Λ and Σ hypernuclei

Nuclear PhysicsA450 (1986) 199c-208 North-Holland, Amsterdam

199c

A and Z HYPERNUCLEI

D. J. MILLENER Brookhaven National Laboratory Upton, New York, USA 11973

The present status of structure calculations including the phenomenological matrix elements~

for p-shell A hypernuclei,

determination of AN effective interaction

is briefly reviewed.

For E hypernuclei

tial model D is used for guidance in constructing

the Nijmegen poten-

EN effective interactions.

The structure of l~Be and I~C, including isospin mixing in the latter case, is discussed,

and a comparison is made with experimental

data.

I. A HYPERNUCLEI The structure of p-shell A hypernuclei

appears to be quite well understood

in the sense that shell-model calculations using phenomenological interactions

can satisfactorily

and spins I'2.

AN effective

account for the measured excitation energies

For a lamhda in the lowest s orbit the spectra are determined by

the PNSA two-body interaction which can be characterized 2 in terms of five radial integrals.

These are conventionally

denoted by V, &, SA, SN and T, and

are associated with the radial dependence of the operators £NA.SN and S12.

i, SN.SA, £NA.SA,

The splitting of doublets based on core levels of non-zero

spin depends on A, S A and T, while SN can affect the separation of states based on different core levels.

V enters only into the binding energy, where a

three-body ANN interaction seems necessary 2'3 to reproduce the smooth behaviour of the A separation energies as a function of A. The spin dependence of the AN interaction required to fit energies of excited states, deduced from the energies of observed ~-rays, can be compared 2 with that deduced from the NiJmegen AN interaction ~.

The calculated non-

central matrix elements are small in agreement with data, especially spin-orbit

interaction.

However,

for the A

in the ease of the central interaction,

the

singlet interaction is found to be stronger than the triplet, in contrast to the free AN interaction.

The effective PNSA interaction is consistent with an

earlier interaction I which worked well for pnpA states observed in (K-,n -) reactions. The cross sections for the formation of A hypernuclear

states in (K-,~-)

reactions are well understood I in terms of the distorted wave Born

0375-9474/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

D.J. Millener / A and ~ hypernuclei

200c

approximation.

The essential ingredients are the use of distorted waves

obtained from fits to elastic scattering data for kaons and pions, the strength of the Fermi-averaged elementary K-n÷~-A forward amplitudes and the structure information contained in the one-body density matrix elements from the shellmodel calculations.

2. Z HYPERNUCLEI Data from CERN, BNL and KEK exist for a number of p-shell targets from 6Li to 160 and for incident momenta between 400 MeV/c and 720 MeV/c.

In most

instances the statistics are rather poor, and the candidates for E-hypernuclear lines are often not easily distinguished from the background.

The backgrounds,

due in part to broad E-hypernuclear excitations and also to the quasi-elastic process, are uncertain.

The theory too is beset with many uncertainties, aside

from limited knowledge of the EN interaction.

For example, our shell-model and

DWBA calculations do not take into account the unbound nature of the observed Z-hypernuclear excitations.

Consequently, we are often reduced to looking for

a certain degree of consistency between the theoretical predictions and the available data.

It should be emphasized that the narrow peaks observed so far,

with the exception 5 of a peak in 6H appear to correspond to E particles Z ' produced in unbound p orbits and that the relatively small widths of the states are not fully understood.

2.1 QUASI-ELASTIC TRANSITIONS In the Fermi gas model one finds 6 an approximately parabolic shape for the quasl-elastic contribution to the AZ (K-,~ +) ~(Z-2) cross section centered at ffiMHy-M A given by = m E_

- mp + Up - U E_ - ( ~

with a full width a half maximum

I

i )kF~ 4 mz-

rl/2

+ q2/2mE -

ffi J~ qkF/mE-"

Both rl/2 and ~ increase with q; at 0 °

q2/2m E_ =

I 1.7 3.4 6.5 12.1

MeV MeV MeV MeV MeV

(400 MeV/c) (450 MeV/c) (550 M~V/c) (713 MeVlc) (at rest)

where the K- lab momentum is given in parentheses. A good candidate for a quasl-elastlc peak is the broad bump (rl/2 - 20 MeV) seen 5 in the 160 (K-,~ +) 16EC spectrum at 713 MeV/c.

