PHYSICA L REVIE% C VOLUME 8, NUMB ER 6 D EC EMBER 1973

Relativistic Description of Nucleon-Nucleon Interaction Using a Complex PotentialT. Ueda, * M. I.. Nack, and A. E. 8. Qreen

Dejartment of Physics and Astronomy, University of F/orida, Gainesvil/e, E/orida 32602(Received 6 July 1973)

Using a three-dimensional reduction of the Bethe-Salpeter equation, we calculate the nucleon-nucleon phase parameters in the energy range of 0-3. GeV of incident laboratory energy. In theelastic energy region the generalized one-boson-exchange potential is assumed for drivingterms and in the inelastic energy region an imaginary potential is added. Ingredients of theone-boson-exchange potential are m', p, cu, g, &, and 8* mesons. The dipole form factor isadopted for all the interactions of mesons with nucleons. The broad mass distribution is alsotaken into account for the p and 8* resonances. Our imaginary potential which is given by asuperposition of regularized Yukawa potentials which have an attractive long-range part and arepulsive core, can describe the peripheral nature of the absorption found in phage-shift analy-ses. Comparisons of our theoretical phase shifts with those derived by the phase-shift analyses,and of our theoretical observables with experiments at 660 and 970 MeV are made in detail andphysical implications are discusssed.

NUCLEAR REACTIONS N -N interaction, 0-1000 MeV; calculated phase pa-rameters, observables.

1. INTRODUCTION

For the past decade it has been shown that theX-N interaction in the elastic energy region eanbe well described in terms of the one-boson-ex-change model. ' An early finding of the validity ofthe model in the intermediate region of the nuclearforce' and subsequent development with introduc-tion of the form factor by Ueda and Qreen, ' or theregularization by Qreen and Sawada4 founded onthe generalized meson theory of Qreen' permittedthe model to describe the whole elastic N-N datavery successfully. Furthermore Qersten, Thomp-son, and Qreen presented a relativistic version ofthe generalized one-boson-exchange potential, 'using a three-dimensional reduction of the Bethe-Salpeter (BS) equation. ' (This work will be re-ferred to as GTG. )

Kith such a background it would appear worth-while to attempt to extend this relativistic versionof the model to the inelastic energy region. Thus,a K-matrix unitarization method'9 and a disper-sion-relation calculation" have been applied byUeda to the 0-1-QeV region. In these works in-elastic effects are treated within a framework ofthe one-boson-exchange model and a good descrip-tion of the elastic X-N data is achieved.

In this paper we extend the QTQ approach to theinelastic energy region up to 1 QeV by utilizing acomplex potential. In the QTQ treatment the BSequation is solved in momentum space with a rela-tivistic two-nucleon propagator derived by Thomp-son" in which the two nucleons are in a positiveenergy state and the relative energy part is equalto zero. The same propagator is used also in thiswork. The real part of our potential is the general-

ized one-boson-exchange potential (OBEP) whoseingredients are m, g, p, w, 8*, and 5 mesons in-teracting with nucleons having a dipole form fac-tor. %'e allow for the broad mass distributions ofthe p and 5* resonances with the help of represen-tations recently obtained by Nack, Ueda, andQreen. "

In the inelastic energy region we arid a phenome-nological imaginary potential which (in configura. -tloll spRce 1RIlguRge) coIlsls'ts of R lollg-1'Rllge Rt-tractive-regularized Yukawa potential. and a short-range repulsive-regularized Yukawa potential.The spin dependence of the imaginary potential isassumed to be J.=O and even-parity exchange type,i.e., analogous to scalar meson exchange. Vhththis phenomenological augmentation we achieve asatisfactory description of the N-N data in the0-1-QeV energy range.

Our three-dimensional equation with the complexpotential ean be derived from the original Bethe-Salpeter equation with a complex interaction ker-nel by an analogous procedure to that worked outby Thompson" for the elastic case. We obtain(suppressing spin indices)

&(p )i) = )'()i, )i) f i( di()'(i&, i()t" (q, )i))((i(I), ((),where p and k are the final and initial e.m. momen-tum of the scattered nucleon, respectively, and

V(p, q) is now a complex potential,

1'(p, q) = 1",(p, q)+ fl'1(p, q)

G(q, k) is a two-nucleon propagator given by

G(q, k) = a(4)1'm'/z'(q) [z(q) -z(k) jj,

UEDA, NACK, AND QHEEN

where E(k) = (k'+m')'~~, m is the nucleon mass,and P denotes the principal value. This propaga-tor can be derived by considering only contribu-tions of two intermediate nucleons which are inpositive energy states and have zero relative en-ergy.

