[ L NUOVO CIMENTO VOL. L A, N. 3 1 o Agosto 1967

The AA-Hypernucleus ~HeAA.

S. ALI (*) ~nd A. R. BOD~'~F~ (*)

International Atomic Energy Agency Tnternatio.~al Ce~trc for Theoretical Physics - Trie.~te

(ricevuto il 28 Dicembre ]966)

S u m m a r y . - - The AA-hypernueleus ~ has been studied in great detail by a three-body ~-A-A model, using the results obtained from the ~-~-A-A model of ~0BeAA. Apart from calculating the binding energy of *HeAA, the shape dependence of various A-A potentials (including both hard-core and pro'ely a,ttractive types) on se,~ttering length, effective range, etc., has also been investigated. Different A-A potentials having different shapes but the same intrinsic range are found to be very nearly equivalent for the AA-hypernuch'i. Our results suggest that the alter- native interpretalion of the event reported by Danysz et al. as l~Be~.~ (all the theoretical analyses of this event have so t~r been based on the ~~ interpretation) deserves serious consideration. The possible implications of our results about the range of ~he A-,W interaction are also discussed.

I . - I n t r o d u c t i o n .

The d i scovery (~) of the double h y p e r f r ~ g m e n t ~~ (or 11BeAA ) h~s

~roused u consider~ble u m o u n t of in teres t in AA-hypevnuc le i . This is because

(*) On leave of absence from the Atomic Energy Centre, Dacca, East Pakistan. (**) Argonne National Laboratory. Argonne, I11., and Department of Physics, Uni-

versity of Illinois, (!hic~go, Ill. Part ly supported by the U.S. Atomic Energy Com- mission.

(1) M. DANYSZ. K. GARBOWSKA, J. I)NIEWSKI, T. PNIEWSKI, ,1. ZOKRZEWSKI, E. J~. FLETCHER, J . LEMONNE, P . ~,Eb,'ARD, J. SACTON, W. T. TONER, D. O~,~ULLIVAN, T. P. SHAH, A. TIIOXIPSON, P. ALLEN, Sr. M. HEERAN, A. 5'[ONTWILL, J. E. ALLEN, )/[. J. BENISTON, D. H. DAVIS, P. A. GARBVTT. V. A. BITLL, R. C. KUM,tR and P.Y. MAI~CII: Nucl. Phys., 49, 121 (1963).

512 s. ALI and A. R, BOD~ER

the s tudy of the AA-hypernuclei, as has already been stressed, amongst others, by IWAO, DALITZ and JEAN, is expected to lead to significant information about the A-A interact ion in the ]So state.

The double hypernucleus I~ has already been analysed by DAT, ITZ and RAJASEKARAI~" (~) and also by other invest igators (3.4) by use of a th ree-body

model. But since the three-body model does not take into account the di-

s tort ion of the 8Be core, a four-body model fo I~ (i.e.,two m-particles and

two A-particles) has also been used by DEL0eF (5) and ~AKA~URA (n) and in a be t t e r way by BODMER and ALI (~) and also by TANG and HERNDO~ (:). Thus determinat ions of s t rength or volume integrals of the A-A interact ions for various A-A potent ial shapes, free f rom uncertaint ies about core distortion,

Lave been m~de. Here we shall use those strengths (or volume integrals, as the case may be)

to est imate the binding energy of the double hypernucleus 6HeAA. Another aim of our work will be to investigate different potent ia l shapes for AA-hyper- nuclei by considering qIenA. In par t icular it is in tended to investigate the effects

of the shape dependence of the A-A potentials on the scat ter ing length, the effective range and the well-depth parameter . We shall use a three-body model

for 6HeAA consisting of an m-particle and two A-particles. Such a model m ay be fairly realistic for 6HeAA because the core nucleus is very t ight ly bound and dis tor t ion by the A-particles is expected to be small. Tha t the dis tor t ion effects in ~HeAA are ra ther sm~ll has been shown b y NAKAMUgA (8) and ,~lso by TANG and H ~ N I ) 0 N (7). Since distort ion effects in the cases of double hyper- nuclei are expected to be at their min imum in ~HeAA it is desirable tha t detai led calculations are made on BHeAA. AS we have ment ioned earlier, we shall assume the s t rength of the A-A interact ion to be reasonably accurately known from

the four-body model of ~~

2. - The potentials and the variational procedures.

We shall carry out detailed calculations on ~IV[eAA with the following types

of A-A potentials:

(I) VAn(r)----- i-rFnA \4u] fir '

(2) R. H. DALITZ and G. RAJASEKARAN: Nucl. Phys., 50, 450 (1964). (3) y . C. TANG. T. C. HERNDON and E. W. SCtIMID: Phys. Lett., 10, 358 (1964). (a) A. :R. BODMER and S. ALI: Phys. Rev., 138, B 644 (1965). (5) A. D]~LO~F: Phys. Lett., 6, 83 (1963). (~) tI. NAKAMI:RA: Phys. Left., 6, 207 (1963). {7) y . C. TANC. and R. C. HERNDON: Phys. Rev.. 138, B 637 (1965). (s) t[. NAKAMURA: Progr. Theor. Phys., 30. 84 (1963).

TUE AA-IIYPERNUCLEUS 6HeAA 513

where UAA is the volume integral of the A-A potent ia l a n d / t -~ its range.

(ii) oo for r < r r

) (~ I ~r

IAh(r ~1~ -- II~XA t t'r for r > to,

where WAA is the s trength of the A-A interact ion, #-~ the range and r c hard-core radius.

(III) c~ for r<~ '~ ,

V~A(r) ~ 3[~. A W(r) for r > re,

the

where J,=A is the E-A-7: coupling constant and r~ the hard-core radius. The explicit expression for the shape funct ion W(r) is given in the Appendix.

The double hypernucleus binding energy B:xA which is exlusive of the core

energy is obta ined by using the trim functions ghr gAo(ro..) gAA(rAA) where r(~ and rc~ are the A-core separations and rAa is the A-A separation, gAo(r) and gaA(r) are trial functions. The core size R~ (which is actually the r.m.s. radius of the core densi ty distr ibution obtained after folding out the proton charge distr ibution from the charge distribution of the core) appears through the A-core potent ia l I ~ which is obta ined by folding a Yukawa A-~W interac- t ion with an intrinsic range b-----1.484 fm appropriate to the two-pion ex-

change mechanism into the core density distribution. The potent ial VA~ ob-

ta ined for R~ = 1.44 fm (*) and fitted satisfactorily by a combination of two exponentials is

(1) l~,.(r) = - - U4(exp [-- 1.9r] - - exp [-- 2.1154r]),

where U4 is four t imes the spin-averaged volume integral of the A-,V interac- tion. For gAr have used a &paramete r trial wave function

(2)

S = 0 corresponds to the 1-parameter gAc. Results obtained with 1-para- mete r gAo and 3-parameter gAr will be called 1-par and 3-par results respectively.

Variational calculations for two-body A-core problem with tile fit ted potent ia l and the trial funct ion (2) have been made. These calculations give an energy which is within ~ 1 ~ of the exact value obta ined by numerical solution of the eigenvalue problem for the original potent ia l VAr

(*) Unless otherwise mentioned, all our results will be giwm for this e-pa,rtiele radius (9) and for the range /t~-~ for the A-0N' interaction.

(3) R. HOFSTADTER: Rev. Mod. Phys., 28, 214 (1956).

