The πσ, πρ and πω meson exchange contribution to the NN interaction

Nuclear Physiw A372 (1981) 349-376 Q North-Holland Publishing Company

THE zrQ, ~rp and zrw MESON EXCHANGE CONTRIBUTION TO THE NN INTERACTION K . HOLINDE and R . MACHLE)DT Institut fûr 7heontische Kernphysik der Unitursititt Bonn, NußaJlce 14-16, D-5300 Bonn, W.-Germany Received 27 May 1981 Abetrad : The fourth-order non-iterative diagrams involving two-nucleon intermediate states and -rm exchange, where a ~ ~, p, u:, are evaluated in momentum space in the framework of non-covariant perturbation theory . Appreciable cancellation is found between the separate contributions . The importance of these diagrams for the medium range part of the tensor force is demonstrated by calculating typical NN scattering phase shifts .

1. Introduction

It is well known that results of nuclear structure calculations depend sensitively on the amount of tensor force present in the underlying nucleon-nucleon interaction. In particular, the binding energy of nuclear systems increases with decreasing tensor force. This is clear from the property of the tensor force being mainly a second-order contribution, which is suppressed in the medium. For example, in a lowest-order Brueckner calculation of infinite nuclear matter, the Reid soft core potential (RSC) t), which predicts a deuteron D-state probability of 6.47%,binds at =10 MeV, whereas one version of the Bonn potential (HM2) 2), with a D-state probability of 4,32%, yields as much as 24 MeV binding. However, the tensor force is an essential quantity not only for the saturation properties of nuclear matter, but also for the binding energies of light nuclei . Namely, two potential models, which differ by 1% in the deuteron D-state probability, predict saturation energies per particle, which differ by about 0.2 MeV in the case of the triton, by roughly 2 MeV for t60, and by as much as 10 MeV in nuclear matter . As expected, the effect depends strongly on the density of the system : for low-density systems like the triton, the suppression of the tensor force due to the medium is small ; it grows with increasing density. Moreover, the amount of tensor force plays an important role in all sorts of few-body reactions like e .g. the photodisintegration of the deuteron 3) . Thus it is of outstanding importance to know even the details of the tensor force in the NN interaction as accurately as possible . The crucial point is, however, that, at present, the empirical information about the two-nucleon system is still too crude to allow for a sufficiently reliable determination of the correct amount of tensor force : the empirical errors in the mixing parameter e, and in the D-state probability PD (which 349

35 0

K Holinde, R . Machleidt / Meson exchange contribution

are quantities most sensitive to the tensor force) are so large that practically any tensor force yielding values of PD from 4 to 7% is allowed by the two-body data . We should mention that the quadrupole moment of the deuteron is only of limited value for pinning down the correct amount of tensor force since meson exchange currents, which until now cannot be reliably calculated, are known to contribute . Thus we are led to ask what the theory can tell us about the amount of tensor force and whether, in the present stage, it is able to sufficiently pin down its behaviour. According to meson theory, most of the tensor force, especially the outer-range part, is given by one-pion-exchange (OPE). The inner part (r < 2 fm) of the OPE tensor force (which influences PD) is cut down (a) by the mechanism of p-exchange and (b) due to OPE vertex corrections represented by the type of diagrams shown in fig. 1 . Here diagram (a) describes the bare vertex whereas the other diagrams build up the -trNN form factor. Process (b) has a low-mass cut and gives the most significant contribution . We refer to the work of Durso et al. °) for details of corresponding calculations .

Fig. 1 . Feynman diagrams contributing to the ~rNN vertex function .

However, if process (b) of fig. 1 is important for a precise determination of the NN tensor force, there is no reason to expect that the diagrams shown in fig. 2 can a priori be neglected since their range is of the same order. In fact, it was already demonstrated by Riska s) that the arm exchange contribution [diagram (c) of fig. 2] yields a tensor force which is of the same order of magnitude as the single-m exchange tensor force. Obviously, because of the small mass of the pion, also the range of such contributions is comparable to that of single-w exchange . Consequently, a meson-theoretic framework should still be reliable for such diagrams . (Because of the quark structure of hadrons, this might well be questioned for extremely short-ranged contributions involving e.g . double-p exchange .)

Fig . 2. 3rrexchange contributions.

K Holinde, R . Machleidt / Meson exchange contribution

35 1

In this paper, we evaluate the diagrams of fig. 2, in the framework of non-covariant perturbation theory, leaving out anti-nucleon lines. This procedure is convenient for a well-defined transition from the two-body to the many-body problem and also for a reliable determination of the modifications of tar exchange contributions in the medium, which are essential for a consistent description of light and heavy nuclei 6). We stay throughout in momentum space and thus avoid characteristic (non-relativistic) approximations, which are necessary in order to obtain an analytic expression in r-space, see ref. s). ['The effect of such approximations, for the case of one-boson-exchange potentials, has been studied in ref.') .] The diagrams are calculated numerically and suitable one-boson-exchange terms are added in order to discuss effects in NN scattering . Sect . 2 contains the underlying formalism. The results are presented and discussed in sect. 3 . A short summary is given in sect . 4. 2. Formalism Our general scheme is to start from a field-theoretic hamiltonian H containing as interaction part not a nucleon-nucleon potential, but nucleon-nucleon-meson and nucleon-isobar-meson vertices. Anti-nucleons are left out from the beginning and H is treated in old-fashioned (three-dimensional) perturbation theory . The reason is the following: first, three-dimensional perturbation theory corresponds to standard non-relativistic many-body theory and will, therefore, allow a direct comparison with the usual procedure. Second, chiral invariance dictates that the nucleon-antinucleon vertex is considerably suppressed compared to the NN vertex . In the two-body case, the corresponding perturbation series for the NN scattering amplitude T can be partially summed by solving an integral equation of LippmannSchwinger type, which contains as driving term an energy-dependent quasipotential V~,r(z) consisting of the (infinite) sum of all irreducible diagrams, i.e. those with at least one meson or one d-isobar present in each intermediate state. We refer to ref. e) for further details. Our aim in this paper is to study a specific group of diagrams oceprring in V~~(z), namely those shown in fig. 3 plus those in which ar and ~ (= Q, p, ~) are interchanged .

n

,n

d

,'

d~

d

5

r

d

2

1 ~s'

~ .n

6

4

3 ,r

rr

d

d

7

8

Fig. 3 . Diagrams of Vay(z) considered in this paper. a = o, p, m.

