
in the lowest s state are just bound and, for typical parameter sets, the first
radial quark excitation already lies in the continuum. In order to resolve
this problem, other degrees of freedomhave to be introduced in the model to
provide binding also at higher excitation energies. The chiral version of the
chromodielectric model (CDM) seems to be particularly suitable to describe
radial excitations of the nucleon since it contains the chiral mesons as well
as a mechanism for confining. The CDM has been used as a model for the
nucleon 1 in different approximations. Using the hedgehogcoherentstate ap-
proachsupplementedbyanangularmomentumandisospinprojection,several
nucleon properties and of the nucleon-delta electromagnetic excitation have
been obtained 1,2,3.
In the present work we concentrate on the description of the Roper reso-
nance. Its structure and the electroproduction amplitudes have been consid-
ered in severalversions of the CQM 4,5,6. The nature of the Roper resonance
has also been considered in a non-chiral version of the CDM using the RPA
techniques to describe coupled vibrations of valence quarks and the back-
ground chromodielectric field 7. The energy of the lowest excitation turned
out to be 40 % lower than the pure 1s–2s excitations. A similar result was
obtained by Guichon 8, using the MIT bag model and considering the Roper
as a collective vibration of valence quarks and the bag.
Our descriptionof baryonsinthe frameworkof the CDM model provides
relatively simple model states which are straightforwardly used to compute
thetransverseandscalarhelicityamplitudesforthenucleon–Ropertransition,
in dependence of the photon virtuality 9. The electromagnetic probe (virtual
photon)couplestochargedparticles,pionsandquarks. However,intheCDM,
baryons have got a weak pion cloud and therefore the main contribution to
the electromagnetic nucleon–Roper amplitudes comes from the quarks.
In Section 2 we introduce the electromagnetic transition amplitudes. In
Section 3 we briefly describe the model and construct model states repre-
senting baryons, using the angular momentum projection technique from co-
herent states. In Section 4 we present the CDM predictions for the helicity
amplitudes for typical model parameters. Finally, in Section 5 we discuss the
contribution of scalar meson vibrations.
emi2001: submitted to World Scientific on February 1, 2008 2
2 Electroproduction amplitudes in chiral quark models
In chiralquarkmodels the coupling of quarksto chiralfieldsis written in the
form:
=gq¯(σˆ+i~τ ~πˆγ )q . (1)
q−meson 5
L ·
Here g is the coupling parameterrelatedto the mass ofthe constituentquark
M = gf . In the CDM the parameter g is substituted by the chromodielec-
q π
tric field which takes care of the quark confinement as explained in the next
section. In the linear σ-model, in the CDM, as well as in different versions
of the Cloudy Bag Model, the chiral meson fields, i.e. the isovector triplet of
pion fields, ~π, and the isoscalar σ field (not present in non-linear versions),
are introduced as effective fields with their own dynamics described by the
meson part of the Lagrangian:
= 1∂ σˆ∂µσˆ+ 1∂ ~πˆ ∂µ~πˆ (σˆ,~πˆ) (2)
Lmeson 2 µ 2 µ · −U
where is the Mexican-hat potential describing the meson self-interaction.
U
Indifferent versionsof the Nambu–Jona-Las´ıniomodel 10 the chiralfields are
explicitly constructed in terms of quark-antiquark excitations of the vacuum
in the presence of the valence quarks.
From (2) and from the part of the Lagrangian corresponding to free
quarks,
=iq¯γµ∂ q, (3)
q µ
L
the electromagnetic current is derived as the conserved Noether current:
Jµ (rrr)=q¯γµ 1 + 1τ q+(~πˆ ∂µ~πˆ) . (4)
e.m. 6 2 3 × 3
(cid:0) (cid:1)
Note that the operbator contains both the standard quark part as well as the
pionpart. Westressthatinallthesemodelstheelectromagneticcurrentoper-
atorisderiveddirectlyfromtheLagrangian,hencenoadditionalassumptions
have to be introduced in the calculation of the electromagnetic amplitudes.
