Weak non-mesonic decays of Λ-hypernuclei

Weak Non-Mesonic Decays of A-Hypernuclei JOSEPH COHEN* Department of Physics, Case Western Reserve Umverstty, Cleveland, Ohio 44106, U S A and Instttute of Nuclear and Pamcle Phystcs, Department of Physw~, Umverslty of Vtrgmta, Charlottesvdle, Vtrgmta 22901, U S A

ABSTRACT We review the current status of studies of weak decays of A-hypernuclei. Both experimental and theoretical aspects are reviewed, with emphasis on the weak non-mesonic decays. Experimental results obtained during the last thirty years are described and discussed in detail. The current theoretical understanding of the subject is analysed, following the various approaches in the pertinent literature. For the most part, available treatments are based on the two-body A+N~N+N weak interaction as the underlying non-mesonlc decay mechanism. Starting with the phenomenological approach of Dalitz et al., and continuing with a description of microscopic models suggested for the A+N~N+N interaction, we review over thirty years of theoretical efforts. The implications of recent studies of weak decays of hypernuclei at Brookhaven where data for, e.g., I~C and ~He have been obtained, are discussed. The need for new and improved measurements is pointed out. Future prospects and selected opportunities for new and further studies are described.

KEYWORDS I Hypernuclei (A=3 to 238); A-hyperon; weak nonleptonic interactions; AI== rule; hypernuelear lifetimes; branching ratios for weak nonleptonle hyper~n and hypernuelear decays: AN~NN; A~N~; non-mesonlc decay mechanisms: meson exchanges; quark models; parity; CEBAF; advanced hadronic facilities.

*Present address: Institute of Nuclear and Particle Physics, Department Physics, University of Virginia, Charlottesville, Virginia 22901, U.S.A.

139

of

140

J. Cohen Table of Contents

1.

2.

3.

4.

Introduction

141

1 . 1 H y p e r n u c l e a r p h y s i c s : Why and how 1.2 Weak i n t e r a c t i o n s in A-hyperons and h y p e r n u c l e i

141 142

Review of c h a r a c t e r i s t i c s and e x p e r i m e n t a l d a t a f o r h y p e r n u c l e a r weak decays

144

2.1 The t o t a l h y p e r n u c l e a r l t f t i m e : P i o n e e r i n g e x p e r i m e n t s 2.2 S p e c i f i c h y p e r n u c l e a r decay modes: E a r l y measurements 2.3 Hypernuclear l i f e t i m e s and decay modes: Recent r e s u l t s

144 147 151

Theory of non-mesonic weak decays of A - h y p e r n u c l e l

161 161

3.1 I n t r o d u c t i o n : An overview of decay mechanisms 3.2 Phenomenology of the A-~'~ n o n l e p t o n i c decay mode 3.3 Non-mesonic decays of l i g h t h y p e r n u c l e i : The phenomenological Approach 3.4 Microscopic models f o r the non-mesonic decay: The one pion exchange mechanism 3.4.1 Why one plon exchange? 3 . 4 . 2 The t r a n s i t i o n p o t e n t i a l 3 . 4 . 3 The t r a n s i t i o n r a t e s in a Fermi gas model; g e n e r a l p r o p e r t i e s of the t r a n s i t i o n a m p l i t u d e 3.4.4 Finite-hypernuclear treatment 3 . 4 . 5 The r o l e of s h o r t - r a n g e two-body c o r r e l a t i o n s , d i s t o r t i o n e f f e c t s and p l o n - b a r y o n - b a r y o n v e r t e x f u n c t i o n 3.5 E l a b o r a t e m i c r o s c o p i c models f o r non-mesonic decay 3.5.1 I n t r o d u c t i o n 3 . 5 . 2 A d d i t i o n a l meson exchanges 3 . 5 . 3 Hybrid quark-hadron model 3 . 5 . 4 Many-body RPA ( r i n g diagrams) r e n o r m a l i z a t i o n e f f e c t s in one pion exchange 3.6 Numerical r e s u l t s and comparison with e x p e r i m e n t a l d a t a

2O4 2O7

Discussion,

222

future prospects and conclusions

161 171 177 177 177 181 185 189 192 192 192 2OO

Decays of A-Hypernuclei I.

141

INTRODUCTION

The interest in hypernuclear interaction issues related to introductory chapter.

physics in general, and particularly in weakhyperons and hypernuclei, is discussed in this

1.1 Hypernuclear physics: Why and how Hypernuclear physics is currently an important and exciting area of nuclear science. (I) An interesting bound nuclear system with strangeness S = -1 is the A-hypernucleus (^A), in which a A-hyperon replaces one of the nucleons. This system is long llved (~lO-1°sec) and provides a variety of fascinating nuclear phenomena. (For reference we note that the free A mass is 1115.6 MeV, and its lifetime is about 2.63x10 -I° sec.) Hypernuclei are interesting because they add another flavor (strangeness) to the traditional electromagnetic, weak and hadronlc probes of the nuclear system. An important goal of bypernuclear physics is the study of the baryon octet, or the u, d and s quark domain, thereby providing an extension and generalization of traditional nucleon (or u and d quark) physics. Furthermore, the hyperon, whose additional quantum number distinguishes it from the rest of the nucleons in the hypernuclear system, is not Pauli excluded from filled nucleon orbitals, and can penetrate deep inside the nucleus. In other words, hypernuclei provide the possibility of differentiating probe interactions from those of the target constituents. We can use hypernuclei to explore such problems as the origin of the nuclear spln-orbit force, short range correlations, relativistic aspects of the many body nuclear dynamics and their extension to hypernuclei, the role of flavor symmetry and the chlral limit, extended models of the strong interaction, weak interactions in the nuclear medium, or possible modifications of baryon properties in the nuclear environment. Many of the phenomena involving strangeness can only be studied using bound hypernuclear systems, as there are no free hyperon targets/beams. Most of the currently available information on hypernuclei comes from the A(K-, ~-)^A* reaction studies at Brookhaven National Laboratory (BNL, N.Y., U.S.A.) and KEK (Japan), and in the past, also from CERN (Geneva, Switzerland). A relatively novel associated production mechanism, the (n+,K +) reaction, provides complementary hypernuclear information. Despite severe shortcomings at these existing facilities, there has been an impressive progress in our understanding of hypernuclei. Some recent highlights include: (i) Clear shell-model A-particle----neutron-hole states, with orbital angular momentum numbers i.=0-4, have been unravelled in (~+, K +) studies at 1.05 GeV/c; (2) (ii) An improved upper limit on the (small) A-nuclear spin-orbit interaction from the observation (3) of a v ray transition in ~Be (This is particularly significant in the context of the relativistic ~-~ or scalar + vector nuclear model, where it is difficult to obtain a vanishing spin-orbit interaction and still get a sufficient central attractive potential in order to bind the A-hyperon.); (iii) Weak-interaction hypernuclear decay modes (pionic and non-mesonic) have been observed: 14-5) these decays are of great interest, and will be discussed in great detail in this review; (iv) The total K+-nucleus scattering cross section currently poses an interesting puzzle: (6) the ratio of the total cross section of K+-carbon to K+-deuteron shows a discrepancy between theory and experiment, with the calculated cross section roughly 10Z-15% below the experimental one. [Although this last issue is not directly related to hyperons and hypernuclei, it is included here because it demonstrates the advantages of using strange probes to study the nuclear many-body system. Indeed, such an effect cannot be convincingly pointed out using pions. Moreover, the K +-nucleus dynamics is an important input in hypernuclear production studies via such reactions as (~I~), (e,e'ICt) or (p,IC~).]

142

J. Cohen

Three theoretical issues of interest include the study of few-body hypernuclei, the role of the Pauli principle in distinguishing between hadronic and quark degrees of freedom in many-body systems, and possible modifications of hadronic properties when embedded in the many-body medium. Thus, because the long range one-pion-exchange is absent from the AN interaction and the AN-EN coupling is strong, three-body ANN forces are predicted to play a crucial role in hypernuclei. ~7~ They are, indeed, essential to the binding of the hypertriton. Regarding the quark structure of hadrons, the view of the A as a baryon distinguishable from both the proton and neutron would allow it to occupy any slngle-particle state in the hypernuclear many-body system (including the ls state in a shell-model picture), while OCD pictures the A as a composite of u, d, and s quarks. In a certain OCD-motivated model of the many-body system based on total or partial deconfinement of the hadrons, one might anticipate that the u and d quarks would experience Paull blocking due to the other u and d quarks when the A is embedded deep within the hypernucleus, while the s-quark could occupy a Is level. This should lead to a distorted A-hyperon inside the nucleus. While existing nuclear data do not unequivocally demand a quark-level description of nuclei, the proposed system offers a unique probe for actually studying such effects. Hypernuclei could also provide, with a A-hyperon in a deep nuclear orbital, a clean probe of possible modifications of hadronic properties in the nuclear medium. ~s'9~ Other important issues we have not discussed include E-hypernuclei, multl-strange systems, or strange dibaryons (in particular the H). 1.2 Weak interactions in A-h~perons and hypernuclei A free A-hyperon decays mostly nonleptonic decay (Fig. 1)

into

pn-

(~64Z)

n~ °

(~36Z)

A ~

a

nucleon

and

a

pion

via the weak

(1)

with an approximate lifetime ~, = 2.63 x lO-1°sec. The energy released in the free decay is 0¢~r*'~ffi 37 Me¢~ and the corresponding c.m. momentum of the nucleon and pion is about 100 MeV/c. H y p e r n u c l e i i n t h e i r ground s t a t e s ( g . s . ) , when they a r e s t a b l e w i t h r e s p e c t to s t r o n g decay modes ( p a r t i c l e e m i s s i o n ) , a l s o decay v i a w e a k - i n t e r a c t i o n mechanisms. The s i t u a t i o n d e s c r i b e d above f o r the weak decay of the f r e e A-hyperon changes d r a m a t i c a l l y when the A i s embedded i n the n u c l e a r medium. I n the n u c l e u s , the A i s bound by some 10 MeV ( f o r medium n u c l e i ) to 25 MeV ( i n heavy h y p e r n u c l e i ) , so the phase space f o r the mesonic decay, Eq. ( 1 ) , i s g r e a t l y reduced. The f i n a l n u c l e o n produced i n the decay has a v e r y low momentum (< 100 MeW/c), and i s c o n s e q u e n t l y P a u l i blocked. The r e s u l t i s an even f u r t h e r s u p p r e s s i o n of the h y p e r n u c l e a r mesonic (A -~ N~) decay mode; more q u a n t i t a t i v e d i s c u s s i o n s w i l l be p r e s e n t e d i n s u b s e q u e n t s e c t i o n s . I t is necessary to invoke a model for a detailed theoretical calculation of the ratio of non-mesonic to mesonic decay rates. One such possible model is based on one pion exchange, as discussed in detail in Sec. 3.4. This measured quantity has not been extensively studied so far (see Ref. 114), and the available experimental results will be discussed theoretically in Secs. 2.2 and 2.3.

Decays of A-Hypernuclei

143

/

J

Fig. 1

The mesonic (nonleptonic) decay A ~ NR.

The nuclear medium affects the weak decay of the A-hyperon by introducing a new, non-mesonic decay mode. This decay is expected to proceed via (Fig. 2):

A + p -~ n + p,

(2a)

A + n -~ n + n.

(2b)

N

Fig. 2.

N

The nonmesonic weak decay mode A+N * N+N.

144

J. Cohen

This decay mode corresponds to an energy release of Q = 176 MeW, leaving each of the final nucleons with a momentum of approximately 417 MeV/c, and an energy of some 88 MeV (to arrive at this estimate we assume the A and the initial nucleon to be at rest and disregard the nucleon binding energy). This decay mode has a much larger phase space (relative to the mesonic one) and the outgoing nucleons are not Pauli blocked, hence is expected to dominate the weak decay process in medium and heavy hypernuclel. This interesting situation has been envisaged a long time ago by Cheston and Primakoff and by Ruderman and Karplus. ~t°) The non-mesonic, two-body mode (AN ~ NN) resembles the weak nucleon-nucleon interactlon--but with a large amount of additional new physics. The nonmesonic decays represent an interesting opportunity for the study of baryonbaryon weak interactions. Some information is available on the nucleon-nucleon (N-N) weak interaction from experiments such as nucleon scattering. The AN NN decay is richer and more interesting since both parity-conserving (PC) and parity-violating (PV) partial rates can be measured, whereas in the weak N-N case the strong interaction masks the PC signal of the weak interaction (in the AN ~ N N processes there is no possible contribution from the strong interaction since strangeness is not conserved). Detailed information on the non-mesonic decay modes would be a u s e f u l test of models of the weak i n t e r a c t i o n . U n l i k e the N-N c a s e , however, t h e r e i s no d a t a on the f r e e AN ~NN r e a c t i o n , and h y p e r n u c l e a r systems a r e the o n l y s o u r c e of i n f o r m a t i o n on t h i s p r o c e s s . This c o m p l i c a t e s the s t u d y of the weak n o n mesonic decays s i n c e , i n a d d i t i o n to the r e a c t i o n mechanism, one has to d e a l s i m u l t a n e o u s l y with c o m p l i c a t e d h y p e r n u c l e a r s t r u c t u r e e f f e c t s . 2. REVIEW OF CHARACTERISTICS AND EXPERIMENTAL DATA FOR HYPERNUCLEAR VEAK DECAYS There is a fair amount of experimental data on the hypernuclear weak decay modes. This body of data is growing continuously~ a number of new measurements have been completed recently or will be available soon. As we point out in the following, these future, improved experimental results will be extremely important in distinguishing between various theoretical models used in the description of the hypernuclear weak decay modes. A critical review of the pertinent data is thus called for, and will be presented in this chapter. 2.1 The total hypernuclear lifetimez Pioneering experiments Early experiments provided data on the total hypernuclear lifetimes, T, based mostly on the observation of an emitted negative plon. The significance of these pioneering measurements is in establishing some (crude) limits on the value of • (see Table 1). Data f o r l i g h t h y p e r n u c l e i , A < 5, have been f i r s t o b t a i n e d i n b u b b l e chamber e x p e r i m e n t s ~111 and e m u l s i o n works. ~121 Only the mesonic decay mode has been observed i n most c a s e s . For example, Keyes e t a l . (1970) (11~ have measured the l i f e t i m e (and o t h e r p r o p e r t i e s ) of the hypern'ucl"eus ~H i n an a n a l y s i s of helium bubble chamber pictures, obtained by exposing the chamber to a stopping K- beam of approximately 650 MeVlc at the Argonne ZGS (Zero Gradient Synchrotron). The

145

Decays of A-Hypernuclei TABLE 1.

Total l i f e t i m e s

Hypernucleus

~H #

z

(%

i n U n i t s of 10 - * ° s e e ) . *

References

Comments

1.23+0.2

Block e t a l .

(1962) (11)

0.95+0.2

Block e t a l .

(1963) (11)

1.05+0.2

Block e t a l .

(1964) (11)

2+o.6 " --0.3

0 6+o.5 • ---0.3

0 0+2.2 • --0.4

3 t.+0.2 " ---1.4

R5+1.27 "---1.05

~+o.4s • ---0.34

~R+°'35 • ----0.26

4 ^H

of h y p e r n u c l e t

1 ~+o.6 -" - - o . 3 +0.0

>_(0.6_0 "2 ) /,+1.9

Heltzer

et al. (11.

Fortney (11) Prem and Stelnberg (12) Kang et al. (~2) Phillips and Schneps 1.2) Keyes e t al. (11) Bohm et al. 112)

a

Crayton et al. (12) Ammar, Dunn and Holland (12) Fortney (11)

• --0.9

R+2.5 • --0.7

&+4.9 • v_l.

Kan g e t

al. (121

3

~+6.o • --1.0

6R+,.66 • -----1.07

~He

Prem and Steinberg (12)

Kang e t a l . (12)

a

Phillips

a

and Schneps (12.

2.0+0.8

Murphy e t a l . (~2)

1 ~+1.o

Ammar, Dunn and H o l l a n d (12)

" --0.4 +0.7

>_(0.9_0. 3 ) /.+1.9 • --0.5

R+I. s " ---0.6

~+z.s • --0.6

,,.o+o.6o " ----0.43

2.74:o: 6

b

Ammar, Dunn and H o l l a n d ~12. Prem and S t e i n b e r g (121 Kang e t a l . c12. Kang e t a l . (12.

b

Phillips

a,b

and Schneps (.2)

Bohm e t a l . (.2)

a

46

J. C o h e n

TABLE 1 (con't)

AeLi

>0.4

Phillips

16

0 RR +0"33 .... -0.26

Nield

Az

Note:

and S c h n e p s 1121

e t al. 113.

The free A lifetime is ~^ = 2.63xI0 -I° sec.

*See also Eqs.(6), (9)-(11), and Table 3 in the next sections. #See text for a discussion of the short lifetimes obtained for Affi3. ( a ) B a s e d o n l y on t h r e e - b o d y ( i n c l u d i n g a It- m e s o n ) f i n a l s t a t e ( s ) decay.

for

the

( b ) Combined 4'ASHe l i f e t i m e . (c) Effective mean lifetime for the mixture of ^1 6 0 and ^1 6 N.

production reaction was K-+ 4He -~ ~H + p + It-, followed by the mesonic decay modes A3H -~ II- + 3He, 1%.3H -~ It- + d + p, A.311 -~ It- + p + p + n. The most reliable (partial) lifetime from this experiment was obtained from the two body (3H -~ It3 . ~ A + He) decay events. The mean Ixfetime of the hypertriton was found to be • = (2.64+0".8. 4) x 10 -1° see using 27 two-body decay events only. A serious source of background was found in the multibody decays, and the combined two- and three-body decays gave a lower limit on ~ of (2.28+o°~) x I0 -~° see. A similar experiment has been carried out by Keyes, Sact'on, Wickens and Block (1973) ¢.I) yielding a similar vahle, x = (2.46~'462) x I0 -I° see; furthermore, the decay branching ratio Y(A3H -~ It- + 3Ile)/F(~B-~"all It- mesonic modes) has been estimated as 0.30 + 0.07, in agreement with values from earlier measurements. I n a n o t h e r e x p e r i m e n t , Prem a n d S t e i n b e r g c12) e x p o s e d a n e m u l s i o n s t a c k a t t h e B r o o k h a v e n AGS t o K- beams a t 2 . 3 GeV/c a n d 2 . 5 G e V / c , d e t e r m i n i n g l i f e t i m e s on the basis of identifying only the negative-pion d e c a y modes (~H ~ 3He + It- a n d H ~ 4He + ~ - ) f o r h y d r o g e n h y p e r n u c l e i , or t h e n- - p r o t o n - r e c o i l f o r ~He. nly very few events have been observed, resulting in large statistical errors. Phillips and Schneps (12), in a later experiment, exposed nuclear emulsions to a I.I GeV/c K- beam at the Lawrence Radiation Laboratory Bevatron. They have considered only those decay modes involving a It- and no neutral particle (3'~H R- + 3 4' He, ~ 4'^H ~ R- + p + 2 3' H, 4 5' He ~ It- + p + 3 4' He, a n d &H ~ It- + H 2 ^ A + H), a n d t h e t o t a l n u m b e r o f r e p o r t e d d e c a y e v e n t s was much l a r g e r t h a n t h a t o f Prem and S t e i n b e r g ( a b o u t 400 l i g h t h y p e r n u c l e i w e r e u n i q u e l y i d e n t i f i e d ) . Phillips and Schneps also reported the observation of the heavier hypernucleus ~ L i , and p r o v i d e d a l o w e r l i m i t on i t s l i f e t i m e (see Table 1). These ('1'121 measurements suffered from a rather low precision, relatively poor statistics, and difficulties with unique identifications. Consequently, large errors were present. Moreover, the short lifetimes(and small error bars) reported from emulsion works for Affi3 were probably due to effects of Coulomb dissociationof the weakly bound A from the deutron core as .~H passed through the emulsion, as discussed by Davis and Pniewski. ~ This effect is expected to be negligible for the helium bubble chamber, where the short lifetimes reported here from Block et al. probably resulted from problems with the data analysis, as pointed out by Keyes et al....~(lt]" - -

Decays of A-Hypernuclei

147

A d i f f e r e n t e x p e r i m e n t f o r m e a s u r i n g the l i f e t i m e o f a h e a v i e r h y p e r n u c l e u s has been carried out ~13~ at the Lawrence Berkely Laboratory Bevatron using heavy ion beams. Mass A=16 hypernuclei were produced by 2.1 GeV/nucleon a~O ions incident on a polyethylene target. Associated with the hypernuclel is a production of low energy K + mesons, which provided an electronic signature by their delayed decay after stopping. Production reactions with two body final states (such as 160 + p + i~0 + K +) were ruled out on the basis of small cross sections obtained in the reaction p+ 12C ~ K + + I~C; reactions with three-body final states are much more likely, since the minimum momentum transfer to the hypernucleus is much lower in this case. With a scanning procedure where only high-Z ions are considered, the assumed important reactions are 160 ÷ p ~ 1~0 + n + K + and *60 + n ~ X~N + n + K +. The decay length of the hypernucleus can be accurately determined by reconstructing its recoil track with spark chambers. The analysis of 22 events yielded a mean lifetime of (0.88 +g']~)x I0 -I° sec for the mixture of x~0 and I~N produced and detected in ~fils experiment. However, estimating sy~tematica~errors in this experiment is difficult. A graphical representation of x against the mass number A will be given in Fig. 3 at the end of this chapter.

2.2 Specific hypernuclear decay modes: Early measurements The pioneering measurements described in the previous section dealt mostly with lifetimes related to the R- mesonic decay mode. It has long been realized that measurements of all possible specific decay modes, and pertinent ratios t h e r e o f , would p r o v i d e i n v a l u a b l e d a t a r e q u i r e d f o r the u n d e r s t a n d i n g o f the decay p r o c e s s . While p o s t p o n i n g a more d e t a i l e d d i s c u s s i o n of the t h e o r e t i c a l i s s u e s to a l a t e r c h a p t e r , we w i l l d i s c u s s e a r l y measurements f o r s p e c i f i c h y p e r n u c l e a r decay modes i n t h i s s e c t i o n .

E a r l y on, ~141 i t has been e x p e r i m e n t a l l y v e r i f i e d i n e m u l s i o n measurements t h a t the nonmesonic decay mode i s r a r e f o r hydrogen h y p e r n u c l e i (~H a n d ~ H ) , but becomes comparable to the mesonic decay mode f o r h e l i u m h y p e r n u c l e i . Mbreover, the nonmesonic mode was found to s t r o n g l y p r e d o m i n a t e f o r h y p e r - n u c l e i w i t h Z > 3. The observed r a t i o of n o n - R - - m e s o n i c to R--mesonic decays f o r Z = 1, 2, 3 and Z > 3 h y p e r n u c l e i r e p o r t e d i n Ref. ( 1 4 ) , a r e shown i n T a b l e 2. F l e t c h e r e t a l . 115) observed mesonic as w e l l as n o n - m e s o n i c decays of a l a r g e v a r i e t y o T hypernuclei in emulsion stacks. They r e p o r t e d a f r a c t i o n of mesonic decays r a n g i n g from a p p r o x i m a t e l y 50Z f o r v e r y " l i g h t " h y p e r n u c l e i to a few p e r c e n t s (namely 2Z - lOg) f o r "medium" s p e c i e s and down to a f r a c t i o n of a p e r c e n t f o r "heavy" ones; " l i g h t " , "medium" and "heavy" a r e d e f i n e d and d e t e r m i n e d by r a n g e distributions in the emulsions. Further experiments ~16-221 provided results for the ratio of neutron-stimulated to total (neutron plus proton stimulated) non-mesonic decay rates, n, as well as for the non-mesonic to ~--mesonic decay ratio, Q- (or, in some of the early experiments, ~2°'2~) for 0'-= the non ~- mesonic to ,- mesonic decay ratio). Attempts at estimating 0, the ratio of non-mesonic to n mesonic decays, were made. ~18. In the ratio n, the proton-stimulated decay is assumed to proceed via the elementary process (2a), while the neutron-stimulated decay mechanism is presumably A + n ~ n + n, Eq. (2b). The ratio n can thus be represented as n=

+

'

(3)

148

J. Cohen

while

rr

(4)

,

where ~(P) is the neutron- (proton-) induced non-mesonlc decay rate, while r.is the n - mesonic decay rate. Note that the total rate is 1

= r=

r.~+rsP+r

r

+ r.o.

(5)

Early experiments confirmed the predominance of the non-mesonic decay mode for heavy hypernucleiCZ6-zs); results have also been obtained for light hypernuclei ~Is-22. where the mesonic and non-mesonic decay rates are comparable. The ratio n, Eq. (3), has been measured, wlth the conclusion that the dominant partial non-mesonlc rate for heavy hypernuclel is r.z (where An ~ nn). Table 2 shows the results of the various experiments, cI¢~2~ Graphical representations of 0- and n against the mass number A will be given in Figs. 4 and 5 at the end of this chapter. As can be seen from Table 2, the early results on the A- neutron stimulation fraction, n, for heavy hypernuclei are in the range 0.6-0.9. For lighter hypernuclei, the ratio n agrees with an average value of 0.3-0.4. Hiller et al. (22) have also obtained an estimate of the hypernuclear nonmesonic-"an~" total decay rates [rN, i r,~ + IT..x, and r, Eq. (5)]. Their estimate is based on previous measurements of the ~ decay rate for .He, on the ratio of non-mesonic to ~- - mesonic decay rates given by Block ~t al. (1964) (H) and Hiller et al. ~22~, and on the ratio = 2.49±0.34

(ro/r-~ L

AH"

from Block et ai.(1964) (H) and from the isospin AT = 2 rule. is discusse~"-in-the next chapter.) Their results are: r,~ = 0.91 x 109 sec -I,

(The AT = ~ r u l e

r = 4.82 x 109 sec -I, for ~He

(6a)

r = 7.01 x 109 s e c - * ,

(6b)

and

ffi 2.11 x 109 s e c -1,

f o r ~He.

[Other decay ratios, which are outside the scope of our present discussion, have also been reported in the literature. Measurement of R +/-, the ~+ to ~mesonic-decay-rates ratio, have been carried out. The n+ production mechanism is presumably A + p ~ n + n + ~+ or pion charge exchange following a n° mesonic decay; the ratio R +/- is therefore expected to be small. Indeed, experimental results for ~He are R +/- = 0.027+0.011 (Benlston et al.(2~*);

R +/- = 0.04 (Block et ai.(22)); R +/- = 0.09+0.03 (Hayeur et a1.(23));

R +I- = 0.043+0.017 (Keyes et a1.(23~).

Decays of A-Hypernuclei TABLE 2.

Decay-rates r a t i o s 0- [Eq.(4)l ( 0 ' - , 0 ) , and n [Eq. ( 3 ) ] .

Hypernucleus

3,4 ^H

0-(0'-;0)

n

Reference

0 Q'-

Schneps, Fry and Swami (14) =

0

0.2

-

0.4 Q=0 O. 26+0._13 0.52+0.10

AH4 ^4He

Silverstein ~2°) GorE& et al. c-a) Bhowmik, Goyal and Yamdagni (Is) Block et al. (1964) ~11.

O. 3+0. I 0 ~g+o.n

0.52+0.13 --

--"

McKenzie (23) Rao et al. (n*

O. 4+0.06

2He

<0.6

1.21+0.19 1.3+0.1 (2.3 0.6
AHe,ALI,ABe (Z = 2,3,4) ALl

^Li,^Be (Z

ABe

=

Block et al. (n)

- ' - 0 . 0 6

0.70+0.19

4,~He

Block et ai.(1964) c11)

0.58

Kenyon et al. (Is* Miller et al. ~22. Coremans et al. (22) Schneps, Fry and Swami (z4) Silversteln (2°~ Sacton (21) Schlein (19) Gorg~ et al. (*s) Bhowmik, Goyal and Yamdagml (Is) Kenyon et al. ~Is) Miller et. al (n) Chaudhari et al. (17) Haldo-Ceolin et al. Its*

<15.5

Schneps, Fry and Swami I*4)

Q'- = 8 - 15 Q'- = 3.6

Silverstein c2°) Sacton (21)

6.6

Gorg~ et al. ~la)

12.8

Bhowmik, Goyal and Yamdagni (re)

2.55+0.66

Chaudhari et al. ~171

2.4+0.7

Holland(17)

3,4) Q'-

38

=

14

-

21

Silverstein (2°) GorE& et al. ~18)

149

J. Cohen

150 TABLE 2 (con't) 4.3+1.1

Montwill et al. (zv)

ABe,A B 11 ^B

64 4.8±1.1

Gorg~ et al. (za)

^B

19

Bhowmik, Goyal and Yamdagni (18)

AB,AC,AN (Z = 5,6,7)

5.9±1.2 5.5+0.5

Coremans et. al. (*~)

>^Li

<70

(Z > 3)

Q'- = 25

Montwill et al. (zv)

0.37+0.06

Montwill et al. I~7*

l

Schneps, Fry and Swami I~) Sacton (~) 0.29

2.4-5 >_ABe (Z > 4)

(z > 5)

Bhowmlk et al. (.8)

3.2+0.7

Kenyon et al. ~ * Holland I-v*

6.6+1.4

Chaudhari et al. (~)

106

Gorg~ et al. (zS)

5.3+1.3

Holland (I~I

"Light"

0.70+0.03

Chaudhari et al. (:8)

0.56

Berkovlch et al. ~18~

0.68 +°'°4 -----0.03

A > I0

Mangotra et al. (.61

0.6-0.9

L a g n a u x e t a l . (16)

"Heavy"

0.85

G a n g u l i and Swami (z61

A > 40

0.9

Cuevas e t a l . ~ 6 )

40
100-200

Reported are also measurements of r(~'2B ~ ~- + 3'4He)/r(all ~- mesonic decays of 3'~H), with results in the range 0.3 - 0.4 for ~H, and 0.7 - 0.8 for ~H; see Keyeset al., (11) Ammar et al.,(2~Ismall et al., ~zl) and Bertrand et ~.(24) This br'anc-~ing ratio was-'i'mp-~rtant for the'--hy-pernuclear spin assig-nment-s-and/or for determining the relative s- and p- wave amplitudes in A ~ N~ decay c2s* (see also Sec. 3.2) atthe early days of hypernuclear weak decay studies]. The early results in Table 2 are subject to large uncertainties and should be viewed with caution. In particular, the analysis of non-mesonic decays presents difficulties, both in their classification as definite hypernuclei and in the subsequent interpretation of their identity from, e.g., decay kinematics. In many cases, the decay involves emission of more than one neutral particle [for example, two or more neutrons, Eq. (2b)] and the unique interpretation of the hypernuclei is very difficult. When using kinematic decay analysis for the hypernuclear identification in experiments involving incident hlgh-momentum mand K- mesons, it is necessary to separate inelastic scattering and m- and Ecaptures from the non-mesonic decays. Some of these problems may be avoided using K- capture, studied, for example, by Schlein, t19) Bhowmik, Goyal and Yandagni, (18) and Holland. (Iv) Identification and interpretation ambiguities

Decays of A-Hypernuclei

151

characteristic of emulsion works do not arise in helium bubble chambers where the ~He hypernuclei can be identified from the kinematics of the two-body production reaction K- + 4He ~ ~He + n-. However, large backgrounds exist for some decay modes due to events not involving hypernuclear production. This situation is a result of low hypernuclear production rates, the short hypernucleus range and the limited spatial resolution of the early detectors, and is particularly true for multibody non-mesonic decay modes. Finally, the statistics of each of the early experiments were not good enough to allow for a clear-cut result. 2.3 Hypernuclear lifetimes and decay modes: Recent results The last five years have seen a revival of interest in the subject of weak hypernuclear decays, especially non-mesonic decays. This rekindled interest is, in the most part, a result of recent and improved experimental measurements of hypernuclear lifetimes and branching ratios. A high-precision lifetime measurement using direct timing techniques has been carried out at the Brookhaven AGS (Alternating-Gradient Synchrotron). ~4'26'2v~ An 800 MeVIc K- beam was incident on a CH target, with a subsequent hypernuclear formation and tagging via the reaction K- + 12C ~ I~C + n-. Detection and momentum analysis of the emitted pions allowed a generation of the I~C mass and excitation-energy spectrum, thus providing a tag for I~C hyper~uclear production. The hypernuclear lifetime is measured directly: t~e start signal is given by the production [via (K-,~-)] and the stop signal by the decay of the hypernucleus. As can be seen from Table 2 and Fig. 4, the non-mesonic decay dominates the weak decay process for I~C (Q- ~ 4 - 5). The charged decay products of the tagged hypernuclei were detected in a scintillator range-time spectrometer, positioned directly below the target; it provided time and energy measurements of the decay process. The spectrometer was able to stop protons up to approximately 120 MeV and pions up to about 50 MeV. The charged particles are well separated using the correlation of energy loss vlth range. In particular, fast high-quality timing scintillators were mounted both in the beam and below the target on the two opposite faces of a multiwire proportional chamber, thereby providing the production and decay times of the tagged hypernucleus, as well as information on particle trajectories. Three d i s t i n c t hypernuclear states were o b s e r v e d i n the e x c i t a t i o n - m a s s spectrum of 12C, i n c o i n c i d e n c e with e n e r g e t i c p r o t o n s from the n o n - m e s o n i c d e c a y s ; theseAmay be i d e n t i f i e d as the h y p e r n u c l e a r ground s t a t e and two excited states. The ground s t a t e of I~C i s bound by a b o u t 11 MeV and i s i n t e r p r e t e d as a s h e l l - m o d e l l a m b d a - p a r t ~ c l e - - n e u t r o n - h o l e c o n f i g u r a t i o n ( A l s , n l p - ). The f i r s t e x c i t e d s t a t e i s 3 MeW above the I~B + p breakuo t h r e s h o l d , and i s r e p r e s e n t e d i n a s h e l l - m o d e l as a p - s u b s t i t u t i o n a l s t a t e (Alp, n l p - ). As t h i s s t a t e i s known to decay s t r o n g l y to I~B + p,<4,2s) the h y ~ r n u c l e a r decays detected in coincidence with this state are associated with .B (rather than I:C) plus a low-energy proton that stops in the target. The t~ird state is unpound by 10 MeV and undergoes a breakup which produces a stable hypernucleus, as established by the detection of energetic protons associated with the weak decays of this state. It is identified c29~ as an ssubstitutional state (Als, nls-1). Based on this assignment and on SU 3 wave functions, BandB, Yamada and ~ofka ~3°~ conclude that the pertinent strong decay channels mostly lead to a ~Be + 3He breakup, and the observed weak-decay is consequently associated wit~ ~Be.

