Production of double-Λ hypernuclei at (K−, K+) reaction points and their pionic decays

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 625 (1997) 107-142

Production of double-A hypernuclei at (K-, K +) reaction points and their pionic decays Yasuo Yamamoto a, Masamichi Wakai b, Toshio Motoba c, Tomokazu Fukuda d a Physics section, Tsuru University, Tsuru, Yamanashi402, Japan b Department of Physics, Osaka University, Toyonaka, Osaka 560, Japan c Laboratory of Physics, Osaka Electro-Communication University, Neyagawa, Osaka 572, Japan d Institutefor Nuclear Study, University of Tokyo, Tanashi, Tokyo 188, Japan Received 13 May 1997; revised 20 August 1997; accepted 26 August 1997

Abstract Starting with the ( K - , K +) reaction on nuclear targets of mass number 9-12 (9Be, l°B, liB, ~2C), the probabilities of producing double-A and single-A hyperfragments at the reaction points have been evaluated on the basis of the statistical model. First, the ~ particle produced abundantly in the quasi-free region is shown to be well absorbed into the nucleus through the ~= slowdown by knocking out a nucleon. Then the formation of a double-A compound nuclear system is assumed together with its subsequent breaking-up into various hyperfragments. The theoretical pion spectra are presented for the weak decays of the double-A and single-A hypemuclei produced efficiently in this scenario, which can be utilized for the identification of their hyperfragments production. It is proposed that the observation of AASHwith the 9Be target is most feasible and promising. @ 1997 Elsevier Science B.V. PACS: 21.80.+a; 25.80.-e Keywords: Double-Acompoundnucleus; Hyperfragmentsformation;Mesonic weak decay The authors would like to dedicate this paper to the memory o f Dr. Carl B. Dover, a prominent contributor to strangeness nuclear physics

1. Introduction The existence of double-A hypernuclei is of very interest because it gives valuable information on A A interactions and is deeply related to other S = - 2 systems such J Present address: Institute for Particle and Nuclear Studies, KEK, Tsukuba 305, Japan. 0375-9474/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PH S0375-9474(97)00483- 1

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as ~ hypernuclei and an H-particle. For a long time there had been reported only two events of double-A hypernuclei in nuclear emulsion ( aABelO [ 1] and 6aHe [2,3] ). After a long break, a new development has been achieved by the KEK-E176 ( K - , K +) experiment with use of the emulsion-counter hybrid detector system [4], where ~ - particles captured in emulsion were confirmed with higher statistics. Among compiled events of ~ - - c a p t u r e at rest, there were confirmed one event of the emission of a double-A hypernucleus [4] and two events of the simultaneous emission of two single-A fragments (twin A-hypernuclei) [5]. These double-A events in emulsion are considered to occur as follows: A ~ - particle is produced via a quasi-free process in the ( K - , K +) reaction and is stopped at an other nucleus after traveling in emulsion. The ='- particle trapped in an atomic orbit and a proton in the nucleus are converted into two A's by the ~ - p - A A strong interaction. Then, the double-A sticking state is formed in the probability of more than ,-~10% [ 6], and finally it decays into some fragments including a double-A hypernucleus or twin A hypernuclei. However, this production process of double-A hypernuclei is not necessarily efficient because most of emitted ~ - particles decay in flight before stopping, which is the reason why double-A events are so rare in the emulsion. It is also natural to consider production of double-A hypernuclei through the direct formation of bound ='- hypernuclear states in the ( K - , K +) reaction. In fact several authors [ 7-10] performed theoretical estimates for the width of ~ - hypernuclear states which comes from the ~ - p --* AA conversion involving various final states of A's and nucleon(s). Here we only note that the cross section for producing the 0sl/2 ~ hypernuclear state on the 12C target is about 0.1 /zb/sr, i.e. about 0.2% of the total cross section [9]. Another idea of producing double-A hypernuclei was proposed [ 11,12], where efficient formation of double-A fragments is expected at the ( K - , K +) reaction points: The ~ - particle produced predominantly in the quasi-free region is absorbed into the target nucleus through the rescattering process with a nucleon, and the double-A sticking intermediate state (double-A compound nucleus) is formed through the Z - p - A A conversion in the nucleus. Final double-A products are obtained as a result of fragmentation of the double-A compound nucleus. The idea of the compound hypernucleus was originally proposed by Yamazaki [13-15]. In the ( K - , K +) reaction with P/¢- ~ 1.66 GeV/c studied at KEK [ 16,17], the ~ - emission probability is about 80% of the total quasifree events, and others are considered to be attributed to secondary processes of ~ inside a nucleus, namely the imaginary part of the Z-nucleus potential. The kinetic energy of the quasi-free ~ - (~100 MeV) is so high that the secondary interaction of in the nucleus is simply related to the Z N elementary cross sections. In the case of using the Z N - A A sector of the Nijmegen SU(3)-invariant interaction [ 19], it is shown that the ~ N total cross section is dominated by the elastic one: the o - ( Z - p --~ Z - p , Z°n) is far larger than o ' ( Z - p --+ AA). Then, it is naturally considered that the main part of the ~-nucleus imaginary potential consists of the nucleon knockout scattering into continuum states, and the slowdown of Z - by the scattering with a nucleon leads to absorption by the nucleus. As a result, eventually the double-A sticking after Z N - A A

Y Yamamotoet al./Nuclear Physics A 625 (1997) 107-142

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conversion takes place with the probability similar to the case of stopped _~- absorption process mentioned above. In Refs. [ 11,12] such a mechanism of double-A sticking was investigated by using a Fermi-gas model. In the ( K - , K +) experiments at KEK so far, the statistics was not high enough to catch double-A signals originated from the quasi-free _~ absorption at the momentum region of Pr~ > 1.0 GeV/c. However, some double-A events have been confirmed at ( K - , K +) reaction points of Px+ < 1.0 GeV/c. Although the detailed tbrmation mechanism cannot always be explained theoretically in the present stage, it is quite interesting to suppose that these double-A fragments are coming from decays of double-A compound nuclei. Now a new experiment (E906) is in progress at BNL-AGS in order to detect double-A hypernuclei by observing characteristic 7"r- mesonic decays at ( K - , K +) reaction points. In fact the major results of the present calculations were reflected in their proposal [ 20]. The formation rate of double-A fragments is expected to be large by two orders of magnitude as compared with the other experiments so far. It is supposed, of course, that the probability of single-A sticking is far larger than that of double-A one [21]. However, one will be able to discriminate double-A hypernuclei by detecting successive weak-decay pions in coincidence. The aim of this work is to put the above-mentioned scenario on the quantitative basis. An important issue which is substantial in actual experiments is how to choose the most suitable target nucleus, because the number of possible fragments from a double-A compound nucleus increases very rapidly with the target mass number and accordingly the resulting pion spectra will become complicated. We will answer to these questions. In Section 2 we describe the basic framework of treating both scattering and conversion of the quasi-free _~- particle in nuclear matter, and then we calculate the formation probability of a double-A compound nucleus. In Section 3 we evaluate the breaking-up probabilities of a double-A compound nucleus into various single-A and double-A fragments by using a statistical model. In the application to some sample nuclear targets (9Be, l°B, liB and 12C), the theoretical formation probabilities of possible hyperfragments are presented in detail for these targets. In Section 4 the pion spectra expected from the produced double-A hypernuclei are calculated, which presents a useful way of their identification in combination with the subsequent pionic decays of the daughter single-A species. Almost all pionic decays have been calculated for the most promising case of the 9Be target. In the final subsection we will add several pionic decay spectra which should be important if one adopts the other three targets. The concluding remarks of this paper will be given in Section 5.

2. Formation probability of a double-A compound nucleus

Let us evaluate the formation probability of a double-A compound nucleus by referring basically to the preceding works [ 11,12]. Here, however, we improved the formulation more reasonably by taking account of the discussion by Koltun [22]. The important point in the present treatment is to study the two types of double-A sticking separately:

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The first is the slowdown and the sticking of the quasi-free ~ through the ~ N ~ ~ N scattering followed by the _~N --~ AA conversion. The second type is the direct ~ N --~ AA conversion between the quasi-free ~ particle and a nucleon. In this paper we are mainly concerned with the first type, while for the second type we will mention the results only briefly. Calculations for the related probabilities are performed in the typical case of 12C target.