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K Holinde, R . Machleidt/ Meson eschange contribution

Fig. 3 contains all possible time-orderings (the four iterative diagrams have to be left out since they are generated by iterating suitable one-boson-exchange parts in the scattering equation) . Diagrams 1, 2 are of stretched-box type, whereas diagrams 3-8 are of crossed-box nature . Basic ingredients are the usual meson-nucleon vertices

rn = ~B+riY S ,

Here, ga, f~ are the usual meson-nucleon coupling constants (including form factors to be specified later) . The tensor coupling fa is known to be small and is, therefore, dropped in the actual calculations . q(q') denotes the four-momentum of the incoming (outgoing) nucleon at the corresponding pNN vertex, m is the nucleon mass (=938.9 MeV) . Starting from these vertices, the evaluation of the above diagrams in momentum space proceeds along the same lines as for the non-iterative 2~r exchange, described in ref. 6).

q k~

~q~ * Qi

q' k-q_q'

q'*q_k

~. i~~ ~

.q -k

_q

Fig. 4. Selected stretched-box (a) and crossed-box (b), (c) diagrams displaying the notation as it is used in the text .

2.1 . +ro EXCHANGE 2.1 .1 . Stretched-box diagrams. One of the two stretched-box diagrams is shown for convenience in fig. 4a, including notation . It is given by (4~~i~1 i~M~ (z)~9AtAa) =

(4-tr) Z z z ~ a Fn~(4' - k)Z~~~(9-k)Z~ eS gQTi ' zz d k 4mq-k wq-k (2~r) x

~nz(-4~)iY SA +( -k)unz( -9)~ni(4~)iY SA+(k)unl(4) Q - Ek - ~â-k) , ( z-E4 .-Ek-taâ.-k)(a-E q' -gQ -mQ,-k-mq-k)(z-E (2 .2)

where we have already replaced the spin sum of the nucleons in the intermediate

K Holinde, R. Machleidt / Meson exchange contRbution

353

states by the corresponding projection operator (2 A+(k) (y~Ek - 1' ' k +m) . 2Ek

.3)

liz Here, Eq =(qz +m z ) and mk =(kz+mâ)'~z; z is the starting energy . The form factor FQ (k z) is parametrized as (2 .4) containing a parameter A~, the so-called cutoff mass . The spinors are normalized such that u +u =1 . In order to abbreviate the formulas, we introduce the following notation : u 1 = u,,,(q), ü1 = û,,;(q'), uz = u,,2(-q), û z = ûn,(-4~)~ We consider the expectation value corresponding to nucleon 2 first. Since we want to do the integration over k using polar coordinates, we must get rid of the Y " k term . Therefore, we expand k in terms of q, q' and q' x q 'x k=aq+bq'+c lg, ql , x

(2 .5)

with a=

. , , 4~ 44~ k -9a4 k -qzq' (4~ ' 4)

b=

z . 4~ . 49 . k=4 4~ k (4~ ' 4) -q 4~z

c

_ q' x q Iq,xgl

k. (2 .6)

Thus y " k can be replaced by q'xq . y k=a7'q+bY . 4'+cY . lq, xgl .

(2 .7)

Using the Dirac equation we then obtain for nucleon 2 i ûziysA+(-k)uz=2Ek [ai~zy SUZ+azûzyS where

Yuz - cûzy Syzuz~ .

(2 .8)

(2 .9)

az=Ek -aEq -bE4- .

The expectation value corresponding to nucleon 1 is given by the same expression as (2 .8) provided we replace c by -c . Thus, taking into account that the terms linear in c disappear after angle integration if we choose q on the z-axis and q' in the xz plane, we get 1 s 4 ~~aiAi+2alazAz+aZA3 - cz Aâj . ~ziy SA+( -k)uzûiiy A+(k)u1= - ~ (2 .10)

354

K Holinde, R. Machleidt / Meson exchange contribution

with

Al =

üzYSUZ~lY5u1

A z = i(ûzYs uzül y s y~ul + ûz y s y ° uzûlY s ul)

S

3

A3 = ~ zY Yuz~lY Yul

(2.11)

,

Aâ = ~ IYSYzuzülYSYzuI Aâ can be replaced by using =y .Y-a~,y,~7'~! -b~(7 .~l7'9+Y'4Y'9')-c~Y'4Y'9,

where

a

_

1

b

q~z sinz B'

, -_ -cos 9 q'q sin z 9'

c

,_

We then obtain

1

q s 2 n g '

(2.12) (2 .13)

with xz = m(EQ -a' - (Eq , - EQ )b' -Eq c') , x3

and

(2.15)

= EQ-a' + 2Eq-Eqb' + EQC' + 1, Aa = ~zYS Y~uz~lY SYwu .

(2 .16)

Consequently, we get u2iY

with

sA

LI b~'1+I bimr+2+I bw~3+I b,.a~4~~ +(- iC)Zlzillly 511+(k)ul = - E 4 (2 .17)

~ = a12 - c 2xl, Lb,l

~ =2(alaz - c 2 zz), L6,2

6,3 2 2 h~ =az-c x3 ,

Eq. (2 .2) can then ultimately be written as

(4,~)z z z ~q~Ai11iIM~(z)~4A1f1z) =- (2~r 8n8oz1 x

Tz

6,4 2 I~ =c . (2.18)

a ~ Ai(q'~lielz :4~1~1z)

i=1

r d3 kI~ (4 ~, 9, k )FÂ L(4' -kb ) z ~Q L(9 -k ) z ~ (2 .19)

355

K Holinde, R. Machleidr/ Meson exchange contribution

where

Dbrro = lz - E4' -Ek

(Vq_k)(Z - E, 4' - E9 - W9-k - W4-k)(z-E Q -Ek - Wq-k)~

(2.20)

The second graph of fig. 3 gives exactly the same result, i.e. the contribution of the stretched-box diagrams is twice the expression (2.19) . For the actual numerical calculations, we need the partial-wave amplitudes (~i11z~M~ (4~~ 4Iz)I~i~1z) = -2~r x 2 x (~SA8oT1 ' Tz a

x~~ xJ

+~

(2~r)

d cos 9 d,;,,~ (e)Ar(9~AiAz ; 411i11z)

d 3 kl~(4~~ 4, k)FÂ[(9~-k)z~ô[(4-k)z]