From the peak position,

= 286 bleV, a value for Up - U E- of 24.6 MeV can be extracted.

After taking

out the Coulomb contributions to Up and U Z the strong E- potential turns out to be slightly weaker than the A potential, a result consistent with that extrac-

D.J. Millener / A and E hypernuclei

ted from analyses of E- atomic data 7.

201c

Using the above value for Up - U E- one

can now predict the energy of the quasl-elastlc peak for other incident momenta.

For example 281.2 MeV for 12C (K-,w +) 12Be at 450 MeV/c E =

or

for 160 (K-,~ +) 16C at 450 MeV/c E

291.6 M~V for 12C (K-,w +) 12Be at rest Z The value of ~ at rest is consistent with the quaslfree background assumed by Yamazakl et al 8, while the value for PK = 450 M~V/c puts a broad quasl-elastlc contribution in the region where narrow structure has been observed 9'I0, thus complicating the extraction of cross sections or even relative intensities for narrow peaks.

2.2 ZN TWO-BODY MATRIX ELEMENTS To discuss the structure seen in recent (K-,w ±) reactlonsS'9'10 on 12C and 160 targets we need EN two-body matrix elements for pNpE.

We follow the pro-

cedure used for AN Interactions 2 and obtain PNSE matrix elements by cutting off the model D potential of the Nijmegen group 4 inside a radius r 0.

The result,

for r 0 = 1.2 fm, is shown in Table i where a comparison is made with our standard set 2 of PNSA two-body matrix elements. Table 1 a

PNSy Matrix Elements AN T=I/2

EN T=I/2

ZN T=3/2

Vs

-1.87

-0.37

-3.06

Vt

-1.37

-2.22

-0.76

Sy SN T

-0.04 -0.08 0.04

-0.04 -0.03 -0.43

-0.06 -0.08 0.21

abased on model D of Ref. 4 vlth r 0 = 1.2 fm. Note that -V t = V - 1/4 A and -V s = V + 3/4 A for V and A defined as in Ref. 2. We notice that the central interaction exhibits a strong spln-lsospln dependence with the strongest attraction being in the triplet channel for T=1/2 and in the singlet channel for T=3/2. slightly stronger than for AN.

The spin orbit interactions are weak, only

The tensor matrix elements are quite large and

exhibit a clear tN.t E dependence associated with pion exchange.

One other

feature of the model D ~N interaction is that space exchange components are very weak. We obtain central and tensor matrix elements for pNpE by choosing the strengths of Yukawa interactions in each channel to reproduce the PNSE matrix elements of Table i.

The range of the Yukawa interaction then fixes the ratio

202c

D.J. Millener / A and Y~ hypernuclei

F(2)/F(O) which characterizes spin-orbit

interactions

p-shell effective interactlons I.

by an effective one-body spln-orblt

terized by the parameter

ep = epl/2 - Cp3~2.

with respect to the case of h hypernuclei" and 4.88 MeV respectively fully compensated E-hypernuclear

We replace the

interaction

charac-

The only essential complication

arises because E- and E 0 are 7.98

heavier than E+.

These mass differences

for by the corresponding

are not

Coulomb energies so that in general

states will not have good isospin.

with respect to a core we use V c = Zeore.C.mtE.

For the Coulomb energy of E Then C = 0.57 MeV if we take,

e.g., the value V c = 2.8 MeV for A=I2 estimated by Batty, Gal and Toker II.