514 s. ALI and ~t. R. BODMER

The calculation of BAA WaS made with the three-body me thod of ref. (t0). I t s appl icat ion to the present p rob lem will be only briefly sketched. For any

given VAA and ]'a~ one obtains a Schr6dinger equat ion for the A-A nmtion w~a) r V a J where I~> with the effective A-A potent ia l V a ~ + ,, AAtgAo, W~A is a func-

t ional of ga~ and VAr and represents the effects due to the presence of the th i rd

part icle, i .e. the core�9 I t is in teres t ing to note here t ha t ~Heaa and ~BeA are

just complemen ta ry sys tems in the sense t h a t one can be ob ta ined f rom the

other by replacing the A by the e and the ~ by the A. One adwnt .age of this

fac t is tha t m a n y of the components of the th ree -body potent ia ls invo lved

in ~Hea~t are the same as those in ~BeA. This can be i l lus t ra ted in detai l as follows :

for the A-A pair in ~ W2~ = W:& + W~(% + W~'k,

for the ~-~ pair in ~BeA, W(~ ) = W(~r~ ) ~- W (~) -~ W <v>

W (~) and W f ' (i s tands or the pair under eonsid- As explMned in ref. (~o),

erat ion, i .e . , o:-c,, or A-A, etc.) are contr ibut ions coming f rom the kinet ic and

poten t ia l energies respect ively for the other two pairs of the re levan t three-

body systems and tu T) is a correction to the ldnet ic energy of the i - th pa i r due

to the presence of the th i rd part icle, w(~) W (~) bo th involve the reduced . r A A * - - a ~

mass m~. a of the e-A system. Because of the e-A s y m m e t r y in bo th 6HeAA

~md 0Be A one has for the same g~x (i.e. , same ~, fl, S) I~(~)=AA ..~W (~) ~nd W (T) w(r)" aA ..=~W (r). Fur the r I,~i(T),, AA and ..~= differ only through the reduced masses

entering. Thus

Wa) �9 r162 I / ~ ' A A

TiT)

- -(T) I wl&] = w2] > w2 b,t since fl= is mostly attractive (for al- u e s of ~, fl, S near the o p t i m u m ones) one has t h a t W~k is more negat ive, i .e.

Ion \ lrlr(T) a t t rac t ive , than W(~r]. The to ta l th ree-body potent ia l W:~ k = (rn~/ AA) *~ a c r -~ WiK) ~(V) -~- ~ -bl~=~ is therefore more a t t r ac t ive than ~ W <a). For a given vMue of U4

and a given gA~ (say e -"~ for simplicity) the over-M1 most a t t r ac t ive poten t ia l

is therefore obta ined for a value of ~ which is sl ightly grea ter t h a n the cor-

responding value for W~I. This becomes appa ren t if one plots W ~ ) and W ~

for the same (74 and the same gAo" The solution of the eigenvalue prob lem for the A-A mot ion with the to ta l

l~ ~ W r gives the 3-body binding energy BAA(~ , fl, S). The maxi- potent ia l , AAT- AA m u m of this as a funct ion of ~, fl, S gives B~A as a funct ion of the s t rength of

the A-A interact ion. I f BA is the ~-A separa t ion energy, then for a r igid.core

(lO) A. R. BODMEI~ and S. Au : Nucl. Phys . , 56, 657(1964).

TIlE AA-I[YI'I'I<NUCLF~U~ q[eaA 515

the additional binding energy

ABAA(q [eaA) : BAA("HeAA ) --- 2 B~\(~HeA)

may be directly related to the strength of tile A-A interaction. In order to

make the results of the variationM eMculation of BAA and B A strictly compa-

rable, we have used the value B A = 3.055 3IeV which is tha t obtained from

the two-body variat ional calculation of lhe A-eore binding energy.

3. - A test of the three-body method.

The quant i ty ABAA(qIeAA) provides us with an interesting test of the

three-body inethod of ref. (~"). Let us consider an a I A t A system but with

an infinitely heavy a-particle, i.e. M ~ - # o o ; then if VAa --- 0 (i.e. no interac-

tion between the A-particles) one has, as expected,

(3) BAA (oo) = '2 B a ( ~ ) ,

i.e. the toted binding enero'y of the three-body system must be just twice the

binding energy of a A with respect to an infinitely heavy a-particle. The above

relation is only strictly true if g:A is the exa(:t solution, however a test of

the method is to ot)tain B a A ( ~ ) - 2BA(oo) for our w~riational solution for 9~A"

Calculations with UAA = 0 (potenti~d (I) with /~-~ =/~-~ = 0.7 f r o ) a n d

,1/~= ~ yield BA. \ =: 11.361 and Ba(oo ) = 5.6766 so that for all practical pur-

poses the relation (3) is fairly satisfied. The maximum parameters associated

with B,A(oc, ) and BA(c~ ) are the s:mm, namely.

c ~ = 0 . 9 5 f m - t , / / = 2 . 0 5 f r o -7 , S = - - 0 . 6 0 5 .

The matz'nilnde of ABAA for UAA = 0 WaS als0 calculated for the aetual

~.-p'~rtiele m~lss. Now, of course, one expects ~l small positive value of ABAA (though there is no A-A interaction) be(.a,use of the A-partieles beint~' ('orrelated

as a result ()f the finite mass of tile eore. The results obtained were

BA. t = 6.382 MeV, and B x = 3.05 .~IeV so that ABAA = 0.282. This is, as ex-

peeted, about twiee the value for ~')BeAA. The values of the parameters asso-

ciated with B x a are c~ = 0.632 fm -~, N - - - 0 . 4 2 , /~ = 2.70 fm -~. These varia-

tional parameters for the three-body 6HexA problem and the best parameters for

the two-body 5He\ problem (namely ~ = 0.68 fm -~, N = - -0 .42 ~ fi = 2.6 fm -~)

~re <'lose to each other and differ lnueh less than the eon'esponding values for

the ease of 9Be A. This is because now we eonsider the A-A pair and the third

particle (i.e. the He-core) is massive compared to each A and, further, also

516 S. ALI and A. z. BODMER

the ~-A in te rac t ions are d o m i n a n t . The reverse is the case for O]3e A where the A

is l ight c o m p a r e d to each ~-part icle a nd the sy s t em is no t d o m i n a t e d b y the

~-A in te rac t ions .

4. - P u r e l y a t t r a c t i v e A - A p o t e n t i a l s .

Before we m a d e deta i led ca lcula t ions on 6HeA,~ we also m a d e a check cal-

cula t ion for V~A(/~2= , ~ = 1.44 fro) a n d UAA = 322 MeV fm 3 for a Gaussi,~n

a t t r a c t i v e VAA h a v i n g the same int r ins ic r ange as t h a t of the Y u k a w a po ten-

t ial (I) (with # =#2=) , i.e, for the po ten t i a l V A n ( r ) = - - U A ~ ( 2 / n ) ~ e x p [ - - 2 r 2]

(2 = 0.935 fm -~) used b y DAL~TZ a nd RAJASEKARA~ ('~). The resul t o b t a i n e d was

BAA = 11.15 MeV (the associa ted p a r a m e t e r s be ing ~ = 0.74 fm -x, fl = 2.0 fm -1,

S = - -0 .498) . This compares v e r y well wi th the value 11.2 o b t a i n e d b y DALrTZ and RAJAS~K_~RAN (~-).

The purpose of doing calcula t ions wi th po ten t i a l (I) (# = / 6 ~ ) was to ob t a in

some idea (by c o m p a r i n g our resul ts wi th those of Dal i tz and lCajasekaran)

abou t the ex t en t to which different A-A in t e rac t ions h a v i n g different shapes

bu t wi th the same in t r ins ic range are equ iva len t for the A-A p rob lems (for

ana logous cons idera t ions for the A-,~ ~ i n t e r ac t i on see, e.g., BODME]~ a n d

SA~PA~TrrER (1~)). The resul ts for the Y u k a w a po ten t i a l (I) a rc shown in Table I . The U A , vs.

TABLE ][. -- 3-par results for 8HeAA ]or the Ytekawa potential (I).

UAA I (MeV fma) I

300

- - 300

BA~ ~ (MeV)

10.60

6.38

<V>* (MeV)

24.50

13.49

(,R~A> (fm2)i Optimum parameters i

I = 0,755 fm -1

6.74 S = - - 0 . 5 1 �9 f l= 1.90 fm -1

~ = 0.63 fm -1 11.91 S = ~0 .42

fl = 2.7 fm -1

4.54 9.65 i ~ = 0.61 fm -1

16.14 ] 8 = --0.36 i f l= 2.80 fm -1 I I

~c3) (R~A) is t h e m e a n s q u a r e of t h c A - A s e p a r a t i o n . (*) I n o u r t a b l e s (V> = ( I 'A . t + I! AA),

(1l) A, R, BODMEI~ ~nd S. SAMPANTHER: Nucl. Phys., 31, 25t (1962).