(2 .21)

where d,',,,- (B) are the usual rotation matrices, A = A 1 -A z, A' = A; -A2. The connection between these helicity-state matrix elements and the corresponding matrix-elements in the common ISJ basis can be found in the review article of Erkelenz 8). The integrations in eq . (2.21) are done numerically. In a second step, we have to consider the corresponding stretched-box diagrams in which ~r and v are interchanged . The diagram analogous to fig. 4a is then given by (9~Ai11z~M~(z)~4A111z)= (4 ~)e8Â8ô?i ~ zz ~ d3k

(2a)

x

FP[(4~-k)z~n[(4-k)z]

4Wq '_kW q _k

ûnf(-9~)11+(-k)~YS Un2( -4)ûnl(9~)~+(k)iY SUn,(4) (Z - E9' - Ek - lD 4'-k ) (z E9' - E9 - W 4'- k - W4- k ) (Z E9 - Ek - W q- k )

(2.22) With the same procedure as before we obtain for the matrix element corresponding to nucleon 2 s ûzA +( -k)iY uz =

where

i s s [biûzY uz+bzûzY Y ouz_cûzysyzuz ] 2 Ek

(2 .23)

(2.24) Again, the expectation value for nucleon 1 is given by the same expression as (2.23) provided we replace c by -c. Therefore, 6,=-m(1-a+b),

6z =a z .

s ûz11+(-k)iY uzûiA+(k)iys ul _-

1 [biAi+2blbzAz+bzA3-czAâ] É

(2.25)

356

K Holinde, R . Machleidt/ Meson exchange contribution

with

bl 2 2 Io:,r =61 -c xl,

b2 2 I~ =2(blbz - c xz) ~

b3 2 2 I~ =6z - c x 3 ~

b4 I~ =c 2 , (2 .26)

and eq. (2.22) becomes

z a (9'Ai~1zIMô;:(z)~4~1~1z)= -(~BASQTI ' Tz ~ A~(q'Aif1i ; 4~l~lz) f-1

xJ

d skl~(4', 4. k)FQL(9~ -k) z ~~ L(q - k)zl

(2.27)

where Dô = ( z - Eq' -Ek - w4-k)(z - Eq' -Eq - ~ Q'-k - wv:k)(z-Eq-Ek-~4 k)

" (2.28)

Again, the second stretched-box diagram gives exactly the same result, i.e. the contribution of both diagrams is twice the expression (2 .26) . Thus we obtain ultimately for the partial-wave amplitudes corresponding to the sum of all stretchedbox diagrams (~1iAZ~MQ~(4~~ 9~z)~A111z)

2 a +1 d cos 6 dnn'(e)At(q'rlilli ; 4~11~1z) = -2~r x 2 x ~aé+réaTl ' Tz ~ ( (27f) i-1 J 1

x~

d 3 k I~(q', 4. k)F~L(g'-k) Z I~PL(g-k) Z l 16É L mQ -kwQ-kD~(z, q', q, k)

+ I~(q'~ 9~ k)FQL(g'-k) z ~ ÂL(q -k) Zl~ . ~q' - klOq - kDmr(Z~ 4', 9, k)

(2.29)

2.1 .2. Crossed-box diagrams. The first crossed-box diagram (3 of fig. 3) is given in fig. 4b, including notation . It is given by d3kF,~L(q' k)z~QL(4-k)z~ (4'Ai~1z~M~(z)~4A1f1z) =(`~8~8QT1 ' Tz (2vr) J 4~q- k mQ- k ~nz(-q')A+(-t)iY s ues(-9)~~ ;(q')IYSA+(k)uel(9) x

(2.30) where t = -k + q + q' and D°~1

= (z-Bq'-Ek-~4-k)(z-Ek-Er-~q-k-m4-k)(z-Eq-Ek-w4-k)

(2.31)

K Hoünde, R. Machieidt/ Meson exchange contribution

357

The matrix element belonging to the first nucleon is the same as in the case of the corresponding stretched-box diagram, see eq. (2.2) . For the second nucleon we get uz~ +( -r)tYSUZ

__i 2Et [ciuzY S U.z+czûzY SY °uz -cûzy5 yz uz]

(2 .32)

where ci=ai,

(2 .33)

Therefore, ûz11+(-t)iySUZûIiy S A+(k)u~ _-

1

4EkE,

[a1c1A1+azczAz+alczAZ+azczA3-czAâ]~

(2 .34)

with Ai = ~ zYSUZÛiYSY°u~ Az = ûzY s

Y uzûiY 3ui .

(2 .35)

In order to symmetrizs this result, we now consider diagram 4 of fig . 3 with ar and tr interchanged, shown in fig . 4c with notation. We get (4~)z z z ~ s Fn[(4~ k)z~ô[(9-k)z] ~4~AiAzIMô; (Z)I9~i~1z)=~B~SaT1 ~ Tz d k ,. (2Tf) 4W Q -_ k mq _k x ue~( -

9~)iY511 +( -k)un~(-9) uAi(Q~)f1+(t)iY5 un~(4)

with

Dâ,.(z, 4~+ 4~ k)

D~.z =D ~.i

(2 .36) (2 .37)

Now the matrix element belonging to the second nucleon is the same as in the case of the corresponding stretched-box diagram, see eq. (2.2) . For the first nucleon we get ûiA +( t)iY

s

ui=2 [c~ûiY s ui+czûlY S Y° ui+cûlys Yzui ] . E,

(2 .38)

Therefore, s ûziy A+( -k)uzûlel +(t)iy5u1 __

1 [aiciAi+alczAz+azclAi+azczAs - c2 Aâ] ~ 4EkE,

(2 .39)

358

K. Holinde, R . Machleidt / Meson exchange contribution

Consequently, the sum of both diagrams [eqs . (2 .30, 2.36)] is given by ~q~~iAi~M ;:ô(z)+M~ (z)I q~lAz) (2a) xJ

~-t

d3 k IQ'(9~, 4~ k)Fn[(4'-k)z~~[(9-k)z]

(2 .40)

with ~,i z IQ =alcl-c x,, IQz c,3

= alcz+azcl-2c zxz, 2

IP =azcz-c x3 ,

(2 .41)

~.a - z IQ -c .