2.3 THE A = 12 E HYPERNUCLEI The 12C(K-,n+)I~Be

reaction 9 at PK = 450 MeV/c shows a peak of width 4±1 MeV

at MHy-M A = 279 MeV followed by a broad shoulder at higher excitation energy. Since pN+PE AL=O transitions experiment,

are strongly favored at the forward angle of this

the peak has been interpreted

configuration. 12C(K-,n-)I~Be

in terms of the lIB gs x P3/2E O+

The spectrum observed at the larger momentum transfer of the reaction with stopped kaons 8 is qualitatively

and possibly three, peaks have been observed,

different.

Two,

and the splitting of 4.6±0.6 MeV

between the peaks at MHy -M A = 277.2±0.3 MeV and 281.8±0.5 MeV has been attributed largely to a spin-orbit

splitting of the P3/2 and Pl/2 Z- orbltals.

rationale for this interpretation

The

can be seen from the pure weak-coupling

results displayed in Table 2. Table 2 l~Be in the weak-coupling Ex(MeV )

limit for ~p = 5 MeV

Configuration

Jn

p2(AS=O)

0

3/2; x P3/2E

O+

0.475

}

2+

0.237

I

i

2

1/2; x P3/2E

2+

0.126

5

3/2~ x Pl/2E

2+

0.237

3/2~ x P3/2E

O+

0.063

I

0.40

2+

0.032

O+

0.126

7

1/2; x Pl/2E

da/d~(rel.)

0.15

In Table 2

(

)

S

P2-AS=O- = 2 ( 2 i N + l )

1 1/2 JA~ 2 1

1/2 JN/

AL 0 AL ]

,

where S = 5.70, 1.51, and 0.76 for the gs, 2.12 MeV and 5.02 MeV levels of liB respectively. In obtaining relative cross sections the result of Gal and Klieb 12 that 0+ is favored over 2+ in the ratio 0.57:0.43 has been used. The existence of the lower two peaks including their ratio 8, but not a third peak, can be explained for Cp - 5 MeV.

D,J. Millener / A and E hypernuclei

203c

Table 3 ZN e f f e c t i v e

Interaction a

T

112

=

Vt GI

-29.0

interactions T

Vs -8.8

312

=

Vt

Vs

-10.8

-45.0

~(fm)

;(0)

1.0

-1.03

G3

-29.0

0.0

0.0

-45.0

1.0

-0.73

MI

-90.0

-30.0

-22.5

-90.0

0.7

-1.03

M2

-94.7

-15.8

-14.3

-114.6

0.7

-1.03

M3

-i00.0

0.0

0.0

-157.5

0.7

-1.03

M4 -50.0 -50.0 -50.0 -50.0 0.7 -i.00 ayukawa forms, strengths in HeV. F(2)/F(O) = 2.43, 3.07 for p = 1.0,0.7. All interactions except M4 contaln a tensor force of range ~=i fm wlth V(T-I/2) = -52.0 and V(T=3/2) = 25.0. We now study the effects of the ZN interaction, several of which are listed in Table 3.

Flg. i shows the distribution of formation strength, In the form

p2(AS=O), as a function of ~p for the interaction M4 which Is central, spln and isospln independent, and similar to the AN effective interaction.

For ep

small there is a strong tendency to concentrate the formation strength in a

i 0.566

A=,Z 0+,Z+,T=~2

~D=9 -

M4 INTERACTION

I

0.569

t

~p=8 -

[ o


I

I

O.573

t (D= 7 -

i 0.578

(p=6 -

0.58,5 ep=5 -

l

i 0.592

• p= 4 -

0.602
I 0,61~ , l I,

~p= 2 i

0'ii8

Jl

0.584

0.25

[,. o

,

~p=O

2

aA 4

6

8

I0

12

Ex (MeV)

FIGURE i Distribution of formation strength, p2(AS=O), as a function of 8p for the M4 interaction. 0+ states are distinguished by arrowheads.

