ΛΛ hypernuclei and the ΛΛ interaction

Nuclear Physics A463 (1987) 221 c - 230c 'North-Holland, Amsterdam

221 c

AA HYPERNUCLEI AND THE AA INTERACTION A. Ro BODMER

Department of Physics, University of I l l i n o i s at Chicago USA, and Argonne National Laboratory, Argonne, IL 60439-4843 USA and Q. N. USMANIt

Department of Physics, University of I l l i n o i s at Urbana-Champaign, Urbana, IL, 61801 USA, and Department of Physics, Aligarh Muslim University, Aligarh, India 6 9 I0 Variational calculations of s - c l u s t e r models for ..He, .Be, A.Be have been made. These calculations require a knowledge o@nthe1~A potential which is obtained in s~veral ways including the use of 5-body Monte Carlo (MC) calculations of A~He. We discus~nthe AA interaction strengths and the relation between ~he A~He and ~Be binding energies and, in p a r t i c u l a r , the dependence of tK~eAon tK~' eA p o t e n t i a ~ For a l l our ~A potentials the binding energy of A~He predicted from '~aBe is 1 MeV or more below the experimental v a l u e . - ~ b r i e f discussion XE given of the implication of the phenomenological strength of AA interaction we obtain and also of the implication of AA hypernuclei for the H-dibaryon. I . ~AHe AND lOB^ AA e The binding energies of AA hypernuclei are conveniently given in terms of ABAA : BAA - 2BA, where BA and BAA are respectively the separation energies of a single A and of two AS from the core nucleus. ABAA is more d i r e c t l y related to the interaction of the two AS than is BAA. There are two reported events: ~AHe with AB~A= 4.68 ± .6 MeV1 and ~Be with A B ~ = 4.29 ~ .I MeV2. We have used BA(~He) ~ B~= 3.12 ± .02 MeV and B(~Be)n a ~ B~A= 6.71 +_ .04 MeV. The lOAABe event has been thoroughly checked and is well-established. For our analysis we thus need to study the A hypernuclei ~He and ~Be as well as ~AHe and I0 AABe, We have made variational calculations assuming appropriate cluster models: ~He ~ a + A,

AHe z a + 2A,

Be ~ 2a + A, AABe ~ 2a + 2A.

For

He w~ shall

Work supported in part by the U. S. Department of Energy, Nuclear Physics D i v i s i o n , under contract W-31-109-ENG-38, and by grants NSF-PHY-84-15064 and NSF-INT-8319827. tAddress a f t e r September 1, 1986: Department of Physics, damia M i l l i a , New Delhi, India. 0 3 7 5 - 9 4 7 4 / 8 7 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

A.R. Bodmer, Q.N. Usmani / A A Hypernuclei and the A A interaction

222c

also use the results of 5-body Monte Carlo (MC) variational calculations3'4. The t r i a l wave functions are products of 2-body correlation functions.

Thus

~(~AHe) = faA(raA~) faA(raA2) fAA(rAA); for ~Be the a and A indices are interchanged; ~(AABe ) L U = faa(r a) fAA(rAA) ~ faA(raiAi) " Following the methods developed by the Urbana group s , the f are obtained from Schrmdinger-type equations which involve variational parameters through suitably chosen potentials.

For central 2-body forces such t r i a l

functions are known to give

excellent results. Details may be found in Ref. 6. Our approach, through our choice of the potentials between the constituents gives the known properties of the component subsystems, e.g. ~He, ~Be and 8Be for i0 AABe, and our choice of wave functions then accurately describes the corresponding correlations. Our aa potential Vaa f i t s the a-a scattering data and also has a shape which is t h e o r e t i c a l l y reasonable 7, and our aA potentials VaA discussed below a l l I0 are then determined by a single give B~= 3.12 MeV. The energies AB~A and ABAA AA interaction strength VAA for an assumed shape of the (1S0) AA potential VAA(r ).