In order to obtain the total sum for all time-orderings (3-8 of fig. 3) we only have to replace D`;~ by D ;~ in eq. (2 .40) with (2 .42) where D;:â = (z - Eq' - Er - m4-k)(z - Ek - Er -lOq, -kwq-k)(z - Eq - Er -~â-k), D;:ô=(z-Eq'-Ek-Wâ=k)(z-Eq ' =Eq-wQ'-k-WQ-k)(z-Eq-Er-~q-k), D;~ _ (z -Eq'-Er-u~Q-k)(z -Eq'-Eq -~â'-k ~â-k)(z - Eq - Ek - ~â-k) , (2 .43) D7c~s , .r-~a ,. o v =(z-Eq -E q- k)(z Ek - Er - (yq'- k - Wq-k)(z E9 - Ek - Wq-k) ~ D;~e (z = Eq' - Ek - wv=k)(z - Ek - Er k - ~4-k)(z - Eq -Er - w4 -k)

w=

Then we get finally for the partial-wave amplitudes corresponding to the sum of all crossed-box diagrams ~~1if1z~MQ (9~~ 9~z)~~iflz) z +i a (( = -2~r x 2 x ~a8n8Qz~ d cos B dnn , (e)t1~(4~AiAi ; 4~if1z) ' Tz ~ (2~r) r=~ J d lô~(4~, q, k)Fn[(9' - k)z~Q[(4-k)z] x ( 3k . 16EkErwQ-k(Oq-kD,Cro(z~ q', 4. k)

(2 .44)

K Holindt, R . Mackleidt / Meson exchange contribution

359

2 .2 . ~rp EXCHANGE

by

2.2.1. Stretched-box diagrams. The stretched-box diagram in fig . 4a is now given

(4~AiAzIMnv (z)~4~illz) F~L(4~ pk)zlFâL(4~ k)zl =-(~gÂ(3-2?i'zz)J d3k (2vr) 4mQ ~_ k mq _ k Dp (z, 4 , 9, k) w w xûef(-4~)~YsA +(-k)[(gn +fP)Y~ -Zm (( - k) +( - 9) )J ue~(- 4) xû,,c(4~)iYsA+(k)[(8P +fa)Yw +4,.)]un,(4) -2m (kw (4Tr) z z (2~rr)

r s

F,z.L(4~ - k)z~FaL(4 - k)z~ 4~Q-_kmQ _kp,m(z, 4 , 4~ k)

s w x{(go +fa)z~ziYSA +(-k)Y uzûiiy A+(k)Ywul -2m

s w (8o+fa)L~2tY 5 ~ +(-k)uzûiiY A+(k)((- k) +( -4)")Yaul

+ûziy5/1 +(-k)(kw +4w)Y~uzüliY SA+(k)uil +4((-k)"+ (-4)~ )( kw+Qw)üz~YSA +(-k)uzûiiYsA+(k)ui },

(2.45)

with

D~ _ (z -Ea '-Ek - w4 -k)(z -Ev'-Ev -wq_ k -w-k)(z -Eq -Ek -~â-k) (2.46)

Note that the minus sign in front of eq . (2.45) comes from the (-g"") term in the vector meson propagator. (We neglect the p"p"/mP term.) ~, runs from 0 to 3. We first concentrate on the (gP +fP)? term. Using the same procedure and analogous definitions we get for the expectation value belonging to nucleon 2 i L2a(Eg8 wo ûziysA +(-k)Y~uz=2Ek +9~g~1)üzY suz s w +diûzy y uz+dzûzY S YY~uz - cûzY 5 Y2Y~ul

(2.47)

with

d1=m(1-a-b), dz=az=Ek _aE4 _bE4 . .

(2 .48)

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K Holinde, R . Machleidr/ Meson exchange contribution

A similar expression is obtained for nucleon 1 ultysfl +(k)Ywul

_i

- 2Ek

[2a (E48w - 9~8a )ülY s ul

+dlûly S Yw ui+dzûlyS y~Yw ui+cûtyS y z Yw u~]

and we get for the product

(2 .49)

ûz~YS~+( -k)Y`~uzüiiY S A+(k)Ywu~

where

4Ek [d

i A i +2d2A z +d ;A 4 +2d,dZAS +dZA b -c zA;],

(2 .50)

(2 .51) A, (i = 1, 2, 4) are defined in eqs . (2.11), (2.16). In addition, As = z[~zY S Y~uz~iY S Y~Ywui+ üzY S Y~Y~uz~iY S Ywui]~ Ae= ~zY S Y~Y~uzüiY S Y~Ywui +

(2 .52)

Ai = ttzYSYZY~uz~iY5Y2Ywu1 A~ can be replaced by A, = YIAI +2yzAz + y 4.9 4 +2y5AS +y~e -A, ,

(2 .53)

with Yi = 4(mz-2Ea'Ea)b~ Ya = mz(a'+2b'+c') ,

(2 .54)

ye = Eq~a'+2Eq~EQb'+EQc'+ 1 ,

and A~ _ ~zY S Y~Y VuzüiY S YwY~u1 .

This leads ultimately to

(2 .55)

s

ûz~Y S A+(-k)Y w uzüiiy A+(k)Ywul 1

=4Ek

[(di - czYi)A~+2(dZ - c zyz)Az+(di - czYa)Aa

+2(dldz -czys)As+(dz-czYe)Ae+c zA, ] .

(2 .56)

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K Holinde, R. Machleidt/ Meson exchange contribution

For the evaluation of the fo(gP+fP) term in eq. (2.45) we need s ûiiY A+(k)((-k) w +(-9)~)Ywul =2(Ek+Eq )û~~Y s~l+(k)Y ° ui - 2mûQiySA+(k)ul and

(2.57)

üziYSA+( -k)(kw +4w)Y~uz

=2(Ek+Eq )ûziySA+( -k)Y uz - 2mûziy SA+( - k)uz .

(2.58)

This requires us to calculate i s ûiiy A+(k)Y °ui ui+cûty Sy °yzui]~ =2 Ek [eiüiY SUi+ez~iY sY° uitYsA

(2.59)

__i [eiuzYsuz+ezûzysy°uz-cûzySy°yzuz ] . +( -k)y uz-2Ek

with e 1 =Ek+aEq -bEq-,

ez=m(1-a-b)=d 1 .