K(p, k) which is determined by Eq. (1) is obvious-ly a complex quantity and related to the 8 matrixin an operator form by means of

1+iK/vp1 —iK/wp

' (4)

where p =E(k)/k.The real part V~ of the complex potential is as-

sumed to be the generalized one-boson-exchangepotential, or the one-boson-exchange potentialmodified by the form factor. For the pseudo-scalar, scalar, and vector meson, respective-ly, they are derived from the interaction Hamil-tonian densities,

H~, = E(k') g~, (j)(p)iy, () (q)(j)~,(k),

&.= +(k') g.0(p)0(q) A.(k),

&.= +(k') g. t(p)iy„t(q)4,"(k)

+ i g(p)o „„(j'(q)[k,—(j)„"(k)—&„(j)("(k)],

where k =p -q, o'~, =(y„y„y,y„)/-2i, and

(5)

(gl)'A„' 1+ (q —k)'/A„'

By choosing A, & A„, we achieve a shape whichcharacterizes the peripheral nature of the absorp-tion. In calculations for the 0-450-MeV energyrange this imaginary potential is deleted. How-ever, it is included in calculations at 660 and 970MeV where inelastic effects are important and itsparameters are adjusted at both energies. In the

The imaginary part Vl of the complex potentialis assumed to be L, =0 and an even-parity exchangetype given by

~i(p, q) = (v(q)(v(k)(v(-q)(v(-k)

&& [v, (q, k) + 7, 7.,v, (q, k)],where v's are the Dirac spinors whose explicitdefinition is given in Ref. ll. vo and v, representcomponents of I=0 and I= 1 exchange, respectively,and are independent of the spin state. We paramet-rize these functions using

actual calculation the factor of [E(k2)]' from Eq.(5), appearing in Vz from using Eq. (5) and in V~

from using Eq. (8), is replaced by II", , [A /(k'+ A, ')], where A, = [1+ (i —1)/100]A and n = (4, 5)for (Vs, Vz), respectively.

Partial-wave expansions of the component one-boson-exchange potentials have been given byUeda' "for the on-mass-shell case and by QTQ'for off-mass-shell cases. Since the partial-waveexpansion of the imaginary potential parametrizedby Eq. (8) may be obtained from similar formulasas the real potential, we will not repeat them here.

The S matrix given by Eq. (4) is expanded intopartial wave components and in the ease of spin-uncoupled scattering it is represented by

~J J'2j6

where 5~ and r~ are a real phase shift and a re-flection parameter associated with a partial waveof total angular momentum J. The correspondingrepresentation for spin-coupled scattering and arelation of r~ to the inelastic cross section aregiven in the Appendix.

3. MODEI. PARAMETERS

The ingredients of the real potential are m, p, cu,

g, 6, and S* mesons. The masses of m, ~, g, and6 are fixed at the experimentally observed reso-nance values. Special attention, however, is givento the p and 8* mass distributions. The mass ofthe p is.taken as 840 MeV with zero w'idth whichis approximately equivalent in effect to the actualp contribution with the mass distribution. " TheI= 0 8-wave phase shfit of the n -m scattering showsa broad mass distribution of the m-m system. Thepresent data have some ambiguity in the 700-1000-MeV region, however, we adopt the 8* solutionwhich seems at this time to have the most experi-mental support, We represent the effect of thebroad mass distribution of the 8* solution by twoI=0 scalar mesons o, and o„." These, in effect,correspond to 0 and q„ in the Ueda-Green I model. 'However, in this case the masses of u, and O„areconstrained to be consistent with the 8* solution.The form-factor parameter A is taken in commonfor all mesons. Adjustable parameters are m(o, ),the coupling constants and the form-factor parame-ter A. As in our previous works we assumed sim-ply (f/g) =0 in accord with the isoscalar part ofthe electromagnetic form factor. However, asmall value for this ratio can be allowed as usedin a recent work by Ueda, "Furuichi, Suemitsu,Yonezawa, and Watari, "and Bryan and Qersten. "The coupling constants and A are adjusted so asto secure fits to the Livermore phase shifts (theenergy-independent solution) at 25, 50, 95, 142,210, and 330 MeV. "

R E LA T I V IS T IC 0 E SC RI P 7 ION OF NUCLEON-NUCLEON. . .