2.1. The _~ particle slowdown due to _~N scattering, the ~ sticking and the double-A sticking The transition rate w~at, and its sticking part w~ c, due to scattering between a quasifree _~ (momentum p z ) and nucleons (momentum PN) in nuclear matter are defined by

wi(p.~) :

d3pN

d3p}N

d3P}N P fN(PN) V_--Nd--~

×6( ( p z s - P ~ N ) / P ~ N ) t~3(p.~u -- P~-=N)~i(P~,fU) ,

(2.1)

where superfix i distinguishes the two rates (scat, stic), and p is the nuclear density. do'/dO is the ~ N scattering cross section from the initial relative momentum pzN to the final one P~N, and v z s is the relative velocity. T h e notations scat and stic mean scattering and sticking, respectively. P.~v and /~E/v denote the initial and final center-of-mass momenta, respectively, f u ( P s ) = (4gqrp F) 3 - 1 0 ( p F -- pN) gives a Fermi distribution of nucleons, PF being the Fermi-momentum. The factors which restrict the final momenta, ~soat = 0(,Olo - PF)

and

~tic = O(p~ - PF) O(qz -- p ~ ) ,

(2.2)

define two kind of transition rates wscat and wS= tic, respectively. O(fu - PF) is the Pauli exclusion factor for the final nucleon state of momentum P~v. The transition rate wS= eat includes only this Pauli factor for nucleon states. On the other hand, another factor O(q= - p ~ ) is included in w~ e, which allows transitions only to ~ states whose momentum (p~) are lower than the critical momentum q~ which is determined by q~ 2M=

U..= = 0,

(2.3)

with U= being a ~ well depth in medium: Thus w~ c is considered as the transition rate to represent the ~ sticking through ~ N elastic scattering in medium. In the following calculations we assume the isotropic angular distributions do'/dO = O'el/4~, O'el being the _~N elastic scattering cross section. Then the above expression, Eq. (2.1), reduces to more compact form as (i = scat, stic) Pv

wi:.pf 0

1

1

f d......., f d¢__-..,. --I

--I

(2.4>

Y Yamamoto et a l . / N u c l e a r Physics A 625 (1997) 107-142

I I1

with IXu = (PN . P--)/PNP~ and I Z IE N = ( P ~ N . PEN) / p - N I' ' = N . Similarly we can obtain the conversion rate w~ in medium by using the conversion cross section O'cv for ~ N --~ AA as follows: PF

1

wCV= 2zrp f dpN PZuf N(PN ) f dt zN v -=N°'cv 0

(2.5)

-1

where the momenta of two A particles in the final state are not restricted. We define the reaction rate of ~ by WrE T M = w~ at + WCE v. One should be careful that the "scat" here means the ~-nucleus inelastic scattering, where a nucleon is knocked out by the elementary ~ N elastic scattering. The ~ N total cross section is given by O'tot = O'el -~ O-cv. The conversion width F ~ is given by hw~, and correspondingly Fs=cat = hWSE TM is called here the scattering width. Then, the total width is given by FtE°t = ]-,~at _1_/"2' The mean free path (MFP) of a in medium is defined by /IE = Vz/wreEac = 1 / ( p O ' t o t ) , v E being a velocity of ~ . Now we represent the MFP's for scattering and conversion cross sections by A~at = 1/(ptr, l) and ACE v = 1/(pO'cv), respectively. Then, we have the relation 1/az = l / a ~ at + 1 / a ~ . The probability of the ~ slowdown processes in a finite nucleus can be estimated on the basis of the eikonal approximation. Assuming the forward scattering OK+ = OE = 0 ° as in the standard way, the ( K - , K +) effective number for the reaction of the produced quasi-free =" and nucleons is given by

N reac=

J i 2¢rbdb

0

dz p(V/-~+ z2)F(b,z)

--oo

{

1 - exp

-

(2.6) z

-

/

with

F(b,z) = e x p

{ : --O'K.-u

p ( V / ~ + Z'2)dz'--grK,u

--00

The total effective number l-exp

- f

J

p(V/-~+ Z12)dz I

Z

Ntota I

}

. (2.7)

is given by omitting the last factor with curly bracket

1---dz'

/

in Eq. (2.6), and the reaction probability of ~ in medium is given by e reae _ NrEeac/Ntotal" Similarly we have the effective number N ~ for the ~ N ~ AA conversion by replacing AE in Eq. (2.6) with A~. On the other hand, the scattering part of the effective number, which is originated from the _~N elastic scattering in medium, is defined as the difference NrEeac -- N~, for which one may refer to Koltun [22]. We can get straightforwardly

_

N~at =

E

/

E 0

--eeo

Y. Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

112

xexp

- [-~c~dz'l

1-exp

-/~dz'

.

(2.8)

If we multiply this N~ at by the ratio w~e/w~at, we can deduce the effective number for ~ sticking, N~ e. Then we obtain the ~ sticking probability as p.fic

Nstic ~-

.... ~

with

NS~C=

stie W ~ Nscat

(2.9)

Ntotal

It should be noted that the probability of conversion is independent of scattering, while the probability of scattering is reduced by conversion, as pointed out by Koltun [22]. Next, we consider the sticking probability of two A particles which should be produced through the ~ particle sticking followed by the ~N-AA conversion in nuclear medium. It is reasonable to assume that a produced A particle sticks to nuclear medium if its momentum is less than a critical momentum

qz

determined by ~

-

UA =

0. Thus

the conversion rates to the AA-sticking states, W~A-~, are calculated by imposing the restriction on the A final momenta in Eq. (2.5). PF

1

W~A-A-) =27rp/dpup~fu(pN) f dlZNV.~l~O'cvO(qa--p& O(qA--PA2) . 0

(2.10)

-I

Here the notation ( - - ) indicates that both A's have negatwe energies. Hereafter, we use also the notations ( + - ) and ( + + ) whose meanings are self-evident. Then, the direct double-A sticking probability is defined by S~AA-) = WtAA-)/W~, which is given as a function of E z and Fermi momentum PF. Since the absorbed E particle has a negative energy in medium, we use the averaged value S~A~-) over the energy region of E~ < 0 (we will see the behavior of this probability in Fig. 3):

S~A-)(pF) = ~

l/

0

S~-~-)(Ez,pF) dEs.

(2.11)

-u_--

The probability of the double-A sticking via the ~ slowdown is finally given by the product of two factors as pAPa--) = p~ic ~(A-).

(2.12)

On the other hand, for the sake of completeness, we briefly mention the direct ~N --* conversion process between the quasi-free ~ and nucleons, where there are three cases for the final two-A states: (i) the direct double-A sticking, (ii) one A is sticked with negative energy and the other is energetic in continuum (QF), and (iii) both A's are in continuum. The first case is unlikely as shown before (cf. Fig. 8 of Ref. [ 12] ). In the latter two cases, the QF A particle(s) can be partially absorbed into medium through the rescattering with nucleons. Then the corresponding conversion rates are given by w~+-~ and •W A~++) which are obtained by replacing the factor O(qa --PAl ) O(qA --PAz ) in AA A

AA

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Eq. (2.10) with 20(pa~ - q A ) O(qA -- PA2) and O(pA~ -- q a ) O(pA2 -- q A ) , respectively. From these rates we estimate the probabilities corresponding to the latter two cases: pC+-) and *pc++) (See Ref. [ I 1 ] ) AA AA

2.2. Results The core nucleus 12C is treated here as a uniform sphere of which radius is determined by R = 5 ~ / ~ ( r 2 ) I/2. The root-mean-square radius is calculated from the experimental matter density distribution in which a proton charge distribution is unfolded. Then we get R = 2.95 fm and the uniform density of/90 = 0.112 fm -3 (pF = 1.18 fm -j ). The potentials of N, A and _~ are represented by square-well forms correspondingly, whose well depths UB (B = N, A, ~ ) are determined so as to simulate binding energies obtained by the respective Woods-Saxon potentials. In the well-established cases of N and A, we have definite numbers: UN = 48 MeV and UA = 24 MeV. On the other hand, the potential depth of _~ is uncertain in the present stage. In the past a fairly deep potential was proposed by Dover and Gal [23] on the basis of the old emulsion data of "~-hypernuclei", while the recent analysis of the events of twin A hypernuclei observed in the KEK-E176 experiment suggests a shallower potential with the well depth of about 2/3 of the deep Woods-Saxon well depth [24]. In the present square-well prescription with radius R, we employ two kinds of potentials: U=- = 18 and 13 MeV corresponding to the deep and shallow cases, respectively. It is noted that both cases here with R = 2.95 fm give the similar ~ single particle spectra as the corresponding Woods-Saxon potentials. Since there is no experimental data of O'el and O'cv, they are calculated with the Nijmegen model D interaction [19]. The hard-core radii are taken to be a common value of 0.5 fm in all channels. Since the quasi-free _~ is of fairly high energy, the cross sections are dominated by p-waves and therefore the choice of the hard-core radii has only a minor effect on the result. In Fig. 1 the scattering and conversion widths, /,~at = hw~at and FC=v = hwC_V-,are drawn as a function of E_= in the cases of U= = 13 (solid) and 18 MeV (dashed), respectively. Here it should be noted that F ~ at dominates over F cv in the high energy region and the dependence on the choice of Uz is very small. In Fig. 2 the sticking probability p_~ic is shown as a function of E z for two cases of U_= = 13 (solid) and t8 MeV (dashed): At E z = 100 MeV, which is a typical value for a quasi-free _~, we obtain pstic

-= =

/ 0.024

for U~ = 13 MeV,

0.035

for Uz = 18 MeV.

(2.13)

The ~ sticking probability turned out to be rather sensitive to the potential depth U= which is related with the critical momentum for ~ sticking through q z = ~ U z . As mentioned in Section 1, it has been theoretically shown [7,9] that the ( K - , K +) direct formation rate of 2 - hypernuclear bound state (0s~/2) amounts to about 0.2-0.4% of the total rate. This suggests that the strength of more than 250 times as large as the

114

Y. Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

20

scat

15

>

s-'""

Y

10

..........

