Volume 24B, number 7 P H Y S I C S L E T T E R S 3 April 1967

6 l 0 THE AA H Y P E R N U C L E I aaHe AND A Be AND

THE A-A AND A-N I N T E R A C T I O N S *

S. ALl ** International Centre f o r Theoretical Physics , Trieste, Italy

and

A. R. B O D M E t t A,'gonne National Laboratory, Argonne, Illinois

and Department qf Physics, University of Illinois, Chicago, Illinois

Received 16 January 1967

6 10 The AA hypernuclei AAtIe and AABe are analyzed with var ious A-A potent ia ls and with two different ranges for the A-N potential . Values of coupling constants a re obtained f rom the A-A potentials: in par t icu la r , val- ues of g2a(yare obtained which are compatible with the values o fg2Nc r obtained from the N-N data.

R e c e n t l y , P r o w s e [1] h a s i d e n t i f i e d an e v e n t a s aA6He wi th a s e p a r a t i o n e n e r g y BaA = 10.8 ± ± 0 . 6 M e V f o r b o t h A ' s . Wi th BA (5He) = 3.04 i ± 0.03 M e V [ 2 ] , t h i s g i v e s A B A a = BAa - 2B a = = 4 .7 =k 0.06 MeV. The e a r l i e r doub le h y p e r n u - c l e u s ( D a n y s z e t a l . [3]) w as i d e n t i f i e d a s m o s t p r o b a b l y 10Be , bu t p o s s i b l e 11 AABe. F o r the f o r m - e r , one hasBA.~ = 1 7 . 2 ± 0.5 MeV, z~Ba^ = 4 . 7 ± ± 0.6 M e V wi th BA(9Be ) = 6 . 2 4 ± 0.25 MeV [2].

a6He m a y to a good a p p r o x i m a t i o n be t r e a t e d a s an c~ + 2A s y s t e m [5-7] . T h i s h a s b e e n s h o w n by Tang and H e r n d o n w i th a s i x - b o d y (2p +2n +2A) c a l c u l a t i o n [4]. T h u s , the c ~ - p a r t i c l e c o r e is on ly s l i g h t l y d i s t o r t e d by the A ' s . R e c e n t l y , T a n g and H e r n d o n [8] u s e d t h r e e - b o d y c a l c u l a t i o n s to o b t a i n the s t r e n g t h of the A-A p o t e n t i a l ~ A f r o m BaA (A6He). They u s e d a VAA wi th a h a r d - c o r e r a d i u s r c = 0.4 fm and a n e x p o n e n t i a l t a i l w i th a r a n g e s u c h t h a t the i n t r i n s i c r a n g e of VAa is b = 1.5 fm . They u s e d two ~ - A p o t e n t i a l s VaA c o r r e s p o n d i n g to G a u s s i a n A - N p o t e n t i a l s of i n - t r i n s i c r a n g e s b a n = 1.5 f m a n d 0.7 fm.

C a l c u l a t i o n s of BAA(A6He) a s a f u n c t i o n of WAA h a v e b e e n m a d e by Al l and B o d m e r [9] f o r

{ oo, r < T C

Vxa= - W A A e - t ~ r / ~ r , r > / rc ,

* Work pe r fo rmed par t ia l ly under the auspices of the U. S. Atomic Energy Commiss ion .

** On leave of absence f rom the Atomic Energy Centre, Dacca, Eas t Pakis tan .

w i th r c = 0, 0 .42, and 0.49 fm and s e v e r a l v a l u e s of /x. S o m e of the r e s u l t s a r e s h o w n in t a b l e 1.

