507
Progress of Theoretical Physics, Vol. 49, No. 2, February 1973
Final-State Interaction between the A0·Hyperon and
Residual Nucleus Ra in the Reaction
K- + 4 He~n + A0 + Ra. I
--Single-Channel Calculation-Y oshihiko IW AMURA
Department of Physics, Science University of Tokyo
Kagurazaka, Shinjuku, Tokyo
(Received April 7, 1972)
For the reaction
K-+ 4 He~7t+A 0 +Rs
(residual nucleus), the investigation of the final
A 0-Ra interaction and its spin-dependence is performed in the framework of the single-channel (i.e., without the 1:-A conversion) Omnes equation whose Born term is given by the im-
pulse approximation. The calculation is carried out for the nuclear capture of K- meson at
rest from both the IS and 2P atomic states. It turns out that the effect of singlet AO-R3
rescattering has considerable influence on the pion momentum distributions, while the triplet
one is small. In the theory used in this article, the refinement is performed so that the
parameter-search may be carried out over all.
§ I.
Introduction
In this paper we report some investigations on the subject of the final-state
interaction and its spin-dependence between the A0-hyperon and nucleus in K+ 4 He~n + A0 +Ra (the residual nucleus, i.e., 3He or 3H) reaction. This problem
seems very important for the search of interaction at low energy between A0 and
nucleus, especially nucleon, and a sufficient information on the interaction, of
course, depends upon the efficiency of final-state interaction in the reaction.
In the experimental data1l-6l accumulated up to the present, the pion momentum or energy distributions show the characteristic shape at higher values of
pion momentum as if we may anticipate the final-state interaction between A0 and
R 3• In fact the higher peak is located near the threshold of K- + 4 He~n + A0
+ R 3 reaction, while the lower peak takes place near the threshold of K- + 4He
~n + .zo + R 3 reaction which is converted into the A0 process through .zo + Ra~ A0
+Ra reaction in the final state. Several papers6l-lll on the basis of the impulse
approximation have been devoted to the explanation of higher peak, from which
one can see that the production process may approximately reproduce the peak
of interest. Thus the authors of these papers are led to the opinion that the
final-state interaction may be negligible and the best-fit curve may be calculated
in terms of the production amplitude alone. However, what reason can allow
508
Y. lwamura
this easy conclusion ? As stated above the region of higher values of pion momentum is essentially related to :final-state interaction . Therefore , in spite of the
smearing by the production process,. the effect of that must be carefully estimated
and the discussion should be given for the possibility of extracting the information about A0-Rs interaction . In the latest repore> on this subject, Said and
Sawicki have concluded that·· the :final-state interactio n is not so important , but
we want to remark that their calculatio n carries several weak points as for the
treatment of the :final-state interaction . Thus it is of particular importanc e to
reexamine this subject on more refined theory of the :final-state interactio n13J-HJ
for the single-cha nnel case. There are several advantage s in our treatment ; (1)
the off-shell theory (dispersiv e approach) is correctly used for two-partic le scattering in the :final state, which may be easily extended to the many-chan nel case,
(2) the spin-depen dence of the A0-R 8 interactio n has been exactly taken into account and (3) the numerical computati on has been carried out analytical ly in all
steps.
Now, three points come into question for the reaction mechanism ; (i) Knuclear capture from a definite atomic state, (ii) A0-hyperon production process
and (iii) interactio n between the A0~hyperon and residual nucleus in the final
state. Our main purpose is directed to the investigat ion of (iii). As for (ii)
we use the impulse approxima tion except for the detailed discussion for elementary process K- + N-'>TC + A0, which is approxima ted by Y1* resonant amplitude ,
considerin g Y 1*-'>TC + A0 process alone. As for the most reliable informatio n related
to (i), the report by Burleson et al. 16h*l should be referred to, the result of which
is as follows: About 80% of K- meson are captured from 2P state and about
20% from lS state. In the present calculation , however, we will not adopt the
mixed initial state of lS and 2P, because we want to see how the particular
initial atomic state influences the pion momentum distributio ns and it is not so
important to seek the best-fit parameter s for the crude calculatio n using the singlechannel theory.
In the present report, since we are concerned with the higher values of pion
momentum , only the pion energy has been considered relativisti cally and Coulomb
distortion for pion is neglected.
In § 2 we briefly discuss the Omnes equation and its well-behav ed solution
at zero energy of the two interactin g particles in the :final state. The pion
momentum distributio ns are explicitly written down in § 3 in terms of the solution of the Omnes equation whose Born term is given by the impulse approxima tion. In § 4 we give numerical results and discussion .