(2.60)

The other matrix elements were already calculated in the foregoing section . Using the corresponding results and definitions we then obtain s üziYSA+(-k)uzûiiY A+(k)(( -k)" +(-9)~)Y"ul +ûziySrl+( -k)(kw +9w)Y"uz~iiy SA+(k)u, ___1 [e'a'A' +( ezal+e'az)Az+ezazA3+4mc zAâ - 4(Ek+Eq)czAs ], 4Ek (2.61) where ei = 4[(Ek+Ea)ei - mai]~

ei=4[(Ek+EQ)ez - maz]

(2 .62)

and As = z[~2Y SY° Yz uzüiYS Yz ui+ ûzySyzuzûly5 y ° yz u i ] . As can be replaced by with

(2.63)

362

K. Holinde, R. Machleidt / Meson exchange contribution

Thus we ultimately get

üzly s~ +(-k)uzûliy S A+(k)(( - k) w +( -4)~)yP.ui +ûziy ' e1+(-k)(kw +4P.)y"uz~iiy S A+(k)ui ___1 ~ElA'+EZAz+E3A3-4mc zA4+4(Ek +EQ )czAS], 4Ek

(2.66)

where E1= eial+4mc zxl - 4(Ek +EQ)czz i , Ez=eZal+eiaz+8mc zxz -8(Ek +Eq)czzz ,

(2.67)

E3 = eZaz+4mc zx3-4(Ek +Eq)cz z3 . The fo term is given by eq. (2.17), multiplied by

Therefore, eq. (2.45) is finally written as (4~Ai~li~MnP (z)~4~iflz) z (4 ,a) z =+(~8n(3-2z1 ' xJ where

Tz)

7 ~ Ar(4~Ai11i ; 4~lAz) r-i

d3k lnv(9~+9, k)FnL(4~-k)z~n~(4-k)z] 16Ek~Q _ktvq_kDnP (z, 4~, 9. k)

(2.68)

Ib,l

+m =Gi(di - c z Yi)+GzEl+G3Fk(ai -czx~) Ib.z =2G 1(di +ro cz Yz)+GzEz+2G3Fk(alaz - c zxz) Ib,3 G2E3+G3Fk(a2 - C Zx3) r ~P Ib,a ~P = Gi(di - czYa) -4Gzmcz +G 3Fkcz , Inn =2G1(dldz -cz ys)+4Gz(Ek +EQ)c z

Inn = Gi(di - cz Ye) b,7 =G,c 2 IP using Gl ~

(ôP

+fP)z ~

G2 ~ - (fPl2rn)(SP +fP)

(2.69)

K. Hoündc, R . Machleidt / Meson exchange contribution

363

Again, the second stretched-box diagram gives exactly the same result. The corresponding diagram, in which ~ and p are interchanged, is given by (Q~~if1z~MPÂ (Z)~4Ai~1z) =-(~gÂ(3-2z,'TZ)x (2~rr) xû,,~(-9~)[(gP +fP)Y" xûn~(4~)[(gP +fP)Y" (4-rr) z z = - ~ga(3- 2T1 ' (2vr) xl

(BP

-2m

r d3~PL(4~ k)b]FnL(4-k) Z] 4w y~-ktyq-kDP ~(Z~ 4 , 9, k)

-Zm

-2m ~

T2)

((-k)" +(-q~)")]~ +( -k)=YSUn2(-4)

(k" +q"),~+(k)tYsun,(4)

(2 .70)

k)z ](4 - k)z] a b r 4m Q~- kty Q-kDP (z, 9 , 4~ k)

d3kF ;L(4~ p

+fP )z~zY"~ +(-k)iYS UZ~iY"~+(k)iyS Ul

(gP

+fP)Ldz~1 +( -k)~YSUZ~i((-k)" +( -4~)")Y"A+(k)iYsul

s +û z(k" +9w)Y"~ +(-k)iY uzûiA+(k)iYS Ui]

with

+4fm

((-k)" +(-9~)")(k" +q~.)üzA +( -k)iYsuzûiA+(k)iYSU1J

DôR = (z -EQ'-Ek - ~Q'-k)(z -Eq'-Eq -m4'-k - ~q-k)(z - Eq - Ek - w4-k) (2 .71)

We have uzY"A

+( -

k)tY s uz

2Er

L(2g_

" o (Ek-aEQ )+2bg"'q'~ - 2cg" z)ûzY s uz

-di~zYS Y"uz - dz~zYS Y~Y"uz+cûzy syz y"uz] .

Similarly lliY"~+(k)tYsui

i =2Ek

(2 .72)

L(2gw(Ek-aEq)-2bgwq~t+2cgw)ûlysu,

-diûiY S Y"ul - dzuiY S Y~Y"ul - cûiy SyzY"ui]

and we get for the product

(2 .73)

ûzY"~ +(-k)iY S UZ~iY"~+(k)iy S Ul 1 +2dZAz +diA a +2d1dzA5+dZA6-czA~] , 4Ek Ld 1 `A l

2.74)

364

K Holinde, R. Machleidt / Meson exchange contribution

where (2.75) This leads to üzY~~+( -k)IYs uzûlYwn+(k)iy sul ___1 _ Yl)A1+2(dz -czYz)Az+(di -czYa)Aa 4Ek [(di - c z +2(dldz - c z ys)As + (di - cz Ye)A6 + czA,] .

(2.76)

Concerning the fP(gP +fP) term we have to evaluate lll(( -k)" +(-9~)")YwA+(k)tysul =2(Ek +EQ~)û 1 y°A + (k)iY s ul -2mûlA + (k)iY su l

(2 .77)

and ûz(kw +4w)Y~A +(-k)1Ys uz =2(Ek +EQ-) ûzY ° A+( - k)iysuz - 2mûzA +(-k)=Ys uz ~

(2.78)

Furthermore, o s __i s syoyz ul ûlY f1+(k)lY ul-2Ek[h1ulY ul+hzûlY s Y° ul+cûly ].

(2.79) __i [hluzY s uz+hzûzys y°uz - cûzy s y °uz - cüzYS Y°Yzuz] uzY °~ +(-k)tYsuz 2Ek with h1=-m(1-a-b),

hz =Ek-aEQ +bEQ~ .

(2 .80)

Using again some results of the foregoing section we get ûz~1+(- k)iY 5 uzû1(( - k)" + ( -q7")YwA+(k)iysul +ûz(kw +qw)Y"~+( -k)iy s uz~lfl+(k)lysul 1 _- ~[h1b1A1+(hZbl+hibz)Az+h26zA3+4mczAâ-4(Ek+EQ~)czAs ], (2.81) where hz hi = 4[(Ek+EQ-)hl - mbl ], =4[(Ek +EQ-)h z -mbz] .

(2.82)

K. Holinde, R. Machleidt / Meson exchange contribution

365

Replacing again Aâ and AS we arrive at ûzA +(-k)iyS Yzùi(( - k) w + (-4~)~)Y~+(k)iYSUi +ûz(k +9w)Y"~1 +(- k)iy'uz~iA+(k)~Y S Ui _ __1 ~H1`4'+HZA2+H3A3-4mczA4+4(Ek +EQ)czAs], 4EÉ where

H1 = hibi+4mx lcz -4(Ek +EQ.)zic z , Hz = hZbl+hibz +8mxzcz -8(Ek +EQ-)z zcz , H3 = hZbz+4mx 3cz -4(Ek +EQ .)zsc z .

