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Medium effects on Δ properties and N-Δ transition form factors
Volume 243, number 4
PHYSICS LETTERS B
5 July 1990
Medium effects on A properties and N - A transition form factors Chr.V. C h r i s t o v a,~, M. F i o l h a i s b, E. R u i z A r r i o l a a a n d K. G o e k e a a Institut~r TheoretischePhysikll, Ruhr-UniversitdtBochurn, D-4630Bochum, FRG b Departarnento de Fisica and Centro de Fisica Tedrica (INIC), Universidade de Coirnbra, P-3000 Coimbra, Portugal
Received 27 November 1989; revised manuscript received 13 March 1990
The properties of the A( 1232) isobar immersed in a nuclear medium are investigated in a chiral quark-meson theory. To this end the Nambu-Jona-Lasinio model is used to evaluate the pion decay constant, the pion mass and the sigma mass at finite medium density. These values serve to fix the parameters of the Gell-Mann-Lrvi sigma model. The latter is then solved in a variational mean-field and projection approach in order to obtain the static delta properties and yNA and nNA vertices. At the nuclear matter density both the mass of the delta and the nucleon-delta mass splitting are reduced by about 20%, the radii show swelling and the form factors get noticeably reduced at finite momentum transfers relevant for charge-exchange reactions and electron scattering.
The idea that the nucleon properties in the nuclear environment should differ from those o f the free nucleon has been suggested in the last years from both the experiment and the theory. The same arguments, at least from the side of theory, should be valid for the properties o f the nucleon resonances, in particular to those o f the A. In contrast to the nucleon, however, there are no such clear evidences for the medium modifications o f the A properties from the side of experiments. Only some charge-exchange experimental data seem to support a possible modification. In particular, in the spectra from (3He, t) reactions with targets heavier than '2C the A peak is shifted [ 15 ] towards lower energy loss than expected from the N - A mass splitting measured for proton target. A similar trend can be seen in the (d, 2p) [ 4 - 7 ] and some heavy ion induced [4,5] charge-exchange reaction data. For (n, p) [8,9] reactions these effects are not observed although even in the last case some speculations about medium effects have been put forward [ 10 ]. A consistent and generally accepted explanation o f the observed energy shift does not yet exist. It seems to be clear, however, that it cannot be explained [4,10 ] as originated from the kinematics and due to the dependence o f the 3He-t form factor on the m o m e n t u m transfer. Chanfray and Ericson [ 11 ] showed that the pion-exchange interaction,
treated in the random phase approximation, may modify the longitudinal response in contrast to the transverse one. Esbensen and Lee [ 10] studied the charge-exchange reactions within Glauber scattering theory with a response of the semi-infinite nuclear matter in which the residual pion-exchange interaction is included in the r a n d o m phase approximation as well. They concluded, however, that the energy shift cannot be explained solely in terms of the pion-exchange interaction and that a reduction o f the nucleon-delta mass splitting due to the nuclear environment has to be included. In contrast to Esbensen and Lee, using a coupled channel formalism, Udagawa et al. [ 12 ] claim that the energy shift can be explained as an effect mainly coming from the A N - i correlations induced by the pion-exchange interaction. It should be noted that in both refs. [ 11,12 ] a possible modification of the delta properties itself by the medium, especially o f the N - A mass splitting, is neglected. In contrast to the charge-exchange reactions the ~/ absorption [ 13,14 ] and the inclusive inelastic electron scattering data [ 15-17 ] show no energy shift of the A centroid or even a small shift in the opposite direction (to higher energy loss) independent o f the target mass. This may be related to the fact that the electromagnetic probes are sensitive only to the
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transverse response function whereas the hadronic probes examines both the transverse and longitudinal spin-isospin response. The experimental information, however, is not detailed enough [ 17] (especially the problem of background contributions is not solved) to make more definite conclusions. Apparently, in contrast to the nucleon the experimental indications for medium effects in the delta properties are rather unclear. Nevertheless, we take the situation as a motivation to investigate the properties of the delta and of the nucleon-delta transitions, if both systems are immersed in a nuclear medium. Our procedure of studying the properties of the delta is similar to the one applied to the medium effects in the case of nucleon [ 18 ]. It is based on the assumption that a nuclear medium can approximately be represented by a quark medium. Furthermore it is assumed that in the framework of the GellMann-Lrvi cr model [ 19 ] the influence of the medium on the delta can be solely expressed by means of modified values of the pion coupling constant f~ and meson masses m~ and rn~. In practice the meson sector of the quark medium is described by the Nambu-Jona-Lasinio model [20]. It is first adjusted for the vacuum to reproduce the experimental value off~ and rn~, the empirical value of the quark condensate and quark bare mass in accordance with the QCD sum rules [ 21 ] and to fulfill the PCAC. In a second step it is solved for a finite medium density yielding modified v a l u e s f ~*,m ~*and rn*o. These meson values actually fix the parameters in the Gell-Mann-L6vi lagrangian [ 19 ] which is then solved in the solitonic sector by mean field and projection methods [22]. The details of the approach are described in ref. [ 18 ] and will here be only sketched shortly. We start with the lagrangian of the Nambu-JonaLasinio model with scalar and pseudoscalar quarkquark couplings [ 20 ]:
= 9i~. 0~~ - m o ~P~+ ½G[ ( ~P~)2+ ( ~5 T~)2]. (1) Introducing auxiliary sigma and pion fields by
a=-g~PT/l~ 2 and ~ t = - g ~ , s r T / ~ 2 and assuming them to be classical (zero boson and one Fermi loop approximation), one can prove [ 18 ] that the total energy density reads 334
E=-4Nc
5 July 1990
f
d3k x/k2+g2(tr2 + ~t2) (2/t) 3
kF<~ lkl <~A
+ ½/t2(cr2+ f t 2 ) - o t a
(2)