§ 2. Formulat ion
To begin with, let us g1ve a simple discussion to the :final-state interac-
*'
The author wishes to thank Professor S. Iwao for the comment related to Ref. 15).
509
Final-State Interaction between the A0-Hyperon
tion12J-l4) by the Omnes equation, which is suitable for the present calculations.
It will be convenient for the latter calculations to define 'here momenta,
masses and other symbols used in this paper. These notations are listed m
Table I.
Let us define the reaction amplitude for the process K- + 'He~n + A0 + Rs as
follows :18>
TJ[= 2ntJ(E, -E1Y:IJ[= 2ntJ(E, -E1 )() (P,- P 1 - p,.) M 8'
(1)
with M 81 =(p,.P1 qAsS1 1 TJP/J!K 4'1f!N 3S,), where Rs is supposed to be an elementary
particle with spin 1/2.
Since the T-matrix for this process contains in general all of the effects of
two- and three-body systems in the final state, it will be very difficult and complicated for the final state to give the exact expression for TJ[. However, in
the maximum region of the absolute value of pion momentum, we can reduce it
to a simpler form from the point of view of final-state interaction, in which a
pion behaves like a free particle, the non-interacting spectator for the other two
particles. With this we will write down the reaction matrix element below.
Table I. Definition of symbols.
The momentum and mass of particle i, where i=K- meson, rr
meson, A0 -hyperon, N (nucleon), R 8 and 4He.
The pion energy (we will take the system of natural unit in this
Wn= (mn 2 +Pn2 ) 112 :
paper).
The total energy in the initial and the :final states, respectively.
E, and E1 :
The total momentum in the initial state.
P,:
The c.m. momentum of A-Rs system in the final state.
P,=pA+pa:
mtmJ :
d
mipJ-mJPt
•
an /.ltJ
m,+m1
m,+m1
The relative momentum and the ·reduced mass between particles i
and j (see Fig. 1).
The spin and its projection of particle i.
and lit:
The total spin and its projection of the N-R 3 system (i.e., 4 He) in
S, and N,:
the initial state.
The total spin and its projection of A-Rs system in the :final state.
S 1 and N 1 :
!rK4(r):
The atomic state wave function of the K--4 He system and r
denotes the relative position vector between K- and 4He.
!rNS(R):
The nuclear state wave function of the N-Ra system and R denotes
the relative position vector between N and Ra.
Pt and m,:
q,,
s,
Fig. 1. A diagrammatic representation of the production process:
K-+ 4He-')rr+A0 +Ra.
510
Y. lwamura
The partial wave decompositi on of the reduced matrix element into the states
spanned by A0-Ra relative orbital angular momenta may be easily carried out to
give
(2)
where lj_Aa denotes the solid angle defined by qAa· We obtain in the same manner an l-th partial wave Bf,f; for the production amplitude, in which all of the
final three particles are in the plane wave states. Then we define the subtracted
amplitude by Af,f; = M/!1- Bf,f;, which is supposed to be a function of energy E = q~3/
2!J.Aa· We may require that Af,f; is an analytic function in the entire E-plane
except for the singularities with respect to the final A0-R 8 system, the two-body
unitarity cut and the bound state poles, while Bf,f;, of course, has singularities
associated with its dynamical structure, but these have nothing to do with the
final-state effects. Furthermor e we assume that Af,f; vanishes on the infinite circle
in the entire E-plane.
According to these statements the dispersion relations for Af,f; are led in the
two separate cases to the following equations: For the case that the bound-state
poles do not occur,
Msr (E) = Bsr (E)
lm
zm
+ ___!_ ("' Disc Af,f; (E') dE'
2ni
Jo
E'-E-ie
'
(3)
where Disc Af,f;(E) =Af,f;(E+) -Af,f;(E-) with Af,J;(E±) =limAf,f;(E ±ie) for
<-->0
real E.
For the case that the bound-state poles occur,
Msr(E) =Bsr(E)
Zm
where E, denote the
lm
+"
£...
•
T(E,)
E - E ,;
bound-state
+-1-
i"'
2 nz· o
Disc Af,f;(E') dE' .
E' - E -ze
·
'
( 4)
energies and T(E,) are given by lim(E
B->Bi
- E,;)AfJ: (E).
