The formation probabilities oF Σ-hypernuclei and the “unstable bound state”

Nuclear Physics A435 (1985) 727-737 @ North-Holland Publishing Company

THE FORMATION PROBABILITIES OF ZE-HYPERNUCLEI AND THE “UNSTABLE BOUND STATE” 0.

MORIMATSU

and K. YAZAKI

Department of Physics, Faculty of Science, University of Tokyo, Bunky&ky

Tokyo 113, Japan

Received 26 June 1984 (Revised 6 September 1984) Abstractz A formalism for treating the processes forming unstable states such as I-hypemuclear states is given. The formation probabilities are shown to be expressed by the Green function for the 1 in the nucleus. The formalism is used for examining the effects of the “unstable bound state” which has been proposed as a candidate for explaining the observed sharp J-hypernuclear states.

1. Introduction

One of the recent exciting topics in hypernuclear physics is the observation of 25hypernuclei, which manifest themselves as peaks in the pion spectra from the (K, T) reaction ‘) and stopped K- experiments ‘). It is well known that all Phypernuclear states are unstable due to the A-conversion process. In addition, many of the observed states are above the E-emission threshold. They are thus expected to have large width, but the observed peaks of the pion spectra seem to be surprisingly narrow. There have been many attempts to explain this puzzle. Most of them try to find some mechanism for suppressing the A-conversion width ‘) but do not consider the E-emission part. A completely different explanation was proposed by Gal and his collaborators “). They studied the S-matrix poles for complex potentials and pointed out the possibility that some of the poles may approach the positive real energy axis, when the potential is absorptive enough, and correspond to sharp states which they called the “unstable bound state” (UBS). It is known that the UBS is not observed in scattering experiments, and a crucial question is whether it shows up in the formation process. The purpose of this paper+ is twofold. We first give a formalism for treating unstable (or continuum) states in the formation processes. It will be shown that the formation probabilities can be calculated from the Green function. We then use this formalism to examine the effect of the UBS on the formation probabilities. The simple case of a complex square-well potential is studied in detail. We show that the UBS gives no observable effect either in the scattering or in the formation process. The puzzle of sharp Zhypemuclear states thus seems to remain unsolved. ’ A preliminary

version of this paper was reported in ref. ‘). 727

728

0. Morimaisu,

K. Yazaki / Formation

probabilities

2. Formation cross sections of Z-hypemuclei

Let us consider the (K, T) reaction forming a 2 in a target nucleus. In the DWIA with a zero-range interaction for the elementary process RN+ ~2, the doubledifferential cross section for the (it, a) reaction is given by “)

where fwz,~N is the T-matrix for the elementary process, UK is the velocity of the incoming kaon and the operator fi is defined by

with F(r) = XC-)*( II r)xE’( r).

(3)

&t(lcIN) is the annihilation operator of Z(N) and x’,-‘(,&‘) is the distorted wave for the outgoing pion (incoming kaon)+. Using the hamiltonian H for the Z-nucleus system, we can rewrite the sum over final states in eq. (1) as Fs(E,+E,-E,-Ei)l(f/e/i)/‘=-fIm(il~+E~+Ei_~~_~+i.oQli).

(4)

Since the hamiltonian contains the interaction giving rise to the A-conversion processes of a 2 in the nucleus, the sum over f should include not only the Znucleus states but also the A-nucleus states. Inserting the complete set of eigenstates {(Y} for the system with one nucleon removed from the target nucleus into the r.h.s. of eq. (4), we have -llm T

(

ilfi+

=-LIm T

1 E-H+iT

C a’,n I

Eli

>

dr dr’E,(r’)G,,,(E;

r’, r)_&(r) ,

(5)

where f, and G,,, are, respectively, defined by

f,(r) = F(r)(4JIdr)lQ9

(6)

’ We have assumed here that P-formation is dominated by the two-body process i(N+ ~2. This is certainly justified for the (K-, n+) reaction where the only two-body process is K-p+ ‘R+X. For the (K-, rr-) reaction, the A-formation two-body process can, in principle, contribute through the A-P conversion interaction in the nucleus. There is, however, no qualitative difference in the experimental data between the (K-, T+) reaction and the (K-, C) reaction in the E-formation region. We thus expect that RN-t XZ also dominates the P-formation in the (K-, T-) reaction.

