ETA photoproduction on the deuteron

Nuclear Physics A501 (1989) 856-868 North-Holland. Amsterdam

ETA PH~TOPRODUCTION Dean Pl~ysic.~ Department.

ON THE

HALDERSON

DEUTERON*

and A.S. ROSENTHAL

Western Michigan

University,

Kalamazoo,

MI 49008,

USA

Received 34 August 1988 (Revised 4 April 1989) Cross sections are calculated for the reaction ‘H( y, n)‘H. The calculation includes an impulse term and rescattering terms which are obtained by a reduction of a covariant formalism. Modern vertex functions and a realistic deuteron wave function are employed. Calculated 90” cross sections are at least a factor of two smaller than the existing data. The calculated angular distributions indicate that future experiments can determine the reaction mechanism and isolate the cause for the disagreement with theory.

Abstract:

1. Introduction Electroproduction and photoproduction experiments on the nucleon have provided valuable information on the electromagnetic transition amplitudes to excited baryonic states ‘). A determination of the transition amplitudes is required for detailed tests of theoretical models of hadron structure. For example, the nonrelativistic quark model with a relativistic correction to the transition operator has been successful in reproducing a broad range of data Im4). However, a number of discrepancies between theoretical predictions and experimental results persist. One outstanding discrepancy between quark model predictions and data is the strength of the S,,( 1535) isoscalar, electromagnetic transition amplitude, To, as measured in the reaction ‘H( y, T)~H. Whereas quark models predict a very small isoscalar as compared to the isovector (T’) amplitude, an experiment “) with the reaction ‘H(y, n)?H was analyzed in terms of the impulse approximation to give To>> T’. Such a result not only disagrees with the quark model, but also with an analysis of data for pion photoproduction from the nucleon in the region of the S,,( 1535) [ref. “)I, This situation is particularly disconcerting, because the TN channel is the dominant decay mode for the S,, , which makes the “H(r, q)‘H reaction a logical choice to determine the electromagnetic transition amplitude to the Sl, from the neutron. With the recent developments in neutral meson spectrometers, now would appear to be an especially appropriate time to begin additional experiments on eta photoproduction from the deuteron in an effort to clarify this discrepancy. The purpose of this article is to demonstrate how these new experiments can distinguish the reaction mechanism for ‘H(y, n)*H and thereby isolate the cause l

Supported

in part by the National

Science

Foundation.

0375-9474/89/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

D. Halde,son and A.S. Rosenthal

for the disagreement

with theory.

/ Eta

photr~pprodu~tion

This is accomplished

by showing

8.57

the results

of a

calculation for ‘H( y, T)‘H which employs modern vertex functions and a deuteron wave function obtained from the Paris potential ‘) with both S- and D-state components.

2. Background The photoproduction amplitude from the nucleon may be written as an isoscalar plus isovector part, T = To+ 7?T’, where r31p) = 1~). The impulse approximation for the deuteron becomes:

(1) Since the deuteron is an isospin-zero particle, the space-spin part of the wave function is symmetric and Td = T,,+ T, = T’,‘+ T(21.Therefore, in the region of the by quark models, one would expect little eta S,, where T,- -T, as predicted production from the deuteron. However, the experimental results of ref. ‘) show substantial eta production. The yield is large enough to conclude that T’ = 0, or that T,,= Tn. An impulse approximation calculation reported in ref. ‘) confirmed this analysis. There a value of To/ T, = 0.84 was required to fit the data. Quark-model predictions of this value for the S,, are approximately zero. Typical values are on the order of -0.02 [ref. ‘)I and 0.10 [ref. ‘)I. These values are in good agreement with those extracted from photo-pion data where the momentum transfers are much smaller. The 1984 Review of particle properties “) gives an average value of -0.02 i 0.16. A possible explanation of this dificulty is given in ref. “‘) where corrections to the impulse approximation were calculated. These corrections are due to rescattering of mesons as represented in fig. lb. In ref. I”) these rescattering terms were calculated to provide approximately five times the strength of the impulse term at 8, m = 90”. Agreement with experiment could then be obtained by assuming an isoscalar amplitude, Al?, = 12 x 10m3 GeV”’ which, using the value of A:?$““= 73 x 10 G GeV”’ from ref. ‘), gives To/ T,, = 0.16 which is acceptable. One might expect the rescattering diagrams to be large, since they allow both nucleons to absorb the

(a) Fig. 1. Graphs contributing

to the reaction ‘H( y, v)‘H:

(b) (a) the impulse term, (h) eta or T rescattering.