Since it may be easily shown that Disc Af,f; (E) =Disc Mi~{ (E), we can calculate this by making use of the unitarity equation16 > for T. Let us introduce
this equation by
(5)
where t is the T-matrix of two-particle subsystem and H 8 is the three-body
energy operator in the intermediate state. In the form of matrix element the
left-hand side of Eq. (5) may be rewritten as
(6)
When we calculate the right-hand side of Eq~ (5); we will assume that there
is no l:-A conversion process in the intermediate state and the final A0-R 8 interaction is spin-indepen dent, i.e., no spin-flip terms. Then we write in conformity
Final-State Interaction between the A0-Hyperon
511
with the conception of the final-state interaction
(p,.P~,~sS1i tip,.' P/ q~aS/) = (J (p,.- p,.')(J (PI-P/)
X
tJs1sr :E Yzm (q,~a) Yzm* (qfa) tz 81 (q,~a, Q~a).
!m
(7)
By using Eq. (7) we have the right-hand side of Eq. (5) in the form of
matrix element as follows:
In the single-channel case we may write the two-body T-matrix element tf!
in the form
1
t 181(E)=-(
)et 8 f 1<8 >sin(J/1(E)l3>
(9)
'TC/.!ABQAB
with a real (J 18 1(E), which is the scattering phase shift for l-th partial wave with
total spin 8 1 in final A0-R8 system. Substitution of Eqs. (6), (8) and (9) into
Eq. (5) leads to
(10)
We are now so near the small relative kinetic energy q~ 3/2ttAs that only the
S wave is needed to be considered in A0-R 3 system. Therefore, for the S wave
we get from Eqs. (3) and ( 4)
Mosr(E) =Bosr(E)
+ _!_ f"' e-ia s~n (JM/~(E') dE'
rc
Jo
E -E-ze
(11)
and
M/r(E) =Bosr(E)
+ :E
Tt
t E - Ei
+ _!_ f"' e-ta s~n tJMos~(E') dE''
rc
Jo
(12)
E - E- ze
where we have set tJ=(J081(E) and Tt=T(Et).
For the triplet case we have to think these two simultaneously, since the
low energy parameters of triplet state are not yet determined and have a chance
to bring forth the bound states.
These type of integral equations are usually called the Omnes equation, the
solutions of which have been studied in detaiJ.12>' 13>' 17> However, the conventional
solutions lack proper care of the delicate singular behavior near the zero-kinetic
energy, generated by the bound-state poles. Therefore we want to give here the
solutions by taking this point into account.
To find the solutions of Eqs. (11) and (12), let us introduce the functions
D(E) =exp[-_!_
rc
and
f"'
Jo
,(J(E') . dE']
E -E-ze
for Eq. (11)
(13)
512
Y. lwamura
D(E) =
II (1-E/Ei) exp[-__!_ f"' ,~(E') . dE']
i
rc
Jo
E -E-ze
for Eq. (12).
(14)
The use of these functions allows us to give the same form of solutions for
Eqs. (11) and (12), i.e., they are
Mos'(E) =Bos'(E)
+
1
rcD(E)
f"' D(E')eia sin ~B/'(E') dE'.
Jo
E'-E-ie
(15)
These solutions have been, of course, taken so as to cure the singular
behavior stated above. 18' The derivation of Eq. (15) will be given in Appendix A.
§ 3.
Impulse approxima tion and pion momentum
distribution s
In this section we calculate the reaction amplitudes by Eq. (15) and the
pion-momen tum distributions . The essential basis of the calculation is the impulse approximati on, in which K- nuclear capture takes place in the interaction
of K- with only one nucleon in 4He.
Since we have now interest only in seeing how the pion momentum distributions behave for captures from the 1S state and 2P one, respectively , we distinguish the production amplitudes form one another, putting B-/1 (A.= 1S, 2P), so
that M/' as well.
To perform the impulse approximati on it is convenient to connect the total
spin state, IS1, N 1) or lSi, N,), to the individual spin state, isN, VN) or iss, Va) or
isA, vA>· Further, in the aspect of the nuclear capture of stopped K- meson
through the atomic states, we can put Pi= 0, then we are in the c.m. system of
the final three particles, i.e., p,. +PA +p 3 = 0. Thus we have
(16)
with
B/1 = L:;C (s..~ssS,;
X
v..~vaN,)
c (sNsa' si;
VNVa' Ni) ~ ••• ,,~••• ,,
sdpKFhK4 (pK) FN3 ( -ps-tpK)( q,;
s..~v..~itiqi;
SNVN).