with a three-momentum cut-off.4 in order to make the integral finite. The a is related to the bare mass mo by a = -lt2mo/g, the G is expressed as G=g2/# 2 and/~ is fixed according to MeiBner et al. [23 ]. The Fermi momentum is given by kF= (3n2p)1/3, where p is the medium density. At the zero density (kF = 0) demanding PCAC and the stationary phase conditions for the energy E with respect to the vacuum values of cr and n one can fix all parameters of the Nambu-Jona-Lasinio model except for the cut-off.4 and the constituent mass M=gdc~. The meson masses are related to the second derivatives of the energy at the stationary point. For f~ and m~ we assume their experimental values. For a given particular value of M t h e A is fixed such that the pion decay constantf~, evaluated by the corresponding Feynman diagram, agrees with the vacuum value of cr which is av=f~ as well. All those conditions together leave M as the only free parameter. For it we use the particular value M=gf~ = 463 MeV obtained in ref. [ 18 ] such that the nucleon mass, evaluated within the Gell-Mann-Lrvi model, is MN=938 MeV. Actually the value of the constituent quark mass used leads to a good reproduction of the structured vacuum. At finite medium densities the symmetry breaking term and the above vacuum values of #, A and g are kept fixed because they define the lagrangian. The medium values of the pion decay coupling constant and meson masses, indicated by an asterisk, can be read of the total energy E* of the system, extracted from eq. (2) with a finite Fermi momentum kF. The medium reduced pion decay constant, f * , is given by the minimum of E* with respect to o-*. The second derivative of E* with respect to o-* and ~r* at or* = f * and at zero, respectively, are used to determine the masses m*~ and m* in the medium. For completeness we present in table 1 the calculated values for meson characteristics at zero, half and full nuclear matter density (Pnm= 0.16 fm-3). One realizes a clear decrease o f f * and m* and an increase of m* with increasing medium density. As already shown in refs. [ 3,18 ] at about 3 times the nuclear matter density, a phase transition from
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the chirally broken Goldstone phase to the restored Wigner phase takes place beyond which f * ~ 0 and m . ~~ mo, • the latter increasing with medium density p. As a next step the f * , m* and m* are used to define a modified Gell-Mann-Lrvi lagrangian 5 e* to describe the delta structure in the medium:
- (½a*)~(o-*~+ n * ~ - ~*~) ~+ ½o~o-* 0~o-* 1 $ + ~0utt • 0utt $ - L m ~ 2o *
(3)
a * = f * , 2 *z =(mo,z - m ~.z )/2f~.2 and u.2 = f ~ : - m ~.2 /2 ,~ fixed by inserting the calculated meson values. For the nucleonic solution of this lagrangian in ref. [ 18 ] we have employed the variational procedure of ref. [ 22 ] based on mean field states with generalized hedgehog structure and projection techniques for spin and isospin. Similarly to the nucleon, the A state appears as a bound state (soliton) in this variational treatment. To be specific, the mean field trial function for the A is assumed to be I ~ ) = t q3) iZ ) IH ). Here ]Z ) and IH ) are quantal coherent Fock states representing the sigma and pion cloud and the Iq3) corresponds to three quarks of different colour in a 1s orbit with the spin-flavour structure of a generalized hedgehog: I u+ ) sin q - Id~ ) cos r/. The actual A state is obtained from I ~ ) by application of projection operators: with
I~JJr) =
Z oJ, ~K, TK TD~ JMK~to TMTKT II ~ . t ) . K,Kr
(4)
The gX.K~, Jr the flavour mixing coefficient q and the orbital degrees of freedom in [q 3), IZ ) and IH ) are determined variationally as described in detail by Fiolhais et al. [ 22 ]. Needless to say that consistency checks like virial theorems are fulfilled in the vacuum as well as at finite density. The projected solution with good spin and isospin numbers obtained for both the nucleon and A can be used for calculations of the static A properties as well as of form factors corresponding to the transition vertices TNA and nNA. To that end we neglect the recoil effects and assume that the projected state of the nucleonic soliton is a good zero momentum state. For the electromagnetic N - A form factors we use, up to a trivial phase, the standard decomposition for the
5 July 1990
matrix element of the electromagnetic current (A(p') IJgmI N ( p ) ) given by Jones and Scadron [24]. The states I A ( p ' ) ) and I N ( p ) ) represent either a A° and a neutron state or else a A ÷ and a proton (momenta p' and p), and for them we use Rarita-Schwinger and Dirac spinors respectively. The nature of the N-A electromagnetic transition is threefold: It might correspond to a transverse electric quadrupole, to a transverse magnetic dipole, or to a longitudinal electric quadruple. The associated form factors are GEa(q2), G ~ ( q 2) and GCA(q2) respectively. Working in the limit of nucleon-delta degeneracy and using the Breit frame for convenience [in which pU= (E, ½q) a n d p ' u= (E, - ½q) ] one can relate the electromagnetic transiton form factors to the corresponding matrix elements: (A(½q) I J°m ( 0 ) I N ( - ½q) ) = ~ 3 2
GCAqkqtf k', (5)
(A(½q) I f i ~ ( 0 ) I N ( - ½q) ) - - - 3i -- 4M N
E
M
( GNa--GNA
3i rz_e {xik_ + 2MN VNA ~k~
) eijkqJg k
qiqk\ qtfkt' ~qT-)
(6)
with the form factors G~a, G ~ and GC~ taken at _q2. In the above expressions,
gk=(1,s'3-s3; ½,s313, s'3) %~-s3,k, fkt= Z ( 1, m; ½, s'3 --ml 3, s'3 ) qlmk(mlatls3 ), m
where q/mk ( m = - 1, 0, 1 ; k = 1, 2, 3) is the transformation matrix from the cartesian to the spherical basis. Considering initial and final states with spin and isospin third components + ½, explicit expression for the electromagnetic nucleon-delta transition form factors can be obtained from eqs. ( 5 ) and (6). Hence, the form factors GNM~,GCNaand G~A read [25]
GMNa =x//6fd3r jl(qr) 2MN qr
( A~-/2I/Jo IN~/2 ),
(7)
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f "3
2MN
zA(qr)
J a rr ~
t"
g~Na = 2 M N "I J d3r z ( A~-/2 [ J~INI/2) "o + .
Y2o(r)
X (A~/21 J°m(r)IN+/2 ), G~Na=0,
(8) (9)
where I A~-/2) and IN~-/2 ) are the projected states I J = T = 3, M = M r = 1 ) and IJ = T = ½, M = M r = 1 ) respectively. In eq. (7), the/1o is the zeroth-zeroth component of the magnetic moment o p e r a t o r / i = ½( r X J ) and ffegm= (J°m, 3) J' the electromagnetic current. Obviously, only the isovector parts of the operators/io and J°m may contribute to G~A and GC~. The vanishing of GEa also occurs in the Skyrme model [26] in leading order in 1~No. Although different authors agree on the smallness of GEa(O)/Gr~a(O), the actual value for this quantity has not yet been well established. The various analyses reported in ref. [24] seem to indicate for that ratio a value less than 0.05. The limit q--,0 of eq. (7) gives the nucleon-delta transition magnetic moment: #N~ = ~ d 3r (A+/2 l/1o IN~-/2 ).