0. Morimatsu, K Yazaki / Formation probabilities

129

(7)

G,,, is the Green function for 2 interacting with the nucleus in the coupled-channels description. We note here that fm includes the spectroscopic amplitude (a[$,.$) for nucleon pickup processes and, therefore, only a limited number of states contribute to the sum over (Yand (Y’in the expression (5). Assuming that the coupling between these states is not so important, we can express the double-differential cross section in eq. (1) in terms of the diagonal Green function G,, which is given by the optical potential U,, for 2 in the nucleus in the state ICY),i.e.

~--l-z

Im

I

drdr’fi(r’)G,,(E;

r’, r)fn(r).

We usually expect that the optical potential does not strongly depend on the states ICY)and take a common potential U for V,,. The state dependence of the Green function G,, then appears only in the energy available for 2 and G,, can be expressed as G,, (E ; r’, r) = G( E - E, ; r’, r) ,

(9)

where E, = E, - Ei is the nucleon separation energy for the state la), and G is the Green function for the optical potential V, satisfying the equation G=G,,+G,UG

(10)

with Go denoting the free Green. function for 2. Taking the imaginary part of eq. (lo), we obtain the following identity: ImG=(1+G+U+)ImG,,(l+UG)+G+ImUG.

(11)

The first term on the r.h.s. of eq. (11) represents the contribution from the escape of the 2 from the nucleus, while the second term is due to the conversion of the 25 into A because the imaginary part of U is due to this conversion effect. Let us define the following quantities: S,,,(E) = -T Im Gf

drdr’e(r’)G(E-E,;

r’, r)f_(r),

1 + G’ U’) Im G,,( 1 + UG)f,

S,,,(E)

= -f(

S,,,(E)

= -fG+

Im UGf.

Using these quantities, we can decompose the double-differential

(12)

(13) (14) cross section into

730

0. Morimatsu,

K. Yazaki 1 Formation probabilities

two parts:

with --

-_ RN-t

~2

ME)

,

L(E)

,

L,(E), where we have introduced elementary process by

the “differential cross section” (da/d~)~N,,~

(17)

(18) for the

(19) Note that this differs from the free cross section by the pion momentum and energy, k, and Em which, in the latter, are determined by the free kinematics. The information about the structure of the Zhypernucleus is entirely included in the strength .functions S(E). In the case of the stopped K- experiment, the energy spectra of the pion emitted after the K- capture is again proportional to the strength function S,,,, except that the kaon distorted wave $’ should be replaced by the bound-state wave function for the K- atom. In the following,‘we will consider these strength functions for the simple case of a complex square-well potential and discuss the problem of the effects of the UBS [ref. “)I. 3. Results for complex square well potentials Since the existence of the UBS [ref. “)I is a general feature of the optical-potential problem with a fairly large imaginary part (or of the coupled-channels problem with a fairly strong coupling), the detailed shape of the potential does not seem to be so important for qualitative considerations. We examine here the effect of the UBS in a simple model as an exercise and leave the realistic calculation for future study. We take a complex square-well potential for the optical potential U, i.e. U(r)=(V+iW)B(R-r).

(20)

Further, we assume that only one state (a) dominates the sum in eq. (8) and fm(r) is a surface-peaked function selecting a definite angular momentum L for Z, i.e.

(21)

0. Morimatsy

The strengths

of the potentials

731

K. Yazaki / Formation probabilities TABLE 1 and the positions

of the poles in the k- and E-planes

Case

V [McYl

W [McVl

Pole position in the k-plane [MeV/c]

Pole position in the E-plane [MeV] “)

:

-26 -16 -9

-2 0 -9

- 16.5 + i88.2 52.3 - i14.8 -144.7+il.l

-3.48-il.35 (I) 1.16-i0.72 (II) 9.70- iO.15 (1)

C

“) (I) ((II))

indicates

that the pole lies on the first (second)

Riemann

sheet in the E-plane.

where the normalization is arbitrarily fixed. Then S,,,(E), S,,,(E) and S,,,(E) defined by eqs. (5)-(7) are reduced to simple forms. It is sufficient to calculate two of them, for instance, S,,,(E) and S,,,(E). By substituting eqs. (20) and (21) into eqs. (12) and (14), we obtain S,,,(E) = -1m G’=‘( E

; l7,

R)

,

(22)

R L”(J9

=

dR’ R’2(G’=‘(E;

-w

R’, R)12.

