PHYSICAL REVIE% C VOLUME 20, %UMBER 3 SEPTEMBER I 979

Resonance degrees of freedom in the two-nucleon system

G. -H. Niephaus, M. Gari, and B. SommerInstitut fur Theoretische Physik, Ruhr-Uniuersitat Bochum, Germany

and Max-Planck-Institut fur Chemic, Kernphysikalische Abteilung, Mainz, Germany(Received S February 1979)

Free nucleon-nucleon scattering is investigated with the explicit treatment of 6(1236) resonances. Thecoupled channel problem (NN + N4 + hA) is solved and compared with the results obtained fromperturbation calculations (impulse approximation). The nucleon-nucleon potential used is the Reid-soft-corepotential which has been modified according to the phase shift and deuteron binding energy requirements.The perturbation treatment turns out to be not a very good approximation as the 6 admixtures are off up toabout 50% in a nonsystematic way.

[NUCLEAB BEACTIONS nucleon-nucleon scattering, resonance admixtures. ]

I. INTRODUCTION

I

It has been recognized for a long time that thenucleon resonances play an important role in theunderstanding of the nucleon-nucleon interaction. 'Especially in the medium range of the interactionwhere the resonances occur as intermediate statesin the two-boson exchange part of the potential,these contributions are able to replace the pheno-menological 0 particle to a large extent. " Re-membering the important role of the resonancesin the NN potential, some workers have tried totreat the nucleon resonances explicitly as addi-tional degrees of freedom in the many-body prob-lem. ~ ' Of course such an explicit treatmentbrings up severe problems for the actual calcula-tion of many-body properties. For example, theBethe-Goldstone equation, which is already verycomplicated in the pure nucleonic case, general-izes to an equation containing additional couplingsto the resonance states. The solution of the coupledchannel problem in the full complexity has beenobtained up to now only in the nuclear matterproblem. " The explicit treatment of resonancesin the case of finite nuclei is still done in a pertur-bation treatment of the resonance parts. ' As faras the two-nucleon system is concerned, thereare several papers in which the effect of reso-nance admixtures in the deuteron is investigated. ' "Some authors also considered NN scattering. 'For a review of the whole subject see Ref. 17 and18.

The present paper is devoted to two subjects:(i) The calculation of the full NN, Nb, , 6h po

tential. This includes the solution of the fullcoupled channel problem for free NN scatteringand the deuteron. Such phase-correct potentialsare needed in order to obtain a realistic treatment

of the many-body problem.(ii) As the & contributions in finite nuclei are

calculated in a perturbative approach, we want tosee how good such a treatment is already in thecase of a two-nucleon system.

We describe the NN part of the interaction bythe phenomenological Beid-soft-core potential. "The reason is that for calculations of many-bodyproperties this potential is the one most widelyused. In the explicit treatment of the resonancesby a coupled channel calculation the potential willbe refitted to give the correct two-body data (phaseshifts and deuteron binding energy). For the com-parison of the solution of the full coupled channelproblem with the perturbation treatment we haveto restrict ourselves to the discussion of the wavefunctions.

In Sec. II we formulate the problem and in Sec.III we present the numerical results and give aconcluding discussion.

lI. RESONANCE DEGREES OF FREEDOM IN THETWO-NUCLEON SYSTEM

A, Schrodinger equation with inclusion of resonances

The conventional treatment of the two-nucleonproblem yields a Schrodinger equation of the fol-lowing form:

r, +V,. IP (l 2)&=~-lg-(i 2)&j=1,2

Here T,. denotes the kinetic energy of the ith nu-cleon and V„ the interaction of nucleon-1 withnucleon 2. In this approach the resonances do notoccur explicitly; they are implicitly taken intoaccount as intermediate states of many mesonexchange processes in phenomenological interac-tions (as the Reid potential, e.g.).

1979 The American Physical Society

20 RESONANCE DEGREES OF FREEDOM IN THE TWO-NUCLEON. . .

Treating the resonances explicitly the Schro-dinger equation reads

where fP „(1,2)) describes the spatial part and

fx „(1,2)& the intrinsic properties (e.g. , spin, iso-spin) of the configuration n. With the normaliza-tion condition

(5)

In contrast to Eq. (1) there is now an additionalterm in the total Hamiltonian, the "intrinsic"Hamiltonian II ',.". This term denotes the intrinsicexcitation energy of the ith particle. The two-body state fP (1, 2)& can now be written as a sumof different intrinsic configurations, namely

(s)

we have

or with the definition

i=lp2

(6)

Configuration n= 1 gives the conventional part ofthe wave function, i.e., both particles are nu-cleons. In the case of n&1 at least one particle isintrinsically excited, i.e., at least one particleis a resonance. The states fg „(1,2)& can be writ-ten as

B. Coupled channel approach

In this chapter we want to express Eq. (I) incoordinate space. In the case of a two-nucleonsystem we may confine ourselves to the solutionof this equation in the "center-of- momentum"(c.m. ) frame. The wave function then reads

UJr&r fg~(1, 2)&= g r(r) = Q Q "~' A~n; [L(s,s2)S]Z; (t~t2)T&.

n L, S(8)

A is an antisymmetrization operator which is necessary if we assume that the interaction V» is sym-metric with respect to the exchange of the two particles, [V,A]=0. Inserting (8) into Eq. (t) we obtainthe following system of coupled differential equations for the radial functions U~~r,(z):

—2p,„(EM„—E) U„~~~(r) =2p„Q Q V„~~~ ~.~. U ~.~.(r).