After determining the real potential the parame-ters of the imaginary potential are adjusted toachieve a fit to the reflection parameters of ppscattering obtained by the phase-shift analyses"or by some theoretical models, ""and also toyield the inelastic cross sections of pp and npscattering at 660 and 970 MeV. As seen in Eq.(8) the imaginary potential has a set of parametersof g~, g»,' g» g» A» A». We determined A, andA» by using the pheripheral-type-reflection param-eters and took them in common at 660 and 970MeV. g'„g'„; g,' and g» are chosen to yield fitsto the pp and np inelastic cross sections as wellas the reflection parameters. Care has been takennot to violate the unitarity limit, since too large avalue for the repulsive part would create such aproblem.

4. DATA FOR COMPARISONS

For experimental data we use the phase parame-ters obtained by the Livermore group through en-ergy-independent analyses at 25, 50, 95, 142, 210,330, 425, and 630 MeV." For the e, parameterwhich has a large ambiguity in the phase-shiftanalysis solutions we also take the energy-depen-dent solution of the I.ivermore group obtained inthe 0-750-MeV energy range. Comparisons arealso made with the Yale phase parameters in the0-350-MeV energy range. " At 660 MeV we usephase parameters obtained by the Kyoto group, "'and at 970 MeV we take those obtained by theKyoto group, "although error bars are not avail-able at this energy.

The reflection parameters to be compared withour theoretical results are of two kinds. The firstones are those obtained by the phase-shift analysesof pp scattering at 660 and 970 MeV by the Kyotogroup. " In their analyses at 660-MeV peripheralabsorption is assumed according to the Mandel-stam model. " Thus the '8, reflection parameteris fixed, to unity and the reflection parameters ofP, D, and E waves were searched as well as thereal phase shifts to make a fit to the elastic andinelastic cross sections. At 970 MeV too, similaranalyses as well as analyses not constrained bythe Mandelstam model have been presented. Thelatter are predictions of theoretical models.Amaldi" calculated reflection parameters in the600-1400-MeV energy range, using the one-pion-exchange model modified by the form factor forthe one-pion production process which is dominantin the inelastic processes of the energy range. TheK-matrix unitarization model given by Ueda andothers also provides the reflection parameters. ''In this model the one-boson-exchange amplitudesfor pp scattering and pp- NN* reaction are simul-

taneously unitarized by the damping relation andthe reflection parameters are obtained by fittingthe model to both elastic and inelastic data, . Un-fortunately, the reflection parameters obtained bythese various methods show considerable ambigui-ties at 660 and 970 MeV.

Inelastic cross sections were evaluated from ex-perimental total cross sections and elastic totalcross sections at 660 and 970 MeV, except for thepn inelastic cross section at 660 MeV." " To ob-tain the pn inelastic cross section at 660 MeV, weused a value derived by integrating the elastic dif-ferential cross section at 660 MeV.

The data for the pp and np observables, as de-fined in the review article by Hoshizaki, "are tak-en from the NN scattering data compilation byBystricky, Lehar, and J'anout at Saclay (1972).~We use their tabulated data which are at timesonly graphically represented in the 34 originalarticles cited under Ref. 28 using their referencecode.

5. NUMERICAL RESULTS AND COMPARISON

PATH DATA

Numerical results of the'phase parameters andreflection coefficients and inelastic cross sectionsare shown in Figs. 1 and 2, and in Tables I-IV.

A. Real Phase Parameters

We present two solutions of our search proce-dure for the real potential. The No. 1 solution re-sembles the U-G model of Ref. 3 in each contribu-tion of its ingredients, and solution No. 2 resem-bles the GTG model. The X' test per 103 phaseparameters with J & 5, using the Livermore ener-gy-independent solution" at 25, 50, 95, 142, 210,and 330 MeV, produced results of 4.5 and 3.9 forthe No. 1 and 2 solutions, respectively. Thus, theNo. 2 solution gives a little better fit in the:elas-tic energy region, and the No. 1 solution gives bet-ter fits in 'D, and E'y at 660 and 970 MeV and alsohas (f/g)~ =4.96, which is nearer the electromag-netic form-factor value of (f/g)~ = 4 than No. 2.