The

fP

(2 .83)

(2.84)

term is given by eq. (2.25), multiplied by

Therefore, eq. (2.70) is finally written as ~q~11 iA2~Mââ (z)Iq~iAz) A(3-2T1'TZ) ~ Aa(q~~iAz+4rliAz) = ~8 (2ar) ,_i

x ~ dsk l âR (9~~ 4~ k)Fv ~(9~ - k)z~Â ~(q - k)z] , 16Ekwq~_kwQ_kD~ (z, 4~~ q, k)

(2.85)

where I~ =Gi(di-czyi)+GzHI+G~k(bi-czxl), I~ =2G1(dz - cz Yz)+GzHz+2G3.Fk(blbz - czxz) I~ =GzH3+G3Fk(bz-czxs)~ I~ =Gi(di-c z Ya)-4G zmcz+G3Fkc z ,

I~ =2G1(dldz-czYs)-4Gz(Ek+EQ')cz . IâR ° Gl(dz - c z Y6) 6.7

IP 

=

Gl c 2 .

Both stretched-box diagrams give twice the expression (2.85).

(2.86)

366

K Holinde, R. Machleidr/ Meson cxchange contribution

2.2.2. Crossed-box diagrams. The crossed-box diagram in fig. 4b is now given by {4~~iAz~M;~ (z)~9~if1z) (4~)z

z

~ s

F; L(4~-k)z~oL(q-k)z] P

xû,,f(-q')[(BP+ÎP)Y~-Zm ((

-r)P.

C.

+(-9')")~A +( -t)IYSU~~(-q)

xûnl(4~)IY S A+(k) (8P +fP)Yw (kw +9P.),un,(q) L -2m -

z z (4 ,a) z ~ s F~ L(9~ - k) Z J~' n L(g - k) ] 68n(3+2T1 " Tz) d k (2~r) 4coq-k~9_kDAO (z, 9~, q, k)

x (BP +JP) z llzY~~ +( -t)iYS UZÛiiy s A+(k)Ywu~ l -2m

(BP +fP)L~zY" (kw +4w)f1 +( - t)lY S UZ~iiy s fl+(k)u~

+ûzA+( -t)iy 5 uzûliy S A+(k)Yw (( -~)" +(-9~)")u~]

+4m (kw +Rw)((-t) " +(-R~)~)ûzA+( -t)iy S UZÛIUy S r1+(k)ul} ~

(2 .87)

where, as before, t = -k + q + q' and

Dar =

9

(Z -

P

Eq' - Ek -~ -k)(Z -Ek - Er - w4 -k - lDy_k)(Z -Ek -Eq - ~q -k )

We start again with the is given by

(gP

(2 .s8)

z +fP) term . The matrix element for the second nucleon

~zY~~ +( - t)ly S Uz i

wo

{L28 =2 E,

(E~ - (1

-

a)Eq )+2g"t (1

_b)q~i +28 P.z

+dlûzy s y"uz+czûzy sy° y w uz-cûzy5 yz y~u2} ,

c]üzY suz (2 .89)

dl and cz have been defined in eqs. (2 .33), (2 .48) . The matrix element for the first nucleon is given by eq . (2 .49) . Therefore, üzY~A +( -t)tYS UZÛliY S ~l+(k)Ywui _-

1

4EkE~

[P1AI+PZAZ+PZAZ-diA4+d,dzAs+dlczAs+dzczA6-czA ; ], (2 .90)

K. Holinde, R. Machleidt/ Meson txchange contribution

367

with Pl =2[di -dzcz +2a(1-b)(EQ-Eq +q' ~ q)-cz], P2 =-2[d i c z -2d 1 (1-b)EQ.+m(1-b)d z],

(2.91)

PZ = 2[2aEgd1 - amcz] , and As = üzY3Y~uzüiYSY°Ywui . As = ~zYsY°Y~uzüiYsYwul~

(2.92)

The other quantities have been defined before . Symmetrizing this result again by adding diagram 4 of fig . 3 with ~ and p interchanged (see fig . 4c for notation) we get for the sum ~zY~A+(- t)tYs uzttliyS A+(k)Ywu~ + ûziySA +(-k)Y~uz~iYwA +( +t)iysul _-

2 [P1AI+PZAz+diAa+dl(dz+cz)AS+dzczA6-czA~] 4EkE

2 = 4EkEr [(Pi - czYi)Ai+(Pz - 2czyz)Az+(di -c zYa)Aa +(di(dz + cz) - 2cz Ys)As + (dzcz - cz Ye)~1s+ c zA,]

(2.93)

with Pz = P2 + PZ .

(2.94)

For the evaluation of the fP (gv +fv) term in eq. (2.87) we need the following matrix elements : üzY~(kw +qw)A+(-t)tYsuz _ i [rluzYsuz+rzüzYsY°uz+2mcûzysyzuz+r3cûz+r3cûzysy°yzuz] 2E, with rl =cicz-2(1-b)EQ-c2+2Egcz-2md1 +4(1-b)q' ~ q, r3 = -c2+2Eq , where

(2.95)

368

K Holinde, R. Machkidt / Meson exchange contribution

Correspondingly, s ûiiY A+(k)Yw (( - t) w +( -Q~)")ui

with

i [s1û~Y5ui+szlliYSY°u~-2mcûlySyzul+s3cûlysy°yzu~ ], =2Ek

(2 .98)

sl =cZaz+2aEgc2 -2E4-a z +2md1 +4aq' ~ q, ss = -ci -2EQ

where

c2 =E,-Ek+Eq+EQ~ .

(2.100)

Using the results of the foregoing section we then obtain ~zY~(kw +4w)ß+( - t)tY S UZ~iiY S A+(k)ui + itz11+(-t)iY S UZÛiiY S11+(k)Yw (( - t)~ +(_4~)")u, _

1 +clsl)A i + (az~i +clsl)Az + (airz+czsi)Az 4EkE~ [(ai~i +(azrz+czsz)A 3 +4mczAâ-s3czAs1 +r3c z As] .