Here p„denotes the reduced mass of the configuration n

n n

M +Mn n

is the expectation value of the intrinsic Hamiltonian. The matrix elements of the interaction are given by

V„z~ ~~. = (n; [I (s,s,)SJJ;(t,t, )T fAVAfm; [I '(s,'s,')S']J; (t,'t,')T&.

The boundary conditions which must be imposed upon Eq. (9) depend on the sign of the c.m. energy. Forscattering states (E& 0), in which we are mainly interested in this paper, they read

(i2)

(Is)

U~~r~(r) „;r~", for all n (we assume that the potential is less singular than I/r')

U„zz(r) = sin(k~ ——,'L w+ 6&~), for n= 1g~Oo

U„zz(r) - 0, for n&1.g~ OO

We stress the fact that the exact asymptotic solution (r-~) of Eq. (9) for n& 1 is not determined by theterm 2p.„(&M„—E) but depends mainly on the term 2g„Zz.z. [V~zrz, z.z. U~~~rz. (r)]. It is therefore rather te-dious to determine it. We found out that the simple form of Eq. (13) is sufficient for all cases under con-sideration.

G. -H. NIEPHAUS, M. GARI, AND 8. SOMMER

C. First order perturbation theory

As an alternative to the approach given in Sec. IIB one might think of a perturbative solution of Eq. (7).As the resonance admixtures are expected to be small, first order perturbation theory should be suffi-cient. One obtains"

1~(z @ )~

-&

&y. .lv,.lx. i&l& i&, for n&1 .Ho —E~)x„ (i4)

Here l(t),& is the wave function of conventional nuclear physics. With the definitions of Sec. IIB and thelast equation we find the following expression for the radial functions U~zre(r):

with

2G„,(r, r, Z) = - rr

71'

f,(kr) j,(kr )k l2g„+ ~M„—E (16)

j~(x) denotes the spherical Bessel function.The integral in Eq. (16) can be solved analytically:

G„,(r, r ', E) = —"e &"& sinh(k„r&),

G„,(r, r', E) = —" 1+ - e &"& cosh(k r&)„——a„(' 1 „sinh(k„r&)a„~, " '

k„~&

Q„3 3 „3 3G„,(r, r', E) = —" 1+ +, „e&& 1+ — —„sinh(k„r&) —— - cosh(k„r&)k„k„r, (k„r )' k„r,)' u„~&

(i7)

a„15 15 6 „„- 15 15 6G„,(r, r', E)= —",

k„+,

k

' „+k

+1 e "~"&, —„-+1 cosh(k„r&) — —,+ — sinh(k„r&)(k„x&) (k„r, ' k„r,

g„ 105 105 45 10"""' "=~„" (a„.,)"(a„,,) "(a.,)"~„., "')

with

105 45 . ' 105 10(k~J&) (A'~'v&J (k~J'&] k~ J&

e. =2~. , k. = [2~.(&~. E)]'"-, r I&] =IM,".I(r, r').

D. Baryon-baryon potentials

The equations given so far are quite general,i.e., they are valid for all types of nucleon reso-nances (N* and &). In our numerical treatmentwe shall restrict ourselves to the inclusion ofd(1236) resonances. Because of general selec-tion rules (parity, isospin, angular momentum,and, additionally, antisymmetry in the case of &4configurations) only specific partial wave com-ponents are allowed. They are summarized inTables I and II for isospin T=1 and T=O. Allpossible components for J & 2 are shown.

For the numerical evaluation of the Eq. (9) and

(15) we need the matrix elements of the potential

V». Ne take into account all possible couplings.The total potential reads

V„=V""[Reid(modified) ]+ V",",""'"+ yNN™hih, + yNh, ~Ad,

fl' p I', p

+ y N 6~N 4 + y 6 5~ iI), 5,ff, P 'l1', p

As already mentioned we use for the NN part ofthe interaction the usual Reid-soft-core potentialin the case of Eq. (15) (first order perturbationtheory), whereas in the case of Eq. (9) (coupledchannel approach) the Reid potential has to bemodified. The reason for this modification is dueto the fact that in the coupled channel approach

20 RESONANCE-DEGREES OF FREEDOM IN THE T%O-NUCLEON. . .

X~(&236) 4(1236) ~(1236)

'So3Po3P

(3P 3E )

5D

3Po3P, , ~P, , SE,3 5 3 5P~, P), E), E~

5 3 5 5Sg, D), D), Gp

i 5So Do

3Po, 7Eo

Pi, Ei3 7 3 7 7

5 i 5 5

(or in perturbation theory of higher order than thefirst) there are recoupling effects from resonance

TABLE I. N& and 4~ partial waves in the isospinchannel 7=1 (J& 2). (The 3D& N& partial wave does notcouple directly to the NN Dz partial wave and was there-fore omitted from our numerical calculations. )

configurations to the NN channel. These effectsare implicitly contained in phenomenological po-tentials. Thus if one wants to deal with resonancedegrees of freedom explicitly, one has to removethe resonance effects from the NK potential toavoid double counting. We managed this problemby a-complete new fit of the potential parameters.This has the advantage of an energy-independentpotential definition and thus allows the conventionaltreatment of the many-body problem. Anotherprescription to remove the double counting effectshas been given by Green and Niskanen. 4 W'ith theirprocedure' they obtain an energy-dependent NN

potential. Our modified Reid-soft-core potentialhas the following form:

e«x e-4x 7x

V, (x) = -h —1650.6a„+6484.2a„'so x' " x

' " x

x 2x e-4x e 7"V, (x)=-h —12.322a» -1112.6a» +6484.2a~s x

4 4 „16 4V, (x)=-h 1+ —+ —, e "——+ —,e 4"

x x' xe-2x e- 4x e «7x

+ 27 133a3y y90.74a32 + 20 662ax

2 2, 8 2V, (x)=h 1+ —+ —, "- —+ ~ e '"

x x x x2x e- 4x e -6x

100.0g + 1000.0a — + 5000.0ax x x

V (x)= V (x)+ Vgx)S + V (x) L ~ S,

e-4x e-6xV (x) = 933.48a„+4152.1a„

3 x x x

1 1 1 „4 1 e(x) = h —+ —+ —,e '- —+ —,e " x —34.925as,

e-6xV~e(x) = -2074.1a„ x (19)

V, (x) =3hx

e «2 x 3x

634 39a6y + 2 163.4a62x x

2 2 „8 2V (x) =-3h 1+ —+ —, e "- —+ —,e

3D2 „X X2 X X

2x e~"—220.12an + 871.0a72x x

V, , (x) = V,(x)+ Ur (x)S„+VJe(x)L ~ S,Sy Dy

x 2x

V&(x) = -h + 105.468a,~

e-4x e -6x—3187.8a82 + 2 '3a83x x

e-4x e-6xVzz(x) = 708.91a8~ —2713.1as7x x

3 3 12 3 e-4x, e-6xV (x)=-h 1+ —+ ~ e "——+ —e " x+351.77a —1673.5a

x x x x' . "x "x'

1100 G. -H. NIEPHAUS, M. GARI, AND B. SOMMEB, 20

TABLE II. A4 partial waves in the isospin channelT=0 (J~ 2).

6(1236) 6(1236)

( Si- Di)

iP3D

3 3 7 7~12 Di~ Di2 Gi

i 5 5Pi, Pi, Ii3 7 7D2, D2, G Ng

yNN~Z yNN dd

Here,fg = 10.463 MeV,g =O.Vy.

The usual Reid-soft-core potential is obtainedfrom Eq. (19) with the following choice for theparameters a, , '.

&4, = 1.352 5

a42 = 0.472 Bj. ,

a43——0,

(20)

W

x', 9a

x' 9

a,, = 1, in all other cases.As indicated in Eq. (18) we calculate the reso-

nance potentials in the limit of w and p exchange.We do not include co exchange because it gives riseto unphysical results (see Ref. 13). Until now twodifferent methods have been used to derive thesepotentials.

We first mention the older method in whichFeynman techniques are used to determine the Mmatrices for the relevant processes (see Fig. 1).The potentials are then given by the Fourier trans-forms of the nonrelativistic approximations of theM matrices. We choose the following baryon-baryon- meson vertex form factor:

A2-m'F(e')= A2™,, (21)A2+ q2

where nz denotes the mass of the exchanged me-

Nq

W

7l 9

Ng

yN4 Nd

FIG. 1. Illustration of the different processes whichare taken into account for the calculation of the reso-nance potentials.

son and q, its four-momentum. A is the so-calledregularization parameter.

The resonance potentials are then given by thefollowing formulas:

+8182 8182(2 ) = (l', ~ p, )t ~ $818281 2+ (8 x 8' ) p81818282 y S818'18282 pB~B18282'B1B1 B2B2 1 g 81B1 B2B2 C 12 T + +2

pBp818282 lY818i828), P(f)xl lx) + 2 fTB1818282, P(M lx)c r c p

+81818282 lYB18)8282, x(M 2 ) p'81818282, P(fYf lx)T

y8181828) + 1 1 Pg82 2 Af lY (M lx) lx (Alx) l + PA

S jr, P

2A2

jr81818282,n(M )P) f 1 1 f82 2 y' (~ 2,) lx (Alx) ] +

A3 AA' M'j»8»282 ~(Af 2) = ~ fB'8'~f828'~ 1'(M 2) Y (Ar) —— ~ A2 1'(A2)Q 0!

I

1 ( 8 8' )(OB 8 )(+8 '8' 8 8)'X 1 3 3Y(x)= ——, Y(x)= )+ — Y(x), Y(x)= —,+ —+l)Y(x).

(22)

20 RESONANCE DEGREES OF FREEDOM IN THE TWO-NUCLEON. . . 1101

TABLE III. The factors &g, G'p, Pg, P), yg, yp of Eqs. (22), (24), and (25) for the differentresonance potentials.

B(B~ BgB~

NN

NNN4N4NA

0

0001

001111

stS'+1slsls's/

602'4~15

6 ~10

S+Js+JS+Js+JS+ J+1s+J

4&3060~30

60& 24V 30

60vY

The factors e, and a, for the different processes and the coupling constants are listed in Tables III and IV.Taking into account the antisymmetrization operators in Eq. (12), the matrix elements of the potentials

read:

v,",„[Ns],, = (&n, ; [I (-,'-,') s]z; (-,'-,')7l v!(2,",].; [I '(s,'s2')s ']J'(t,'t 2')» —,

' [1 —(-1)1"r][1—(-1) ""']with s,'= s,'=t,'=t,'= f',//2'],

v„",„ps],,s = ~2(iv~; [L,(22)s]z;(22)rlv!Q",); [I.'(s,'s2)s ]z;(t,f,)» —,' [1 ( 1)"s"']

with s,' = s2 = t,' = t ' = ('/')

v„„,„„,=(xn; [1.(-', -',)s]z'(22)rlvl&n' [I.'(-'2)s']~'(- )»+(/~;[I (;'.')s]z;(2,')-Tlvlx/; [I, (-', —,')s ]z;(-',—,')z)( I)"2'&".