At 660 MeV the phase-shift analysis solutionshave large error bars. Our result for the 12phase parameters shows that deviations are lessthan about 3 standard deviation except for the 'B,phase shift. The phase-shift analyses give con-siderable ambiguity for the e, parameter at thisenergy. In particular it is noted that the energy-dependent analyses yield values of opposite signfrom those of energy-independent analyses. Ourtheoretical prediction is nearer to the energy-de-pendent results.

At 970 MeV the phase-shift analyses by the Kyotogroup give two sets of solutions for pp scattering

2064 UEDA NARK, AND GREEN

O.OOO-«DCX)

OO-CO-

oo—ONOCD .

OO NOD-OCXI

C+

O

ooONOO

O-CDCCI

OED

ooCD

CDCD

DD NOO

R:m+

DCD

& O—

DH—O

CDD

O ONI

OOO

coI

- O

LA

N

ON

LC)

CL

- O

OO-O-OCO

O

ONI

O o

OO

O O DI

00000

00000

0

000

CQ

08

C

N CXI

I I'

I

N

NI

E

ECL

- O

I

N NONI

O

DI

D ONI

CDO-O-OO

CDCDCD

ION ~

O-QP

'CI

oo—O

I

E

E

OOD:O"tf)-

I

ON O

O

I, , I

D Ogp ED

OOO

I

cDN

OO O O ON

~, I

cf'I

I

I I I

-O:rr)

-NID

—N

CL

CDCDDO-OCO

OO

CDOCP

oo—O

CDO

ONO

I

0000

00

0000

O

CDN

I

I

N cO ~ D

O

ON

DN

(7) 'cI O

O

CIJ ONI

OOOOCDCO

ION

LJCL

O

JDooOCDD

N

IDN

ECL

O CDD

I, I

Oco

OGD

I

O O O ONI I I

O O N g P0

(g ~ NI

I I I OOI I

I

O O oNO Ia

I

0m

8M

o0 ~0

+~gyaw

4Q

o

'lSQ 0

q)

o rM cd

Q C4

II

0

4e 0 m

0Q

cd

M

Q Qm A '~cd Q m

tQ

Q

m s.Q s+' + o0

I M

Q

ba

q) Qaf ~ g0

Q) eW

0 Qm 4 m

4

m W

Q St

RELATIVISTIC DESCRIPTION OF NUCLEON-NUCLEON. . . 2065

phase parameters and no error bar is available.Our theoretical values of the six phase parametersfor the PP scattering are consistent qualitativelywith those of the phase-shift analyses, except inthe case of the 'D, state where a rather large dif-ference develops.

B. Low-Energy and Deuteron Parameters

We constrained our solutions to fit the (n, P) en-ergy-dependent phase shifts, 'S, and 'S„of TableVII of Ref. 16 at 1, 2, 3, and 10 MeV. Using ef-fective range theory"" the singlet and tripletscattering lengths (&„&,) and the effective ranges(r„r,) are calculated by using

k cotd = ——+ —, rk'+ O(k')1a

I

at 1 and 2 MeV, where k is the c.m. momentum ofthe nucleon of mass m. The minus root of the re-lated quadratic equation

~ =(mE, )'"/ac= + —,' r, y'—

gives the deuteron radius y ' and binding energyE„. The quadrupole moment of the deuteron Q isobtained at 2 MeV using our values of r, and y,and

lane, = I&2(1 -yr, )q]k'+ 0(k4) .The values of a„r„a„~„E„andQ are shown

TABLE I. Parameters of the generalized OBEP.

MesonMass(MeV)

No. 1g 2

No. 2

g 2

6

P

(fig) p

rl

A(Me V)X2/data

138.7960.0840.0

782.8790.0352.0548.8

14.6001.3450.7514.9638.8008.8140.7382.486

2075.04.5

14.2000.3690,5006.3769.917

10.9600.7003.091

1860.03.9

C. Imaginary Phase Parameters or ReflectionParameters

in Table V as determined by d('S, ), d('8, ), and e„of Ref. 16 and our solutions 1 and 2, and they canbe compared with the experimental values. A bet-ter experimental determination of the low-energybehavior of e, is necessary for an improved pre-diction of Q from these values as k'- 0, and for ameans of properly constraining theoretical calcu-lations of c, for k'-0. A more direct method ofexamining the nonrelativistic deuteron low-energyparameters for generalized OBEP models has beengiven by Qersten and Green. "

I,O'

0.8

0.6

0.4—

660 MeV

---- SOL. I

SOL. 2

At 660 MeV our imaginary potential gives quanti-tative agreements with the experimental values ofthe inelastic cross sections of both of PP and nPscattering. In addition the peripheral nature ofthe absorption is consistent with results of thephase-shift analyses or the predictions of theoreti-cal models. At this energy we need both compo-nents of the I=0 and I= 1 exchange and both withattractive long-range and repulsive short-rangepotentials. Figure 3 presents a Fourier trans-form of the imaginary potential into configurationspace illustrating these features of the potentialshape.