TltE AA-IIYI'ERNUCLEUS ~HeaA 517

B x~x plot is shown in Fig. l together with the (.orresponding plot obtained by

I)ALITZ and RAJASEKARAN. A comparison of the two plots does indeed reveal

the equivalence of different A-A interactions having different shapes but the

same intrinsic range. For UA.x = (330~25) McV fm a (corresponding to ABAA----

-- (4.5~0.5) MeV for ~nB^ -- eaA) obtained from mBeAA for a rigid core, we have

BAA(6HeAA) ~ 11.2 ~0.6, which corresponds to ABAA ~ 5.1 ~0 .6 MeV for qIeha .

The corresponding scattering length a~x~x, the effe(.tive range r%A and the (1 7~ ~~ f~n, (o 54+02q fm and well-depth parameter ~AA a re aA. \ - - - - - ~ . . . . . . o.3o! roAa ~ . . . . . o.2o1

SAA = 0.64~0.05. These values compare extremely well with the ones obtained

by ])ALITZ and I~AJASEKARAN for a rigid (.ore. However, if we accept the value

UA,,~ = (233=t=43)McV fm 3 (which is the estimate derived from ~~ after

distortion of the SBe core was taken into account and hence is likely to be more

= ~ fc~ (corresponding to ABA~ = reliable) we obtain BAA(qIeAA) (9.35~0.7) ~ o.3~ (.a -n ~ o.8q fm and t~AA = (3.25=}=0.7) MeV) with aAA = - (0.89_o ~8 ) fin, roA, ' ----- t -" '"-0.ss! ----

= 0.45-[=0.08. I t should be mentioned in this connection that TANC, and HERN-

DON have cM(,ulated the binding energy of ~Hex~ for a hard-core A-Apoten t ia l of exponentiM form (7) using the

wflue of the interaction strength

which they determined from the

fom'-body model of ~"Beaa. Their

value is 9.33 which ~grees very

well with our results. This :tgree-

ment is of ('ourse ,~ reflection of

the fact tha t Tang and Herndon 's

calculations of distortion effects

in ~~ agree in all essentials with ours. The opt imum values

of the wuve-function parameters

quoted in Table I indi(,ates the

two-body (x-A) behaviour in the

three-body system and hcm.e are

a measure of the g-A correlations

involved in ~HcA: x. As pointed

out earlier, the A-A correlation

has been treated exactly by our

three-body method. The change

in the ~-A wave-function para-

meters is in the expected direction.

Thus, as we increase the strength

of the interaction, the wave rune

lion contracts. The reverse is

the case for strongly repulsive

400, //~

.~" 200 i

-200 ,J

!

400 IL ~ .... j ...... ~ 4 6 8 10 12 14 16

B^,] MeV)

Fig. l. - Tim volume integral [I~,~ of the A-A pot(mtiM (D is shown plotted as a funet4on of BAa (for 3-par g~l~)" Also given arc the results obtained by DAI,ITZ and ~{AJASEKARAN (2). -- - Yukawa potential (I) (b=l .484fm); . . . . . . . Gaussian potential of DMitz and Raja-

sekaran (b = 1.484 fro).

5 1 8 S . A L I and A . ~ . B O D M E I ~

s t reng ths (for which the A-A w~ve func t ion becomes smal l in the region of

the A-A potent ia l ) .

Wi th po ten t i a l (I), calculat ions were also done wi th a different va lue of the

size of the g-part icle. The w d u e used was R~. - -1 .54 fm and was chosen on

the b~sis of the expe r imen ta l resul ts of Bur leson and Kendal l ('-+), who ob t a ined

for the e-par t ic le an r .m.s , c tmrge radius of (1 .68~=0.04)fm which is s o m e w h a t

larger t h a n the value r epo r t ed b y HOFSTAOTER (9), n a m e l y (1 .6t4:0.05)f lu- .

The basic ~-A po ten t i a l gene ra t ed for R~ = 1.54 fm was aga in f i t ted s~tisfac-

tor i ly b y the fol lowing c o m b i n a t i o n of two exponen t i a l s :

I~, A (#~ , R~ = 1.54 fro) = - - U~(exp [ - - 1.86r] - - exp [ - - 2.049r])

wi th U4 (the vo lmne in tegra l of the ~-A in te rac t ion) = 1 11,1.5 MeV fm'X The

va lue of l'~ was f o u n d b y a d j u s t m e n t su(,h t h a t we still have B:~(~tIeA) ---- 3.] 5[eV

for the new ~-p~trticle radius . The c,~lculations for a different rad ius us ing the

appropr i a t e U4 which gives us the expe r imen ta l B x(.SHeA) for this rad ius is

expec ted to tell us the effect of an error in R~ on the value of B\: t when d i s t o f

t ion is a s sumed negligible.

TABLF [I. - l -par results ]or potential (I) ]or ,li]]erential ~-partiele radii but same B t (~fIeA).

U~ A (MeV fm 3 i

!

R~ (fm)

200 1.44 1.54

0 1.44 1.54

300 1.44 1 . 5 4

500 1.44 1.54

B A A ( ] ~ [ e V )

4.30 4.45

5.97 6.025

9.75 9.72

15.65 15.50

()otm~mn para- i �9 T / 2 2 : ~ - - ,

<V> (M(~) , I~A ) (fnl) 'meter ~ (fm-1)i

8.93 8.96

12.45 12.45

22.52 22.17

40.85 40.12

17.90 0.54 18.15 0.525

13.2 0.56 13.3 O.56

7.68 0.6o 7.87 0.59

4.43 0.625 4.54 0.61

The resul ts shown in Table I I for R ~ - - 1.54 fm were o b t a i n e d for 1-para-

m e t e r g~A, i.e. the t r ia l f unc t ion (2) w i th S = 0. The co r re spond ing 1-par

resul ts for R~----1.44 fm are shown in tile same Table. I t is easi ly seen t h a t

the UAA VS. BAA resul ts ~re v e r y nea r ly tile same for b o t h R~---- 1.44 fm and

(12) G. I{. BURLESON and H. ~V. KENDALL: Nucl. Phys., 19, 68 (1960).

] ' l ie AA-nYPERNUOLlXUS qIea. 519

R~ = 1.54 fm (*). This is sa t i s fac tory since one, in fact;, does no t expe(.t m u c h

difference for different /?~ so long as V~A in each case reproduces the cor-

fee t BA('~HeA). However , such a change of ra-

dius is essent ial ly equ ivMent to a

change in the range of the A - . \ '

i n t e rac t ion so long as the t w o - b o d y

A-eore binding' ene rgy is again giv-

en correct ly . I n o ther words, for

a g iven o~-A b i nd i ng energy, a

change in the e-A p o t e n t b d can

be b r o u g h t a b o u t in two equiv-

a lent ways , n a m e l y b y chang ing

the core size (keepin~ /~a-.v,-~ the

range of the A-A" in te rac t ion , con-

s tunt) or b y chang ing #A-~,v (keep-

ing the core size cons tan t ) . We

have a l ready d o n e e,fl(.ulations

for different core radii (for same --1 --1) t~A-a', i.r #z~ �9 I t now seems desir-

able to do ,some ca.Ieula/,ions for a

fixed /r (say R~ = 1.41 fin) bll t

for a range different f rom #7~-

We have chosen this t o be /t~ t

( = 0.4 fro), the range eom'espond-

in~ to the K - m e s o n exchange

400 "

i / / ,_ / /

~ 0 / /

--200 L / 1 L z_ . . . . j _ l 4 6 8 10 12 14 1

B A ; ( M e V ?

Fi~'. 2. - Tim volume inl,.~ral U.. rs. B~z for 3-i)ar .%.t a, nd fro" the A-A pott'nl;iM ([) for l~ a 1.44 hn a, nd for K-mesott r~(,ng'e for the A-~N ~ inter~mtion. Als,, .~hown ~u'e the c o l responding 3-par results for 2~z rang(~.