D. Halderson

858

and A.S. Rosenthal

/ Eta photoproducrion

momentum transfer, whereas in the impulse approximation, it seems unlikely that the deuteron could hold together if only one nucleon is struck. However, in the present work the contribution of the rescattering diagrams insufficient to account for the observed cross section.

is determined

to be

3. Theory In this paper the impulse and rescattering terms are calculated which are nonrelativistic reductions of a covariant formulation.

3.1. THE

IMPULSE

from expressions

APPROXIMATION

We use the methods of relativistic scattering theory developed by Arnold, and Gross “) for the deuteron and by Celenza and Shakin “) for general reactions. The S-matrix for deuteron photoproduction may be written as (rl; p’~ MrlSlk, h; pi =

Mi)

-(2x-)4j[16kEn(P)En(P’)w(q)]-“‘{,$’;,(M,)(q,

The term in curled A= j

Carlson nuclear

brackets

represents

d”p’ ~[:,(M,)[C’~(p’, I (27r)4 -

P’IA~“lk,

a Lorentz

invariant

quantity.

P’)]~:V,[P’-~‘-M,+je],~~,(q,

P’IM,$,,lk,

X&.(A)[~‘-M,+~]~~,[~-~‘-MN+~F]~~[~(~’, In eq. (2), M is the photoproduction

(2)

P)&‘I(A)[,,(Mi)}.

P-p’)

P)C]:y[u(Mi).

(3)

operator M = ; sP,A, , ,=I

where ~~~=2y,(~.p,k.p,-~.plk.p,),

.hz, = Y&, .tiJ =

rs(k . pid-

Eq. (2) is evaluated [p’Following amplitude

by placing

M,+je],,~+-2rrj

the notation by

of Arnold,

[7-@‘-

the spectator *

N

S(Ph-

Carlson,

M,+i~l,i$fi(~‘,

E [(27~)‘2P,j]“‘lL6(

(4)

-UZ,=y,(k.pze’-E.p&).

.z. P&L

nucleon E,(p’))

on the mass shell c U(P’, m)/AP’, ,n

and Gross I’), define a covariant

~),[T(P’,

m),

(5)

deuteron

P)ClLySv(Mi)

p’, m; P)&v( Mi) .

(6)

D. Halderson

The Lorentz

invariant

and A..!? Roenrhal

amplitude

/ Eta photoproduction

8.59

is therefore

A = [2f,,2P;,]“‘C

dp’

m; P’)

x (9, P’-p’lM&,Ik, P-p’l~,(A)+~(p’,

m; P)(lr(Mi).

W1I (7)

The wave function $i(p’, m; P) gives the amplitude for finding a nucleon of momentum p’ when the deuteron has momentum F? It is related to a similar amplitude evaluated in the deuteron rest frame:

where iii, = (0, MC,1 ,

_I,n;i, = P,

.I,(’

.l,P, =p’,

as well as the definition of the rotation angle R, are discussed boost 11: relates &$( p’, m; P’) to a rest-frame wave function

These relationships, in ref. ‘I). A similar and we have A=2M,

J

dp

x[S~ ‘(.Wln&,

P’-p’lM;&,

x ~~(h)[S(-,~,)l~~[~,;;‘(p,,

~[S-%i~)],y,~(q, x

ml;

P-p’)

~~,)~l,lDf!~,,,(R,RZ’)

P’-p’jM;,,Ik,

[S(.~~,)l,,~~,,,,,,,,[p,,5’lh(-P,,

In the last line we approximated the single-particle negative energy (antinucleon contributions): L~,,“(P,

m;

fi,)

= C

I

*,,d+(-P,

P-p’)e,,(h)

4

.

(9)

wave function

MN = ~ &-P~, EN( P’)

(10)

invariant

amplitude

in familiar

m4)[Sm’(A12)18.PXq, P’-p’lM$,/k,

x [S(.l,)lP8u,(-p,, The Lorentz

by neglecting

r).

Explicit forms for +,,,,. are given in ref. ‘I). Eq. (9) can be written by defining a generalized photoproduction operator H&,,,

= 5.

P-p’)

4)

takes the form of a nuclear

form

(11) matrix

element

(12)

860

0. Naldrrson

and AS.