(17)*'
*> The C's denote the Clebsch-Gordan coefficients,19> 2J the summations over the individual
-spin states and (t) is the T-matrix element of the elementary process K-+N-7~r+A0, in the c.m.
system of the two particles :
q{==qNK=pK+
:K
mK
ps, q,==qAa,
mN
513
Final-State Interaction between the A0-Hyperon
As for 1S and 2P wave functions of K- mesonic atom, we can take their
values at the origin to a good approximation from the point of view of Knuclear absorption, furthermore the 0'-function approximation in the momentum
representations in conformity with the zero energy capture. 8l-11 l· 20 l
With this
we can put
for 1S capture
(18)
and
7f!K4
2P
C )~
Px
1
(25nax5Yf2
i (e ,
. Px
) 0' C )
for 2P capture ,
px
(19)
where e specifies the orientation of the 2P Bohr orbit and ax is the Bohr radius
of K- meson. As the state of 4 He as the N-R 3 bound system must be the singlet
S state (i.e., S,= 0), we choose the Hulthen S state wave function for 'IJfN8 (R),
then
'1]/'Ns(v)
=4nNu[
1
V 2
+ti
-
1 ].
v 2 +v 2
(20)
where .a= 0.8 Fm- 1 and J.l= 1.25 Fm- 1 so that this parametrization assures approximately the correct nuclear mean-square radius of 4He and N 9 is a normalization constant. For the reaction matrix element of K- + N~n + A0 process we
use only the Y1* resonant amplitude in the Breit-Wigner form~ which gives main
contribution9l in the region of interest. For convenience, let us write the Y1
resonant amplitude as the form introduced in Ref. 9). *l
Then, we have from Eqs. (16)'"'"' (20)
*
x(s [2q·q'+i(1'·[qxq'J[s)
s -M1* + iT1/2
'
I
for IS capture
(21)
,.;
and
1 )
v 2 + v2
1
(v 2 +v2)2
(s [2e. q' + i(l'. [ e x q'J [8 )
1
-./
s - M1 * + iT1/2
'
} (2e·v)(s~[2q·q' +i(l'·[q xq']
f i -M1*+iT1/2
for 2P capture,
[s,)].
(22)
*>
~
2q,·q1 +iu· [q,x q1 ]
(q1sAitlq,sN>=<sAig ..; s -M1*+irl!Z isN>•
where v's = (m" 2 +ql) 1' 2 +mA+ql/2mA, M 1*=1385 MeV,
ing constant.
r 1 =36 MeV21l
and g is the effective coupl-
514
Y. lwamura
where
with M =m,(mA+ms)
" m,+mA+ms·
As a result, the pion-momentum distributions are written as, except for the
numerical constants,
for lS capture
(23)
and
for 2P capture,
(24)
where E·C= p,2/2(mA+ mz) +q~a/2ttAs+ w,+ ms+mA-m4 -mx and 1/4nf de denotes
an average over the direction of K- nuclear capture from the. 2P Bohr orbit.
M0~{8 and M0~{p are calculable from Eq. (15) by using B~~ defined above, then
we ought to carry out the angular integration of qAs with respect to the S wave
projection of the A 0-R~ system. This may be easily done by taking the axis of
the integration over qAa as the direction of p,.. The transformation into the bodyfixed system whose z-axis is directed to p, is
(25)
where ¢Pn• f)Pn, r are the Euler angles associated with rotation to the body-fixed
coordinate system, Dmm' are the rotation matrix element/9l p,.= (sin f)P cos ((Jp,
sinfJp sin ((Jp, cos f)P) and cos fJpq=p ..:qAs/P,qAs· Then it should be admitted to replace
f dqAs by f drd cos fJpq·
Thus, we can readily perform the integration over qAs to get
for lS capture
(26)
for 2P capture,
(27)
and
where T 2'p,B, etc., have been given in Appendix B.
Then, M0~{8 and M0~{p are easily shown to have the following forms:
(28)
and
Final-State Interaction between the A0-Hyperon
515
where
T.•.t=
1
f"'D,,t(E')eia,,t sin tJ.,tTN(E') dE'
nDs,t(E)Jo
E'-E-ie
and tJ,,t are the phase shifts for singlet (S1 = 0) state and triplet (S1 = 1) state,
respectively; D,,t are given by Eqs. (13) and (14) in terms of tJ,,t.
The matrix element of spin-flip term with respect to total spin state may be
easily calculated in terms of the Wigner-Eckart theorem. Thus, if we substitute
Eqs. (28) and (29) into Eqs. (23) and (24), we obtain finally
(30)
and
(31)
where
§ 4.
Numerical results and discussion
r),.