U_=I8 U_=13

5 -~cv

E -----=-_ ................ 0

~

20

I

40

,

I

60

i

I

~

80

I

1 O0

,

I

120

i

140

E= (MeV) Fig. 1. Scattering and conversion widths, Fs=cat a n d Fc~v, of the ~ particle in the nuclear medium calculated as a function of E~. The solid and dashed curves correspond to the cases of U~ = 13 and 18 MeV, respectively. --~ bound state strength is in the quasi-free region. A rough estimate assuming constant quasi-free strength gives 250 × p s=~c ~ 6 . 0 - 8 . 8 , so that the rescattering/absorption of the quasi-free _~ provides at least 6 times more efficient way of producing double-A hypernuclei than the direct production through a E hypernuclear bound state. Fig. 2 and Eq. (2.13) update the previous estimates [ 11,12] in the sense that they are based on the Koltun's prescription (Eq. (2.8)) and, in addition, more reasonable inputs are employed here for the E well depth U~ and the nuclear radius R. In the previous estimates we employed the formula N~ c = (w~C/w~=eac)Nx_= eac in place of the latter relation of Eq. (2.9). We found, however, that the difference between two prescriptions is very small (typically less than 2%), since for example we obtain p ~ c = 0.02434 (previous) vs. 0.02395 (present: cf. Eq. (2.13) for Uz = 13 MeV) with use of the same parameters. It should be noted here that the two different formulae lead to these similar values because F ~ is much smaller than Fs=TM in this particular case as seen in Fig. 1. In Fig. 3 the direct double-A sticking probability SfAA-) of Eq. (2.11 ) is shown as a function of PF, where the well depth for PF is assumed to be U n ( p / p o ) , with B = N, A and ~ . The solid and dashed curves are for Uz = 13 MeV and 18 MeV, respectively. S~A -) is found to be strongly dependent on PF. There is no clear reason, however, that the value of PF = 1.18 fm - l used in our uniform sphere model should coincide

Y. Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

115

0.20 (MeV . . . . . . . . . U_=18

U_=13 0.15 x x

0.10

x ",x

~

p

stic

I

,

0.05

0.00 20

I

40

,

I

60

,

80

I

1 O0

,

I

120

,

140

E= (MeV) Fig. 2. Sticking probability p ~ c of the ~ particle calculated as a function of E=. The solid and dashed curves correspond to the cases of U_= = 13 MeV and 18 MeV, respectively.

with the one used in the nuclear matter estimation of the A A sticking probability. Only a qualitative correspondence is meaningful, so that we take the flexible choice of PF appropriate to leC: 1.1 fm -1 < PF < 1.2 fm - l . Corresponding to this typical range of PF, we get the following estimates for the double-A sticking probability 0.23 < S~A- ) < 0.47

for U= = 13 MeV,

0.27 < S~A-) < 0.52

for U=- = 18 MeV.

(2.14) Using the values of p~ic, we finally obtain the formation probability of the double-A compound nuclear system as { 0.006 < P I ~ -) < 0.011

for Us = 13 MeV,

0.010 < p~A--) < 0.018

for U_~ = 18 MeV.

(2.15) Though it may be difficult to discuss the result of S(A~-) more definitely within the present formalism, we remark an interesting fact that the different approach of Ref. [9] supports to the present estimates: They calculated the double A sticking probability from a _~ hypernuclear state within the microscopic framework for a finite hypernucleus. Their values obtained for the 0 s - ~ - in E12- Be are 0.18 and 0.43 for the shallow (U ws E 16 MeV) and deep (24 MeV) Woods-Saxon potentials, respectively, which correspond

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Y. Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

1.0 .........

U_=l 8(MeV) u_=13

0.8

/ /

0.6

4. /

0.4

0.2

0.0 0.95

1.00

1.05

1.10

1.15

1.20

1.25

PF (fm'l) Fig. 3. Double-A sticking probability ,~(AA-), defined by Eq. (2. l l ), calculated as a function of the nuclear Fermi momentum PF. The solid and dashed curves correspond to the cases of U= = 13 MeV and 18 MeV, respectively. to our two square potentials. These values are in fair agreement with the above-mentioned values of S(a~-}. The large values of double-A sticking probabilities were also predicted by Dover et al. [8,10] with the different model. Finally let us discuss briefly on the other process of double-A sticking. In our model the ~ reaction probability in nuclear medium is composed of the scattering part pscat = Ns~at/Ntotal and the conversion one p gv = N ~ / N t o t a l , where the latter comes from the d i r e c t _ ~ N ---, A A conversion process between the quasi-free ~ and nucleons. Then the probability p gv of producing two A's can be divided into three parts as mentioned at the end of the preceding subsection. The double-A sticking probabilities for these ( + - ) = 0.3 × 10 -3 [ l l ] ) . The processes have been estimated to be very small ( e . g . .P AA recalculation with more careful treatment gives typically •P (AA + - ) = (0.21--0.23) × 10 -3 and the similar values for "P{++) AA , SO that all the obtained probabilities for the d i r e c t conversion are more than one order-of-magnitude smaller than the values of Eq. (2.15). In conclusion, we can expect large amount of double-A sticking via the slowdown process of quasi-freely produced ~ particle, which leads to a double-A compound state discussed in the next section. The present calculation confirms the conclusion of the previous works [ 11,12], although numerically we have still large allowable range for

Y Yamamom et al./Nuclear Physics A 625 (1997) 107-142

117

the physical values concerned. In this section, following Koltun [22], the framework of the calculation has been improved, although we found no remarkable change of the result. Another improvement has been achieved, because the consistency between the adopted uniform-sphere model and the well depths of N, A and ~ has been treated more carefully than before. Owing to these new ingredients, the ambiguity of the results has been reduced considerably in comparison with the previous works [ 11,12].

3. Fragmentation of a double-A compound nucleus

3.1. Application of the statistical model In order to treat the fragmentation process of the double-A compound nucleus, we apply the statistical model developed in Refs. [ 11,21]. A double-A compound nucleus is assumed to decay via processes of breaking up into particles and/or fragments. There are many kinds of possible breaking-up channels, which satisfy the conservation of baryon numbers, strangeness and total energy. This process is described here on the basis of a statistical view point. [25] Though possible sets of fragments compose a macro-canonical ensemble, we approximate it with a canonical ensemble for simplicity and calculate the breaking-up probability P,~ into a specified channel of fragmentation ot from the initial double-A compound nucleus. In this work the breaking-up is taken into account up to four-body decay channels. The total energy (Hamiltonian) of the system is decomposed into a sum of proper energies of fragments: Il rr

net

H , = ~-~Hi,, + Z i.

Vi, i,,,

(3.1)

i,,. <.j.

where Hi. and Vi°j, are the energy of Gth fragment and the interaction energy between i,~- and j,~th fragments, and n,~ denotes the number of fragments in the channel or. Hi, is expressed as a sum of the kinetic energy Ki, and internal (binding) energy ni. of the fragment. Only the Coulomb force is taken into account for V/,,.i.. Then, the distribution function of the system (temperature T) is given by (3.2)

Z,~(T) = exp

Bi,, -

Vi,j,,

(3.3)

i. <,j,~

n Tr e x p ( - ~

] Ki,,) = N -l

(2~T) 3

//~r-- l

H n~ mi. ) 3/2 (3.4)

i.

In the above equations, V is the spatial volume of the system and mi is the mass of ith fragment. The factor N = H~ (n~!) is to avoid double counting in the calculation, where

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Y. Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

k n a,

is the number of kth kind of fragment in the channel a and the relation n,, ~ n k 4 3 holds. The volume V for the mass number A is taken as gTrr v A with rv = 2.0 fm. The probability for breaking-up into a special channel of fragmentation a is given by P~ = Z ~ ( T ) / Z ( T ) . The temperature T is determined by energy-conservation condition

no-l>

<35

where E0 is the total energy of the original double-A compound-nucleus.

3.2. Parameters involved in the model calculation

It is vital in the present treatment how to choose E0, which is related closely to the model of double-A sticking. As mentioned above, we consider the two physical elements: (i) the ~ absorption followed by a nucleon knockout and (ii) the _~N-AA conversion in the nucleus. The energy deposit AK to the nucleus caused in the former process is related to nucleon-hole excitation energy which comes mainly from the knockout of a s-shell nucleon. We estimate simply with AK = l e s - e p l ( A - 4 ) / A , where es and % are the s- and p-shell single-particle energies, respectively. Referring the data of nucleon knockout reactions, AK is taken as 5, 6 and 7 MeV for the targets 9Be, l°B and 12C ( l l B ) , respectively. On the other hand, the energy deposit due to the second process is the mass difference M ~ - + Mp - 2MA = 28 MeV. Thus, the total deposited energy is given by AK + 28 MeV. Though the present estimation may be rather rough, it should be noted that our result does not change much even if AK is treated as an arbitrary parameter within the reasonable range 0~20 MeV [ 11 ]. In order to perform the model calculation, we need the two-A binding energies BAA of possibly produced double-A hypernuclei. In contrast to many single-A binding energy data, of course, BAA are mostly unknown, and we have not much data for this quantity. The E176 event [4] was identified as either AaBe to 13 or AAB, for which B4A = 8.5 ± 0.7 MeV or BAA = 27.6 i 0.7 MeV are deduced respectively. The two candidates lead to the opposite interpretation on the A - A bond energy: ABAA = BAA --2BA = --4.9 MeV(if it is ]°Be) or 4.9 MeV (~3B). However, the former interpretation, which leads to abnormally strong AA repulsion, is quite doubtful from the theoretical side [26,27]. On the other hand, the attractive AA interaction obtained from the latter 10 B e [1]. In Ref. [27] the values of BAA is at least consistent with the old data of AA for various double-A hypernuclei were calculated systematically by using the attractive AA interaction which reproduces both the known experimental data. There was another attempt of predicting A4H to be the lightest possible bound system, suggesting the value of BAa = 0.5 MeV [34]. In this paper, we take three options for the AA interaction: Case A: use the calculated BaA values [27] - the attractive AA interaction; Case B: a s s u m e ABAA = 0 MeV - very weakly attractive AA interaction;

Y Yamamom et al./Nuclear Physics A 625 (1997) 107-142

119

Case C: assume ABAA = --4.9 MeV - repulsive AA interaction.