In Ref. 6, several d i f f e r e n t shapes were considered.

Here we give

results for only a 2x Urbana-type potential8: VAA: Vc - VAA T2(r)~ ,

(1)

where Vc is a fixed repulsive (Woods-Saxon) core close to that of the Is O NN potential, and T is a tensor form factor with cutoff 6. mA potentials.

VmA plays a central role for all four a-cluster hypernuclei.

We consider five different Va .

For the f i r s t we use a Yukawa AN potential

VAN e- ur/~r with ~ = 2.788 fm-~ corresponding to a (~ meson) mass of 550 MeV. This is then folded into the a - p a r t i c l e density Pa to obtain a VaA denoted by YUK. The strength VAN : 366.5 MeV is such that VaA gives B~= 3.12 MeV. Such potentials were extensively used in the older calculations. Our other VaA are based on a AN potential of the form of Eq. ( I ) , but with In our approach an a t t r a c t i v e , spin average, strength VAN (= #1 vAN s + # v~N) to the binding energies of hypernuclei 3'4 we also include a dispersive type ANN potential VANN = W T2(rlA) T2(r2A) .

(2)

This is repulsive for W> 0 and is required to avoid overbinding of BA for A 5, in particular for ~He and the well depth, i . e . to achieve consistency between the BA for A ~ 5 and Ap scattering. Corresponding effective interactions are then obtained by using AN correlation functions fAN determined by nuclear matter calculations3'9:

A.R. Bodmer, Q.N. Usmani / AA Hypernuclei and the AA interaction

2

223c

}i2 MN + MA

~'AN = fAN [VAN - T- (M - - ~ A - ) v21n fAN] '

(3)

AN, = W(f2ANT2.)(f2AN ) "

(4)

which is the sum These are then folded into p~ to obtain V A = V(AN) ~A + V(^NN) ~A of AN and ANN contributions.

Such first-order use of ~AN corresponds to the

lowest order in the cluster expansion and neglects higher-order correlation contributions.

I t is the variational analog of using the Brueckner G-matrix as

an effective interaction in a Hartree-Fock calculation.

Since for ~He AN

exchange effects are negligible, our VaA obtained from ~AN' ~ANN thus correspond to Hartree potentials.

We consider two cases: EFF2obtained with only a

VAN potential (V AN = 6.006 MeV, W= O) and EFF2+3 obtained with both VAN and VANN (V AN = 6.217 MeV, W= 0.02 MeV). Both V A give B~ = 3.12 MeV. VAN for EFF2 and for MC2 (see below) gives much too small Ap cross sections (overbinding problem), whereas VAN for EFF2+3 and MC2+3 gives only s l i g h t l y too large cross sections (VANN is somewhat too repulsive).

EFF2+3and MC2+3 correspond

to AN and ANN forces which are reasonably representative of interactions with a strongly repulsive ANN force and with a VAN consistent with Ap scattering, and which are also consistent with both B~ and with the well depth D. Finally, from our 5-body variational MC calculations (which include both 2body and ANN correlations) we have obtained the " t r i v i a l l y " equivalent local V A: The MC results for the A density PA (relative to the ~) give the relative aA wave function p~/2 from which V . is obtained by solving the Schrodinger equation for VaA = [- BA +

/2)/dr2]/p /2.

These MC aA poten-

t i a l s include many-body effects not included in the Hartree-type effective interaction potentials, since they are obtained from complete correlated wave functions without any truncation.

For given VAN, VANN the MC calculations are

then expected to give the best V A, appropriate for a r e a l i s t i c correlated wave function for ~He.

In particular, these V A are then also expected to provide

the preferred V A for use in the ~-cluster model calculations. Again we have considered two cases, close to the corresponding effective interaction potentials:

MC2 (VAN = 6.0 MeV, W= O) and MC2+3 (V AN = 6.23 MeV,

W= .02 MeV) (These potentials have been s l i g h t l y adjusted by a multiplicative factor : l.Ol to give exactly B~ = 3.12 MeV.) The ~A potentials are shown in Figs. l and 2.