C a l c u l a t i o n s w e r e a l s o m a d e f o r the t w o - p i o n - e x c h a n g e p o t e n t i a l [7, 10], w i th r c = 0.42 and 0.49 fm; the a t t r a c t i v e t a i l b e i n g p r o p o r t i o n a l to f A 4 , w h e r e f A E is the p s e u d o - v e c t o r A)Z~ c o u p l -

ing c o n s t a n t . F o r r c = 0.42 f m the t w o - p i o n - e x - c h a n g e p o t e n t i a l h a s b = 1.5 fm - the s a m e as a Yukawa p o t e n t i a l w i th r c = 0.42 fm and p -1 = = 0.22 f m = (6.1 m ~ ) - l . The t h r e e - b o d y m e t h o d d e v e l o p e d e a r l i e r [11], a n d a l r e a d y a p p l i e d to AA h y p e r n u c l e i in r e f . 7, was used . M o s t l y a p u r e l y a t t r a c t i v e A-N Yukawa p o t e n t i a l VAN wi th p ~ =

= 0.7 fm (baN =1.5 fm) was u sed . F o r VAa wi th r c = 0, /~-1 = 0.7 f m a l s o /x~ 1 = 0.4 f m (baN =

=0 .85 fro) was c o n s i d e r e d . (The s h a p e of Vc~ a i s o b t a i n e d by fo ld ing VAN in to the c~ -pa r t i c l e d e n - s i t y , and the s t r e n g t h i s t h e n f ixed by BA(5He). )

The r e a s o n f o r c o n s i d e r i n g v a l u e s tt~ 1 < 0.7 f m i s t ha t ( for a g i v e n a - p a r t i c l e s i z e ) the r a n g e of V a a i s e x p e c t e d to be d e t e r m i n e d p r i n c i p a l l y by the r a n g e of the a t t r a c t i v e tail of VaN [12 ,13] . H o w e v e r , t h i s r a n g e i s e x p e c t e d to be l e s s t h a n 0.7 f m on t h e o r e t i c a l g r o u n d s - e i t h e r the two- p i o n - e x c h a n g e p o t e n t i a l o r a o n e - b o s o n - e x c h a n g e model [14]. Thus witha(r meson (I=0, J=O +) dominating, and with mcr ~ 3m~ (consistent with analyses of the N-N data) one has tz ~1 ~ 0.47 fm $. A range considerably shorter than 0.7 fm receives

See footnote next page.

343

Table 1 Results for A-A potentials for BAA (A6He) = 10.8 ± 0.6 MeV

Shape Yuk. Yuk. Yuk. Yuk. 2 p i o n expon, expon. 2 pion exch. b) b) exch. c) Yuk. Yuk.

Potent ia l -shape pa ramete r s

b (fm) 0.99 1.484 1.484 1.5 1.5 1.5 1.5 1.5 2.11 2.67 ~-1 (fm) 0.47 0.7 0.7 0.22 0.47 0.7 r c (fro) 0 0 0 0.42 0.42 0.4 0.42 0.42 0.42 0.42 bA~ l (fro) 1.5 1.5 0.85 1.5 1.5 1.5 0.7 0.7 1.5 1.5 /.tA~ q (fro) a) 0.7 0.7 0.4 0.7 0.7 "0.7" "0.33" "0.33" 0.7 0.7

WAA (MeV) 169 71.8 ±15 :L 6.95

+0.33 +0.41 -aAA (fm) 1.2_0.25 1.56 0.31

1.74 2.71 7~OA A (fm) ±0.17 ±9.28

0.624 0.6 SAA :L 0.057 ~ 0.055

AbAA( I0 Be)(MeV) 4.0 i0 .6

AB~ (~0 B+)(MeV) 5.95 ±0.8

Resul ts of calculations

62.5 20600 fA~=0.2725 1049 1003 fAE=0.27 1100 315 • 6.9 ±600 ±0.002 ±34 ±32 ±0.002 ±50 ±17

+0.31 _~+0 4 +0.4 1.84 1.40 1.40 +1 1 1.16_0.24 1 .~_0:3 1.8_0.3 ±0.37 ±0.26 ±0.26 3"17-0:7 "~='~-1~+2"2.3

3.14 2.26 2.35 2.24 2.50 2.67 3.0 3.55 ±0.38 ±0.18 ±0.2 ±0.16 =e0.21 ±0.25 ±.0.25 ±0.3