We can now readily write down the amplitudes for the electroexcitation
of nucleon excited states in terms of the EM current (4). Let us denote by
N˜ and R˜ the model states representing the nucleon and the
| M,MTi | J,T;M,MTi
resonantstate,respectively(theindexesM andM standfortheangularmo-
T
mentum and isospin third components). The resonant transverse and scalar
helicity amplitudes, A andS respectively,defined in the restframe ofthe
λ 1/2
resonance, are
2πα
A = ζ d3rrr R˜ JJJ (rrr) ǫǫǫ eikkk·rrr N˜ (5)
λ − rkW Z h J,T;λ,MT| em · +1 | λ−1,MTi
emi2001: submitted to World Scientific on February 1, 2008 3
2πα
S1/2 =ζrk Z drrrhR˜J,T;+21,MT|Je0m(rrr) eikkk·rrr|N˜+12,MTi, (6)
W
where α = e2 = 1 is the fine-structure constant, the unit vector ǫǫǫ is the
4π 137 +1
polarizationvectoroftheelectromagneticfield,k =(M2 M2)/2M isthe
W R− N R
photon energy at the photon point (introduced rather than ω which vanishes
at Q2 = M2 M2) and ζ is the sign of the Nπ decay amplitude. This sign
R− N
has to be explicitly calculated within the model; from (1) in our case. In
the case of the ∆ resonance (T = J = 3), λ takes two values λ = 3 and 1,
2 2 2
while for the Roper state (T = J = 1), only one transverse amplitude exists
2
(λ= 1).
2
The photon four momentum is qµ(ω,kkk) and we define Q2 = q qµ. In
µ
−
the chosen reference frame the following kinematical relations hold:
M2 M2 Q2 M2 +M2 +Q2 2
ω = R− N − ; kkk2 k2 = R N M2 . (7)
2M ≡ (cid:20) 2M (cid:21) − N
R R
Theelectroexcitationamplitudesforthe∆resonancehavebeenanalyzed
in the framework of chiral quark models 3,11,12. They are dominated by the
M1 transition but contain also rather sizable quadrupole contributions E2
and C2. The CQM model predicts here too low values for the M1 piece and
almostnegligiblevaluesforthequadrupoleamplitudes. Inchiralquarkmodels
there is aconsiderablecontributionfromthe pions (i.e. fromthe secondterm
in (4)): up to 50 % in the M1 amplitude, and they dominate the E2 and
C2 pieces. The absolute values of the amplitudes and their behavior as a
function ofthe photonvirtualityQ2 is wellreproducedin the linearσ-model.
Though the ratios E2/M1 and C2/M1 are also well reproduced in the CDM,
thismodelgivessystematicallytoolowvaluesfortheamplitudes,whichcould
be attributed to its rather weak pion cloud. As we shall see in Section 4, this
might also explain the small values of the Roper production amplitudes at
low Q2.
In the next section we construct the states N˜ and R˜ for the Roper in
| i | i
theframeworkoftheCDMand,inSection4,wepresentthemodelpredictions
for the amplitudes.
3 Baryons in the CDM
The Lagrangianof the CDM contains, apart of the chiral meson fields σ and
π, the cromodielectric field χ such that the quark meson-interaction(see (1))
is modified as:
g
= q¯(σˆ+i~τ ~πˆγ )q . (8)
q−meson 5
L χ ·
emi2001: submitted to World Scientific on February 1, 2008 4
The idea behind the introduction of the χ field is that it acquires a nonzero
expectation value inside the baryon but goes to 0 for larger distances from
the centerofthebaryon,thuspushingthe effectiveconstituentquarkmassto
infinity outside the baryon. In addition, the Lagrangiancontains kinetic and
potential pieces for the χ-field:
1
= 1∂ χˆ∂µχˆ M2χˆ2, (9)
Lχ 2 µ − 2 χ
where M is the χ mass. In this work we consider only a simple quadratic
χ
potential;otherversionsoftheCDMassumemorecomplicatedforms,namely
quartic potentials.
The free parametersof the model have been chosenby requiringthat the
calculated static properties of the nucleon agree best with the experimental
values2. IntheversionoftheCDMwithaquadraticpotential,theresultsare
predominantly sensitive to the quantity G = gM ; we take G = 0.2 GeV
χ
(and g = 0.03 GeV). The model contains othper parameters: the pion decay
constant, f = 0.093 GeV, the pion mass, m = 0.14 GeV, and the sigma
π π
mass, which we take in the range 0.7 m 1.2 GeV.
σ
≤ ≤
The nucleon is constructed by placing three valence quarks in the lowest
s-state,i.e.,thequarksourcecanbewrittenas(1s)3. FortheRoperthequark
sourceis(1s)2(2s)1,i.e.oneofthethreequarksnowoccupiesthefirst(radially)
excited state. The quarks are surrounded by a cloud of pions, sigma mesons
and chi field, described by radial profiles φ(r), σ(r) and χ(r) respectively.
The hedgehogansatzisassumedforthe quarksandpions. Thequarkprofiles
(described in terms of the upper, u, and the lower component, v) and boson
profiles are determined self-consistently.