(2 .101)

Adding again diagram 4 of fig . 3 with a and p interchanged we obtain ~zY~(kw +R~)~+( -t)iY S UZüiiy S A+(k)ui +ûzA+( - t)iY S UZll~iy SA+(k)YK(( -t)" +(-q~)")ui +ûziy SA+( -k)(tv. +Rw)Y~uzûill+(t)iY S UI +ûziys A+(-k)uzüi(( - k)w + (-9)")YwA+(t)iySUi __

2 ~[QiA1+QzAz+Q3A3-4mc zA4 - 4(EQ-+EQ)c zAs], 4EkE

(2.102)

where Qi =al rl+c lsl+4mc z x1+4(Eq .+EQ)czzl , Qz = azrl+clsz+alrz+czsl +8mczxz +8(EQ, + EQ)c zzz ,

(2.103)

Qs = azrz+czsz+4mc zx3+4(EQ-+EQ)c zz3 . Here, Aâ and AS have again been replaced by using eqs . (2.14), (2.64). The fo term of eq. (2 .87) is given by eq. (2.34) multiplied by

K. Holinde, R. Mackleidt / Meson ezckange contribution

369

If we add diagram 4 of fig. 3 with ~ and p interchanged, the corresponding term becomes ~ 4FrLûzrl+( -t)ly suz~ii?'S~1+( k)ui+ûziyS A+(-k)uzüiA+(r)iysui~ __ _

2

4EkE~

L(aici-c zxl)A1 +(alcz+a zcl -2czxz)Az

z +(azcz-CZX3)A3+czA,~4m Fe~

(2.104)

Thus we get ultimately for the sum of both diagrams ~4~~iAz~M~iâ (Z)+1Kââ (Z)~4~if1z) z 2 (2~r)~8~(3+2T1 ~ zz) ~ A~(4~AiAi ; 4f1~Az) = x t

where

x d3 k lv~(9~~9,k)F~L(4~-k) ZIFPL(4 - k)z~ J

(2.105)

i Iv =Gi(Pi - czYi)+GzQi+G3Fr(alcl - c zxi)~

IPZ = G,(Pz-2czyz ) +GzQz+G3F',(alcz+azci-2czxz) IPa

= Gi(di - cz Ya) - 4Gzmcz+G3F'~cz ,

s Ip =G~(di(dz+cz)-2c z ys) - 4Gz(EQ ~+Eq)c z ,

6

Ip I c.~

(2.106)

=Gi(dzcz - c z Ye)

p = G1 cz . The total sum of all time-orderings (3-8 of fig. 3) is obtained by replacing D;;P in eq. (2.105) by D;p defined by (2.107) are given in eq . (2.43) if v is replaced by p. Consequently we ultimately get for the partial-wave amplitudes corresponding to the sum of all crossed-box diagrams D;~

~Ai~z~MP (4~~4~Z)~~i~1z) 4~r z =2-rrx2x~8Â(3+2z1~TZ) ~ dcos9din~(B)Aa(4~Aie1i ;4~if1z) a_i J x j d3k lp i (Q~.~h k)FÂf(4'-k)ZIFPL(9-k)Zl 16EkE~mq-kWq-kD .~np(Z, q', 9~ k)

(2.108)

370

K Holinde, R. Machleidt / Meson exchange contribution

2.3 . ~rrw EXCHANGE

The corresponding results for vr~ exchange are simply obtained by replacing in eqs. (2.68), (2.85), (2.108) the index p by ~ and (3 t 2T, ~ Tz) by Ti ~ Tz. In practice, we will choose fd to be zero so that the formulas simplify considerably . 3. Results aad discussion The values for the meson-nucleon coupling constants gâ, meson masses mQ and cutoff masses AQ [eq. (2.4)], used throughout this work, are given in table 1 . They lie in the range of those values used in one-boson-exchange models z.a.9) of nuclear forces . Furthermore, the value for A is consistent with model calculations of the ~rNN form factor, see ref. 4). TABLE 1 Meson-nucleon coupling constants gâ, meson masses ma (in Mew and cutoff masses Aa (in Meld used for the vertices of the diagrams in fig. 2

a rr

p oa

ga 14 .4

0.5 (6) 6 23

ma

Aa

138 712 500 782 .8

1000 1300 1300 1300

The number in brackets denotes the tensor-to-vector cqupling ratio f l go~

Fig. 5 shows the matrix elements M(q', q ~ qo) (z = 2~) for the sum of stretched-box and crossed-box diagrams in the 1So partial wave and as function of q', setting q = qo = 250 MeV. In addition to the contributions relevant in this paper (-rro, ~rp, orw) we show also the corresponding -rra contribution, see e.g. ref. 6), for comparison . There is a remarkable cancellation between air and ~rp, and also. between ~ and vrw, leading to a total result which is small compared to the separate contributions. The same is true for the 3S1 state, as can be deduced from figs. 6, 7 . Fspecially in the non-diagonal part (fig. 7), which characterizes the strength of the tensor force, the cancellation is very pronounced . (The vr~r contribution in the diagonal part (fig. 6) is exceptionally small, due to strong cancellation between stretched-box and crossed-box contributions.) This special interplay between the separate contributions is not limited to the S-states shown here, but is present in all partial waves. In fact, this is not surprising since the underlying physical mechanism is the well-known strong cancellation between single ~r and p, or v and w exchange contributions to the NN interaction .

K. Holinde, R. Mackleidt/ Meson ezchang~ contribution

371

1S 0

N

W

\~ Q

lD Ô

500

. q

1000

MeY

Fig. 5 . Matrix elements M(q', q ~ qu) for the sum of stretched-box and crossed-box diagrams in the 1 So partial wave and as function of q', setting q = qu = 250 MeV. The dashed lines show the ar~r ("+r") and ~rrp ("p") wntributions, whereas the dash-dot lines refer to the ~ ("o") and arm ("~") contributions. The solid line gives the sum of all four wntübutions . In case of rrrr and ~rrp, the dotted lines originate from the use of A 4 =1 .2 GeV and Ao =1 .5 GeV.

2

0 g i

_1

_2

0

500

4

1000

MeV

Fig. 6. Matrix elements M(q', q ~ qu) for the sum of stretched-box and crossed-box diagrams in the diagonal part of the 3 S 1 state and as function of q', setting q=qu=250 MeV. The dashed lines show the ~rrr ("a") and trp ("p") contributions, whereas the dash-dot lines refer to the ~rrQ ("Q") and +rw ("d") contributions . The solid line gives the sum of all four contributions.

372

K. Holinde, R. Mackleidt/ Meson exchange contribution 0.4

N W

35- ~i

-0.4

500

q

1000

MeV

Fig . 7 . The same as fig . 6, but for the 3 51 -3 D1 transition amplitude .