Rosenthul

This form contains

the effects of Fermi motion

/ Eta ~~7~to~~o~z~~tio~

and the kinematic

modifications

of

the mechanical variables which follow from special relativity. A detaiied evaluation of these non-static effects will be reported on elsewhere. In this paper we have used eq. (9) as our starting point and made a frozen nucleon approximation, as discussed in ref. 13). In this approximation duction

operator

is written

the boosts are set equal to unity and the photopro-

in CGLN

= (4nWcl

Wxin,

form:

([j,

m])h~

where fir=a*EI,

f22=ia+rGx~,

and WC is the y-nucleon eq. (9) then becomes

&=0*k;jG,

center-of-mass

A=8xW,JE,(k)E,(q)/E,(tk) where

$(p)

is the Fourier

(2 c.m.) total

dpd,“(-tq-p) transform

n,=cr*@*G

[

energy.

The amplitude

121 1 i

of the nonrelativistic

4(x) = u(_X)[ Y”X’]fh,f

(13)

4(-;k-p),

Fin,

deuteron

w(x)[ Y’($x’]f,

in

(14)

wave function

.

(151

By evaluating the F amplitudes at the nN 2 c.m. energy and 0” and then transforming to coordinate space, one obtains the T-matrix in the yd center-of-mass (ACM),

where

t = k -4.

This expression

At this point it may appear a previous result. However, a rescattering terms. One must approximations in both terms

The Lorentz

invariant

B = t-11

d4Q (2n)4

agrees with the results that a great deal of major concern is the start with the same in order to compare

rescattering d"Q' (2T)4

[:

operator

of ref. 13).

work has been done to arrive at relative size of the impulse and formalism and make the same their relative contributions.

(to be added

to A above)

is

(Mfj

1,’

x[@(-Q’,

P’)]~::,,[P’+fZ(‘-MN+i~]y.:.[-O’-hilN+i&],,,l,,

x (4”-p“t

in)-

‘(q’, -Q'(ML(k

x [,Q’- MN+ i~l$[P-@-

MN+

P- QNs, P’+Q’I&,lq’,

Q)

i&)ii[r(Q, P)CII;yCa(Mi> 1

(17)

D. Halderson

In this equation,

and A.S. Rosmtkal

N is the meson-nucleon

/ Eta photoproduction

scattering

861

vertex:

N ==A+;(g,&)B. We again [-o’-

evaluate MN+

[Q-M,+

by placing

kg;:,,+

iP]&

the spectator

Ed Q')

-27ri *

x *I,,,,

,,I

on the mass shell:

s(Q~~- E(Q)) 1 u(Q, m)pfi(Q, ml,,. ,,l

N similar

to those

of the preceding

~“‘]D’1/~‘*(R_)D”‘“(R

[Q

I

*

,,I

3

E,(Q)E,(Q’)

,,I2

The same approximations

+(-Q,,

P-Q)&,(A) m,)S-‘(%)
(21)

eq. (14) to eq. (16) plus the identification

m3)(9, P’+ Q’INfi,,19’,

= (477W:lM)x:,J.f;

B can be

operator

Q)u(Q, m), .

that converted

(19)

(20)

1

,HVl,

m,)(9’~-~“+j~)-‘Un,(-QQz,

x (9, P’+ Q’\N,,,19’,

sections.

). ?

CC-Q’, m’)<,Jq’,-Q’(Whlk

xS(.l,),,,+,(-Q,,

m’),,,,

,,I’

photoproduction-rescattering

M2, K. ‘-

nucleons

-Z?ri--MU s(Qb+E(Q')) C u(-Q', m’),,u(-Q’,

After making substitutions reduced to the form

Here K is a combined

(18)

+.fi(o

Q)u,(Q, m) $)(a . 4)lxm

lead to

X(q”-p’+i~)~’

;

[

,=I

F,O,

@(-Q--;k)

(23)

I

This expression agrees with Lazard, et al. 15) and ref. “‘) up to factors of 2 Mhr - Md. Refs. I”) and IX) differ in their treatment of the energy denominator appearing in eq. (21), 9” -k + ie = 9h’- 9’” -EL’+ is. The prescription used in this paper is similar to that in ref. I’). It consists of placing the intermediate eta on shell by setting 9; = 9, (approximation #I). (For the pion rescattering 9;, is replaced with 9(, from the reaction yd + 7rd). In ref. I”) the denominator is simplified by approximating 9{, = W-

EN(Q) - EY( Q’) by W - E,($q)

- E,(:k)

(approximation

#2).