The potential YUK is of

r e l a t i v e l y short range and increases monotonically outwards from r = O.

The

potentials EFF2, EFF2+3 have a central hump. Thus, even EFF2 (only VAN) has a slight hump and is in any case much shallower and of longer range than YUK, although the range of T2 is not very different from that of the Yukawa

A.R. Bodmer, Q.N. Usmani / AA Hypernuclei and the AA interaction

224c

0

ANL-P-18,39S

ANL-P-18,398

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r , fm FIGURE 1 mA potentials obtained with a AN potential only

potential.

i

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...//;~"---Y U K ,

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4

FIGURE 2 aA potentials obtained with AN + ANN potentials

This reflects the effect of ~ . for a repulsive core potential.

For EFF2+3, the large central hump is dueA{omVNN)~ which is of quite short range (because of the double folding of T~). The MC potentials have an even larger central hump than the corresponding EFF potentials.

This hump is quite

pronounced even for MC2 (only VAN). The difference between the MC and EFF potentials thus reflect the very significant higher-order effects of correlations.

These are mostly due to the AN forces; thus the difference between MC2

and EFF2 is about the same as between MC2+3 and EFF2+3. The analog of our MC and EFF potentials were already previously obtained from G-matrix calculations 10 and show q u a l i t a t i v e l y similar behavior. There are significant differences in observable consequences associated with different VaA which all give the same B~; in particular the MC potentials can lead to significantly different results than the corresponding EFF potentials. An example is the pionic decay of hypernuclei 11

Herewe discuss the effect of

different VaA on the m-cluster hypernuclei. The results for the different VaA are shown in the Table.

The parameter

which is most significant in ordering the results is rmin, the distance where VaA is a minimum. The values of rmi n show a trend consistent with Figs. l and 2 for ~He. This trend is also shown by 1/2, the rms value of @~A

2.

AB~A for a given strength (V AA = 6.24 MeV; c a l c u l a t i o n s were also made f o r 6.1 and 6.5 MeV) decreases

to a good approximation l i n e a r l y ,

with 1/2

This is because the two As have on the average a l a r g e r separation i/2 , - . and hence have a reduced i n t e r a c t i o n energy <- VAA>, as l/2 increases.

"

A.R. Bodmer, Q.N. Usmani/AAHypernucleiandthe AA interaction

225c

TABLE. For explanation, see text. (Lengths in fm, energies in MeV.)

rmi n

YUK

EFF2

MC2

EFF2+3

MC2+3

0

0.76

1.41

1.27

1.60

1/2

2.66

3.00

3.13

3.07

3.25

AB~A(6.24)

4.65

3.51

3.07

3.27

2.71

1/2(6.24)

2.8

3.18

3.28

3.21

3.38

<-VAA>(6.24)

8.15

6.10

5.45

5.74

4.93

B~

5.93

7.51

7.87

6.56

7.26

0

0

0

1.16

0.82

6.29

5.52

4.49

3.99

3.34

0

0

0

3.42

2.06

vAA(~AHe)

6.25

6.46

6.57

6.52

6.69

vAA(~Be)

6.0

6.06

6.20

6.29

6.43

-aAA(~Be)

1.6

1.8

2.45

3.06

4.5

6 10 ABAA(AABe)