152

J. Cohen

With t h e s e i d e n t i f i c a t i o n s , in T a b l e 3.

the lifetimes

obtained in this experiment are given

Continued e x p e r i m e n t a l e f f o r t s a l o n g t h e s e l i n e s c4'26°2~ have f o c u s e d on s p e c i f i c p a r t i a l d e c a y modes and on t h e l i g h t e r h y p e r n u c l e i 4'^SHe. I n o r d e r to a c c o m p l i s h t h e s e measurements, a n e u t r o n s c i n t i l l a t o r d e t e c t o ~ a r r a y was p l a c e d above the t a r g e t . Neutron e m i s s i o n was s t u d i e d by a t i m e - o f - f l i g h t method f o r e n e r g i e s E > 10 MeV. The r a n g e s p e c t r o m e t e r was used a s e x p l a i n e d a b o v e , to i d e n t i f y a~d measure t h e y i e l d o f e n e r g e t i c d e c a y p r o t o n s ( w i t h e n e r g i e s 30 MeV < E < 120 HeV) and p i o n s (15 MeV < E~ < 50 MEW). The d e c a y o f f r e e A ' s (fo~loving a quasi-free A production in the target) resulted in a high pion y i e l d ; however, a f r e e A decay c a n n o t p r o d u c e a h i g h e n e r g y p r o t o n , and r e s t r i c t i n g t h e a n a l y s i s t o p r o t o n decay e v e n t s w i t h E > 30 MeV e x c l u d e s contributions from free A decay. P The analysis of yields for protons, neutrons and n-'s emitted from the tagged hypernuclei provided values for n and Q-, Eqs. (3), (4). Limited statistics in the neutron time-of-fllght spectrum and uncertain background subtraction from the ground state region of the coincident plon excitation energy spectrum resulted in relatively large error bars for these quantities. Results for n were also corrected for nuclear multiple scattering and charge exchange using an intranuclear cascade calculation. The extracted values are given in Table 3. Using the results for 12^C along wlth

r.o/r.-

= ~

,(,z}

the following partial decay rates are obtained (4'26} [see Eq. (5)]: rNpM/rA = 0.49 -0.2+°3 ~./r^

= n ~+02 ~"

-----0.3

(r.~ + r.~)/r^ --

;.14_+o.2

(for I~C),

(7)

r~-/r^ = 0.05 -0o3+°'°s r~o/r^ = 0.06 -0.0s+°'°s

where r^ is the free h total decay rate, r. = 3.80 x 109 see -I. Thus, the mesonic decay rates are very small for I~C; t~he partial non-mesonic proton and neutron stimulated decay rates are comparable, although much less well determined than their sum. Helium h y p e r n u c l e i were produced by u s i n g t h e 6Li(K-, n - ) ~ L i , w i t h t h e ~Li ground s t a t e b e i n g p r o t o n unbound by a p p r o x i m a t e l y 0 . 9 HeV and t h e r e f o r e d e c a y i n g s t r o n g l y v i a p r o t o n e m i s s i o n to 5He; t h e p r o p o s e d ~He s t u d i e s w i l l use a l i q u i d 4He t a r g e t to d i r e c t l y p r o d u c e ~ 4^He. So f a r , t h e ^ e x c i t a t i o n s p e c t r u m o f ~Li was o b t a i n e d w i t h a decay p r o t o n c o i n c i d e n c e r e q u i r e m e n t ; i t s ground s t a t e ( A l s , n l p -1) d e c a y s s t r o n g l y to p + ~He, and s u b s e q u e n t l y gave e n e r g e t i c p r o t o n s from the weak non-mesonic d e c a y o f ~He. An e x c i t e d s t a t e a t a b o u t 18.3 MeV above the ground state is identlfied as (Als, nls-1), i.e., an

153

Decays of A-Hypernuclei

TABLE 3.

Summary of hypernuclear weak decay rates from Brookhaven experiments E759 and E788. (~'26'2~)

Hypernucleus

x(lO -1° s e e )

12 AC

2.11+0.31 --

11 AB

1.92+0.22 _

~Be

2.01£0.30

ASHe

2.56+0.21

n

O~2+43 ---12

O 57 +0"14 . . . . -0.23

O 51 +°"11 v'--_0.15

0 57 +0"43

1 03 +0.34

--

" ----0.72

....

-0.57

s-substitutional state. It i s a l s o expected to decay to a s t a b l e h y p e r n u c l e u s and s u b s e q u e n t l y g i v e e n e r g e t i c p r o t o n s from the weak decay p r o c e s s . Another excited state, namely, the p-substitutional (Alp, n l p -1 ) e x c i t a t i o n , (29) a broad e x c i t e d s t a t e at about 8.5 MeV above the ground s t a t e , i s suppressed by the proton c o i n c i d e n c e requirement, i m p l y i n g a p o s s i b l e A emission decay mechanism ( t h e r e s u l t i n g f r e e A cannot p a r t i c i p a t e i n a nonmesonic weak decay). The ~He e x c i t a t i o n spectrum was taken w i t h no decay c o i n c i d e n c e r e q u i r e m e n t , and shows the ~He ground s t a t e (Als, n l s - 1 ) . Results f o r ~He are g i v e n i n Table The v a r i o u s p a r t i a l widths are:

3; a l l

these r e s u l t s are p r e l i m i n a r y c4'26)

r.~/r^ = 0.19£0.07 n r../r^

O ~5 +0"11 . . . . . -0.30

(r.~ r.~)/r^ +

. . . . .

-o.31

(for ~He).

(8)

r - / r ^ = 0.43£0.10 r ~ o / r h = 0 . I- ~ ÷°-34 v-0.21 These e x p e r i m e n t s (BNL E759 and E788) aimed a t p r e c i s i o n measurements u s i n g traditional methods of h y p e r n u c l e a r p h y s i c s w i t h s t a t e - o f - t h e - a r t timing t e c h n i q u e s . A d i f f e r e n t approach had been f o l l o w e d a t CERN LEAR (Low Energy A n t i p r o t o n R i n g ) , where the mass r e g i o n A = 200 was e x p l o r e d . (5) Such heavy h y p e r n u c l e i were produced i n h e a v y - e l e m e n t s (23eU and 2°9Bi) t a r g e t s f o l l o w i n g antlproton annihilation. The h y p e r n u c l e i a r e i d e n t i f i e d t h r o u g h t h e i r f i s s i o n decay, which i s d e l a y e d by the l i f e t i m e f o r weak decay. The a n n i h i l a t i o n p r o c e s s a l s o i n d u c e s prompt f i s s i o n ; the d e l a y e d f i s s i o n e v e n t s a r e presumably induced by the e n e r g y r e l e a s e d i n the n o n - m e s o n i c decay ( i . e . , some 170 HEY), and a r e s e p a r a t e d from the prompt ones u s i n g a r e c o i l - d i s t a n c e method. (Note, however, t h a t t h e s e measurements d i d not p r o v i d e d i r e c t e v i d e n c e f o r s t r a n g e n e s s p r o d u c t i o n i n the i n v e s t i g a t e d p r o c e s s . ) This technique, developed for s t u d i e s of f i s s i o n i s o m e r s , i s based on two p o s i t i o n - s e n s i t i v e d e t e c t o r s of

154

J. Cohen

fission fragments, placed symmetrically and perpendicularly to the target plane at a large distance (27 cm) comapred with the target width (2 mm). The detector geometry enabled one to distinguish between events with different decay points, by measuring the impact of the decay products on the upstream parts of the detector (namely, areas mostly in the shade of the target). Only delayedfission fragments (presumably resulting from hypernuclear non-mesonic decays) could reach those detector areas, by travelling a few tenths of a millimeter outside the target, while prompt ones could not. The impact position at the detector depends on the lifetime and on the velocity of the recoiling hypernucleus. Momentum distribution of hypernuclei in the downstream direction is provided by the distribution of the angles between the trajectories of the delayed-fission fragments. In this experiment, no preferred direction of the recoil nuclei exists, and their angular distribution is isotropic. The following information was obtained for each two fragments detected in coincidence: (i) the position of each fragment in the detector; (ii) the timeof-flight value (where the zero-time signal was provided by the antiprotons); (lii) the ionization produced in the counter. The results obtained for the lifetimes are = (I " ~o+I"% x I0 -I° sec f o r 230U -0.5" +2.5 x i0-I0 sec f o r 209Bi" = (2.~.0) Note that these results are similar to the low-A Fig. 3a); however, one should be cautious in especially since there is no direct evidence that from non-mesonic decays of hypernuclei produced in process.

(9a) (9b)

hypernuclear lifetimes (see interpreting these results, the delayed fission results the antiproton annihilation

Another experiment relying on delayed fission was carried out at the linear electron accelerator in Kharkov. c32~ The pertinent production reaction is A(e,e'K+)AA ", although no direct evidence was obtained for strangeness production (as in the CERN experiment). The underlying idea for detecting hypernuclear weak decays was similar to the one used at CERN, and the recoil-distance method was utilized. The electron beam impinged upon a bismuth target deposited onto a support of thickness sufficient to absorb fission fragments, at electron beam energies of 1.2 and 0.7 GeV. Mica detectors placed above and below the target were used to record the fission fragments. The detected fragments resulting from prompt fission processes could be emitted from the target only into the hemisphere downstream of the target plane, with those traveling in the other directions being absorbed by the target support. The only fragments which could be detected at the other hemisphere (namely, corresponding to the upstream area of the detector) resulted from a delayed fission at a certain distance from the target. This e x p e r i m e n t was d i f f i c u l t to p e r f o r m , and s o u r c e s o f l a r g e e r r o r s o c c u r r e d . The c r o s s s e c t i o n f o r p r o d u c t i o n o f h y p e r n u c l e i i s s m a l l (some 7 to 8 o r d e r s o f magnitude below the antiproton-annihilation experiment), while background conditions were severe. Moreover, detection efficiency was low. Background was estimated by placing another target-detector system in a mirror-picture position relative to the original system in the upstream direction. This way, recoil nuclei cannot be emitted into the space in front of the target, and the

Decays of A-Hypernuclei

155

d e t e c t o r s in t h i s e x t r a system a r e e x p e c t e d to r e c o r d o n l y background e v e n t s . Delayed f i s s i o n r e s u l t i n g from d - d e c a y s o r shape i s o m e r s c o u l d n o t be r u l e d out. However, no t r a c k s o f ( d e l a y e d ) f i s s i o n f r a g m e n t s were o b s e r v e d in t h e "shadowed" a r e a s o f the d e t e c t o r s at 700 MeV beam e n e r g y , which i s below threshold for hypernuelear production. Finally, in t h i s experiment the a c c u r a c y o f measurements f o r th e r a n g e t r a v e l e d by t h e h y p e r n u c l e i between the p r o d u c t i o n and d e l a y e d f i s s i o n p r o c e s s e s ( s p a t i a l r e s o l u t i o n ) was ab o u t 0 . 5 mm, more than t h r e e t i m e s th e e x p e c t e d a v e r a g e r a ng e f o r h y p e r n u c l e i w i t h l i f e t i m e s x=x.A ( t h e CERN e x p e r i m e n t , on th e o t h e r hand, was c a p a b l e o f m e a s u r i n g s h o r t lived fission fragments). Thus, th e s h o r t - l i v e d f r a g m e n t s r e c o r d e d a t CERN could not be measured a t g h a r k o v . A lifetime o f = (2.7±0.5)

x 10 -9 sec

for Bi

(I0)

has been reported, ~32~ exceeding z^ and other measured values of hypernuclear lifetimes for weak decays by an order of magnitude. A new experiment

was carried out

at Kharkov {33) using the same recoil-dlstance

t e c h n i q u e , but w i t h a d r a s t i c a l l y improved s p a t i a l resolution. In t h i s e x p e r i m e n t , t h e mica d e t e c t o r s were a r r a n g e d on t h e i n t e r n a l s i d e of the c y l i n d r i c a l s u r f a c e o f a f i s s i o n chamber, and used o n l y i n o r d e r to d e t e r m i n e the t r a c k c o o r d i n a t e s i n the d e t e c t o r . However, no f r a g m e n t - t r a j e c t o r i e s were c o n s t r u c t e d in t h i s ease. With t h i s experimental set-up, t h e r e was no p o s s i b l i l i t y o f d e t e c t i n g random t r a c k s by t h e i r d i r e c t i o n s . In the data analysis, total track-density distribution in the detectors against the distance to the target plane was obtained. The difference between the "real" and "mirror" results in the fission chamber appears as a sum of two exponents. A least-squares fit to the data thus provides two lifetimes ~a3~

(11a)

z(short) = (0.6±0.1) x 10 -1° sec , for Bi •(long)=

(1.6±0.4) x 10 -9 sec.

(11b)

These r e s u l t s have been i n t e r p r e t e d in Ref. (33) as e v i d e n c e f o r t h e e x i s t e n c e o f two d e c a y i n g components, one w i t h a s h o r t e r l i f e t i m e , Eq. ( l l a ) , and t h e o t h e r l o n g e r - l i v e d , gq. ( 1 1 b ) . The l a t t e r i s c o n s i s t e n t , w i t h i n the e x p e r i mental e r r o r , w i t h t h e p r e v i o u s v a ] u e o b t a i n e d by t h e a u t h o r s c32~ a t Kharkov; z ( s h o r t ) , Eq. ( 1 1 a ) , i s about a f a c t o r o f 3~5 l o w er than t h a t o b t a i n e d a t CERN. The s i t u a t i o n r e g a r d i n g heavy h y p e r n u c l e a r l i f e t i m e s i s u n c l e a r a t t h i s t i m e . Concluding t h i s c h a p t e r , we p r e s e n t in F i g s . 3-5 g r a p h i c a l r e p r e s e n t a t i o n s o f the r e s u l t s r e p o r t e d h e r e f o r ~, Q- and n as a f u n c t i o n o f A. The amount o f available data is reasonable, however h i g h e r - q u a l i t y and more p r e c i s e r e s u l t s are evidently called for. Data r e l e v a n t to heavy h y p e r n u c l e i i s p a r t i c u l a r l y important. As we p o i n t out in t h e n e x t c h a p t e r , such f u t u r e r e s u l t s w i l l be crucial for further theoretical progress.

156

J. Cohen

5

'

CJ (L) GQ

4

I

3

'

'

'

'

I

'

'

'

'

'

I

I

'

'

'

I

i

I

o B

)

0

)

L--

.11

+

-ii •

II

o

0

i

I

I

I

I

5

I

i

I

I

50

10

A Fig. 3: (Caption on next page)

i

i

100

(a)'

Decays of A-Hypernuclei

157

3.0

C~ (~

.

.

.

.

'

.

I

,

.

.

tb)'

.

2.5

CO

O

2.0

I (~

1.5

v-H

1.0

0.5

o.o

o ,

,

,

I

,

,

,

,

5

10

,

,

,

I

,

15

A Fig. 3. (a) Results for • as a function of A from Table 1 and Eqs. (6) (octagons), Table 3 (diamonds), and Eqs. (9) ( s t a r b u r s t s ) . (b) The lov-A lov-X region of Fig. 3(a). (For more d e t a i l s , see the Tables and gqs.)

158

J. Cohen

!

i

,

i

!

!

!

J

!

!

(a)

20 o

15

O O

I

CY

10

5 A..L r'1" 0

+ ~_2

l-.--.J

15

10

A Fig. 4 (Caption on next page)

20

Decays of A-Hypernuclei

i

,

,

!

|

I

I

159

!

I

|

t

0

II 0

I

. . . .

10

i

15

,

,

,

,

1

20

A Fig. 4

(a) Results for O- as a function of A from Tables 2 (octagons) and 3 (diamonds). (b) An enlarged representation of the data at the lower part (0-<8) of Fig. 4a. The rise of 0- with A is clearly seen. (For more details see Tables 2 and 3.)

160

J. Cohen

1.0

I

I

I

I

. . . .

I

. . . .

I

. . . .

I

. . . .

Ii

0 0.8

I

0.8

0.4

I

I

m

( 0.8

0.0

-

8

-

I

O

,

i

i

i

,,,

I

....

I , , , ,

80

30

I

40

,,,

I

50

A

Fig. 5

Results for n as a function of A from Tables 2 (octagons) and 3 (diamonds). The large error bars at A = 5 come from the ~He results of Table 3. (For more details see Tables 2 and 3.)

161

Decays of A-Hypernucle, 3.

TREORY OF NON-MESONIC WEAK DECAYS OF A-HYPERNUCLEI

In this chapter we review and discuss theoretical approaches and efforts aimed at understanding the non-mesonic weak decay processes in A-hypernuclei. Recent theoretical progress, which has not been previously reported, is also discussed. 3.1

Introduction: An overview of decay mechanisms

A brief introduction to and motivation regarding the subject are given in Sec. 1.2. In studying the weak non-mesonic decays, our purpose is to gain insight into the fundamental aspects of the pertinent four-Fermlon, strangenesschanging weak-interaction processes. There are several theoretical approaches to the problem o f hypernuclear weak decays in the literature. ~1°'2s'31'34-4s~ They will be reviewed and discussed in this chapter. The range is broad: the extensive and successful phenomenological analyses of Dalitz et al. carried out in the 1960's <2s'34-3s~ where the starting point is the gene'-ra~---nonrelativistic forms of the matrix elements M(Adp~-), M(Ap-mp), and M(An-mn), provided an insight into the empirical structure of the elementary amplitude; early studies by Adams ~39~ aimed at a microscopic description of the non-mesonlc decay mechanisms based on one-pionexchange; these were followed by modern'studies in the 1980's, where additional mesons are considered, ~4°-42~ effects of the many-body nuclear medium on the exchanged propagating pion are calculated, ~43~ or a better treatment of the one-pion-exchange mechanism within the shell model is attempted; ~441 a hybrid quark-hadron model for the non-mesonlc weak decay process was suggested by Kisslinger et al. <4s~ in an attempt to study short-range correlations based on quark dynam~--cs-~n nuclear matter and in finite nuclei, while keeping the simple n-exchange model for the long-range part of the decay mechanism. These various approaches will be discussed here. The theoretical situation regarding the non-mesonic decay mode is rather uncertain. Furthermore, each approach should be tested against the experimental data. As we proceed, we shall point out that the data are insufficient at this time, and are unable to confirm or reject the available calculations. Requirements for new and improved data will be pointed out. 3.2

Phenomenology o f the A-#N~ nonleptonic decay mode"

Theoretical hints and proposals (46 '47) for parity violation (46) and non-lnvarlance under charge conjugation ~47~ were promptly verified experimentally ~4a'49~ in A~p~- decay. These effects, characteristic of weak-interaction processes, are now well-established and accurately measured, and will be crucial for our theoretical discussion. Parity nonconservation in the decay of the A, a spln-~ hyperon, allows for both s- and p-wave (i~=0,1, respectively) outgoing pions. Denoting the amplitudes of these two states by s and p, the decay is characterized by three real constants: Isl, {Pl, and the relative phase between the two amplitudes. (Note that the s-wave violates parity conservation; in contrast, the p-wave is parity conserving.) The angular distribution o f formlS~34) * I am i n d e b t e d of this section

the

pions

emitted

in polarized-A decays has the

to Terry Goldman f o r u s e f u l d i s c u s s i o n s on t h e s u b j e c t m a t t e r and for constructive c o m m e n t s on a p r e l i m i n a r y version thereof.

162

J. C o h e n

1 + oeA cos O,

(12)

where 8 is the angle between the pion and the initlal spin direction, average polarization of the decaying A-hyperons, and ~=2Re

P^ is the

sp

is12 + ip12

(13)

is a constant characterizing the degree of mixing of parities in the decay. The possible range of a is -l
IHwIA> =


s

o'q

+ p qA

(14)

'

where S and P are the respective = above, ~ is the pion momentum in the I"A value f o r a f r e e (unbound) A decay. be more convenient to write in

0 and i,= 1 amplitudes as explained rest frame, and q^ = I00 MeV/c is its I n the f o l l o w i n g

[see Eq. ( 1 8 ) ]

it

will

terms of Dirac operators I and "fs' Note that the parity conserving" and "parity violating" amplitudes need not have equal strength at this hadronic-penomenologlcal level. W

W

The v a l u e of a d e p e n d s on t h e d e t a i l e d mechanism r e s p o n s i b l e f o r t h e A-~Nn decay p r o c e s s . E a r l y e x p e r i m e n t s (4s~ d e t e r m i n e d o n l y t h e c o m b i n a t i o n ~ ^ ( s e e Table 4). With t h e u p p e r l i m i t o f some 70% p l a c e d on ~^ i n t h e s e e x p e r i m e n t s , Table 4.

E x p e r i m e n t a l r e s u l t s r e l a t e d to the measurement o f a [Eq. ( 1 3 ) ] .

Measured quantity

Pmax

Measured Value

Reference

+0.40 + 0.11

Eisler

+0.51 + 0 . 1 5

Cravford et al.

+0.55 ± 0 . 0 6

Cork e t a l . ¢49~

+0.73 + 0 . 1 4

Crawford e t a l .

(3 6 7 +°-18

e t a l . ~4s~ ( 1 9 5 7 ) ¢48}

( 1 9 5 8 ) ~481

-~" ---0,24

B e a l e t a l . c51~

-0.45 ± 0.4

B i r g e and F o w l e r (s2~

o 75 +0. 50

---* - - - - 0 . 1 5

L e i t n e r e t a l . ¢52)

-0.65 ± 0.02

0 v e r s e t h and Eoth (53b)

-0.64

Particle

+

0.01

Date Group (6°)

(average result)

Decays of A-Hypernuclci

163

a lower l i m i t of about 0 . 6 - 0 . 7 could be p l a c e d on I~1. Moreover, p r o t o n s r e coiling from unpolarized A-~p~- decay are longitudlnally polarized; ~°'3s~ the value of the mean longitudinal polarization for an unpolarized A, speelfJed in the A rest frame and averaged over all directions of decay is _a.(50,3s) Studies of such protons ~51-5~ by means of, e.g., scattering asymmetry in ~12C scattering in a carbon-plate spark chamber ls1~ or proton scatterin~ in a propane chamber c52~ established that the sign of ~ is negative; ~=1's~ furthermore, they also provided values for a itself; results are given in Table 4. By the middle 1960's, the value of a was well known and given by Dalitz I~s~ as a = -0.7±0.1. A more recent figure, ¢ = -0.64±0.01, is given by the Particle Data Group [see Ref. (60)]. Assuming tlme-reversal invariance and in the absence of final-state interactions, the coefficients s and p are real. ~s4~ Thus, the phases for these amplitudes arise only from the (strong interaction) final-state pion-nucleon scattering. Indeed, the explicit forms for s and p in A~pR- decay are ~34'ss~

slexp(i

+

s exp(i ) , (15)

P = ~

Plexp(i61) + ~

P3exp(i6~) •

In Eq.(15) we use the notation s 2 , P i for the partial amplitudes of final isospin I. = ~ and ~ (as we ~ a l f see shortly, the I, = ~ amplitude is relativel~ sm~ll); the phases 6~ are the (experimentally'det~rmined) ~ - N scattering phase shifts. (Ss) At'{he c.m. kinetic energy of interest here (some 37 MeV) they are all very small (a mere few degress for I~=0, and approximately 0 °- -1 ° for 1 =1). The coefficients s and p can therefore be taken to be real numbers. (~ common overall phase is unimportant here.) Indeed, the experimental results (st's3) for = 2 Im

s p IsI2+ [pl 2

yield small values(S3): 6 = 0.18±0.24 (s3a) and 8 = - 0.10±0.07; (s3b) and = - 0.13±0.07 (s3cl and ~ = - 0.094±0.060. (53d) In Eq. (15), both I, L and I f = z~ partial amplitudes for s and p were included. Early on IsT~ it became clear that the experimental data relating to strange-particle decays demand an approximate al = ~ isospln selection rule. This empirical rule has, in the meantime, become well-established. (ss-6°) The consequences of the AT = ~ rule are extremely simple for the decay of the A-hyperon, which has I=0. It~implies that the pion and nucleon are left in a final I t = ~state, so that both s 3 and P3 [Eq. (15)] vanish. Furthermore" comparing t ~ e - X ' ~ ° decay with the A~p~- one, the appropriate IK=O,1 amplitudes [see Eq. (14)] satisfy

s(O)

.(o)

1

(16)

164

J. Cohen

or

s(_)2 + p(_)2 s(_)2+ p(_)2+ s(O)2+ p(O)2

= 2~

,

w h i l e from gq. (13) =(o)

(17)

= =(-)

As a consequence of t h e s e r e l a t i o n s , th e decay r a t e s s h o u l d s a t i s f y ( t h e phase space f o r t h e two r e a c t i o n s i s almost i d e n t i c a l , and r a d i a t i v e c o r r e c t i o n s a r e small)

r(^~..o) = L r(^~pn-) or

r(^-~ charged h a d r o n s ) / r ( A - ~ a l l h a d r o n s ) = ~ . These r e l a t i o n s were e x p e r i m e n t a l l y v e r i f i e d to w i t h i n a few p e r c e n t by Brown e t a l .
and the polarizations, suffices to determine the magnitudes of s and p and their relative sign. Cronln and Overseth, and Overseht and Roth, found
)

I

=

os 0.36 -+o~,.^=, v~

( ) IPI/s ( - )

I

ffi 0 . 3 8 + 0 . 0 1 ;

p -(

)

and

s -(

)

have

opposite signs. ~Ith t}lese measurements, the relative magnitude of p is <6 3) , ~p ( - ) I 2

Is(-,I Hore

+ Ip(-)l

= 0.125+0.01 2

r e c e n t l y , Band5 and Takakl (64) used the values I s ( - ) 1 2 = 0.86 x 10 - 1 4 ,

I p ( - ) l 2 = 0 . 1 0 x 10 - 1 4 ,

Decays of A-Hypernuclei (note that s and p al. (66) use

are

dimensionless),

s c-~ = 0.96 x 10 -7 ,

while

Motoba

165 at al. ~6s~ and Mach et

p~-~ = - 0.35 x 10 -7

Our ensuing discussion will rely on Feynman diagrams techniques; towards this end, it will be more appropriate to write H w [ E q . (14)] in terms of Dirac spinors and operators. Assuming only invariance under proper Lorentz transformations, the most general structure of the matrix element for the decay is

=

CgwuN(PN)(1 + X75)uA(P A) ,

(18)

where p^ and PN are the four-momenta of the decaying A-hyperon (of mass MA) and the produced nucleon (of mass M,). The constant C is for A~p~C = (

(19) -I

for A-~nR°

]See Eq. (16)], X is a real number, (s4) and 75 is the usual Dirac matrix. (67) Other scalar and pseudoscalar terms constructed from the available kinematic quantities appear possible at first glance, but each of those terms is already included in Eq. (18), as can be seen by a suitable application of the Dirac equation (this is true on-shell). The amplitude , Eq. density L

= - gw % ( 1

+

(18),

may

XTs) % v/^.~,, ,

be

derived

from the effective Lagrangian (20a)

where ~, is the nucleon field, ~_ is the pion field (a vector in isospin space), and % is the isospin (Pau~i) operator for the nucleon. It is also possible to express L in terms of a current and the derivative of the pion field, L = - gw~,Tu(A+BTs)%~ A. auth.

(20b)

However, since the pertinent constants (gw,X; A,B) cannot be calculated at present from first principles, the two forms Eqs. (20a) and (20b) are equivalent on-shell by virtue of the Dirac equation. [Note that the parityconserving, or B-part of Eq. (20b), is equivalent to the pseudovectorderivative coupling of the strong rtNN vertex, c6v)] Since the pion is a pseudoscalar, the unit term in Eq. (20a) gives rise to a pseudoscalar component of the amplitude and corresponds to a final-state swave, while the other term yields a scalar and corresponds to a p-wave. The "field" ~^ in Eqs. (20) is a spurion containing the A field,

•^ I°^l

166

J. Cohen

(or a product of the i s o s i n g l e t A s p i n o r with the I I , I z > = {2~ - 2~> i s o s p i n o r ) , and t h i s form takes c a r e of the 61 = ~ r u l e . [The term " s p u r i o n " was o r i g i n a l l ~ used by Wentzel to d e s c r i b e a s p u r i o u s " p a r t i c l e " t h a t t a k e s c a r e of the 61 = ~ r u l e and i s o s p i n c o n s e r v a t i o n , t*s~ I n a p u r e l y h a d r o n i c weak decay, i t was imagined t h a t the n e u t r a l s p u r i o n i s e m i t t e d , t r a n s f e r r i n g no e n e r g y or momentum but c a r r y i n g i s o s p i n ~ and whichever z-component (namely, -2 +~) needed to b a l a n c e I i n the r e a c t i o n . With the s p u r i o n i n c l u d e d , i t was p o s s i b l e to assume i s o s p ~ n c o n s e r v a t i o n i n the decay p r o c e s s . This d e v i c e i s e q u i v a l e n t to the 61 = ~ r u l e . ] Since L w i l l be used o f f - s h e l l i n a p p l i c a t i o n to n o n - m e s o n i c decays ( s e e Sec. 3 . 4 ) , a v e r t e x f u n c t i o n , or form f a c t o r , w i l l be r e q u i r e d . We p o s t p o n e a d i s c u s s i o n of t h i s e x t r a i n g r e d i e n t to $ubsec. 3 . 4 . 5 . Other c o n t r i b u t i o n s , such as anomalous and a n a p o l e moment terms [ i . e . , V a q~(l+X y . ) + . . . ] may a l s o contribute off-shell. Nontheless, these contrlbutlons are not determined from experimental data for free A-~N~ decay, and the pertinent parameters cannot be calculated theoretically in a reliable fashion. Under such circumstances it is customary to start with a "minimal" model, represented by Eq. (18). Comparing Eqs. (14) and (18), we find that the amplitudes s and p are related to gw and X: I/e first carry out a nonrelativisitc reduction of Eq. (18). Using the form Eq. (20a), the Y5 term gives an approximate

~'q

operator,

24 since the initial and final baryons have different masses (HA and )IN, respectively). Here, M is a mass that presumably lies between /4. and M^, and may be taken to be ~ = 2~ M A + M r) " As an alternative one m1~ht use, for • . . . , instance, lq = 2~HAM.)112; however, the A--N mass spllttlng is relatlvely small [(MA-H~)/(M^+M ~) = ~.O85], and the two averages give almost identical results (~I = 1027.5 ~4eV or 1023.8 MeV, respectively)• Note, however, that the pseudoscalar. -u~s ^ Yu may be replaced by the pseudovector ~Ys~UAl(H~+ M A) where q is the four-momentum transfer. Using the Dirac equatiori for the free nucleon and A, we see that this replacement is exact on-shell. A non-relativistic ~" q reduction can now be carried out to lowest order, 0(1), yielding a HA+H N operator exactly to this order. Based on this argument, we shall adopt the choice of I~=~H^+MN). [Note, however, that off-shell (e.g., in a relativistic scalar + vec-to~ndclear model) the pseudoscalar and pseudovector choices are no longer identical.] This gives for, e.g., the A-~p~- decay," s (-) = ~'2" g w

(21a)

sP•=

(21b)

and

X q A = O.05X

2ff

m~ )2.

* S t r i c t l y s p e a k i n g , t h e s e r e s u l t s a r e c o r r e c t to o r d e r ( ~

Exact e x p r e s s -

ions can be found, for example, in Refs. (69) and (62), but they are not required for our purposes here.

Decays of A-Hypernuclei [the factor (4n) -I/2 comes from the o r b i t a l state wave functionC69)]. Thus, the numerical values of the constants

167

angular-momentum p a r t of the final-

gw

and k in Eq. (18) may be taken

as

..Igwl = 2.35 x 10 -~,

X

= - 6.9

(22)

These values are consistent with the compilation of Overseth in the Review of Particle Properties. 16°I The phase of gw is not determined experimentally from A-decay measurements. We note that the value of X (or, equivalently, the determined ratio p/s) implies that the second (p-wave, or XY5), parity-conservlng term of the decay amplitude contributes only about 12% of the free total decay rate (r^). of course, the existence of the p-am-~itude is crucial for the existence-of the decay asymmetry [see Eq. (13)]. Furthermore, we shall see that the p-wave becomes very important in a non-mesonlc decay mechanism based on the one-pion exchange model (Sec. 3.4). The theoretical hadronic phenomenology discussed above provides an adequate starting point for the description of A-decays which we shall adopt as a possible means for dealing with hypernuclear decays. It can also be extended to other hyperons and other flavors. 17°I One would naturally like to be able to relate the weak nonleptonic A decay to other weak-interaction processes, and eventually have a microscopic approach whereby this process will be theoretically understood. These subjects are outside the scope of the present discussion, and will only be mentioned briefly here: ~vl) (i) Weak IASI = 1 transitions are suppressed by a factor of approxiamtely 20 relative to AS = 0 ones. A unified scheme for IASI = 0,1 weak interactions was suggested by Cabibbo (1963). In the model, the weak nonleptonic and leptonic currents are governed by the same weak coupling constant, but the former is split into AS = 0 and IASI = 1 components with• appropriate factors of cose C and sine C , respectively, where e is the cablbbo angle• At the lepton-quark level, the leptons participating i ~ chargechanging weak interactions are organized in lepton-neutrino doublets; similarly the u, d, and s quarks are also organzied in a doublet

(;c) =

U

( d cOSec + s sine c )

,

after the d and s quarks participating in the weak interactions are "rotated" by a Cablbbo angle, , ce . (The pertinent Cablbbo mixing matrix is a rotation operator, where ec xs the rotation angle.) The resulting coupling constants are reduced by factors of, e.g., cose c for AS = 0 processes involving u and d quarks, and sine for IASI = 1 interactions, involvlng the s quark. (ii) All neutral-current ~eak processes observed are characterized by AS = O. In 1970, Glashow, Iliopoulos and Maiani introduced the c quark, and proposed a further doublet for weak interactions: C

With t h i s second d o u b l e t , no I~sl = 1 n e u t r a l c u r r e n t s a r i s e ; I n d e e d , they a r e e x p e r i m e n t a l l y known to be s u p p r e s s e d to v e r y low l e v e l s . ( t i i ) I n the g e n e r a l

168

J. Cohen

c o n t e x t , i n v o l v i n g s i x quark f l a v o r s , t h e weak c u r r e n t s w i l l be d e s c r i b e d by u n i t a r y t r a n s f o r m a t i o n s among t h r e e q u a r k d o u b l e t s , c h a r a c t e r i z e d by t h r e e Euler angles and six phases and providing a 3x3 generalization of the Cabibbo mixing matrix, as suggested by Kobayashi and Haskawa (1972). In addition to a unified interaction strength, these models also retain the simple V-A structure of the weak interaction. (iv) The origin of the aI = ~ r u l e is not completely understood at present. There are theoretical indications that it may be rooted in strong interaction corrections to the Cabibbo theory, including gluon exchange such as one-loop gluon radiative corrections (box and penguin* diagrams) and renormalization corrections. ()2-v4~ [It should be emphasized, however, that these methods deal only with "hard gluons", since only at high energy and momentum is the strong interaction (at the quark-gluon level) coupling constant small enough to justify a pertubative calculation. Nonleptonic weak decays are probably governed by "soft gluon" effects, which are much more difficult to handle. Thus, the theoretical situation is rather uncertain at this time; see also the review and critique by Pakvasa (v3~ and Hill (v3)] (v) Finally, there may be possible paths from the quark-model amplitude to the hadron level phenomenology used here. (75) However, the combination of strong-interaction effects at short distances, with the quark model for description of large distances might not be exact since it involves extrapolations from small couplings up to very strong ones. This is why the effective Hami]tonian used here, Eq. (18), cannot be obtained at present from calculations based on certain aspects of QCD (e.g., perturbative ones). We can regard the analysis of nonleptonic weak decays as a three-stage process. It starts with some definite assumptions, or a model, about the structure of the "fundamental", bare Hamiltonian. In the present case it is customary and reasonable to take the Standard Model as a starting point. The Standard Model incorporates the Glashow, Weinberg and Salam theory of electroweak interactions, as well as Quantum Chromodynamics (QCD), which is currently widely accepted as the fundamental theory of strong interactions. The next stage is to modify this Hamiltonian to take into account strong-interaction corrections at short distances (at the sub-hadronic level). This yields the so-called effective weak interaction Hamiltonian. In the third step, a scheme has to be devised to evaluate the matrix elements of the effective Hamiltonian in order to obtain information directly related to physical amplitudes. The emerging picture, though incomplete, leads from the Standard Hodel of strong, weak and electromagnetic interactions to the A-~N~ weak decay phenomenology described in this Section. The Standard Model incorporates the electroweak theory of Glashow, Weinberg and Salam (1961-1970), wherein charged-current weak interactions are associated with the exchange of the charged W boson (Fig. 6). In the low-energy and low momentum transfer region (note that the mass of the W is about 80 GeV) of interest here, the result is a zero-range (contact) interaction. Quantum Chromodynamics (QCD), combined with the Cabibbo theory (or extensions thereof) presumably results in an effective V-A weak Mamiltonian which "explains" the relative strength of AS = 0 and JAS I = 1 transitions. Introduction of strong-interaction corrections within the theoretical framework of QCD might eventually explains the origin of the AI = ~ rule. At the end of this chain, we show in Fig. 7 the low-energy quarkZmodel decay diagrams, corresponding to the hadronic phenomenology used here and depicted in Fig. 1. *Penguin diagrams represent an interplay between the electroweak and strong interactions. An s-quark is changed into a d-quark by emitting and reabsorbing a W boson (i.e. s-~d by a W loop), while interacting wlth another quark (or an antiquark) via gluon exchange in the intermediate state. These are pure aI=~ diagrams.