In the last section the pion-momentum distribution
(p,.) has been g1ven
by solving the Omnes equation in terms of impulse approximation. Next, we
perform the numerical computations of Eqs. (30) and (31) by taking appropriate
parametrization of the phase shift for the final-state scattering of A0-R 8 system.
To seek the reasonable values of parameters we must scan over the wide range
of these, therefore it will be needed to transform the expressions of Eqs. (30)
and (31) into forms more suitable for computation. We now consider the effective-range theory for single-channel case, 12 ).1 3l i.e., in S wave,
q• COt tJs,t= -1/as,t+rs,tl/2.
Then, we obtain
12 l' 13l' 22 l
D
(q= lqAsl)
from both Eqs. (13) and (14)
( ) _ q+if3,,t _ -1/a.,t+r.,tl/2-iq
B,t q •
q+zar,t
rs,t ( q2 +a2B,t )/2
with r,,t(a,,t-f3s,t)/2=1 and r,,tas,tf3r,t/2= -1/as,t·
Using this parametrization, the integrations in Eqs. (30) and (31) along the
real E-axis from 0 to oo may be carried out exactly with the transformation
into momentum variable. One type of these integrations is easily performed, while
another has the typical form
J"'
-ex>
dq'
{(q'
+B}
f(q') ·In
+Ap)2
,
q' -l-ie
(q'-Ap) 2 +B
2
where A and B are the positive numerical constants and B is, in addition, to be
the complex constants; f(q) is an odd function of q. This type of integration
5I6
Y. lwamura
is also easily calculable by the method of separation of the integrand so as to
avoid the logarithmic branch-cuts. 22 l' 28l As a result, we can perform the bulky
parameter-scanning without time-losing numerical integration.
In order to see how the final-state interaction has an effect on p,..-distribution, we first evaluate TA(p,..) using only the production amplitude, in which
only Eq. (26) or (27) is in need of calculation. The results are shown in Figs.
2(a) and (b) with arbitrary normalization. The experimental histograms are
taken from Refs. 3) and 5), in particular, one of Fig. 2(a) has been converted
into the momentum distribution from energy distribution in the original report. 5l
For the case of IS capture both of the curves show large deviations from
experiment, on the other hand for the case of 2P capture, a preferable shape.
Thus it seems that the main process of K- nuclear capture is not from the IS
Bohr orbit, but preferably from the 2P one. Next, to examine the effect of finalstate scattering we compute T>..(p,..) from Eqs. (30) and (3I) by changing the
effective-range parameters. For singlet state the parameters a, and r, are in a
one-to-one correspondence through binding energy of the hypernucleus, A'He
(2.3I MeV) or A4 H (2.02 MeV), 24l and some of these are shown in Table II, while
number of events
number of events
300
K- •
-
4
20
He
K- • 4 He
rr- • 11.0
• 3He
-
rr 0 •
/\ 0 •
"H
15
10
5
(a)
(b)
Fig. 2. The pion-momentum distributions r, (p") used for the production amplitudes:
For (a) K-+'He~l!'-+AO+SHe reaction and (b) K-+ 4He~~+A 0 +3H reaction
with arbitrary normalizations.
(PV) phase space; (IS) case of IS capture; (2P) case of 2P capture.
517
Final-State Interaction between the A0-Hyperon
Table II.
For ,lHe (2.31MeV)
a,(Fm)
4.49
4.69
4.91
·5.15
5.34
5.71
r 8 (Fm)
1.80
2.00
2.20
2.40
2.60
2.80
a,(Fm)
4.69
4.88
5.08
5.31
5.55
5.82
r,(Fm)
1.80
2.00
2.20
2.40
2.60
2.80
For
,~4H
(2.02MeV)
number of events
number of events
15
10
5
(b)
Fig. 3. The pion-momentum distributions r 1 (p") for (a) K-+ 4He~rr-+AO+BHe
reaction and (b) K-+ 4He~n-0 +A0+ 8H reaction with the same normalization
as in Fig. 2 and with several sets of effective-range parameters:
(B) curves for the production processes :
(a) --{a,=5.71Fm and {at=-0.50Fm
rt=l.50Fm,
r,=2.80Fm
----{a,=5.03Fm and {at=5.03Fm
rt =2.30Fm,
r, =2.30Fm
(for the assumed binding energy 2.31 MeV)
------{a• =4.49Fm and {at= -0;50Fm
rt =1.80Fm.