(23)

I 0

explicit expression of the Green function for the complex square-well potential is given in the appendix. In order to simulate the case of 12C as a target, we take L = 1, R = 3 fm and the reduced mass p = 1080 MeV, while V and W are varied so as to give several cases with bound states, resonances and/or the UBS. We first study three typical cases. The values of V and W together with the positions of the relevant poles in the k- and E-planes are given in table 1. The poles in cases a, b and c are the bound-state pole (which has become unstable,due to the imaginary part of the potential), the resonance pole and the UBS pole, respectively. Figs. l-3 show S,,,(E) and S,,,(E) for these cases. As is clearly seen, in cases a The

Fig. 1. S,,,(E)

(the solid curve)

and S,,(E)

(For E
(the dashed

S,,,(E)

=S,,(E).)

curve)

for case a in arbitrary

units.

732

0. Morimatsu,

K. Yazaki / Formation probabilities

S(E)vc.bcC-_

---__ :::

5

----A

10 E (MA5

Fig. 2. The same as fig. 1 for case b.

Fig. 3. The same as fig. 1 for case c.

and b there are remarkably sharp peaks near the energies equal to the real parts of the pole positions in the E-plane with the half-widths fairly well corresponding to twice the imaginary parts. On the contrary, in case c there is no visible structure around 9.7 MeV, the real part of the pole position, although there should be one if the UBS has the nature which was expected by Gal and his collaborators ‘). Secondly, the effect of the UBS is more thoroughly investigated. This time V is fixed to be -16 MeV and W is varied from 0 to -6 MeV. The motions of poles in the k- and E-planes as W changes are shown in fig. 4 and fig. 5, respectively. Figs. 6a-6d show S,,,(E) and S,,,(E) for each case. When W= 0, the resonance pole

-50

Fig. 4. The positions

of the poles

I

xa .b xC

‘d

in the k-planes (V= -16 l&V; W = -3 MeV; d: W = -6 MeV).

a: W = 0; b: W= -2 MeV;

c:

0. Morimatsu,

A6 WV) 5--

K

733

Yazaki / Formation probabilities

physical sheet

6

5

(MeV)

T

unphysical sheet

a

ER (@3v) 5’

0

o_b

0

xa

CO

5

$ b

t

do

d”

-5.-

-5.-

Fig. 5. The positions

Eq (B&V)

of the poles

in the E-planes (V= -16 MeV; W = -3 MeV; d: W = -6 MeV).

a: W= 0; b:

W= -2 MeV;

c:

and the partner pole are located symmetrically with respect to the imaginary k-axis. Since the resonance pole is sufficiently close to the positive real k-axis, S,,,(E) has a very sharp peak at the resonance energy (- 1.2 MeV) and its width corresponds to twice the imaginary part of the pole position in the E-plane (- 1.4 MeV). As 1WI increases, the resonance pole moves down away from the real k-axis while the partner pole moves up and eventually crosses the negative real k-axis. Accordingly, the peaks of S,,,(E) and S,,,(E) become lower and broader and the ratio S,,,(E)/S,,(E) increases. The resonance energy and the width for each case also seem to be determined comparatively well by the pole position in the E-plane. The UBS appears for W = -3 and -6 MeV but no corresponding anomaly is observed in the behaviour of S,,,(E) and S,,,(E). For W = -3 MeV, the UBS should have shown up at the energy -2.2 MeV with the width -0.7 MeV. The reason can be traced to the fact that though the pole position of the UBS looks similar to that of the usual resonances in the E-plane, it actually does not lie on the same sheet and is too far from the physical region to produce any observable effect. The UBS, however, might enhance the so-called threshold effect, when it is close to the threshold. In the cases discussed above the corresponding Xnucleus scattering parameters and cross sections are also calculated. The results are consistent with those of refs. 4P7):The phase shift changes rapidly at the UBS energy, but the elasticity is very small there, and therefore there are no peaks in either the total cross section or the reaction cross section.

134

0. Morimatsy

Fig. 6a. S,,,(E)

K. Yazaki / Formation probabilities

for W = 0 in arbitrary

units.

V is fixed to be -16 MeV.

S(E)

T

I

-5

0

.

----.

.

5

1

ld

E (MeV) Fig. 6b. S,,(E)

(the solid curve)

and S,,(E) (the dashed curve) (For J? CO, &G) = L(E).)

for W = -2 MeV in arbitrary

units.

0. Morimatsu,

K Yazaki / Formation probabilities

135

A

S(E),

S(E)

1

Fig. 6c. The same as fig. 6b for W = -3 MeV.

Fig. 6d. The same as fig. 6c for W = -6 MeV.