Decays of A-Hypernuclei

169

U

W

U

Fig. 6

Lovest o r d e r s h o r t range AS = I i n t e r a c t i o n picture.

i n a S t a n d a r d Model

170

J. Cohen

7/'-

P

7r-

r

°t s

dJ/

° ul

d/

d u

d

u

,u

d

A Fig. 7

u d

A

Low-energ7 quark-model d i a g r a m s f o r t h e 16S1=1 A~p~- weak d e c a y p r o c e s s . The weak i n t e r a c t i o n v e r t i c e s a r e e f f e c t i v e l y p o i n t l i k e ; the s t r o n g q u a r k - a n t i q u a r k p a i r p r o d u c t i o n r e q u i r e s a d y n a m i c a l nonp e r t u b a t i v e model [ f o r a r e c e n t d i s c u s s i o n s e e , e . g . , H . J . Weber, Phys L e t t . B218, 267 ( 1 9 8 9 ) ] .

Decays of A-Hypernuclei 3.3 Non-mesonlc d e c a y s o f l i g h t

hypernuclei:

171

The p h e n o m e n o l o g i c a l a p p r o a c h

An e x t e n s i v e t r e a t m e n t o f A - h y p e r n u c l e a r weak d e c a y modes has been p r e s e n t e d a t an e a r l y s t a g e by O a l i t z and h i s c o l l a b o r a t o r s , t~5'~4-37~ Our main i n t e r e s t h e r e i s in t h e i r i n v e s t i g a t i o n s o f t h e n o n - m e s o n i c d e c a y mode, t371 where t h e underlying elementary process is A+N-~n+N (N=n,p; see Fig. 2). No particular basic reaction mechanism was suggested for this process; the investigation of such reaction mechanisms will be described in the following sections.

The properties of interest of the A.N-~N+N weak interaction are its isospin and spin dependence• As a result of the fairly low AN relative momenta anticipated for light hypernuclei, only the relative two-body s-state is expected to contribute to the transition. This is expected to be a good approximation at least for light hypernuc]ei. Indeed, with a A-hyperon and a nucleon in a Is harmonic-oscillator she]l-model state the relative A-N state is also necessarily an s-wave (i.e., IAls>INls> = [Is> [Is>r.l, where cm and tel represent the center-of-mass system and the relati~ state, respectively). Under these conditions, the possib]e AN-~NN transitions are listed in Table 5, where R are the partial A-N capture rates for total angular momentum J, per unlt nucleon densxty at the posltlon of the A. Moreover, RNj = r

R(a'-~8) ,

(23)

where the individual transition rates from an s-wave initial state =' to a final state ~, are denoted by R(~'-~). From Table 5 the requried rates for Ap-~np are (assuming only relative s-states for the Ap system) R(Iso-~Is0 ), R(Iso->3po ), R(3Sl-~3pl ), R(3Sl-~Ipl ) and R(3sI-~3Sl + 3dl). These rates are then used to get R a using Eq. (23) [for instance, R = R(isn-~iSn) + R(Is.-~3p^)]. For An-~nn thee final state has only isospin I. =~%, so t~ans~itions a{e n6t possible to the states 3si, 3di, Ipi. TABLE 5.

The p a r t i a l r a t e s c o n t r i b u t i n g to A - h y p e r n u c l e a r n o n - m e s o n i c d e c a y s s t a r t i n g from an i n i t i a l r e l a t i v e s - s t a t e f o r t h e A-N p a i r .

Initial State

Iso(L=O, J=0)

Final State

!s o

3po

3Sl

!P!

1

1

0

0

If AD-~np

~

Rpo

~

An-~nn

~

Rno

~

3Sl(L=O, J=l)

÷

0

3d__.!

3pl

0

1

0

Rnl

Rpl

0

172

J. Cohen

It is not known whether the AT = I. rule is satisfied for the AN -~ NN interaction. Given its validity in ot~er [6S[ = I transitions as discussed in Sec. 3.2, we shall follow Block and Dalitz (37~ and assume the AI = ~ rule for non-mesonlc decays. With this rule, it is possible to relate the neutron- and proton-induced partial decay rates for the final (NN) I. = I state. Using the Wigner-Eckart theorem, (76) a general AI = ~ o p e r a t o r R~y2) yieldsi31)


If z [Q(~ - )

I=V2,1z>=

l-lfz (_Ifz_ V 5 2i~z)(_l)



The difference between the neutron- and proton-induced

i n the 3j symbol, 1

( 0

f o r An-mn we f i n d

) -- _

_l z

(24)

processes is embedded

( 1 - ~½ - 5 ) = _ ~ _.! ' w h i l e f o r Ap-~np

We conclude,

therefore,

that for the I f -- 1 trans-

s i t i o n s , the n e u t r o n induced p a r t i a l decay r a t e i s twice as l a r g e as the p r o t o n induced one. Note t h a t t h i s c o n c l u s i o n i s t r u e i r r e s p e c t i v e of the decay mechanism. Thus the t r a n s i t i o n r a t e s to the i s o s p i n I . = 1 s t a t e s lSo, ~Po, 3 Pl a r e • twice the c o r r e s p o n d i n g r a t e s f o r Ap-mp by the ~ aI = L~ r u l e , as shown above. • The 6I = L r u l e can be used to, o b t a i,n Rn o = 21~p o ; , f o r the R ~ transitlons all that can be obtained is the i n e q u a h t y R n l . -<- 2R p i . ,slnce not a ~ f i n a l s t a t e s allowed f o r the p r o t o n - i n d u c e d decay a r e p o s s i b l e f o r the n e u t r o n - i n d u c e d p r o c e s s (see T a b l e 5 ) . Assuming, t h e n , the u n d e r l y i n g AN-NqN e l e m e n t a r y p r o c e s s , Block and D a l i t z (37~ wrote the g e n e r a l n o n - r e l a t i v i s t i c forms f o r the AN-~nN(N=p,u) t r a n s i t i o n matrix elements: M(Ap-~p) = apPo+bpPl+ ~ .

_)

_)

.

_)

~N (~¥" ~'*~N" q-~-

~ "~ "~ 2. oy. ON q )

e

and M(An-~nn) = anP o +

en

.~

-~ .

-~

dn (~Y+ ~N )'~ + '~-MN(°¥-°N; ' q PO

(26)

I n Eq. (25) ~y d e n o t e s the s p i n of the h or the f i n a l n e u t r o n , and %H i s the s p i n of the p r o t o n ; i n Eq. (26), %~ r e p r e r s e n t s the s p i n of the h and one of the f i n a l n e u t r o n s , w h i l e %~ c o r r e p s o n d s to the o t h e r n e u t r o n . I n Eqs. (25) and ( 2 6 ) , ~ i s the momentum of the f i n a l n e u t r o n , P and P. a r e p r o j e c t i o n o l L o p e r a t o r s f o r S=O and S=1 t o t a l s p i n s t a t e s , respectively [ e . g , Pv = ~ (1 ~y'%,)]. The v a r i o u s terms i n Eqs. (25) and (26) c o r r e s p o n d to the Lg L~S~ t k a n s i t i o n s of T a b l e 5:

Decays of A-Hypernuclei

a.: Is 3

b.:

o

sz 's c.: 3 I d.-" Iss~ e,: f,: sl

-~ ts 3 ° s. + 'd: -~ 33pI" e lp o -~ Pl

•

17~

(N = p,n) (N = p only) (tensor term; N = p only) (N = p,n) (N = p , n ) (N ffi p o n l y ) .

[Note t h a t t h i s n o t a t i o n d i f f e r s s l i g h t l y from t h a t of Ref. ( 3 7 ) . Thus, a here i s a of Ref. ( 3 7 ) , b i s c of Ref. ( 3 7 ) , e t c . The p r e s e n t n o t a t i o n t~ adopted from Dover and ~;a~ker. cl)] As i n T a b l e 5, no b - , Cn- , or f - t e r m s a r e p r e s e n t i n Eq. (26) because they i n v o l v e P a u l t - f o r b t c f ~ l e n nn f i n a l s t a t e s . P a r i t y - c o n s e r v i n g t r a n s i t i o n s a r e induced by the a..-, b~- and c N terms [ t h e f i r s t t h r e e terms i n Eq. (25),. and the f i r s t t e r m " i n ~ , ~ . ' ( 2 6 ) ] , whereas the d - , e - and fN-terms [ t h e r e m a i n i n g ones i n Eqs. (25) and ( 2 6 ) ] i n d u c e p a r i t y v~olat~on. Rate measurements in non-mesonlc decays are not sufficient for a unique determination of the various coefficients in Eqs. (25) and (26). Measurements involving polarizations would be required as well. We have already seen in Chapter 2 that such data are not available and the pertinent measurements probably lie well in the future. As already explained, it is possible to eliminate the neutron coefficients in favor of the proton ones if the al = L is valid here: 2 x

Xpn _ ~/

;

x = a,d,e

(27)

[Eq. (27) follows directly from Eq. (24).] Block and Dalltz (37) showed that a number of important properties of the AN+NN amplitudes can be extracted from the existing data for light A-hypernuclear non-mesonic weak decays. In their study, the A decay by different nucleons is treated as incoherent. This procedure neglects final state interactions for the two energetic outgoing nucleons, and interference effects arising from antisymmetrizatlon of the final state. ( For coherenttransltlons, Gal and Dover have explored the effects of spin and isospln selectivity on the decay rates, as done for the strong-lnteractlon conversion process ZN~AN in Ehypernuclei, cTvl) In this model, the elementary rates RNj (defined above) are

=

RpO lapl2

Rno

=

+

)2

lepl2 ~N'N

lanl 2 + 1%12 ~.._)2 N

Rnl = Idn {2

~g~N )2

(by the 6I= ~ rule, Rno = 2Rp0 and Rnl < 2Rpl as shown above).

(28)

174

J. Cohen

The non-mesonic decay r a t e f o r a h y p e r n u c l e u s ~ i s g i v e n i n t h i s model w i t h a l o c a l ^-N i n t e r a c t i o n by r . . ( ~ ) ffi p~(A~Z)~ ~ ' ~ s the s p i n - i s o s p i n a v e r a g e of the R.j for t h i s hypernuc~eus, and o~=(A-1)~u~(~)p.(~)d~ i s the mean n u c l e o n d e n s i t y a t the ^ p o s i t i o n , with u . b e i n g the K w a v 4 f u n c t i o n ( s - s t a t e ) and p., the n u c l e o n d e n s i t y . D a l i t z and "'Rajasekharan ¢35) c a l c u l a t e d p_ and found p=

0.038 fm -3 (for ASHe), P4 ffi0.019 fm -3 (for ~H), and P3 1.9~B^ x I0 -3 fm ~3 (for ~H, where B~ is the A binding energy, of the order of a quarter MeV). For some speciflc cases, Dalltz et al. wrote. (35'37) =

FNM(~He) = osR(~ He) ffi~ P5 (3Rpl + Rpo + 3Rnl + RnO)

rNM(~He) =

P4R(~Be) P4R(~H)

=

= ~ P4 (3Rpl + Rpo + 2Rno)

(29) = ~ P4 (2Rpo + 3Rnl + Rno)

= P3R(~ H) = ~ P3 (RpI + 3Rp0 + Rnl + 3Rn0) " /.

(Note that in ~H , for example, only the singlet i n t e r a c t i o n i s e f f e c t i v e f o r Ap-~np.)

F u r t h e r m o r e , d e f i n i n g the r a t i o of p r o t o n - to n e u t r o n - i n d u c e d n o n -

mesonic p a r t i a l rates [see Eq. (3)],

%$

=

l-n n

=

FNMp

rN ~

'

we find v(~Be)

3RpI+ Rp0 = 3Rnl + Rno

v(~Be) =

3RpI+ RpO 2Rn0

(30)

~(~H)

2Rp0 = 3Rnl+ Rno

v(~B) =

RpI+ 3RpO Rnl+ 3Rno

To d a t e , we do not have b e t t e r d a t a than those used by D a l i t z e t a l . ~35'371 (almost t h i r t y y e a r s a g o t ) . I t i s l i k e l y , however, t h a t new d a t a - - w i I I soon be a v a i l a b l e from Brookhaven (see Sec. 2 . 3 ) . On the b a s i s of the ( o l d ) a v a i l a b l e d a t a , Block and D a l i t z found R ^ = ( 7 . 4 + 2 . 4 ) L f m 3 and (3R .+ R . ) / 4 = ( 8 . 2 + 2 . 0 ) F , fm3 ( n o t e t h a t Block a n d " ~ a l i t z c 3 7 1 u s e d a ~ a l u e of F , ~ h i c ~ U i s h i g h e r by some ~2~ than the c u r r e n t l y measured v a l u e of 3.80 x 109~ec-1; t h e i r r e s u l t s have been r o u g h l y a d j s u t e d here to the more r e c e n t v a l u e ) . Using R_, = 2R_n

from the 6I = ~ rule,

Rpl

=

(9.6

±

3.3)rAfm3

was obtained.

The'b~serve~

Decays of A-Hypernuclei

175

value for Q-(.4H) (see Table 2) along with an estimate of r~-(4e) = r^ [see Refs. (25), (~5) and Table 1], gave R i~ (17 + 6)r^fm 3. Summarizing these results, we find that R I/2R i is in tlie- range-of O.3-1.5 and R I/R o lies between 1.3 and 5.2, wh~le l~./R . is in the range of 1.1-4.6. Althougla it is difficult to reach definite ~nc~Uusions given the wide ranges quoted above, we note that the conceivable R n . ~ 2R p . implies the dominance of transitions to I. = 1 final states, while R I~>R 0 ~mplies that the spin-triplet (J=l) channe~ dominates over the spin-si~glet p (J=O) state. If both conditions are valid, then the strongest AN-~NN transition is the 3s1-*3pz, parlty-changlng If= 1, and the next strongest are the Is o -+Iso , 3p_, I.r = 1 transitions. The corresponding , , , o dominant transltxon matrlx element would then largely correspond to the d,- , a N- , and e N -terms of Eqs " (25) and (26) [note that this result corresponds, of course, to the complete An-~'nn amplitude, Eq. (26), since the final nn state has only If= 1]. The conclusions of the preceding discussion rely on the 6I = ~ rule. From Eqs. (29) and (30) it is clear that a datum for ASHe would make it possible to derive values of R j which are independent of the AI = _I-rule. Indeed, such measurements have geen recently attempted at Brookhaven, z as described in Sec. 2.3 [see especially Table 3 and Eq. (8)]. Existing error bars for r..(~He) and especially for n(ASHe), are too large at present to permit a definive discussion, but the followlng observations are of interest. Using the central value for r..()He) in addition to the data used by Block and Dalitz, yields 3R I1 . + R 110 4 ---~5~1 8 r fm s and (3R . + R 110.)/(3R 1. + R , ) ---1.8. Consequently, the " ° A ratio r...(.He)/r_ (.~) ylelds ~ p .IR_ -- 1 " ~, vhlch is consistent with the NM /X A ~U results o b t a i n e ~ y Block and bahtz, but does not imply a dominant spintriplet channel in the AN-~NN transition. Moreover, we find that R _/R_ = 1 nu t~o 1.32, while the AT = . rule implies R _/R . = 2. We can also use the measdred ~0 5 H

-i

n

5

p

~

+ 0 65

(^ e ) = r N . ( A H e ) / r . M ( A H e ) = 1 . 3 0 1"6 •

nu

If

pu

we c h o o s e a g a i n

to use the central

value (despite the extreme uncertainty represented by the large error bars), we find that (3R n A. +, R 110_)/(3R pJ.. + R 0) -" 1.3. [Along with the non-mesonic decay rate , for ASHe, this gives a cePntral value (3R p l_ + R 0)14 = 8.9r.A fm 3, still c o n s a s t e n t w i t h the one g i v e n above from Block and DalP~tz.] T h i s r a t i o , w i t h the d a t a used i n Ref. ( 3 7 )• , yields Rp l. / R pO_ -- 0.33 ' i . e . , t h e AN-)NN p r o c e s s would have a large spin slnglet, J=O component, unlike the previous analysis. Also, the resulting Rno_/Rpv. = 0.45 implies a major violation of the AI = L rule. Similar results may be obtained by using both F,M(~He)and ~(~HeJ together in the same analysis. Although the results are intriguing, we emphasize once again that the large error bars do not allow for any definite conclusions to be drawn at this time. More precise experimental measurements are evidently called for and eagerly awaited. Quality data for ~He and other light nuclei would clearly be useful in providing better estxmates for the various RNj rates [see Eqs. (29), (30)]. As shown above, these are extremely useful quantities for determining the structure of the ANdNN interaction. Unfortunately, no definite statements of this kind can be made at this time. We have a l r e a d y n o t e d t h a t n, Eq. (3), has r e m a r k a b l y d i f f e r e n t v a l u e s f o r l i g h t and heavy s y s t e m s ( s e e T a b l e 2 ) . While n ~ 0 . 3 f o r ~He, i t was found t h a t n - 0 . 8 f o r heavy h y p e r n u c l e i . The f o r m a l i s m o f D a l i t z e t a l . i s h e l p f u l in u n d e ~ s t a n d l n g t h i s t r e n d as w e l l . The An i n t e r a c t i o n i n the--0~-ground s t a t e of ~He must be in a spin-singlet state. Thus, ~(~He) is given in Eq. (30) as (3Rp1+Ep0)/2R 0, while for the heavy species

176

J. Cohen 3Rpl+ Rpo ~(heavy) = 3Rnl + Rn 0

The two expressions yield the measured values complete agreement with our earlier discussion.

provided

that

Rnl/Rn0

~

3,

in

A different phenomenological approach to the problem was suggested by Filimonov and Potashev. (Ts~ In their model, the details of the non-mesonlc decay process were not considered, and the calculation was based on a direct transition from a A-hyperon into a neutron. (The non-leptonic A-~'qHdecay would then occur as a sequence of a A-~n decay in a pole model, {~9) along with a strong pion emission.) The hypernuclear non-mesonic decay was calculated starting from an effective Hamiltonian for the direct A-~n weak transition: H^~, = GMN~n(aA~n+ bA~,~s)~^d} .

(31)

In Eq. (31), G = I0 -? is a dimension]ess constant which takes care of the order-of-magnltude of the transition rate; a^~n and barn are dimensionless coefficients which are of order unity and were taken from an analysis of nonleptonic and radiative decays of hyperons in the pole model. It is clear, however, that this analysis did not yield unique values for these coefficients. A simplified nonrelativistlc reduction of Eq. (21) was used, and the final state interaction of the outgoing neutron wlth the nucleus was neglected in the calculation. ~Ts) The authors calculated the two-body neutron decay rate, r ( ~ -~A-Iz + n), as well as the total non-mesonic rate obtained by applylngthe closure approximation to the final nuclear states; results were given for A4H, ^4He and A~He. They claimed that their results were in good agreement wit~ the available experimental data, provided that appropriate choices are made for the coefficients and for the range of the AN force (the non-mesonlc decay rates turned out to depend strongly on the latter). It was found that the contribution of the parity conserving (a_~) term in Eq. (31) dominates, with the parlty-violating contrJbutlng as Io~ n a s 8Z to the decay rate. This result is in contrast to the earlier discussion based on the elementary JtN-)nN process, where we have shown that parity-violating transitions are expected to dominate the decay process. The reason for this discrepancy is not clear and may be rooted in the pole model used by the authors. We concede with the conclusion of Dover and Walker, cI) that it is difficult to judge the merits of the direct A-~n transition model on the basis of a single calculation. Horeover, this model is somewhat more difficult to test than the other one, since in its current form it is unable to make predictions for n, Eq. ( 3 ) . More r e c e n t works ~39-45) have emphasized the s t u d y of m i c r o s c o p i c models of the

non-mesonic decay mechanism; most of our s u b s e q u e n t t h e o r e t i c a l review w i l l focus on t h i s l i n e of s t u d y , t r y i n g to u n d e r s t a n d the u n d e r l y i n g r e a c t i o n mechanism. The s t a r t i n g p o i n t of most of the p e r t i n e n t models i s the e l e m e n t a r y AN-~IN p r o c e s s . I t would, i n d e e d , be n a t u r a l to s t a r t w i t h the l o n g - r a n g e r e a c t i o n mechanism, which i s presumably dominated by o n e - p i o n exchange, and then to move on to s h o r t - r a n g e mechanisms.

Decays of A-Hypernuclel 3.4

177

Microscopic models for the non-mesonic decay: The one pion exchange mechanism

3.4.1 Why one pion exchange? In attempting a more basic and microscopic et~ e t i c a l model for the four-Fermion decay mechanism , it is natural to start with a one-pion-exchange mechanism, using the pion emitted in a free A decay (Fig. I) to construct a pion-exchange diagram (Fig. 8). This mechanism may be expected to describe the long-range part of the AN-~NN interaction, and is sometimes called the Karplus-Ruderman mechanism. (*°) [As we shall point out in the following, the pion exchange mechanism yields an isospin dependence proportional to %1"%2, while the data might be indicating that the Isospin I=1 channel dominates. ~owever, values (Tables 2 and 3) for n [Eq. (3)] are poorly determined at this time, and a critical test of any theoretical model based on such data is premature. In any case, the one-pion-exchange mechanism is worthy of study, as it may provide a convenient starting point for more elaborate mechanisms. 14°'41 Indeed, it has been studied extensively in the literature. ~39-44~] With these limitations and motivation in mind, we now turn to the pertinent formalism and calculations. 3.4.2 The transition potential The weak (v)- and strong (s) interaction ver---~ces in Fig. 8 are described in terms of appropriate weak and strong effective vertex operators. These are purely phenomenological interaction Hamiltonians, and are not derived from an underlying fundamental form. The strong vertex is the one traditionally used in describing the rtN'Nvertex. ~67~ The weak vertex is identical to that used in a phenomenological description of the free A decay, Eqs. (18)-(20). The assumption is that this is a valid approximation, although the pion is off the mass shell. Weak interaction phenomenology and experimental evidence from hyperon and kaon decay show that AI = ~ transitions are much stronger than AI = ~ ones: this is the 6I = ~ rule, whichZhas alread[ been discussed in Sec. 3.2. z In the present calcul~tion we use a pure 6I = t transition operator which should be valid to within a few percent for the A-~Nn decay. The situation is less clear, however, for AN-~NN as no high-quality data on partial decay rates are available (this point has already been demonstrated in Sec. 3.3, and will be discussed again later). Using the pseudoscalar coupling scheme (all authors seem to choose this coupling, Is°l but they use a nonrelatlvistic model anyway), the weak-interaction vertex, H w, of Fig. 8 is given by Eqs. (18), (20), and the strong-interaction vertex operator is H = g~N.~.TS%*.'~,

(32)

The s t r o n g ra~ c o u p l i n g c o n s t a n t i n Eq. (32) i s g . ~ = 1 3 . 4 . I n Sec. 3 . 2 , we have remarked t h a t most of the c o n t r i b u i t o n to the f r e e A-~N~ decay width comes from the p a r i t y - v i o l a t i n g , s-wave component of H . The s i t u a t i o n i s v e r y d i f f e r e n t f o r n o n - m e s o n i c decays v i a o n e - p i o n exchange~ s i n c e the momentum t r a n s f e r i s very l a r g e ( o v e r 400 MeV/c, as m e n t i o n e d i n Sec. 1.2 and shown i n the f o l l o w i n g . For the moment we n o t e t h a t a s i m p l e n o n r e l a t i v i s t i c e s t i m a t e , with the i n i t i a l particles a t r e s t and b i n d i n g e n e r g i e s ignored, gives qO = ~M^- Ms)

and

1~1C [M~(M^- MN)]1/2 )

Consequently, the parity conserving component of the interaction is important here. Moreover, teh~trong interaction Hamiltonian H , Eq. (32), is a purely p-wave, momentum-transfer dependent, parity-conserving operator, enhancing the effect of the high momentum-transfer. Note that the foregoing discussion is,

178

J. Cohen

N

P'

_

Fig. 8

N

Hs

A plon-exchange mechanism for th e non-mesonic weak decay AN~NN.

o f c o u r s e , based on

the

one

pion

exchange

model

f o r t h e n o n - m e s o n l c decay

mechanism. Using the s t a n d a r d Feynman r u l e s (6~) F i g . 8: -

9r

. ?. FF(a ~ )

~

M~ = [UN(E2)gXNNY5~XuN(P2)]

we

~

w r i t e the amplitude for t h i s diagram,

_

9t

+

[uN(Pl)gw(l + ky5)~kuA(Pl)]

(33)

A,N) spinor, m i s t h e pion mass, and F(q 2) has been i n s e r t e d to a c c o u n t f o r a p o s s i b l e momentum dependence a t the v e r t i c e s . This form i s s i m i l a r to an OPEP a m p l i t u d e . where u_B is the baryon (B =

In t h i s work, we use a n o n r e l a t t v i s t i c r e d u c t i o n o f M , gq. ( 3 3 ) . T h i s does not n e c e s s a r i l y y i e l d the s i m p l e r e s u l t that depends o n l y on t h e t r a n s f e r r e d momentum ~ (which i s the c a s e w i t h 0PEP), as a l r e a d y m en t i o n ed in Sec. 3.2 f o l l o w i n g Eq. ( 2 0 ) ; we adopt h e r e the same R. [We n o t e t h a t a r e l a t i v i s t i c nuclear calculation avoids this difficulty easily and n a t u r a l l y . A similar s i t u a t i o n i s e n c o u n t e r e d i n h y p e r n u c l e a r p r o d u c t i o n p r o c e s s e s such as t h e ( e , e ' K +) c a s e , (81* where r e l a t i v i s t i c as w e l l as n o n r e l a t i v i s t i c s t u d i e s have been c a r r i e d o u t . However, t h e n o n r e l a t i v i s t i c c a l c u l a t i o n s a r e s u b j e c t to other, severe uncertainties, resulting from the p r e s e n c e of t h e p i o n - b a r y o n c o u p l i n g . ( s l ' s 2 * ] Another d i f f i c u l t y o c c u r s as a r e s u l t o f t h e l a r g e e n e r g y t r a n s f e r , qO~ 88 MeV, r e s u l t i n g m a i n l y from the mass d i f f e r e n c e M - . MN " Most -. .A s t u d i e s of A - h y p e r n u c l e a r non-mesonic decays d i s r e g a r d t h i s c o m p l i c a t i o n ; a g a i n , t h i s i s not a r e a l c o n c e r n in a r e l a t i v i s t i c treatment.

We shall use, therefore, a nonrelativistic form of M R which r e a d s (34)

Decays of A-Hypernuclei

179

= ½(MA+ MN) is used in Eq. (34). |A different satisfactory The average mass nonrelativistic reduction, directly obtained from the v 5 pseudoscalar vertex operator, gives k~l" (

Pl

Pl M~N ).

This form depends on both the momentum

transfer ~ and the sum ~i+~, unlike Eq. (34). Since (MA-MN)/2M<
the effect

In Eq. (34) m is effective pion mass introduced here in order to take care of the finite, relatzvely large, energy transfer in the pion propagator. (This energy transfer is roughly constant if the initial A-N pair has a relatively low kinetic energy.) It is evidently a poor approximation to assume a static propagator as conventionally done in the nucleon-nucleon domain. At the present time we have no guidance as to the appropriate treatment of the energy transfer in Eqs. (33) and (34). We shall therefore use m K as a constant, and explore the effect of varying its value within what would appear to be reasonable limits. (44) In the present review, as in virtually all published theoretical works, we are primarily interested in a nonrelativistic description of the decay processes. Towards this end, a transition potential is obtained from M,, Eq. (33). the coordinate-space potential is a Fourier transform of M (~), Eq. (34): Va(~ ) = ~ d~23-~) Ma(~)e i~'~

(35)

It has three distinct components, two of which are parity-conserving and one is parity-violating. The former are analogous to the nucleon-nucleon one pion exchange potential, and consist of a central piece with the 8(~) part removed" ~

c * grLNN Va(r) = - gw H ~N

-mKr -3 1 e ~ * * + m~ ~ 4rtmar ~1"=2xi'~2

X ~

'

(36)

and a tensor part, t ~ grtNN X -3 1 .1 i V,(r) = - gw M ~N ~ff mK(~ + m~r + ~ ^

"

"

9

where $12(r ) = 3(~'r)(~2"r ) - oz'~ 2. In a form factor F(q'-) = I in Eq. (34), with F(q ) ~ 1 (see Subsec. 3.4.5). the central potential, Eq. (36), is reasonable two-body wave function should potential is 2

(40)

e -m~r " ) 4nm,r~ S12(r)~l't2

(37)

obtaining (36) and (37) we have used but it is straightforward to calculate Note also that the 6-function part of omitted, since we believe that any vanish as r-~0. The parity-violating

-mnr =

ig w

grLNN m~(1

+ ~

1

m.r

) e

4=~ r

~ "~ o2"rx1-x2

.

(38)

"This is a model-dependent result which may not work equally well for all partial waves; care must be exercised in an uncorrelated plane wave calculation.

180

J. Cohen

Thus, v(~)

= v~(~) + vt(~) + v~v(~) .

A discussion of the properties of the various components of the transition potentials may prove beneficial. Of the three components VC(2), vt(~) and V~V(2), the former two are similar to the corresponding components of the strong-lnteraction OPE. All three types of potentlals are scalars in the combined, spin and orbital angular momentum, space, and therefore leave the angular momentum J of the relative baryon-nucleon wave function unchanged. Furthermore, the matrix elements of these potentials between twobody initlal and final states are independent of J , the z-projection of J The three potentials have, however, distinctive properties in the separate, spin 9_£ orbital angular momentum, spaces. ~/e are interested in the matrix elements of the three types of potentials V~(~) (where C = c, t, or pv) between the initial and final states characterizea by the relative orbital angular momenta L and L', the total spins S and S', and the combined total angular momenta and their z-projectlons, J,J and J', J'. z . z . The central potential [ E q . (36)] is a scalar zn tSe total-spzn space (as well as an overall scalar in the combined J-space). Thus, the above quantum numbers are left unchanged between the initlal and final states: L = L', S = S', J = J', and Jz = J'., The matrix element of the %z.%2 operator depends only on S, and has the well-known values , ~

~

j,-3,

= 6jj,6jzj'z6SS ,6LL , x \

I,

S = 0 S=

1

(39)

In the spin-space itself (with the two individual spins coupled to a total S, S ) this operator is independent of S as a result of its scalar properties. The tensor interaction [Eq. (37)] is proportional to2the scalar product of two second-degree tensors, [%1 x %2] (21 and [2 x 2[. ~ ~ (It could also be thought of as a scalar built from the individual scalars r 2, ~ -2, % .2 and -~ , . 1 2 ~z'5' but the former representation is more convenzent for our purposes). Thus, S12 = 3~'9 [[51x 52](2) x [r x r](2)] (0)

= 4~([51x 521(2). [~ix ~11(2)),

(40)

where the round brackets represent a scalar product, and YI is an 1 = 1 spherical harmonic. ~v6) Being a non-central interaction, the tensor potential does not remain Invariant under a rotation of the space or the spin coordinates separately~ the orbital and the spin angular momenta are not constants of the motion. However, the magnitude of the total spin is still constant for a twobody system in the presence of the tensor force. Furthermore, the expectation value of the tensor operator, $I~ , vanishes in the singlet spin state (S = 0). (Only the central and parity-vlolating potentials operate in this spin state, resulting in S = 0 ~ S' = 0 for the central and $ = 0 ~ S'= 1, S = I ~ S' = 0 for the parity-violating interactions.) In the orbital angular momentum space the presence of the tensor interaction implies 6L = 2, since this interaction is proportional to T 2. Thus, one finds for the matrix element that J = J', J z = J'z and S = S' = i in order to obtain a nonvanls~Ing contribution, furthermore, for L = 0 we have L' = 2 and J = J' = I. For future reference we note that

Decays of A-Hypernuclei

181

L' 1 J = -30 & j j , & j z j ~ S , 1 6 S , , I ( - 1 ) J ( 1 L 2 )

x ~(2L'+l)(2L+1)

L'I L'' 0 0 0

E (-1)L''(2L''+1) L''

)( 0L "

1 L

1 1 )( 0 L L'

2

0

L*'

)

(41)

The p a r i t y - v i o l a t i n g i n t e r a c t i o n [Eq. (38)] is the s c a l a r product of two v e c t o r s , one in the spin space ( ~ ) and the o t h e r in the o r b i t a l space (r or 71). This i n t e r a c t i o n r e s u l t s in L' = h+l (h' = L is forbidden) f o r the r e l a t i v e o r b i t a l angular momentum, and t h e r e f o r e changes the p a r i t y of the r e l a t i v e two-body wave function. Since i t c o n t a i n s the i n d i v i d u a l spin operator ~2' t h i s p o t e n t i a l changes both the t o t a l spin of the two-body system and i t s z - p r o j e c t i o n (S and S ). The e x p e c t a t i o n value of ~2. r in the spin slnglet (S = 0) state vanishes, while S = 0 ~ S' = I, 1 ~ D and 1 ~ 1 are allowed. In these transitions, some limitations on the values of S and S' apply. For instance, S = 1, S = 0 + S' = 1, S' = 0 is forbidden [~ince %" operating on [1,0> = 4 ~ t # + #~) cannot give bac~ the same state. It could~ however, yield the antisymmetric state [00> = 4 ~ t # - St)]. The parityviolating interaction produces changes in both S and L such that 6 S = S S'S = - AL g • - (L E - L'), or AJ Z = 0 " The matrix element of %_.r between t~e ~ Z initial and final state is

= _ ~" 6jj,Sjzj~(_I)J[(2L,+I)(2L+I)(2S,+I )

x (2S+1)]112( L' I L

L'S'J'

½

S'

I

3.4.3 The transition rates in a Fermi ~as model; General properties of the transition amplitude It is now straightforward using the transition amplitude M (~), Eq. (34), or the transition potential VK(~ ) in Eqs. (35)-(38), to obtain t~e non-mesonic decay rate for very heavy A-hypernuclei, treated as a A-hyperon embedded in nuclear matter. Details are given by Adams ~39~. The actual calculations use a coordinate-space transition potential [as in Eqs.(35)-(38)]. The rate is then obtained by inserting initial (AN) and antisymmetyrized final NN states, squaring the matrix elements, summing over all possible final states, integrating over initial undetected momentum variables, and averaging over initial polarizations. It is convenient to transform to relative and c.m. three-momenta for the initial and final pair of particles. The pertinent transition matrix element is

<¢(A-2)X(NN) IVl ~(~Z)> , where ~(A~Z) i s the i n i t i a l hypernuclear ground s t a t e , ¢(A-2) i s the (bounds t a t e ) wave f u n c t i o n of the f i n a l nuclear system of A-2 nucleons, and X(NN) i s the f i n a l - s t a t e wave f u n c t i o n of the two nucleons knocked in the AN-~TN decay process. The t r a n s i t i o n p o t e n t i a l V to be considered i n t h i s s e c t i o n i s V~, Eq. (35), however other choices f o r V w i l l be discussed l a t e r .