At 970 MeV the imaginary potential leads to in-

0.20 I

I I I

2 5 4 5 0 I

L

I I

2 3 4 5 TABLE II. Parameters of the imaginary potential.

I'IG. 2. The PP reflection parameters &1. of fixedI-averaged over J at 660 and 970 MeV, where r& =N"~

xQ f ixz z&. The solid lines go through past phase-shiftanalyses t Ref. 17a (cross) at 660 MeV, and Bef. 17b(cross and plus at 970 MeV)] and also past-averagedtheoretical predictions [Ref. 8 (up-triangle), Bef. 9(down-triangle), and Bef. 18 (circle)], and the dashed(sol. 1) and broken (sol. 2) lines go through our theoret-ical predictions. Only one dashed curve indicates oursolutions coincide at these points.

(g0)2

(g'.)'(g i)2

(g f)2

W, (MeV)A p, (Me&)

660 MeVNo. 1 No. 211.50 11.02

31.822.318.16

1100.01980.0

970 MeVNo. 1 No. 218.70 18.40

43.100.003.00

1100.01980.0

2066 UEDA, NACK, AND GREEN

elastic cross sections which are fairly good. Tomatch the observed l dependence of the reflectionparameters our imaginary potential has been madeto be ot peripheral type. At this energy the I=0exchange part dominates and the I= 1 part is verysmall.

D. Observables

To deal with the ambiguity of the phase-shiftanalysis solutions we have made direct compari-sons with experimental observables at 660 and970 MeV. In Fig. 4 we show the theoretical pre-dictions of solutions 1 and 2 for the PP and nP ob-servables which are the differential cross sectiondo jdQ (mb/sr), polarization I', depolarization D,the spin rotations A, A., and the spin correlationparameters C», CEJ, compared with the data. "

6. DISCUSSION AND CONCLUSION

relation with inelastic effects in the 0-3-GeV en-ergy range. " Comparisons of the 'S„'P„,„and'D, phase shifts in the 0-1-GeV energy range showthat both calculations give qualitatively similar re-sults. However, an appreciable difference is notedin the 'D, phase parameter where the present onedecreases with increase of energy more rapidlythan that of the dispersion relation calculation inthe 400-1000-MeV energy range.

The present calculation shows that the inelasticeffect on the real phase shifts are not so large andin most cases makes the real phase shifts go down-ward, i.e., it acts repulsively. However, the dis-persion relation calculation indicated that the in-elastic effect on the real phase shifts are more ap-preciable and act attractively in the sub-GeV re-gion, although at very high energy it acts repul-sively. These basic differences must be resolvedby future study.

A. Comparison with Result of the Dispersion

Relation Calculation

First we compare the present calculation withthe result given by Ueda, using the dispersion

B. Energy Dependent Parameters

of the Imaginary Potential

We have chosen the strength parameters g, andg„of the imaginary potential at each energy and

TABLE IlI. Real phase parameters (6 or e) in degrees for solutions 1 and 2 computed usingV = V&+ i VI at 660 and 970 MeV compared with computation using only Vz to show the quan-titative effect of adding V& at these energies.