(10)
For the nNA form factor G~Na we assume the simplest coupling [27,28 ] associated with the transition: ( A ( P ' ) [ 3~ I N ( p ) )
G~Na(q 2) -- - a~(p', s' ) pUUN(p, S) 2MN X (~ta 17~q ½tN ),
( 11 )
where ]~ is the pion current source operator, the 7~ is the isospin transition operator and the a~(p', s' ) and the uN(P, s) are the Rarita-Schwinger and Dirac spinors, respectively. Working in the Breit frame one can obtain for the GnNa in the limit of nucleon-delta degeneracy M~
GnNa = 6 M N
¼qZ+M2
x3 f d3rJl(qr) z(A+/2 [ J~IN~/2). % + qr
(12)
The limit q ~ 0 ofeq. (12) gives the pion-nucleon delta coupling constant:
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(13)
In eqs. (12) and (13) the factor ~ comes from the matrix elements of the spin and isospin transition operators. Our results for the static A properties at vanishing medium density are presented in the second column of the table 1 in comparison to some available experimental estimates (first column). They are very similar to the numbers of ref. [22] which are obtained with a slightly different sigma mass. The A mass appears to be smaller by 11% than the corresponding experimental values and hence the calculating N-A splitting exhausts only a half of the experimental value. In the model the N-A splitting originates basically from the difference of pion kinetic energies in the respective clouds. It may possibly be improved by incorporating an additional chromomagnetic interaction between the quarks due to one gluon exchange. Despite of the fact that the charge radii and the magnetic moments are not measurable we present them for completeness. More interesting are the observables connected to the N-A transition since there is some experimental information available. Both the magnetic moment and the RMS radius are in a good agreement with the experimental estimates [26,3,29]. The ratio #,6/(/zp-/~n)=0.67, however, is close to 1/x/~ which follows from the collective quantization [ 30,26 ] and deviates a bit from the experimental ratio 0.76. Similar to the g~NN calculated in ref. [22 ], g~aN is noticeable larger than the experimental number. The latter, following Jones and Scadron [24 ],
1
F(A--,Nn) = ~-~np
,3E'+MN(g~Na~ 2 Ma ~,2MN,I '
(14)
is related to the experimental value of 120 MeV for the A width. The p' and E' in eq. (14) are the momentum and the energy of the outgoing nucleon. The ratio g~r~a/g~NN=2.65 is larger than the empirical value of 2.18 (we take g~NN= 13.5 ) and also exceeds the value of 3/x/~ expected [30] from 1~No expansion in the Skyrme model as well as the value of 2.15 obtained by Kaiser et al. [ 31 ] in Skyrme-type calculations with vector mesons. At finite medium density all A properties except
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Table 1 Medium effects on delta properties. For finite medium densities the values are given relative to the theoretical numbers at vanishing density. Quantity
Absolute values experiment
f~ (MeV) rn~ (MeV) mo (MeV)
Ea (MeV) Ezx-EN (MeV) (r~ >a~++> (fm 2) (rez )a~+~ (fm 2) ( r 2 )~0~ (fm 2) (r2)a~-~ (fm 2) /tac++~ (n.m.) /~a~÷~ (n.m.) /tA~0>(n.m.) ga~-) (n.m.) ~/( r2 )Na (fm) /~Na (n.m.) g~Nzx
93 139.6 1232 295
-
Relative values p = 0.0 fm - 3
p = 0.08 fm- 3
p = 0.16 fm - 3
93 139.6 937.7
0.89 1.06 0.93
0.77 1.14 0.85
1094 156 1.44 0.67 -0.10 -0.87 5.57 2.21 - 1.14 -4.50
0.92 0.92 1.16 1.19 0.94 1.11 1.04 1.06 0,98 1.02
0.82 0.79 1.41 1.41 0.80 1.30 1.10 1.15 0.92 1.03
0.89 3.42 45.4
1.10 1.05 1.01
1.25 1.11 1.04
0.84 3.67 29.6
for the g~Na show a clear modification. The relative n u m b e r s for half a n d full nuclear m a t t e r density are s u m m a r i z e d in the last two columns in table 1. The energy o f the delta as well as the n u c l e o n - d e l t a mass splitting decrease with the m e d i u m density resulting in E*~/E~ = 0 . 8 3 and E~N/EAN = 0 . 7 9 at Pn~. The reduction o f the N - A splitting is very close to the shift o f the energy loss centroid observed in (3He, t) charge-exchange reactions [ 1-5 ]. Such a direct comparison, however, is not straightforward since the particular reaction m e c h a n i s m m a y also play an important role. Nevertheless the reduction is rather large and hence should be included in the considerations dealing with the excitation o f the delta in nuclear medium. Indeed, this has been done by Esbensen a n d Lee [ 10 ] who in their a p p r o a c h need a reduction o f about 10% o f the N - A splitting ( s i m i l a r to our n u m ber at half nuclear m a t t e r density) in o r d e r to explain the shift o f the A peak position. The b e h a v i o u r o f the charge MS radii o f the delta at finite m e d i u m density are similar to those o f the nucleon. F o r the N - A transition both the MS radius and the magnetic m o m e n t show an increase a n d it is reflected in the b e h a v i o u r o f the form factor G~a at small transfer m o m e n t a . This form factor is plotted as a
function o f the f o u r - m o m e n t u m transfer q2 for various m e d i u m densities in fig. 1. The experimental data [ 32 ] shown corresponds to zero density. A p p a r e n t l y the present results are in fairly good agreement with the experimental estimates. In particular it is valid for the value o f the form factor at the origin (the transition magnetic moment/tNa) a n d for its slope at the same p o i n t (related to the magnetic transition R M S radius) both very close to the experimental numbers (presented in table 1 ). At increasing med i u m density because o f the increase o f the transition magnetic m o m e n t the GNMashow a slight increase at low m o m e n t u m transfer q2~<0.1 G e V 2 a n d a decrease at larger transfers. The reduction is less than in the case o f the other two transition form factors a n d at q 2 ~ 0.4 G e V 2 consists o f about 20% at Pnm. Fig. 2 displays the longitudinal electric quadrupole transiton form factor for several m e d i u m densities. Although the experimental d a t a are very scarce, at least the m o d e l predictions seem to be in qualitative agreement with estimates [24] on the ratio o f the electric q u a d r u p o l e to the magnetic dipole amplitudes in n u c l e o n - d e l t a transitions. In contrast to the transverse magnetic transition form factor, the longitudinal electric one at finite m e d i u m density shows 337
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E. X
2
©
0 0.00
0.20
0.40
0.60
0.80
1.00
1.20
q2 (GeV 2) Fig. 1. Magnetic form factor for the N - A transition as a function of the momentum transfer q2 for various medium densities: 0.0 (solid curve), 0.08 (dashed curve), 0.16 (dotted curve) and 0.24 fm -3 (dash-dotted curve). The experimental data (11) [31 ] correspond to zero medium density.
b 0
E
35 3O
0 M.-
25
N
20
(9 -(3
15
c 0 ~)
10
\
":tX '. \ ~x x
O c
5
cO c)
0 0.00
0.20
0.4-0
0.60
0.80
1.00
1.20
q2 (GeV 2) Fig. 2, Longitudinal electric quadrupole form factor for the N - A transition versus momentum transfer q2 for different baryon densities: 0.0 (solid curve ), 0.08 (short-dashed curve ), 0.16 (long-dashed curve ) and 0.24 fm-3 (dotted curve ).