182

J. Cohen

For a A-hyperon a t r e s t ( i n t h e l o w e s t momentum s t a t e o f t h e Fermi g a s ) and a nuc l eo n w i t h momentum between 0 and t h e Fermi momentum k_, t h e r e l a t i v e A-N momentum ~ in t h e i n i t i a l h y p e r n u c l e a r bound s t a t e v a r i e s From k=O ( w i t h both the A and the n u c l e o n a t r e s t ) to about k = krMA/M+, where M. = MA+ M. ( t h i s upper l i m i t i s o b t a i n e d when c o r r e c t i o n s of order (k~/MN) ~ and h i g h e r a r e neglected). (For k r = 270 MeV/c, t h e l a t t e r l i m i t i s 146 ~eVTc.) The r e l a t i v e N-N momentum I[I ( i n th e f i n a l s t a t e ) i s o b t a i n e d as a s o l u t i o n o f t h e e n e r g y conservation equation

2(~

+ t2) ~/2 = ( ~

+ k2) ~/~ + (.~ + k~) ~/~,

and v a r i e s from about t = 417 MeV/c ( f o r k = O) to 4 4 1 M e V / c ( f o r k = krHA/M+). In o r d e r to f a c i l i t a t e the e x a m i n a t i o n o f two-body c o r r e l a t i o n e f f e c t s , most a u t h o r s have c o n s i d e r e d i n i t i a l and f i n a l relative wave f u n c t i o n s i n s t a t e s which a r e a s y m p t o t i c a l l y s p h e r i c a l ( r a t h e r than p t a n e w a v e s ) . (Note t h a t t h i s s e c t i o n d e a l s w i t h a Fermi gas model and, i n t h e a b s e n c e o f such c o r r e l a t i o n s , the p l a n e - w a v e s t a t e s would have been more c o n v e n i e n t i n t h e c a l c u l a t i o n s . ) The i n i t i a l s t a t e s , c h a r a c t e r i z e d by the quantum numbers o f t h e t o t a l s p i n S, the r e l a t i v e o r b i t a l a n g u l a r momentum L, and t h e t o t a l a n g u l a r momentum and i t s z-component J and J z , a r e d e n o t e d by ~; th e f i n a l s t a t e s by 8. The r a t e f o r the non-mesonic decay s t i m u l a t e d by a n e u t r o n (n) o r a p r o t o n (p) assuming a two-body p r o c e s s AN-mN (Nffin,p) i s , then ¢39)

kFMA/M+ ~0

dkk2(RN0+ 3 R N 1 ) "

The p a r t i a l A-N c a p t u r e r a t e s R the s t a t e a) a r e t h o s e i n t r o d u c ~

RNj =

f o r t o t a l a n g u l a r momentum J ( p e r t a i n i n g i n Sec. 3 . 3 ; they depend on k v i a :

E R(~'~6), R(~*8) ~t ~

8

]<8,t[Ml~,k>[ 2

to

(44)

We n o t e the u s u a l (39-41) a p p r o x i m a t i o n o f k e e p i n g i n R- N J o n l y t h o s e s e t s ~' o f i n which t h e r e l a t i v e a n s u l a r momentum i s z e r o ( s - w a v e ) . This approximation, which could be j u s t i f i e d more e a s i l y in Sec. 3 .3 f o r few-body h y p e r n u c l e i , i s used f o r r e a s o n s o f s i m p l i c i t y and a l s o s i n c e t h e r e l a t i v e A-N momentum i s q u i t e low. Then, th e i n i t i a l s t a t e i s L = O; J = 0, 1; I = ~ t h e p o s s i b l e t r a n s i t i o n s c o n s i d e r e d a r e s-~s, p, d. T h i s s-wave a p p r o x i m a t i o n i s n o t necessarily reliable. I n d e e d , we s h a l l a r g u e t h a t t h e r e a r e l a r g e r e l a t i v e pwave components in the i n i t i a l A-N s t a t e , and p r e s e n t new e s t i m a t e s f o r a f i n i t e n u c l e u s w i t h L = 1 components i n c l u d e d . T h i s a p p r o x i m a t i o n has a c o n s i d e r a b l e e f f e c t on t h e r e s u l t s d e s c r i b e d i n t h i s sec--'~on, as made c l e a r by the f o l l o w i n g d i s c u s s i o n . The r e l a t i v e w e i g h t o f t h e i n i t i a l J = 0 and J = 1 p a r t i a l r a t e s i n Eq. (43) (1:3, respectively) is a direct c o n s e q u e n c e o f t h e s c a l a r p r o p e r t i e s o f the t r a n s i t i o n p o t e n t i a l s i n the t o t a l a n g u l a r momentum ( J ) s p a c e [Eqs. ( 3 6 ) - ( 4 1 ) ] . S i n c e the t r a n s i t i o n o p e r a t o r i s I n v a r i a n t under r o t a t i o n s [ n o t e t h e Gj j , f a c t o r s in Eqs. ( 3 9 ) , (41) and ( 4 2 ) ] , th e w e i g h t f a c t o r s a r e o b t a i n e d Z a ~ 2J + 1. S i m i l a r remarks a p p ly to Eq. (29) as w e l l .

Decays of A-Hypernuclei

183

Using the AT = ~ r u l e i t i s p o s s i b l e to r e l a t e the n e u t r o n - and p r o t o n - i n d u c e d p a r t i a l decay r a t e s f o r the f i n a l NN I~ = 1 s t a t e . A general result, i n d e p e n d e n t of the decay mechanism, has been p r e s e n t e d i n gq. ( 2 4 ) . We now show t h a t t h i s c o n c l u s i o n can a l s o be reached f o r the o n e - p i o n exchange decay mechanism. I n the n o t a t i o n of Eqs. (20), (32) and ( 3 4 ) , the c o n t r i b u t i o n to the m a t r i x element from the i s . s p i n o p e r a t o r i s

=
½,- ½; ½,~Nz>] =

1

1 , n e u t r o n (~Nz = - ~, p r o t o n (ZNz= + ½,

I --

'

-

1)

Ifz- O ) -

Ifz--

(45) •

Here ZNz= ± ½ f o r a p r o t o n or a n e u t r o n ( r e s p e c t i v e l y ) ,

the A i s r e p r e s e n t e d

by a {½ , - ½ > state in this formalism, and Ifz= O, -I for a final pn state or nn state, respectively. We have used the fact that ~I.~T2 is independent if Ifzand is diagonal in the total is.spin space (with =I ). We thus conclude that for the I~= I transitions, the neutron induced partial decay rate is twice as large as the proton induced one. We mention again that this conclusion, derived here for the special case of the one-pion exchange mechanism, was shown in Eq. (24) to be a general result, independent of the intermediate decay mechanism (however, the AN-~nN process is assumed to take place). We recall that the AI = ~ rule can now be used to substitute R = 2R , and the inequality R , < 2R ~ is obtained. We thus obtaln' fromn~q. ( ~ ) the total rate for the non-~esonlc decay in the Fermi gas model:

r ~ = r~M + ~M=

3

~0

dkk2(RpO+Rpl+Rnl)"

(46)

Roughly s p e a k i n g , R~j(N = n , p ; J = 0 , 1 ) i s p r o p o r t i o n a l to the decay p r o b a b i l i t y f o r an s-wave A-N system with a t o t a l s p i n J , per u n i t n u c l e o n d e n s i t y , a t the p o s i t i o n of the A. A l l decay r a t e s a r e p r o p o r t i o n a l to ~2~2 The r a t e s f o r p a r i t y - c o n s e r v i n g vw~ffNN* t r a n s i t i o n s (s-as or s-)d) a r e p r o p o r t i o n a l to X2, w h i l e the r a t e s of the p a r i t y v i o l a t i n g t r a n s i t i o n s a r e i n d e p e n d e n t of X. (This i s e a s i l y i n f e r r e d from Eqs. (20) and (32) or Eqs. ( 3 6 ) - ( 3 8 ) . The term p r o p o r t i o n a l to k i n the weakinteraction Hamiltonian H [Eq. (20)] has the same form as the s t r o n = i n t e r a, c t i o n v e r t e x HS [Eq " w ( 3 2 ) | , and, combined w i t h the l a t t e r , the two Y i e l d a parity-conserving amplitude similar to t h a t from s t r o n g one p i o n exchange. The p a r i t y - n o n c o n s e r v i n g ( u n i t o p e r a t o r ) term i n Hw does n o t c o n t a i n the

184

J. Cohen

o p e r a t o r Vs.} Note t h a t b o t h t h e p a r i t y - c o n s e r v i n g and p a r i t y - v i o l a t i n ~ transitions are experimentally accessible (unlike t h e N-N c a s e , where t h e s t r o n g i n t e r a c t i o n masks t h e p a r i t y - c o n s e r v i n g weak i n t e r a c t i o n ) . The weak d e c a y s o f the A - h y e r n u c l e u s t h e r e f o r e provide unique information~ not a v a i l a b l e i n the n u c l e a r c a s e . As we s h a l l emphasize in the f o l l o w i n g ( S e c . 3 . 6 , T a b l e 6 ) , most d e c a y mechanisms p r o p o s e d i n the l i t e r a t u r e are reported to r e p r o d u c e t h e e x p e r i m e n t a l l y o b s e r v e d t o t a l decay r a t e . A more r e s t r i c t i v e t e s t f o r any o f the models would be the c o n s i d e r a t i o n of the ratio of proton-to-neutron-induced r . j r . . [Eq. ( 3 ) ] . T h i s p a r t i a l r a t e s , ~ • F.p / r ".. o r , a l t e r n a t i v e l y , r a t i o can be e x p r e s s e ~ i n terms of t h e p a r t i a l r ~ t e : a s

.

. rNSp

rN~

=

rpo+ 3rpl rn0+ 3rnl

(47)

or n =

2 ~ rpO + r n l

(48)

,

rpo 4- rnl4- rpl

f o l l o w s from Eq. ( 4 6 ) . An a l t e r n a t i v e where the s e l f - e v i d e n t n o t a t i o n ~ (F.j) way of expressing t h i s r a t i o i n terms o f t h e i s o s p i n s t a t e s i s e a s i l y o b t a i n e d u s i n g , e . g . , T a b l e 4 and t h e r e l a t i o n R o ffi 2 Rpo:

•

rsP

3FI f=O+

r~(Pf_)_ 1

(49)

or

2r(P) If=l

(50)

n ffi 3r(P )

If=l + 3FIf=O where F (p)

i s o b t a i n e d from the f i n a l np s t a t e .

[ I n d e r i v i n g Eqs

(49) and

If=l (50) it is crucial to include the 2J+l factors, as in Eq. (43). For example, F(P)If=I= rp0 + 3r (3si~ 3pl); but rif=o • Fpl - F(3Sl~ 3pl).] Using Table 5

we can w r i t e e x p l i c i t l y ~n =

[r(lso~ I%)

4-

r(Iso_~3po) + 31-(331.~331) 4- 3r(3sl,3dl) + 3F(3si_~ ipl )

+ 3F(3Sl~3p1)]/[2r(1s0~lso)+ 2F(ls0,3po ) + 6r(3s1,3p1)].

(51)

Decays of A-Hypernuclei

185

It is important to note that, assuming an initial relative AN s-wave, the one pion exchange mechanism would sreatl 7 suppress the neutron-induced (relative to the proton xnduced) partial rate. This observation follows from the fact that the major contribution to the decay rate results from the tensor interaction [Eq. (37)] while the central-interaction contribution is only marginal~ Antisymmetry (Pauli principle) considerations dictate that the tensor force does not contribute to the An-mn transition [see Eq. (41) and Table 5], hence the predicted suppression. Moreover, the preceding discussion implies that the dominant contribution to the A+n * n+n transition in the one-pion exchange model comes from the parity-vlolating component of the transition potential, V~.v [Eq. (38)]. Therefore, the nn final state in this model is mostly the r~sult of a parity-violatin$ transition. Ne note, however, that retaining ~igher possible partial waves of the initial relative AN wave function in the calculation may lead to modifications in this predictions. As noted earlier, this will be discussed in Sees. 3.4.4 and 3.6. It is, indeed, clear that a useful comparison of theory and experiment can be obtained by considering the parity-violating and parity-conserving rates. A handy quantity may be the ratio of pv and pc transition rates, rDv = r(ls0~ 3p 0) + 3r(351 ~ 3p 1) + r(3sl ~ Ip I) E

r(lso~ is0) + r(3s1~ 351) + r(3s1 ~ 3dl)"

(52)

Some approximations are useful when calculating the decay rates, as discussed by Adams (39) and McKellar and Gibson. ( 4 0 ) In particular, the matrix element could be calculated (4°) at a fixed final relative momentum t = 420 MeV/c (the matrix elements vary by approximately lOZ over the range of variation of t). Furthermore, the combination R p _ + R B t. + R n_ [Eq. (46)] is a weak function of k, the relative A-N momentum. ~hanglng k ~rom 0.1MeVlc to 125 MeVlc does not affect the results by much. It is therefore useful to approximate

30

rNM = 8

(Rpo + Rpl + Rnl) '

2k~ P = 3~ 2

(53)

This expression is equivalent, in the Fermi gas model, to the form given by Block and Dalitz, Eq. (29), and using the /~I = ~ rule.

3.4.4 Finite-hypernuclear treatment I n t h e p r e v i o u s d i s c u s s i o n , Subsec. 3 . 4 . 3 . , a s i m p l e h y p e r n u c l e a r model was used where t h e A-hyperon i s embedded i n nuclear matter. This model has some obvious drawbacks. For example, if the variation in the nuclear density is taken into account, a nucleon produced near the nuclear surface with a momentum larger than the local Fermi momentum at that location should be able to leave the nucleus. - - - ~ c e the local Fermi momentum, k F (f) at the vicinity of the nuclear surface can be much lover than k_ at central nuclear densities (kr(f) < kr), the modified local-density p~cture allows for higher rates of hypernuclear mesonic decays, especially in light species (where most of the nucleons are at densities much lover than the nuclear matter density). Even for medium hypernuclei, one should not a priori discard the possibility of finite-nucleus effects, especially when dealTngw---'~ the one-pion exchange mechanism which is characterized by long-range (~i - 2 fm) e f f e c t s . *This fact, which is a well-known property of the OPE, is clearly demonstrated in Table 8 of Sec. 3.6 (See. 3.6 contains all the numerical results for this review).

186

J. Cohen

A r e l a t i v e l y s i m p l e and f r u i t f u l e x t e n s i o n from a n u c l e a r - m a t t e r to a f i n i t e n u c l e u s f o r m a l i s m may be o b t a i n e d by u s i n g t h e Lo cal D e n s i t y A p p r o x i m a t i o n (LDA) a l l u d e d to above, whereby l o c a l n u c l e a r d e n s i t i e s and Fermi momenta a r e used i n s t e a d o f the c o r r e s p o n d i n g c o n s t a n t n u c l e a r m a t t e r or c e n t r a l v a l u e s . This approach was f o l l o w e d by Oset and S a l c e d o ; (43) their work w i l l be d e s c r i b e d i n more d e t a i l i n t h e n e x t S e c t i o n . Our main i n t e r e s t h e r e w i l l be in a more r e a l i s t i c , shell-model finite-nucleus calculation. However, we r e s t r i c t our a t t e n t i o n to a s i m p l e s i n g l e - p a r t i c l e model, where t h e i n i t i a l h y p e r n u c l e a r s t a t e i s d e s c r i b e d as a A in t he l s s i n g l e - p a r t i c l e l e v e l coupled to the ground s t a t e o f the n u c l e a r c o r e : ,(~Z)

=

,core(A-Iz)g[Als> .

(54)

( Thi s i s known to be a good a p p r o x i m a t i o n f o r d e s c r i b i n g hypernuclei.) The decay dynamics i s c o n t a i n e d in the m a t r i x e l e m e n t • -)r _) .-)t .) T
; ½ sl

I

t h e ground s t a t e

z

s.

>

of A-

(55)

where the h-hyperon i s assumed ( f o r r e a s o n s d i s c u s s e d a b o v e ) to d ecay from an i n i t i a l Ihls> s t a t e ; f o r s m a l l and medium n u c l e i the i n i t i a l - s t a t e n u c l e o n has n.=l. The o u t g o i n g n u c l e o n s a r e s c h e m a t i c a l l y r e p r e s e n t e d h e r e by p l a n e waves; d i s t o r t e d waves may be s u b s t i t u t e d instead, however o u r e n s u i n g d i s c u s s i o n i s more e a s i l y apprehended in terms o f the p l a n e - w a v e s t a t e s . F i n a l l y , in t h i s S e c t i o n we a r e p r i m a r i l y i n t e r e s t e d i n V(~) = VK(~), Eq. ( 3 5 ) , n o n e t h e l e s s the present discussion is quite general in t h i s respect and a l l o w s f o r o t h e r c h o i c e s o f V to be d i s c u s s e d l a t e r . As noted e a r l i e r (Subsec 3 . 4 . 3 ) , it i s d e s i r a b l e to d e a l w i t h s p h e r i c a l twobody r e l a t i v e s t a t e s . A transformation from h a r m o n i c - o s c i l l a t o r (HO) s h e l l model s i n g l e - p a r t i c l e s t a t e s to a c o m b i n a t i o n o f r e l a t i v e U c.m. wave f u n c t i o n s i s o b t a i n e d i n t h e LS c o u p l i n g by u s i n g Hoshinsky b r a c k e t s . ~ e 3 ) For l i g h t h y p e r n u c l e l , where 1 N does n o t e x c e e d a few u n i t s , and w i t h t h e A r e s i d i n g i n the I Als> s t a t e , the t r a n s f o r m a t i o n is directly o b t a i n e d a f t e r some s i m p l e algebra. It is useful to recall that, transforming from the single-particle states with quantum numbers n 1 1 1 ; n i 2 , into c.m. and relative states with corresponding NL; nl, the quantum numbers satisfy the relation 2n,+ 11 + 2n 2 + 12 = 2N + L + 2n + 1 . For completeness, we note that

1 (.2r~+~2r22) =

=

. . Yo0(rl)Yo0(r2)

1 (2~x2R2+ ~ =2r2 ) . . N Rls N rls e - ~ Y00(R)Yoo(r) . {~ls>(a ~-~)[~is>(Cd~/)

(56a)

Decays of A-Hypernuclel

187

and _ 1 (=2r2 + 2 r 2 ) [AIs>INIp> = ~Is~ip "'A"N = r 2 e -

½(2a2R2+ ½ a2r 2)

1 [NISNRIPc~-~- R e P~ _

r

.

R

~

.

- [ (2a2R 2 + ½ a2r 2) r e

1 [l~ls>(~J~Z)lfflp>(=~Z) P~

.

.

Y1x(R)Yoo(r)

1

NlPN ls =

.

~ YoO(rl)YIk(r2 )

. . Y1x(r)Yoo(r)]

- i~lp>(~J~Z)lffls>(=~Z)

1.

(56b)

Here, the N"1 are a p p r o p r i a t e n o r m a l i z a t i o n c o e f f i c i e n t s , and the c m and r e l a t i v e c o o r d i n a t e s a r e d e f i n e d in terms o f ~1 and i~2 as ~ = ~ , + i~2) and ~ = i~ -i~ , assuming N = M... A s i n g l e H0 parameter a i s used, but the q u a n t i t y to = x

"

"

" I ~B differs for the nucleon (B=N) and the A This is done in the absence of any other, well-established theoretical guidance. Note that Eq. (56b) goes beyond the relative A-N s-wave approximation discussed so far [see Eqs. (43), (44) and the pertinent discussion]. Using final (56), (55),

(i)

plane waves, it is easy to carry out the separation of the two-nucleon state into c.m. and relative components. Using Eqs. (36)-(42), (55)the following expressions are obtained for the matrix element T, Eq. in the case of a one-pion-exchange mechanism and a 1s-nucleon:

For V(r~) = Vii(r), c -~

T~I = P Yoo(P+)Yoo(I~)f(P+) m2 I ~ r d r J o ( t r ) g ( r ) (57a)

SSz where ~+ and ~ a r e the c.m. and r e l a t i v e p+ m~ f(p+) = exp[(~ )2] e x p [ ( - ~ ) 2 ] ,

g(r)

=

and

2 m~ e x p [ - ~ - (r + 2 - ~ )2] ,

-3 , S=O ~S= (

1 , S=I

momenta f o r the f i n a l s t a t e ,

188

J. Cohen

(ii)

For the t e n s o r p o t e n t i a l ,

~(~) = Vn(r )t ~

Tnt = 4nYoo(P+)f(p+) mll "2 ~:rdr J2(tr)] 3 +

(-~)

~. ~-

m' SzS~ Y2m'

x

,

1=_ + 1 mnr ( m n r ) 2 ] g ( r )

<~)< ½ sl z ~ s~" ~ ~ ' 1

,

(57b)

where the allowed values f o r the angular momentum and spin quantum numbers have been used. (iii) For V(~) = v~V(~) we find T~v= - 4hi ¥oo(P+)f(p+)mn[®rdr ~o

Jl(kr)(l + I )g(r) mKr

~4 ~ ~ ~'~z ~ ~ ~'~>4 ~ ½ ~1~ ~ ~z><~'~o' I~'~ ~z>

(57c)

[In Eq. (57a)-(57c), coupling constants have been suppressed.] Similar, but more complicated expressions are obtained for a 1p-nucleon. (44) The full matrix element T obtained this way can then be used for calculating the finitehypernucleus non-mesonic transition rate. Note that the LS basis is used in Eqs. (55)-(57), while JJ coupling is implied by gq. ( 5 4 ) . This r e q u i r e s a t r a n s f o r m a t i o n from the l a t t e r to t h e former basis. F u r t h e r m o r e , i n c a l c u l a t i n g t h e h y p e r n u c l e a r n o n - m e s o n i c d e c a y b a s e d on the AN-~NNmechanism, i t i s n e c e s s a r y to t a k e i n t o a c c o u n t t h e o v e r l a p o f .~ with the AN p a i r c o u p l e d to the r e m a i n i n g n u c l e a r s y s t e m ^-2Z o r ^ - 2 ( Z - 1 ) I th~s i n f o r m a t i o n i s c o n t a i n e d in the s p e c t r o s c o p i c factors, which depend on t h e quantum numbers o f both the n u c l e o n s t i m u l a t i n g t h e d e c a y and t h e r e m a i n i n g nucleus. (The l a t t e r p r o c e d u r e can be thought o f a s p r o v i d i n g t h e e f f e c t i v e number o f s t i m u l a t i n g n u c l e o n s i n t e r a c t i n g w i t h t h e d e c a y i n g A - h y p e r o n . ) The p e r t i n e n t f o r m a l i s m was d e s c r i b e d in d e t a i l by Heddle and K i s s l i n g e r . 145''41

Decays of A-Hypernuclei

189

3 . 4 . 5 The r o l e o f s h o r t - r a n g e two-body c o r r e l a t i o n s , d i s t o r t i o n e f f e c t s and p i o n - b a r ~ o n - b a r y o n v e r t e x f u n c t i o n The e x t r e m e s i n g l e - p a r t l c l e model d o e s n o t i n c l u d e any two-body r e p u l s i v e c o F r e l a t i o n s , and, i n t h e p r e s e n t c a s e o f a AN p a i r , even t h e P a u l i p r i n c i p l e has no e f f e c t (at least at the level of the b a r y o n s ) . Under t h e s e c i r c u m s t a n c e s , it is possible f o r t h e n u c l e o n and ^ hyperon to i n t e r a c t a t v e r y s h o r t d i s t a n c e s (r-~O). F o l l o w i n g t h e t r a d i t i o n a l n u c l e a r - p h y s i c s a p p r o a c h , most a u t h o r s c39-41'45'84) add by hand an ad hoc short-range repulsion effect. Note t h a t we have a l r e a d y assumed t h e effl-st'-~-'ce of such r e p u l s i o n i n S u b s e t . 3.4.2, where t h e b - f u n c t i o n p a r t o f t h e c e n t r a l OPE p o t e n t i a l i s o m i t t e d i n Eq. ( 3 6 ) ; a s i m i l a r s t e p i s t a k e n i n many n u c l e a r physics studies. Two a p p r o a c h e s a r e found i n t h e p e r t i n e n t l i t e r a t u r e regarding the treatment of s h o r t range c o r r e l a t i o n e f f e c t s in hypernuclear non-mesonic decays. In the first, a phenomenological coorelation function, f(r), i s i n s e r t e d by hand, m u l t i p l y i n g the u n c o r r e l a t e d r a d i a l i n t e g r a n d o r t h e s h e l l model t w o - p a r t i c l e wave function [e.g., in Eqs. (56) and (57)] in the two-body transition matrix element, Eqs. (44) and (55). The second approach, originally used by Adams, c39~ uses an approximate form of the solution to the Bethe-Goldstone equation with a hard core. Various forms were used in the literature for f(r) in the first of the two approaches alluded to above. McKellar and Gibson ¢4°I multiplied the potential (or the radial integrand) by

-r2/r~ f(r) = 1 - e where r = 0.70 fm, 0.75 fm or 1 fm; a similar form was used by Dubach. (4xl Heddle ~nd Kisslinger (4s,e4) following Miller and Spencer (85) used a slightly different _oct2 f(r) = 1 - e (I - ~r2), with ~ = 1.1 fm-2 and 8 = 0.68 fm -2. These values for ~ and 8 were tested in nuclear calculations of parity violations by Haxton et al. and Brown et al. (86) The phenomenological form is known to yield results-w~ich are consistent with much more elaborate and sophisticated treatments of correlations. (86) (Note, however, that the resulting correlation effects have not been previously tested experimentally for the AN system.) The short range correlations present a potenetial source of problem regarding the proper normalization of the two-body wave function. It was argued (87) that one may have to renormalize the relative two-body wave function once a correlation function is inserted into the two-body integral. (The c.m. part is unaffected by the short range correlations, which depend only on the relative coordinate, r.) The question has been addressed by Cohen and Walker. 144~ Normalizing the HO relative (two-body) wave function in the presence of the -r2/r~ correlation function f(r) = 1 - e , we find modified relative normalization constants. (For example, the relative ls constant is replaced by N~ l s = where b 2 = ½ 2 + _~ ; e v i d e n t l y Nr, l s >Nrl s ,

2.-l/4[(P//a) 3- 1/b31-1/2,

R0 which might compensate for the effect of the correlation function.

Similarly,

190

J. Cohen

N~IP/N~P= [I - ~5/(~/'b)5]-I/2.)

We find, however, that the renormalization

effect is completely negligible, and amounts to about 2Z for matrix elements in the ls case with rn = 0.75 fm, because the normalization at short distances is weighted with r2[ Nevertheless, the short range correlations are very important for the calculation of the transition matrix element, Eqs. (55) and (57), having a crucial effect once the transition potential is included in the radial integrand. (Detailed numerical zesults will be discussed in Sec. 3.6.) The second approach for the treatment of A-N short range correlations was originally used in the calculation of Adams ~39) and later also by Heddle, Cheung, and Kisslinger. c45'84~ These are nuclear matter calculations, where the A and the stimulating nucleon are assumed in an initial relative s state. In the absence of initial state interactions, the relative two-body wave function is just a spherical Bessel function, ~i(kr) = jo(kr), where k is again the relative AN momentum. Assuming a hard-core A-N potential (making no distinction between slnglet and triplet states), Adams (39) used an approximate solution of the Bethe-Goldstone equation for the (initial) wave function: (sa) sln(krc)Si(Sr) kr ] 0(r-re)

Oi(k'r) = [3o (kr) -

(58)

I n Eq. ( 5 8 ) , r c i s the core r a d i u s t a k e n as 0 . 4 fm by Adams, the s i n e - i n t e g r a l function S i ( x ) = - r®sin----~Udu Jx u and the parameter ~ = 1.633 fm-1 i s i n v e r s e l y p r o p o r t i o n a l to the h e a l i n g d i s tance. A c r i t i c a l s t u d y of the r o ] e of p h e n o m e n o l o g i c a l s h o r t r a n g e c o r r e l a t i o n s has been p r e s e n t e d r e c e n t l y by Moszkovski and Noble, ~89~ who a d v o c a t e a s y s t e m a t i c t r e a t m e n t of n u c l e a r o b s e r v a b l e s s t u d i e d on a c a s e - b y - c a s e b a s i s . The decay p r o c e s s r e s u l t s , f o r the most p a r t , i n t w o - n u c l e o n k n o c k o u t . I n the absence of f i n a l s t a t e interactions, the r e l a t i v e p a r t i a l waves a r e s i m p l y g i v e n by the s p h e r i c a l B e s s e l f u n c t i o n s , ~L)(tr)

= JL(tr)

,

where I~| is the re]ative NN momentum in the final states. (For a relative swave AN initial state, we see from Table 5 that possible values of L are L = 0,1,2.) Distortions of the outgoing nucleons will change this simple result. The effect may be estlmated (4s'e49 by using an elkonal approximation along with

Decays of A-Hypernuclei

191

the central part of a nucleon-nucleus optical potential. Thus, assuming a back-

t o - b a c k emergence of the two f i n a l s t a t e nucleons the d i s t o r t e d r e l a t i v e NN p a r t i a l waves a r e

:

~

_

(namely, ~'1 = - ~'2 • ~ ~ ) '

Uopt( ~, ~, ~idz']}

PL(COsO)d(cosO). (59)

Cheung, Heddle and Kisslinger (45's4) used the eikonal approximation, Eq. (59), in their nuclear matter and finlte-nucleus calculations. In Refs. (40) and (41), final state wave functions were obtained from a solution to the Schr~dinger equation with a Reid soft core nucleon-nucleon interaction. As a final ingredient included in the AN-~NN weak decay calcu]ations based on a one-pion exchange mechanism, we consider the effect of the nNA and ,NN vertex functions, or form factors. These should be included in order to take into account off-shell and nonzero momentum-transfer effects. A form factor simulates the fact that the pertinent hadrons are extended objects rather than pionlike. In the absence of indications to the contrary, we follow conventional procedures cd°'ds) and combine the weak and strong vertex functions into a single form factor. This form factor was already included in our original expressions for the transition amplitude Hx, Eqs. (33) and (34). It may be parametrized in the convenient form

F(~2) : ~

z2

(60)

,

where K is the cutoff parameter. Phenomenological studies (9°l indicate that K = 1 - 2 GeV, while quark models are characterized by much softer vertex functions. (911 Nevertheless, the hard hadronic phenomenological form factors are not firmly established and are subject to uncertainties. Since the nonmesonic decay process involves relatively high momentum transfers, we expect the form factor to have a substantial effect on the numerical results. In deriving Nonetheless, form factor changes the

the transition potential V~(~), Eqs. (35)-(38), we used F(q 2) = 1. all our previous results can be easily extended to accommodate the of Eq. (60). Indeed, including F(q2)~l in Mn, Eqs. (33) and (34), pion propagator from (~2 + ~)-i to

I

K2

K2

Changing V~(~), Eqs. ( 3 5 ) - ( 3 8 ) ,

V~(~) ~ ~

K2

~

into

[V~(~,m~) - VK(~,K)]

(62)

will therefore include the effect of the form factor in the modified transition potential, using our existing formulas [Eqs. (35)-(38)]. In Eq. (62), V,(~,Q) is just the one-pion-exchange transition potential with the (constant) m replaced by the constant c. Including the form factor F(~), Eq. (60), require~ therefore that the potential be calculated twice (once with mx, and again with

K).