Becauseofthehedgehogstructure,the solutionisneitheranangularmo-
mentumeigenstatenoranisospineigenstate,andthereforeitcannotberelated
directlywithaphysicalbaryon. However,the physicalstatescanbe obtained
from the hedgehog by first interpreting the solution as a coherent state of
three types of bosons and then performing the Peierls-Yoccozprojection1,13:
1 1
|N12,MTi=N P212,−MT|Hhi, |R′21,MTi=N′P212,−MT|Hh∗i, (10)
whereP istheprojectorandweintroducedthesymbol*todenotetheRoper
intrinsicstate. Becauseoftheirtrivialtensornature,theχandtheσ-fieldsare
not affected by projection. This approach can be considerably improved by
determining the radial profiles φ(r), σ(r) and χ(r), as well as the quark pro-
files,using the variationafterprojectionmethod1,separatelyforthe nucleon
and for the Roper.
emi2001: submitted to World Scientific on February 1, 2008 5
0,3
2,5 Nucleon f Nucleon
0,2
-3/2u,v / fm 0112,,,,5050 v u 1s -1f, s, c / fm --0000,,,,0121 c
0,0 -0,3 s
-0,5 -0,4
-1,0 -0,5
0,0 0,5 1,0 1,5 2,0 0,0 0,5 1,0 1,5 2,0
r / fm r / fm
0,3
2,5 Roper 0,2 f * Roper
-3/2u,v / fm 0112,,,,5050 v* u* 12ss -1f, s, c / fm --0000,,,,0121 c *
0,0 -0,3 s *
-0,5 -0,4
-1,0 -0,5
0,0 0,5 1,0 1,5 2,0 0,0 0,5 1,0 1,5 2,0
r / fm r / fm
Figure 1. Quark and meson radial wave functions for the (1s)3 (Nucleon) and (1s)2(2s)1
(Roper) configurations. The vacuum expectation value of the sigma field is −fπ. Note
that the effective quark mass is proportional to the inverse of the χ field. We use the
symbol*todenotetheRoperradialfunctions. Themodelparametersare: Mχ=1.4GeV,
g=0.03GeV,mπ =0.14GeV,fπ =0.093GeV,mσ =0.85GeV.
Figure 1 shows the radial profiles for the (1s)3 and (1s)2(2s)1 configu-
rations. Those corresponding to the Roper extend further. The strength of
the chiral mesons is reduced in the Roper in comparison with the nucleon.
Another interesting feature is the waving shape acquired by the Roper chro-
modielectric field, χ∗. A central point in our treatment of the Roper is the
freedom of the chromodielectric profile, as well as of the chiral meson pro-
files, to adapt to a (1s)2(2s)1 configuration. Therefore, quarks in the Roper
experience meson fields which are different from the meson fields felt by the
quarks in the nucleon. As a consequence, states (10) are normalized but not
mutually orthogonal. They can be orthogonalizedtaking
1
R = (R′ cN ), c= NR′ . (11)
| i √1 c2 | i− | i h | i
−
A better procedure results from a diagonalization of the Hamiltonian in the
emi2001: submitted to World Scientific on February 1, 2008 6
subspace spanned by (non-orthogonal) R′ and N :
| i | i
R˜ =cR R′ +cR N , N˜ =cN R′ +cN N . (12)
| i R| i N| i | i R| i N| i
In Table 1 the nucleon energies and the nucleon-Roper mass splitting
aregiven. Theabsolutevalueofthenucleonenergyisabovetheexperimental
valuebutitisknown2thattheremovalofthecenter-of-massmotionwilllower
those values by some 300 MeV (similar correction applies to the Roper). On
the other hand, the nucleon-Roper splitting is small, even in the case of the
improved state (12). The smallness of the spitting is probably related with a
much too soft way of imposing confinement.
Table 1. Nucleon energies and nucleon-Roper splittings for two sigma masses. EN is the
energy of the nucleon state (10), ∆E was obtained using (11), E˜ and ∆E˜ are calculated
usingthestates(12). TheothermodelparametersareinthecaptionofFigure1. Allvalues
areinMeV.
mσ EN ∆E E˜N ∆E˜
700 1249 367 1235 396
1200 1269 354 1256 380
4 Amplitudes
OurresultsforthetransversehelicityamplitudesareshowninFigure2forthe
parametersetusedforFigure1. Theexperimentalvaluesatthephotonpoint
50
40
-1/2c) 20 -1/2c) 40
V/ V/
Ge 0 Ge 30
310 / ( -20 310 / ( 20
pA 1/2-40 nA 1/210
-60
0
0,0 0,5 Q2 / (G1e,0V/c)2 1,5 0,0 0,5 Q2 / (G1e,0V/c)2 1,5
Figure2. Nucleon-Ropertransverseamplitudes. TheexperimentalpointsatQ2=0GeV2
arethe estimates of the PDG14. The solidsquares 15 andthe open circles16 resultfrom
theanalysisofelectroproduction data.