We adopt Case A throughout of this paper as the most likely choice. At the same time, in order to see how the uncertainty in the AA interaction strength affects our result, we investigate the weaker attractive (B) and the repulsive (C) cases. For reference, the lightest double-A hypernucleus predicted in Case B is AASH,while n6He in Case C. 3.3. Results of the calculation f o r the 9Be target

In the ( K - , K +) reaction, double positive charges are eliminated so that we choose the Z = 4 - 6 nuclear targets considering possible formation of light bound systems with S = - 2 . Since a double-A compound nucleus is formed after one-nucleon (N) knockout in the present conjecture, the target nuclei 9Be, l°B, llB and 12C lead to the mixed systems o f [aBHe *, ASH*], [A9Li *, AAgHe*], [10aLi*' AAHclO . .~ . and . tAAt~e,ll n * AA ~ l lIl: * ], respectively. Then, we have the cascade processes as follows: 9 B e ( K - , K +) __~ { ~ - +8 Li*}

~ Nemit + [AABHe*, AA8H * ]Compound,

(3.6)

lOB(K- ' K +) ___~ {,~- +9 Be*} --~ Nemit + [A9Li * , AA 9He ]Compound,

(3.7)

tlB( K - , K + ) _ _ _ ~ { ~ - + 1 0 B e . }

(3.8)

*

I z C ( K - , K + ) _ _ ~ { ~ - +ll B*}

,Nemit+[~OLi*, AA 10u_* "*c ]Compound, 1~* ,AA 11 LI.* ]Compound~ Nemit + rll [AA ~

(3.9)

In the case of 9Be, for instance, a neutron(n) emission with resulting AA sticking leads to a compound system AASHe*, while a proton (p) emission to ABH*. As for the relative production ratio of these two types of compound nuclei, we assume that it is proportional to the ratio of neutron and proton numbers in the core nucleus which the E - particle hits. Thus the relative production ratios are 5:3 (9Be), 5:4 (l°B), 6:4 (liB) and 6:5 (12C), respectively. In the following tables of this subsection, we present the calculated results of the hyperfragment formation probabilities for the four targets, 9Be, l°B, liB and lZc, respectively. For the convenience of expressing various breaking-up channels, we classify them into three categories as follows: Type 1: A + A Z + others (including A + A + others), Type 2: A~ZIA+AA2 Z2 + others,

(3.10)

T y p e 3- aA Z + others.

The emission channels of one or two free A's are simply included in Type 1. First let us look at Table 1 for the case of hyperfragment production in the cascade process (Case A) after the 9 B e ( K - , K +) reaction. All the possible channels are classified into three types, where the notation x stands for nucleon(s) and/or nuclear fragments. For example, the channel A+SaHe+x on the fifth line is composed of A+~He+3H and A + ~ H e + e H + n in detail and each number in the table gives the summed probabilities over the component channels. For Type 3 which is mainly concerned here, we list all

120

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Table 1 Formation probabilities of single-A and double-A hyperfragments as calculated for the cascade process Eq. (3.6) after the 9Be(K-, K+) reaction. The intermediate compound systems are represented in [ It. The attractive AA interaction (Case A) is employed. The notation x stands for remaining nucleon(s), nuclear fragment(s) and/or their combination.For Type 3 the details are given instead of x. In parenthesis the summed value for each type is listed Channel Type 1

Type 2 Type 3

Total sum

IAaHe8*lc

lASH*It

Average

A+ A + x ?tH+A + x

0.169 0.026

0.123 -

0.152 0.016

4H+A + x

0.059

-

0.037

5AHe+A+ x

0.073

6tHe+A + x

0.074

-

0.046

7AHe+A

0.098

-

0.061

3H+~AH+X

0.018

-

0.011

4H+41H

0.055

-

0.034

AAnH+3H+n A4H+4H

0.014 0.012

-

0.009 0.007

ASH+p + 2n

0.009

-

0.006 0.016

0.046

AaSH+2H+n

0.025

-

A+~H+3H

0.074

-

0.046

AASH+3n AA 5He+3n

0.009

0.877 -

0.329 0.006

A6He+2n

0.128

-

0.080

AA7He+n

0.157

-

0.098

1.0

(Sum)

1.0

(0.358) (0.045)

(0.597)

1.0

the detailed channels explicitly instead of using x. The numbers in the "average" column mean the formation probabilities which are obtained by averaging two compound hypernuclear systems with the ratio(5:3) proportional to the core nucleons as mentioned above. It is quite interesting to find that A5H will be produced with exclusively high probability among all possible double-A hypernuclei. This result is clearly displayed in Fig. 4 ( l e f t ) . As seen in Table l (Type 3), the reason is because the breaking-up channels from A8H * are very limited so that only one proton remains to bind neutrons and A's to form A5H. (It is noted again that the breaking-up channels taken in our calculation is up to four-body decay channels.) One notices that the large probability of AASH in the former is also because there is only one channel (a5Hq-3n) in the decay of A8H *. This prediction seems helpful in selecting the suitable target and preparing the experiment, since the successive pionic decays of A5H involve both very characteristic pion spectra as discussed in the next section. In Table 2 and Fig. 5 we show how the formation probability of double-A hyperfrag-

Y Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

O. 40

--

_

(9Be)

~

121

( 10B )

O. 35 O. 30 O. 25 oz

O. 20 O. 15 O. I0 O. 05

0. 00 5

7

7

.8

i,

~H ~SH^~Ie~He~He ~L1 ~Lz•

~H 5H SHe ,~e ~He

6

Fig. 4. Formation probabilities of double-A hyperfragments as calculated for the cascade process after the ( K - , K+) reaction on the °Be target (left), and those on the mB target (right). Table 2 Theoretical formation probabilities grouped into three types of fragmentation channels. Type 1 includes also the "A + A + x" channel. The notation x stands for remaining nucleon(s), nuclear fragment(s) and/or their combination. They depend on the attractive AA interaction (Case A), the repulsive one (Case C) and the intermediate (Case B) AA interaction

Target

Channel type

9Be

I I ) AZ+A + x

0.358

0.434

0.828

12) Az+A'z' + x

0.045

0.054

0.068

(3) ,Az +x

0.597

0.512

0.104

(I) AZ+A + x

0.403

0.490

0.613

(2) Az+A'z' + x

0.104

0.141

0.214

(3) j~Z +x

0.493

0.369

0.173

WB

~2C

Case A

Case B

Case C

(I) AZ+A + x

0.315

0.381

0.478

(2) AZ-~-ACz t -}" X 4 A

0.153

0.198

0.271

(3) A~Z +X

0.532

0.421

0.251

ments (Type 3) changes depending on the different AA interactions employed. For the 9Be target, the group probability for Type 3 (59.7% in Case A) decreases appreciably and then rapidly as the AA interaction becomes less attractive (Case B) and the repulsive (Case C), respectively. One should note that, in Case C which might not be the real case, AASHbecomes unbound, resulting in the small probability of double-A sticking

122

Y. Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

lOO

cn 80 I.u 60 I.U "1I---

u. 40 O ILl

2: 20

0

(A) (B) (C)

(A) ( B ) ( C )

9Be

l0B

(A) (B) (C)

12C

Fig. 5. Calculatedshare of three types of the fragmentationchannelsare shown as a functionof the changing AA interaction: (A) attractive, (B) weaklyattractive,and (C) repulsiveAA interactions.For the classification into three types, see Eq. (3.10). especially in the process starting with 9Be. Table 2 and Fig. 5 contain also the cases of l°B and ]2C targets. We note first that the share of producing double-A hypernuclei (Type 3) remain large except Case C for every target. Secondly, it is remarked that the events of Type 1 may be separated well by tagging emitted A particles [20]. On the other hand, the events of Type 2 become background for detection of double-A hypernuclei, In this respect, the 9Be target which involves very small number of breaking-up channels of Type 2 is again most advantageous for the experiment. 3.4. Results f o r the other targets

The calculated formation probabilities of hyperfragments are summalized in Tables 3 and 4 for the 1°B target, and in Tables 5 and 6 for the liB and ~2C targets, respectively. One sees that the addition of proton(s) leads to remarkable increase of the breaking-up channels from the strange compound systems. It is notable, however, that the number of Type-2 channels remain still not large in the l°B and liB targets. Here we are mainly concerned with the double-A hypernuclear production. Their formation probabilities are visualized in Fig. 4(right), Figs. 6 and 7 for the I°B, liB and 12C targets, respectively. In every case, one notices that three species AASH,A6He and AATHeare equally well produced with high probabilities. Starting with the l°B target