Finally we give the matrix elements of the operators occurring in the potentials:

S(B,'B,', (IS)Z!(o, , a, , )!B,B,;(L, 'S'}Z)=( 1)'1P,

2

(23)

(24)

and s, s, SI' 2 L S J(B'B' (I-s) Jlss 121'"222 !BB ' (f. 's') J}= (-1)"1y I L'ss'

1 29!L00 OiS' I' 2

sg s~ S|1 2(26)

where A: =(2A+ 1)'/'. The factors p„p„y„and y, are listed in Table 111.It should be mentioned that as far as the potentials V"""~~ and V~~"~~ are concerned, Eq. (22) in this

paper should be identical with Eq. (11)—(18) in Ref. 13. Unfortunately there are two misprints in Eqs. (16)and (1V} of Ref. 13, the sign of the last term in Eq. (16) and the overall sign of Eq. (IV) is opposite.

The second method to determine the transition part of the resonance potentials has been described byDurso et al." The main idea of this method. is to identify parts of a two-pion exchange graph with anintermediate resonance with a twice iterated one-pion exchange process (see Fig. 2). This procedureyields a one-pion exchange transition potential which is proportional to (if we forget about spin and iso-spin)

ySN~N 5 ~ 1 1 Iq2+M2+(q2+M2)1/2((q2+M2)1/2M)q2+M2+(1|2+M 2)1/2(($2+M 2)1/2M) (26)

We do not want to show the calculations which lead to Eq. (26) here; all the details may be found in Ref.21. It should however be pointed out that in the limit M~-M„Eq. (26) does not yield the usual one-pion

TABLE IV. The two sets of baryon-baryon-meson couplirg constants used in this work. Set (a) corresponds to aratio of 3.7 for the p-meson tensor to vector coupling, set (b) to a ratio of 6.0.

~Neap /4+ g /4 f~„, /4~ f /4 f~~,~/4' f/' /47l

(a)(b)

0.0810.077

3.214.5

0.890.55

0.350.35

9.1313.0

0.0030.003

0.890.55

0.1270.18

1102 20N I EPHAU GARI, AN I

u

SOMMER

D

q ~(M~ —M

urso et al." themthemselves haveve given tw ations:

v""""'o furtheei app I"oxlDl

2 q+M,1

q +M, '+(iP+M ' '~'

yNN ~Eh ~

'(M, M„) '

q2+~ 2 2 2q'+M, '+M (M ~ —M~)

(26a)

(26b)

exchane

p«entiaalculations

Soino r opinion i

Several auth

nconsiste t t in coupled channel

s imoivp i y t e handling with the

Saarel

2 q+M

n term in Eq. (26

q'+M '+ [(q' M,'+, )( '+M

very welP'

M„/M, ]"' ' (2V)

It should be me mentioned that

potentials V"""~~e used to derive the tr

o Durso

~ andtoiransition

(26)-(26bthe fsrst t p- g po

e substitution M -M .M, . n

ange (OPEot tialll the new are quite different from

expression (26'erence between

arge; especiallppr oximations

very goodappi oxlmatlo

veryion of Saarela is

Unfortunately the Foy nrier tra fo~- 28a) cannotbe er

ca culations this iy ically.

is is no princiormation can be done

numerically. Th is proce4

t ti 1 iththus an important pro-

2, 0-

1.8-

1,4-

1.2-

&.0-

+ .. + 0.8-

~ yNN~B yNN~N0

N 2MN 2M

0.6-

+ & W +terms vvi t hnegati ve e nergyi ntermedi atestates

0.2-

0.8 2,4 3.2 4 0 4.8q(fm ~)

FIG. 2. ConstrFI . truction of the tro-pion exchan e

e transition potennge Feynman gr

ance.graph with an ian intermediate

FIG.. 3. Comparisop rison of the differentp ent~a1s are b

d th f to r p.ing the

20 RESONANCE DEGREES OF FREEDOM IN THE TWO-NUGI, EON. . . 1103

TABLE V. The parameters a;; of the modified Heid potential [transition potential (22), set (a) of couplirg constants,~= 5.0 fm ]. In the case of the P2- I"'& NN partial wave the 44 admixtures were not taken into account for the fit.

12345678

0.287 47-0.499 98

1.681 270.422 300.527 751.114061.229 691.056 00

0.903 330.573 291.049 540.639 471.283 131.192 141.400 110.906 03

2.922 161.483 671.426 44

-0.573 54

1.054 55

1.127 59

1.003 42 1.002 45 1.002 63 1.001 18

1$

D23Pp3P3 '3P2- I'2

1P3D

23 3

perty of the new potentials is the smaller longrange part.