a stronger reduction at zero as well as at finite momentum transfer. The calculated nNA vertex form factor G,,~a for different medium densities is depicted in fig. 3. The value at the origin (the coupling constant g~NA) as well as the ratio g,,,,,/g,~,N stay practically constant with the medium density. For very low momentum transfers up to 0.05 GeV 2 there are no medium ef338
fects. At larger momentum transfers the G,Na shows a significant reduction increasing with both the medium density and the momentum transfer. In particular, for P,m and q2 ~ 0.15 GeV 2 (which is relevant for the A excitation in the charge-exchange reactions) the reduction is about 15%. The A width for the A ~ xN decay, calculated from eq. (14) using the corresponding medium values, show a reduction of
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0.60
0.40
E.r-v
C~
©
0.20
"'.% N \-,\ -. \ -, "... \ x x
0.00 0.00
0.20
0.40
0.60
0.80
1.00
1.20
q2 ( G e V 2) Fig. 3. Pion nucleon delta form factor versus momentum transfer q2 for different baryon densities: 0.0 (solid curve), 0.08 (short-dashed curve), 0.16 (long-dashed curve) and 0.24 fm-a (dotted curve).
about 20% with respect to its vacuum value. This contradicts the general believe that the delta width should increase in the medium. One should note, however, that the reduction is related to the width due to the direct A--,nN decay since the large contribution to the A width in medium coming from the AN---,NN channel is not taken into account in our calculations. Actually after submitting the paper the present authors were confronted with the results of Meil3ner [ 33 ] concerning the properties of the delta and the N-A transition form factors in baryon medium. There a generalized skyrmion model including dynamical vector mesons was used with medium modified meson values evaluated within a suitably formulated Nambu-Jona-Lasinio model. His numbers, however, show a strong dependence on the parameter sets used. The reduction of the nucleon-delta mass splitting predicted in the medium is smaller than ours as two of the parameter sets give no change and the others a reduction of about 9%. Unfortunately, as far as the N-A transition form factors are concerned the resuits in ref. [33 ] are not detailed enough in order to allow a direct comparison with our numbers. The pion nucleon-delta transition form factor shows a q dependence similar to ours at nuclear matter density and finite momentum transfers.
To summarize, using the Nambu-Jona-Lasinio model for the description of the meson sector of the medium with a finite baryon density we are able to evaluate the solitonic sector of a delta embedded into this continuum. We find a noticeable decrease of both the energy of the delta and the nucleon-delta splitting at finite baryon density. The nucleon-delta transition form factors, namely the electromagnetic and the pion nucleon delta ones, get strongly reduced at finite momentum transfers whereas the pion-nucleon-delta coupling constant itself stays nearly constant. Valuable discussions with F. Osterfeld are gratefully acknowledged. The work has been supported by the JNICT, Lisbon, by the Bundesministerium ftir Forschung und Technologie, Bonn (Internationales BiJro and contract No. 06-BO-702), by the KFA Jiilich (COSY Project) and the Deutsche Forschungsgemeinschaft.
References [ 1 ] c. Elegaard et al., Phys. Lett. B 154 (1985) 110. [ 2 ] D. Contrado et al., Phys. Lett. B 168 (1986) 331.
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[3] L. Bergqvist et al., Nucl. Phys. A 469 (1987) 648; V. Bernard, U.-G. MeiBner and I. Zahed, Phys. Rev. D 36 (1987) 819; Phys. Rev. Lett. 59 (1987) 966. [4] C. Elegaard, Can. J. Phys. 65 (1987) 600. [5] C. Gaarde, Nucl. Phys. A 478 (1988) 475c. [6] C. Elegaard et al., Phys. Rev. Lett. 59 (1987) 974. [7] C. Elegaard et al., preprint (October 1988). [ 8 ] B.E. Bonner et al., Phys. Rev. C 18 ( 1978 ) 1418. [9] D.A. Lind, Can. J. Phys. 65 (1987) 637. [ 10 ] H. Esbensen and T.-S.H. Lee, Phys. Rev. C 32 ( 1985 ) 1966. [ 11 ] G. Chanfray and M. Ericson, Phys. Lett. B 141 ( 1984 ) 163. [ 12] T. Udagawa, S.-W. Hong and F. Osterfeld, Phys. Rev. Lett. (1990), in print. [ 13 ] J. Ahrens, Nucl. Phys. A 446 ( 1985 ) 229c. [14] J. Ahrens, L.S. Fereira and W. Weise, Nucl. Phys. A 485 (1988) 621. [ 15] P. Barreau et al., Nucl. Phys. A 402 (1983) 515. [ 16] J.S. O'Connell et al., Phys. Rev. Lett. 53 (1984) 1627. [ 17 ] R.M. Sealock et al., Phys. Rev. Lett. 62 (1989) 1350. [ 18 ] E. Ruiz Arriola, Chr.V. Christov and K. Goeke, Phys. Lett. B 225 (1989) 22; Nucl. Phys. A 510 (1990) 689. [ 19] M. Gell-Mann and M. Lrvy, Nuovo Cimento 16 (1960) 705. [20] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 ( 1961 ) 354. [ 21 ] M.A. Shifman, A.J. Vainstein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385;B 163 (1980)43.
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[22 ] M. Fiolhais, K. Goeke, F. Griimmer and J.N. Urbano, Nucl. Phys. A 481 (1988) 727. [23] Th. MeiBner, E. Ruiz Arriola and K. Goeke, Z. Phys. A (1990), in print. [24] H.F. Jones and M.D. Scadron, Ann. Phys. 81 (1973) 1. [25 ] P. Alberto, E. Ruiz Arriola, M. Fiolhais, F. Griimmer, J.N. Urbano and K. Goeke, Phys. Lett. B 208 (1988) 75; P. Alberto, E. Ruiz Arriola, M. Fiolhais, F. Griimmer, K. Goeke and J.N. Urbano, submitted to Z. Phys. A. [26] E. Braaten, S.-M. Tse and C. Willcox, Phys. Rev. D 34 (1986) 1482. [27] R. Machfleidt, K. Holinde and Ch. Elster, Phys. Rep. 149 (1987) 1. [28 ] T.E.O. Ericson and W. Weise, Pion and nuclei (Oxford U.P., Oxford, 1988). [29 ] W.W. Ash, K. Berkelman, C.A. Lichtenstein, A. Ramanauskas and R.H. Siemann, Phys. Lett. B 24 (1967) 165. [30] G.S. Adkins, Ch.R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552. [ 31 ] N. Kaiser, U. Vogl and W. Weise, Nucl. Phys. A 490 ( 1988 ) 602. [32 ] E. Amaldi, S. Fubini and G. Furlan, Pion-electroproduction, Springer Tracts in Modem Physics, Vol. 83 (Springer, Berlin, 1979). [33] U.-G. MeiBner, Nucl. Phys. A 503 (1989) 801.