--I --1 /%..v=:f~,,~::0.7 fm; - - ,~lV=:uTL= ) 4 I'm; bolh curves axe for potential (D,

mechan i sm. F r o m a compar i son of the plots of UA.~ vs. B~.~ for ff2-~ and #7 '

(both for R = = 1.44 and for 3-par Ga(**)) one can easi ly see t h a t the change

from/&~-~ to/~,~-~ for the A-,~',' i n t e rac t ion seems to m a k e an appreei~d)le change

in the results Baa(6Hea .O (see Fig. 2). This seems a p p a r e n t l y r a the r incon-

s is tent wi th the above results for different R~ b u t can t)e exp la ined in the foI lowing wa,y.

Let l'~ be the a-A po ten t i a l gene ra ted for R~ 1.11 fm and -~ i.e. ~2~ lrl = - - [ [ l V l ( ~ 2 -1, . s 1.44 fm) where U~(~-Ud is the vo lume in tegra l of the

(*) A similar situation also occurs for 1~ for which calculations of BaA with different core radii (a = 1.65 fm and a = 1.5 fm for SBe) yielded very nearly the same results (4). The calculations for 6HeaA (for different core radii) give added support to the conclusion made from the analysis of I~ that BAA is insensitive to the details of the core size.

(") The 3-par results for p~l were obtained from the 1-par results (shown in ]'able I I I ) by making the fairly reasonable assumption that the percentage improvement of 3-par results over the 1-par ones for #K l is about the s~me as that in the case of --~ It2 ~ �9

5"20 s . ALI and A. m BODI~IER

UAA (MeV fm~) '

3O0 0

300 5011

TABLE [[[. - 1-par results Jor K-meson range at~d ]or the A-A

J B , ~ (MeV) (V} (MeV) (R~\A} (fm 2)

I 4.02 9.18 I 16.20 6.05 13.8 I 11.20

10.70 26. l I 6.33 17.70 46.5 ! 3.78 I

~otential ([).

Optimum para- meter a (fin-')

0.575 0.625 0.66 0.69

e-A in terac t ion ( = 1038 MeV fm 3) and vt is the shape function. Let V2 be the

potent ia l ob ta ined for a different radius, e.g., R~----1.54 fm bu t for the same range -~ -~ #2~, i.e. V2 = - - U2v~. (#2n, R~ = 1.54 fro) where v2 is the new shape func- t ion and U2 is the new volume integral ( = 1111.5 M e V f m 3) again adjus ted

to give BA(.~HeA)= 3.1 MeV. I f now we define a potent ia l Vs which is equi-

va lent to V2 (in the sense tha t i t reproduces the value of BA(hHe,x)= 3.1 ~IeV)

but which is obta ined b y using a range #'-~ different f rom /~-~ (keeping the

e-part icle size fixed at 1.44 fm), then V~ = U~va(# '-1, R~ = 1.44 fro) where v3 is

again a new shape function. The way to find re' is to assign different values to it and pick the one which gives the r ight BA(hHeA). I t is found f rom actual

calculations tha t # ' - 1 > / ~ by only 0.075 fro. This difference is much less in absolute value titan the change -~ -~ #.2~-+/~I~ . Hence it is plausible t h a t the results for ~ t ( R ~ = 1.44 fro) can differ apprec iably f rom the results for

300

6 E 10 12 14 16 B^^(NeV)

200 E

q_

>

100

Fig. 3. - The vo lume integral Ua, ~ vs. B,, ,~ for 3-par g~A and for the range lean-1

for the A A potential (I).

R~ 1.54 fm (#- ' -1 The = = #~n)- reason why in fact the results for #K are different f rom those for #2= (both for

R~ = 1.44 fm) can be unders tood f rom

the fact tha t the e-A potent ia l for the K-range is deep near the origin and shal- low outside while tha t for the 2r: range

is less deep near the origin bu t more ex tended outside. Since the A-A wave

funct ion falls off ve ry sharply because

VAA(r ) drops very rapidly, the Kqneson

potent ia l contr ibutes p ropor t iona te ly

more to the b inding energy BA.~ than

the 2r~ potent ial .

Some 3-par results for a range #3-~

(which corresponds to the exchange of a

scalar T--- 0 part icle with m3= 420 MeV) for the Yukaw~ A-A in terac t ion (I)

are shown in Table IV. These results (shown in Fig. 3) were obta ined to ex-

THE AA-1tYFERNUCLEUS 6HcAA 521

TABLE IV. - 3-par results tot pote~tial (I) with the rar~ge /xa~ (intrinsic range = 0.989 fm).

]UA. ~ (MeV fm a) BAn (MeV) ( V ) (~IeV) (R~A) (fm z) : Op t imum parameters

100

200

7.67

10.04

14.84

16.30

22.80

40.60

i

I

]0.10

7.64

4.3O 300

ce = 0.70 fm -~ S . . . . 0.46 fl =- 2.15 fm -I

u = 0.74 fm -I S = - - 0.49 f l = 2.0 fm -1

= 0,90 fm -~ S = - - 0.64 fl = 1.70 fm -1

a m i n e , b y c o m p a r i s o n of t h e r e s u l t s we a l r e a d y h a v e fo r # ~ ( for t h e Y u k a w a

p o t e n t i a l (I)) , w h a t , i f a n y , a r e t h e s h a p e - i n d e p e n d e n t p a r a m e t e r s . H o w e v e r ,

i t a p p e a r s f r o m t h e r e s u l t s t h a t n e i t h e r t h e v o l u m e i n t e g r a l n o r t h e s c a t t e r i n g

l e n g t h c a n b e s t r i c t l y r e g a r d e d as a s h a , p e - i n d e p e n d e n t p a r a m e t e r fo r p o t e n t i a l s

of d i f f e r e n t i n t r i n s i c r a n g e s .

5. - Phenomenologieal hard-core A - A potentials.

W e sha l l n o w p r e s e n t s o m e r e s u l t s fo r p h e n o m e n o l o g i c a l h a r d - c o r e A - A p o -

t e n t i a l s . I n f a c t , i f f o r t h e u h a r d - c o r e p o t e n t i a l ( I I ) , w i t h r~ = 0 .30#~ ~ =

= 0.42 f m o n e p l o t s A B v ~ = B A A - - 2 B A , t h e n o n e o b t a i n s v e r y n e a r l y t h e

TAnLE V. - 3-par res~dts for the Y u t a w a hard-core A-A potential (II) with re : 0 .3 /~L

i Wan (MeV) '~ BAA (=~[(~V) <'V> (]~If~ v) ';rll,'l~2~> (fm 2) Opt imum paramete, rs

150 6.31 15.47 12.62

250 8.875 21.78 9.46

365 12.72

: 0.60 fm -1 N . . . . 0,375 /;~ = 3.0 fm -x

32.70 6.68

= 0.65 fm -1 N : - -0 .415 fl : : 2.325 fm -1

~ : 0.75 frn -1 S : - - 0.52 fl : 1,85 f m - '

34 : - l l Nuovo Cimen~o A .

522 s. ALI and a. R. BODI~ER

same 3-body model results (see Fig. 4) for bo th q~eAA and ~~ (The same is also t rue for the Yukawa potent ia l (I).) Thus ABAA is r a the r insensit ive

to the value of B A. This is in agree-

2

1 O0 200 300 400 W ̂ (MeV)

Fig. 4. - The energy ABAa as a fltnction of the strength WA, ~ of the A-A potential (II) with ro= 0.3#~ ~ shown for both 6IteAA and l~ for 3-par g~A: 6He~A;

_ __ ~0Be~A.

men t wi th the conclusions of Dal i tz

and I~ajasekaran (~). I f we take the

s t rength WAA as de te rmined f rom the

analysis of ~~ , namely B ~ x = = ( 2 6 0 • MeV, we get B A A ~

4-0.8 (9.0 o.7) MeV which is quite com- parable wi th the results for the po-

tent ia l (I). The associated scat ter ing length, effective range and well-

{2 .~+o.s% fro, dep th p a r a m e t e r are - - ~ .=-o.5,, +1.07 (4.93 o ~ ) f m and 0.675:~0.065 re-

spectively. The value of aAA for po-

tent ia l (II) is considerably larger than

for po ten t ia l (I) and is due to the

larger intrinsic range of (II) (2.67 fm).