123

Y. Yamamoto et aL/Nuclear Physics A 625 (1997) 107--142

Table 3 Formation probabilities of single-A and double-A hyperfragments as calculated for the cascade process Eq. (3.7) after the roB(K-, K+) reaction. The intermediate compound systems are represented in [ ]c. The attractive AA interaction (Case A) is employed. The notation x stands for nucleon(s), nuclear fragment(s) and/or their combination. See also Table 4 for details of the Type-3 channels with large probabilities Channel

Type l

Type 2

Type 3

Total sum

IAA9 L1. , ]c

9 He • Ic [AA

Average

A+ A+ x

0.077

0.138

0.104

3~l-l+A+ x

0.032

0.014

0.024

4H+A + x 5~t-te+A+ x

0.063 0.084

0.056 -

0.060 0.047 0.043

6~l-le+A+ x

0.026

0.065

6Li+A + x

0.006

-

0.003

7He+A + x

0.009

0.098

0.049

7~Li+A + x

0.009

0.017

-

8He+A

-

0.125

0.056

8Li+A

0.014

-

0.008

31tt+4~H+x

-

0.011

0.005

~1It+5 He+x

0.024

-

0.013

31t+6 He+x

0.027

~tt+4H+x

-

4H+51He

0.095

Aj~H+x A,~H+x ,l~He+x

0.035 0.144 0.006

0.024 0.130

0.030 0.137 0.003

A~iHe+x A,i~He+x

0.196 0.071

0.125 0.173

0.165 0.117

A~'Li+x ,l~Li+x

0.018 0.056 1.0

(Sum)

(0.402)

0.015 0.042

0.019 0.053

0.010 0.031 1.0

(0.104)

(0.493)

1.0

(Table 3, Fig. 4 ( r i g h t ) ) , the Z = 3 species AATLi and AASHe are newly produced, but their f o r m a t i o n probabilities are small. For the production of the d o m i n a n t three species, we list in Table 4 the detailed channel probabilities in the case o f the standard attractive A A interaction.

If the target n e u t r o n n u m b e r increases as in liB (Table 5, Fig. 6), the calculation suggests a n e w aspect o f p r o d u c i n g a neutron-rich d o u b l e - A hypernucleus a 9 H e with c o m p a r a b l y high formation probability ( 0 . 1 3 ) . As a result we expect production of four d o m i n a n t d o u b l e - A hypernuclei. As shown in Table 6 for the 12C target, the addition of o n l y o n e proton to JiB causes drastic increase o f available channels. It is interesting to find the h i g h - p r o b a b i l i t y production o f a different new d o u b l e - A hypernucleus A~0Li, the rate o f which is as m u c h as the c o m m o n three d o m i n a n t products (Fig. 7 ) .

124

Y Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

O. 20

( 11B ) 0.15 O Z

o

0

O. 10

O. 05

O.O0 ~H 5I-I 5He 611e 7He 7Li MSLi~Be 9He 9El 9Be 1°El Fig. 6. Formation probabilities of double-A hyperfragments as calculated for the cascade process after the ( K - , K +) reaction on the lib target.

O. 20

( 12c ) O. 15

O. i0

O. 05

O. O0

~n ~ ~H~ ~o J~e ZLi ~Li ~Be~.e 3Li 3~ ~Li ~oB~

Fig. 7. Formation probabilities of double-A hyperffagmenks as calculated for the cascade process after the ( K - , K +) reaction on the 12C target (left).

125

Y Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

Table 4 The detailed formation probabilities of the Type-3 final products (az+others) as coming from the breakingup of the intermediate compound systems represented in square bracket [ ]c. Two cases correspond to (a) the InB(K-, K+) reaction and (b) the J2C(K-, K+) reaction. In the latter, minor channels with respect to double-A hypernuclei are not shown. The sum within each group is listed in parenthesis. The attractive AA interaction (Case A) is employed (a) Target: I°B

191Li* Ic

(Sum)

,11~H+3H+p ,~,~H+3He+n a,2H+4He

0.006 0.006 0.132

,116He+p + 2n a6He+2H+n

0.012 0.039

a 6He+3H

0.145

A~Heq--p + n aTHe+2H

0.017 0.054

(0.071)

AABe* lc

(Sum)

(b) Target: 12C

111

9 • Ic [aaHe A5H+2H+2n ,t,~H+3H+n

(Sum)

0.017 0.113

(0.144)

(0.130) A6He+3n

0.125

(0.196)

(0.125) A~He+2n

0.173 (0.173) II L]"* h" IAA

AaSH+3He+3H

0.003

,t,~H+3H+3H

0.013

AaSH+eaHe+p + n a,~H+'aHe+2H ASH+6Li

0.016 0.014 0.007

aaSH+4He+2n A12H+6 He

0.067 0.051

a6He+ 2H+2H+n A6He+3H+p + n ,i6 He+3H+2H

0.007 0.011 0.015

A,6He+ 3He+2n ,16He+4He+n

0.007 0.133

17He+2H+p + n +7He+ZH+ZH

0.003 0.006

a ,~He + 3H+p ~tJ~He+3 He+n A,~He+4He

0.006 0.006 0.071

(0.092)

A')t'Li+p

0.008

( 0.008 )

(0.039)

(Sum)

(0.131) ,t 6He+3H+2n

0.106

(0.173)

(0.106) .t.~He+2H+2n +7He+3H+n

0.013 0.102

(0.115) at~'Li+n

O.190

(0.190)

Before closing this section, we remark that the ( K - , K +) reaction should bring about valuable i n f o r m a t i o n also on single-A hypernuclei if the observation o f m e s o n i c - d e c a y pions from the reaction points is exploited. Furthermore it is interesting that n e u t r o n rich A h y p e r n u c l e i are p r o d u c e d favorably, because this reaction changes the target charge by - 2 . B y l o o k i n g at the tables already shown above, this feature is easily seen e s p e c i a l l y in the cases of 9Be and lIB targets. Here we do not go into the details, since s o m e u n k n o w n A hypernuclei are not taken into account as p o s s i b l e fragments

126

Y. Yamamoto et aL/Nuclear Physics A 625 (1997) 107-142

Table 5 Formation probabilities of single-A and double-A hyperfragments as calculated for the cascade process Eq. (3.8) after the l i B ( K - , K+) reaction. The intermediate compound systems are represented in [ lc. The attractive AA interaction (Case A) is employed. The notation x stands for nucleon(s), nuclear fragment(s) and/or their combination Channel Type 1

Type 2

Type 3

I n~Li* I,.

Iou,, 1/ c

AA H~

Average

A+ A+ x

0.068

0.155

0.102

3~H+A + x

0.017

-

0.010

4~H+A + x

0.059

0.052

0.057

5~He+A + x

0.057

-

0.034

6aHe+A + x

0.043

-

0.026

7~He+A + x

0.018

0.088

0.046

7tLi+A + x

0.008

-

0.005

8,He+ A + x

0.005

0.091

0.040

~Li+A + x

0.006

-

0.004

9ALi+A

0.047

-

0.028

3,H+SHe+x

0.011

-

0.006

3H+61He+x

0.012

-

0.007

3H+7,He+x

0.018

-

0.011

4~H+n~H+x

0.005

0.042

0.020

4H+5He+x

0.045

-

0.027

4tH+6He

0.049

-

0.029

A4H+x

0.026

-

0.016

A5H+x

0.139

0.143

0.141 0.002

A,~ 5He+x

0.003

-

A,6He+x

0.147

-

0.088

A7He+x

0.133

0.150

0.140

-

A7Li+x

0.008

a~ Li+x

0.028

AAgHe+x

0.025

0.278

0.126

a~9Li+x

0.026

-

0.016

Total sum

1.0

(Sum)

(0.352)

(0.100)

0.005 0.017

1.0

(0.549)

1.0

in the calculational procedure. For instance, the p r o d u c t i o n probability o f 9 H e b e c o m e s c o m p a r a b l e to that o f 7aHe in the case o f ' J B target, if it is incorporated by extrapolating the b i n d i n g energy. T h e r e is another process to p r o d u c e s i n g l e - A fragments at the ( K - , K +) reaction points: T w o A ' s can be p r o d u c e d by the direct conversion o f the quasi-free ~ -

and a

proton in the nucleus. T h e probability o f d o u b l e - A sticking in this case is small, because the kinetic energy o f the ~ -

is transferred to two A's. It is fairly probable, however,

that one o f t w o A ' s sticks to the nucleus [ 1 1 ]. Thus,for instance, 8aHe* f o l l o w e d by a

Y. Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

127

Table 6 Formation probabilities of single-A and double-A hyperfragments as calculated for the cascade process Eq. (3.9) after the 1 2 C ( K - , K +) reaction. The intermediate compound systems are represented in [ h.. The attractive AA interaction (Case A) is employed. The notation x stands for nucleon(s), nuclear fragment(s) a n d / o r their combination Channel Type 1

Type 2

[AABe 1~"