D.Z Millener / A and ~ hypernuclei

204c

small energy region as we know experimentally

for the A hypernuclear

Ep increases,

and the strength associated with the Pl/2 E

the spectrum,

the possibility

a third peak for Cp = 7). model D E N strength

As

orbit moves through

of a second peak occurs for ~p ~ 6 MeV (and maybe

However,

this interaction does not resemble

interaction and, even for ~p large,

tends to concentrate

too much in the lowest group of states.

T 0,507

I

In Fig. 2 results

G3 INTERACTION

I

,

t

i

I

t

T 0.507

~p=5

,

,

IJ

t

0.508

I

for inter-

~p=6

I

I

the

the

~p= 7

,I

0.507

I

<3

case.

~p=4

,

I

I

t

A:I2 0+2 + T=3/2

<3 0.536

I

GI INTERACTION

J

I

(p=7

,,

t

0,538

I

eP=6

,

I

I

,t

0.541

I 0.25

I

%:5

,

,

To.~; ~ 0

2

I

I, 4

I

t %;,

t

6 Ex (MeV)

8

I0

FIGURE 2 Same as Figure I for the GI and G3 interactions. actions GI (based closely on model D) and G3 are shown.

For ~p = 4 the 2 +

strength is very much spread over the lowest four 2+ states, but for Ep = 5 + + + + + the centroid of the 23, 24, 25 group is about 4.7 MeV above that of the 21, 01, 2~ group with relative intensities However,

there is considerable

data 8 can be convincingly

of 31% and 34% for GI and G3 respectively.

fragmentation,

explained

and it is not clear that the KEK

by these calculations.

The CERN data 9 for PK = 450 MeV/c show a single peak in the 12C(K-,w+)12Be spectrum at AM = MHy -M A = 279 MeV and two peaks in the 12C(K-,~-)I~C

spectrum

at AM = 270 and 275 MeV, while at PK = 400 M~V/c only the lower of the two peaks appears clearly.

The three peaks should correspond

x P3/2 ~ 0+ configurations.

In order of increasing

11>,12> with T z = 1/2 and 13> with T z = - 3/2. good isospin be labelled as Ii'> and 12'>, i.e.,

to basically 3/2- gs

AM denote the states as

Let the T z = 1/2 states with

D.J. Millener / A and E hypernuclei

I1,> - I ~ - 1/2 ~

12.>-IT-

205c

- 1/2 > - , , r r s I ~'~ . ~o> _ , , - , ~ 1 , , ~ x ~->

3/2 , z -

"cx

+

If the strong interaction conserves isospin its contribution, binding energy for states

12'> and 13> is the same.

B3/2, to the E

The binding energy, BI/2,

l

l

for state II'> is in general different from B3/2.

Dover, Gal and Millener 13

showed that, for any reasonable value of Vc, a value of AB not much smaller than E3-E I is required to reproduce the observed separation of states and, furthermore,

that the relation E3-E 2 < 4.93 - 4/3 V c (MeV)

should hold.

A consequence of the large value of AB is that the isospln mixing

in states Ii> and 12> is small. the differences

If, instead, we start with the charge basis

in diagonal energies are E(Z-) - E(E 0) ffi 3.41 - V c + 1/3 AB E(E 0) - E(E +) ffi 4.57 - V c + 1/3 AB

and the off-dlagonal matrix element between the E 0 and Z+ states is /~ AB/3. The large value of AB causes considerable mixing and obviously leads to wave functions with close to good isospin. full shell-model

This is evident in the results of the

calculation presented in Table 4. Table 4 Interaction Vc(MeV )

M1

M3

2.0

2.85

2.0

2.85

% T ffi 3/2 in 0t "T ffi 1/2"

5.8

3.2

2.0

1.0

% T ffi 1/2 in 03 "T ffi 3/2"

6.1

3.3

2.4

i.i

AE ( 0 3 - 0~) MeV

4.92

4.37

8.27

7.84

AE (0~- 0~) ~eV

2.01

1.00

2.15

1.08

AE (0~ - 0~) MeW

6.9

5.4

10.4

8.9

For the interactions MI, M2, and M3 of Table 3, the force in the two weakest central channels is progressively weakened while maintaining of F(0) -1.03

f o r T ffi 1/2 and T ffi 3 / 2 ;

M~V.

actions

F o r c e M1 i s

similar

produce almost identical

t h e l o w e s t 0+ s t a t e

increasing AB.