No. 1660 MeV

No. 2 No. 1970 MeV

No. 2

So:P

0iPSiPiDiE'i

DP

F

iF

D3

3Q

E3iG

3Q

3H4

E4iH53Q

H5I5

-38.9—39.3—34.4-39.1-41.0

21.74.75 4

11.420.9-5 7

0.7—9.5

3.2-8.1-5.0

7.22.63.8

12.70.7

—1.9-2.4

0.1

2 Q 14.6

37+1-38.7

22 y3

-39.7-40.5

19.S6.55.9

14.821.0-5.6

1.0—953.1

-7.9—5.0

7.72.74.0

12.70.72 ~ 1

—2.40.12 ~ 12y 14.6

-41.9-41.0-38.6—33.8-39.1

3.0-1.94.3

10.514.9-6.9

1.1—10.2

5.9-8.3-4.8

5.92.63„6

11.60.5

-1.8—2.5

0.4201

-2.04.4

-39.7-40.4-35.8—34.2-38.7

3.0-1.5

4.613,614.9—6.8

1.3—10.1

6.0—S.l-4.9

6.22.63.8

11.60.5

-1.9—2 5

0.42y 1

-2.04.5

—57.6-57.1—31.9-57.2—49.3

40.5-1.7—0.4

5.07.9

—14.32.1

-15.5—1.9

-12.01.84.62.63.1

12.0—0.7-1 5—3 5-0.2—3.0-2.4

4.9

-56.1-56.2—3.9

-52.4—48.8

31.8-1.4—0.2

9.18.2

-14.32.5

—15.3—1.911~ 71.75.22.83.6

12.1—0.7

1+ 7-3 5

0.0—3.0-2.4

5.2

-61.0-58.3-41.9—52.0—46.8

11.4-4.7-2 5

3.11.1

—16.02.3

—16.51.7

-12.40.03,22.42.6

10.01+3

—1.3—3.9

0.43Q 12 y3

4.6

—58.9-57.2-36.7—51.8—46.6

11.3-6.3—2.4

6.51.3

-16.02.7

—16.21.9

—12.1-O.l

3.12.63.0

10.11Q 3

-l.5-3.8

0.63y 12 y3

4.9

RELATIVISTIC DESCRIPTION OF NUCLEON-NUCLEON. . . 2067

TABLE IV. Reflection parameters (p, r) for phase shifts 6, and (0., $) for mixing parame-ters e or p as defined in our Appendix according to the (Livermore, Kyoto) definitions, re-spectively, and inelastic cross sections.

State

660 MeVNo. 1 No. 2 No. 1 No. 2

p (deg)

970 MeVNo. 1 No. 2

p (deg)

-No. 1 No. 2

~$0

3Sg

'&oiy)

3p'I'23Dg

'D23D

3D

3+1j3+

3G

~G4

3G

G5

84B53+3I5

16.731.142.134.947.78.8

39,342.925,128.520.714.024.430.910.116.19.8

10.36.84 47.62.1

8.324.541.629.047.25.8

28.641.322.727.819.913.123.830.29.1

15.89.09.86.64.1

2.0

0.96 0.990.86 0.910.74 0.750.82 0.880.67 0.680.99 0.990.78 0.880.73 0.750,91 0.920.89 0.890.94 0.940.97 0.970.91 0.910.86 0.860.99 0.990.96 0.960.99 0.990.98 0.990.99 0.991.00 1.000.99 0.991,00 1.00

25.4133.648.863.455.58.1

66.350.147.354.228.133.784.742.530.026.025.828.812.214.314.27.8

22.7141.449.156.755.215.653.048.746.654.127.533.334.241.826.725.624.828.912.014.114.07.7

0.90 0.920.69 0.790.66 0.650.46 0.550.57 0.570.99 0.960.56 0.610.64 0.660.68 0,700.60 0.590.88 0.890.83 08.40.82 0.830.74 0.750.87 0.890.90 0.900.90 0.910.88 0.880.98 0.980.97 0.970.97 0.970.99 0.99

State 0. (deg) = Q (deg) 0. (deg) = P (deg)

3$3D3 3

D3- G33 . 3E4- H4'G5-'I5

0&+& (mb)0~&& (mb)

-14.0—3 4-8.6-2.0-0.119.416.2

14.6-3,6-4.2

2 p2

-0.118.314.3

Exp.

18.6~1.113.6+ 3.2

-102.2

-28.1-7.8

1y3

21.625.7

-100.9-2.9

-39.2-8.6-1 5

21.424.5

Exp.

21.7+1.423.8 +3.5

found that the best values for the parameters dif-fered at 660 and 970 MeV. The possibility of find-ing an imaginary potential which describes inelas-tic effects with a single analytic function (possiblysmoothly energy-dependent} over the entire energyregion of interest also remains for future study.

C. Reflection Parameters

It is noted that the reflection parameter of unityis obtained for the'S, state at 660 MeV, thoughthe imaginary potential produces an appreciableeffect on the real phase shift. This comes obvious-ly from the following reason. The reflection pa-rameter results from some averaging over the at-tractive part and repulsive part of the imaginarypotential and can portend no effect from the imagi-nary potential. However, the effect of the imagi-nary potential on the real phase shift originates

TABLE V. The low-energy and deuteron parameters.