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Medium effects on A properties and N - A transition form factors Chr.V. C h r i s t o v a,~, M. F i o l h a i s b, E. R u i z A r r i o l a a a n d K. G o e k e a a Institut~r TheoretischePhysikll, Ruhr-UniversitdtBochurn, D-4630Bochum, FRG b Departarnento de Fisica and Centro de Fisica Tedrica (INIC), Universidade de Coirnbra, P-3000 Coimbra, Portugal
Received 27 November 1989; revised manuscript received 13 March 1990
The properties of the A( 1232) isobar immersed in a nuclear medium are investigated in a chiral quark-meson theory. To this end the Nambu-Jona-Lasinio model is used to evaluate the pion decay constant, the pion mass and the sigma mass at finite medium density. These values serve to fix the parameters of the Gell-Mann-Lrvi sigma model. The latter is then solved in a variational mean-field and projection approach in order to obtain the static delta properties and yNA and nNA vertices. At the nuclear matter density both the mass of the delta and the nucleon-delta mass splitting are reduced by about 20%, the radii show swelling and the form factors get noticeably reduced at finite momentum transfers relevant for charge-exchange reactions and electron scattering.
The idea that the nucleon properties in the nuclear environment should differ from those o f the free nucleon has been suggested in the last years from both the experiment and the theory. The same arguments, at least from the side of theory, should be valid for the properties o f the nucleon resonances, in particular to those o f the A. In contrast to the nucleon, however, there are no such clear evidences for the medium modifications o f the A properties from the side of experiments. Only some charge-exchange experimental data seem to support a possible modification. In particular, in the spectra from (3He, t) reactions with targets heavier than '2C the A peak is shifted [ 15 ] towards lower energy loss than expected from the N - A mass splitting measured for proton target. A similar trend can be seen in the (d, 2p) [ 4 - 7 ] and some heavy ion induced [4,5] charge-exchange reaction data. For (n, p) [8,9] reactions these effects are not observed although even in the last case some speculations about medium effects have been put forward [ 10 ]. A consistent and generally accepted explanation o f the observed energy shift does not yet exist. It seems to be clear, however, that it cannot be explained [4,10 ] as originated from the kinematics and due to the dependence o f the 3He-t form factor on the m o m e n t u m transfer. Chanfray and Ericson [ 11 ] showed that the pion-exchange interaction,
treated in the random phase approximation, may modify the longitudinal response in contrast to the transverse one. Esbensen and Lee [ 10] studied the charge-exchange reactions within Glauber scattering theory with a response of the semi-infinite nuclear matter in which the residual pion-exchange interaction is included in the r a n d o m phase approximation as well. They concluded, however, that the energy shift cannot be explained solely in terms of the pion-exchange interaction and that a reduction o f the nucleon-delta mass splitting due to the nuclear environment has to be included. In contrast to Esbensen and Lee, using a coupled channel formalism, Udagawa et al. [ 12 ] claim that the energy shift can be explained as an effect mainly coming from the A N - i correlations induced by the pion-exchange interaction. It should be noted that in both refs. [ 11,12 ] a possible modification of the delta properties itself by the medium, especially o f the N - A mass splitting, is neglected. In contrast to the charge-exchange reactions the ~/ absorption [ 13,14 ] and the inclusive inelastic electron scattering data [ 15-17 ] show no energy shift of the A centroid or even a small shift in the opposite direction (to higher energy loss) independent o f the target mass. This may be related to the fact that the electromagnetic probes are sensitive only to the
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transverse response function whereas the hadronic probes examines both the transverse and longitudinal spin-isospin response. The experimental information, however, is not detailed enough [ 17] (especially the problem of background contributions is not solved) to make more definite conclusions. Apparently, in contrast to the nucleon the experimental indications for medium effects in the delta properties are rather unclear. Nevertheless, we take the situation as a motivation to investigate the properties of the delta and of the nucleon-delta transitions, if both systems are immersed in a nuclear medium. Our procedure of studying the properties of the delta is similar to the one applied to the medium effects in the case of nucleon [ 18 ]. It is based on the assumption that a nuclear medium can approximately be represented by a quark medium. Furthermore it is assumed that in the framework of the GellMann-Lrvi cr model [ 19 ] the influence of the medium on the delta can be solely expressed by means of modified values of the pion coupling constant f~ and meson masses m~ and rn~. In practice the meson sector of the quark medium is described by the Nambu-Jona-Lasinio model [20]. It is first adjusted for the vacuum to reproduce the experimental value off~ and rn~, the empirical value of the quark condensate and quark bare mass in accordance with the QCD sum rules [ 21 ] and to fulfill the PCAC. In a second step it is solved for a finite medium density yielding modified v a l u e s f ~*,m ~*and rn*o. These meson values actually fix the parameters in the Gell-Mann-L6vi lagrangian [ 19 ] which is then solved in the solitonic sector by mean field and projection methods [22]. The details of the approach are described in ref. [ 18 ] and will here be only sketched shortly. We start with the lagrangian of the Nambu-JonaLasinio model with scalar and pseudoscalar quarkquark couplings [ 20 ]:
= 9i~. 0~~ - m o ~P~+ ½G[ ( ~P~)2+ ( ~5 T~)2]. (1) Introducing auxiliary sigma and pion fields by
a=-g~PT/l~ 2 and ~ t = - g ~ , s r T / ~ 2 and assuming them to be classical (zero boson and one Fermi loop approximation), one can prove [ 18 ] that the total energy density reads 334
E=-4Nc
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f
d3k x/k2+g2(tr2 + ~t2) (2/t) 3
kF<~ lkl <~A
+ ½/t2(cr2+ f t 2 ) - o t a
(2)