F~

to that

~o)

s u g g e s t e d by mode l D. The t h r e e

T = 3 / 2 Affil2 s p e c t r a .

becomes p r o g r e s s i v e l y

more b o u n d .

F o r T ffi 1 / 2 , This

ffi

inter-

however,

is equivalent

to

for e = 5 MeV, V = 0 and mass differences set +p c+ to zero, the separation between 01; T ffi 3/2 and 01; T - 1/2 are 3.52, 4.91, and 7.13 MeV.

Specifically,

the average values

(o)

- - 1 . 5 1 MeV, F 3 / 2 ffi - 0 . 7 8 MeV,

D.J. Millener / A and E hypernuclei

206c

~ O,8

r

I I

0,4

T

° Z C ( K ~

I

Vc=2.B5 MeV

Pl/2 i

P312

-I

i

Vc 2 M eV Z

0.4 - I

I

I

-0.4

I

I '270

272

274

zzs'¢~°'*~+'

276 278 MHy-M A(MeV)

280

282

284

FIGURE 3 AS=O one body density matrix elements for the 12C(K-,~-)I~c reaction calculated using the MI interaction. Yamazaki (Ref. 14) has reported peaks at MHy-M A = 271.2, 277.6 and 282.1 MeV. 0+ states correspond to dashed lines and 2+ states to solid lines. The full distribution of formation strength for 12C(K-,~-)I~c is shown in Fig. 3 for two values of Ve, where 01/2 and 03/2 are the AS=O one-body density matrix elements (not reduced in isospin). considerable isospln mixing.

Evidently there is, in general,

Also shown are the positions of three peaks

reported by Yamazakl at this conference 14.

It may be possible to explain the

lowest two peaks, but not the third.

~(3/2) The (K-,~ +) and (K-,~-) formation amplitudes are proportional to i L

. (0)

03/2 and /2/---~f~i/2)(0°)Pl/2 - /i-7~ f~3/2)(0°)P3/2 where f~AT)(o°) is the Fermi averaged 0 ° elementary amplitude.

The free space K-N÷~Z lab cross sections at

0 ° from the analysis of Gopal et a115 are displayed in Figure 4.

They clearly

display the effect of Y* resonances, most notably the Y*(1520) at about PK = 390 MeV/c.

Averaging dilutes the effect of the sharp Y* resonances.

In Fig. 5

we plot the ratio of cross sections for I~C (A=II gs x P3/2 Z 0+ wave functions) for the isospin and charge bases after Fermi-averaging the amplitudes.

In the

isospin basis the ratio of the two peaks 9 at AMffi270 and 275 MeV changes more between PK = 400 and 450 MeV/c than it does in the charge basis. correct trend to explain the data 13.

This is the

Detailed comparisons are not possible

because cross sections have not been extracted from the data.

D.J. Millener / A and E hypernuclei

207c

2.4 THE A = 16 ~ HYPERNUCLEI In the case of the 160(K-,~+)I~c reaction I0 at PK = 450 MeV/c the exlstenc of two peaks about 6.5 MeV apart has been claimed.

A spin-orbit splitting of

12 MeV has been deduced I0 in disagreement with the value of 5 MeV claimed fro the l~Be data 8.

For our ZN interactions the off-diagonal matrix element, v,

between the p~)2 N Pl/2 E and P3/2 N-I

P3/2 Z 0+ states is large.