D. Halderson

862

and A.S. Ro.wnthal/

Eta photoproduction

A reasonable test of these approximations can be made with s-state, harmonic oscillator wave functions. If one considers only the S,, resonance, then the terms F, and f, are the contributors

to B. Therefore,

the amplitude

B is proportional

to

the integral: BK

I

dQ dQ’@(-Q’+;q)(q”-p’+i&)-‘@(-Q--_k) .

(24)

With the substitutions q’ = Q’- Q, s = $( Q + Q’), 4,(q) = (2p/ 7~)~‘~exp (-j3q2), and EN(Q) + M + Q’/2M, the integral can be reduced to a two-dimensional integral, which can be performed easily. The use of a non-relativistic nucleon energy is a very reasonable approximation for the momenta involved. In fact, more than 90% of the principal part of the integral comes from the region where the percentage difference between EN(Q) + EN( Q’) and 2M + (Q’+ Q’2)/2M is less than 0.5%. Eq. (24) becomes [-$(k”+q’)](2//3’73)

B~(2~)‘(2P/r)“~exp X

2s exp [-2pf]

x 32.&2q’exp

sin (/3st)

(-$q”)

(q’4+

sinh ($Tq’)

bq’2+ c)

dq’ ds

(25)

where T=-k-q, b=8(s’-MW), c=16{[s’-M(W-2M)]‘-M’p’)}, and W is the total ACM energy. The real and imaginary parts and the sum of their squares are plotted in fig. 2 for the quantity in curled brackets in eq. (25) at 0,, = 90” and p = 1.29 fm’, along with the results of approximations # 1 and #2. From this figure one sees little evidence for choosing one approximation over the other, but one does see that both approximations over predict the magnitude-squared of the integral by at least a factor of two through most of the energy range of interest. If one continues with approximation # 1, then the propagator in eq. (23) becomes space much simpler and a Fourier transform of the wave functions to coordinate yields

a T-matrix. 1 Tba = - (2~)’

1

w, w:. (kE,(q))“2

h&7)K&k)

“k+“-r’2&,(r)Lfi

+.&I (T)

[T F,Q,] dM,(r) .

(26)

In the derivation of eq. (26), it was assumed that the meson-scattering amplitude is evaluated at 0” and that i’= i in the F’s. The neglect of the angular dependence is again justified, as in writing eq. (24), by the dominance of the S,, in the energy

D. Haldemon and A.S. Rosenthal

640

660

660

863

/ Eta photoproduction

TOO

720

740

E’F (MeV) Fig. 2. The quantity

region of interest

in curled brackets in eq. (23). The solid line is the exact result. The dashed approximation #I. The dotted line is approximation #I.

and the near threshold

production

of the intermediate

line is

eta. In fact,

the I = I partial wave scattering ampIitudes for elastic eta scattering remain negligible for this energy region. The energy at which ,f; i-.h and the Fi’s is to be evaluated is chosen so that the elementary scattering amplitude can be placed on shell. Therefore-f, +.fz is evaluated at the n N 2 cm. energy; and, since we have q{, = q. in approximation # 1, the Fi’s are also evaluated at the nN 2 c.m. energy and 0”. The amplitudes, F;, for y+p+ n +p are taken from the global analysis of Hicks et al. Id) the nN elastic and GTN+ TN amplitudes are taken from the coupled channels analysis of ref. “) and the (y, 7~) amplitudes are taken from the analysis

864

D. Halderson

and A.S. Rosenthal

/ Eta photoproduction

of ref. “). The phases between refs. ‘“) and “) are fixed by demanding for the S,, contributions at resonance.

consistency

4. Results Shown

in fig. 3 as a solid line are the impulse

approximation

results

for the 90”

excitation function with To/T,= 0.84. This confirms the conclusion in ref. ‘) that the impulse approximation requires a value of TO/T,,which is inconsistent with the quark-model predictions and photopion results. Also included in fig. 3 is the impulse approximation result with To/T,, = 0.84 when the S,, contribution is omitted from the Hicks amplitudes. This demonstrates that the S,, , with its branching ratio for decay into 7 + N of about one-half, dominates the yd + qd yield in this energy region. The (7, 77) reaction should, therefore, be just the reaction to isolate the properties of the S,, . In fig. 4 are plotted the impulse approximation results, the contribution of the eta-rescattering diagrams, and the contribution of the pion-rescattering diagrams,

Fig. 3. Cross sections at 90” for the reaction ‘H(y, q)‘H. The solid line is the impulse term. The dashed line is the impulse term without the S,, contribution to the elementary amplitude.