3.06

2.65

2.90

3.53

3.50

9 AB~(6.24) 10(6.24)

Conversely, to obtain the (given) experimental AB~A requires an interaction strength V^A which increases with 1/2 i . e . in going from YUK to MC2+3. For ~Be and 10 AABe a new feature is the presence of a repulsive mmA potential V~mAdUe to the interaction of a A, via ~ . . . . with a pair of nucleons each in a different ~ (and not included in v~NN)).AN~ A is obtained by appropriate folding of VANN and is proportional to W. The same folded V A (for W= .02 MeV) was used with both EFF2+3 and MC2+3. The contribution is quite significant, : 1MeV for ~Be. Without t h i s , i . e . with only VaA, B~ would be : 8 MeV for both EFF2+3 and MC2+3. We again emphasize that all our V A reproduce B~ = 3.12 MeV. As already reported earlier 3'4, i t is which brings the calculated values close to the experimental value of 6.7 MeV. For those V A obtained with only AN potentials, YUK gives too l i t t l e binding whereas both EFF2 and MC2 give too much; the l a t t e r which includes many-body effects gives a significantly larger B~ than EFF2. The contribution of a weakened p-state AN potential Vp = Xp Vs with Xp : 0.5

226c

A.R. Bodmer, Q.N. Usmani / A A Hypernuclei and the AA interaction

was estimated both with shell model wave functions using an equivalent spaceexchange force 3'12 and also from calculations with the aaA model 13.

(Xp = 1 -

2E where ~ is the c o e f f i c i e n t of the space-exchange part of VAN. The r a t i o of the forward/backward cross sections gives ~ : .25, i . e . Xp : .5.) c a l c u l a t i o n s give the same reduction of 0.4 MeV for Xp = 0.5.

Both

This value is in

good agreement with that of Ref. 14 where a microscopic c l u s t e r approach is used.

Allowing for uncertainties in Xp, would bring the calculated value of B~

for MC2+3 into excellent agreement with the experimental value.

Of our f i v e

VaA, MC2+3 represents the most r e a l i s t i c VaA (using MC calculations for ~He) for those AN, ANN p o t e n t i a l s most consistent with Ap s c a t t e r i n g , with ~He, and also with the phenomenological well depth D3'4.

Thus for VAN with a reasonable

repulsive core (also needed for Ap scattering) our results strongly indicate the need for a repulsive aaA c o n t r i b u t i o n and hence of dispersive type ANN forces. For 10ReAAv, is somewhat more than twice that for ~Be, because the extra binding due to the second A reduces the size of the system.

Thus 1/2 =

3.85 fm for ~Be and = 3.5 fm for 10AABe. To be noted is that even the l a t t e r value is larger than twice the arms radius of 1.47 fm, thus supporting the internal consistency of our a-cluster models. The calculated values of ABI0 are obtained using the appropriate calculated values of B~. The trend of AB!~ ( f o r VAA= 6.24 MeV) with I/2 is s i m i l a r to that for ABe^ although the decrease is not as uniform. The strengths vAA(~Be) (errors : + .02 MeV) are those required by the experimental -BI0 a AA" The values AB~A(~Be) (errors ~ ± .I MeV) are then the corresponding predicted values of AB~A. AB~A(~Be) for a l l our VaA are s i g n i f i c a n t l y less than the experimental value. This "underbinding" was alreadying previously obtained for an EFF2+3 type of VaA6.

The values for EFF2+3 and MC2+3 are seen to be

e s s e n t i a l l y the same, : 3.5 MeV, whereas with only AN forces (YUK, EFF2, MC2) the values are even smaller: ~ 3 MeV. The r e l a t i o n between AB~A and AB~ for a fixed VaA, and hence the predicted underbinding of ~AHe, is almost independent of the shape of VAA. The larger values of VAA obtained from ~AHe than from 10Re of course j u s t r e f l e c t t h i s underbinding. AU i0 should reduce many of the Use of the calculated value of B to obtain ABAA t h e o r e t i c a l uncertainties since these are expected to be s i m i l a r (per A) for ~Be and 10 AABe- However, such cancellations w i l l not be complete. Thus the reduction due to a weakened p-state VAN is expected to be s l i g h t l y larger per A for 10AABethan for ~Be because of the smaller size of the former.