r 8 =1.80Fm
{at=5.94Fm
and
(b) -{a,=5.82Fm
rt =2.80Fm,
r 8 =2.80Fm
(for the assumed binding energy 1.80 MeV)
----{a,=5.19Fm and {a,=-0.50Fm
r, =1.50Fm,
r, =2.30Fm
------{a1 =4.69Fm and {a,=4.69Fm
r,=l.SOFm.
r 1 =1.80Fm
(for the assumed binding energy 2.02MeV)
518
Y. lwamura
for triplet state, at and rt can be changed arbitrarily . As in our previous paper25l
we wish to examine the possibility of triplet bound states in addition to the spindependenc e of A0-R 8 interaction , therefore we also assume two kinds of series of
parameter s determine d by the supposed binding energies of the hypernucl eus,
,,/He (2.00 MeV and 2.31 MeV) or A'H (1.80 MeV and 2.02 MeV).
The change in the theoretica l curves depends essentially on the singlet parameters. The maximum of peak is gradually raised as the singlet effective-r ange
r, decreases, and at an acceptable value of it (e.g., about 1.5 Fm) the peak of
curve is excessivel y enhanced. Figures 3 (a) and (b) show the dependenc e of
pion-momentum distributio ns on several sets of parameter s for reactions K- +'He
~rc- + A0 + 3He and ~rc 0 + A0 + 8H, respective ly.
Here the figures are presented
only for the case of 2P capture. The appreciab le difference of triplet parameter s
does not change the shape of the curve determine d from a pair of singlet parameters. This insensitiv ity for the triplet ones comes from the small mixture of
triplet part. To see this point, let us estimate the ratio R(p,.) =21TiP,B+Tipl 2/
ITiP,B+T:pl 2 • The p,.-dependence of R(p,.) is shown in Fig. 4 with the same
parameter s as in Fig. 3 (a) and another set of triplet ones, only for K- + 4He
~rc- + A0 + 8He reaction.
We see from Fig. 4 that the triplet part is rather small for negative scattering-len gth parameter s (for triplet state) and quite small for positive ones,
9 R(P,.)
R(P,.)
400
8
7
6
5
300
I
9 R(Ptr)
8
7
6
~
4
\
3
2
\
100
MeV/c
210 220 230 240
(a)
MeVIc
210 220 230 240
(c)
Fig. 4. The ratios R(p~) of the singlet events to the triplet ones for final A0-R 3 interaction
in the reaction K-+ 4He-7x-+AD +3He:
(a) {a,=5.71Fm .
- { a1 = -0.50Fm
r, =2.80Fm'
r 1 = 1.50Fm,
at= -l.OOFm
{
---- r 1 =2.80Fm.
(b) {a,=5.03Fm .
{a 1 =5.03Fm
r,=2.30Fm '
r 1 =2.30Fm,
(for B. E. 2.31 MeV)
at=5.21Fm
---- {r1 =2.30Fm.
(for B. E. 2.00 MeV)
(c) {a,=4.49Fm ;
-{at= -0.50Fm
r 1 =1.80Fm
rt=1.80Fm,
a
1 =-0.75Fm
---- {r 1 =2.30Fm.
Final-State Interaction between the A0-Hyperon
519
further for the respective case of these, the main behavior of curves is again
determined by singlet parameters. From these facts we can say, within our
model, that in spite of the masking by the production process the effect of final
A0-R 8 rescattering is never negligible and in particular the dependence on scattering in singlet state is appreciably large, contrary to the assertion in Refs. 9)
and 11). This means that there is a good chance of extracting the info~mation
a·bout A0-R 3 singlet-state interaction from the experiment. If we consider a more
correct amplitude for the elementary process K- +N~n + A0 and moreover use
the mixed initial state, the singlet scattering would be more efficient, against the
triplet one. We think that the over-enhancement in the single-channel calculation is not necessarily troublesome, since the 2-A conversion process may largely
lower the height of the peak. 9> At any rate, if we confine ourselves only to the
production process for A0-channel, the theoretical curves may fail to reproduce
the maximum, slope and width of a peak appeared in the experiment, because
of inflexibility of production amplitude which is independent of parameters. Thus
we stress the importance of the final-state interaction for two-channel case and
then the parameters of interest must be determined by considering the absorption
into 2°-Rs channel. Our calculations, taking the 2-A conversion into account,
are now in progress by means of the coupled-channel Omnes equation and these
results will be reported in a subsequent paper.
Acknowledgements
The author is deeply indebted to Professor N. Mishima and Professor Y.