IAl,~Ll• * ]c

11

*

Average

A+ A+ x

0.087

0.061

0.075

3~Hq--A + x

0.019

0.009

0.014

4~H4-A + x

0.024

0.036

0.029

4He+A + x

0.007

-

0.004

51He4-A + x

0.116

0.027

0.076

6He+A + x

0.056

0.031

0.045

6Liq-A 4- x

0.007

-

0.004

71He4-A + x

0.006

0.034

0.019

7~Liq--A + x

0.010

-

0.005

81He4-A + x

0.001

0.011

0.005

81Liq--A + x

0.002

0.003

0.003

8~Be-I-A 4- x

0.003

-

0.001

9~Li4-A + x

0.003

0.028

0.014

91Be4-A 4- x

0.018

-

0.010

ll°Be + a

0.020

-

0.011

3,H4-3,H+x

0.007

-

0.004

O.OLO

-

0.005

3~H+SAH e + x

0.015

-

0.008

3 H + 6 He4-x

0.005

0.005

0.005

3, H4-6,Li4-x

0.002

-

0.001

3~H+7He+x

0.002

0.009

0.005

3H+7,Li+x

0.003

-

0.001

~H+SAHe

-

0.012

0.005

~H+8,Li

0.001

-

0.001

4H+SAHe+x

0.015

0.023

0.019

4H+6,He+x

0.004

0.025

0.014

4H+6Li+x

0.004

-

0.002

4tH+7~He+x

-

0.042

0.019

41H+7,Ci+x

0.006

-

0.003

4 He+51He+x

0.005

-

0.003

4He+OlHe+x

0.004

-

0.002

4~He+TAHe+x

0.004

-

0.002

~IHe4-5 He + x

0.056

-

0.031

5,Heq-6AHe

0.042

-

0.023

(Sum)

(0.315)

(0.153)

128

Y Yamamotoet al./Nuclear Physics A 625 (1997) 107-142

Table 6--continued

Type 3

Channel

11 [ AA Be* Ic

11 ' * I a,1 L1 I c

Average

(Sum)

A4H+x

0.018 0.039 0.020 0.173 0.092 0.014 0.021 0.001 0.002 0.005 0.006 0.008 0.037

0.013 0.131 0.106 0.115 0.015 0.062 0.015 0.190 -

0.016 0.081 0.011 0.142 0.103 0.008 0.018 0.001 0.029 0.010 0.003 0.091 0.020

(0.532)

1.0

1.0

1.0

A5H+x A5He+x A6He+x A7He+x ,L7Li+ x AASLi+x a,~Be+x a9He+x a9Li+x ,L9Be+x al~Li+x a~)Be+x Total sum

A emission is generated in the case of 9Be target, the decay of which also leads to a single-A hypernucleus. Anyway, it is promising to search for neutron-rich A hypernuclei as well as double-A species with the same experimental setup.

4. Successive pionic decays of double-A hypernuclei 4.1. Framework o f the calculation

If a hypernucleus has a two-body final state in the pionic weak decay, AZ .___+A Z ~ + 7r, then the energy-momentum conservation leads to a definite (monochromatic) pion momentum so that this characteristic pion has been often used to identify the particular hypernucleus [28,29]. Similarly the most promising way to confirm double-A hypernuclear production is to detect successive pions from their weak decays. The typical 7"r- decay spectra to be used for such purpose have been demonstrated theoretically in Ref. [ 30] for the following cases: A6He:==> 5 H e + p + , r r 2

~

5AHe

,4He+p+rrl.

(4.1)

Hereafter ~2 and vrl often denote the pions emitted from double-A and single-A hypernuclei, respectively. In the first and subsequent stages of the above series, both pions (~2 and 7rl ) have continuum momenta corresponding to the three-body final states. However, each pion spectrum was shown to have a characteristic peak of which the width is sharp enough to be utilized as an indicator [30]. (As the pion kinetic energy

Y YamamoW et aL/Nuclear Physics A 625 (1997) 107-142

129

Table 7 Calculated pionic decay rates of light double-A hypernucleito be produced in the ( K-, K+ ) reaction on 9Be. The calculations are made for the two-body and three-body final states. DW denotes the use of pion distorted waves described in the text. All decay rates are given in units of the free-A decay rate l'a zr- DW .t4H

A~H

,l~He

a6He

a7He

~.l~ DW

==~ 4,He+~--

0.25

==> 41H +Tr °

0.13

=:~ !IH + p + "n'-

0.52

~

0.28

==~ 51He+rr-

0.38

==~ (No 2-body)

-

==¢, 4H + p + ~'-

0.61

==~ 4H + n + T r °

0.31

=:=> 51He+Trll

0.18

=:~ 41He+n + rr°

0.22

==> ~lHe+Tr°

0.23

==~ 51He+n + ~o

0.13

~

(No 2-body)

-

41He+p + r r -

0.48

==~ (No 2-body)

-

==¢, 51He+p + 7r-

0,60

31H +n + ~.o

==~ 71Li+Tr-

0.26

==~ 71He+~-°

0.22

:=¢- 5~He+d + ~ ' -

0.06

==~ 6He+n + 7"rI~

0.03

:==> 61He+p + T r -

0.21

T~ and the momentum q~ in Ref. [30] are based numerically on the averaged pion mass, here we give the corrected values for them): T~r2 -~ 29.5 with ATe, 2 ~_ 0.10 MeV T~., _~ 31.7 with AT~r~ ~_ 1.0 MeV

(q~r2 ~-- 95.5 with Aq~r2 ~ 0.45 M e V / c ) .

(4.2)

(q,r, "~ 99.2 with Aq,r, ~- 1.7 M e V / c ) .

(4.3)

In this section we estimate pion spectra of double-A and single-A hypernuclei relevant to the processes discussed in the preceding section. In this and the next subsections, we confine ourselves to the case of the 9 B e ( K - , K +) reaction products of Eq. (3.6). As shown in Table 1 (Type 3), the process is expected to yield five kinds of double-A hypernuclei: A4H, A~H, AASHe, A6He and A]He. In their pionic decays we take all twobody final states and important three-body final states into account as listed in Table 7. As for the decays of single-A hypernuclei we also follow the same prescription. It is remarked that particular three-body and four-body or more final states can be neglected in the practical estimates as known from typical emulsion data of event-rates shown below: ~He ---~ 4He + p + ~ - ( 1012 ev.), 3H+p+p+Tr-(1 4H---+4He+Tr-(914ev.), d+d+rr-(12ev.),

ev.),

3He + d + r r - ( 1 2 ev.), 3He+p+n+rr-(Noev.)

3H+p+rr-(301

ev.),

[31].

(4.4)

3He+n+Tr-(88ev.),

f o u r b o d y final states ( N o e v . ) [32].

(4.5)

Y Yamamow et al./Nuclear Physics A 625 (1997) 107-142

130

Table 8 Pionic decay rates of light A-hypemuclei calculated for important final states. The empirical non-mesonic decay rates are also listed in the third column to show the total and relative (in %) rates. All decay rates are given in units of the free-A decay rate 1",t "rr- DW 31H

~

3He+~--

----*d+p+zrFNM "~ /'tot

4H

4He

5He

~

71Li

0.34 ( 3 4 % )

~

0.19 ( 1 8 % )

--~d+n+zr

3H +or°

0.17 (17%)

°

1.02 (100%)

4He+or-

0.42 ( 3 6 % )

~

0.30 ( 2 6 % )

----* 3H+n+'n'°

FNM "-'

0.29b(25%)

/'tot "-'

1,16 (100%)

~

0.10 (10%)

0.21a(21%)

~'

----* 3H + p + ' n ' -

(No 2-body) 3He+p+cr-

0.30 ( 3 5 % )

I'NM ~--

0.19c(22%)

/'tot "~

0.85 (100%)

(No 2-body) 0.15 (13%)

- - ~ 4He+~°

0.23 (27%)

~

0.13 (15%)

3He+n+Tr°

---* (No 2-body)

-

~

(No 2-body)

-

----+ 4He+p + ¢r-

0.39 (39%)

~

4He+n + ~o

0.22 (21%)

FNM "~

0.41d(40%)

""

1.02 (100%)

6Li+'n'-

0.32 ( 2 6 % )

-----* 6He+'tr°

0.17 (14%)

---* 4 H e + d + z r -

0.16 ( 1 3 % )

------~5He+n+~'°

0.08 ( 6 % )

FNM '~

0.52a(41%)

/'tot ~

1.26 (100%)

7Be+~'-

0.30 ( 2 4 % )

---* 7Li+qr°

0.16 (13%)

4He+3He+¢r-

0.11 (

~

0.06 ( 5 % )

I1NM '~

0.63a(50%)

Ftot ""

1,38 (100%)

/'tot

6AHe

7r° DW

~

~

8%)

4He+3H+'n'°

a Assumed Fp/Z = 0.11 and Fn/N = 0.10 from •,~exp,5., NMLQae), (Ref. [27]). b Taken from ~'Mp (4H)=0.29+0.14 (Ref. 128[). c Taken from -r exp (4He)= 0.19 =k 0.04 (Ref. 129]). NM d Taken from ~

(~ne)=0.41 =[=0.15 (Ref. I27]).

In g e n e r a l the p i o n s p e c t r u m c o n s i s t s o f d i s c r e t e p e a k ( s ) and c o n t i n u u m part corr e s p o n d i n g to t w o - b o d y and t h r e e - b o d y final states, etc., respectively. In o r d e r to treat b o t h f e a t u r e s o n e q u a l f o o t i n g , the K a p u r - P e i e r l s m e t h o d o f treating the c o n t i n u u m final states w a s s u c c e s s f u l l y a p p l i e d in Ref. [ 3 0 ] w h i c h s h o u l d b e r e f e r r e d to for the n e c e s sary f o r m u l a s . H e r e w e a p p l y the s a m e theoretical f r a m e w o r k to the f o l l o w i n g t y p e s o f s u c c e s s i v e p i o n i c d e c a y s (¢r2 G 7rl ).

E Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

'

I

'

\ >

7.

5He \

5

I

i .-~

\

4

<3

)

~

2

16

-,;'-

k

k

\,I

\

^LII \

'

\~ I I

I

'

I 130

I

1He

3H

114.31 I I

\1o8.o,=-

k

i

I

131

11 )8,4

II

\I,ii

1 !

1 ~1 1

0

-1 i 90

100

110

120

140

(MeV/c) Fig. 8. A-A binding energy ABAA v e r s u s discrete pion momenta q~r- (solid lines), which are expected for the two-body 7r- decays of the AA-hypernuclei produced after the compound state formation reaction 9Be(K-, K + ) ---+ I aa8 He * 'AA8 H * ]compound+N. The vertical dashed lines indicate the characteristic pion momenta known for the two-body ~ - decays of light single-A hypemuclei, with the numbers being monochromatic q~r in MeV/c. In the figure are also included the continuum but very sharp pions from the three-body decays of Arile and 5,He (AqTr "~ 0.45 MeV/c and 1.7 MeV/c, respectively).

(4.6)

AZ ~ (AzI-Jr-TT2, A-IAZt + N-+-~2) , AA a'7t >( A'z/t + ¢ r l , A'--I zll + N + ~ r j ) . a --

(4.7)

In the process of Eqs. (4.6)-(4.7), we are concerned with the weak decays of single-A hypernuclei of 3'4H, 4'5'6~7He and 7Li in addition to the five double-A hypernuclei. The employed decay modes of them will be listed in Table 8. On the basis of the experimental A binding energies B(A1) o f these single-A hypernuclei, we display in Fig. 8 the monochromatic 7r- momenta which comes from the definite two-body decays, respectively. On the other hand, the single-A binding energy /~A in a double-A hypernucleus aAz is defined with the A-A bond energy ABAA by BA( AAZ)

= BA( A-1z)

(4.8)

EXP -Jr- ABAA .

Here BA w i t h o u t tilde refers to the A binding energy in the single-A hypernucleus. Then, the m o n o c h r o m a t i c pion m o m e n t u m q is given as a function o f A B A A based on the e n e r g y - m o m e n t u m relation: q2 M a - B 4 = M N -- BN(.t') + ~

+ O.)q,

wq = ~

q- q2 ,

(4.9)

132

Y. Yamamotoet al./Nuclear Physics A 625 (1997) 107-142

w h e r e MA denotes the mass of the residual single-A hypernucleus AZ~. For the threebody final state, Eq. (4.9) should be modified accordingly by replacing --BN(f) with EN(f) and M a with MA-I + MN, respectively. In Fig. 8 one sees how q~,~- depends on ABAA. It is interesting to note that the pion momenta from these double-A hypernuclei are sufficiently different from each other (cf. Fig. 8) and the q,~- measurement might yield nice information on the A - A interaction. Since there is no data of the above five double-A hypernuclei (except A6He [2,3] ), in the present estimation we employ the following A-binding energies arbitrarily as trial inputs, but we take some theoretical works [27,33,34] into account:

AA4H: BAn = 0.5 MeV (ABAA=0.24 MeV; BA= 0.37 MeV), A5H:

BAA -- 5.0 MeV (ABAA=0.92 MeV; Ba= 2.96 MeV),

AASHe: BAA = 5.0 MeV (ABAA=0.22 MeV; BA= 2.61 MeV), a6He:

BAn = 10.9 MeV (ABAA=4.66 MeV; BA= 7.78 MeV),

A7He:

BAA = 12.5 MeV (ABAA=4.14 MeV; BA= 8.32 MeV).

It is remarked that the different input of ABAa causes certain change of the 7r-decay rates but the pattern of the emitted pion spectrum does not change so much.

4.2. Results and discussion on the pion spectra relevant to the 9Be target

As the double-A hypernucleus A/iSH is produced most abundantly in the process concerned (about 40% of the products, cf. Table 1 and Fig. 4), first we show the predicted ¢r- decay spectrum in Fig. 9. The discrete peak is attributed to the two-body decay mode which yields very high momentum pion due to the large proton binding energy in ~He (Bp = 20.9 MeV): A~tH ~

~He + cry- :

T~2 = 53.9 MeV (q,~2 = 133.9 MeV/c).

(4.10)

There is no other case except 4H(q~t = 132.9 MeV/c) that emit such a "high momentum" pion in the weak decays. Therefore the detection of such a pion itself confirms the formation o f AASH. Here we employ the theoretical value BAA = 5 MeV [25] so t h a t ABAA = BAA -- 2BA(4H) = 0.92 MeV. See Fig. 8 for the variation of q~ as a function of ABAA. The continuum part of the spectrum in Fig. 9 comes mainly from the three-body mode ( 4 H + p + 7"r-) which has a broad peak around T7/"~2 = 30.8MeV ( q ~ = 97.8MeV/c). The 7r- decay rates for the discrete and continuum final states are estimated as shown in units of the free A decay rate: F '°r g - - = 0.38FA vs. F 7/"c = 0.61FA. The F F spin=0 structure of the initial AASH state unfavors the transition to the twobody channel. Anyway the pattern of the pion spectrum is most important for the present purpose. Thus, in order to identify the AASHformation, one should combine this monochromatic pion (q~-2 = 133.9 MeV/c) with ~'1 from the 5He decay of which the spectrum has a very sharp peak given by Eq. (4.3) and shown in Fig. 10(left). Another

133

Y. Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

^5H 0.5 I--

0

jJ

kz~

r~ >-

i

i

i

[

i

20 PION

i

i

i

~

40

3O

KINE

IC ENERGY

[

50

T~

60

( MeV )

Fig. 9. The calculated ~'- spectrum from the weak decay of AA5H,which consists of the monochromatic peak for the two-body decay (SAHe+~-) and the continuum part for the three-body decays (~H+p+~r-).

group of pions coming from the decay of 4H, which is a product of the three-body decay mode, should also join the 7r2 ® ¢rl combination. As shown in Fig. 11, 4AH itself is well known to emit the characteristic pion as 4H

,4 He + 7ri- "

T~r, = 53.2 MeV (qrr, = 132.9 MeV/c).

(4.11)

In the present case, however, this pion momentum might be smeared because 4H is a product of the three-body decay and hence it should have a similar momentum distribution to the 7r2 continuum spectrum (T~2 ~- 30.8 MeV) displayed in Fig. 9. The expected branching ratios in the decay of double-A hypernuclei are estimated, including the 7r° decay and the non-mesonic decay. From the estimates we expect that about 20% of the total decay leads to the two-body mode involving the monochromatic pion. The population of the lightest double-A hypernuclei A4H is estimated to be small in the cascade process (cf. Table 1). This is partly due to the small total binding energy assumed in the calculation. A4H may be a loosely bound system with BAA ~-- 0.5 MeV as suggested theoretically in Ref. [34]. If we follow this BAA value, the 7r- decay spectrum is predicted to have a discrete peak at A4H==~ 4AHe+~- :

T~-2=43.4 MeV (q~r2=ll8.3MeV/c),

(4.12)

and a broad continuum part of the 3 H + p + 7r- mode. In the subsequent process 4He leads to continuum 7r-- spectrum, but in the ~.0 sector it emits a monochromatic pion.

134

Y. Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

A

I-

0

0

Z~ ,,i p--

0

0

/

>-

L i

J

J

*

I

'

q

,

,

,

30

40

PION KINETIC

ENERGY

20

,

i

,

I

20

;

'

T.

'

'

40

30 (MeV)

Fig. 10. The continuum pion spectrum calculated for the weak decay 51He---~4He+p + zr-(left) and that for the two-body and three-body ¢r- decay of 71Li (right).

i

,

,

i

i

,

,

,

,

i

i

,

,

i

,

i

,

r.-,

~_, o I:

°

I.LI t',,-"

0

>-

J

i

20

t

,

30

PION

KINETIC

I

r

1

50

4O

ENERGY

T

1T

60

( MeV )

Fig. 1 1. The calculated ~r- spectrum from the weak decay of 4 H, which consists of the monochromatic peak for the two-body decay (4He+~--) and the continuum part for the three-body decays (3H+p÷q'r-).

Y. Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

i

i

i

,

i

i

~

J

135

i

^~H e

^6H e

.25

el

d

0

>-

ill

S

i

20

20

30 PION

KINETIC

ENERGY

3O

T 11"

( MeV

40 )

Fig. 12. The continuum pion spectrum calculated for the weak decay a 6He--~SHe +p + 7r-(left) and that for the two-body and three-body ~'- decay of A,~He (right).