Some results 0 -pa r ) are also given

for r~ ~ 0 . 3 5 / ~ . These are shown in Table V I together with the corresponding 1-par results for r~ = 0.30#~ t.

I t was found by compar ing the results for BAA and aA~ for these two hard-core

radii t ha t the different s t rengths WAA which are needed for different % to give the same BAA give about the same scat ter ing length and, as expected, for a given BaA the well-depth p a r a m e t e r was found to increase wi th the hard-core radius.

TABLE V[. - 1-par results ]or potential (II) for re= 0.30#~ 1 and r~= 0.35/~ 1.

lV~a (MeV) ro(#~ 1) a (MeV) <V> (MeV) <R~A> (fm2) meter a (fm -1)

I

- -1

150

250

365

500

0.30 I 4.00 0.35 3.74

0.30 6.02 0.35 5.325

0.30 8.15 0.35 6.9

0.30 11.91 0.35 9.55

0.35 14.06

9.40 18.80 8.97 19.95

14.50 12.98

20.3 17.25

30.66 24.39

36.8

13.54 15.13

0.525 0.52

0.56 0.55

10.35 I 0.575 12.16 i 0.56

7,36 9.26

6.6

0.60 0.575

0.61

T I I E AA-IIYPERN[~CLEUS 611e~.~ 5 2 3

I t seemed in te res t ing howeve r to explore the equ iva len t Yukaw,~ range #-i of the hard-core potential (II) (with r~ = 0 .3~ I) which h~s the same intrin- sic r ange as the Y u k a w a po ten t i a l (I) of range -~ /~2~. F r o m the ha rd -co re resul ts

(re = 0 . 3 # ~ ) for -1 tt,~ i t appea red t h a t such a #-~ m u s t be cons ide rab ly smaller

t h a n -1 t t~ . I n f ac t the r equ i red range was even shor te r t h a n -1 tt4~ and was f o u n d

to be # - ' = 0 . 1 5 7 / ~ = 0.22 fro. Table V I I shows the 3-par resul ts for the

Y u k a w a hard-core po ten t iM ([I) for this r :mgc ~md for r. = 0 .3#~ ~. F r o m

TABLE VII. - 3-par re.~ults /or the hard-core potential (II) with # - 1 = 0 . 2 2 fm and % = 0.3#~L

WAA (MeV) - BAA (MeV) <V} (MeV) (/~:~n} (fm~) Optimum p~_rameters

a = 0.66 fm -1 1.40-104 6.70 18.96 i ll .06 S = --0.435

! fl = 3.00 fm -1

1.86" 104 9.165 29.21 8.155

2.32.104 13.82 52.645 5.21

---- 0.71 fm -1 S = - - 0.440 fl = 1.70 fm -1

= 0.85 fm -1 S = --0.62 fl = 1.50 fm -1

Fig. 5, 6, 7 ~nd 8 one hns for A B A A ~ 2.4

(3.1 =~0.5) MeV the wflues WAA ~ (1.86 :~ , , + 0 . 2 5 :t: 0.08)-104 MeV, aAA ~ --- (0.9]3_0. 2 ) fro, 2.2

, + 0 5 ~ r o ~ (3.08_o14o) fm a nd ~S'AA= 0.744 :~ 0.032. These v'flues comp~re ex t r eme ly well with ~o 2.0 the ones ob tMned b y T ~ G and HERN- •

~0~ (~) f rom the ~malysis of ~~ wi th ~" 2.2

the use of a ha rd -core A-A po ten t iM of exponen t i a l fo rm which h,~d the same in- k< 1.6

t r ins ic r ange ,~s our p resen t h~rd-eore

potent ia l . Thus, different phenomeno lo - 1.4

gicM hard-core po ten t ia l s h a v i n g the same

to ta l in t r ins ic range are f o u n d to be equiv- 1.21

Ment. I t is also seen t h a t w i th the ~bove

Y u k ~ w a h~rd-core A-A po ten t iM a nd the

Y u k a w a a t t r ac t ive A-A po ten t i a l (I) (both

of which have the same intr insic range,

n a m e l y b-----1.484 fro), v e r y nea r ly the same

8 10 12 14 16 s~ (̂MeV)

Fig. 5. - The s~rength W~A vs. BAA shown (for 3-par g~A) for the range #-1-- 0.22 fm for the A-A potential

(II) for rc = 0.3/~ 1.

524 s. ALl and A. R. BODM~R

- - 2

- - 4

- - 6

- - 8

--1 2

!

I -1 6 - .

!

- - 1 8 ; �9

e)

I

- - 2 0 I i i _ i i i i 3 5 7 9 11 13 15 17

B ^ ^ ( M e V )

19

Fig. 6. - The scattering length aAA as a function of BaA for 3-par g~A for the. various potentials: a) potential (I), # = # 2 ~ ; b) potential (I), #=/~a=; c) potential (II), /~-1 = 0.22 fm, r~ = 0.3/~1; d) potential (II), # = tt2r~, r~ = 0.31~1; c) potential (III),

]-~z = 0, r~ = 0.3/~ 1.

resul ts are ob ta ined . However , t he wel l -depth p a r a m e t e r for the fo rmer is sub-

s tan t ia l ly larger t h a n t h a t for the la t ter . This is r easonab le because , as d iscussed

in ref. (13) for the A-,,V po ten t ia l , the ha rd -co re A - A po t en t i a l wh ich fits BAA a n d

has a specified in t r ins ic range is expec t ed to h a v e ~ g rea te r we l l -dep th p a r a m e t e r

t h a n a soft A-A po ten t i a l of the same in t r ins ic range. Howeve r , f r o m the a b o v e

(13) R. H. DALITZ: The n.~clear interactions o] the hyperons, Enrico Fermi Insti tute for Nuclear Studies, University of Chicago, Rept. EFINS-62-9 (~arch 1962), p. 33.

TH]~ AA-nYP~RNUCLEUS 6tI%~ 525

14.

12

1o

~" 8 q- v <<

F e)

b) C) I , la2 I _ _ 8 10 14

~^(Mev) 16

1.2

1.0 a')~-" e)

O. 8 J c~~

m << 0.8

0 - - _L__ i _ I 6 8 10 12 14 18

B^ (̂NeV)

Fig. 7. - The effective range r%.~ vs. BAA plot for 3-par ga~ for the potentials: a) potential (I), / t = & = ; b) potential (I). /t = #at:; c) potential (II), /~-* = 0.22 fro. .r~= 0.3M~; d) potential (II), /t = t~2~, r~ = 0.3#~1; e) potential (III), ],:,= = 0,

ro = 0.3p~ 1.

Fig. 8 . -The we]]-depth parameter is plot- ted against BAA for 3-par g~x for the potentials: a) potential (I), # = ~ 2 n ; b) potential (I), / ~ = / ~ ; c) potential (II), /~-1 = 0.22 fm, r~ = 0.3/~7~; d) potential (II), #=/t2~, ro= 0.3~1; e) potential (III),

]~z=0 , T r ~.

results i t is clear tha t , for a g iven intr insic range, the ha rd -core and the pu re ly

a t t r a c t i ve po ten t ia l s are more or less equ iva len t (in the sense t h a t b o t h give al-

mos t the same b i nd i ng ene rgy and abou t the same sca t t e r ing length) . The ha rd -

core po ten t ia l s are, however , expec ted to be more real ist ic t h a n the pure ly a t t r ac

t i r e ones. This is because the presence of a shor t - range repuls ion in the nucleon-

nuc leon po ten t ia l , usua l ly r ep resen ted b y a h a r d core, suggests t h a t ~ h~rd

core of s imilar size m a y also be p resen t in the A-A po ten t i a l (14). l=[ence we

have also done ex tens ive calcula t ions wi th the meson theore t ica l A-A po ten t i a l s

wh ich have , however , some theore t ica l basis.

6. - Meson theoret ical hard-core A - A potent ia ls .