192

J. Cohen

Numerical r e s u l t s based on the one-pion exchange mechanism will be presented in Sec. 3.6. In the next Section we discuss other, extended and more elaborate microscopic models for the AN-~NN weak interaction (as the underlylng mechanism in the non-mesonic decay process). For the most part, these models are based on the one-pion exchange as a starting point. 3.5

Elaborate microscopic models for the non-mesonic decay

3.5.1 Introduction. The previous section has focused on a partlcular microscopic model for the AN-~NN weak interaction, namely the one-pion exchange. A second ingredient in each calculation is the hypernuclear and nuclear structure input, described in detail in Subsecs. 3.4.3-3.4.5. The purpose of the present section is to elaborate on the first ingredient, i.e., the microscopic AN-~-N interaction model. The ideas reviewed here (¢°-42'45) rely on one-pion exchange as a starting point, describing a part of the interaction. In all but one of these models, (4s) only meson exchanges are used to describe the interaction. A somewhat different approach is studied in Ref. (43), where the nuclear structure (going beyond a simple shell-model description) and the microscopic interaction mechanism (based on just one-pion exchange) are interwoven, and should be discussed as a single entity. 3.2.5 Additional meson exchanges. In an attempt to improve upon the one pion exchange model, a reasonable approach would be to add heavier mesons as possible mediators of the AN-~NN interactions• Since the pion is by far the lightest of all the exchanged mesons, this approach deals wlth the short-range effects of the non-mesonic decay process. McKellar and Gibson (4°) calculated the total non-mesonlc decay rate for nuclear matter, based on g and p meson exchange mechanisms. The pion contribution has been described in Sec. 3.4. Turning now to the p contribution, the weak ANp vertex is not determined from experiment, and must be estimated theoretically. Assuming the validity of the AI = ~ rule as well as the absence of second class currents to eliminate a possible ~ q 7- term, and using ~ pP = 0 in order to // // ellmlnate q and 7sq terms, Mc~e~lar and Gibson wrote the effective Hamiltonian for this vertex as: •

*

Heff(.)= G~ m2 % [ ~ 7 ~

~i atJvqv

~ "A'~Ij

(63)

In Eq. (63), G F = 1.02 x 10-s/m 2 (m = the mass of the proton) is the Fermi constant, mp is the p-mass (mp ~ 7 7 0 pHeV), ~^ is the spurion introduced in Eq. (20), and ~,8,~ are constants. Note that H .... has a vector, tensor and axial vector components. The validity of the ~ = 2 rule implies [recall Eq.

(16)]

= _ _!I .

In order to obtain a t r a n s i t i o n

amplitude for the ~-~lN weak i n t e r a c t i o n based

on p-exchange, Fig. 9 (similar to Fig. 8, but with meson), the strong pNN vertex is also required. Using

F2

~ s , p ) = gp[F1vu + ~

%~qV] ~ ,

the

• replaced by a p

(64)

Decays of A-Hypernuclei

193

where ~ is directed towards the pNN strong vertex, the p--exchange contribution to the (nonrelatlvlstlc) AN-~NN transition potentlal In momentum space was given by HcKellar and Gibson as: GF Hp(q) + = ~

~

m2

1

P~-~p [F1 ct -

FI+F2 + i¢ ~

(a+~)(FI+F 2)

(2MN)2

_> -~ -~ -~ (olxq) • (o2xq)

~ ~ ~I~ alXQ2.qjxl.~ 2 .

(65)

In Eq. (65), which is the counterpart of H,(~), Eq. (34), we again identify a parity-conserving as well as a parity-violatlng component. For the former, the identity =

-

Hefflo~Hslo)

Fig. 9

A

p

meson

exchange

mechanism

for

the weak interaction

AN-~NN.

194

J. C o h e n

may be used to facilitate the Fourier transformation to configuration space, along the lines of Subsec. 3.4.2 [see Eq. (35)]. The dependence of parityviolating component of M (~) on the momentum variable ~ is similar to that of @ , o the corresponding ~-exchange component in M (~), and our previous experlence in obtaining V (~), Eqs. (35)-(38), applies in this case as well. Note that a single mass M = M = M^ is used in Eq. (65). Although M may be substituted [as in Eq. (3~)1, it is impossible to assess its effect since the p-contribution is subject to considerable theoretical uncertainties and ambiguities, (4°) as discussed in the following. Furthermore, since m is much larger than both m K and the pertinent energy transfer along the mesonic propagator ~m p >>mK, m p >>q o ), we use the free mass (m_) in Eq. (65) [cf. Eq. (34), where mx was introduced]. (We believe that the~effects of the energy transfer are again negligible relative to the pertinent theoretical uncertainty.) The resulting configuration-space transition potential for consequently similar in structure to its ~-exchange counterpart,

v(t)

= v2(t)

+ v~(t)

+ v~(t)

,

p

exchange is (66)

where V~(~) -- [V~'°(r) + V~'"(r)~1"%]%1"% 2 ,

= V

V(r)

2

,

(67)

(68)

and

V2(~) = V2(r) S12(r)%1"%2 .

(69)

In Eqs. (67)-(69), V~ and Vt d e p e n d on ~ a n d 8 o f Eq. ( 6 3 ) , w h i l e Vpv d e p e n d s on c. N o t e t h a t t h e r s p i n 2 i n d e p e n d e n t t e r m ~67F'=)in Eq. ( 6 5 ) g i v e ~ r i s e t o a spin-independent central potential v c ' ° i n Eq. o

I t i s now n e c e s s a r y t o s p e c i f y t h e p a r a m e t e r ~ , ~ a n d e c h a r a c t e r i z i n g t h e weak ANp v e r t e x , a s w e l l a s t h e form f a c t o r s F. a n d F_ u s e d a t t h e s t r o n g v e r t e x . As was made clear in Sec. 3.2 (dealing with ~ the A-~N~ decay), a direct calculation from first principles (i.e., based on the Standard Model) is impossible at present. Moreover, even simpler quark-model procedures do not seem to provide satisfactory results. (92) In the absence of experimental constraints, it is necessary to rely on (simplified) models. Two models suggested and used by McKellar and Gibson (4°I are the factorization approximation and the pole model. Both have been used for a long time in weak interaction physics, and are known to be essentially unsatisfactory. It was hoped (4°), however, that these models should provide an adequate guide for studying the order of magnitude of the p-meson contribution.

In using the factorization approximation, Fig. 10, McKellar and Gibson 14°~ enhanced the resulting A-~Np amplitude by a factor of sinS cos8 (i.e., they omitted the sin8 . ¢ cos8 factor which appears in the calculatlo~) 162~and treated the relative slgn of ~he parity conserving and parity violating terms as an unknown parameter. This was done in order to minimize the effects of established shortcomings of the model. The resulting parameters ~,~ and c in

Decays of A-Hypernuclel

Fig. I0

195

The factorization approximation for the weak A-~Np vertex [used in Ref.

(40)1. H ff(p), Eq. (63), a r e (4°)

= U

DI(q 2) ,

tB = ~-3"~. .~ D2(q2) , = 0.71 ~ D A ( q 2 )

(70) ,

where D (q2)

F~ (q2) m 2 gp mp-~

'

~ = 1,2,A .

(71)

In Eq. (71), F.(q 2) and F.(q 2) were defined for the strong pNN vertex, Eq. (64), and FA(q~) is an Axial form factor. In deriving Eqs. (70), SU(3) symmetry for the weak currents was used with D/F ratios of 0 (vector), (tensor) and 2 (axial vector), along with vector meson dominance for the vector form factor. The second model used in Ref. (40) is based on the observation that the parity conserving component of the A->N~ decay amplitude in the current algebra approach to a pionic decay is dominated by the pole graphs of Fig. iI. (62~3~ This so-called ~ model applies only to the parity conserving weak A-)N~ amplitude and yields for the pertinent coefficient in Eq. (18):

J. Cohen

196

%2

(1-=)(D~-F ') ~ M E - MN + 15

gw =

F'+ D~I3 MA_ MN ] .

(72)

In Eq. ( 7 2 ) , ~ i s the f r a c t i o n of F - t y p e c o u p l i n g , o r t h e F/(F+D) r a t i o , i n t h e o c t e t s t r o n g b a r y o n - b a r y o n - m e s o n c o u p l i n g s o f SU(3) and from SU(6) ( 1 - ~ ) / ~ = ~. The reduced matrix elements of the AS=I weak Hamiltonian, D' and F', were treated in Ref. (40) as free parameters, determined from a fit to available pwave hyperon decay amplitudes. The parity conserving part of the weak A-~Np amplitude was then approximated by a baryon pole picture similar to Fig. 11 but with a p replacing the R meson, with the vector and tensor ( y and op V k v coupling) DIF ratios of O and 3/2, respectively, and the same weak ~amiltonlan parameters D' and F' as in Eq. (72), the parameters ~ and 8 of the parity conserving part in H ~z(p), Eq. (63), are given by 4F'/3

= hF 1 ~

' (73)

F' ~ D ' / 3 = hF2 MA

MN

'

i@" i

T I

/

/

/

/

/

I

N

A

F i g . 11

The baryon p o l e g r a p h s f o r t h e p a r i t y A-~Nn v e r t e x .

A

N

N

c o n s e r v i n g p a r t o f t h e weak

Decays of A-Hypernuclei

197

where h = (3/2) x/2 (mR/mp)2. Although there seems to be a numerical inconsistency in Ref. (40), it Is clear that the q2 dependence in Eqs. (73) is somewhat different from that of Eqs. (70). More importantly, the dimensionless parameter measuring the strength of the B-+B'+m weak interactlon, where B is a baryon and m stands for a final state meson with mass m m, was found to be Grm2 in the factorization model and Grm ~ (independent of m) in the pole model fitte~ to the B~B'+~ data. (Further conclusions cannot be established at present, however a new estimate for the p-exchange contribution in the pole model has appeared recently and will be discussed here.) For the form coupling ~gsl F2(q2

Fl(q2

factors

FI

and

F 2,

McKellar

and

Gibson

chose

= m~)

m~); = 6.6 ,

the strong

(74)

and Fl(q 2)

A2

m2

(75)

where Ap = 1.5 mp, r e f l e c t i n g

a r a p i d v a r i a t i o n of F2(q 2) r e l a t i v e

to F l ( q 2 ) .

With the form f a c t o r s s p e c i f i e d [Eqs. (70), (71) and ( 7 3 ) - ( 7 5 ) ] , i t i s now p o s s i b l e to o b t a i n an e x p l i c i t e x p r e s s i o n f o r the c o n f i g u r a t l o n - s p a c e t r a n s i t i o n p o t e n t i a l Vn(~), Eqs. ( 6 6 ) - ( 6 9 ) . This i s done i n complete a n a l o g y w i t h V~(~) f o r the "pion c a s e , Eqs. ( 3 5 ) - ( 3 8 ) and ( 6 1 ) , (62); the r e s u l t i n g e x p r e s s i o n s a r e g i v e n i n Ref. (40) and d i f f e r , of c o u r s e , i n the two models c o n s i d e r e d . The combined t r a n s i t i o n potential, based on ~+p exchange, i s o b t a i n e d by a d d i n g the r e s u l t i n g p c o n t r i b u t i o n to the p i o n - e x c h a n g e p o t e n t i a l ( d e s c r i b e d i n Sec. 3 . 4 ) V_ = V +V . I t may then be used to c a l c u l a t e rate), o b s e r v a b l e q u a n t x t l e s (sue~ as the t r a n s , t l o n However, the r e l a t i v e s i g n of the • and p p o t e n t i a l s was n o t d e t e r m i n e d i n the c a l c u l a t i o n ; t h e r e f o r e , c a l c u l a t i o n s were c a r r i e d out ~4°~ based on the t r a n s i t i o n p o t e n t i a l s V~ + [Vp[ as w e l l as V~ - [Vp[. •

•

÷P

•

p

.

°

N a r d u l l i has r e c o n s i d e r e d , i n a r e c e n t work, ~961 the p - c o n t r i b u t i o n i n the pole model. I n h i s c a l c u l a t i o n , the ANp weak v e r t e x was e v a l u a t e d from the p o l e model based on both baryon and meson p o l e s . The b a r y o n p o l e g r a p h s were s i m i l a r i n n a t u r e to the ones d e p i c t e d i n F i g . 11, however b a r y o n s b e l o n g i n g to the g r o u n d - s t a t e baryon o c t e t and to the ~+ f i r s t e x c i t e d b a r y o n o c t e t [ t o which the P11(1440) b e l o n g s ] were c o n s i d e r e d . Moreover, a K* meson p o l e (where the weak t r ~ h s i t i o n i s K +p) was a l s o c o n s i d e r e d . For the p a r i t y - v i o l a t i n g ANp c o u p l i n g , ~- l o w - l y i n g f i r s t e x c i t e d n e g a t i v e p a r i t y b a r y o n s were c o n s i d e r e d . The v a l i d i t y of the AT = ~ r u l e was assumed. A s i m i l a r model had been t e s t e d f o r hyperon n o n l e p t o n i c and r a d a a t i v e decays ( w i t h ~ and y r e p l a c i n g the p meson) and found to be s a t i s f a c t o r y ; 197~ t h i s f a c t p r o v i d e d f u r t h e r m o t i v a t i o n f o r and c o n f i d e n c e i n the p e r t i n e n t model s t u d y of the ANp weak c o u p l i n g .

198

J. C o h e n

Nardulli t96'97) considered both the parity conserving and parity violating arts of the weak A-~Np interaction. The former was studied from the baryon and pole terms mentioned above. Numerical values for the pertinent parameters of the weak-decay amplitudes were chosen to fit the experimentally-derived amplitudes, while baryon-baryon-vector meson matrix elements were constrained, based on vector meson dominance model, by the known electromagnetic couplings of nucleons and resonances. The resulting expressions (96) for the parityconserving ANp weak vertex are generally similar to thoseof Eq. (72) for the A and E poles, t4°) however other baryon resonances and the K contribute as well. As mentioned above results are also given for the parlty-violating ANp coupling, based on negative-energy baryon poles in the SU(6) limit; the contribution of the axial vector meson pole [K,(1280)] was estimated and found relatively small. With these new values for t~e ANp weak vertex, Nardulli 196* then repeated the non-mesonic hypernuclear decay rate calculation of McKellar and Gibson. t4°* However, it is worthy of note that all relative signs were determined consistently within the pole model in Nardulli's calculation. Furthermore, Nardulli considered only the parity-conserving part of TVh~and Vp, which are known ~4°) to dominate the non-mesonlc decay rate. tensor components of the separate n and p transition potentials are opposite in sign, and the total rate obtained in this model (the combined ~ and p contribution) is therefore expected to be lower than the ~-exchange contribution. As we shall see in Sec. 3.6, this also has important consequences for the calculation of ~, Eq. (47), as reducing the tensor part allows for a relatively large An-~nn contribution.

~

These calculations (4°,96) were further carried out based on the following assumptions and approximations: (i) only a relative AN s-state contributes (see the discussion of Subsec. 3.4.3 and Table 5); (ii) The transition matrix element could be calculated at a fixed final N-N relative momentum t = 420 MeV/c (this matrix element varies by only about I0~ over the possible range of t); (iii) the spin dependence of the initial- and final-state wave functions was n o t t a k e n i n t o a c c o u n t . M c K e l l a r and G i b s o n (4°) a p p l i e d t h e ~+¢ t r a n s i t i o n potential to a non-mesonic d e c a y o f a A h y p e r o n embedded i n n u c l e a r m a t t e r ; t h e i r n u m e r i c a l r e s u l t s are given in Sec. 3.6. A ftnite-hypernuclear calculation a l o n g s i m i l a r l i n e s was carried out by Takeuchi et al. (98[ for A = 4,5 hypernuclei. For the K-exchange transition potential, these authors used the complete V K discussed here in Sec. 3.4, while for the c-induced transition potential they chose the factorization model and kept only the tensor part Vp, t Eq. (69), which is numerically the largest term in V . The hypernuclear wave functions used in these calculations were similar to E~. (54) with the ground state of the nuclear core described in terms of gaussians. The A-hyperon s-state wave function was obtained by solving the equation of the folding potential model derived from the AN Gmatrix for a Nijmegen interaction; (9~) this wave function was expanded in terms of H0 wave functions so that a Moshinsky transformation could be carried out [as in Eqs. (56)]. Furthermore, initial (AN)- and final (NN) -state realistic correlations were introduced by solving the AN Bethe-Goldstone equation and the NN scattering equation, respectively, with a Nijmegen interaction. Soft form factors, similar to those used by McKellar and Gibson, were introduced in the calculation of Takeuchi et el. as well.

A further elaboration on the meson-exchange model for the AN~NN interaction and non-mesonic decays was presented by Dubach et el. (4*'99) and is based on higher meson exchanges (K,p,~,~,K,K',a). Unfortun'ate--i-y, no detailed account of this calculation is available at the time of this writing, and our ensuing discussion is based on brief accounts presented at conferences in recent years.

Decays of A-Hypernuclei

199

The new f e a t u r e s o f t h e model a r e c o n t a i n e d i n t h e t r a n s i t i o n p o t e n t i a l . As already noted, this i s a m u l t i - m e s o n - e x c h a n g e model, which i n c l u d e s s t r a n g e mesons (K,K*) in addition to the nonstrange ones (n,p,~,~,a). For the former case, the AN*NN diagrams require a strong ANm and a weak NNm vertex, while for the nonstrange meson exchanges the situation is reversed (as in Fig. 8). The limitation on the types of mesons considered to pseudoscalar and vector exchanges was based on earlier works in the nucleon-nucleon sector. ~99) Both parity-conserving and parity-violating ANm and NNm weak couplings were requried. The former were evaluated from the pole model with baryon (N,A,£) and meson poles. With the AT = ~ rule enforced into the calculation and from SU(6)w symmetry, all the weak meson-~neson amplitudes required for the pole model-were determined from the weak K ~ amplitude (). The remaining necessary two-hadron weak vertices were calgulated ~ia current algebra with PCAC assumptions, which relate these amplitudes to experimentallydetermined, parity-violating three particle weak vertices. Thus, the K ~ amplitude was obtained from the physical K4x~ decay:

= , while the baryon*baryon amplitudes were related that =

to

Y*Nn decays (Y = A,E) such

.

Using the model to evaluate a measureable amplitude for the A-~n~° decay, the authors (41'99) concluded that it works at the level of about 25Z accuracy (a similar figure was given by Nardulli~g7~). The strong couplings were obtained by means of SU(3) symmetry, PCAC and the Goldberger-Treiman relation. Finally, the exchange of the scalar-isoscalar meson was also included, since it is known to be important in certain models for nucleon-nucleon and pion-nuc]eon interaction. The a mass was taken to be 760 MeV, and its strong interaction with the nucleon (i.e., the strong ~NN coupling) was obtained from phenomenology. The weak oNN coupling was described by a two-pion exchange model with a weak baryon-baryon-pion vertex. No further details regarding the construction of the transition potential are available! however the methods used appear to be similar to those of Refs. (96) and (97). Based on the available reports, ~41'99) McKellar ~42) has raised some doubts regarding the construction of the transition potential. In particular, the agreement between the model results and the empirical ANx amplitude would, according to McKellar, contradict other state-of-the-art calculations wherein it is difficult to obtain a satisfactory and consistent description of both parity-conservlng and parity-violating vertices with the same matrix elements without adding higher ones (e.g., from N" or K* poles). F~rthermore, the K-pole tends to cancel the baryon poles in the parity conserving amplitude. These concerns were later addressed by Nardulli, c96'97) who included the excited baryon poles in his calculations. (Nardulli's work is discussed above.) McKellar has also pointed out that SU(6) w weak couplings to vector mesons require amplitudes not determined by free -hyperon decays, and that meson-pole contributions to the AN*NN decay are actually double-pole terms involvlng a double propagator, which may be slgnificantly affected by the introduction of form factors. The latter point would seem to apply to Nardulli's calculation as well; note that McKellar and Gibson did not consider at all the meson pole contribution in their pole model calculation for the ANp vertex.

200

J. Cohen

From this model, Dubach et el. {41'99~ constructed a momentum space transition amplitude. An effective"t'vo---~ody configuration space transition potential was then obtained by means of a Fourier transformation [see Eq. (35)]. Non-mesonic decay rates were first calculated for A at rest in nuclear matter with equal number of neutrons and protons and a Fermi momentum k_ = 268 HeV/c. The pertinent formulas for the decay rate were similar to those presented here [Eqs. (43)-(46) of Subsec. 3.4.3]; only an s-wave was considered for the relative AN state. Results were given for both a plane-wave initial state (spherical Bessel function, no correlations) and the phenomenological two-body correlation function used in Ref. (40) and discussed in Sec. 3.4.5. Likewise, both a plane wave state and a solution to the Schr~dinger equation with a Reid s o f t core NN p o t e n t i a l were i n t r o d u c e d and s t u d i e d f o r the f i n a l s t a t e .

Dnbach e t e l . r e p o r t e d a l s o f i n i t e - h y p e r n u c l e a r s h e l l model r e s u l t s f o r [He and 12^C i n an extreme s i n g l e - p a r t i c l e model ( w i t h no c o n f i g u r a t i o n m i x i n g ) , (gsl but no d e t a i l s on the c a l c u l a t i o n a r e c u r r e n t l y a v a i l a b l e . Numerical r e s u l t s w i l l be p r e s e n t e d i n Sec. 3.6. I n a d d i t i o n to t o t a l decays r a t e s , the r a t i o s ~, Eq. (49) and ~, Eq. ( 5 2 ) , were a l s o c a l c u l a t e d ( t h e i m p o r t a n c e of t h e s e r a t i o s has been d i s c u s s e d i n Subsec. 3 . 4 . 3 ) . It i s worthy of n o t e t h a t the n o t a t i o n used by Dubach i n t h i s c o n t e x t has caused some ( u n n e c e s s a r y ) d e b a t e i n the r e c e n t literature. However, the i s s u e has been r e s o l v e d r e c e n t l y . (1°°) Thus, i n Dubach's n o t a t i o n the v a r i o u s p a r t i a l r a t e s I F ( i s ~ 1So), e t c . ] were m u l t i p l i e d by a f a c t o r of 3 r e l a t i v e to the s t a n d a r d n o t a t i o n i n t r o d u c e d by Adams (39) and used i n t h i s work, except for r ( 3 s . ~ 3pl) which was i n s t e a d m u l t i p l i e d by a f a c t o r of 9. [The r e a s o n b e i n g t h a t ~ i n t h i s n o t a t i o n the t o t a l decay r a t e , Eq. (46), is simply a sum of the various modified partial rates, with no weight factors. This is, however, no longer the case when the ratios ~ or n, Eqs. (47) and (48), are considered.] 3.5.3. Hybrid q u a r k - h a d r o n model A s t u d y of s h o r t - r a n g e c o r r e l a t i o n s i n n u c l e a r m a t t e r and f i n i t e n u c l e i based on quark r a t h e r than n u c l e o n d e g r e e s of freedom was p u b l i s h e d by Cheung, Heddle a n ~ - ~ ' s s l i n g e r . (4s's4) Using a h y b r i d q u a r k - h a d r o n model i n t h e i r work, the decay p r o c e s s was d e s c r i b e d by two s e p a r a t e mechanisms: one f o r the l o n g - r a n g e i n t e r a c t i o n ( s e p a r a t i o n s over a p p r o x i m a t e l y 0 . 8 fm) and the o t h e r f o r s h o r t r a n g e s ( l e s s than a b o u t 0 . 8 fm). For the l o n g - r a n g e , h a d r o n i c s e c t o r , the a u t h o r s c o n s i d e r e d o n l y the o n e - p i o n exchange mechanism, w h i l e a s i x - q u a r k model w i t h W boson exchange was used f o r the s h o r t range i n t e r a c t i o n . (In Sec. i.I, we have pointed out that existing nuclear data do not unequivocally demand a quark-level description of nuclei. Horeover, the shell model, with purely hadronic degrees of freedom, has recently been shown (2~ to provide an excellent description of hypernuclear states. However, the possibility of explicit quark degrees of freedom in nuclei continues to be an exciting subject of research, and hypernuclei may provide a unique possibility for actually studying such effects. The work of Cheung, Heddle and Kisslinger (4~84) should be examined along these lines.) We begin with a description of the hadronic sector. Many of the details have already been discussed in Sec. 3.4. The hypernuclear system is represented in this sector by baryonic (nucleons plus a A-hyperon) degrees of freedom, and the shell model is used; there is no mention of quarks or subnucleon degrees of freedom in this region. A transition potential was derived based on Fig. 8, see Eqs. (33)-(38), and an effective pion mass m~ was introduced [see Eq. (34)]. Initial-state, two-body c o r r e l a t i o s f o r n u c l e a r m a t t e r were i n c l u d e d

Decays of A-Hypernuclei

201

following the procedure of Adams, ~39~ while a phenomenological two-body radial correlation function (ss) was used for finite nuclei (see the discussion in Subsec. 3.4.5). For the final (N-N) state interaction, an eikonal model with the central part of a p-*2C nuclear optical potential was used (also discussed in Subsec. 3.4.5). The pertinent nuclear structure input has been outlined in Subsec. 3.4.4. After separation of relative and center of mass coordinates, only those components with c.m. angular mometnum L=0 were retained. The short-distance interaction responsible for the non-mesonic decay process was assumed to be the s+u-~u+d transition mediated by W boson exchange between two of the six quarks contained in the A-N system; see the pertinent discussion in Sec. 3.2 and Fig. 6. The I~51=1 Hamiltonlan for the interior region was chosen along the lines described at the end of Sec. 3.2. Three models were suggested, denoted hereinafter by CHK1, CHK2 and CHK3 as follows. In the absence of strong interactions a Standard Model V-A interaction Mamlltonian plus a Cabibbo rotation was used, which reduces to a polntlike interaction for the energies and momenta of interest here. This will be called model CHK1. As explained in some more detail in Sec. 3.2, the empirical AT = L rule is badly violated by this Hamiltonian. The authors (4s) adopt the viewpoint that stronginteraction corrections should indeed produce the desired effective weak Hamlltonian obeying the AT = ~ rule, and write for the AS=I case Gw = ~ sinScC°SSc[ClUVp(1-Y5)savP(1-vs)u + c2avp(1-vs)s~v~(1-v5)u ]

=

Gw Cl+ c 2 c I- c 2 ~sinScCOSSc[---- ~ Q3/2 + " ~ gl/2 ] •

(76)

In Eq. ( 7 6 ) , glZ2 and g3/2 are pure I = £L, I = ~ isospln operators, respectively, and c i , c_z are constants, zrom the values of Gilman and Wise cv3), for example, the AT = ~ amplitude is suppressed by a factor of only c 1- c 2 RG_ w = Cl + c2 = 3.6, (77) which is far from the empirical result (represented by the /11 = ~ rule), wherein R =20. Cheung, Heddle and Kisslinger c4s.e4) suggested the possibilit~x%f implementing the AT = ~ rule in an ad hoc manner by-using c I + c 2 = 0.37 c I - c 2 = 7.38 ,

(78)

yielding coefficients c. and c . w h i c h a r e s i m i l a r i n m a g n i t u d e and o p p o s i t e i n sign. The o n l y j u s t i f i c a t i o n ~or varying c I and c 2 o v e r such a v i d e r a n g e i s t h e r e s u l t i n g s u p p r e s s i o n f a c t o r R=20. I n t h e e n s u i n g discussion, model CHK2 refers to a Standard Model-Cabibbo Hamiltonian with strong interaction c o r r e c t i o n s as i n Eq. ( 7 7 ) , w h i l e Eq. ( 7 8 ) r e p r e s e n t s model CHK3.

Nith the models for the effective weak interaction Hamiltonian chosen, transition amplitudes may be obtained once the model wave functions have been

202

J. Cohen

specified. Cheung, Heddle and g i s s l i n g e r f u n c t i o n f o r the two-baryon system i s ¢ *BB(r), = ~ *6q'

rYRo

chose a model ¢84'x°z) where t h e wave

(79)

r
Here, r is the distance between the centers of mass of the two (composite) baryons, ,s,(r) is the exterior (conventional) wave functions expressed in terms of ha~ronic degrees of freedom, and ,~q is a six-quark wave function. The boundary between the quark and hadron reglons was set at r = R , with R chosen to be about 0.8 fm. Since there is a number of configurations allove~ for the slx-quark system, ,~q is given in a quark shell model by *6q = iE a i * ~ )

'

(80)

where a. are spectroscopic amplitudes and ~i~ are wave functions constituting a complete set of orthonormal slx-quar~ q states. In their work, the authros ~s4'x°11 chose a single-partlcle model whereby each ~il is represented by a product of slngle-quark MIT bag wave functions with m~%sless quarks. It was assumed that the AN-~NN interaction could be described in the short range region in terms of four spectators (inert quarks) and two quarks interacting weakly via W exchange as in Fig. 6. While the inert quarks were all restricted as to welan~i/2 state, the model space for the interacting quarks included p-states The i n i t i a l - s t a t e six-quark probability, P~q, was r e g a r d e d as a p a r a m e t e r in the n u c l e a r m a t t e r c a l c u l a t i o n . The numerxcal v a l u e P~q = 0 . 1 5 was chosen i n Ref. (4 5 ) . For the f i n i t e - n u c l e u s case, on t h e o t h e r hand, c o n s e r v a t i o n of p r o b a b i l i t y y i e l d s i n the i n i t i a l s t a t e 2

Pi6q = 1 - ~RoI*AN(r)

r2dr ,

(81)

where ,^~(r) is the normalized relative A-N wave function. For the final N-N scattering states, the six-quark probability was defined, following the methods of Henley et al., cx°l~ as that probability excluded from the hadronic (longrange) description by the presence of the quark (short-distance) region, i.e., the missing probability. This definition is somewhat arbitrary and requires a division by a constant having the units of a volume. Significance cannot be ascribed to the absolute size of p6q but relative sizes thereof can be meaningfully comapred. With the weak i n t e r a c t i o n Hamiltonian specified f o r both t h e r>R and r, f o r t h e two r e g i o n s . In a d d i t i o n to t h e p o r e i n t e r i o r and e x t e r i o r H a m i l t o n i a n s , B< ! P, r e s p e c t i v e l y , s u r f a c e terms which c o u p l e the i n t e r i o r and e x t e r i o r r e g i o n s , H~<~ P~H P< and

Decays of A-Hypernuclel

203

H<>E P, may a l s o a p p e a r . These l a t t e r terms were n e g l e c t e d by Cheung, Heddle and K i s s l i n g e r (4s) (an e x p l i c i t c o n s i d e r a t i o n of such terms would r e q u i r e u n d e r s t a n d i n g the dynamics of the t r a n s i t i o n from q u a r k - g l u o n to h a d r o n i c d e g r e e s of freedom, and r e p r e s e n t s a v e r y d i f f i c u l t and c o m p l i c a t e d problem). This d i s c u s s i o n a p p l i e s to the t r a n s i t i o n m a t r i x e l e m e n t as w e l l . I t was written in Ref. (45) as a sum T = T6q *

<#,(NN) IHI#(AN)>

,

(82)

where the f i r s t p a r t c o r r e s p o n d s to rR ° ( t h e e x t e r i o r r e g i o n ~ . No s u r f a c e terms a r e c o n s i d e r e d . The p i o n i c c o n t r i b u t i o n ( i n the l o n g - r a n g e r e g i o n ) was c a l c u l a t e d a l o n g the l i n e s d e s c r i b e d i n Sec. 3 . 4 . For the quark c o n t r i b u t i o n to the n o n - m e s o n i c decay, Heddle and K i s s l i n g e r found (451 r6q=~ 2.73(ci+c2)2

p6q F(al ,a2,. .. )r^

•

(83)

I n Eq. ( 8 3 ) , F i s a f u n c t i o n which depends on the c h o i c e of the s i x - q u a r k model wave f u n c t i o n . Also i n c l u d e d i n Eq. (83) i s a s p e c t r o s c o p i c f a c t o r of 2.5 a r i s i n g from assuming e q u a l c o n t r i b u i t o n s from Ap-mp (a system w i t h t h r e e u q u a r k s ) and An-mn (two u q u a r k s ) . The main p o i n t emerging from Eq. (83) i s that f

1

rNM6q = (el+ c2)2 = ~ 0.42 0.14

,

, ,

CHKI CHK2 CHK3

(84)

for the v a r i o u s models (CHK1, CHK2 and CMK3) mentioned above. No p l a u s i b i l i t y arguments were p r o v i d e d f o r the r e s u l t i n Eq. ( 8 3 ) . Note t h a t cz+c. i s the c o e f f i c i e n t of the aI = ~ o p e r a t o r i n Eq. ( 7 6 ) . Thus, (42) the s i ~ - q u a r k c o n t r i b u i t o n i s g i v e n s o l e l y by the AI = ~ p i e c e of the e f f e c t i v e weak Hamiltonian, Eq. (76). Since the 6I = ~amplitude is so much smaller than its 61 = ~counterpart, the six-quark contri6ution is Greatly suppressed. Moreover, it would not have been present at all if only the AI = ~ amplitu--~e had been taken into account, as done in the hadronic sector; the two sectors appear to be in complete contrast in this respect. These intriguing observations are left unexplained at present. The six-quark contribution (short-distance sector) was slml]ar for the nuclear matter and finite-nucleus calculations, the only change being the different values for P~q. In these calculations, the authors (45'841 found that there is no phase difference between the hadronic and quark regions, and thus stated that the proper way to combine the pion-exchange contribution and the slx-quark (with W boson exchange) one is to add the square roots of the individual c o n t r i b u t i, o n s to r N ( t h e a m p l i t u d e s f o r the i n t e r i o r and e x t e r i o r s e c t o r s add c o n s t r u c t i v e l y i n t h i s work). I n a r e c e n t paper, 11°2) Heddle and K i s s l i n g e r c a l c u l a t e d the r a t i o ~, Eq. ( 4 7 ) . I n the quark s e c t o r , they assumed the v a l i d i t y of q u a r k c o u n t i n g . S i n c e the t r a n s i t i o n a t the quark leve~i i s g i v e n by us-md v i a W boson exchange, the a u t h o r s wrote f o r the r a t i o ~ i n the i n t e r i o r r e g i o n : F~M-6q = 3 ~6q = "NrnM-6q ~

(85)

204

J. Cohen

This result is based on counting three u quarks in the Ap system and two in the An system. Note that this ratio needs to be calculated, in fact, using the six-quark wave functions and the quark-level weak interaction Hamiltonian with W boson exchange. In such an improved calculation, the matrix elements of spin operators are not expected to scale linearly with the number of u-quarks. It is therefore important to verify the quark counting approximation using a more detailed calculation. Indeed, this is particularly interesting because the one pion exchange mechanism would greatly suppress the neutron-induced (relative to proton induced) partial rate, in contrast with Eq. (85). The ratio ~6 has to be combined nov with the corresponding one in the hadronic sector, b a s ~ on one-pion exchange mechanism. We expect, therefore, a combined ratio, ~, lying well above unity. Numerical results are given in Sec. 3.6. 3.5.4. Many-body RPA (ring diagrams) renormalization effects in one pion exchange A different approach to the problem was suggested by Oset and ~ ( 4 3 ) Their approach is based on one-pion exchange as the underlying mechanism for the elementary AN-R~N process, nonetheless they adopted an improved many-body model whereby the strong pion interaction with the nuclear medium modifies its propagation in this multinucleon system. Interestingly, this method is able to deal simultaneously with both the mesonic and nonmesonic decay rates, since both the real pion from the mesonic (A-~In) decay and the virtual plon in the nonmesonlc mode are allowed to interact with the nuclear medium. The authors (43; considered the part of their work pertinent to the nonmesonic decay as complementary to the complete boson-exchange model, (411 since it deals with the additional strong-interaction many-body effects, modifying the pion exchange piece of the decay mechanism. This propagator modification is described diagramatically in Fig. 12(a), and its effect on the A self-energy is described in 12(b). The nonmesonic decay rate can be thought of as a cut in Fig. 12(b) through the intermediate nucleon line and a p-h bubble. Likewise, the mesonic decay is obtained by cutting through the intermediate nucleon and a pion line. The added RPA (ring) diagrams represent corrections to the free-A mesonic decay and to the elementary nonmesonic decay mechanism induced by a one plon exchange (Fig. I). Thus, the A-wldth is given to the lowest order by the imaginary part of the A self energy, ~^, represented by the first diagram (A~[Nn loop]~A) in the expansion Fig. 12(b): r = -2

Im E^ .