In spite of the advantages of the new method todetermine transition potentials, the situation isnot yet satisfactory. Durso's method is not ableto give predictions for the interaction betweenresonances, so one is forced to use the Feynmanprescription for the interaction between reso-nances. Of course this is somewhat inconsistentand the results obtained with this procedure shouldnot be looked at as final results. It is known that

especially in 8 states changes in the interactionbetween resonances may give rise to great changesin the admixture probabilities. Therefore we donot think it would be worthwhile to calculate all KNscattering states with the expressions (26) or (2V).In this work we will use the approximate expres-sion (28b} which is much easier to handle. Theeffect of the different approximations (26)-(28b}will be discussed in the case of the deuteron. In-cluding the form factor (22), the potentialsVNN

"N~ and VNN" ~~ based upon Eq. (28b) read

(&) (f~ rZ )t(C rp' )rprNBlNB2+$NB1NBp lrrNBiNB2]+ (] 2 On]y 1f + + +n)12 T & 1 2

l M 'A' M2'-yNNN6 1 jrNNNk, r(M +)+ ii

i

r pNNNk, n(M +)2 M, & A2-Si 2

+ yNNNLR, P(M ~)+ P P PNNN6, P(M ~)M P2 M2 2

C M A -19 pP P

2 P2 M 2 2 M A-MyNdlN rrI ir P jrNdN 5, ir(M /) y 2 P P i y/NPNk, P(M ~)p)

2 $2 M 2 2yNNNE 1 gNNN1rr, r(M ') + ri p jrNNN

harp(M

)2 M h.'-Si'2 +2 M 2 2

1 jrNNN Er p(M ) p p g NNN kr p(M )T p 2 M p2 gj 2 T pP P

. yNM'b, if r @NENES ff M ~ p p yNENh p M

(29)

TABLE VI. The parameters a;; of the modified Reid potential [transition potential (22), set (b) of coupling constants,A= 5.0 fm-'j.

0.359 53-1.318 17

1.909 040,596 050.275 171.111271.211611.008 91

0.932 640.794 971.205 811.024 241.291 111.188 411.374 850.857 39

3.038 421.522 551.053 91

-0.035 996

1.008 76

1.05272

1.008 79 1.008 88 1.008 07 0.995 92

'L

Sp~D2

3Pp3p3'3P2- E2iP3D3 3

G. -H. 5IEI'HAUS, M. GARI, AND B. SO MMER

TABLE VII. The parameters a;, of the modified H, eid potential [transition potential (29), set (b) of coupling constants,&= 5.0 fm ~].

123

567-8

0.550 24-1.872 13

1.590 130.677 350.714 361.122 061.258 590.879 29

1.142 791.381371.373 930.704561.613 181.179631.357 580.924 02

3.667 391.569 051.226 560.137 83

1.083 05

1,11984

1.046 02 1.02981 1.007 32 1.024 47

'sD2

3Pp3p3 3P2- I2Pg

3D3 3Sg- Dg

Here,

M—:[M +M (M~ Mg}]~-~

Vs|sis& 2 ~(M r), and V~/ I 's'~(M r) are thesame as in Eq. (22). As already mentioned, forthe other parts of the resonance potential(VN6 6k VNrh N d V Ak kk) E (22)

III. NUMERICAL RESULTS AND DISCUSSION

One aim of this paper is to compare the resultsof the coupled channel approach and first orderperturbation theory. To get reliable results inthe case of the coupled channel approach, one isforced to remove the aforementioned implicit reso-nance effects from the Reid potential. %e there-fore made new X' fits of our modified Reid poten-tial to the NN scattering data and the deuteronbinding energy. As we did not include the Coulombforce in our calculations, we could not use theexperimental proton-proton scattering data. Sowe used the phase shifts calculated with the usual

Reid potential to determine the potential parame-ters a,,. These parameters are summarized inTables V-VII; they are valid for a regularizationparameter A = 5.0 fm-'. %e first discuss the re-sults which are obtained with the potential (22) andset (a) of the coupling constants given in Table IV,that is, we use the potential parameters a, , of Ta-ble V for our modified Reid potential.

In Figs. 4-6 we present some phase shifts forthe conventional calculation (only nucleonic de-grees of freedom} and for the explicit treatmentof the & resonance. These figures show thequality of the fit. In Figs. 7-10 we give the mostimportant N4 and 4~ wave functions of the scat-tering states for 30 MeV laboratory energy and thedeuteron &4 wave functions. In addition we havelisted the integral N (square of the norm) of theresonance wave functions for the perturbationtreatment and the exact solution

N = [U„~q (r) ]2 dy .0

5(ra d)

1,0.

6 (rad)il

0

—0,08-

0.6- -0.24-

0.4- -0,32-

0.2- -0,40-

—0.48-

-0.20

I S I

40 80 120 160 200 240 2 80'EL AB (Me V)

—0.56 I

80I I I I

120 160 200 24 0 280EL AB (Me V)

FIG. 4. S p phase shifts calculated with the usualReid potential (no resonance admixtures) and with ourmodified Reid potential. [Eq. (19) and Table VJ.

FIG. 5. 3P& phase shifts calculated with the usual Beidpotential (no resonance admixtures) and with our modi-fied Reid potential [Eq. (19) and Table U].

. 20 RESONANCE DEGREES OF FREEDOM IN THE TWO-NUCLEON. . . ll05

5 (rod)

2.0-S,

1.2-

0.8-

0.4-

00 40 80 120 160 200 240 280

EL AB(Me V)

FIG. 6. S f phase shifts calculated with the usual Beidpotential (no resonance admixtures) and with our modi-fied Reid potential [Eq. {19)and Table V].

We realize that in some cases both perturbationtreatment and exact solution agree quite well.However, just for the largest components at 30MeV, the perturbation treatment is off by =20%up to =60/o compared to the exact solution. It isinteresting to see that there is no general trendin these deviations. In some cases the perturbationtreatment gives larger results, in some cases theresults are smaller. This means that if we do a

comparable treatment in finite nuclei or nuclearmatter we obtain only very rough information. Forexample, the most important ~ contribution ex-ists in the 'S,NN channel. The Nb 'D, partialwave integral N differs by 58% compared to theexact solution. The perturbation calculation over-estimates this contribution considerably. This istrue also for other energies.