The use of the meson theore t i ca l ha rd -core A-A poten t ia l s is necessary if

one is to get a n y in fo rma t ion abou t fSA (the E-A-r: coup l ing cons tan t ) . The

behav iou r of ]sA as a func t ion of the ha rd -core rad ius ro and also as a func t ion

(14) j . j . D): SWAnT: Phys. Lett., 5, 58 (1963).

i

526 s. ALl and A. R, BODMER

of / ~ ~nd aAA has been considered b y DALr~:Z (~s) and by DE SWAItT (t~). For tuna t e ly for the 1So s ta te (which is the only one re levant for the

AA-hypernuclei) the coupling with the Z E channel is weak and we m a y

jus t use the lowest ~th-order potent ia ls (the second-order potent ia ls being

zero because T = 0 for A) corresponding to the stat ic l imit of the graphs.

A A

/ " I / / / /

/ / Z ' / /

A A

A A

\ z /

+ " Z

/ / / \ x \

A

These contr ibut ions are ~ J~A- So the poten t ia l will be (for the 1S0 state)

VAA(r) = ~ , r < ro,

= 31~A W(r ) , r > ro,

- lO ~ I

<

2 lO

3 - - 10

-10 O,Z+ .8 1.2 1.6 2.0 r" ('Pm)

Fig. 9. - The meson theoretical poten- tial (III) is plotted as a function of the distance between the A-particles outside a hard-core raditls ro = 0.3p~ 1 for the

coupling constant ]~A = 0.267.

The expression for W(r) given in t e rms of the Bessei funct ions K, , K: (see Appendix) -*'as calculated numerical ly b y using series expansions for Ko, K1. Some idea abou t the values to consider for

J~A (i.e. about the depth of the a t t rac- t ive par t ) was obta ined f rom the work of DE SWART (~4). The meson theoret ical

po ten t ia l when p lo t ted as a funct ion of the dis tance r be tween the A-part ic les

has an ex t remely deep, narrow and rap-

idly va ry ing outer a, t t r ac t ive par t . One

e~n easily see this f rom :Fig. 9, in which

the a t t r ac t ive pa r t outside :~ hard-core

radius of 0.3#~ 1 has been p lo t ted for

J-~A = 0.267. For shor t - ranged a t t rac-

t ive wells outside a ha rd core, the value

of BAA is expected to depend strongly on the depth of the a t t r ac t ive well.

Bu t since the depth itself, being ~"J~A,

(15) R. I-[, DALITZ: Phys. Lett., 5, 53 (1963).

THE AA-ItYPERNUCLEUS 6HCAA 527

is v e r y s ens i t i ve to t h e v a l u e of JxA, one expe c t s t h a t BA~ wil l d e p e n d v e r y

s e n s i t i v e l y on /'~A. This in f ac t is t h e case, as can be seen f r o m the r e su l t s for

r~ = 0 . 3 # ~ ~ (shown in T a b l e V I I I ) . The J~,-BAA cu rve shoots u p v e r y r ap -

"/,/~A> (fm2)

TAULE VII I . - 3-par res~tlts ]or the mesov~ theoretical A-A potential (III) with re = 0.3/,~ -1.

]XA BAA (}IeV) ~ (V> ()IeV) I Optimum parameters '~

0.250 6.80 19.31

0.270 9.97 35.09

11.17

7.38

4.745 0.2825 15.12 63.725

= 0.64 fm -~ S = 0.41 fl = 2.55 fm - t

= 0.75 fm -1 S = - -0 .51 fl = 1.75 fm -1

cr = 0.95 fm -1 S = --0.77 fl = 1.41 fm -1

as we h a v e seen, on ly be a t t r i b u t e d

to t h e i r h a v i n g t h e s ame i n t r i n s i e r ange ,

i t s e e m e d v e r y l i k e l y - - a n d i t was

i n d e e d f o u n d so b y c a l c u l a t i o n s - - t h a t

t he m e s o n t h e o r e t i c a l h a r d - c o r e A-A p o t e n t i a l for ro = 0.3 #~t has an i n t r i n -

sic r a n g e ~ 1 . 5 fro, c o m p a r e d to 1.484

for t he (( e q u i v a l e n t )) Y u k a w a h a r d -

core p o t e n t i a l w i t h r r ~. The

aAA VS. BAA p l o t for t h e m e s o n theo -

r e t i c a l p o t e n t i a l ( I I I ) w i t h ro = 0 .3#~ ~

i d l y as ]~A inc reases (see F ig . 10) a n d one can eas i ly ge t A - A b o u n d s t a t e s , e tc . ,

if ],:A is chosen too large .

F o r t h e v a l u e of t h e coup l ing c o n s t a n t ]~I~ = 0 . 2 6 7 • d e r i v e d f r o m t h e

a n a l y s i s of I~ one o b t a i n s BAA(~HeAA ) = (9.4-~0.8)MeV~ which c o r r e s p o n d s

to ABAA = ( 3 . 3 4 - 0 . 8 ) M e V w i t h a~.~ ~ - (1.132~o~ fm, r%A = (3.08_+]i~ ~ fm a n d

SAA = 0.77~:0.0~. These resu l t s ,~re

in exce l l en t a g r e e m e n t w i t h t h e ones 16

o b t a i n e d w i t h t he (( e q u i v a l e n t )~ Yu- ~[

k a w a h a r d - c o r e A - A p o t e n t i a l (wi th 1~ i

r c = 0.3/z~ 1 a n d #-~ = 0.22 fm). S ince !

t he olose a g r e e m e n t of t he r e su l t s ob- 12

t a i n e d w i t h two d i f fe ren t p o t e n t i M s can, ~ l

]

5^

Fig. 10. - BAA as a function of tim coup- ling constant f~A (for 3-par gaA) for the meson theoretical A-A potent ial (III)

with % = 0.3/~ 1.

528 8. ALI and a . ~. BODMER

is, aS expected, very similar to tha t for the Y u k a w a poten t ia l (I), because bo th

have more or less the same intrinsic range. The scat ter ing lengths ahA are,

however, ve ry sensit ive to JrA. Thus if one wants to calculate the n u m b e r of bound states wi thin a cer tain range of values of JZh one has to proceed care-

fully in ve ry small increments of JzA, otherwise there is every possibi l i ty t h a t

they might be missed. For Jz,t-->o, i.e. when the a t t rac t ion tends to zero,

aAA mus t approach the hard-core radius re (this has been checked for several

values of re). As ]xA is increased, corresponding to more and more a t t rac t ion ,

aAA at first decreases to zero and then becomes ve ry large and negat ive as the

a t t rac t ion becomes sufficient for binding.

Some discussion abou t the in te rpre ta t ion of the scat ter ing length ahA in

t e rms of the meson theoret ical po ten t ia l (for even EA par i ty) has been given in ref. (~.~4.~) and we shall not enter into it here. However , to explore the

sensi t ivi ty of /ZA to the value of the hard-core radius, i t was though t worth-

while to inves t iga te the effect on BAA Of changing rr Accordingly, we made some calculations (1-par) for a different hard-core radius, namely re ~ 0.35/,~z = - -0 .49 fm. Table I X shows the results of these calculations toge ther wi th

some 1-par results for ro ~ 0.3/~; ~.

TA]3L~ IX. - 1-par results/or the A-A potential (IlI) /or re : 0.30p,~ z and ro = 0.35/t~ z.

/zA rc(//~ 1) I BAA ( ~ o V ) /V),, (MeV)

! 0.30 ' 4.61 11.1

l Optimum para-, (~kA) (fro2) :meter ~ (fm-1) :

. . . .

0.20 16.65 0.54

0.25 0.30 6.425 17.75 12.27 0.56 0.35 4.77 12.01 16.16 0.54

0.275 0.30 10.745 39.8 7.22 0.58

0.30 0.30 33.45 161.89 2.28 0.64 0.35 7.98 25.63 9.95 0.575

0.325- 0.35 15.96 67.16 4.89 0.625

0.350 0.35 46.26 203.80 1.96 0.64

I t is seen f rom Fig. 11 t ha t as we increase JZA, the value of BAA depends very sensit ively on the value of the hard-core radius used. However , the

BAh-rE A curves for the two radi i reveal the feature, namely t ha t the different s t rengths of the A-A potent ia ls which are needed for the two radi i considered

to produce the BAA values obta ined with other A-A potentials , give ve ry

g

TH~ AA-HxP]~RNUCL)~US 6HeAA 529

r o u g h l y the same sca t t e r ing lengths. As we have seen earlier, a s imilar sitm~tion

is also obse rved in the resul ts of the Y u k a w a hard-core po ten t i a l for these two

radii .