emi2001: submitted to World Scientific on February 1, 2008 7
are the PDG most recent estimate 14 Ap = 0.065 0.004 (GeV/c)−1/2
1/2 − ±
andAn =0.040 0.010(GeV/c)−1/2. The pioncontributionto the charged
1/2 ±
states only accounts for a few percent of the total amplitude. The discrepan-
cies at the photon point can be attributed to a too weak pion field, which we
already noticed in the calculation of nucleon magnetic moments 2 and of the
electroproduction of the ∆ 3. Other chiral models 12 predict a stronger pion
contributionwhich enhances the value ofthe amplitudes. If we calculate per-
turbatively the leading pion contribution we also find a strong enhancement
at the photon point; however,when we properly orthogonalizethe state with
respect to the nucleon, this contribution almost disappears.
50 3,0
-1/2c) 40 -1/2c)
V/ 30 V/2,0
e e
G G
30 / ( 20 30 / (
pS 11/2 10 nS 11/21,0
- -
0
0,0
-10
0,0 0,5 Q2 / (G1e,0V/c)2 1,5 0,0 0,5 Q2 / (Ge1,V0/c)2 1,5
Figure3. Nucleon–Roper scalarhelicityamplitudes(seealsocaptionofFigure2).
In Figure 3 we present the scalar amplitudes. For the neutron no data
are available which prevents any judgment of the quality of our results.
InCQMcalculations4,5,6whichincorporateaconsistentrelativistictreat-
ment of quark dynamics, the amplitudes change the sign around Q2 0.2–
∼
0.5 (GeV/c)2. The amplitudes with this opposite sign remain large at rel-
atively high Q2, though, as shown in 5,6, the behavior at high Q2 can be
substantially reduced if either corrections beyond the simple Gaussian-like
ansatz or pionic degreesof freedomare included in the model. Other models,
in particular those including exotic (gluon) states, do not predict this type
of behavior 17. The present experimental situation is unclear. Our model,
similarly as other chiral models 12,18, predicts the correct sign at the photon
point, while it does not predict the change of the sign at low Q2. Let us also
note that with the inclusion of a phenomenological three-quark interaction
Canoet al.6 shiftthechangeofthesigntoQ 1(GeV/c)2 beyondwhich,in
∼
our opinion, predictions of low energy models become questionable anyway.
emi2001: submitted to World Scientific on February 1, 2008 8
5 Meson excitations
The ansatz (10) for the Roper represents the breathing mode of the three
valence quarks with the fields adapting to the change of the source. There is
another possible type of excitation in which the quarks remain in the ground
state while the χ-field and/or the σ-field oscillate. The eigenmodes of such
vibrationalstatesaredeterminedbyquantizingsmalloscillationsofthescalar
bosonsaroundtheirexpectationvaluesinthegroundstate19. Wehavefound
that the effective potential for such modes is repulsive for the χ-field and
attractive forthe σ-field. This means thatthere areno glueballexcitationsin
whichthe quarkswouldactasspectators: the χ- fieldoscillatesonly together
with the quark field. On the other hand, the effective σ-meson potential
supports at least one bound state with the energy ε of typically 100 MeV
1
below the σ-meson mass.
We can now extend the ansatz (11) by introducing
R∗ =c R +c a˜† N , (13)
| i 1| i 2 σ| i
wherea˜† isthecreationoperatorforthislowestvibrationalmode. Thecoeffi-
σ
cientsc andtheenergyaredeterminedbysolvingthe(generalized)eigenvalue
i
problemsinthe2 2subspace. ThelowestenergysolutionistheRoperwhile
×
its orthogonalcombination could be attributed to the N(1710), provided the
σ-mesonmass is sufficiently small. In sucha casethe latter state is described
aspredominantlytheσ-mesonvibrationalmoderatherthanthesecondradial
excitationofquarks. Thiswouldmanifestinverysmallproductionamplitudes
since mostly the scalar fields are excited.
The presence of σ-meson vibrations is consistent with the recent phase
shift analysis by Krehl at al. 20 who found that the resonant behavior in
the P channel can be explained solely through the coupling to the σ-N
11
channel. In our view, radial excitations of quarks are needed in order to
explain relatively large electroproduction amplitudes, which would indicate
that the σ-N channel couples to all nucleon 1+ excitations rather than be
2
concentrated in the Roper resonance alone.
This work was supported by FCT (POCTI/FEDER), Portugal, and by
TheMinistryofScienceandEducationofSlovenia. MFacknowledgesagrant
from GTAE (Lisbon), which made possible his participation in EMI2001.
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