Assuming the

AN exchange force correction to scale approximately with < r ~ > - I , as suggested by shell model estimates 12, gives approximately .1 MeV more reduction per A for i0 by ~ .2 MeV below our 10RveAA than for Be ( f o r Xp = .5). This would reduce ABAA

A.R. Bodmer, Q.N. Usrnani / A A Hypernuclei and the A A interaction

c a l c u l a t e d values.

This w i l l ,

t h e r e f o r e not s i g n i f i c a n t l y

227c

6 10 in t u r n , increase ABAA(AABe) by = .15 MeV and

change our conclusions about the underbinding of

AB~A. (The corresponding increases in vAA(~Be) are : .035 MeV).

Wang Xi-

chang et a l . 14 using a very d i f f e r e n t microscopic s - c l u s t e r approach also obtain underbinding by about the same amount as we do. J

In view of the l a r g e quoted errors of AB~A and the possible doubtfulness of

the^~He~.., event, the discrepancy with the values of AB~A(~Be ) .. predicted from 10 AABe should not be considered as too significant. Perhaps this discrepancy should rather be taken as reinforcing the importance of obtaining new data on ~AHe (and other AA h y p e r n u c l e i ! ) .

Our analysis shows that such new data would

be of importance not only f o r the A^ i n t e r a c t i o n but also f o r the AN, ANN forces through the mA p o t e n t i a l . 2. THE AA INTERACTION With our 2~ p o t e n t i a l , Eq. ( l ) , we obtain a strength VAA : 6.5 ± .02 MeV 10 from AABe with MC2+3 which is our favored aA p o t e n t i a l . This is a q u i t e l a r g e v a l u e , almost as large as the comparable NN strength (see below), and is considerably l a r g e r than = 6.25 MeV obtained e a r l i e r with an EFF2+3 type VaA potential. negative.

The corresponding s c a t t e r i n g length aA^ = -5 fm is large and (A bound AA state is obtained for VmA = 6.75 MeV when aAA . . . .

)

Smaller strengths ~ 6.2 MeV would be obtained with VmA obtained with only a AN force (YUK, MC2, EFF2).

Of course the values obtained from the quoted e x p e r i -

mental AB~A are even l a r g e r than those obtained from i0 AABe. The r e s u l t s of Ref. 6 for d i f f e r e n t VAA shapes i n d i c a t e t h a t for VAA with a r e p u l s i v e core the i n t r i n s i c

range b was the s i g n i f i c a n t parameter determining

-aAA, and t h a t t h i s increases with b.

Our 2~ p o t e n t i a l , Eq. ( I ) ,

gives b : 2.0

fm.

Thus, a VA^ f o r which b = 2.44 fm (a meson Yukawa with Wc) gives aAA : - 6

fm.

Larger r e p u l s i v e cores Vc give l a r g e r b and hence l a r g e r -aAA.

Thus with

MC2+3 the AA p o t e n t i a l could give a bound AA state i f the r e p u l s i v e core is increased only s l i g h t l y . I t is i n t e r e s t i n g to make some naive comparisons with the IS 0 AN and NN potentials. particular,

To make t h i s comparison s i g n i f i c a n t we must use the same shape, in the same r e p u l s i v e core Vc f o r a l l three p o t e n t i a l s .

We use the

shape of Eq. ( I ) with Vc the same as for VAA and VAN. Note t h a t VNN then represents the e f f e c t i v e strength of VNN - V , i . e .

a f t e r the OPE p o t e n t i a l has

been subtracted (aNN : -17.6 fm for VNN and -3.5 fm for VNN - V ). 6.60, VAN = 6.35 (Ref. 4), and VAA : 6.5 MeV.

Then VNN =

I t is q u i t e remarkable t h a t

these strengths are so close to each o t h e r , i n d i c a t i n g t h a t excluding OPE the three IS 0 i n t e r a c t i o n s are very s i m i l a r .