(86)

For a free A decay, Oset and Salcedo obtained r^ by placing the n and N on their mass shell; the self energy for the free case was determined solely by the weak ANn interaction, (43) given phenomenologically in Eqs. (14) and (18). A monopole form factor with cutoff K = 1.2 GeV was used at the weak vertex. In the nuclear medium, the nucleon and plon propagators of the Nn loop were modified as follows. For the nucleon, the Pauli exclusion principle and binding corrections for an uncorrelated Fermi gas of nucleons gave the modification

GN(P)

=

1 + i~ pO_ c(~)

.

n ( ~ ) _ vN+ ic pO_1 -~(~)

n(~)

+ pO_ ~ ( ~ )

_ VN-

i~

,

(87)

where e(~) i s the n u c l e o n k i n e t i c e n e r g y , V. i s i t s p o t e n t i a l e n e r g y , and n(~) i s the n u c l e o n o c c u p a t i o n number f u n c t i o n " i n momentum space (which may be taken, f o r s i m p l i c i t y , to r e p r e s e n t a Fermi gas a t zero t e m p e r a t u r e ) . The

Decays of A-Hypernuclei

7r

P ;- ,'

=

/

7r

.p,p ~

A

+""

205

A

=N ..:°T+N'

!

I

!

!

!

71"

7r

7r

7r

A

A

(a) Fig 12.

A

A

:0

+---= N

A

A

(b)

Many-body e f f e c t s m o d i f y i n g the R - h y p e r n u c l e a r weak decay r a t e s : (a) The r e n o r m a l i z e d pion p r o p a g a t o r , or p i o n s e l f e n e r g y (H) due to p a r t i c l e (N,6) - h o l e ( p - h ) e x c i t a t i o n s . The wavy l i n e r e p r e s e n t s the T=I p-h i n t e r a c t i o n . (b) The A s e l f e n e r g y (E^) i n the n u c l e a r medium.

s o - c a l l e d r e n o r m a l i z a t i o n of the p i o n p r o p a g a t o r f o l l o w s a p a r t i c u l a r model used by Oset e t a l . (le3) f o r s t u d y i n g ~ - n u c l e a r p h y s i c s and 6 p r o p a g a t i o n i n the n u c l e u s * . I n the model, the pton p r o p a g a t o r was m o d i f i e d by the i s o v e c t o r (T=I) p-h i n t e r a c t i o n as shown i n F i g . 1 2 ( a ) :

DIt(q)

=

qO2_

~1_m2 +

tc

-~

qO2_ ~ 2 _

21

mn -

,

(88)

n ( q ° , ~ ')

where H(q°,~) i s the pion s e l f e n e r g y . With the m o d i f i e d p r o p a g a t o r s , Eqs. (87) and ( 8 8 ) , the A s e l f e n e r g y i n the n u c l e a r medium i s r e p r e s e n t e d by F i g . 1 2 ( b ) . Using the decay r a t e as d e f i n e d i n Eq. ( 8 6 ) , EA now c o n t a i n s two s o u r c e s of i m a g i n a r y p a r t s c o n t r i b u t i n g to the w i d t h . The f i r s t one i s o b t a i n e d from the p i o n - n u c l e o n p o l e , p l a c i n g both p a r t i c l e s on the mass s h e l l , and d e s c r i b e s the A - h y p e r n u c l e a r mesonic decay ( t h e e l e m e n t a r y p r o c e s s i s A-~Nn). The summation of a l l the mesonic c o n t r i b u t i o n s i n F i g . 12(h) as w e l l as ~ - n u c l e a r i n t e r a c t i o n s v i a the s-wave rdq i n t e r a c t i o n (which was d e s c r i b e d by Oset and P a l a n q u e s (1°])) y i e l d s the c o n t r i b u t i o n to ImE^ [Eq. ( 8 6 ) | from the modified pion pole a t qO = ~(~), where ~)2

= q2 + m2 + n ( ~ q ) , q )

.

-

S

(89)

The second c o n t r i b u t i o n to the i m a g i n a r y p a r t of E^ comes from the pion s e l f e n e r g y , H(q°,~), which has an i m a g i n a r y p a r t f o r certain values of qO,~; I t c o r r e s p o n d s to p-h e x c i t a t i o n s on the n u c l e o n mass s h e l l , y i e l d i n g the h y p e r n u c l e a r non-mesonic decay v i a a 2 p - l h e x c i t a t i o n of the n u c l e a r medium ( t h e e l e m e n t a r y p r o c e s s i s AN-~NNv i a one pton e x c h a n g e ) . For the s t r o n g ~ N v e r t e x , the a u t h o r s c43) used an i n t e r a c t i o n H a m i l t o n i a n m a t r i x element s i m i l a r to Eq. (32) ( i n the n o n r e l a t i v i s t i c a p p r o x i m a t i o n ) , modified by a monopole form f a c t o r with a c u t o f f p a r a m e t e r K = 1.2 GeV ( a s d i s c u s s e d i n Sec. 3 . 4 . 5 , t h i s v a l u e i s c h a r a c t e r i s t i c of p h e n o m e n o l o g i c a l s t u d i e s f o r the N-N i n t e r a c t i o n ) . I n o r d e r to c a l c u l a t e the r e n o r m a l t z a t i o n of the pion p r o p a g a t o r , the s t r o n g rLN6 v e r t e x i s a l s o r e q u i r e d . This is taken in the u s u a l n o n r e l a t i v i s t t c form[1°3'1°4~

* S i m i l a r approaches e x i s t i n the l i t e r a t u r e , cx°4) however the p r e s e n t model r e n o r m a l i z e s the pion p r o p a g a t o r and not the s p i n - i s o s p i n o p e r a t o r .

-

206

J. Cohen

.xN Ss

fl = mK ~'~ TX

,

(90)

where g and T a r e t h e s p i n and i s o s p i n transition o p e r a t o r s , and f*= 2f = g m /MN. In a d d i t i o n , the i s o v e c t o r p-h i n t e r a c t i o n , c o n n e c t i n g t h e RPA b~l%s, i s needed. This i n t e r a c t i o n , a l r e a d y used in o t h e r s t u d i e s , ¢I°~'I°4~ was based [ i n Ref. (43)] on n+o+g' terms ~I°3-I°5~ and has a t r a n s v e r s e (~-xq) as w e l l as l o n g i t u d i n a l (%.q) component:

The RPA rings can be summed to all orders using techniques in the pertinent literature ~I°3'I°41 (specifically, the Lindhard function is introduced). Note that since the energy transfer for non-mesonic decays is large, renormalization effects for this decay channel are expected to be suppressed relative to situations where the energy transfer is much lower ~I°4. (including hypernuclear pionic decays, which, however, are greatly suppressed due to the Pauli principle). Using this formalism, it is possible to do better than infinite nuclear matter. Oset and Salcedo (431 were able to employ a local-density approximation (LDA) to approximately include the e f f e c t s of f i n i t e nuclear size. I n s t e a d of the n u c l e a r - m a t t e r ( o r the f r e e - p a r t i c l e ) e x p r e s s i o n f o r the d ecay r a t e ( o r w i d t h ) , r = - 2ImE^ [Eq. ( 8 6 ) ] , the l o c a l d e n s i t y a p p r o x i m a t i o n r e a d s r = - 2 ~ d 3 r [ ~ ^ ~ ) [ 2 ImEA[(0(~)]. As p o i n t e d out in Subsec. 3 . 4 . 4 , t h i s has an i n t e r e s t i n g consequence r e g a r d i n g the mesonic d e c a y , s i n c e a n u c l e o n produced c l o s e to the n u c l e a r s u r f a c e w i t h a momentum l a r g e r than t h e l o c a l Fermi momentum a t a c e r t a i n l o c a t i o n ~, w i l l l e a v e the n u c l e u s . S i n c e kF(~) can assume any v a l u e ~k~, t h i s LDA p i c t u r e a l l o w s f o r mesonic h y p e r n u c l e a r d e c a y s , e s p e c i a l l y in l i g h t e r n u c l e i (where most o f the n u c l e o n s a r e a t d e n s i t i e s much lower than the c e n t r a l , o r n u c l e a r m a t t e r d e n s i t y ) . As a r e s u l t , one s h o u l d e x p e c t a r e d u c t i o n in the n o n - m e s o n i c decay r a t e r e l a t i v e to t h e i n f i n i t e nuclear matter calculation, and th e mesonic decay mode (which i s t o t a l l y suppressed for i n f i n i t e nuclear matter) would become p o s s i b l e . A further i n c r e a s e in the p i o n i c decay r a t e in t h i s p a r t i c u l a r model r e s u l t s from t h e o b s e r v a t i o n t h a t , in Eq. ( 8 9 ) , ~(~)<(~2+ m~)l/2, p r o v i d i n g more e n e r g y (and momentum) to the n u c l e o n and i n c r e a s i n g the a v a i l a b l e phase s p a c e f o r t h i s decay mode. For the l o c a l - d e n s i t y potential e n e r g y VN, Oset and S a l c e d o (43) wrote VN = - 8F . -

k2M~N = - M~N [ ~ m2p(r)] 2/3 ,

(92)

where o(r) is the local nuclear density. (Note that a comparison of the local density approximation for the renormalization effects, to an exact finitenucleus treatment, revealed cI°4) that the latter is, in general, not welldescribed by the former. Bowever, until a finite-nucleus calculation is available, the local density approximation is an important step in the right direction.)

Decays of A-Hypernuclei

207

In the next section, See. 3.6, we present numerical results obtained in the models described above (Secs. 3.4 and 3.5). These results will be discussed, and compared to each other and to the available experimental data. 3.6

Numerical results and comparison with experimental data

In this section, we present and discuss numerical results obtained from the models described in Secs. 3.4 and 3.5. Some comparison with recent experimental results is given where such results are available; see, however, Tables 1 and 2 as well. All the pertinent numerical results are summarized in Tables 6, 7 and 8, dealing with the total non-mesonic decay rate r,~, the ratios [Eq. (47)] and ~ [Eq. (52)], and with the details of the one-pion exchange contribution to the decay process, respectively. We begin with models based on only one-pion exchange as the pertinent decay mechanism. Following the pioneering investigations, ~I°) Adams (3~) carried out a detailed calculation of non-mesonic decay rates in nuclear matter, based on this interaction mechanism. In his work, Adams assumed the A to be at rest in nuclear matter; only the relative s-wave was considered for the initial A-N state. The relative A-N wave function for a given A-N potential was described by the Bethe-Goldstone equation. The potential had a hard core, with a core radius of r = 0.4 fm, and no distinction was made between the A-N slnglet and triplet sta~es [see Eq. (58)]. For the nucleon-nucleon interaction, Adams used central and tensor potentials. Due to the large energy release in the nonmesonic decay, the outgoing nucleons were treated irrespective of the spectators in the Fermi sea, so the Schr~dinger equation was used for their relative wave function. The 3s and 3d I final NN triplet states were coupled by the strong tensor force (aI weak transition 3s1~3d I followed by a strong ~d1*3sl); the same radial function was used for the 3p0 and 3p, states. As pointed out by McKellar and Gibson, (4°~ the coupling constant gw used by Adams ~39) is a factor of 2.61 lower than the value derived from free-A lifetime, r^. Since Adams gave his decay rates in units of the measured free-A lifetime (and does not calculate it using the coupling constants chosen in his work), his rates are too small, and should be multiplied by 6.81. In the following discussion and Tables 6,7 and 8 we always give the corrected results. Detailed numerical values were given in Tables I and II of Ref. (39) for a relative AN momentum k = 0.1 MeV/c for various calculations such as (i) no interbaryon potentials in the initial or final states, using free-particle wave functions, or (ii) with central and tensor potentials included (called "standard correlation"). According to Adams, (39) the interactions in the initial and final states had a dramatic effect (which persisted even when the input parameters and the A-N wave function were varied): a large reduction of the transition rates from J = 0 states (R,o), and a suppression of transitions to I = 0 states (as compared to I_ = 1). Most of the gross features are due to the hard core, apart from th~ tensor-force-induced suppression of the dwave. For the total nonmesonic decay lifetime, Adams found r = 3.5r^ for the case (i) (no correlations), and claimed that this quanti~, being an order of magnitude lower for case (ii) (standard correlations), was a sensitive measure of the short range correlations in nuclear matter. Likewise, the ratio (Table 7) is also sensitive to the short range correlations. Note from Table 7 that the tensor-force-induced suppression of the d-wave, results in a lower

208

J. C o h e n

Table 6.

Theoretical

results

for the total

hypernuclear

non-mesonic decay rate

r,M (in units of r^)- A comparison with experimental results is given where such data are available from recent experiments; see also Table I. Meson exchanges Ref.

~ only

p only

R,O

K, pl~,w K,K ,(a)

6-quark ")

R and 6-quark

Remarks b)

NUCLEAR MATTER:

(39)

(40) c)

3.5 4.4

no correlations

1.57

no tensor

1.02

no i n i t i a l correlations standard

no c o r r e l a t i o n s , kr= 270 MeV/c

0.38 4.13 0.97

0.52

no c o r r e l a t i o n s , no form f a c t o r factorization m o d e l : n+p previous line plus tensor correlation

2.91 2.33

0.97

0.52

0.1

factorization model: ~ - p previous line plus tensor correlations

0.71 0.97

(96)

-0

-1.0

P o l e model VDM strong coupling

0.7 2.1

(6.6) (41,99)c)3.89

(45,84)

(43)

1.82 0.99

no correlations 1.55

1.23(1.23)

no c o r r e l a t i o n s , no form f a c t o r , r>R °

0.77

5.19

9.96

CHKI

0.77

2.18

5.54

CHK2

0.77

0.73

3.0

CHK3

4.3

no c o r r e l a t i o n s , no form f a c t o r

2.3 2.0

g'

=

g'

= 0.52

0.13

209

Decays of A-Hypcrnuclei 12 ^C.,

(41,99)c)3.4 2.0

no c o r r e l a t i o n s

1.2 no correlations, no form factor

(45,84)c)0.48 1.76 0.74 0.24

0.41 0.41 0.41 (43)

3.87 2.25 1.28

CHK1 CHK2 CBK3 g' = 0.13,0.52

1.5

Experiment: (4,26) (1.14±0.20)r^

(98)

SHe : -0.5 0.144

no c o r r e l a t i o n s

0.097

0.450 O. 033

(41,99)=*1.6

no c o r r e l a t i o n s

0.9 (43)

0.5

1.05

g'

1.15

g ' = 0.52

=

0.13

Experiment .(4'26) (0.44 -+°'lS~r " 0.31 " -A ~He:

(98)

0.126

Experiment(from

(98)

0.013

0.077

0.369

n+p

0.038

x-p

Block and D a l i t z ( 3 7 ) ) : (0.16±0.03)F^

0

0.013

Experiment (from Block and Dalitz(37)):(0.23 -+°°%r 0.07"-A a ) C o n t r i b u t i o n from the quark ( s h o r t range) s e c t o r in the h y b r i d q u a r k - h a d r o n model c a l c u l a t i o n s of Refs. ( 4 5 , 8 4 ) . b)Most remarks a r e d i r e c t l y r e l a t e d to d i s c u s s i o n s p r e s e n t e d in the t e x t . c)"Best" values quoted by authors: Ref. (40), 1+1.0. ~ s , Ref. (41,99), 1.23 (nuclear matter), 1.2 (*~C), 0.5 ( ~ H e ) ; (45,84), 1.28 (I~C).

210

J. Cohen

Table 7. Theoretical results for the ratios v - r~./I~..=, Eq. (47), and K -= r v/rp¢, Eq. (52). A comparison with th~"ex~%rimental results is glven where such data are available from recent experiments; see so Table 2. Meson Exchanges Ratio

Nucleus

Nuclear matter

Refs.

(39)

n

~, p

~, p,~]., 6-quark a) ~ and (~, K, K 6-quark

7.0

no correlations

2.3

no initial correlations no tensor correlations standard VDM strong coupling (6.6) no correlations

14 2.6 (96)

1.5 0.8

(41,99) 11.2 16.6 13.1 12

^C

(41,99)

(I02)

ASHe

Remarks

4.6 5.0

2.9 1.6

2~-

5.3

2.7

with correlations no correlations with correlations CHK3

Experiment: (4,26) .... 0 74 -0.3 +1'2 [see also Eq. (7) ] K+ p (98) 17.2 58.8 0.43 (41,99) 15 19

Jl-p

no correlations with correlations

2.1

Experiment: (4'26) v>0.5 [see also Eq. (8)] Nuclear matter

12

AC

( 3 ~ b)

0.23

no correlations

0.56

no i n i t i a l correlations standard no c o r r e l a t i o n s w i t h correlations

2.93 (41,99) 0.14 0.18 0.21 (41,99) 0.1 (41,99) 0.I

0.90 1.1 0.8

a ) C o n t r i b u t i o n from t h e s h o r t - d l s t a n c e b)Approximate r e s u l t s

s e c t o r in t h e h y b r i d q u a r k - h a d r o n model.

based on Adams' R ( a ' ~ )

v a l u e s [ s e e Eq. ( 2 3 ) ] .

211

Decays of A-Hypernuclel T a b l e 8. D e t a i l s of t h e o r e t i c a l c a l c u l a t i o n s f o r the n o n - m e s o n i c decay r a t e r ~ based on the o n e - p l o n exchange mechanism o n l y . R e s u l t s a r e glven in units of rA.

McKellar and Gibson: 14°l'a~

s+s I) No short range c o r r e c t i o n s

Partial rates s+p s-~d

F~ )

D--.-.U1

i-7".50

3~.2

4.~

2) Initial two-body f(r), ro= 0.75 fm

0.001

0.44

1.87

1.87

3) As 2), but ro= 1.0 fm

2x 10 -4

0.25

1.31

1.56

0.031

0.005

1.03

1.06

4) Form factor, K= 624 MeV; correlations, ro= 0.75 fm 5) Form factor, K= 624 MeV; correlations, r ° = 1.41 fm

0.75

6) no form factor

2.03

7) form factor with K=624 MeV

0.97

a) The r e s u l t s ( 1 ) - ( 5 ) do not i n c l u d e t e n s o r - f o r c e c o r r e l a t i o n s . In obtaining (6) and (7)9 McKellar and Gibson used the Reid s o f t core p o t e n t i a l to g e n e r a t e the s c a t t e r i n g s t a t e s and a n u c l e a r m a t t e r r e a l i s t i c c o r r e l a t i o n f u n c t i o n . R e s u l t s (6) and (7) i n c l u d e c o r r e l a t i o n s and t e n s o r - f o r c e effects. All the results of McKellar and Gibson were calculated for nuclear matter. Takeuchi, Takaki and BandS: 1981'bI Partial rates 1So~is °

iSo~3Po

3SI~3S 1

l):He

4.4xi0 -4

8.3xi0 -3 0.092

2)A4Be

5.5XI0 -4

3)4He

4.4xi0 -4

3Sl~3d I

3Sl~Ipl

3Si~3pl

r~ )

0.038

2.8x10 -3

3.2xi0 -3

0.144

9.1xlO -3 0.072

0.028

0.013

2.8x10 -3

0.126

7.2xi0 -3 0

0

0

5.6xi0 -3

0.013

b) We give the results of the full calculation with short range corrections included.

212

J. C o h e n

Dubach et al. (41'99~'¢~ (nuclear matter):

lSo~lS °

lSo~3Po

3S1@3S1 3S1~3d1

1) No c o r r e l a t i o n s 0.01

0.156

0.01

2.93

0.468

0.312

3.89

0.00

0.037

0.789

0.751

0.128

0.117

1.82

results

as

3sldlpl

3S1~3pl

r~ ~

(plane waves) 2) W i t h all correlations

c) We provide the

Dubach's notation at the end

published of

should be devided by 9, the other

Subsec. partial

by

Dubach; recall the discussion of

3.5.2.

Thus, the3s,~3plpartlal rate

rates by 3, in order to convert the

numbers to our notation. Cheung, Heddle and Kisslinger: (45's4) s~s

s~p

s~d

r~'

6.1x10 -s

0.18

0.81

0.99

Nuclear Hatter: 1) No short range correlations 2) As 1) but R ° = 0

3.9

3) Fermi Averaging

7.7x10 -s

0.19

0.85

1.04

4) Initial Adams correlations

1.3x10 -4

0.19

0.92

1.12

5) Form factor, K = 624MeV

1.2x10 -3

0.17

0.76

0.93

6) Final eikonal distortions

9.8x10 -4

0.11

0.66

0.77 d)

12 ^ C ..

1) No correlations

0.48

2) Initial HillerSpencer f(r)

0.49

3) Form factor, K = 624 HeV

0.45

4) Eikonal distortions

0.41 a~

d) "Best results" for these calculations .

Decays of A-Hypernuclei r a t i o ~ (by i s o s p i n c o n s i d e r a t i o n s f o r the f i n a l 3s1~3d I t r a n s i t i o n d o m i n a t e s the decay p r o c e s s ) .

213 NN s t a t e ,

~>>1 when the

The nonmesonie decay r a t e s vould be s l i g h t l y h i g h e r i f one used k F ffi 270 HeV/e i n s t e a d of the number k r = 250 HeV/c used by Adams. Using Eq. (53) ve f i n d t h a t the decay r a t e s f o r k r a n- 270 MeV/c, v h i c h i s a more a p p r o p r i a t e Fermi momentum f o r heavy n u c l e i d nuclear matter, v o u l d be a b o u t a f a c t o r of (270/250) 3 = 1.26 h i g h e r then the one f o r k r = 270 MeV/e. T h i s b r i n g s Adams' "standard" r to 0 . 4 8 r , and the u n c o r r e l a t e d r a t e to 4 . 4 r . . The r a t i o s ~ and a r e not a p p r e c i a b l y a ~ f e c t e d by t h i s m o d i f i c a t i o n ( i n d e e d , to the e x t e n t t h a t the r a t e s a r e p r o p o r t i o n a l to k~, t h e s e r a t i o s a r e t o t a l l y i n d e p e n d e n t of k r ) . As shorn i n T a b l e s 6 , 7 , and 8, o t h e r vorks have n o t been a b l e to v e r i f y Adams' r e s u l t s r e g a r d i n g the r o l e of c o r r e l a t i o n s . While Adams' u n c o r r e l a t e d r e s u l t s a r e i n rough but r e a s o n a b l e agreement v i t h o t h e r a u t h o r s , h i s f u l l y - c o r r e l a t e d c a l c u l a t i o n s c o n t a i n e d too much s u p p r e s s i o n of FNN ( i . e . , r N was reduced by an o r d e r of magnitude as a r e s u l t of e o r r e l a t i o n s ) ~ - a n d the b ~ a v i o r of the r a t i o once c o r r e l a t i o n s a r e i n c l u d e d i s i n q u a l i t a t i v e d i s a g r e e m e n t v i t h some of the o t h e r a u t h o r s . B a r r i n g a c a l c u l a t i o n a l e r r o r , a l l t h i s might i n d i c a t e t h a t the c o r r e l a t i o n s used by Adams v e r e too l a r g e . I n t h a t s e n s e , the c a l c u l a t i o n reached i t s g o a l of p r o b i n g the s h o r t range c o r r e l a t i o n s . ( I t may be t h a t the n u c l e a r many-body model vas too c r u d e , but a c c o r d i n g to Adams' arguments t h i s p a r t of the i n p u t i s r e l i a b l e . ) Cohen and Walker ~44~ a r e c u r r e n t l y s t u d y i n g the r o l e of o n e - p i o n exchange i n non-mesonic decays of medium h y p e r n u c l e i , v h e r e r e l a t i v e AN p - s t a t e s (1=1) a r e i n c l u d e d i n the c a l c u l a t i o n , and the e f f e c t of the l a r g e e n e r g y t r a n s f e r i n the pion p r o p a g a t o r i s c o n s i d e r e d . The i m p o r t a n c e of s t u d y i n g t h e s e e f f e c t s has a l r e a d y been a l l u d e d to i n See. 3 . 4 . Although no f i n a l r e s u l t s a r e a v a i l a b l e , the f o l l o v i n g c o n c l u s i o n s can be reached a t t h i s time. S i g n i f i c a n t c o n t r i b u t i o n s a r e expected from the t e n s o r and p a r i t y - v i o l a t i n g components of the o n e pion exchange p o t e n t i a l , s t a r t i n g from a r e l a t i v e AN p - r a v e ( v h e r e the s i n g l e particle initial states a r e Als and N l p ) . T h i s i s m o s t l y c o n t r i b u t e d by r e l a t i v e p - v a v e ~ p - v a v e f o r the t e n s o r , and p-vave-~i-vave f o r the p a r i t y violating interaction. No a p p r e c i a b l e c o n t r i b u t i o n i s found from the c e n t r a l p o t e n t i a l , and the t e n s o r c o n t r i b u t i o n to the decay r a t e i s s t i l l d o m i n a n t . These r e s u l t s a r e p r e l i m i n a r y , but i f confirmed by f u r t h e r york they could b r i n g a change i n the r a t i o ~ o b t a i n e d under the a s s u m p t i o n of an i n i t i a l relative s-rave only. Only a modest r o l e i s played by l a r g e e n e r g y t r a n s f e r ~ the p i o n p r o p a g a t o r . We f i n d very s i m i l a r r e s u l t s f o r m = 107 MeV(ffi ~m~-q~, q~ = 90 MeV) and m. = 139.5 MeV. The r a t i o s v and ~ a r e not e x p e c t e d to be s i g n i f i c a n t l y a f f e c t e d by the v a l u e chosen f o r m . A f i r s t s t e p beyond the one pion exchange model c o n s i s t s of a d d i n g the o meson exchange, as d i s c u s s e d i n S u b s e t . 3 . 5 . 2 . R e s u l t s from R e f s . ( 4 0 ) , (96) and (98) a r e shorn i n T a b l e s 6, 7, and 8. The t e n s o r component of the t r a n s i t i o n p o t e n t i a l i n t h i s model vas found to be the dominant one. I n the c a l c u l a t i o n s of NcKellar and Gibson, ~4°~ most of the decay p r o c e s s proceeded through the s-~d t r a n s i t i o n , induced by the t e n s o r p a r t of the p o t e n t i a l . In addition, tensor c o r r e l a t i o n s played an i m p o r t a n t r o l e (though n o t as i m p o r t a n t as found by Adams(391). The form f a c t o r , v i t h K = 624 MeV, reduced the t o t a l n o n - m e s o n i c r a t e i n the p i o n - e x c h a n g e model by about 50~, and reduced s h a r p l y the s~p c o n t r i b u t i o n to the t o t a l r a t e , so t h a t most of the decay p r o c e s s proceeded through the s-~i t r a n s i t i o n . This d r a s t i c s u p p r e s s i o n of the s~p p a r t i a l r a t e vas not v e r i f i e d by Dubach e t e l . ~41'99~

214

J. Cohen

These f e a t u r e s of the non-mesonic decay p r o c e s s a r e s u p p o r t e d by the work of Takeuchi e t a l . , (gs) where i t was l i k e w i s e found t h a t the n o n - m e s o n i c decay of l i g h t A-hy-per-'nuclei i s dominated by the t e n s o r i n t e r a c t i o n , l e a d i n g to very s m a l l r a t e s f o r A4H compared with A4He and A5He. I n t h e i r c a l c u l a t i o n s , 19s) the 3 ~3 s. d. t r a n s i t a •o n s t r o n g l y dominated the decay r a t e . With t e n s o r c o r r e l a t i o n s inclu~ed, this channel decreased s i g n i f i c a n t l y in importance (in qualitative agreement with the r e s u l t s of Adams). Instead, the 3s.~3s1 t r a n s i t i o n was g r e a t l y enhanced due to the induced f i n a l - s t a t e NN 3d. r e l a t i v e s t a t e . This is a r e s u l t of the i n t e r p l a y between the NN t e n s o r ~ n t e r a c t i o n and the t e n s o r component of the AN-R~N t r a n s i t i o n p o t e n t i a l . The n and p c o n t r i b u t i o n s were found to be comparable i n magnitude i n the dominant 3s.~3s. c h a n n e l . Thus, the t o t a l decay r a t e based on n+p i s l a r g e , w h i l e n-p l e ~ - ~ o an a l m o s t complete mutual c a n c e l l a t i o n . I n the ~H h y p e r n u c l e u s , the i m p o r t a n t t r a n s i t i o n s 3sx~3sl,3d. a r e m i s s i n g because t h ~ I . = 0 f i n a l s t a t e s can o n l y be reached from the Ap i n i t i a l p a i r , w h i l e the ( s i n g l e ) Ap p a i r i n ~H i s i n a r e l a t i v e i s ° state. This dominance of the t e n s o r - i n d u c e d t r a n s i t i o n s r e s u l t s i n a g r e a t l y enhanced p a r t i a l r a t e f o r the Ap-mp c h a n n e l r e l a t i v e to An-mn. With s-~d t r a n s i t i o n o n l y [as assumed i n p a r t of the c a l c u l a t i o n s of Ref. ( 4 0 ) ] , no An-mn t r a n s i t i o n s occur. A p r a c t i c a l l y similar situation o c c u r r e d i n Ref. ( 9 8 ) , where the ~s ~3s ,3d. t r a n s i t i o n s were found to s t r o n g l y dominate over o t h e r o n e s . An i n t e r e s t i n g e x c e p t i o n i s the work of N a r d u l l i ~ 6 1 who found ~ = 1.5 f o r n u c l e a r m a t t e r , i n much b e t t e r agreement with the d a t a (though such d a t a a r e r a t h e r inaccurate). N a r d u l l i , (961 r e l y i n g on Ref. ( 4 0 ) , kept o n l y the p a r i t y - c o n s e r v i n g components of the t r a n s i t i o n . p o t e. n t i a l s VR and Vp . The c o r r e s p o n d i n g t e n s o r terms were found to be o p p o s i t e xn s i g n , l e a d i n g to a reduced n o n - m e s o n i c decay r a t e i n t h i s model (of combined n and p exchanges) r e l a t i v e to the n - e x c h a n g e model result. An i m p o r t a n t consequence of the reduced t e n s o r c o n t r i b u t i o n i s the g r e a t l y improved r a t i o ~. Note t h a t i n i t s p r e s e n t form, N a r d u l l i ' s c a l c u l a t i o n would g i v e a r a t i o n = 0 [ see Eq. ( 5 2 ) ] ; even with the ( r e l a t i v e l y s m a l l ) p a r i t y - v i o l a t i n g components of the t r a n s i t i o n p o t e n t i a l r e t a i n e d , the e x p e c t e d r a t i o i s n<
rNM=

i 3.52r A , n+p 0.72r A, n-p

Decays of A-Hypernuclei f o r the VDH v a l u e o f 3 . 7 ,

rNH = for a ratio

6.13 r h ,

,+p

0.06 r h ,

n-p

215

and

o f 6 . 6 . (gs)

While d i s c u s s i n g N a r d u l l i ' s work, i t i s i n t e r e s t i n g to p o i n t o u t t h a t h i s weak ANp c o u p l i n g was c a l c u l a t e d by u s i n g a p o l e model. U si n g a s i m i l a r model, H c K e l l a r and Gibson (4°~ found a n e g l i g i b l e c o n t r i b u t i o n from p e x c h a n g e ; t h i s i s not a t a l l t h e c a s e in Ref. ( 9 6 ) . I t i s worthy o f n o t e t h a t N a r d u l l i ' s p o l e model was based on both baryon and meson p o l e s , and b a r y o n s b e l o n g i n g to t h e g r o u n d - s t a t e baryon o c t e t and to t h e ~÷ f i r s t e x c i t e d baryon o c t e t were considered. Moreover, N a r d u l l i pointed out that t h e f a c t o r x z a t i o n model o f Ref. (40) i s s i m i l a r to th e K pole contribution i s h i s (961 p o l e model. Finally, McKellar and Gibson i n d i c a t e d that their p-exchange t r a n s i t i o n p o t e n t i a l in t h e f a c t o r i z a t i o n a p p r o x i m a t i o n goes to a c o n t a c t i n t e r a c t i o n c o n s i d e r e d by Block and D a l i t z , (37r as m -~. P

Dubach e t a l . (41'99) a l s o p u b l i s h e d r e s u l t s o f a s t u d y o f A - h y p e r n u c l e a r nonmesonic-~e~ys based on ~ and p exchanges. ( T h e i r f u l l model i n c l u d e d o t h e r mesons as w e l l , and such r e s u l t s will be d i s c u s s e d shortly.) These c a l c u l a t i o n s g e n e r a l l y a g r e e w i t h t h o s e o f R e f . ( 4 0 ) , however some d i f f e r e n c e s exist. While H c K e l l a r and Gibson o b t a i n e d a l m o s t a l l o f t h e i r d ecay r a t e from the s->d t r a n s i t i o n , Dubach found an a p p r o x i m a t e l y 20~ c o n t r i b u t i o n to T,M from s~p. In g e n e r a l , 0-exchange had a r e l a t i v e l y s m a l l e f f e c t on D u b a c h ' s r e s u l t s , whereas H c K e l l a r and Gibson d e m o n s t r a t e d t h e wide v a r i a t i o n i n t h e p - e x c h a n g e c o n t r i b u t i o n o b t a i n a b l e from d i f f e r e n t o p t i o n s f o r the ANp v e r t e x . This point was f u r t h e r pursued by N a r d u l l i (96), who a l s o found, however, a r a t i o ~ = 1.5 or 0.8 as d i s c u s s e d above; such a m o d i f i e d r a t i o is in disagreement with Dubach's numbers. On the o t h e r hand, N a r d u l l i ' s a s s u m p t i o n t h a t no c o n t r i b u t i o n a r i s e from t h e p a r i t y - v i o l a t i n g p a r t s o f V~ and V i s r o u g h l y s u p p o r t e d by the r e l a t i v e l y s m a l l v a l u e o f ~ = 0 . 2 r e p o r t e d by Du~ach e t a l . (As we s h a l l s e e s h o r t l y , a d d i n g o t h e r meson e x c h a n g e s to t h e d ecay mechanism ch an g es t h e situation substantially.) The o v e r a l l p i c t u r e r e s u l t i n g from th e c a l c u l a t i o n o f R e f s . ( 4 0 ) , (96) and (98) i s t h a t o f d i s a g r e e m e n t and c o n f u s i o n . I t would be e s p e c i a l l y i m p o r t a n t and i n t e r e s t i n g to c l a r i f y the r o l e o f t h e t e n s o r i n t e r a c t i o n and c o r r e l a t i o n s i n the n and p exchange model, l o o k i n g s i m u l t a n e o u s l y i n t o t h e s~p d ecay c h a n n e l . A much improved u n d e r s t a n d i n g of t h e hNp and NNp i n t e r a c t i o n would be r e q u i r e d in o r d e r to r e s o l v e t h e t h e o r e t i c a l a m b i g u i t i e s s u r r o u n d i n g the p exchange contribution. ~ i t h t h e s e problems c o n c e r n i n g the 0-exchange c o n t r i b u t i o n to t h e t r a n s i t i o n p o t e n t i a l s t i l l u n r e s o l v e d , we t u r n now to r e s u l t s from t h e c a l c u l a t i o n s o f Dubach e t a l . (41'991 where more meson e x c h a n g e s were i n c l u d e d ( f o r d e t a i l s s e e Subsec.-~.~..2). For the p i o n - e x c h a n g e c o n t r i b u t i o n Dubach found a v e r y s m a l l i s o ~ls o t r a n s i t i o n r °a t e , which became n e g l i g i b l e i n t h e c o r r e l a t e d c a l c u l a t i o n . The t e n s o r c o r r e l a t a o n s i n t h e NN f i n a l s t a t e , r e s u l t i n g from t h e t e n s o r component o f the Reid p o t e n t i a l used i n t h e c a l c u l a t i o n , mixed t h e 3s I and 3d 1 f i n a l s t a t e s as shown i n T a b l e 8. F u r t h e r m o r e , t h e NN f i n a l s t a t e c o r r e l a t i o n s r e s u l t e d i n a r e d u c t i o n o f the t o t a l 3 s l ~ 3 s l , 3 d 1 s t r e n g t h by a f a c t o r o f about two, and of the s~p p a r t i a l rates by a f a c t o r o f 3 - 4 . I n c l u d i n g the p