As far as the deuteron is concerned there is al-so an overestimation of the large &&'D, compo-nent in the perturbation treatment: 0.76% admix-ture compared to 0.58%. The ~4'S, componenton the other hand is underestimated: 0.08% com-pared to 0.11%. The total ~ probability calculatedwith perturbation theory differs by about 20% fromthe exact value.

Concerning a perturbation treatment of 4 de-grees of freedom in finite nuclei or nuclear mat-ter, we do not expect this to be a good approxima-tion because of the bad reproduction of the largestcomponents.

To get an idea of the energy dependence of theresonance admixtures, we show in Fig. j.l the in-tegral N of some resonance partial waves (exactsolutions) as a function of the laboratory energy.W'e have selected those partial waves which aremost important at 30 MeV and the largest ones at300 MeV. The figure demonstrates an interestingproperty. The resonance partial waves which cou-

u(r)0.16-

0,08-E =30 MeV

—0.08-

—0, 16-

-0.24-

—0.32-

Nd O.

- - ——perturbation theory

exact soluti on

x 10

per. th.

7. 724

S0 0, 347 0. 1271

D0 0. 794 1.132

—0.4 0 I

0.5 1.0f.

1.5l

2.0I

2.5 3.0I

3.5l

4.0 4.5 5.0r(fm)

FIG. 7. Radial functions of the N4 and ~ admixtures to the NN 8 0 partial wave calculated with set (a) of the couplingconstants and the resonance potential Eq. (22). The results of perturbation theory [Eq. (15)] are compared with theexact solution lEq. (9)]

1106 G. -H. NIEPHAUS, M. GARI, AND B. SOMMER 20

1 O X U (r) if

0.24-

0.16- ELAN= 30 Me V

-- - --perturbation theory

0.08-

—0.08-

-0.16-

-0.24-

-0,320

I

4.5S

4.0I

2.51.0I l

0.5 1.5 2.0 3.0 3.5 5.0r(fry))

FIG. 8. Radial functions of the N& and && admixtures to the NN P& partial wave calculated with set (a) of the couplingconstants and the resonance potential Eq. (22). The results of perturbation theory [Eq. {15)]are compared with theexact solution [Eq. (9)].

pie to NN S waves show quite a different behaviorcompared with all other waves. The integrals Nof the former increase rapidly for very low ener-gies (& 10 MeV) and then increase only very littleup to 300 MeV, whereas the integrals N of all u(r) Ji

0.10- Deuteron

other resonance partial waves show somethinglike a quadratic dependence on the energy in the

U(p) ( 3g 3Q )

0.12-

0.08-

0.04-

tegrals il x

ex. sol

S1 0.206

D1 0.079

D1 1.203

0. 123

10

per. th.

0. 166

0.060

1.46o

0. 141

0.08-

0.06-

0.04-

0.02-

10

per. th.

~. 031

0. 766

0.067

J ~A i331

-0.04-

-- - - —perturbation theoryLI ~ lj 41

exact s oluti on

ELAN = 30 Me V-0.02-

-0.04 -.

on theory

ti on

-0.120 0,8 1.6 2.4 32 4.0 4.8 56

r(fm)

-0.06 I

0.8I

1.6 2.4I

3.2l

4.0 4.8 5.6r(fm)

FIG. 9. Radial functions of the 4A admixtures to theNN($ ~- D&)~ eigensolution calculated with set (a) ofthe coupling constants and the resonance potentials Eq.(22). The results of perturbation theory [Eq. (15)] arecompared with the exact solution [Eq. (9)].

FIG. 10. Radial functions of the 4A admixtures to thedeuteron calculated with set (a) of the coupling constantsand the resonance potential Eq. (22). The results ofperturbation theory [Eq. (15)] are compared with the ex-act solution [Eq. (9)].

20 RESONANCE DEGREES OF FREEDOM IN THE TWO-NUCLEON. . . 1107

0.07-

o.og-

NZ D, ("S )Ng P„( P„)

0.05-

0,03-

0.02-

Nd P» ((P~-F»)~)

-N 8 'S, ('D, )

~N d'P» ((~g-F» )„)=N4 P~( +~)

FIG. 11. The integralsR of the largest resonanceadmixtures as functions ofthe laboratory energy. Theresonance admixtures arecalculated with Kq. (9),set {a)of the couplingcon-stants and the resonancepotential {22).

0.0 t-

I

50 100 150 200I

250 300„a™v)

u (r)

0.8-

0.4-

----- without

wi th 8 admi xtures

—0,80

I

0.8I

1.6I

2.8I

3.2I

4.0r(fm)

FIG. 12. Comparison of the NN ~S0 radial function

calculated without explicit resonance admixtures (usualReid potential) and the radial function of the coupledchannel calculation (modified Beid potential with thepa, rameter values of Table V).

whole region. Because of this behavior the mostimportant partial waves in the low energy region((50 MeV) are different from those which are im-portant for higher energies (& 200 MeV).

In Fig. 12 we present the NN 'S, wave functioncalculated without 4 admixtures (usual nuclearphysics) and the wave function which follows fromthe coupled channel calculation. The latter func-tion is smaller for distances up to 2 fm. This is

true in all NN partial waves. Of course this changeof the NÃ wave function does not occur in first or-der perturbation theory. %e think it would beworthwhile to examine the effect of this wavefunction renormalization in electromagnetic pro-cesses, for example.