This is perhaps a p r i o r i

quite

suprising on the basis of meson-exchange models, although by no means incon-

A.R. Bodmer, Q.N. Usmani / AA Hypernuclei and the A A interaction

228c

sistent with these even when these are strongly contrained by available data as the Nijmegen potentials15 are.

We note that factorization for the attractive

part of the potentials, i . e . the assumption VAN = (VNN vAA)I / 2 , gives VAA = 6.0 MeV with the above values of VNN, VAA. Of course, factorization is expected to be poor for interactions at low energies. The phenomenologicalAA interaction is in fact consistent with existing meson-exchange models. Thus, Bando and collaborators 16 have investigated the consequences of the D and F Nijmegen potentials15 for VAA. They find that model D gives a reasonable value for AB~Awhereas model F gives a negative (repulsive) contribution.

They also find from reaction-matrix calculations17

that D gives a better f i t than F to the binding energies of A hypernuclei (F has a too strong AN, zN transition potential and gives too small BA.) 3. A^ HYPERNUCLEI AND THE H DIBARYON We make some comments on the implications of AA hypernuclei for the H dibaryon (0+, S = -2). I.

For a recent review of H dibaryon physics, see Ref. 18.

Since the phenomenological AA interaction is consistent with at least one

existing meson-exchangemodel, the large value of -aAA cannot be considered as evidence for the existence of an H just unbound with respect to 2MA. 2.

The observation of A^ hypernuclei (at least one event:

taken as excluding the possibility of a bound H.

~Be!) is usually

The identification of a A^

hypernucleus in emulsion requires that both As decay pionically.

Since in

heavier A hypernuclei (A > 5) pionic decay is strongly suppressed, this implies that AA hypernuclei may be produced quite abundantly in emulsion but not identified.

Since the mere existence of AA hypernuclei is relevant to the

question of a bound H, i t would be most interesting i f some signature of ^A hypernuclei which decay nonpionically in emulsion could be established. 3.

Since BAA ~ lO MeV for A ) 6, this allows the possibility of an H bound

by - I0-20 MeV with respect to 2MA, since i f MH > 2MA - BA^ the hypernucleus cannot decay strongly, e.g. 10AABe+ H + 8Be would be energetically forbidden Since BAA increases with A, i t would be very interesting to look for the existence of l i g h t AA hypernuclei, since this could then place limits on MH. Thus the observation of 10AABepermits an H with MH ~ 2MA - 18 MeV, i . e . bound by less than BAA - 18 MeV. I f thenA~He were not observed, this could be interpreted as due to an H with lO ~ 2MA - MH ~ 18 MeV, since then the decay ~AHe + H + a would be allowed but 10 AABe ÷ H + 8Be would be forbidden.

In any

case, i t is interesting to take seriously the consequences of an H which is weakly bound by : 10-20 MeV, since this seems the only possibility of a bound H consistent with the existence of AA hypernuclei. 4.

One such interesting consequence is that of bound H-nucleus states with a

A.R. Bodmer, Q.N. Usmani / AA Hypernuclei and the AA interaction t o t a l energy below t h a t of the ground state of the AA hypernucleus.

229c

I f we

suppose the H-nucleus p o t e n t i a l is s i m i l a r to the A-nucleus p o t e n t i a l , then t h i s could lead to H-nucleus states bound by several MeV with respect to the ground state of the AA hypernucleus, since the l a r g e r MH gives a reduced k i n e t i c energy and hence l a r g e r b i n d i n g . MeV and then : 4 MeV and BH = i i

MeV.

Thus, for ~Be, = B MeV, BA : 7 I f MH = 2MA - 10 MeV t h i s w i l l

give

an energy f o r l~Be of : 2 MA - 21 MeV (with respect to 8Be), which is then bound by = 3 MeV with respect to I0 AABe. The AA hypernucleus could then r a d i a t i v e l y decay i n t o the H-nucleus.

Since both are 0+ states ( f o r IOAABeand

l~Be) such a 0+ ÷ 0+ monopole t r a n s i t i o n would be by e+e- or 2y decay.