216

J. Cohen

exchange in the calculation enhanced the Is -~3po partial rate by some 50~ (from o.037rA to O.052rA, using Dubach's definitions for the partial rates) and reduce~ other par'~ial rates (especially the 3s1-~3sI +3di), reducing somewhat the total rate as shown in Table 6. Inc]uding "all 'r the mesons (considered in the calculation) contributed to a further . reduction of rN M , (the a meson exchange did very little except for enhanclng the Iso-~Iso partlal rate from 0.001r^ to 0.004r^, and thus had a completely negligible effect in these calculations). The main effect of considering "all" the mesons was a reduction in the s-~s,d rate (from 1.54rA in the correlated R-exchange to 0.647r.) and a substantial enhancement of t~e 3s.-~3p. transition (from 0.1171". to 0.~36r., respectively) ~ • (the 3 s I_ ) 1 Pl transition is~ only sl;ghtly changed ~n the comp~[ete calculation relativ~ t-o the correlated• pion-exchange one, from 0 " 128r A to 0 * 110r., i~ respectively ). Dubach attrlbutes ~41'99) this effect to the K meson exchange. This is especially important when the ratio ~, Eq. (52), is considered, as in these calculations it went up from 0.14 (for an uncorrelated one pion exchange interaction mechanism) to 0.18 (for a correlated one pion exchange) to 0.21 (for ~ and p exchanges), and up to 0.90 for the full calculation with "all" mesons considered. These are the only calculations yielding such a large ratio which, if measured, would provide an important test of this model. Moreover, heavy meson exchanges were also crucial in obtaining ,0 = 2.9 in this model, while ~ and p exchanges alone tend to yield (except for Nardulli's (96) calculation) a much larger ratio (~Yl0). The experimental situation is still unsettled, nonetheless available results do seem to support the lower values for the ratio ~ (see chapter 2, especially Tables 2 and 3). Note that the ratios v and • seem to be (at least in this calculation) somewhat less sensitive to correlations, but vary strongly with the model. This shows once again that they are especially attractive measurable quantities for a meaningful comparison between theory and experiment. In the hybrid quark-hadron model calculations, (4s's4) discussed in Subsec. 3.5.3, Heddle and Kisslinger chose an initial six-quark probability P,6 q = 0.15 for nuclear matter. For the finite-hypernuclear case, PI6 q was calculated from Eq. (81) and found to be strongly-dependent on the relative A-N partial wave and on the presence of correlations. Thus, for the s-wave P,6 q = 0.078 (uncorrelated) or 0.051 (correlated), while for the p-wave P~q = 0.007 and 0.005, respectively. (Recall that only those components with center-of-mass angular momentum L=0 were retained by the authors for reasons of calculational simplicity.) The finite-hypernuclear numbers for p6q are very different from the nuclear matter one; no explanation is available for these large differences. In the absence o f a reliable calculational scheme for the six-quark wave function in the short-distance region, it was impossible to determine the amplitudes of the various configurations. Instead, Heddle and Kisslinger studied the variation of r,, over a range of ten different choices for these expansion coefficients. TS~ largest contributor is the configuration where all six quarks (including the two quarks interacting weakly in the reaction s+udu+d) are in the sl, = state; this is also the only configuration considered in the initial study F f Cheung et al. (4s) The smallest contribution results from the state where the inte'ract--fng quarks are occupying, the p3/2pl/2 configuration. In order to estimate the A-hypernuclear non-mesonlc oecay rate,

Decays of A-Hypernuclei

217

Heddle and Kisslinger (4s''4) averaged r__ over the seven largest eontributions~ ignoring the three configurations w i ~ the smallest calculated r_,. This yields the 6-quark results in Table 6 for their three different mode~s (CHK1, CHK2 and CHK3, as introduced in Subsec. 3.5.3). As explained above, the combined (x and 6-quark) decay rate was obtained by adding the square roots of the two individual contributions to r from the two sectors, and then squaring the result [for example, (40.77 + 42.~) 2 = 5.54 for model CHK2; the amplitudes for the interior and exterior sectors add constructively in this work]. We recall that the six-quark contribution (from the short-distance sector) was similar for the nuclear matter and finite-nucleus calculations, the only change being the different values for P~q. Note also that the contribution to the non-mesonic decay rate from this sector, Eq. (83), is proportional to P~q. On the other hand, the pionic contribution (from the long-distance sector) was treated differently for the nuclear matter and finlte-hypernuclear (shell model) calculations, as explained in Sec. 3.4 (see also Subsec. 3.5.3). Detailed results of the pionic calculation by itself are presented in Table 8; note that the s-~d transition rate is dominant, while the s-~s one is negligible again. The role of short-range two-baryon correlations ]s somewhat less crucial in the hybrid quark-hadron model. It affects both the pionic contribution (at the level of some 20Z-30%, as shown in Tables 6 and 8) and the initial six-quark probability P~q (as discussed above). The reason for the more modest role of the short-range correlations in this case is the basic premise that the behavior of the A-N and N-N systems at r
"

"

218

J. Cohen

V i t h the u n c e r t a i n t i e s and a p p r o x i m a t i o n s made in c a l c u l a t i n g t h e A - h y p e r n u c l e a r non-mesonic decay r a t e s w i t h i n t h e q u a r k - h a d r o n h y b r i d model, i t becomes increasingly difficult to distinguish the pertinent numerical results from those of other models. The results of Ref. (45) are very sensitive to the choice of the weak-interaction Hamiltonian (i.e., in models CHK1, CHK2 and CHK3) for the short-distance sector. The theoretical construction of this Hamiltonian is still an unsolved problem at present. Furthermore, the results are strongly dependent on p6q, the initial six-quark probability, which cannot be precisely determlned In these calculations. There is, however, one feature that distinguishes this model from all other calculations, namely, the AI = 2 nature of the six-quark contribution. This is an important characteristic which should be tested experimentally (a possible test may be devised using the approach discussed in Sec. 3.3). The last approach discussed in Subsec. 3.5.4 is that of Oset and Salcedo. 143~ Results were given in the local density approximation for a range of nuclear species A=5-208, where the radial dependence of the density was taken in the Fermi form

p(r) = 3 4Ac 3

((1

nZt2/c2)[1 + exp(~-q)]} -1,

(93)

+

with t(=4.4a)=2.3 fm and c=I.IA I/3. For the A wave function, the authors used the Is. 2 harmonic oscillator wave function with parameter ~=(45A-I/3-25 A-2/3)~eV; the A binding energy was taken as the lowest eigenvalue of the potential V(r) = -32/[1 + exp[(r-c)/a]} Hev which is a reasonable approximation for the A-hypernuclear central potential. CI°7) The value of g' is unknown, and results were given for g'=O.13 and 0.52. Results are again shown in Tables 6, 7 and 8. For the highest values of A (2°~Pb) the results are very close to nuclear-matter ones calculated at a central constant density. The form f a c t o r and n u c l e a r s t r u c t u r e e f f e c t s reduced the n u c l e a r m a t t e r r e s u l t by a f a c t o r of about 2. The r a t e s r.m v a r y smoothly from 1 . 1 r A f o r SHe, to 2 * Or A f o r 4°Ca, to 2 . 3 r f o r 2°~Pb, f o r g/'-L0.13 ^

(the respective numbers ~or g'=0.52 are LI5 L,

1.8#^ and 2.0r^). Thus, the

local density approximation brings about an a~preciable reduction in the nonmesonic decay rates relative to the infinite nuclear matter rate, especially for the lighter species (A<40). The sensitivity to g' is modest; as already explained in Subsec. 3.5.4, this is expected on the basis of the large energy transfer in the non-mesonic decay process. In general, Fermi gas calculations reveal a quenching of the Isovector spln-transverse (~xq) response and an enhancement of the spinlongitudinal (%.q) one relative to a free calculation, as a result of the spinisospin correlations. For g'=O.13, the longitudinal response dominates and the ne t r e s u l t i s a s l i g h t i n c r e a s e in t h e decay r a t e , w h i l e the r e v e r s e i s t r u e for g'=0.52. The s m a l l e f f e c t o f th e s t r o n g - i n t e r a c t i o n m o d i f i c a t i o n s o f t h e pion p r o p a g a t. o r on r f f c l e a r s the s c e n e f o r s t u d y i n g g e n u i n e e f f e c t s o f o t h e r decay mechanxsms, suck as the m u l t i - m e s o n e xch an g es o r q u a r k - q u a r k i n t e r a c t i o n d i s c u s s e d above. Although a d e t a i l e d s t u d y o f the ~ h y p e r n u c l e a r decay mode i s o u t s i d e the scope o f t h i s Review, we n o t e t h a t th e r e l e v a n t r a t e s in R e f . (43) were found to be v i r t u a l l y i n d e p e n d e n t o f g ' . However, they i n c r e a s e d d r a m a t i c a l l y as a r e s u l t o f the m o d i f i e d p i o n i c p r o p a g a t o r . As e x p l a i n e d i n Subsec. 3 . 5 . 4 , t h i s may be u n d e r s t o o d as a c o n s e q u e n c e o f the m o d i f i e d r e l a t i o n s h i p between t h e

Decays of A-Hypernuclei

219

pion e n e r g y and momentum i n the n u c l e a r medium, Eq. ( 8 9 ) . We o b s e r v e t h a t f o r the mesonic decay ~(~)<(~2 + m~)1/2, p r o v i d i n g more e n e r g y and momentum to the n u c l e o n and i n c r e a s i n g the a v a i l a b l e phase space f o r t h i s decay mode. ( I n d e e d , the mesonic decay r a t e t u r n s out to be v e r y s e n s i t i v e to the e f f e c t i v e p i o n mass a t t y p i c a l n u c l e a r - s u r f a c e d e n s i t i e s . ) The r e s u l t i n g mesonic decay r a t e s i n t h i s • model a r e very l a r g e : , 0.4F.A . f o r '2C or 0.7F^ f o r ~He; a s u b s t a n t i a l t h e o r e t a c a l u n c e r t a i n t y a s s o c x a t e d with th~se r e s u l t s was e s t i m a t e d by the a u t h o r s to be l a r g e r than 3OZ. This u n c e r t a i n t y comes a b o u t b e c a u s e an a v e r a g e momentum was chosen ~43~ f o r the A-hyperon. V a r y i n g t h i s momentum, the mesonic r a t e s could go down by as much as a f a c t o r of t e n . T h e r e f o r e , a comparison with e x p e r i m e n t i s r e n d e r e d m e a n i n g l e s s . This i s , of c o u r s e , u n f o r t u n a t e , s i n c e the l a r g e e f f e c t s of the m o d i f i e d p i o n i c p r o p a g a t o r i n the mesonic decay case p r o v i d e a good o p p o r t u n i t y f o r a c t u a l l y testing the model ( s u c h an o p p o r t u n i t y i s not p r o v i d e d by the n o n - m e s o n i c decay p r o c e s s ) . N o n e t h e l e s s , we n o t e t h a t a v a l u e of 0 . 4 5 r . f o r the mesonic decay r a t e does n o t a g r e e w i t h the e x p e r i m e n t a l r e s u l t s i n Eq~ ( 7 ) . S i n c e one s h o u l d a l s o a c c o u n t f o r p i o n t r u e a b s o r p t i o n and charge exchange p r i o r to comparing w i t h e x p e r i m e n t a l data, Oset and Salcedo ~43~ e s t i m a t e d t h a t t h e s e e f f e c t s y i e l d an o v e r a l l a t t e n u a t i o n f a c t o r of about 0.7 ( i . e . , a 30Z r e d u c t i o n i n the number of p i o n s o b s e r v e d w i t h r e s p e c t to those o r i g i n a l l y produced by the mesonic decay p r o c e s s ) i n a n u c l e u s such as 1~C. Note t h a t c a r e must be taken i n d e a l i n g w i t h such e f f e c t s as the absorbed ~ i o n s may i n d u c e a n o n - m e s o n i c decay s i g n a l by m u l t i - n u c l e o n k n o c k o u t , n o n e t h e l e s s t h i s c o m p l i c a t i o n has not y e t been looked i n t o . The r e s u l t i n g e s t i m a t e f o r the mesonic decay r a t e i s c o n s e q u e n t l y 0.2F^-O.3FA, which i s much c l o s e r to the numbers i n Eq. ( 7 ) . R e s u l t s o b t a i n e d with the f r e e ( n o n r e n o r m a l i z e d ) pion p r o p a g a t o r , namely O.1Fa, a l s o f a l l w i t h i n the r a n g e of the e x p e r i m e n t a l e r r o r b a r s . The e x p e r i m e n t a l e r r o r b a r s on the measured mesonic decay r a t e s f o r A~He, gq. ( 8 ) , a r e p r e s e n t l y too l a r g e to a l l o w f o r a comparison with t h e o r y . H i g h e r - p r e c i s i o n r e s u l t s a r e e v i d e n t l y c a l l e d f o r . The f o l l o w i n g i n g r e d i e n t s i n the c a l c u l a t i o n s had a n e g l i g i b l e e f f e c t on the non-mesonic mode and r e s u l t e d i n a s m a l l r e d u c t i o n of the mesonic decay r a t e s : ~43~ ( i ) the s-wave p a r t of the pion s e l f - e n e r g y , r e s u l t i n g from the r e p u l s i v e s-wave rtN i n t e r a c t i o n ( s e e a l s o Oset and P a l a n q u e s ~ l ° 3 ~ ) ; ( i i ) an average of the decay r a t e c a r r i e d out over the momentum d i s t r i b u t i o n of the A wave function (~;p^(~)r(~)d3k); (iii) using the c.m. pion momentum for ~ [in places such as Eq. (14)]. (State-of-the-art treatments of hypernuclear pionic decays are available in the literature. ~1°s~ Such calculations, which use realistic wave functions, DWIA effects with distortion and absorption and strong-interaction correlations, are outside the scope of the present work.) Where both nuclear-matter and finite-hypernuclear results were given in the same work, the above works revealed a reduction of the non-mesonic decay rates in going from nuclear matter to finite nuclei. The ratios r (nuclear matter)/rNM(finite nuclei) vary with the model, as shown in Table ~ Table 9 demonstrates the importance of a finite-hypernuclear treatment in a realistic many-body model. Note that the six-quark model calculations of Refs. (45,84) yield much larger ratios than the other models. This is partly a result of the substantially different initial six-quark probabilities used for nuclear matter and finite nuclei (SZ and 15Z, respectively, as discussed above). From the calculations of Dubach, ~99~ who is the only author to provide (preliminary) results for the ratios ~ and ~ in both nuclear matter and finite nuclei, we conclude that these ratios are also modified by going from nuclear matter to finite nuclei (see Table 7).

220

J. Cohen

Table 9.

R a t i o s of h y p e r n u c l e a r non-mesonic t o t a l decay r a t e s r ( n u c l e a r NM matter)/F.M ( f i n i t e h y p e r n u c l e u s ) i n the v a r i o u s models d i s c u s s e d i n the t e x t .

Hypernucleus

References

Ratio

12C

(99)

1.0

(45,84)

2.6

~He

Comments

model CHKI

2.5

model CHK2

(43)

2.3 1.5

model CHK3 g'=0.13

1.3 2.5

g' = 0.52

(99) (43)

2.2

g' = 0.13

1.7

g' = 0.52

We f e e l c e r t a i n t h a t , a t the end of t h i s c h a p t e r , the r e a d e r i s aware of the t h e o r e t i c a l complexity involved in the n o n - m e s o n i c weak decay p r o c e s s . Moreover, the l i t e r a t u r e c o n t a i n s w i d e l y d i f f e r e n t t r e a t m e n t s of the problem. Under such c i r c u m s t a n c e s , i t is useful to p r o v i d e s i m p l e e s t i m a t e s , e s t a b l i s h i n g l i m i t s on the magnitudes of the p e r t i n e n t p h y s i c a l q u a n t i t i t e s (such as r ~ . ) . Such an argument was g i v e n by McKellar and G i b s o n , <4°~ r e l y i n g on the one-pion exchange mechanism for AN-~IN and the optical-model expression ¢ 7 ~ rsM = P •

(94)

I n Eq. (94), d e r i v e d by Gal and Dover f o r the l o w - e n e r g y I~I+AN c o n v e r s i o n , the bra and ket (<>) denote Fermi a v e r a g i n g , v i s the r e l a t i v e v e l o c i t y of the AN p a i r c o n v e r t i n g with t o t a l c r o s s s e c t i o n ~^s+s,' and p i s the n u c l e a r d e n s i t y a t the A (and s t i m u l a t i n g n u c l e o n ) p o s i t i o n . (Note t h a t the c r o s s s e c t i o n i s i n v e r s e l y p r o p o r t i o n a l to v . ) Using 14°> o . . + . . ~ 10 -12 mb, = 3k./4M_, and p 0.17fm -3, McKellar and Gibson found r,~" ~ 0 . 3 r ^ . Among othe~ th~ngs t h i s e s t i m a t e n e g l e c t s , of c o u r s e , the e f f e c t s of s p i n and i s o s p i n s e l e c t i v i t y <~7> on the decay r a t e s . =

Oset e t a l . (*°9) have r e c e n t l y e s t i m a t e d r NM based on t h e i r s e l f - e n e r g y approach. For the t o t a l decay r a t e they used Eq. ( 9 4 ) ; was c a l c u l a t e d from = 7~"~nMN q ~ZlTI2PF(q)

•

(95)

Decays of A-Hypernuclei

221

In Eq. (95), T is the AN+NN transition amplitude, ZZ is a sum over final spins and an average over initial spins, the average momentum ~ = [HN(HA-H,)]z/2~ and PF(~) is a Pauli blocking factor, given by

PF( ) =

J(2 rd+-p-)3

=1-

- n(; +

0(2- ~F)li - ~ ~F + ~ (~F)3I .

In obtaining Eqs. (95) and (96), ITI 2 was assumed direction ~, angular dependence was averaged nonrelativistic approximation was assumed. Based exchange, Oset et al. ~z°9~ found 4.41r A , rNM =

4.04r A , 2.62r A ,

no Paull blocking with Pauli blocking with form factor, K = 1.25 GeV.

(96)

to be independent of the over, and an extremeon lowest-order one-pion

(97)

These numbers a r e i n v e r y good a g r e e m e n t w i t h t h e n u c l e a r - m a t t e r r e s u l t s p r e s e n t e d in T a b l e 6} 3 9 ' 4 ° ' 4 z ' 4 ~ The e f f e c t o f P a u l i b l o c k i n g i s s m a l l , P,(~) = 0 . 9 2 , due to t h e l a r g e momentum t r a n s f e r (q = 407 MeV/c). The s u b s t a n t i a l role played by the form factor is clearly demonstrated. This is again a result of the large momentum transfer involved. {Note that identical form factors were used at both the weak and strong vertices, yielding a factor of [F(~2)] 4 = 0.63, and thereby creating a softer combined weak-strong effective form factor. The latter type of form factor is the one included in the calculation of HcKellar and Gibson ~4°~ and in Eq. (33) above.} There is no satisfactory dynamica] model for the elementary AN+NN as yet. We have discussed in detail a few possible approaches to the subject. Although they all have their shortcomings, we expect future studies to rely on the earlier works. There are many possible improvements in various directions. Some examples are briefly mentioned here. Following Ferrari and Fonda, (zi°~ Band5 et al. ~zil~ considered the AN+EN-~NN process, induced by ~ exchanges. The pertinent Feynman graph involves a box diagram with three strong and one weak rd~B' vertices. Combined with the onepion exchange mechanism, Band~ et al. found that the AN-EN coupling increased r__(PHe) by some 50%, but at the same- time had also increased the ratio ~ from 7~o 27.2, relative to just one-pion exchange. [Note that the one-pion exchange results differ somewhat from those given in Ref. (98) for the same interaction mechanism[. This increase in ~ is not supported by the data; however the box-diagram representation considered in Ref. (111) is only one example of A-E coupling effects. Other diagrams exist, and should be included in such a calculation before any conclusions can be drawn. G r o u p - t h e o r e t i c a l methods [based on SU(6) and SU(3) s y m m e t r i e s ] , have a l s o been a t t e m p t e d ,
222

J. Cohen

In conclusion, we note that the subject of weak non-mesonic decay in ,%hypernuclei is not properly understood theoretically at present. There is not much justification for hopes of using this decay in order to obtain information on quantities that cannot be measured in free processes (such as the ANp coupling constant) in the near future. It is necessary, instead, to gain better understanding of the elementary and nuclear decay process, in a tedious job of model-calculations. Several models have been applied to it so far. All are based to a certain extent on a one-plon exchange as a starting point; all have their shortcomings. It is possible to obtain results in good agreement with the experimental data for the total non-mesonic decay rates (although results for heavy hypernuclei are not firmly established at this time). However, the ratio ~, Eq. (47), is difficult to understand. For example, we have shown that a very large value for ~ (typically around 10) is obtained in the one-pion exchange model, while more elaborate models lower this result. Consequently, this is a favorable quantity for testing the models experimentally. lit is most important that a precise final state interaction (FSI) analysis be applied to the experimental ratio before a meaningful comparison with the theoretical results is made. Such an analysis is necessary in order to distinguish between the properties that the elementary interaction mechanism should obey, and the consequences of the FSI of the ejected nucleons. This problem bears a lot of similitude to other processes involving high-energy release, such as a-absorption or ~ annihilation in a nucleus.] Likewise, the ratio x, Eq. (52), may also provide important experimental constraints on the available theoretical models. Experimental results for are available or expected in the near future; no measurements of a have been carried out so far, and none is expected at present. With improved and new data, it would also be fruitful to carry out an updated phenomenological analysis as described in Sec. 3.3. In particular, Block and Dalitz ~3v~ suggested that the strongest AN-~NN transition is the 3s,-~3p., parity-changing I. = 1, and the next strongest are the *s_-~*s., 3p^, ~, =~1 transatxons. Once verified on the basis of new experxmental results, these constraints would be extremely important theoretically, as none of the suggested models has been able to yield such results (Dubach's calculation goes in the right direction, increasing the ratio ~ to about 1.) Interestingly, almost thirty years after the original work of Block and Dalitz ~3v~ it still seems to be the most appropriate means for obtaining information on the characteristic properties of the AN-~'N interaction. ,

4.

,

Z

,

v

v

DISCUSSION, FUTURE PROSPECTS AND CONCLUSIONS

We have presented here a review of non-mesonic decays in the general context of hyperon and hypernuclear physics. A great deal of experimental (Chapter 2) and theoretical (Chapter 3) effort has already been invested in the subject, and the research efforts continue. Indeed, exciting physics is involved in every aspect of the problem: weak and strong interactions, models of hyperon decay (A-~Nn and AN-~'N), hypernuclear structure, the modification of weak currents in the strongly-interactlng many-body system, to name just a few. The non-mesonic decay mode offers a unique opportunity for studying the full structure of the (strangeness-changing) weak interaction, including both the parity conserving and parity violating parts thereof. Unlike the case in weak nucleon-nucleon (N-N) interactions, the strong interaction does not mask the parity-conserving part of the AN-~'N process.

Decays of A-Hypernuclei

223

The non-mesonic decay process is not properly understood at present. Little is firmly established about the fundamental aspects of the pertinent four-Fermion, strangeness-changlng weak-interaction process. Unlike the N-N case, there is no experimental data on the free AN-*NN reaction. Hypernuclei are the only source of information at this time, making the study of this subject particularly difficult, as one needs to deal simultaneously with the reaction mechanism, hypernuclear structure and final state interaction effects. Most phenomenological decay mechanisms proposed in the literature, and discussed in detail in Chapter 3, are able to reproduce the experimentally observed total decay rate. Of great importance are the ratios of protoninduced to neutron-induced, or parity-conserving to parlty-violating, partial decay rates, where the various models differ. Although virtually no data exist for the latter, and large error bars exist on the former, the available information seems to rule out the simplest models (such as one-pion exchange, or n- and -p meson exchanges), demanding more complex, often complicated, models. Many exciting open questions exist. The theoretical calculations rely heavily on the bl = ~ rule. Its applicability to the non-mesonic decay mode is an open question. This question is difficult to answer at present from first principles in the absence of a satisfactory understanding of the subject w ~ the Standard Model of electroweak and strong interactions (see Sec. 3.2). However, a phenomenological analysis along the lines discussed in Sec. 3.3, with new and improved data, is likely to clarify the situation from this point of view. Another open question involves the theoretical implications of possible changes in the nucleon structure inside the nucleus. (a'9) Ne also know that the relativistic nuclear a-~ model has interesting predictions for hypernuclei (9'a21 because the A and the nucleon have different Isospins and they couple differently to the mean scalar and vector fields. The relevant implications on the weak decay modes of A-hypernuclei will be of great interest, albeit seriously complicated by the relevant pion dynamics, t811 An intersting conjecture has been pointed out in the literature. (26) Using the central values of the experimental result for r ~ in Eqs. (7) and (8), we obtain:

r H(l c)= O.lOrA, r H(Z--------= He) O.llr A

(98a)

.

(98b)

If this ratio is, indeed, a constant for light nuclei (which still remains to be examined and verified), it would imply that the non-mesonic decay ratio divided by the number of stimulating nucleons available for the non-mesonic decay process, or r~M per a A-N pair, is a constant for light hypernuclear species. This does not seem to be the case for heavy hypernuclei so that a saturation mechanism may be in effect, with the decaying A-byperon interacting mostly with s- and p-shell nucleons. Such observation would, indeed, provide important constraints on the theoretical models of hypernuclear non-mesonic decays. This provides the justification for state-of-the-art measurements of non-mesonic decays for heavy hypernuclei. Moreover, it would be very desirable to obtain the A-dependence for these decays.

224

J. Cohen

Unlike the theoretical situation for non-mesonic decays, a relatively good understanding of the pionic decays has been achieved. 11°.'113~ This decay mode is directly connected with the known A-~Na amplitude in free space, and the pion distortion can be phenomenologically accounted for using low energy pion optical potential. The pionic decay probes the hypernuclear short-range correlations and the propagation of low energy pions deep inside the nuclear medium. The pionlc hypernuclear decay rates seem to be consistent with the established theory and phenomenology of the nuclear structure, plon-nuclear optical potential and the free A-~Nn decay amplitude. Of related interest are calculations of O-, Eq. (4). It14~ this quantity, for which some experimental data exist (see chapter 2) has not been extensively studied so far. Bycbkov and ghafizov ~11s~ suggested using pionic hypernuclear decays as a means of observing the proposed ~us) surface A-hypernuclear states. Such states are based on a Als wave function localized on the surface of a heavy hypernucleus, which could be triggered by a relatively large gradient of the density and a small surface potential necessary to trap the A-hyperon on the surface of a heavy hypernucleus. Pionic weak decay rates in heavy hypernuclei are expected to be sensitive to the existence of such states, since the Pauli suppression would be less effective at surface densities. Polarized hypernuclei have been the subject of recent research. ~116~ Possible future production of such polarized hypernuclei by means of the (a+,K ÷) reaction would have important implications on studies of hypernuclear weak decays, both in the pionic ~117~ and non-mesonlc ~11.~ decay modes. Asymmetry of pions from the weak decay could provide information on hypernuclear structure and on pion distortion at central densities, c117~ The angular distribution of outgoing protons resulting from non-mesonic decay of polarized hypernuclei is sensitive to the elementary decay mechanism ~118~ and should provide further constraints on such possible mechanisms; it might also be possible to measure the ratio ~, Eq. (52). Nith improved precision and experimental techniques it would hopefully be possible to eventually measure the magnetic moments of hypernuclei along with the weak-decay observables. Such measurements could throw new light on our understanding of the hypernuclear many-body system and of the behavior of baryons embedded in the nuclear medium, cs'9) Another subject of interest involving weak decays is doubly-strange nuclear systems ~Ix91 and the H-particle, ~12°~ namely, a symmetric quark state uuddss, with spin and isospin zero, and strangeness -2. If the predicated H were sufficiently light, the transition AAN...N~HN...N could occur in the doublystrange hypernucleus. These processes conserve strangeness and could proceed via strong interactions at a rate of -1023sec -I, completely dominating the weak decay processes. Thus, a re-investigation of the existence of AA-hypernuclei stable to the strong interaction, unstable only to the weak interaction, has a timely importance because it could establish strong limits on the mass of the suggested H-particle. An active search for the H based on its weak decay, H*Apn-, with the resulting A-hypernucleus subsequently decaying weakly, has been proposed by the Heidelberg group. A search for the H could also be carried out on the basis of an inverse-non-mesonic decay in a A-hypernucleus (provided that the H is light enough), namely, ANN~AAN-*HN, where the first transition proceeds via the weak interaction. A better understanding of the weak-decay process is evidently desirable for such studies.

Decays of A-Hypernuclei

225

Future studies of such exciting problems will depend, of course, on available facilities. For the time being, these are at Brookhaven National Laboratory and KEK, along with their possible upgrades. In the longer term, it is expected that KAON, lz2x~ CEBAF, and the proposed AHF ~z22~ would provide the new and high-quality data which is nov required. Of special interest would be a study of this decay in a strong magnetic field, vlth a measurement of the resulting asymmetry in the angular distribution of the emitted nucleons; perhaps the all-important hypernuclear magnetic moments could also be obtained. Electromagnetic production of hypernuclei (sz~ via A(e,e'K+)AA holds some promise for a study of weak decays. Mecking lxz3~ suggested an experimental scenario suitable for such investigations at CEBAF. An incident electron beam of extremely high quality (as will be available from the superconducting CEBAF accelerator) is essential for the success of the experiment. High flux of quasi-real tagged photons produced by electrons scattered at small angles will be used. The produced K + mesons will be detected in a high resolution spectrometer vlth a moderate maximum momentum, after separation of outgoing electrons and kaons in a common transverse magnetic field. This set-up requires a short large-solld-angle spectrometer for K + detection as well as a broad-band, low-energy electron spectrometer (the primary beam is assumed to be relatively low in energy for the purpose of obtaining high absolute energy resolution and therefore the K ÷ energy will also be low). The expected counting rate is 300/day, with a signal-to-noise ratio of approximately 20. The total hadronic production rate in the target is expected to be 2x105 particles/sec, making feasible the identification of coincident decay products in a large acceptance detector. We feel, therefore, that the subject of non-mesonic weak decays is likely to shed light on a large variety of fundamental problems in weak- and stronginteraction physics. Despite some thirty years of research, a lot remains to be done, both experimentally and theoretically. This subject will undoubtedly require considerable efforts and ingenuity, and the pertinent studies would encounter stupendous difficulties, nonetheless the payoff is very promising. ACKNOWLEDGEMENTS I t is a pleasure to thank my collaborator, George Walker, and the Nuclear Theory Center at Indiana University for enjoyable visits and support. Discussion with Avraham Gel during and after a visit to the Racah Institute of Physics at the Hebrew University were very helpful. Thanks are due to Hans Weber for reading a preliminary version of this work. I am especially grateful to Terry Goldman for his useful comments and suggestions regarding Sec. 3.2. I have benefited a great deal from discussions and exchanges with members of the Carnegie-Mellon University experimental group, in particular Peter Barnes, Gregg Franklin, Reinhard Schumacher and John Szymanski, and with John Dubach, Carl Dover, Ch. Elster, Judah Eisenberg, David Heddle (who has provided his Ph.D. thesis), P.K. Kabir, Leonard Kisslinger, John Millener, Julian Noble and Eulogio Oset. I sincerely thank Idit Gil-Cohen for much needed help at the proof-reading stage. This work was p a r t l y s u p p o r t e d by the U.S. Department of Energy (DOE) through CWRU. P a r t of t h i s work was c a r r i e d out a t the Los Alamos N a t i o n a l L a b o r a t o r y , which i s o p e r a t e d by the U n i v e r s i t y of C a l i f o r n i a f o r the U.S. DOE. Support from the I n s t i t u t e of N u c l e a r and P a r t i c l e Physics at the U n i v e r s i t y of V i r g i n i a has e n a b l e d me to c a r r y out t h i s work to c o m p l e t i o n , and i s t h e r e f o r e especially appreciated.

226

J. Cohen

REFERENCES 1.

2.

C.B. Dover and G. E. Walker, Phys. Rep. 89, 1 (1982); A. Gal, in: Adv. in Nucl. Phys., (ed. M. Barenger and E. Vogt-~ 8, 1 (Plenum, New York, 1975); C. B. Dover and A. Gal, in: Prog. Part. Nucl. Phys., (ed. D.H. Wilkinson) 12, 171 (Pergamon, Oxford, 1984); B. Povh, Ann. Rev. Nucl. Sci. 28, 1 (-'f978);D.H. Davis and J. Pniewski, Contemp. Phys. 27, 91 (1986); R.E. ChrienandC-~. Dover, Annu. Rev. Nucl. Part. Sci. 39,113 (1989). R. Chrien et al., Nucl. Phys. A478, 705c (1988).

3.

M. May et al., Phys. Rev. Lett. 51, 2085 (1983); D.J. Millener et al., Phys. Rev. C 31, 499 (1985).

4.