In addition to the vector-dominance model valuesfor the p-meson coupling constants [set (a) in Ta-ble IV], we also use the values corresponding tothe coupling constant analysis of Ref. 23 and. 24.These values are summarized in Table IV [set (b)].%'e made two further fits of our modified Reid po-tential to the scattering data with these couplingconstants and the resonance potential Eq. (22) (Ta-ble VI) and alternatively the Durso potential Eq.(29) (Table VII). Some resultant resonance wavefunctions are shown in Figs. 13-16. Comparingthe two calculations with the resonance potentialEq. (22) and the different sets of coupling con-stants, we recognize that there is no generaltrend. The change in the integrals N varies from-7.2% up to 33.3% compared with the old values.The change in the largest admixtures to each NNpartial wave however is smaller: —7.2% up to I%%uo.

The total admixture probability in the deuteronis increased from 0.768 to 0.814.

The resonance wave functions calculated with theDurso potential are also shown in Figs. 13-16. Inthe asymptotic region they are generally smalleror at least not larger than those calculated withpotential Eq. (22). This effect is caused by the

I l 08 G. -H. NIEPHAUS, M, GAB I, AND B. SOQMER 20

I U(r)0,10-

g.05-

-0,05-

—0.10-

- 0.1 5"

x 10

r) pot. (r9)

3. 972

0. 523

0.624

-0.20-

potentia-l (29)-0.25

0,5 1.0 1.5 2.0 2,5i

3.0 4.0 50r(fm)

FIQ. 13. Hadial functions of the Nd and hA admiztures to the NN S 0 partial wave calculated with set (b) of the cou-pling constants. The full curves show the wave functions obtained with potential Eq. (22), the dashedones those calcu-lated with the transition potential Eq. (29) and the other parts of the resonance potential given by Eq. (22).

dimj. nishing of the long range part of the potentia1. .The change of the integrals N is quite large in thiscase: -55.5% up to 50% compared with the results

from the calculation with potential Eg. (22). Theoverall deuteron resonance admixture is decreasedfrom 0.814 to 0.708, that is a change of -13%.

E~AE =30 Me V

0.08"

—0.08-

—0.16-

—, 0.24-

-0.4 00.5

C

1.0I

2.0

N

potential (22)

potential (29-)-t

2,5 3.0t

3.5

5tie

3P1

aF

4,0

0.044

0.004

0.018I

4,5

0.031

0.003

0.00v

5.0r(fm)

FIG. 14. Radial functions of the Nn, and bA admixtures to the iVN3I'& partial wave calculated with set (h) of the couplingconstants. The full curves show the wave functions obtained with potential Eq. (22), the dashed ones those calculatedwith the transition potential Eq. (29) and the other parts of the resonance potential given by Eq. (22).

20 RESONANCE DEGREES OF FREEDONI IN THE TWO-NUCLEON. . . 1109

( ) (3g 3Q )0.12-

0.08- 51 0.315

D1 0.082

0, 426

0.094

0.927

integrals &I x 10

pot. (22) pot. (29)

u(r)

0.08-

0.06-

0euteron

'1

1 s N x 10"

ot. I22) pot, f29)

. 139 0. 187

0.04-01 0. 122 0.092

0.04-

0.02-

. 03o 0.042

. 583 0.439

. 054 0.040

. 814 0, 708

-0.04-

-0.08-

-0.120

I

0.8I

1.6I

2.4 321

4.0 4.8 5.6r(fm)

EL Ag = 30 MeV

potential (22)--- ——pot e nti a l (29)

—0.02-

-0.04-

te nti al (22)

enti a( (29)

FIG. 15. Radial functions of the AA admixtures to theNN(3S 1 6f) ~ eigensolution calculated with set (b) of thecoupling constants. The full curves show the wavefunctions obtained with potential Eq. (22), the dashedones those calculated with the transition potential Kq.(29) and the other parts of the resonance potential givenby Zq. (22}.

—0.06 V

0.8l

1,6l

2,4'I

3.2 4.0I 1

4.8 5.6r (fm)

FIG. 16. Radial functions of the 4d admixtures to thedeuteron calculat|:d with set (b) of the coupling constants.The full curves show the wave functions obtained withpotential Eq. (22), the dashed ones those calculated withthe transition potential Kq. (29) and the other parts ofthe resonance potential given by Eq. (22).

~~ u'dr0.0 7-

potential (22)

0.06-

0.05-

N8 D, ( S, )

N8P, ( P)

0.04-

0.03- P ( P )

0.02-

0.01-&& 0, (('S„-0) )

~8M'D, ("S )

0 100 1501

200I I

250 300L. Az (&ev)

FIG. 17. The integrals N of some large resonance admixtures as functions of the laboratory energy. The resonanceadmixtures are calculated with Eq. (9) and set (b) of the coupling constants. Full curves: resonance te t 1 E '22' ~po n la q. ( );as ed curves: transition potential Eq. (29) and the other parts of the resonance potential giveri by Eq. (22).

1110 G. -H. NIEPHAUS, M. GARI, AND B. SOMMER 20

TABLE VIII. &4 admixtures to the deuteron calculated with different transition potentials(7t. + p, A= 5.0 fm ~). The interaction V~~ +~ was set equal to zero.