Thus it is of outstanding importance to include all four contributions at the same time. For example, the comparatively strong effect of non-iterative vrw exchange, as pointed out in ref. s), is strongly reduced by the analogous ~v contribution . Furthermore, corresponding -rr~r and -rrp contributions should be grouped together despite belonging to different pieces if a classification scheme according to the number of exchanged pions is used (~rtr belongs to the 2~r exchange, whereas vrp belongs to the 3a exchange). This was pointed out already a long time ago in the lo) . context of transition potentials leading to nucleon-isobar intermediate states In other words, there are good physical reasons to demand that potentials containing 2~r exchange contributions should also contain the corresponding ap contributions. As expected, the results depend on the choice for r1Q : For example, the dotted lines in fig. 5 refer to the ar~r and arp contributions using ~ln = 1.2 GeV and AP =1.5 GeV, i.e. both cutoff masses are increased by 200 MeV . Especially the ap contribution is considerably increased, and so is the total result. Fortunately, this has no drastic consequences because the total sum is small in any case. The effect on important NN scattering phase shifts is demonstrated in figs . 8-17, adding the contributions separately to a one-boson-exchange potential 9) and calculating the phase shifts by solving the scattering equation . Clearly, the above mentioned cancellation between the separate contributions, demonstrated before for the potential matrix elements, persists for the case of NN scattering phase shifts . Note that the effect on 3S1 is surprisingly small, due to an interplay between the diagonal (fig. 6) and non-diagonal part (fig. 7). There is a sizeable net increase of tal, especially for higher energies (see fig. 13), which can be mainly traced back to the fact that here the effect of the a-rr and ~rp contribution go into the same direction, i.e. both increase the mixing parameter.

K Holinde, R. Machleidt / Meson exchange contribution

373

S 1 .0 rad 0.5

0

-0.5 0

100

200

MeV

Elab

Fig. 8. Nucleon-nucleon nuclear bar phase shifts (in radians) as a function of the nucleon lab energy (in Mew, in the 'So partial wave. The error bars are taken from the energy-independent Livermore analysis 11). The solid line denotes the result obtained from a one-boson-exchange potential 9) ( Vose). The dashed lines are obtained by adding the (stretched-box plus crossed-box) aror ("~") and ~rp ("p") contributions to VoaE. The dash~ot lines denote the results when the oorreaponding zrv ("Q") and ~nw ("ar") non-iterative diagrams are added to VaHE. The dotted line refers to the result when all four (~mr, gyp, ~, arm) non-iterative contributions are added to Vage .

Fig. 9. The same as in fig. 8, but for 3Po.

Fig. 10. The same as in fig. 8, but for'Pl.

374

K. Holindt, R . Machkidt / Meson exchange contribution 0 d -0.1 rad

1 .5

-0.2

rad 1 .0

-0 .3 0 .5 -0 .4

0

100

E lâb

200

MeV

Fig. 11 . The same sa in fig. 8, but for 3 P1 .

0

100

200

E lab

MeV

Fig. 12 . The same as in fig. 8, but for'S l .

On the first glance, this is quite surprising since e 1 should mainly depend on the (S-D) transition matrix element, see fig. 7, where, as expected, the ~~ and ~rp contributions have opposite behaviour. In fact, fig. 7 suggests that e, should decrease when including the or~r contribution, opposite to fig. 13 . However, e 1 depends not

,, 100

200

MeV .

E lab Fig. 13 . The same as in fig. 8, but for el .

0

100

E lab

200

MeV

Fig. 14. The same as in fig. 8, but for 3 D 1 .

K Holinde, R. Mackleidt/ Meson exchange contribution

375

b 0.5 rad 0.4

0.3

0.2

100

200 MeV Elab Fig. 15 . The same as in fig. 8, but for'DZ.

00

100

MeV 200 E lab Fig. 16 . The aeme as in fig. 8, but for 3D2.

100

200 M eV E lab Fig. 17. The same as in fig. S, but for 3PZ. only on the non-diagonal amplitude R(S-D), but also on the diagonal amplitudes R(S-S), R(D-D) :

The arTr contribution strongly decreases R(S-S)-R(D-D), as can be deduced from the corresponding phase shifts, see figs. 12, 14 . This effect results in a net increase in el .

376

K Holinde, R . Machkidt / Meson exchange contribution

This result leads us to the following speculation: In OBE models, the cutoff mass A has to be chosen larger than =1300 MeV in order to obtain a reasonable description of el. This value is larger than A =1 GeV suggested from model calculations °). [Note that A =1 GeV in the non-iterative diagrams, whereas A _ 2.5 GeV in Voss 9) .J The use of A~ =1 GeV in VoHS would make et drastically too small, see fig. 6 of ref.b). Consequently, the inclusion of the non-iterative diagrams discussed here (which increase e l) might perhaps make it possible to use a more realistic value for A in Voss, too. 4. Sammary In this paper, we have evaluated the non-iterative as meson-exchange diagrams, ~ = v, p, m and compared them with the corresponding contributions arising from ~r~r exchange . It turns out that the separate contributions are appreciable; there is, however, a strong cancellation between ~rr-rr and ap, and ~ and ~ exchange diagrams, making the net result relatively small. Therefore, it is absolutely essential to consider all contributions together. Although the total result is small, it is non-negligible . In fact, the inclusion of such non-iterative diagrams might make it possible to use a sufficiently low value for the cutoff mass A in the OBE vertices, which is consistent with theoretical model calculations. References 1) R.V. Reid, Ann. of Phys . 50 (1968) 411 K. Holinde and R. Machleidt, Nucl . Phys . A256 (1976) 479, 497 H. Arenhbvel and W. Fabian, Nucl . Phys . A282 (1977) 397 J.W. Durso, A.D. Jackson and B.J. Verwest, Nucl. Phys. A282 (1977) 404 D.O . Riska, Nucl. Phys. A274 (1976) 349 K. Holinde, Phys. Reports 68 (1981) 121 K. Holinde and H. Mandelios, Nucl . Phys . A364 (1Q81) 365 8) K. Erkelenz, Phys. Reports 13C (1974) 191 9) K. Kotthoff, K. Holinde, R. Machleidt end D. Schütte, Nucl . Phys . A242 (1975) 429 10) A.M. Green, Rep. Progr. Phys. 39 (1976) 1109 11) M. MacGregor, R. Arndt and R. Wright, Phys. Rev. 182 (1969) 1714 2) 3) 4) 5) 6) 7)