Takahashi for frequent and stimulating discussions and encouragements throughout
the course of this work .. He thanks Mr. S. Shioyama for the help in computation and the hospitality extended to him at Engineering Computation Consultant
Co., Ltd.
Appendix A
We consider following the single-channel Omnes equation
T(E) =B(E)
+ ,E
t
Ti
E-Ei
+_!_ f"'ru<E'> s'in8(E?T(E')dE',
7C
Jo
E -E-ze
(1)
where T(E) = T(E +) (see § 2).
As well as Omnes/7> we define an analytic function of the form
F(Z)
= ___!___ f"' e-iB<E'> sin 8 (E') T (E') dE' :
2ni Jo
E' -Z
(2)
then
T(E) = B(E)
+ ,E
Ti +2iF(E+ ),
• E-E,
(3)
520
Y. lwamura
where
F(E±) =lim F(E±ie).
S--+0
Now we want to manage Eq. (2) in a way slightly different from Omnes'
to keep the outlook visible. From Eqs. (2) and (3) we have
e-iB<E>F (E+)-eiB< E>F(E-)= sin!J(E)( B(E)+ L;
i
Ti ).
E-Ei
(4) ·
Let us multiply Eq. ( 4) by the function
g(E)exp (- P
n
f"' IJ(E') dE'),
Jo E' -E
where g (Z) is an entire function of Z (carrying no cut) and P means that it
takes a principal value of integratio n. We further define the function
D(Z)
=g(Z)e~u<ZJ,
(5)
where
u (Z) = _!_
n
f"' IJ(E') dE'.
Jo E' -z
Then Eq. (4) leads to
D(E+ )F(E+) -D(E- )F(E-) =D(E+ )etB<E>sin IJ(E) (B(E)
+ L;
t
Tt ).
E-Et
(6)
An analytic function of Z satisfying Eq. (6) is given by
D(Z)F(Z ) =~
2m
f"' D(E+ )etB<E> sin IJ(E) (B(E) + :EtTt/(E -Et))dE +P(Z),
Jo
E-Z
(7)
where P(Z) is an arbitrary polynomia l of Z and of course has not any cut.
The function g (Z) is an arbitrary function introduced so as to cure the
singular behavior of eu<ZJ and would be cancelled in Eq. (6), if it does not show
any singular behavior. In fact, from the Levinson's theorem
IJ(O)=nn
and
IJ(oo)=O ,
where n is the number of bound states, then
e-u(Z) ~
zn.
2:-+0
'
thus g (Z) must be determine d to cancel the zn_singula r behavior of eu<ZJ at the
origin.
Next, consider Cauchy's integral formula to a function TtD (Z) / (Z- Et), then
we have
521
Final-State Interaction between the A0-Hyperon
_ _!_
n
f"' T~.D(E' + )eiB<E'J sin o(E')dE'
(E' -E~.) (E' -Z)
Jo
r~.D
(Z)
TtD (Et)
(8)
E~.-z
Z-E~,
By using Eqs. (3) and (7) the solution of Eq. (1) can be written as
+ :E r~,
T(E) =B(E)
E-E~.
t
+
f"" D(E')etacE'J sino (E') (B(E') + :EtTt/ (E -Et) )dE'
1
nD(E)
Jo
(9 )
'
E' -E-ie
where D(E) =D(E+) and P(E) has been neglected. 18 l
Thus the insertion of Eq. (8) into (9) gives a varying solution
+ :E D(E~,) · r~,
T(E) =B(E)
t
1
+ nD(E)
D(E)
E
-E~.
i"" D(E')etacE'l sin o(E')B(E')dE'
.
(10)
E' -E-ze
o
Now, we put, in particular, 9(Z)
=IIi (1-Et/Z),
cancelling zn_singular
behavior of eu<ZJ at origin. Then D(E) takes the correct form of the Jost function in the case that bound-state poles occur13 l' 22 l and then D(E~.) =0. In this
case the solution of Eq. (10) is rewritten by
f"' D(E')etacE'l sin o(E')B(E')dE'
1
T(E) =B(E)
+ nD(E) Jo
E' -E-ie
'
which is just the same form as in the case of no-bound state.
Appendix B
The terms Tfs,B, etc., have the forms
_
T.1s,B-.
1
2
(/f. -
-
•
T 2P,B=
Q)
q
A3
2-
CAs
21
+
c- 9s}-Is
9! + -C
c- } -p· J
{29! +-9t
(p. -Q)
+
[ {2
ip
-
[p.~v],
q
+4C AsP9! +2C-P92"']fu
c-2- -(9t)2
2
[ --9192-1
-P
CAa
CAs
919a _ 9! _ c- 9a] Cis_ R')
1 [-12CAa q
2CAa p q q
(p. 2 -Q?