with a three-momentum cut-off.4 in order to make the integral finite. The a is related to the bare mass mo by a = -lt2mo/g, the G is expressed as G=g2/# 2 and/~ is fixed according to MeiBner et al. [23 ]. The Fermi momentum is given by kF= (3n2p)1/3, where p is the medium density. At the zero density (kF = 0) demanding PCAC and the stationary phase conditions for the energy E with respect to the vacuum values of cr and n one can fix all parameters of the Nambu-Jona-Lasinio model except for the cut-off.4 and the constituent mass M=gdc~. The meson masses are related to the second derivatives of the energy at the stationary point. For f~ and m~ we assume their experimental values. For a given particular value of M t h e A is fixed such that the pion decay constantf~, evaluated by the corresponding Feynman diagram, agrees with the vacuum value of cr which is av=f~ as well. All those conditions together leave M as the only free parameter. For it we use the particular value M=gf~ = 463 MeV obtained in ref. [ 18 ] such that the nucleon mass, evaluated within the Gell-Mann-Lrvi model, is MN=938 MeV. Actually the value of the constituent quark mass used leads to a good reproduction of the structured vacuum. At finite medium densities the symmetry breaking term and the above vacuum values of #, A and g are kept fixed because they define the lagrangian. The medium values of the pion decay coupling constant and meson masses, indicated by an asterisk, can be read of the total energy E* of the system, extracted from eq. (2) with a finite Fermi momentum kF. The medium reduced pion decay constant, f * , is given by the minimum of E* with respect to o-*. The second derivative of E* with respect to o-* and ~r* at or* = f * and at zero, respectively, are used to determine the masses m*~ and m* in the medium. For completeness we present in table 1 the calculated values for meson characteristics at zero, half and full nuclear matter density (Pnm= 0.16 fm-3). One realizes a clear decrease o f f * and m* and an increase of m* with increasing medium density. As already shown in refs. [ 3,18 ] at about 3 times the nuclear matter density, a phase transition from
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the chirally broken Goldstone phase to the restored Wigner phase takes place beyond which f * ~ 0 and m . ~~ mo, • the latter increasing with medium density p. As a next step the f * , m* and m* are used to define a modified Gell-Mann-Lrvi lagrangian 5 e* to describe the delta structure in the medium:
- (½a*)~(o-*~+ n * ~ - ~*~) ~+ ½o~o-* 0~o-* 1 $ + ~0utt • 0utt $ - L m ~ 2o *
(3)
a * = f * , 2 *z =(mo,z - m ~.z )/2f~.2 and u.2 = f ~ : - m ~.2 /2 ,~ fixed by inserting the calculated meson values. For the nucleonic solution of this lagrangian in ref. [ 18 ] we have employed the variational procedure of ref. [ 22 ] based on mean field states with generalized hedgehog structure and projection techniques for spin and isospin. Similarly to the nucleon, the A state appears as a bound state (soliton) in this variational treatment. To be specific, the mean field trial function for the A is assumed to be I ~ ) = t q3) iZ ) IH ). Here ]Z ) and IH ) are quantal coherent Fock states representing the sigma and pion cloud and the Iq3) corresponds to three quarks of different colour in a 1s orbit with the spin-flavour structure of a generalized hedgehog: I u+ ) sin q - Id~ ) cos r/. The actual A state is obtained from I ~ ) by application of projection operators: with
I~JJr) =
Z oJ, ~K, TK TD~ JMK~to TMTKT II ~ . t ) . K,Kr
(4)
The gX.K~, Jr the flavour mixing coefficient q and the orbital degrees of freedom in [q 3), IZ ) and IH ) are determined variationally as described in detail by Fiolhais et al. [ 22 ]. Needless to say that consistency checks like virial theorems are fulfilled in the vacuum as well as at finite density. The projected solution with good spin and isospin numbers obtained for both the nucleon and A can be used for calculations of the static A properties as well as of form factors corresponding to the transition vertices TNA and nNA. To that end we neglect the recoil effects and assume that the projected state of the nucleonic soliton is a good zero momentum state. For the electromagnetic N - A form factors we use, up to a trivial phase, the standard decomposition for the
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matrix element of the electromagnetic current (A(p') IJgmI N ( p ) ) given by Jones and Scadron [24]. The states I A ( p ' ) ) and I N ( p ) ) represent either a A° and a neutron state or else a A ÷ and a proton (momenta p' and p), and for them we use Rarita-Schwinger and Dirac spinors respectively. The nature of the N-A electromagnetic transition is threefold: It might correspond to a transverse electric quadrupole, to a transverse magnetic dipole, or to a longitudinal electric quadruple. The associated form factors are GEa(q2), G ~ ( q 2) and GCA(q2) respectively. Working in the limit of nucleon-delta degeneracy and using the Breit frame for convenience [in which pU= (E, ½q) a n d p ' u= (E, - ½q) ] one can relate the electromagnetic transiton form factors to the corresponding matrix elements: (A(½q) I J°m ( 0 ) I N ( - ½q) ) = ~ 3 2
GCAqkqtf k', (5)
(A(½q) I f i ~ ( 0 ) I N ( - ½q) ) - - - 3i -- 4M N
E
M
( GNa--GNA
3i rz_e {xik_ + 2MN VNA ~k~
) eijkqJg k
qiqk\ qtfkt' ~qT-)
(6)
with the form factors G~a, G ~ and GC~ taken at _q2. In the above expressions,
gk=(1,s'3-s3; ½,s313, s'3) %~-s3,k, fkt= Z ( 1, m; ½, s'3 --ml 3, s'3 ) qlmk(mlatls3 ), m
where q/mk ( m = - 1, 0, 1 ; k = 1, 2, 3) is the transformation matrix from the cartesian to the spherical basis. Considering initial and final states with spin and isospin third components + ½, explicit expression for the electromagnetic nucleon-delta transition form factors can be obtained from eqs. ( 5 ) and (6). Hence, the form factors GNM~,GCNaand G~A read [25]
GMNa =x//6fd3r jl(qr) 2MN qr
( A~-/2I/Jo IN~/2 ),
(7)
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f "3
2MN
zA(qr)
J a rr ~
t"
g~Na = 2 M N "I J d3r z ( A~-/2 [ J~INI/2) "o + .
Y2o(r)
X (A~/21 J°m(r)IN+/2 ), G~Na=0,
(8) (9)
where I A~-/2) and IN~-/2 ) are the projected states I J = T = 3, M = M r = 1 ) and IJ = T = ½, M = M r = 1 ) respectively. In eq. (7), the/1o is the zeroth-zeroth component of the magnetic moment o p e r a t o r / i = ½( r X J ) and ffegm= (J°m, 3) J' the electromagnetic current. Obviously, only the isovector parts of the operators/io and J°m may contribute to G~A and GC~. The vanishing of GEa also occurs in the Skyrme model [26] in leading order in 1~No. Although different authors agree on the smallness of GEa(O)/Gr~a(O), the actual value for this quantity has not yet been well established. The various analyses reported in ref. [24] seem to indicate for that ratio a value less than 0.05. The limit q--,0 of eq. (7) gives the nucleon-delta transition magnetic moment: #N~ = ~ d 3r (A+/2 l/1o IN~-/2 ).