&m=90”

-

Fig. 4. Cross sections at 90”. The dashed line is the impulse term. The dotted line is pion rescattering. The dot-dashed line is eta rescattering. The solid line is all terms contributing. Data are from ref. ‘).

D. Halderson as

well as the contributions

and A.S. Rosenthal

/ Eta photoproduction

86.5

of all those terms. These curves require

fit the data. This value is still far from acceptable. terms in this work does not produce

The inclusion

a factor of five enhancement

To/T, = 0.6 to

of the rescattering as it did in ref. I”),

but more like a factor of two. There are three reasons

for the difference

between

these results for eta rescattering

and those of ref. I(‘). The first is the difference in the approximations for the energy denominator as shown in fig. 2. The second is the evaluation of the off shell yN + 77N amplitudes. This effect is illustrated in fig. 5a where the magnitudes of the isoscalar F, used by ref. lo) and the isoscalar magnitude of Hicks, assuming T”/ T,, = 0.16 are plotted. The choice was made in this present work to place all elementary amplitudes on shell by evaluating them in the nN 2 c.m. system. For example at Eyh= 700 MeV one has an 7N 2 cm. energy of 7.76 fin-‘. Therefore F, has the same value for the impulse term as it does in the eta rescattering term. In ref. “‘) the prescription for the impulse and rescattering F,‘s is such that they are evaluated at 2 c.m. energies of 7.72 fin-’ and 7.50 fin-’ respectively for El;” = 700 MeV. These energies are shown in the figure where one can see an enhancement of 1.2 of the rescattering F, over the impulse F, . Third, the main difference between these results and those of ref. “I) is the difference in the nN elastic amplitudes employed. The Liu and Haider by a coupled channel analysis which amplitudes I’) used in this work are determined includes recent data. They produce an nN total cross section which is consistent Wc (MeV)

1.2 I.0 ZO.8

-

,I’ ‘\ /’

‘\\

_,=

Wq)N ‘\

o”

‘\ ‘.

ZN 0.6 r

_

e=

‘\ ‘.

‘.._

0.4 _

. .

t

0.2 0 7.4

‘--__

-

I

I

1

7.6

,

,

7.8 WC

, 8.0

-

, 82

(Cm-')

Fig. 5. (a) lsoscalar IF,) for N(y, q)N. The dashed line is from ref. “‘I with only S,, contributing; solid line - ref. “1 with T”/ T, = 0.6. (b) 1.f;+,/?I for N( 7, 7 IN. The dashed line is from ref. “‘) with only S,, contributing; the solid line is from ref. I”).

866

with recent

D. Halderson and A.S. Rosenfhal ,I Era ~hoi[)produ~fion

extractions

of this quantity

from the (n; v) reaction

on nuclei

I’). The

0”nN elastic amplitudes for both refs. “) and Ih) are plotted in fig. 5b. Here one can see that the amplitude in ref. ‘“) is at least a factor of two greater than that of ref. ” over the energy range of interest. This difference in 77 elastic amplitudes reflects the reduction of estimates of the S,, + v N branching ratio based on data accumulated since 1978. Of course the phases of the amplitudes will determine the exact overall factor distinguishing the calculation of ref. lo) and the present one, but by squaring the product of 1.2 and two, one arrives at a factor of approximately six for the difference between this work and ref. ‘O). This, plus the probable factor of two from use of approximation #2 for the energy denominator would make a total reduction of about a factor of 10 for the eta rescattering contribution reported in ref. lo). Although it would have been gratifying to see the rescattering terms produce agreement with data, the existence of large rescattering contributions would make determination of T, from experiments on deuteron targets very complicated. Such extractions rely on the impulse approximation. It is, therefore, cruciai to determine the size of the rescattering terms from experiments. Some experimental evidence does exist concerning the size of the rescattering terms. In ref. 18) it was reported that eta photoproduction experiments on heavier targets showed little evidence for reseattering. For the deuteron it is possible to directly distinguish the size of rescattering contributions by measuring appropriate anguiar distributions. This can be seen from the calculated angular distributions shown in figs 7 and 8 of ref. ‘“) or fig. 6 of this present work. In fig. 6 are plotted the cross sections at E.,ah= 675 MeV for the impulse approximation (solid line), eta-rescattering (dashed line), and pion-rescattering (dotted line). The impulse contribution is large at forward angles and small at backward angles, while the eta-rescattering contribution has just the opposite behavior. The pionrescattering is intermediate between the two. With these characteristic behaviors for

0

40

00 8c.m.