0.52 0.823 0.837 0.812 0.776 0.805 0.8 0.82 ±0.055 ±0.025 ±0.025 ±0.026 ±0.025 ±0.025 ±0.04 ±0.045

3.35 4.1 3.45 4.4 +0.6 ±0.6 ±0.6 ±0.6

5.35 6.15 6.3 5.45 6.25 ±0.8 ±0.8 ±0.6 ±0.8 ±0.8

a) Values in quotes are the Yukawa range which gives the same bAN. b) Resul ts f rom ref . 8. c) Resul ts obtained by adjusting f A ~ to give the same value of aAA as is calculated f rom the exponential potential

with b AN = 0.7 fm.

t e n t a t i v e c o n f i r m a t i o n f r o m c o n s i d e r a t i o n of 9Be [13, 16], and 1~C [16].

The r e s u l t s f o r the t h r e e h a r d - c o r e p o t e n t i a l s ( exponen t i a l , Yukawa, and t w o - p i o n - e x c h a n g e ) w i th b = 1.5 fm a r e in v e r y s a t i s f a c t o r y a g r e e - m e n t . F o r a g iven r c (=0.42 fm) , the s c a t t e r i n g l eng th laAA/ i n c r e a s e s wi th ~ - 1 , whi le the w e l l - d e p t h p a r a m e t e r SAA r e m a i n s about the s a m e . In

-1 p a r t i c u l a r , f o r P 3 r = 0.47 fm (b =2.11 fm) a c o n - s i d e r a b l y l a r g e r va lue of laAAI i s o b t a i n e d than f o r b =1.5 fro. F o r a f i xed va lue of g , bo th [aAA I and SAA i n c r e a s e wi th r c. -1

The only c a l c u l a t i o n s f o r PAN < 0.7 fm a r e -1

t h o s e fo r PAN = 0.4 fm fo r a Yukawa p o t e n t i a l w i t h r c = 0 , g -1 = 0.7 fm and t h o s e f o r bAN = = 0.7 f m fo r the e x p o n e n t i a l p o t e n t i a l of Tang and He rndon . T h i s l a t t e r i s , h o w e v e r , e f fec t±-

Among the l igher J= 0 mesons , the 77 and K can also contribute to the tail of VAN and the 7/ to the tail of VAA . A further contribution to both VAN and VAA may come from heavier J = 0 + mesons , e . g . , the s meson with rn s ~ 700 MeV [15]. Since the ~, s, and K have m a s s e s grea te r than m6, their inclusion can only read to a shor te r range.

ve ly e q u i v a l e n t , wi th r c = 0.42 f m , to the Yukawa p o t e n t i a l wi th ~ - 1 = 0.22 fm and a l s o to the t w o - p i o n - e x c h a n g e p o t e n t i a l . U s i n g th i s e q u i v a l e n c e and tak ing ~ = 0.4 f m , one o b t a i n s W =

AA

= (19 900± 600) MeV fo r the f o r m e r a n d f A ~ = -1

= 0.27 ± 0.002 fo r the l a t t e r . F o r ~AN = 0.4 fm , -1

and w i t h r c = 0.42 fro, ~ 3 s = 0.47 f m , one e s t i - m a t e s WAA ~ 1035 + 50 MeV, and a x A ~ -2 .35 =~ i 0.6 fro, rOA A ~ 3.35 ± 0.3 fm , and SAA ~ 0.755 ± ± 0.36. The s t r e n g t h WAA d e c r e a s e s wi th g ~ ,