4. Summary and conclusion

We have shown that the formation probability of a Zhypernucleus can be calculated from the Green function for the X in the nucleus. Assuming that the Z-nucleus interaction is described by a simple optical potential, we have examined the effect of the UBS on the formation probability. The results of our study can be summarized as follows: (i) The UBS does not show up in either the scattering process or the formation process. It is thus unlikely that the UBS corresponds to a sharp Z-hypernuclear state. (ii) The UBS together with its partner pole may give an enhanced threshold effect if they are close to the threshold. Even in such a case, the width of the enhancement has nothing to do with the “width” of the UBS. The optical potentials are assumed to be local and energy independent in this study. It is, however, expected that the A-conversion process gives rise to a non-local and energy-dependent term in the optical potential for 2. A strong energy dependence or non-locality may explain the puzzle of the sharp Zhypernuclear states. A coupled-channels calculation which explicitly includes the A-channels will clarify the situation. We would like to thank A. Gal for valuable discussions and useful suggestions, and K. Shimizu for discussions about the formalism.

736

0. Morimatsu, K. Yazaki / Formation probabilities

The UT-MSL VAX-l l/Hongo computer system and the M280H computer system at the Computer Center, the University of Tokyo were used. The computer calculation for this work has been financially supported in part by Research Center for Nuclear Physics, University of Osaka.

Appendix

In this appendix we will briefly review the general procedure for obtaining the Green function in the one-body problem and give the explicit expression for the complex square-well potential. Let us consider the following single-channel Schrodinger equation: -$@+

U(r)

I

@I(r)= E+(r)

(A.11

with U(r) denoting a spherically symmetric potential. The Green function for this problem is obtained as G(E;

r’, I)=

C Y~‘~(~~Y~)(~)G(~~(~; LA+

GCL’(E ; r’, r) = -2pku,(k,

r.&(k)

r’, r), I;) ,

(A.3

where k = G and ZQ,(or)) is the radial part of the regular (outgoing) solution of eq. (A.l) satisf~ng the following boundary condition: + crL dk,

(r+O) G4.3)

r) +--$I’,-‘(kr)-S,hi+‘(kr)] i 2i

&‘(

(r-fm), (r+cO).

k3 r) + hr’( kr)

(A.41

In eqs. (A.3) and (A.4) we use the spherical Hankel functions h(L+)and hr’ which are related to the usual ones as kg’(x) = &y’(x) f

kg’(x)

= -ihf’(x).

In the case of the complex square-well potential (es. (20)), ut. and or’ can be analytically obtained as follows:

(r
ML(Kr)

uL(k;r) = where

K =

J2p(E

-$h!_‘(kr)

-bh’,i’(kr)]

(r> RI,

(A.3

- V- i W). The coefficients a and b are obtained by the boundary

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K. Yazaki / Formation

probabilides

737

condition at the surface as 1 W[h’,-‘(kr), h(L+)(kr)],zR a = -u w[j~(Kr), h(L+)(kr)]r=R ’

(A.61 (A.7)

with

In the same way,

04.8)

‘=

Wh’,+‘(kr), dKr)LR WjdKr), e(Kr)ld

= -d~h(L+‘(kr),

d = Ul[_k.(Kr), h(L+)(krLR = -KR2Ul[jL(Kr) W_k(Kr), dKr)Id Therefore, the Green function (AS)-(A.lO).

is also obtained

nL(Kr)],=R,

(A.9)

, h(L+)(kr)lreR.

(A.lO)

analytically

by eqs. (A.2) and

References 1) R. Bertini et al., Phys. Lett. 90B (1980) 375; 136B (1984) 29 2) T. Yamazaki et al., Proc. Int. Symp. on electromagnetic properties of atomic nuclei, November 1983, Tokyo 3) S. Wycech, W. Stepien-Rudzka and J.R. Rook, Nucl. Phys. A324 (1979) 288; A. Gal and C.B. Dover, Phys. Rev. Lett. 44 (1980) 379, 962 (E); J. Dabrowski and J. Rozynek, Phys. Rev. C23 (1981) 1706; R. Brockmann and E. Oset, Phys. Lett. 118B (1982) 33; Y. Yamamoto and H. Bando, Prog. Tbeor. Phys. 69 (1983) 1312 4) A. Gal, G. Toker and Y. Alexander, Ann. of Phys. 137 (1981) 341; C.J. Batty, A. Gal and G. Toker, Nucl. Phys. A402 (1983) 349 5) 0. Morimatsu and K. Yazaki, Proc. Int. Symp. on nuclear spectroscopy and nuclear interactions, March 1984, Osaka, ed. H. Ejiri (World Scientific), to be published 6) J. Hiifner, S.Y. Lee and H. Weidenmiiller, Nucl. Phys. A234 (1974) 429 7) J.A. Johnstone and A.W. Thomas, Nucl. Phys. A392 (1983) 409