120

160

beg)

Fig. 6. Angular distributions for the reactions “H(y, q13H at E,, ,&-=675 MeV. The dashed line is the imp&e term. The dotted line is pion rescattering times five. The dot-dashed line is eta rescattering times five. The solid line is all terms contributing.

D. Haldemon

the three terms,

one could

and A.S. Rosenthal

determine

/ Eta photoproduction

the dominant

reaction

867

mechanism.

One can

also see from the angular distributions that 90” is not the angle for maximum confidence in extracting the isoscalar amplitude; not only is the predicted cross section larger at forward angles, but the contributions from rescattering are smallest there.

In addition,

approximately

30% of the 90” cross section

is from the deuteron

D-state wave function. The pion-rescattering angular distributions were not isolated in ref. ‘(‘), because they were calculated to be much smaller than the eta-rescattering terms. They are also small in the present calculation. This is in part due to the stronger damping of the exponential term in eq. (24), as pointed out in ref. “‘). However, the most notable reason for their being small is again the contribution of only the isoscalar electroproduction amplitude. Indeed, it would require only an isospin-violating term of 15% (J= P+ vn- -0.85,jV+n + 77~) at the rrN+ TN vertex to reproduce the magnitude of the experimental 90” cross section. However, the energy dependence of the 90” excitation function would be wrong, if pion-rescattering were the dominant reaction mechanism. Again, the measurement of angular distributions would determine, if the yd+ vd yields were due to pion-rescattering. 6. Conclusion This paper has presented the results of a calculation for yd+ vd based on a reduction of a covariant formulation. Modern vertex functions and a realistic deuteron wave function with S- and D-state components were employed. The calculation included impulse approximation and rescattering terms. Although previous calculations “) of such effects were successful in reproducing ‘H(Y, n)‘H data, their sum is not able to account for the experimental ‘H(y, n)‘H yield. Rescattering terms were calculated to be small and produce an energy dependence of the 90” excitation function which is not in agreement with experimental results. Angular distributions were calculated and shown to be critical in determining the reaction mechanism for photoproduction of etas from the deuteron. Experimental cross sections are, therefore, eagerly awaited. References 1) 2) 3) 4) 5) 6) 7) 8) 9)

F. Foster and G. Hughes, Rep. Prog. Phys. 46 (1983) 1445 F. Foster and G. Hughes, 2. Phys. Cl4 (1982) 123 R. Koniuk and N. Isgur, Phys. Rev. D21 (lY80) 1868 T. Kubota and K. Ohta, Phys. Lett. B65 (1976) 374 R.L. Anderson and R. Prepost, Phys. Reb. Lett. 23 (1969) 46 W.J. Metcalf and R.L. Walker, Nucl. Phys. B76 (1974) 253 M. Lacombe, ef al., Phys. Lett. 101 (1981) 139 Particle Data Group, Rev. Mod. Phys. 56 (1984) 5216 Y.N. Krementzova and A.I. Lebedev, in Meson-nuclear physics-1976, (Carnegie-Mellon, 1976) p. 634 10) N. Hoshi, H. Hyuga, and K. Kubodera, Nucl. Phys. A324 (1979) 234

AlP

Conf.

Proc.

no. 33,

86X 11)

D. Halderson

R.G. Arnold,

/ Eta photoproduction

C.E. Carlson, and F. Gross, Phys. Rev. C21 (1980) 1426 and C.M. Shakin, Relativistic nuclear physics (World Scientific, 1986) A.S. Rosenthal, D. Halderson, K. Hodgkinson, and F. Tabakin, Ann. of Phys. 184 (1988) 33 H.R. Hicks, et al., Phys. Rev. D7 (1973) 2614 C. Lazard, R.J. Lombard, and Z. Marie, Nucl. Phys. A271 (1976) 317 L.C. Liu and Q. Haider, Phys. Rev. C34 (1986) 1845 J.C. Peng, Excited baryons-1988, Rensselaer Polytechnic Institute Conference, unpublished C. Bacci, et al., Phys. Lett. B28 (1969) 687

12) L.S. Celenza

13) 14) 15) 16) 17) 18)

and A.S. Rosenthal