The decay branching ratios are listed in Table 7. As the production rate of aASHe is estimated to be very small in the process of Eq. (3.6), here we skip the display of the calculated pion spectrum. It has not the two-body rr- decay mode (there is a channel 5AHe+~0), but the continuum spectrum has a sharp peak at T~.2 ~- 30.3 with AT~,:. ~- 2.2 MeV (q~'2 ~- 96.9 with A q ~ 2 ~_ 3.7 MeV/c),

(4.13)

which can be utilized when this hypernucleus is produced under some circumstances. If 1r° detection is available, the two-body decay mode emitting a monochromatic pion is also helpful to identify the AASHeformation, since the pion energy is expected to be also exceptionally large: A,~He ~

~He + 7r2° :

T~2 = 57.8 MeV (q,~z = 137.7 MeV/c),

(4.14)

The predicted rr- spectrum from A6He is shown in Fig. 12(left). This was first presented in Ref. [30] and the very sharp peak in the continuum is characterized by the process of Eq. (4.1) having a nearly monochromatic pion with the narrow momentum width given by Eq. (4.2). The sharpness is attributed to the effect of the very sharp p-state resonance between a proton and 5He (Refs. [30,35] ). In Fig. 12(right) is shown the calculated rr- decay spectrum from AVHe of which the population amounts to about 10% in the process concerned (cf. Table 1). Two discrete

136

Y. Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

20 18 AA

8Lii(l+)

~-

'

DECAY

16

09

14 []

O < ~D

6

4 _

2

2-

~e

0 0. 00. . . .

0. 02. . . .

0. 04. . . .

DECAY RATE

0. 06. . . . 0.. 08 . . .

0. 10

F~(Jf)/F h

Fig. 13. The calculated ~'- spectrum from the weak decay of A~Li( 1+ ) leading to low-lying states in ~Be. The solid lines indicate the monochromaticpion, while the wide histograms the continuum part.

peaks correspond to the two bound states of 7Li and for the ground state the pion energy is A7He ~ 7

Li + ~ 2 "

T~2 : 34.6 MeV (q~2 = 104.3 M e V / c ) .

(4.15)

The continuum pion seems to be negligible. The combination with the subsequent twobody 7r- decay of 7Li is promising (see Fig. 10(right)) to identify the formation of aATHe. One may understand the usefulness by looking at the calculated spectrum for 7ALi shown in Fig. 10(right). 4.3. Some pion spectra relevant to the other targets As discussed in Section 3.4, when we adopt the JOB, IJB and t2C targets instead of 9Be, new double-A hypernuclei are produced. In this subsection we add theoretical pion spectra for such typical S = - 2 species relevant to these targets (AASLi, a9I"Ie, A~Li) and their daughter A-hypernuclei.

E Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

137

DECAY

~Be ~-.

(2-)

4

M

3

2 z 8B 2+

2+

0

0. 0l

0.02

Fro (Jf)/I"

0.03

A

0

0.01

0.02

0.03

0.04

0.05

0.06

1-'~( Jf ) / F A

Fig. 14. The calculated 7r- spectrum from the weak decay of 81Be( 1-, 2 - + ) . Comment as for Fig. 13.

As for the double- and single-A hypernuclear states concerned in this subsection, we employ the [ s4p n (Sl/2) 2 ] and [ s4p " (Sl/2) 1 ] shell-model wave functions which are obtained within the different framework from the preceding subsection. This prescription of wave functions is an extension of the previous work done successfully for the pionic decays of p-shell hypernuclei [38,39]. In general the estimates of pionic decay rate are sensitive to the initial and final state wave functions. Here, however, the pattern of the pion spectrum corresponding to the final bound state region is most important, since monochromatic pions coming from the two-body final states are useful for the identification of hypernuclear production. In this respect, therefore, we take fully care of the energy relation which is consistent with the experimental threshold energies. Fig. 13 shows the calculated pion spectrum for the ~ - decay of AASLi( 1+) to 8Be. Two sharp peaks corresponding to the 8 B e ( 2 - , l - ) final bound states are clearly expected. They correspond to the pion momenta of q~; ~_ 102 and 101 MeV/c, respectively. They themselves serve as an identifier of the ASLi production. The pion spectra of subsequent 7r- decay of 8Be are also calculated as shown in Fig. 14. The lowest particle threshold in 8B is the 7Be+p channel at 0.14 MeV just above the ground 2 + state, so that the unique sharp peak is also useful in identify the species. The similar theoretical estimates are made for the 7r- decay of a9He which is produced with the ]JB target. The calculated spectrum (Fig. 15) consists of several discrete peaks, which combination may be used to trace the double hypernucleus. However, the corresponding pion momenta are rather small so that continuum background coming from other double-A hypernuclei may disturb it. The daughter hypernucleus decays

138

Y Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

10

DEOAY

% 8

L 6 ma z

m 4

z

5/2 +

[.-, [-

2

r.D x [.T.a

5/2 + 0 3/2 +

,

0.00 . . . .

O. 02. . . .

O. 04.

DECAY RATE

.

;

i

O. . 06.

i

h

.

.0.08.

0. I0

r~(Jf)/rA

Fig. 15. The calculated cr- spectrum from the weak decay of A9He(3/2-). Commentas for Fig. 13. subsequently to two distinctive bound states in 9Be as shown in Fig. 16. The last example is the pion spectrum from AI°Li which is produced with the 12C. As shown in Fig. 17, it consists of two discrete peaks corresponding to IAOBe(2-, 3 - ) . The energy separation between two bound states is about 3 MeV, and the lowest threshold for three-body decay is 9 B e + n + ~ - which is 5 MeV above the 2 - state. Two monochromatic pion momenta are predicted to be q~- -~ 115 and 111 MeV/c, respectively. Fig. 18 shows the two pion spectra from the daughter hypernucleus, for which there are two candidate spins ( ! - or 2 - ) and the different pattern is useful in determining the real ground state spin.

5. Concluding remarks A new experiment at BNL-AGS is now in progress to detect double-A hypernuclei produced at the ( K - , K +) reaction points by observing successive weak-decay pions in

139

Y Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

DECAY 16

(5/2 +)

14 E

/

12

lO m 8 z o

6 9Be

4 m

LD

5/2-i

m m

2 3/2m

0 i 0.02

0. 04

0. 06

0.08

r=(Jf)/ra

0. 10

0. 12

0

O. 02

r

O. 04

(Jf)/rA

Fig. 16. The calculated ~ ' - spectrum from the weak decay of 9Li(3/2+ 5/2+). Comment as for Fig. 13.

coincidence. The early calculations continued in this paper has started at the same time of the proposal. We emphasize that the scenario of producing more double-A hypernuclei through the quasi-free ~ - rescattering has been now confirmed on the quantitative basis, as demonstrated theoretically in this paper. First, by using a statisticalmodel, the production rates of various double-A and singleA hypernuclei have been calculated starting with the ( K - , K ÷) reactions on appropriate nuclear targets (9Be, l°B, liB, 12C). We treated the fragmentation process of the doubleA compound nucleus which we assume to be formed by the slowdown of the ~ particles produced abundantly in the quasi-free region. The nucleon-knockout process plays an important role in the ~ - slowdown. It is shown in this paper that several particular double-A hypernuclei are produced with high probabilities: AASH, a6He and A7H are commonly produced, and one more is added depending on the ltB or J2C targets. In the case of the 9Be target, especially, the number of breaking-up channels is small, and the production probability of AASHis distinctively large. This feature gives a suitable and feasible condition of experiment with this target. Next the 7"r- spectra from the weak-decay of the produced double- and single-A hypernuclei were calculated to be utilized for the identification of parent hyperfragments. In the case of the 9Be target, almost all available species have been investigated. Among possible products with S = - 2 (AaH, AASH,A~iSHe,A6He and A7He), the most promising is to observe characteristic weak-decay pions from AASH.They are monochromatic with

140

Y. Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

20 [] 18 []

IAOLi (2+)

~-

DECAY

16 [ ] 14 ~ . 12

Z 0

8

P

I

m

r.~ X

4

_

2

2 0

0.00'

. . . . . . . 0.02

I .... 0.04

DECAY RATE

I ,

0.06

,

0.

~08 . . . .

0. 10

F~(Jf)/FA

Fig. 17. The calculated 7r- spectrum from the weak decay of aaLK2 l0 • + ). Comment as for Fig. 13. distinctively large momentum and isolated from the background due to other single-A and double-A hypernuclear decay. It is also expected to identify other double-A hypernuclei. Finding of light double-A hypernuclei and their A A binding energies will bring about the decisive information on the elementary interaction between two A particles.

Acknowledgements

This work has been done as one of the projects supported by the Grant-in-Aid for Scientific Research on Priority Areas (Theoretical Study of Nuclei with Strangeness) from the Ministry of Education, Science, Sports and Culture in Japan. The authors are

141

Y Yamamoto et al./Nuclear Physics A 625 (1997) 107-142

DECAY

1°Be i(1-)

6

lOBe (2-)

m

5 4

m

t~

3 1÷ o

2

0+

p-. ~D

1

1+

ioB 3+

0

N

0

•

0.02

0.04

Frc(Jf)/FA

0

0, 02

0, 04

0. 06

0.08

0.

F~(Jf)/FA

Fig. 18. The calculated ~-- spectrum from the weak decay of ~l~lBe(I - , 2 - ). Comment as for Fig. 13.

very grateful to A. Gal for his helpful comment to improve the theoretical framework adopted in Section 2. They like to express sincere thanks to R.E. Chrien, K. Ikeda, K. Imai, K. Itonaga, M. Sano and Y. Akaishi for their discussions and comments.

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