For the inter-

action GI v ffi Ps + Pt + T -2.57 = -1.66 + 0.30 - 1.21 ; T = 1/2 1.47 = -0.61 + 1.45 + 0.63 ; T = 3/2 , where v is broken down into contributions from slnglet and triplet central forces and from the tensor force.

I

[

I

I

A consequence of the large v is that the

I

FREESPACEKN--~Tr~DIFFERENTIAL CRI_p_~III;ECTIONS IN LAB (0°) _

~o ~ 0

b~z v

)l°(E°) /

\ ~0

-

400 500 700 LAB MOMENTUM(GeV/c}

FIGURE 4 Free space K-N÷~z differential cross sections at 0 ° as a function of lab K- momentum.

900

I

I~ ' ~ I i 600 800 LABMOMENTUM( MeV/c)

FIGURE 5 Ratio of cross sections for I~C 0+ states (see text) in the isospin and charge bases as a function of lab momentum using Fermi-averaged amplitudes.

(K-,~) formation strength tends to be predominantly in the upper state for T ffi 3/2 and the lower state for T - 1/2, more or less independently of the spin-orblt splitting.

If the two peak structure in I~C were confirmed it

would be a strong indication that the ~N interaction based on model D is inadequate, as would the empirical need for a large Z spin-orbit splitting.

I000

D.J. Millener / A and E hypernuclei

208c

3. DISCUSSION It is clear that we have difficulties nuclear data for A=I2 and A=I6. is not unambiguously differentiate

determined.

in understanding

In particular,

Such a determination

between the predictions

the existing

the E spin-orbit

E hyper-

splitting,

is important in order to

of ~p based on one-boson-exchange

models 16, and those of quark models 17 inspired by quantum chromodynamics. number of (K-,~) experiments

~p,

A

better suited than A=I2 and A=I6 for a determina-

tion of Cp have been suggested 12'18'19

ACKNOWLEDGEMENT. This paper is a report on work done in collaboration with C.B. Dover and A. Gal. This work was supported by the U. S. Department of Energy under contract DE-ACO2-76CHO0016. REFERENCES i. E.H. Auerbach et al, Ann. Phys. (NY) 148 (1983) 381. 2. D.J. Millener, A. Gal, C.B. Dover and R.H. Dalltz, Phys. Rev. C31 (1985) 499. 3. A.R. Bodmer, Q.N. Usmani and J. Carlson, Phys. Rev. C29 (1984) 684. 4. M.M. Nagels, T.A. RiJken and J.J. deSwart, Phys. Rev. DI2 (1975) 744; 15 (1977) 2547. 5. H. Piekarz et al, Phys. Lett. IIOB (1982) 428. 6. R.H. Dalitz and A. Gal, Phys. Lett. 64B (1976) 154. 7. C.J. Batty, Phys. Lett° 87B (1979) 324. 8. T. Yamazaki et al, Phys. Rev. Left. 54 (1985) 102. 9. R. Bertini et al, Phys. Lett. 136B (1984) 29. I0. R. Bertini et al, Phys. Lett. 158B (1985) 19. Ii. C.J. Batty, A. Gal and G. Toker, Nucl. Phys. A402 (1983) 349. 12. A. Gal, this conference. 13. C.B. Dover, A. Gal and D.J. Millener, Phys. Lett 138B (1984) 337. 14. T- Yamazaki, this conference 15. G.P. Gopal et al, Nucl. Phys. BII9 (1977) 362. 16. C.B. Dover and A. Gal, Progress in Particle and Nuclear Physics 12 (1984) 171; R. Brockmann, Phys. Lett. I04B (1981) 256; A. Bouyssy, Nucl. Phys. A381 (1982) 445. 17. H.J. Pirner, Phys. Lett. 85B (1979) 190; 0 Morimatsu, S. Ohta, K. Shimizu and K. Yazaki, Nucl. Phys. A420 (1984) 573. 18. C.B. Dover, A. Gal, L. Klleb and D.J. Millener, BNL preprlnt 19. J. Zofka, contribution to these Proceedings.