Solution 1 Solution 2 Ref. 16 Experiment

a (fm)(fm)

a ] (fm)r& (fm)Ez {MeV)

(fm2) e

-25.452.665.501,872;230.263

-26.062.685.211.822.560.216

-23.682.515.401.792.280.053

23 ~ 722.73 a

6.41'1.712.22 d

0.282

H. P. Noyes and H. M. Lipinski, Phys. Rev. C 4, 996(1971).

L. Koester and W. Nistler, Phys. Rev. Lett. 27, 956(1971).

H. P. Noyes, Phys. Rev. 130, 2026 (1963).R. Wilson, The Nucleon-¹cleon Interaction (Inter-

science, New York, 1963).Note that the units of Q are fm, not fm 2 as quoted

in Refs, 31 and 15.

2068 UEDA, NACK, AND GRE EN

from some other t eype of averaging because of acomplicated cou linp 'ng between the real potentialand imaginary potential. Theerefore the reflectionparameter of unity does tthe imaginary potential h

s no necessaril im 1' y' py that

p ase shift.o en lal has no effect on the real

20-

IO

I

Y(T=O, E=660 MeV)————Y(T=I, E =660 NleV)——Y(T=0, E=970 MeV)—- —- —Y(T=I, E= 970 MeV)

I

V

I

LLII

I

-I 0 I.50.5 I.O

r (fm)

FIG. 3. Fourier transforms (E) ore s(see Ref. 3) of th T=

or x space potentials

~ b.l':.Il. b, -(. 'e =0, 1parts of E . 7

y =0, E) =F(vo+v&) and F(T=]., Evo ave), where v; are defined by Eq. S

li htl 0. A(~ ~ - ~ y

xpanded) scale is usedcompressed e& 3, respectively, to best sh

-d ion --.b.h. '-

D. Mass Distributed Propagat f hors o t en'-n'System

It has been shown that the effect of the mass dis-

approximately represented by a renormalizationof the cou lin cp g onstant ' or an appropriate dis-32 I 33

placement of its mass from the b

ue with thee o served mean val-

he zero-width approximation. " In this

paper the mass of p was chosen to be 840 MeV

rather than its usual mean value of 750 MeV in an

effort to allow for its mass-d' t b tI

- is rl ution effect.The p propagator corresponding to the actual massdistribution found from the I= 1 I'--wave m-w phase

shift and the one-po1. e approximation with 840 MeV

are shown in Fiin Fig. 5. The comparison shows that

this one-pole approximation is fairly good.In Fig. 5 the S* propagator corresponding to the

broad mass distribution I=0 S-wave n-m

n y e solid line. The c

constructed bse ln he presentp ent work which is

c e y the 0, and a„contrig the N-Ã data thee ines. In fittin

lative coupling c tr mass component o, requires a

'g constant than is ex e

a greater re-g expected from the

ossl ly this difference ma y b a ribion rom I=0 5 =0

correlated tw-=0 part of the un-

wo-pion exchange. This a rtion of identifying th

is approxima-'ng e uncorrelated two- io-

change contribution with a'

n wl a fictitious low masssca ar-meson-exchange contrib

'n as een suggested by Furuichi. "

E. orm Factor Parameters

Few of the particles listed in stand d tmentary —i.e., specified onl b the p

numbe Fs an conserved char eges or quantum

p , esonances require atrs. or exam le reast a width ar

detaiparameter F, if not actu ll th

led phase-shift behavio " Ita y emore

r. was noted bUeda and Qreen' that the 1e nucleon is not elemen-ary, and that its virtual pion cloud structure

polarity parameters are (A = Ac/A, N res

a dlse = of quadrupole regularization or

a lpole nucleon-meson for f trm ac or results in anexponential pion densii e electromagnetic data as well as the strong

interaction data of this paper Ws ependin on the ''

g interaction we use to probthe nucleon. The h

o pro e

=0.095 fme adrons see the strong ~' = h

of solution 1 of this wo k 'thg s= cA

re 'gxon, and Xz= hcjA, = 0.179 fm in the inel

660eV. The larger size parameter at

MeV could reflect the excited natu

with a die electromagnetic interaction probe ths e nucleon

lfferent size parameter X = kc 84= 0.235 fm as '

r ~M- c 840 MeVm, as it is a long-range force.