R. Grace et al., Phys. Phys. A47~, ~-~c (1988).

5.

J.P. Bocquet et a l . , Phys. L e t t . B182, 146 (1986); J . P . Bocquet et a l . , Phys. Lett. Bi-~2~--312 (1987). Some f u r t h e r d e t a i l s are provide~--b~-S. Polikanov, Nuc--~. Phys. A478, 805c (1988).

6.

P.B. Siegel, W.H. Kaufmann and W.R. Gibbs, Phys. Hey. C 31, 2184 (1985); J. A l s t e r , in: Nuclear S t r u c t u r e s with Medium Energy ProGes, Proceedings of the APS/DNP Workshop, Santa Fe, October 1988, (ed. J.N. Ginocchio and S.P. Rosen) p. 277 (World S c i e n t i f i c , Singapore, 1989).

7.

A.R. Bodmer and O.N. Usmani, Nucl. Phys. A477, 621 (1981); B.F. Gibson, Nucl. Phys. A450, 243c (1986); Nucl. Phys. A ~ , , 115c (1988); A.R. Bodmer, Nucl. Phys. A--4"~, 257c (1986); B.F. Gibson et al., Phys. Rev. C 27, 2085 (1983); Yu You-Nen, T. Motoba and H. Band~, Prog. Theor. Phys.-7"6, 861 (1986).

8.

The idea is due to J.V. Noble; see Joseph Cohen and R.J. Furnstahl (9) for a relativistic shell-model analysis of hypernuclear currents.

9.

Joseph Cohen and R.J. Furnstahl, Phys. Rev. C 35, 2231 (1987).

10.

N. Cheston and H. Primakoff, Phys. Rev. 92, 1537 (1953); H. Primakoff, Nuovo Cim. 3, 1394 (1956); M. Ruderman and R. Karplus, Phys. Rev. 102, 247 (1956).

ii.

M.M. Block et al., in: Proc. 1962 Int. Conf. on High Energ7 Physics at CERN, Geneva-~ I-~62, (ed. J. Frentki) p. 458 (CERN, Geneva, 1962); M.M. B-~ et al., in: Proc of the Sienna Int. Conf. on Elementary Particles and Hig-]~-E-nergyPhysics~ Sienna, 1963, (ed. G. Bernardini and G.P. Puppi) Vol. I, p. 62 (Societa Italiana de Fisica, Bologna, 1963); M.M. Block et al., in: Proc. of the Int. Conf. on Hyperfragments, St. Cergu~, Switzerland, March 1963, (ed. D. Evans and W.O. Lock) p. 63 (CERN Report 64-01, CERN, Geneva, 1964); L.R. Fortney, ibid, p. 84; C. Meltzer et al., Bull. Am. Phys. Soc. 2, 49 (1962); G. Keyes et al., Phys. Rev. Let-t.-~O, 819 (1968); Phys. Rev. D I, 66 (1970); G. Keyes, J. Sacton, J.H. Wickens and M.M. Block, Nucl. Phys. B67, 269 (1973).

12.

N. Crayton et al., in: Proc. 1962 Int. Conf. on High Energy Physics at CERN, Geneva--~ I~62, (ed. J. Prentki) p. 460 (CERN, Geneva, 1962); R.J. Prem and P.H. Steinberg, Phys. Rev. 136B, 1803 (1964); R.G. Ammar, W. Dunn and M. Holland, Phys. Lett. 3, 340 (-~3); Y.W. Kang, N. Kwak, J. Schneps and P.A. Smith, Phys. Rev. Lett. 10, 302 (1963); Phys. Rev. 139B, 401 (1965); R.E. Phillips and J. Schneps, Phys. Rev. Lett. 20, 1383 (1968);

Rev.

Lett.

55,

1055 (1985); P.D. Barnes, Nucl.

Decays of A-Hypernuclei

227

Phys. Rev. 180, 1307 (1969); G. Bohm et a l . , Nucl. Phys. B16, 46 (1970) (for ~H life-~me); Nucl. Phys. B23, 9-3- ~-[970) (for ~He li-[e-time); C.I. Murphy et al., in: Proc. Int. C-onf. on Nationa[-La-~., p. 438--['~gonne, 196-9"~?.

H~pernuclear Physics, Argonne

13.

K.J. Nield et al., Phys. Rev. C 13, 1263 (1976).

14.

J. Schneps, W.F. Fry and M.S. Swami, Phys. Rev. 106, 1062 (1957).

15.

E.R. Fletcher et al., Phys. Lett. 3, 280 (1963).

16.

J.P. Lagnaux et al., Nucl. Phys. 60, 97 (1964); J. Cuevas et al., Nucl. Phys. BI, 411--[1~'~7); S.N. Ganguli--and M.S. Swami, Proc. Indian Acad. Sci. 66A, 77-(1967); L.K. Mangotra et al., Nuovo Cim. 13A, 826 (1973).

17.

M.W. Hol]and, Nuovo Cim. 32, 48 (1964); A. Montwill et al., Nucl. Phys. A234, 413 (1974); K.N. Cha--udhari et al., Proc. Ind~-an--Acad. Sci. 69A, 78 Ff9-69).

18.

M. Baldo-Ceolin et al., Nuovo Cim. 7, 328 (1958); V. Gorg& et al., Nucl. Phys. 21, 599 (1~0~-[ I.B. Berkovich- et al., Soviet Physics JETP 11, 311 (1960)--[J. Exptl. Theoret. Phys. (U.S.S.R.) 38, 423 (1960)]; l.R.--Kenyon et al., Nuovo Cim. 30, 1365 (1963); B. Bh--6wmik, D.P. Goyal and N.K. Y-~m~i'agni, Phys. Lett.--3, 13 (1962); Nucl. Phys. 48, 652 (1963); K.N. Chaudhari et al., Proc. Indian Acad. Sci. 65A, 240 (19-67); G. Coremans et al., Bull. Inst. de Phys. de l'Universit-%'- Libre de Bruxelles, no. 4--6 (--f971).

19.

P.E. Schlein, Phys. Rev. Lett. 2, 220 (1959).

20.

E. Silverstein, Nuovo Cim. Suppl. iO, 41 (1958); mesonic to n- mesonic decay rates was measured.

21.

J. Sacton, Nuovo Cim. 18, 266 (1960); measured.

22.

M.M. Block et al., Nuovo CJm.

as

the ratio Q'- = non K-

in Ref. (20), the ratio Q'- was

28,

299 (1963); H.G. Miller, M.W. Holland, Rev. 167, 922 (1968); J. McKenzie, in: Proc. Int. Conf. on H~pernuclear Ph_~/~., Argonne National Lab., p. 403 (Argonne, I--9-69);G. C~emans et al., Nucl. Phys. BI6, 209 (1970); N.K. Rao et al., Proc. Indian Acad. Sc~. 7-fA, i00 (1970). J.P. Roalsv~-g a-nd R.G. Sorensen?'-Phys.

23.

M.J. Beniston et al., Phys. Rev. 134B, 641 (1964); C. Mayeur et al., Nuovo Cim. 44A, 698 (-f9~'6); G. Keyes, J. Sacton, J.H. Wickens and--M?'M. Block, Nuovo Cim. 31A, 401 (1976).

24.

R.G. Ammar et al., Nuovo Cim. 19, 20 (1961); A.Z.H. Ismail et al., Nuovo Cim. 27, 12~'8 (-[963); D. Bertran'd et al., Nucl. Phys. B16, 77-[1~'70).

25.

R.H. Dalitz and L. Liu, Phys. Rev. 116, 1312 (1959).

26.

P.D. Barnes, Nucl. Phys. A450, 43c (1986); J.J. Szymanski, in: ProceedinGs of the Second Conference on the Intersections between Particle and Nuclear P~,sics, Lake Louise, Alberta, 1986, (ed. D.F. Geesaman) AIP Conf. Proc. 934 (AIP, New York, 1986); J.J. Szymanski, The Weak Decay of Lambda Hypernuclei, Ph.D. Thesis, Carnegie Mellon University, 1987.

228

J. Cohen

27.

P.D. Barnes and J . J . Szymanski, i n : Hypernuclear P h y s i c s 1986, P r o c e e d i n g s of the 1986 INS I n t e r n a t i o n a l Symposium on Hypernuclear P h y s i c s , Tokyo, 1986, (ed. H. BandS, O. Hashimoto and K. Ogawa) p. 136 ( I n s t i t u t e f o r Nuclear Study, U n i v e r s i t y of Tokyo, 1986).

28.

T. Cantwell et e l . ,

29.

R. B e r t i n i et e l . , Nucl. Phys. A360, 315 (1981); A368, 365 (1981); V.N. F e t i s o v et a-I.~-Z. Phys. A314, 239--~983).

30.

H. BandS, T. Yamada and J. ~ofka, Phys. Rev. C 36, 1640 (1987).

31.

A. Gel, unpublished ;quoted by Barnes and Szymanskiin Refs.(26,27).

32.

V.I. Noga et e l . , (1987)J.

33.

V.I. Noga e t e l . , "Delayed F i s s i o n Physics and"Tech--6ology-Kharkov r e p o r t .

34.

R.H. D a l i t z , Phys. Rev. 112, 605 (1958).

35.

R.H. D a l i t z and G. Rajasekharan, Phys. L e t t . 1, 58 (1962).

36.

R.H. D a l t t z and F. yon Hippel, Nuovo Cim. 34, 799 (1964).

37.

H.H. Block and R.H. D a l i t z , Phys. Rev. L e t t . 11, 96 (1963).

38.

R.H. D a l i t z , i n : Proceedings of the I n t e r n a t i o n a l School of P h y s i c s "Enrico Fermi", Course 38, Varenna, 1966, (ed. T.E.0. E r i c s o n ) p. 89 (Academic, New York, 1967); R.H. D a l i t z , S t r a n g e P a r t i c l e s and Strong I n t e r a c t i o n s ( P u b l i s h e d f o r the Tara I n s t i t u t e of Fundamental Research, Bombay, by Oxford U n i v e r s i t y P r e s s , London, 1962); R.H. D a l i t z , Nuclear I n t e r a c t i o n s of the Hyperons ( P u b l i s h e d f o r the Tata I n s t i t u t e - ' ~ Fundamental Research, Bombay, by Oxford U n i v e r s i t y P r e s s , London, 1965).

39.

J.B. Adams, Phys. L e t t . 22, 463 (1967).

40.

B.H.J. HcKellar and B.F. Gibson, Phys. Rev. C 30, 322 (1984).

41.

J . F . Dubach, Nucl. Phys. A450, 71c (1986).

42.

B.H.J. HcKellar, i n : Hypernuclear Physics 1986, P r o c e e d i n g s of the 1986 INS I n t e r n a t i o n a l Symposium on Hypernuclear P h y s i c s , Tokyo, 1986, (ed. H. BandS, O. Hashimoto and K. Ogawa) p. 146 ( I n s t i t u t e f o r Nuclear Study, U n i v e r s i t y of Tokyo, 1986).

43.

E. Oset and L.L. Salcedo, Nucl. Phys. A443, 704 (1985); A450, 371e (1986).

44.

Joseph Cohen and G.E. Walker, and work in p r o g r e s s .

45.

C.-Y. Cheung, D.P. Beddle and L.S. K i s s l i n g e r , Phys. Rev. C27, 335 (1983); D.P. Heddle and L.S. K i s s l i n g e r , Phys. Rev. C33, 60~'- (1986); L.S. K i s s l i n g e r , i n : Proceedings of the Second Conference on the I n t e r s e c t i o n s between P a r t i c l e and Nuclear P h y s i c s , Lake Louise, A l b e r t a , 1986, (ed. D.F. Geesaman) AIF Conf. Proc. 150, p. 940 (AIP, New York, 1986).

Nucl. Phys. A236, 445 (1974).

Yad. F i z . 46,

1313

(1987) [Soy. J . Nucl. Phys. 46, 769 of

(1966);

Bull.

Bismuth

Nuclei", Institute

of

J.B. Adams, Phys. Rev. 156, 1611

American Phys. Soc. 32, 1576 (1987);

Decays of A-Hypernuclei G.

46.

T.D. Lee, J. Stelnberger, Rev. 106, 1367 (1957).

47.

R. Gatto, Phys. Rev. I0__88,1103 (1957).

48.

F.S. Crawford et al., Phys. Rev. 108, 1102 (1957); in: Proc. 1958 Annual International C-6nir?.on High ~ h y s . at CERN, Geneva, 19b~-,(e--d'~.B. Ferretti) p. 3 ~ ( C E ~ , Geneva, 1958); F. --Eisel-e-{-et al., Phys. Rev. 108, 1353 (1957); L.B. Leipuner and R.K. Adair, Phys. Rev. 109, 1358 (195~'8~. [Note that the existence of large angular asymmetry in A~p~- decay also establishes that the associated production reaction, n- + p ~ A + K °, yields highly polarized A hyperons. It is believed that the A polarization is due to polarization of the strange quark in its production; the underlying mechanism is poorly understood. For a review, see T. De Grand, in: Proceedings of the Symposium on Future Polarization Physics at Fermilab, Batavia,--Ill-i~ois, 1988, (ed. E. Berger et al.) ~T-f~-i-[ab-?, Batavia, 1989).]

49.

B. Cork et al., Phys. Rev. 120, i000 (1960).

50.

T.D. Lee and C.N. Yang, Cim. 8, 68 (1958).

51.

E.F. Beal et al., Phys. Rev. Lett. 7, 285 (1961); 8, 75 (1962).

52.

R.W. Birge and W.B. Fowler, Phys. Rev. al., Phys. Rev. Lett. 7, 264 (1961).

53.

(a)J.W. Cronin and O.E. Overseth. Phys. Rev. 129, 1795 (1963); (b) O.E. Overseth and R.F. Roth, Phys. Rev. L e t t . 19, 39i--((1967); (c) W.E. Cleland et a l . , Phys. L e t t . B26, 45 (1967); (d) W-?E. Cleland et a l . , Nucl. Phys. B-~07--221 (1972). [Not--e-that the e a r l i e r r e s u l t s of B-61~-[ et a l . , Phys. Rev. L e t t . 1, 256 (1958) are in disagreement with Refs. (51-5~')).]'-

54.

G. Takeda, Phys. Rev. I01, 1547 (1956); W. Pauli, in: Niels Bohr and the Development of Physics~--~ed. W. Pauli) p. 30 (McGraw-H~-l-I, New Yor~,an--~ Pergamon, Lon--~on,--~); R. Gatto, Nuovo Cim. 3, 318 (1956); L.B. Okun', Weak Interactions of Elementary Particles (Pergamon, Oxford, 1965) chapter 13.

55.

M. Gell-Mann and A.H. Rosenfeld, Ann. 5.

56.

L.D. Roper, R.M. Wright and B.T. Feld, Phys. Rev. 138B, 190 (1965).

57.

M. Gell-Mann and A. Pals, in: Proceedings of the 1954 Glasgow Conference on Nuclear and Meson Physics, Glasgow, 1954, (ed. E.H. Bellamy and R.G. M-6o~)--~.~/--[Pergamon, London, 1955).

58.

For earlier evidence, see M. Gell-Mann and A.H. Rosenfeld, Ann. Rev. Nucl. Sci. 7, 407 (1957), especially Sees. 3,5,6; R.H. Dalitz, Rev. Mod. Phys. 31, 8~3 (1959); and references therein.

Phys.

Feinberg,

Rev.

P.K.

229

108,

Kabir and C.N. Yang, Phys.

1645 (1957); J. Leitner, Nuovo

Lett. 5, 254 (1960); J. Lietner et

Rev. Nucl. Sci. Z, 407 (1957), Sec.

230

J. Cohen

59.

For experimental evidence not cited in the reviews of Ref. (58), see Appendix I and Appendix II in Review of Particle Properties, Particle Data Group-R.L. Kelly et al., Rev. Moa'? --Phys. 52, $1 (1980) and references therein; see also G. Wentzel, Phys. Rev. 10__i-?,1215 (1956); J.L. Brown et al., Phys. Rev. 130, 769 (1963); F. Zisler et al., Nuovo Cim. 5, 170-0 (--i957); B. Cork e-t-al., Phys. Rev. 120, 1000--[196-'0);J.L. Brown et al., Phys. Rev. Lett.-~,-563 (1959); S. Osl-~n et al., Phys. Rev. Lett. ___.~,--'~43 (1970) [not a complete list].

60.

F.S. Cravford et al., Phys. Rev. Lett. 2, 266 (1959); early results are compared by F.S. Crawford, in: Proc. 1§62 Int. Conf. o_nnHigh Ener~ry at CERN, Geneva, 1962, (ed--T-J. Prent--~T) p. 827 (CERN, Geneva, th-l's p-~er also discusses other decays and the aI=½ rule); also M. Schwartz, in: Proc. of the 1960 Annual International Conference on H ~ Energy P h s ~ at Roc--~ester, Ro--~este--~,1960, (ed. E.C.G. Sudarsha~, J.H. Tinlot and A.C.-Melissimos) p. 726 (The University of Rochester, dist. Interscience, New York, 1960); and in Proc. 1958 Annual International Conf. on ~ Energy Phys. at CERN, Geneva,---1-~8, (ed. B. Ferretti) p. 273 (-~N, Geneva, 1958); W.E. Hump-'h-r'-ey and R.R. Ross, phys. Rev. 127, 1305 (1962); M. Chr~tien et al., Phys. Rev. 131, 2208 (1963); C. Baltay et al., Phys. Rev. D 4, 670~1971); for recent figures, see Review of Pa-rt-~le Properties, Particle Data Group - R.L. Kelly et al., Rev. Mod. Phys. 52, S1 (1980), or M. RooM et al., Phys. Lett. Blll (1982).

~

61.

This point was raised in early discussions of week hyperon decays [R.E. Marshak, S. Okubo and E.C.G. Sudarshan, Phys. Rev. 113, 944 (1959); it was also discussed by V.I. Zakharov and A.B. Kaidalov, Zh. Eksp. Teor. Fiz. Pis'ma 3, 459 (1966) [Soy. Phys. JEPT Lett. 3, 300 (1966)]; O.E. Overseth, Phys. Rev. Lett. 19, 395 (1967); G. K~ll~n, Elementary Particle Physics (Addison-Wesley, R-eading, Massachusetts, 1964), Sec. 17-3~ S.P'?Rosen and S. Pakvasa, in: Adv. in Particle Phys., (ed. R.L. Cool and R. E. Marshak) Vol. 2, p.473 (Interscience, New York, 1968); for a related dicussion concerning E decay, see S.P. Rosen, Phys. Rev. Lett. 6, 504 (1961)].

62.

D. Bailin, Weak Interactions (Sussex University Press, 1977) Sec. 5.1.3.

63.

R.H. Dalitz, in: Nuclear Physics, Les Houches 1968, (ed. C. DeWitt and V. Gillet) p. 701 (Goro-ro'~n~-'Br--e-~h,New York, 1969).

64.

H. Band5 and H. Takaki, Prog. Theor. Phys. 72, 106 (1984).

65.

T. Motoba, K. Itonaga and H. BandS, Nucl. Phys. A489, 683 (1988).

66.

R. Mach et al., Z. Phys. A331, 89 (1988).

67.

J.D. Bjorken and S.D. New York, 1964).

68.

H i ~ Energy Nuclear Physics - Proc. of the Sixth Annual Rochester Conf., 1956, (ed. J . B - ' ~ m et al.) p. VII'Y-Ib'---(~-i-~.Interscience, New York, I~"~); M. Gell-Mann and A.H. Rosenfeld, Ann. Rev. Nucl. Sci. Z, 407 (1957), Sec. 3.8.

69.

G. K&ll~n, Elementary Particle Physics (Addison-Wesley, Reading, Massachusetts, 1964), Sec. 17-4; S.P. Rosen and S. Pakvasa, in: Adv. in Particle Phys., (ed. R.L. Cool and R.E. Marshak) Vol. -'~-, p.473 (Interscience, New York, 1968).

Drell, Relativistic Quantum Mechanics (McGraw-Hill,

Decays of A-Hypernuclei

231

70.

J.D. Bjorken, Phys. Rev. D 40, 1513 (1989).

71.

I n t r o d u c t o r y d i s c u s s i o n s and r e f e r e n c e s may be found in D.H. P e r k i n s , I n t r o d u c t i o n to HiglaEne_BE~Ph__.~, 2nd E d i t i o n (Addison-Wesley, Reading, Massaehusetts~-1982), Chapter 6; or in E.D. Commins and P.H. Bucksbaum, Weak I n t e r a c t i o n of Leptons and Quarks (Cambridge U n i v e r s i t y P r e s s , Ca--~ridge, 1983), Ch'-apters--rff-T~,6~

72.

M.K. G a i l l a r d and B.W. Lee, Phys. Rev. L e t t . 33, 108 (1974); G. A l t a r e l l i and L. Maiani, Phys. L e t t . B52, 351 (1974).

73.

M.A. Shifman, A . I . V a i n s h t e i n and V.J. Zakharov, Nucl. Phys. B120, 316 (1977); A . I . Va{nshte~n, V.J. Zakharov and M.A. Shifman, P i s ' m a z-h-~-.Eksp. Th~or. F i z . 22, 123 (1975) [Sov. Phys. JEPT L e t t . 22, 55 (1975)]; A . I . Vainshte{n, ~ . I . Zakharov and M.A. Shifman, Zh. g k s p . - ' T e o r . F i z . 72, 1275 (1977) [Sov. Phys. JEPT 45, 670 (1977)]; M.B. Wise and E. W i t t e n , Phys. Rev. D 20, 1216 (1979); F---~J. Gilman and M.B. Wise, Phys. Rev. D 20, 2392 (1979); R.D.C. H i l l e r and B.H.J. McKellar, J. Phys. G8, L1 (198~); s e e , however, a l s o F . J . Gilman and M.B. Wise, Phys. L e t t B83, 83 (1979), and P. Ginsparg and M.B. Wise, Phys. L e t t . B127, 265 (1983). For a c r i t i q u e , see S. Pakvasa, in High Energy Physics - 1980, XX I n t e r n a t i o n a l Conference, Madison, Wisconsin, (eds. L. Durand and L.G. Pondrom) AlP Conf. Proc. 68, p.1165 (AIP, New York, 1981); C.T. H i l l , i b i d , p. 386.

74.

J . F . Donoghue e t a l . , Phys. Rev D 21, 186 (1980); H. G a l l S , D. Tadi~ and J. Trampeti~, N-'ucl-7. Phys. B158, 30-6 (1979); D. Tadi~ and J. Trampeti~, Phys. Rev. D23, 144 (1981).

75.

One method is developed in B. Desplanques, J.F. Donoghue and B.R. Holstein, Ann. Phys. (NY) 124, 449 (1980); for a brief outline of the approach, see B. Desplanq-'~s, in: High Eenergy Physics and Nuclear Structure, Proc. of the Eighth Internat--i-~al-Con--'~ence, V a n c o u - - ~ r ~ , (ed. D.F. Mea--a~ayand A.W. Thomas) Nucl. Phys. A335, 147 (1980) (North Holland, Amsterdam, 1980).

76.

A. de-Shalit and 1963).

77.

A. Gal and C.B. Dover, Phys. Rev. Lett. 44, 379 (1980); __44, 962 (1980); A. Gal, in: Proceedings of the International Conference on Hypernuclear and Low-Energy Kaon Physics, Jablonna, Poland, 1979: Nukleonika 25, 447 (1980).

78.

V.A. Filimonov and A.G. Potashev, Yad. Fiz. Phys. 10, 76 (1970)I.

79.

G. Feldman, P.T. Matthevs and A. Salam, Phys. Rev. 121, 302 (1961); V. Singh and B.M. Udgaonkar, Phys. Rev. 126, 2248 (1962); J.C. Pati, Phys. Rev. 130, 2097 (1963); S. Kawasaki, M. Imoto and S. Furui, Prog. Theor. Phys.-~, 90 (1966); K. Fujii et al., Prog. Theor. Phys. 38, 388 (1967); B.W. Le-~ and A.R. Swift, Phys. R'ev.-'I36H, 228 (1964)i B.W. L'-~e, Phys. Rev. 140B, 152 (1965).

80.

See also S.P. Rosen and S. Pakvasa in: Adv. in Particle Phys., (ed. R.L. Cool and R.E. Marshak), Vol. 2, p.473 (lnterscience, New York, 1968): In Sec. III.1 the pseudovector-derivative coupling is rejected on the basis of its behavior in the exact SU(3) limit, where the parity-violatlng amplitudes vanish.

I.

Talmi,

Nuclear

Shell

Theory, (Academic, New York,

10, 130 (1969) [Soy J. Nucl.

232

J. Cohen

81.

For a review, see Joseph Cohen, (1989); and references therein.

82.

Joseph Cohen, Phys. Rev. C37, 187 (1988).

83.

T.A. Brody and M. Moshinsky, Table of Transformation Brackets for Nuclear Shell Model Calculations, 2nd Edition (Gordon and Breach, London, 1967); T.A. Brody and M. Moshinsky, Table of Transformation Brackets (Monograflas del Instituto de Fisica, Mexico, 1960); M. Moshinsky, Nucl. Phys. 13, 104, (1959).

84.

David P. Heddle, Lambda Nonmesonic Decay in the Hybrid 0uark-Hadron Model, Ph.D. thesis, Carnegie-Mellon University, 1984.

85.

G.A. Miller and J.E. Spencer, Ann. Phys. 100, 562 (1976).

86.

W.C. Haxton, B.F. Gibson and E.M. Henley, Phys. Rev. L e t t . 45, 1677 (1980); W.C. Haxton, Phys. Rev. L e t t . 46, 699 (1981); B.A. Brown, W.A. R i c h t e r and N.S. Godwin, Phys. Rev. L e t t . 45, 1681 ( 1 9 8 0 ) .

87.

R.J. McCarthy and G.E. Walker, Phys. Rev. C 2, 809 (1974); Paul Goldhammer, Phys. Rev. C 9, 813 (1974); R.J. McCarthy and G.E. Walker, Phys. Rev. C 11, 383 (1975); for an earlier discussion see C.M. Shakin, Y.R. Naghmare and M.H. Hull, Phys. Rev. 161, 1006 (1967).

88.

L.C. Gomes, J.D. (1958).

89.

S. Moszkowski and J.V. Noble, in: Relativistic Nuclear Many-Body Physics, Proc. of the Workshop, the Ohio State University, Columbus, June 1988, (ed. B.C. Clark, R.J. Perry and J.P. Vary) p.566 (Norld Scientific, Singapore, 1989); and private communication.

90.

R. Machleidt, K. Holinde and Ch. Elster, Machleidt, Adv. in Nucl. Phys., (ed. J.N. p.189 (Plenum, New York, 1989).

91.

For example, M. Bozoian and H.J. Neber, Phys. Rev. C 28, 811 (1983) use F(~ 2) = exp(-~2/6~), where ~i/2 =K0.32 GeV; or 3jx(qR)/qR'-~ith R = 0.8 fm, which is roughly equivalent to = 0.78 GeV, from A.W. Thomas, Adv. in Nucl. Phys., (ed. J.W. Negele and E. Vogt), Vol. 13, p.1 (Plenum, New York, 1983).

92.

An example of a quark-model study is R.D.C. Miller and B.H.J. Mckellar, Aust. J. Phys. 35, 235 (1982); this procedure is unsatisfactory when tested for hyperon decay, S. Pakvasa, in: High Energy Physics - 1980, Proc. of the XX International Conference on High Energy Physics, Madison, Wisconsin, 1980, (ed. Loyal Durand and Lee G. Pondrom) AIP Conf. Proc. 68, p. 1104 (AIP, New York, 1981). See also Ref. (75) and the pertinent discussion in E.G. Adelberger, Comments Nucl. Part. Phys. 11, 189 (1983).

93.

Y. Hara, Y. Nambu and J. Schechter, Phys. Rev. Lett. 16, 380 (1966); L.S. Brown and C.M. Sommerfield, Phys. Rev. Lett. 16, 751 (1966); C. Bouchiat and Ph. Meyer, Phys. Lett. 20, 329 (1966).

Valecka

and

International Jour. of Modern Phys. A4, 1

V.F.

Veisskopf,

Ann.

Phys.

(NY) 3, 241

Phys. Rep. 149, 1 (1987); R. Negele and E. Vogt), Vol. 19,

Decays of A-Hypernuclei

233

94.

M. Bozoian, J.C.H. van Doremalen and H.J. Weber, Phys. L e t t . 122B, 138 (1983); M.M. Nagels, T.A. R i j k e n and J . J . deSwart, Phys. Rev. D - - ~ , 2547 (1977).

95.

G. HShler and E. Pietarinen, Nucl. Phys. B95, 210 (1975).

96.

G. Nardulli, Phys. Rev. C 3.88, 832 (1988).

97.

G. Nardulll, Nuovo Cim. IOOA, 485 (1988).

98.

K. Takeuchi, H. Takaki and H. BandS, Prog. Theor. Phys. 73, 841 (1985); H. BandS, Prog. Theor. Phys. Suppl. 81, 181 (1985).

99.

J . F . Dubach, i n : Proc. of the I n t e r n a t i o n a l Symposium on Weak and E l e c t r o m a g n e t i c I n t e r a c t i o n s in Nuclei (WEIN), H e i d e l b e r g , J u l y 1986, (ed. H.V. Klapdor) p.576 ( S p r i n g e r - V e r l a g , 1986).

100. J . F . Dubach and A. Gal, p r i v a t e communication. 101. L.S. K i s s l i n g e r , Phys. L e t t . Bl12, 307 (1982); E.H. K t s s l i n g e r and G.A. M i l l e r , Phys. Rev. C 28, 1277 (1983). 102. D.P. Heddle and L.S. K t s s l i n g e r , Nonmesonic neutron s t i m u l a t e d r a t e s , unpublished (1987).

decay

of

Henley,

L.S.

12^C.. p r o t o n vs

103. L. Tauscher, C. Garc~a-Recio and E. Oset, Nucl. Phys. A415, 333 (1984); E. Oset and A. Palanques, Phys. Rev. C 29, 261 (1984); E. Oset and L.L. Salcedo, Nucl. Phys. A468, 631 (1987). 104. See, e . g . , N.C. Mukhopadhyay, H. Toki and W. Weise, Phys. L e t t . B84, 35 (1979); Joseph Cohen and J.M. E i s e n b e r g , Phys. Rev. C 29, 914 ('1-~84); Jospeh Cohen, Phys. Rev. C 30, 1238 (1984). 105. For a d i s c u s s i o n of the p-exchange Cohen, Phys. Rev. C 30, 1573 (1984).

in

this

context, see, e.g.,

Joseph

106. D.J. M i l l e n e r , as quoted in Ref. (45); P . J . B r u s s a a r d and P.W.M. Glaudemans, S h e l l Model A p p l i c a t i o n s in Nuclear S p e c t r o s c o p y ( N o r t h Holland, Amsterdam, 1977). 107. A. Bouyssy, Phys. L e t t . B84, 41 (1979). 108. Y. K u r i h a r a , Y. Akaisht and H. Tanaka, Phys. Rev. I t o n a g a , T. Motoba and H. BandS, Z. Phys. A330, 209 H. Takaki, Phys. L e t t . B150, 409 (1985); H. Band6 Theor. Phys. 72, 106 ( 1 ~ - ~ ; R. Mach e t a l . , Z. Phys. 109. E. Oset e t a l . ,

C 31, 971 (1985); K. (19-~8); H. Band5 and and H. Takakt, Prog. A331, 89 (1988).

MIT p r e p r i n t (1989).

110. F. F e r r a r i and L. Fonda, Nuovo Cim. 2, 320 (1958). 111. H. BandS, Y. Shono and

(1988).

H.

Takakt,

Int.

Jour.

of

Mod. Phys. A 3, 1581

234

J. Cohen

112. T. Tamiya et al., Prog. Theor. Phys. 34, 833 (1965); V.A. Filimonov and B.P. Nikiforov, Yad. Fiz. 8, 71 (!968) [S-6v. J. Nucl. Phys. 8, 40 (1969)]; see also a review, dealing with hyperon decay, by S.P. Rosen and S. Pakvasa, Ref. (80). 113. J.T. Londergan, Nucl. Phys. BI6, 109 (1970); H. Bandb, Czech. J. Phys. B36, 915 (1986); H. Bandb, Prog. Theor. Phys. Suppl. 81, 181 (1985); T. Motoba, H. Bandb, K. Ikeda and T. Yamada, ibid, p.-'~2; T. Motoba, K. Itonaga and H. Bandb, in: Hypernuclear Physics-~986, Proceedings of the 1986 INS, International Symposium on Hypernuclear Physics, Tokyo, August 1986, (ed. H. Bandb, 0. Hashimoto and K. Ogawa) p.160 (Institute for Nuclear Study, University of Tokyo, 1986); V.A. Lyul'ka, Soy. J. Nucl. Phys. 44, 37 (1986) [Yad. Fiz. 44, 57 (1986)]; T. Motoba, K. Itonaga and H. Ban~-6, Nucl. Phys. A489, 683 (--i-988). 114. F. Cerulus, Nuovo Cim. 5, 1685 (1957); (1986). 115. A.S. Bychkov and R.U. Khafizov, Yad. Phys. 48, 259 (1988)]; A.S. Bychkov, Nucl. P--6ys. 40, 259 (1984)].

H. BandG, Czech. J. Phys. B3_66, 915 Fiz. Yad.

48, 65 (1988) [Sov. J. Nucl. Fiz. 40, 408 (1984) |Sov. J.

116. H. Ejiri et al., Phys. Rev. C 3__66,1435 (1987); H. Band5 et al., Phys. Rev. C 39, 587-[1~-~9). 117. T. Motoba, K. Itonaga and H. Band~, Nucl. Phys. A489, 683 (1988). 118. T. Fukuda et al., in: Hypernuclear Physics 1986, Proceedings of the 1986 INS International Symposium on Hypernuclear Physics, Tokyo, August 1986, (ed. H. BandS, 0. Hashimoto and K. Ogawa) p.i70 (Institute for Nuclear Study, University of Tokyo, 1986). 119. M. Danysz et al., Nucl. Phys. 49, 121 (1963); J. Prowse, Phys. Rev. Lett. 17, 782 (19"~6~-[ R. H. Dalitz e}"-al., Proc. R. Soc. Lond. A426, 1 (1989). 120. R.L. Jaffe, Phys. Rev. Lett. 3--8, 195 (1977). 121. Kaon, Antiproton, Vancouver, 1985).

Other

Hadrons,

Neutrino

Factory

Proposal

(TRIUMF,

122. The Physics and a Plan for a 45 GeV Facility that Extends the HighIntensity Capability in Nuclear and Particle Physics, Los Alamos National Laboratory Report LA-10720-MS/UC-28 and UC-34 (1986). 123. B.A. Mecking, Experimental Aspects Report CEBAF-R-86-013 (1986).

of

Hypernuclear

Physics

at CEBAF,