-1 b e c a u s e a d e c r e a s e in ~tAN r e d u c e s the r a n g e of VC~A; and th i s p u l l s in the ~ - A wave f u n c t i o n s , t hus a l lowing the A ' s to i n t e r a c t m o r e e f f e c t i v e l y . The r a n g e of VC~A can a l s o be c h a n g e d by chang ing the ( ~ - p a r t i c l e s i z e . The r e s u l t s s h o w n a r e f o r an r . m . s . r a d i u s Rc~ = 1.44 fro. Some c a l c u l a t i o n s [9] w e r e a l so m a d e f o r R a = 1.54 fm and U~ 1 = = 0.7 fm. Such an i n c r e a s e c o r r e s p o n d s to 5 R / R = = 0.049 and changesBAA by only 5BAA = 0.03 MeV, w h e r e R is the r . m. s . r a d i u s of V~A. Any u n c e r - t a in ty in Rff is thus qu i t e u n i m p o r t a n t . Chang ing ~ ] 1 f r o m 0.4 to 0.7 fm f o r Rc~ = 1.44 fm is e q u i -

v a l e n t to the l a r g e ch an g e 5 R / R = 0.203 (for a

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Volume 24B, number 7 P H Y S I C S L E T T E R S 3 April 1967

f i x e d p 2 1 ) and l e a d s to 6BAA = 0.8 MeV. This e q u i v a l e n t change was o b t a i n e d wi th G a u s s i a n VAN of the s a m e i n t r i n s i c r a n g e s as the Yukawa

p o t e n t i a l s c o n s i d e r e d . If the cr m e s o n d o m i n a t e s VAA *l , then [14]

(1 m2~-~exp(-mar)o ! = . . . . . , 7/" ~ T" C VAA (r) "(mc~)g2~ m r

4m ~,/ A

w h e r e gcr : gAACr is the AAcr coup l ing c o n s t a n t . F o r bAN = 1.5 fm and r c = 0.42 f m , one ob -

2 t a i n s gcr = 2.75 ± 0.125 f o r m a = 3ran; i n t e r p o l a - t ion of our r e s u l t s fo r ~'c = 0.42 fm g i v e s g2 =

2 = 6.32 ± 0,25 f o r mcr = 4m n and gcr = 13.6 ± 0.46 f o r mcr =5 ran ; f o r m e r =2mTr one h a s g 2 = 1.16 ± ±20.063. F o r the s m a l l e r va lue s of bAN one has gcr ~ 2 . 5 6 ± 0.125 f o r ; n c y = 3 m T r . F o r r c = 0 . 4 9 f r o , b~N = 1.5 fm and tncr = 3ran one ob t a in s WaA

2 ~ 3.42 ± 0.16. The va lue s of 1370 ± 65 MeV, gcr 2 2

gcr s e e m c o m p a t b i l e $$ wi th t h o s e of gNNCr o b t a i n e d

f r o m the N-N i n t e r a c t i o n [17-19] , and a r e thus c o n s i s t e n t wi th a ~ which is a SU(3) s i n g l e t , un i - v e r s a l l y coup led to the b a r y o n s . F o r the t w o - p i o n - e x c h a n g e p o t e n t i a l , f A N is v e r y p r e c i s e l y d e t e r -

$$ The U and s mesons also can contribute to the tail of V,/~A. Analysis of N-N data [17-19] indicates that 2 gNNu is quite small and consistent with zero. How-

ever, for VXA it is gAAU which is relevant and x2 =

= (g;~Azl/gNNU) z eouldbe very different from unity. Thus 8U(3) predic ts x 2 = 4(1 - 002/(40~- 1) 2, where ot = F/(F+D) and one could have very large xa; for the value o~ = 0.4 predicted by SU(6), x 2 = 4. P e r - haps the s imi lar s t rengths of VxA and I%x N (both 1S o and 3S1) may be taken as a fur ther indication that the U is unimportant for both Vx\ and VxN , the relevant quantity for V,u N being gNNT~xA??" This is especial ly suggestive in view of the x4-eak'coupling of the K; thus, for VaN(1So), the square of the pseudoveetor coupling constant is f2KN =

2 _2 = gA~N(rnK/4m raN) ~ 0.245 ~: 0.05 [20] and is di- 2 f rect ly comparable with g = g~ g i . . . . . . . 2 Cr ,~; 2~0" NNO"