If this

r a t e is comparable with the weak decay rate of the AA hypernucleus (T = Io-IOs) then i t could compete with the l a t t e r

and one could look for r a d i a t i v e t r a n s i -

t i o n s followed by the weak decay of the H-nucleus.

(If

the r a d i a t i v e rate is

much l a r g e r than the weak decay rate then the AA hypernucleus would not be observed.)

To make an estimate of Fra d we use the 0+(7.65) ÷ O+(grd)

t r a n s i t i o n in 12C (Ref. 19). Trad = 2 × 10-13s.

For this t r a n s i t i o n rrad(12C) ~ 4 x 10-9 MeV and

To obtain the corresponding rate for the AA hypernucleus we

multiply this by the p r o b a b i l i t y (preformation factor) that the two As overlap to form the H, i . e . by (~H/~A)2 : 2.5 × 10-3 , where ~H : 1 fm3, ~A : 20 fm3 are the volumes of the H and of the A wave function, respectively.

Then rra d

10-11 MeV and Trad ~ 10-10s, i . e . comparable to the weak decay time, thus indicating that radiative decay could indeed be competitive with the weak decay!

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A.R. Bodmer, Q.N. Usmani / A A Hypernuclei and the A A interaction

230c

9) Q. N, Usmani, Nucl. Phys. A340 (1980) 397. lO) Y. Kurihara, Y. Akaishi and H. Tanaka, Prog. Theor. Phys. 71 (1984) 561. l l ) Y. Kurihara, Y. Akaishi and H. Tanaka, Phys. Rev. C 31 (1985) 971; E. Oset, L. L. Salcedo and Q. N. Usmani, Nucl. Phys. A450 (1986) 67c.

12) R. H. Dalitz and B. W. Downs, Phys. Rev. 111 (1958) 967; A. R. Bodmer and J. W. Murphy, Nucl. Phys. 64 (1965) 593. 13) To a very good approximation the effect of the AN exchange interaction can be incorporated in the o-cluster model calculations of ~Be by writing the AN interaction as VAN(r) = V~_o(r)_ ( I - ½ L+2) + V = l ( r ) 21 [+2, where the 1 - TI L+2 and ~I L+2 operators project out .the ~ = 1 and ~ = 0 states, respectively. The ~A potential V A = f¢ ( r l , . . r 4) ~ VAN(rAi) @( r l . . r 4) d3r1..d3r 4 then acts as an operator in the calculation of the ~A potential energy
f f LV + V i f f f > for ~Be. The o-particle wave ~a ~A obAl o~A %Ai ~ ~A ~A A function @~ iS th~ product oT the sihgle-particle oscillator functions with the oscillator parameter fixed to give the ms radius. For an exchange interaction with C 0.25 the reduction in BA(~Be), is then 0.37 MeV in close agreement with the earlier shell model estimates 3.

(To be submitted

for publication, M. Shoeb, Q. N. Usmani and A. R. Bodmer). 14) Wang Xi-chang, H. Ta~aki Bnd H.~Bando " ~ Exchange-Force Effects on the Binding Energies of ~He, ~He, A~He and ~Be Hypernuclei", to be published in Prog. Theor. Phys7 15) M. M. Nagels, T. A. Rijken and J. J. deSwart, Phys. Rev. D 15 (1977) 2545; 20 (1979) 1633. 16) K. Ikeda, H. Bando and T, Motoba, Prog. Theor. Phys. Sup. 81 (1985) 147. 17) Y. Yamamoto and H. Bando, Prog. Theor. Phys. Sup. 81 (1985) 9; Yamamoto, Nucl. Phys. A450 (1986) 275c.

Y.

18) C. B. Dover, Nucl. Phys. A450 (1986) 95c; G. B. Franklin, ibid 117c. 19) F. Ajzenberg-Selove, Nucl. Phys. A433 (1985) 1