(10)
For the nNA form factor G~Na we assume the simplest coupling [27,28 ] associated with the transition: ( A ( P ' ) [ 3~ I N ( p ) )
G~Na(q 2) -- - a~(p', s' ) pUUN(p, S) 2MN X (~ta 17~q ½tN ),
( 11 )
where ]~ is the pion current source operator, the 7~ is the isospin transition operator and the a~(p', s' ) and the uN(P, s) are the Rarita-Schwinger and Dirac spinors, respectively. Working in the Breit frame one can obtain for the GnNa in the limit of nucleon-delta degeneracy M~
GnNa = 6 M N
¼qZ+M2
x3 f d3rJl(qr) z(A+/2 [ J~IN~/2). % + qr
(12)
The limit q ~ 0 ofeq. (12) gives the pion-nucleon delta coupling constant:
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(13)
In eqs. (12) and (13) the factor ~ comes from the matrix elements of the spin and isospin transition operators. Our results for the static A properties at vanishing medium density are presented in the second column of the table 1 in comparison to some available experimental estimates (first column). They are very similar to the numbers of ref. [22] which are obtained with a slightly different sigma mass. The A mass appears to be smaller by 11% than the corresponding experimental values and hence the calculating N-A splitting exhausts only a half of the experimental value. In the model the N-A splitting originates basically from the difference of pion kinetic energies in the respective clouds. It may possibly be improved by incorporating an additional chromomagnetic interaction between the quarks due to one gluon exchange. Despite of the fact that the charge radii and the magnetic moments are not measurable we present them for completeness. More interesting are the observables connected to the N-A transition since there is some experimental information available. Both the magnetic moment and the RMS radius are in a good agreement with the experimental estimates [26,3,29]. The ratio #,6/(/zp-/~n)=0.67, however, is close to 1/x/~ which follows from the collective quantization [ 30,26 ] and deviates a bit from the experimental ratio 0.76. Similar to the g~NN calculated in ref. [22 ], g~aN is noticeable larger than the experimental number. The latter, following Jones and Scadron [24 ],
1
F(A--,Nn) = ~-~np
,3E'+MN(g~Na~ 2 Ma ~,2MN,I '
(14)
is related to the experimental value of 120 MeV for the A width. The p' and E' in eq. (14) are the momentum and the energy of the outgoing nucleon. The ratio g~r~a/g~NN=2.65 is larger than the empirical value of 2.18 (we take g~NN= 13.5 ) and also exceeds the value of 3/x/~ expected [30] from 1~No expansion in the Skyrme model as well as the value of 2.15 obtained by Kaiser et al. [ 31 ] in Skyrme-type calculations with vector mesons. At finite medium density all A properties except
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Table 1 Medium effects on delta properties. For finite medium densities the values are given relative to the theoretical numbers at vanishing density. Quantity
Absolute values experiment
f~ (MeV) rn~ (MeV) mo (MeV)
Ea (MeV) Ezx-EN (MeV) (r~ >a~++> (fm 2) (rez )a~+~ (fm 2) ( r 2 )~0~ (fm 2) (r2)a~-~ (fm 2) /tac++~ (n.m.) /~a~÷~ (n.m.) /tA~0>(n.m.) ga~-) (n.m.) ~/( r2 )Na (fm) /~Na (n.m.) g~Nzx
93 139.6 1232 295
-
Relative values p = 0.0 fm - 3
p = 0.08 fm- 3
p = 0.16 fm - 3
93 139.6 937.7
0.89 1.06 0.93
0.77 1.14 0.85
1094 156 1.44 0.67 -0.10 -0.87 5.57 2.21 - 1.14 -4.50
0.92 0.92 1.16 1.19 0.94 1.11 1.04 1.06 0,98 1.02
0.82 0.79 1.41 1.41 0.80 1.30 1.10 1.15 0.92 1.03
0.89 3.42 45.4
1.10 1.05 1.01
1.25 1.11 1.04
0.84 3.67 29.6
for the g~Na show a clear modification. The relative n u m b e r s for half a n d full nuclear m a t t e r density are s u m m a r i z e d in the last two columns in table 1. The energy o f the delta as well as the n u c l e o n - d e l t a mass splitting decrease with the m e d i u m density resulting in E*~/E~ = 0 . 8 3 and E~N/EAN = 0 . 7 9 at Pn~. The reduction o f the N - A splitting is very close to the shift o f the energy loss centroid observed in (3He, t) charge-exchange reactions [ 1-5 ]. Such a direct comparison, however, is not straightforward since the particular reaction m e c h a n i s m m a y also play an important role. Nevertheless the reduction is rather large and hence should be included in the considerations dealing with the excitation o f the delta in nuclear medium. Indeed, this has been done by Esbensen a n d Lee [ 10 ] who in their a p p r o a c h need a reduction o f about 10% o f the N - A splitting ( s i m i l a r to our n u m ber at half nuclear m a t t e r density) in o r d e r to explain the shift o f the A peak position. The b e h a v i o u r o f the charge MS radii o f the delta at finite m e d i u m density are similar to those o f the nucleon. F o r the N - A transition both the MS radius and the magnetic m o m e n t show an increase a n d it is reflected in the b e h a v i o u r o f the form factor G~a at small transfer m o m e n t a . This form factor is plotted as a
function o f the f o u r - m o m e n t u m transfer q2 for various m e d i u m densities in fig. 1. The experimental data [ 32 ] shown corresponds to zero density. A p p a r e n t l y the present results are in fairly good agreement with the experimental estimates. In particular it is valid for the value o f the form factor at the origin (the transition magnetic moment/tNa) a n d for its slope at the same p o i n t (related to the magnetic transition R M S radius) both very close to the experimental numbers (presented in table 1 ). At increasing med i u m density because o f the increase o f the transition magnetic m o m e n t the GNMashow a slight increase at low m o m e n t u m transfer q2~<0.1 G e V 2 a n d a decrease at larger transfers. The reduction is less than in the case o f the other two transition form factors a n d at q 2 ~ 0.4 G e V 2 consists o f about 20% at Pnm. Fig. 2 displays the longitudinal electric quadrupole transiton form factor for several m e d i u m densities. Although the experimental d a t a are very scarce, at least the m o d e l predictions seem to be in qualitative agreement with estimates [24] on the ratio o f the electric q u a d r u p o l e to the magnetic dipole amplitudes in n u c l e o n - d e l t a transitions. In contrast to the transverse magnetic transition form factor, the longitudinal electric one at finite m e d i u m density shows 337
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E. X
2
©
0 0.00
0.20
0.40
0.60
0.80
1.00
1.20
q2 (GeV 2) Fig. 1. Magnetic form factor for the N - A transition as a function of the momentum transfer q2 for various medium densities: 0.0 (solid curve), 0.08 (dashed curve), 0.16 (dotted curve) and 0.24 fm -3 (dash-dotted curve). The experimental data (11) [31 ] correspond to zero medium density.
b 0
E
35 3O
0 M.-
25
N
20
(9 -(3
15
c 0 ~)
10
\
":tX '. \ ~x x
O c
5
cO c)
0 0.00
0.20
0.4-0
0.60
0.80
1.00
1.20
q2 (GeV 2) Fig. 2, Longitudinal electric quadrupole form factor for the N - A transition versus momentum transfer q2 for different baryon densities: 0.0 (solid curve ), 0.08 (short-dashed curve ), 0.16 (long-dashed curve ) and 0.24 fm-3 (dotted curve ).