=~v'Cr= fffK = LinK- (ma-mN) ]=" When both the s

and (y are included [181, analysis of N-N data gives g2Nc r ~ 1.7, g~qNs ~ 9 ± 4. Because m s is large,

2 gNNs is quite poorly determined. Ear l i e r analysis 2 ~ 3 for mcr --~ 3mTr and [17] without the s gave gNNCr

2 g N N ~ 9 for m(y = 4rnTr. A recent analysis [19], without the s, gives the values g2NNCr= 2.33 to 2.79 for m(y = 3mrr to 3.4mTr. Thus, our values ofg2Aff are

2 quite compatible with those of gNN(~ (especially if allowance is made for the fact that inclusion of the s will reduce 2 gNNcr )"

m i n e d f o r a g iven r c ( b e c a u s e of the h a r d c o r e and the s h o r t a t t r a c t i v e ta i l p r o p o r t i o n a l to f ~ 2 ) . Indeed , wi th r c = 0.42 fm one f inds aAA = 0, 5BAA ~ 0 f o r f A 2 = 0 . 2 3 andaAa =-°o ( i . e . , j u s t a AA bound s t a t e ) f o r f A ~ = 0.285. H o w e v e r , f A 2 is qui te s e n s i t i v e to r c [ 2 0 ] ; t h u s f A ~ = 0.31 f o r r c = 0.49 fm.

F o r 12Be , one m u s t i nc lude d i s t o r t i o n of the 8Be c o r e by the A ' s , a s done in r e f s . 6 and 7 wi th a 2a + 2A mode l . The t r i a l wave f u n c t i o n s u s e d in r e f s . 6 and 7 a r e s o m e w h a t d i f f e r e n t , but the a g r e e m e n t is v e r y good. One may w r i t e [7]

ABaa :AbAA + [Ec~a(RA) + EAA] ,

w h e r e AbAA i s the value ob ta ined wi th a t h r e e - body c a l c u l a t i o n f o r a 8Be c o r e of the s a m e r . m . s , r a d i u s R A a s i n 9 B e , Eaa(R A) i s the

c~-a e n e r g y in 9Be, and EAA i s the additional c o r e d i s t o r t i o n e n e r g y due to the s e c o n d A. F o r a g iven Va~ and &BAA, the d i s t o r t i o n e n e r g y E a a + EAA i s a p p r o x i m a t e l y i n d e p e n d e n t of the s h a p e of VAA [7]. F o r Vua which g ive equa l ly good f i t s to the a - a s c a t t e r i n g the r e s u l t s a r e the s a m e wi th in about 0.25 MeV.

10 The r e s u l t s f o r AABe, wi th the s t r e n g t h s WAA o b t a i n e d f r o m h6He and wi th the v a l u e s of E a a + + _~Aa f r o m re f . 7, a r e shown in the tab le . F o r tAAN : 0.7 fm one ob ta ins too m u c h b ind ing f o r 10 A,~Be. Our r e s u l t s c o n f i r m Tang and H e r n d o n ' s c o n j e c t u r e tha t a s h o r t e r A-N r a n g e g ives a s m a l l e r ABAA(IOBe) m o r e c o n s i s t e n t wi th e x p e r i -

men t . Thus , f o r p ~ l = 0.4 fm the v a l u e s of ABAA(10Be) a r e r e a d i l y r e c o n c i l a b l e wi th the e x - p e r i m e n t a l one. The c a l c u l a t i o n s in re f . 7 w e r e m a d e wi th #~1 = 0 . 7 fm a n d R x =2 .33 fro; how- e v e r , the r e s u l t s f o r AbAA w e r e i n c r e a s e d by about 0.15 MeV to c o r r e c t fo r u se of p ~ l = 0.4fro. Thus the s e n s i t i v i t y wi th wh ich AbAA d e p e n d s on R A was i n v e s t i g a t e d f o r ~t~ 1 = 0.7 fm by d e c r e a s - ing RA to 2.25 fm. Such a change c o r r e s p o n d s to 5R/R = 0.023, and g i v e s 6B = 0.04 MeV, w h e r e R is the r . m. s . r a d i u s of the A - c o r e po ten t i a l . H o w e v e r , the change /x~ 1 = 0.4 to 0.7 fm is e q u i - v a l en t to 5R/R = 0.082. This j u s t i f i e s our c o r r e c - t ion f o r Ab~x. The d e p e n d e n c e o f AbxA on p - 1 is