One can ob t a in :m u n d e r s t a n d i n g of these results if one refers to the re la t ion

(see ref. ('~))

b ~ = b - 2 r . ,

where

b = in t r ins ic r ange of the ent i re po ten t i a l (i.e. h a r d core plus a t t r a c t i ve well),

b ~ in t r ins ic r ange of the a t t r a c t i ve well t r ans l a t ed to the origin.

The r ange tt of the po ten t i a l is re la ted to b ~ in some fashion (e.g., for expo-

nent ia l ha rd -core po ten t i a l tt = b~ etc.). :Now when one changes the

ha rd -core rad ius re (keeping # cons t an t

and hence b ~ cons tan t ) the v,~lue of b

changes . However , the change in b is

of the order of twice the difference be-

tween the h~rd-core radi i (b = b ~

which is our p resen t ,:ore (i.e. for

ro = 0.3 tt~ ~ and r~ = 0.35 # ~ ) is equal to abou t 0.14 fro. (The ca lcu la ted in-

t r insic ranges for the meson theore t ica l

po ten t i a l for the two radii considered

do ac tua l ly show the same order of dif-

ference.) This is no t a big change and

so b can effect ively be r ega rded as con-

s tunt . Thus we conclude t h a t for a g iven

intr insic range, the sca t t e r ing leng th is

no t expec ted to v a r y v e r y r ap id ly wi th

the hard-core radius . The same con- clusion h a d been an t i c ipa t ed b y DALITZ

and RAJASEKARAN, who e m p l o y e d a

pure ly a t t r ac t ive A-A po ten t i a l in the i r

40

30

:z 2o

Io

.... .1 2

I] J ' -- ' '. ~ 6

.20 0.25 0 30 0.35

Fig'. l l . - The energy ]3AA vs. lEA (for 1-par g~A) shown for the hat'd-cor(~ radii "re= 0.3/t~ 1 and ,'~== 0.35fl~ 1 for t;he meson ~heorctical potential (II[). Also plotted arc the sc~Lttering lengths a.~A as a function of ]XA for both these ra.dii: --- - BAA; . . . . a~a; curves 1) re--

== I ) . 3 / t ~ l ; c u r v e s 2 ) 'r e ~ 0 . 3 5 / t ~ 1.

analysis (5) of A A b ind ing energies bu t , however , suggested extens ive (.ah.ula-

t ions wi th ha rd -core A-A potent ia ls .

(1G) T. 0tIMURA, •. /~ORITA and M. YAMADA: Progr. Theor. Phy.% 15, 222 (1956); 17, 326 (1957).

-v

530 s. ALI and A. R. ]3ODMER

7 . - D i s c u s s i o n .

The results of the present invest igat ion based on a var ie ty of A-A inter- actions show the following interest ing featm'es. Different a t t rac t ive A-A poten- tials having different shapes bu t the same intrinsic range are found to be equivalent for the A-A problem, i.e. the binding energy, the scattering length, the effective range and the well-depth paramete r obtained with these potentials are ve ry nearly the same. The same (( equivalence )) is also found for hard-core A-A potentials again having different shapes bu t the same intr insic range. However, an a t t rac t ive A-A potent ia l and a hard-core A-A potent ia l (both possessing a specified intrinsic range) are almost equivalent because they give

very nearly the same values of BAh and aA:~, bu t the well-depth paramete r for the hard-core A-A potent ia l is, us expected, greater thun tha t for the a t t ruct ive

A-A potential . Thus, for Yukawa ut t rae t ive A-A potent ia l (I), the (( equivulent )) hard-core A-A potent ia l (II) (with to-- 0 . 3 / ~ and #-~---- 0.22 fm) und the meson theoretical A-A potent ia l (III) (re = 0 .3/ t~) , all of which have ulmost the same intrinsic range (~1 .5 fro), one obtains approximate ly the same bind- ing energy and scattering length, namely BAA ~ (9 .3 • and aAA= = (--1=]=0.3)fm. For a given ABAA, potentials with larger intrinsic ranges (> 1.5 fm), e.g. potent ia l (II) with r~ ~ 0.3 tt~ ~ and /t -~ = #~-~ give ra the r larger values of aAA.

I t is found from the calculations with different core sizes tha t the value of ABA~ ~ is ra ther insensitive to the size of the core so long us the A-core binding energy is given correctly. However, for a given core size, ABAA is seen to be quite sensitive to the range of the A-A ~ interuction.

For the meson theoretical potent ia l (III) , BAh is found to be strongly sensitive to ]~A and ro. However, from the BAA VS. ]'ZA curves for different ro, it is seen tha t the different strengths which are needed for different ro to give the same BAA give roughly about the same scat ter ing length. I t is also seen tha t as long as the intrinsic range remains effectively constant, the sc~ttering length aAA does not depend very sensit ively on the hard-core radius.

After this work was completed, a very recent paper (~7) by PROWSE cume

to our u t tent ion in which it wus repor ted tha t an event had been found in an emulsion stack exposed to about 106 4 to 5 GeV K - mesons which appeared

to be consistent with the product ion and decay of u 6HeAA double hyperfrag- ment . PROWSE repor ted a value of (4.6• MeV for ABAA(6HeAA) which is

almost the same value as obta ined by DhNYsz et al. (1) for "~ A. The disagree- ment of our value of ABAA ~ (3.2~0.7) 3IeV (obtained for a var ie ty of A-A potentials having the same intrinsic range ~ 1 . 5 fm and for the range -~ t t~ of

(17) I). J. lh~owsl,:: Phys. Rev. Lett., 17, 782 (1966).

TIIlg A A - I I Y I ' I ~ R N t I C L E U S qteA~ 5 3 I

the A-.~ ~ interaction) with the experimental value repor ted is ra ther interesting. I t might lead one to th ink tha t one should also perhaps consider the distort ion

of the He core. Bu t TA~c~ and HERNDON ]lave shown (,s) by a six-body varia- t ional calculation of qteA~ that such distortions do not make any significant contr ibut ion to ABAA(6HeAa). The possibility then remains tha t the al ternat ive in terpre ta t ion of the 1~ event, i.e. the nBeAA interpre ta t ion (which gives almost the same AB~xA as lOBeaA) deserves more consideration than it had formerly received. The significance of the '~Beaa in terpre ta t ion has been discussed by DALITZ (,5,~9). I t has also been pointed out in ref. (4) tha t if the ~'BeAA inter- pre ta t ion is accepted, then the results obtained for the A-A in terac t ion will be roughly in termediate between those obta ined for ~Beaz t with ,~ rigid core on the one hand and wi~h core distort ion in(4uded on the other. Thus the

strengths of the A-A potentials which were obta ined from the four-body model

of ~OBeA. x and used in cMculating the binding energy of 6Hera ~ will now be en- hanced and the value of AB;xA will also increase and will be closer to the exper- imental ly observed value. A second likely explanat ion seems to exist in the considerations of the range #-J of the A-,V interact ion which enters through the A-core potential . For a given A-A potent ia l the calculations with #~; are seen to give compara t ive ly more A-A binding than those with/~2~ (see Fig. 2). Thus for the same potent ia l (I), one has for Uaa= (233~43) ~IeV fm ~ (obtained from the analysis of ~~ (~) ABA~ = (3.25::~0.7) ~[eV for #a.~-~ =l~7~ and A B \ a :~ (4 0 + l ' o ] MeV for - t #a_ao =/~2J. This suggests theft if we take the s t rength - \ " --0.92

of the A-A interact ion as tha t given by the four-body analysis of ~~ then one clearly needs to consider a shorter range (of the order of # ~ or even less) of

the A-.V interact ion to reproduce the experimentM value of AB55. I t is worth mentioning here tha t such a shorter range of the A-.V interact ion was also needed in an analysis (~~ of the first ex,.ited slate of ~BeA.