2
2C~a~~-Q?[(9a):fa+ (P. -Q-9s)9l"fl']- [/f.~V]
2
(for 8 1 =0)
and
. (11)
522
Y. lwamu ra
T'
_
2P,B-
c+ { 1 (gt)2
'} _..,.
(p.'-Q) 2~a-P-- 2:pq Jl
+
c+
{.. f:"}
2C,~.a(P.'-Q)' Ja- '
sea (c; Q)' +{(ga) ':fa+ (p.'+Q -ga)Yl 'fl}- [p.-v],
AsP.:pq
(for S1 =1)
C± = IJ.KNIJ.wA ± IJ.KNIJ.A.sP. ...~.
mNM.,
with
M.,= m.,(m,~.+ma)
mNm}
m.,+m,~.+ma
and
with
M= ma(m,~.+m.,)'
ma+m,~.+m,.
ft
ft(p, q) = [ {(q-C,~.apY+ /1.'}
ft
ft(p, q) =In[ { (q+C,~.ap)'+ /1.2} /
{(q+C.~..pY+ p.'} ]- 1 ,
{(q-C,~.ap)'+ /1.2} ]
and
g(=a g1 (p, q) = _ IJ.KNIJ.As/1..,,1. p'
+ IJ.KNIJ.wAq2 '
mNm,~.m.,
mNm,.
and
gs=gs(p , q) =q'+C~aP'+Q;
[p.-v] are the terms with v in replace ment of p..
References
1) Helium Bubble Chamber Collaboration Group, Nuovo Cim. 20 (1961),
724.
J. Auman et al., Proceedings of the 1962 Annual Internati onal Conference on
High-En ergy
Nuclear Physics, edited by J. Prentki (CERN, Geneva, 1962), p. 330.
3) P. A. Katz et al., Proceedings of the International Conference on Hypernu
clear Physics,
edited by A. Bodmer and L. G. Hyman (Argonne National Laboratory,
Argonne, ill.)
(1969), p. 862; Phys. Rev. D1 (1970), 1267.
4) J. G. Fetkowich, in Ref. 3), p. 451.
5) K. Bunnell et al., Phys. Rev. D2 (1970), 98.
6) J. Leitner and S. Lichtman, Nuovo Cim. 15 (1960), 719.
7) M. M. Block, Nuovo Cim. 20 (1961), 716.
8) J. Sawicki, Nuovo Cim. 33 (1964), 361.
9) P. Said and J. iawicki, Phys. Rev. 139 (1965), B991.
10) J. Sawicki, Phys. Rev. 152 (1966), 1246; NucL Phys. Bl (1967), 183.
11) J. Uretsky and K. Bunnell, Phys. Rev. D2 (1970), 119.
12) J. Gillespie, Final State Interactions (Holden-Day, Inc., 1964).
13) M. L. Goldberger and K. M. Watson, Collision Theory (John Wiley
& Sons, Inc., 1964),
2)
Final-State Interaction between the A0-Hyperon
523
p. 540.
14) I. J. R. Aitchison, Nuovo Cim. 35 (1964), 434.
15) G. R. Burleson et al., Phys. Rev. Letters 15 (1965), 70.
16) R. ]. Eden, P. V. Landshoff, D. I. Olive and ]. C. Polkinghorne, The Analytic S-Matrix
(Cambridge University Press, 1966), Chap. 4.
17) R. Omn~, Nuovo Cim. 8 (1958), 316.
18) G. Alberi and P. ]. R. Soper, Proceedings of the Topical Seminar on: INTERACTION
OF ELEMENTARY PARTICLES WITH NUCLEI (Trieste, 1970), p. 21.
19) M. E. Rose, Elementary Theory of Angular Momentum (John Wiley & Sons, Inc., 1957).
20) A. Fujii and R. E. Marshak, Nuovo Cim.' 8 (1958), 643.
21) Particle Data Group: A. Rittenberg et al., Rev. Mod. Phys. 43 (1971), No. 2, Part II.
22) R. Jost and W. Kohn, Phys. Rev. 87 (1952), 977.
23) J. B. Bronzan, Phys. Rev. 134 (1964), B687.
24) D. H. Davis and J. Sacton, in Ref. 3), p. 159.
25) Y. Takahashi et al., Nucl. Phys. B17 (1970), 472.