• ' ~ A N

c o n s i d e r a b l y s m a l l e r f o r 10 6 b e - AABe than f o r aAHe c a u s e the 8Be c o r e is c o n s i d e r a b l y l a r g e r than 4He. The change ~t~ 1 = 0.7 to 0.4 fm i n c r e a s e s Eota by about 0.1 MeV [11,13] .

Thus if the e v e n t of Dm~ysz et al, [3] i s ]0Be A A

then one has f u r t h e r s u p p o r t f o r a s h o r t r a n g e of VaN, and h e n c e a l s o of VAA, c o n s i s t e n t wi th t h e o - r e t i c a l e x p e c t a t i o n s . Such a s h o r t r a n g e then t e n - t a t i v e l y i m p l i e s a r e p u l s i v e c o r e in l(~ N [12]. If

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Volume 24B, number 7 P H Y S I C S L E T T E R S

11 the e v e n t of r e f . 3 s h o u l d b e AABe, t h e n t h i s wou ld 5. b e c o n s i s t e n t wi th b o t h v a l u e s of -1 T h u s the ttAN.

6. e x p e r i m e n t a l v a l u e of ~BA,(A1A1Be) would b e abou t the s a m e as f o r AABel0 [21], bu t the d i s t o r t i o n e n - 7. e r g y i s e x p e c t e d to b e on ly s o m e w h a t m o r e t h a n

10 8. ha l f of t h a t f o r AaBe [7]. 5 B ( ) } B e ) would t h e n AA . , ,~ 9 .

be i n t e r m e d i a t e b e t w e e n the c a l c u l a t e d v a l u e s of A/ )AA(~Be ) ~ - and ABAA(~0Be) and t hus in a g r e e m e n t 10.

w i th t he e x p e r i m e n t a l v a l u e f o r b o t h ~t-1 = 0.4 AN 11. and 0.7 fro.

The d e t e r m i n a t i o n of BA, f o r a h e a v i e r AA 12. h y p e r n u c l e u s w h o s e c o r e i s f a i r l y r i g i d , s u c h as 13. 14C cou ld g ive r a t h e r d e f i n i t e i n f o r m a t i o n a b o u t AA ' 14.

tt-1-'AN T h u s fo r 14C the v a l u e of ABA, would , to AA ' 15.

a good a p p r o x i m a t i o n , be t h a t o b t a i n e d by u s e of a c o r e + 2A m o d e l and would be c l o s e to the v a l - u e s AbAA(~}°Be). Wi th the r e s u l t s f o r , 6 H e , the

16. v a l u e s p r e d i c t e d fo r hBA:~ (1A4C) would t h e n d i f f e r by abou t 0,7 M e V f o r # ] 1 = 0.7 fm and 0.4 fm. 17.

One of u s (S. Al i ) w i s h e s to a c k n o w l e d g e the k i n d h o s p i t a l i t y of P r o f e s s o r s A. S a l a m and P. B u d i n i and of the IAEA at the I n t e r n a t i o n a l C e n t r e f o r T h e o r e t i c a l P h y s i c s , T r i e s t e , I ta ly . 18.

R e f e r e n c e s 19. 1. D.J . Prowse, Phys. Rev. Le t t e r s 17 (1966) 782. 2. C .Mayeur et a l . , Nuovo Cimento 43 (1966) 180. 20. 3. M.Danysz et a l . , Nuclear Phys. 49 (1963) 121. 4. Y . C . T a n g a n d R . C . H e r n d o n , Phys. Rev. Le t te rs 21.

14 (1965) 991.

3 April 1967

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