In final conclusion'on this work indicates that thegeneralized one-bosima lnal'

-boson-exchange potential 'th-bos — ia wl an

the 0-1-g' ry potential can describe th N-Ne e -N data in0-1-QeV energy ran e. It i0-1- g . ls especially note-

or y at our 'S hamost se

', p ase shift, which represe tnsitively the interaction in the i

a ,' sen s

region, fits the h

'n ln e innermost

i s e phenomenological phase shiftsthe entire energygy range.

e s l s over

The authors would like to thank Dfol' u

an r. T. Sawadauseful conversations about the imaginar o-

anks Dr. A. Salam fortential. One of us (T.U. ) th . . or

~ ~ ~

his hospitality at the International C t' na enter for Theo-retical Physics, Trieste. A fthe

lnal thanks goes tohe North East Regional Dat Phe ' a a rocessing Center

a the University of Florid G'' a, alnesville, for their

computational support.

RELATIVISTIC DESCRIPTION OF NUCLEON-NUCLEON. . .

2QQ t I t t I I 1 t 1 I I t I ~ t 1 t 2Q

IQQ

u& 50

E20

~~ IQ

5

'IO

50

20 E

I.O

Q5 ~

Q4

0.2

t I I I ~~~ I r tI

I ~I I I

Q, I

-Q6-

Q6 r ~~ ~

Ir t

I r t Ir r

It I I r

0.2

0

0I.O

QB

-0.4I.Q

~PP

Ir r

II ~

l I Ir ~ I

' r l 0

CPP

02- DPP

50 60 90 i20 l50 l80 (g )8, „(deg) at 660 MeV

Q —00 X) 60 &0}O 50 60 90

8, (deg) at 660 MeV

IQ-

—50~CA

20E

p~ 0.5-G

Oj

0.6-I

t tI

I 1 l t

-Q2

02

0

-06-~ I I I I r I I I t + r r

pPP

II r

I it I IQ

PP -O.e

04-PP

02-

0 X) 60 90 l20 l50 le0

8, , ~, (deg) at 970 MeV

Q

8,.~, (deg} at 970 MeV

FIG. 4. Ob8eX'VRMG8 Rt (R) 660 MGV R118 g) 970 MGV RS P16810488 bg 801. 1 (801l6 CUZVe) RQQ 801. 2 (410kGIl OQX'Ve).

20VO UEDA, MACK, AND GEE EN

2.0—

where

S~„~= [1—(r~„z/r~„~)pz']'"r~„~exp(2it)~„~),

I.G g =t p (r, ~r~„~)"'exp[i(&~ „~+5~„J + Q~),

(A2)

09

(cosp cos2ee"—s=l

(i sin2e e'( ' ~+ 'z sin2e e'"-' "'"'

cosp+ cos26 8 +

(A4)

0.4

I

0.4I

Q8

k (GeV )

I

l. 2I

l.6

FIG. 5. The distributed. S* and p propagators d (I')(smooth curves) of Ref. 12 compared with their approxi-mations d~ (dashed curves) used in this paper, wheret)(ttt mt) = (t)2+ m2) t, 6(I')= fd (ttt, t')p~(t')dt' fort' = m~, and p~(t') is the spectral function of the 8* or p.The approximations are 4' =PE(k m ) + (1 —P) 6(k, m2&)

for sols. 1 and2of the 8 whereP g2(ph)/'[g2(oh)+g2(0))jp= 1 for the p, and mI, = ~ or the mean mass for both the8* and p as determined from p~(t').

APPENMX

8 =1-1.I

SJ'+ L J

For the spin-coupled scattering with the inelasticeffect we used the two following pararnetrizationsof the S matrix:

In these expressions the sign + denotes a relation/= J+1. Inelastic cross sections are related withthe reflection parameters r as follows:

tt~~ =, Q (2Z+ 1)[1—('r~)']J:even

L+1~ g p (2J ~ 1)!1 —('~„)*]},

L:add J= L -j.(A5)

o P= 2 2J+1 1 — JJ

~ Q Q !2z~ 1)!1-('r„)']},(A6)

where k is the c.m. momentum of any nucleon,and 'xJ and 'r«represent the reflection parame-ters for the singlet and triplet scattering, respec-tively.

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