We are grateful to the Manchester Universi ty Computing Labora tory for the use of their <(Atlas ~ computer . One of us (S. A.) wishes to acknowledge the k ind hospi ta l i ty of Profs. A S,',r,?,g, P. B~mINT and IAEA at the In terna- tiona,1 Centre for Theoretie 'fl Physics, Trieste, where this work w~s completed.

S. AH was supported in most par t of this work by the Manchester Universi ty.

(1~) y. C. ':['~(; and R. C. I[]31r Phys. 1iev. Lett., 14, 991 (1965). (,9) R. H. D.~Lrrz: Hypernuclear interactions, an invited paper presented a~ the

7'ol)ieal Co.,~]erenee on the Use o] Elementary Particles in Nuclear Structure l~esearch, held Scptemb(~r 14-1(;, 1965, at the Inst. Intcrtmivcrsitaire des Sciences Nucl5Mres, Universily of Bru~(,ls.

(~o) S. ALI, J. W. i~r and A. R, Bo~M~'~: Phys. Rev. Lett., 15, 534 (19(15).

5 3 2 S. A L I and A . R . B O D M E R

While this paper was on its way to the journal, a note by TA~G a n d

HERNDON on (( A-A interact ion from "HeAA )) (2~) came to the a t tent ion of the

authors. The conclusions of Tang and Herndon are almost in the same direc-

tion as those reported in this paper. Their results, however, were not as

comprehensive as ours.

A P P E N D I X

The meson theoretical A-A potential for may be wri t ten (.02)

where

even EA par i ty and for f~z~-0

VAA(r) = oo , r < 'rr

4 = 3]ZA W(r) , r > rr

iV(r) = (X] ' ; 4) -- 3X V2 ) + I I r(4)_ 3I 1,. ) I "('> .

X refers to crossed graphs and I I to uncrossed graphs (shown earlier). ~ refers to the spin-dependent contribution.

The component parts of the shape function W(r) are given by

xvi~' (*) = - ~ L L . ~- ' > <(~'") + 7 + . v

r l~

. , ) ] -LW ~ ~ + ~ , , - ,s : , (~,)(!+~§ ~ ,-'~'o(,,) i t . x ~

where x - - / ~ = r and K,,(x) is defined 1)y

co

K~(~,,) - - F ( u + {) 1 % . P cos 1,'x

l)

(21) y . C. TANG and R. C. IIFRNDON: NUOVO Cimento, 4 6 B , 117 (1966). (~2) j . j . T)~ SWART "md C. IDmNGS: Phys. Rev., 128, 2810 (1962).

TIIE AA-IIYPERNUCLE['S 6I[eA,t 533

A D I i E N 1) U_-~[

I t s e e m e d i n t e r e s t i n g to also cons ide r some resu l t s for t h e A - A p o t e n t i a l ( I I ) for # - ~ = f f g ~ a n d r ~ = 0 . 3 f f g ~ ( b = 2 . 1 1 fro). F i g u r e 12 shows t h e r e l e v a n t

1.41 ~ ,~ 1.73

1.32 ! 1.50 1.655 i 1.80 I ~ -T r - - T T - r T T/-]I /* I 1.58 / / i

/// I :"

2" i l l d oa : / i

o L" �9 . . . . . . . . . . / 700 900 1100 1300

Vy. MeV

Fig. 12. - The scattering length aaA and the binding energy BSA as a function of the coupling constant g A z , a and the strength W t, of the A-A potential ([[) for re = 0.3F*~ 1

alld /t-1 =/~3~-1, II~A(S~A ~ l ) - - 1.375 ~eV. . . . . a A . ~ , - - - B ~ . ~ .

resu l t s for t h i s p o t e n t i a l n a m e l y aa.a a n d BAA Vs. WAA 0rod gAA~). The re su l t s for BAA are 3 - p a r a d j u s t e d ones b a s e d on l - p a r cMcula t ions . The a b o v e p o t e n t i ' f l is i m p o r t a n t b e c a u s e i t c o r r e s p o n d s to a o n e - b o s o n e x c h a n g e n m d e l wh ich is d o m i n a t e d b y ~ e x c h a n g e for t i le a t t r a c t i v e t ,d l of I ~ A w i t h mo = 3m~, wh ich seems a r e a s o n a b l e vah tc on t h e bas i s of t he ~mMysis of t h e dg'-,N ) d a t a . gAAo is t h e A A ~ coup l ing c o n s t a n t . ~ho r e l a t i o n (2a) b e t w e e n IVAA a n d .qAA~ is g iven b y

/

(=>a) B. W. D(,wxs and It. II. N. I'I[1LLII'S: N l t o ~ ; o C i m e n t o , 33, 137 (1964); 36, 120 (1965).

R I A S S U N T O (*)

8i g studiato molto det tagl iatamente il AA-ipernueleo ~HeAx con un modello a tre eorpi a-A-A, scrvendosi dei risulLati o t tcnut i dal mod(,llo a-e-A-A per il l~ Oltre a eMeolare l 'cnergia di legame del GHeAa, si ~ anehe s tudiata la dipendenza della

(*) T r a d u z i o n c a c a r a de l la R e d a z l o n c .

534 s. ALI and A. R. BODMER

formt~ dei var i potenzial i A-A (compresi i til0i a nocciolo d u r o c pu ramcn te a t t r a t t iv i ) da]l,% hmghezza di scat ter ing, range effet t ivo eco. Si t r o w che diffcrenti potenzia l i A-A che h~mno forms diverse ma lo stesso range intr inseeo sono con mol ta approssimazione cqu iwden t i per i AA-ipernuclei . I nostr i r i sul ta t i suggeriscono che l ' in te rpre taz iom; a l t e r n a t i w del l ' cvento ripor~ato da Danysz et al. come ~Bcaa ( tu t te lc analisi tecniche di qucsto cvento sono sta~e basatc sinora sul l ' in terpre taz ionc come ~~ ) merit~t s t r ia considcrazione. Si disctttono anche lc possibili implicazioni dei nostr i r i su l | a t i sul range del l ' interazionc A-~ ~

AA-runepaapo ~HcAA.

Pe3mMe (*). - - l/IclIOJIb3yil pe3yabTaTbI, noayqenHbIe n3 ~-~-A-A MO~e~t4 )laSt ~~ M~t nO~po6ao nccae~oBaar~ AA-ranepv~po 6HeAA c IIOMOlI~blO TpexaacTaqaox~ a-A-A MoAeJ1H. I-IOMHMO BblqHc31eHl4~t 3HepFHH CB~IBH 6~CAA ~ TaK~e H3yqeHl, l 3aBI~CHMOGTb d~OpMbt ~:I~ pa3naqimlx AA-nOTem~naSIOB (BgaIOqa~i H TBep~oe s~po n THI'IbI xfkICTOFO ~pnTaxenrIa) OT ~ann~I paccezrm~, 3qbqbeKTrlBltOR 06aacTrt rt T.~. O6HapymeHo, a~o pa3nnqrIsie AA-nOTeHttgaYmI, nMeiou/He pa3JIHqUbIe qbOpMb~, He O~HHaI<OByR) Bi~yTperm~o~o o6~aCTb, IIOqTH 3IgBHBaIIeHTHbI ]UDt AA-runepa~ep. Harem pe3ynbTaTbI npe~irioaaratoT, qTO aJIbTepHaTHBHa~ i4HTepnpeTa~/r~fl C06blT~4fl, yKa3aHHOrO ~[~aHbe3eM 14 )lp., TaK ~re KaK ~BeAA (BCe TeopeTttqCCKHe atta~Lq3bI 3TOre CO61MTH~I ~O CHX Hop OCItOBblBaYLI4Cb Ha X~ nHTepnpeTat~r~n) 3ac~ymnBaeT BHHMaTeJIbttO1-O pacCMOTpenrla. TaK~re 06cy~r)la- tOTC~I BO3MO~Gtble BbtBO~bl I43 HalHHX pe3yJIbTaTOB OTHOCriTeYlbHO o6aac lH A-A ~ B 3 aHMO)~el~CTB!4.q.

(*) HepeeeOeno peOaKque(t,