No. Type of potentialx (%) x (%)

total

Eq. (22)Eq. (28b)Eq. (28a)Eq. (27)Eq. (26)

0.0820.1410.1620.1470.142

0.0280.0320.0260.0220.021

0.6300.4950.2690.2120.205

0.0500.0400.0220.0180.017

0.7900.7080.4790.3990.386

In Fig. 17 we have calculated the energy depen-dence of some large resonance admixtures usingthe experimental coupling constants [Table IV, set(b)]. It can be seen that the integrals N of thelargest resonance admixtures to the 'S„'P„and('S, —'D, ) NN partial waves are decreased when

the Durso potential is used.Finally we have used Eqs. (26)-(28b) and the

transition part of potential Eq. (22) to calculatethe integrals N of the resonance partial wavesin the deuteron. The coordinate-space potentialswere obtained from Eqs. (26)-(28a) by performingthe Fourier transformation numerically. Again theform factor Eq. (21) was included in the same wayas in the potentials Eq. (22) and (29) (A = 5.0 fm-').m and p exchange was taken into account with thecoupling constants of Table IV [set (b)]; the inter-action between resonances was neglected. Weadjusted the binding energy of the deuteron to theexperimental value by modifying only the parame-ter a» in our modified Reid potential Eq. (19).The results of these calculations are shown in Ta-ble VIII. First of all we recognize that the total

u(r)0.12-

0.10-

A= 80 fm"/1 70 fm"

Deu teron

dd Of

0.08-

0.06-

0.04-

0.02-

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 r(fm)

FIG, 18. The radial functions of the AA ~D& admix-tures to the deuteron calculated with the transition poten-tial Eq. (22) (no 44 interaction) for different values ofthe form factor parameter A.

admixture probability is decreased continuouslyfrom 0.79'%%u~ to 0.39%%uo when going from potential(22) via the different approximations to the exactexpression (26). At a first glance it is somewhatsurprising that the change from potential Eq. (22)to Eq. (29) does not bring up such a great effect ascould be expected from Fig. 3. But a more ca.re-ful investigation of Fig. 3 shows that the maindifferences between Eq. (22) and Eq. (29) appearfor small values of the momentum transfer q,for higher values of q the two curves coincide.Thus only the long range part of the coordinate-space potential is modified. In this region the NN

wave function is small too, so the effect is notvery important. Remembering that the deuteronS wave ha.s its maximum value at short distances(=1.5 fm), one should expect that a change of thepotential at large values of q will also produce aconsiderable change in the admixture probabilities.This effect shows up if one varies the potentialfrom Eq. (28b) to (28a) or from Eq. (28a) to (26)as should be expected from Fig. 3. As a resultof these calculations one can say that it is quiteimportant to use a "correct" transition potentialbecause the admixture probabilities sensitively de-pend on this potential. On the other hand the useof the correct transition potential will not giveany final result unless a prescription for the de-termination of the interaction between resonancesis found and the choice of the form factor is lessarbitrary. To demonstrate the last point we showin Fig. 18 the wave functions of the && 'D, ad-mixture to the deuteron calculated with the transi-tion potential Eq. (22) (no && interaction) for dif-ferent values of the parameter ~. The admixtureprobability varies from 0.428/~ (A = 4.0 fm-') up to0.875% (A=8.0 fm ').

We are especially grateful to Prof. H. Waffler.for his generous help. This work was partiallysupported by Deutsche Forschungsgemeinschaft.

20 RESONANCE DEGREES OF FREEDOM IN THE T%0-NUCLEON. . . . llll

H. Sugawara and F. von Hippel, Phys. Rev. 172, 1764(1968).

~K. Holinde and B. Machleidt, Nucl. Phys. A280, 429(1977).

A. M. Green and P. Haapakoski, Nucl. Phys. A221, 429(1974).

4A. M. Green and J. A. Niskanen, Nucl. Phys. A249, 493(1975).

:5B.D. Day and F. Coester, Phys. Rev. C 13, 1720 (1976).G. Horlacher and H. Arenhovel, Report No. KPH 19/77{unpublished) .

~W. Manzke, Z. Naturforsch. 33a, 861 (1978).8W. Manzke and M. Gari, Nucl. Phys. A312, 457 (1978).9P. Haapakoski and M. Saarela, Phys. Lett. 53B, 333

(1974).' H. Arenh5vel, Phys. Lett. 53B, 224 (1974)."H. Arenh6vel, Z. Phys. A275, 189 (1975).

E. Rost, Nucl. Phys. A249, 510 (1975).' M. Gari, H. Hyuga, and B. Sommer, Phys. Rev. C 14,

2196 (1976).~4S. Jena and L. S. Kisslinger, Ann. Phys. (N. Y.) 85,

251 (1974).P. Haapakoski, Phys. Lett. 48B, 307 {1974).

~6R. A. Smith and V. R. Pandharipande, Nucl. Phys.A256, 327 (1976).

~A. M. Green, Bep. Prog. Phys. 39, 1109 (1976).H. J. Weber and H. Arenh5vel, Phys. Rep. 36C, 277(1978).

98. V. Beid, Ann. Phys. (N. Y.) 50, 411 (1968).H. Arenhovel, M. Danos, and H. T. Williams, Nucl.Phys. A162, 12 (1971).'J. W. Durso, M. Saarela, G. E. Brown, and A. D.Jackson, Nucl. Phys. A278, 445 (1977).

~~M. Saarela, Ph. D. thesis, Oulu, 1977 (unpublished).36. Hohler and E. Pietarinen, Nucl. Phys. B95, 210(1975).

~4W. Grein, Nucl. Phys. B131, 255 (1977).