a stronger reduction at zero as well as at finite momentum transfer. The calculated nNA vertex form factor G,,~a for different medium densities is depicted in fig. 3. The value at the origin (the coupling constant g~NA) as well as the ratio g,,,,,/g,~,N stay practically constant with the medium density. For very low momentum transfers up to 0.05 GeV 2 there are no medium ef338
fects. At larger momentum transfers the G,Na shows a significant reduction increasing with both the medium density and the momentum transfer. In particular, for P,m and q2 ~ 0.15 GeV 2 (which is relevant for the A excitation in the charge-exchange reactions) the reduction is about 15%. The A width for the A ~ xN decay, calculated from eq. (14) using the corresponding medium values, show a reduction of
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0.60
0.40
E.r-v
C~
©
0.20
"'.% N \-,\ -. \ -, "... \ x x
0.00 0.00
0.20
0.40
0.60
0.80
1.00
1.20
q2 ( G e V 2) Fig. 3. Pion nucleon delta form factor versus momentum transfer q2 for different baryon densities: 0.0 (solid curve), 0.08 (short-dashed curve), 0.16 (long-dashed curve) and 0.24 fm-a (dotted curve).
about 20% with respect to its vacuum value. This contradicts the general believe that the delta width should increase in the medium. One should note, however, that the reduction is related to the width due to the direct A--,nN decay since the large contribution to the A width in medium coming from the AN---,NN channel is not taken into account in our calculations. Actually after submitting the paper the present authors were confronted with the results of Meil3ner [ 33 ] concerning the properties of the delta and the N-A transition form factors in baryon medium. There a generalized skyrmion model including dynamical vector mesons was used with medium modified meson values evaluated within a suitably formulated Nambu-Jona-Lasinio model. His numbers, however, show a strong dependence on the parameter sets used. The reduction of the nucleon-delta mass splitting predicted in the medium is smaller than ours as two of the parameter sets give no change and the others a reduction of about 9%. Unfortunately, as far as the N-A transition form factors are concerned the resuits in ref. [33 ] are not detailed enough in order to allow a direct comparison with our numbers. The pion nucleon-delta transition form factor shows a q dependence similar to ours at nuclear matter density and finite momentum transfers.
To summarize, using the Nambu-Jona-Lasinio model for the description of the meson sector of the medium with a finite baryon density we are able to evaluate the solitonic sector of a delta embedded into this continuum. We find a noticeable decrease of both the energy of the delta and the nucleon-delta splitting at finite baryon density. The nucleon-delta transition form factors, namely the electromagnetic and the pion nucleon delta ones, get strongly reduced at finite momentum transfers whereas the pion-nucleon-delta coupling constant itself stays nearly constant. Valuable discussions with F. Osterfeld are gratefully acknowledged. The work has been supported by the JNICT, Lisbon, by the Bundesministerium ftir Forschung und Technologie, Bonn (Internationales BiJro and contract No. 06-BO-702), by the KFA Jiilich (COSY Project) and the Deutsche Forschungsgemeinschaft.
References [ 1 ] c. Elegaard et al., Phys. Lett. B 154 (1985) 110. [ 2 ] D. Contrado et al., Phys. Lett. B 168 (1986) 331.
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[3] L. Bergqvist et al., Nucl. Phys. A 469 (1987) 648; V. Bernard, U.-G. MeiBner and I. Zahed, Phys. Rev. D 36 (1987) 819; Phys. Rev. Lett. 59 (1987) 966. [4] C. Elegaard, Can. J. Phys. 65 (1987) 600. [5] C. Gaarde, Nucl. Phys. A 478 (1988) 475c. [6] C. Elegaard et al., Phys. Rev. Lett. 59 (1987) 974. [7] C. Elegaard et al., preprint (October 1988). [ 8 ] B.E. Bonner et al., Phys. Rev. C 18 ( 1978 ) 1418. [9] D.A. Lind, Can. J. Phys. 65 (1987) 637. [ 10 ] H. Esbensen and T.-S.H. Lee, Phys. Rev. C 32 ( 1985 ) 1966. [ 11 ] G. Chanfray and M. Ericson, Phys. Lett. B 141 ( 1984 ) 163. [ 12] T. Udagawa, S.-W. Hong and F. Osterfeld, Phys. Rev. Lett. (1990), in print. [ 13 ] J. Ahrens, Nucl. Phys. A 446 ( 1985 ) 229c. [14] J. Ahrens, L.S. Fereira and W. Weise, Nucl. Phys. A 485 (1988) 621. [ 15] P. Barreau et al., Nucl. Phys. A 402 (1983) 515. [ 16] J.S. O'Connell et al., Phys. Rev. Lett. 53 (1984) 1627. [ 17 ] R.M. Sealock et al., Phys. Rev. Lett. 62 (1989) 1350. [ 18 ] E. Ruiz Arriola, Chr.V. Christov and K. Goeke, Phys. Lett. B 225 (1989) 22; Nucl. Phys. A 510 (1990) 689. [ 19] M. Gell-Mann and M. Lrvy, Nuovo Cimento 16 (1960) 705. [20] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 ( 1961 ) 354. [ 21 ] M.A. Shifman, A.J. Vainstein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385;B 163 (1980)43.
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[22 ] M. Fiolhais, K. Goeke, F. Griimmer and J.N. Urbano, Nucl. Phys. A 481 (1988) 727. [23] Th. MeiBner, E. Ruiz Arriola and K. Goeke, Z. Phys. A (1990), in print. [24] H.F. Jones and M.D. Scadron, Ann. Phys. 81 (1973) 1. [25 ] P. Alberto, E. Ruiz Arriola, M. Fiolhais, F. Griimmer, J.N. Urbano and K. Goeke, Phys. Lett. B 208 (1988) 75; P. Alberto, E. Ruiz Arriola, M. Fiolhais, F. Griimmer, K. Goeke and J.N. Urbano, submitted to Z. Phys. A. [26] E. Braaten, S.-M. Tse and C. Willcox, Phys. Rev. D 34 (1986) 1482. [27] R. Machfleidt, K. Holinde and Ch. Elster, Phys. Rep. 149 (1987) 1. [28 ] T.E.O. Ericson and W. Weise, Pion and nuclei (Oxford U.P., Oxford, 1988). [29 ] W.W. Ash, K. Berkelman, C.A. Lichtenstein, A. Ramanauskas and R.H. Siemann, Phys. Lett. B 24 (1967) 165. [30] G.S. Adkins, Ch.R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552. [ 31 ] N. Kaiser, U. Vogl and W. Weise, Nucl. Phys. A 490 ( 1988 ) 602. [32 ] E. Amaldi, S. Fubini and G. Furlan, Pion-electroproduction, Springer Tracts in Modem Physics, Vol. 83 (Springer, Berlin, 1979). [33] U.-G. MeiBner, Nucl. Phys. A 503 (1989) 801.