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A one-boson-exchange potential for ΛN, ΛΛ and ΞN systems and hypernuclei
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 642 (1998) 483-505
A one-boson-exchange potential for AN, AA and systems and hypernuclei K. Tominaga a, T. Ueda a, M. Yamaguchi a, N. Kijima a, D. Okamoto a, K. Miyagawa b, T. Y a m a d a c a Faculty of Science, Ehime University, Matsuyama, Ehime 790-8577, Japan h Faculty of Science, Okayama University of Science, Okayama 700-0005, Japan c Laboratory of Physics, Kanto Gakuin University, Yokohama 236-8501, Japan
Received l 1 May 1998; revised 26 August 1998; accepted 9 September 1998
Abstract N N OBEP due to Ueda, Riewe and Green is extended to A N and AA systems in the framework of the nonet mesons and the SU(3) invariance. The data for Ap cross sections and hypertriton are reproduced. The A N ~So phase shift is found to be more attractive than the 3S~. Furthermore, the data for the separation energies of 6aHe, la°aBeand AA t3 B a r e reasonably reproduced. No bound state exists for the AA system. The AA ~So phase shift is similar to the A N I So. The AA total cross section is predicted. ~ N phase shifts are also predicted. (~ 1998 Elsevier Science B.V. PACS: 13.75.Ev; 21.80.+a; 13.75.Cs; 27.20.+n Keywords: A - N OBEP; A - A OBEP; ~ - N OBEP; Hypertriton; Double A hypernuclei
1. I n t r o d u c t i o n In recent years various theoretical approaches to hyperon-nucleon and h y p e r o n hyperon interactions have been presented. The Nijmegen group has presented OBE models with soft or hard cores [ 1 ]. The Jiilich group has presented an OBE model combined with the two-meson-exchange contribution due to the isobar degree of freedom [2]. The N i i g a t a - K y o t o group has proposed a hybrid model where pseudoscalar and scalar meson exchanges are taken into account in hybrid with the quark exchange for short distance [ 3 ] . However, none of these attempts is fully consistent with the experimental facts concerning the A N scattering and the single and double A hypernuclei. Historically, the one-boson-exchange potential ( O B E P ) for the N N interactions has been advanced by several theoretical groups independently since 1962, including one of 0375-9474/98/$ - see front matter (~) 1998 Elsevier Science B.V. All rights reserved. PII S0375-9474(98) 00485-0
484
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
the present authors (T.U.) and Bryan [4-7]. A realistic N N OBEP was first proposed by Ueda and Green [7]. This potential involves 7r, r/, p, to, t3, S* and o- exchanges with the vertex form factors which are introduced to represent the nucleon structure phenomenologically and describe the scattering data below EL ~< 330 MeV as well as the deuteron data. The vertex form factor regularizes the Yukawa potential. This model employs no additional phenomenological core. The short-range repulsion due to the to and due to the velocity-dependent terms from the vector and scalar mesons provides sufficient and necessary repulsion for a realistic potential. In this paper we extend the potential to A N and A A systems, l SU(3) invariance was first applied to hyperon-nucleon systems by the Nijmegen group [ 1 ]. Here, an SU(3) nonet meson scheme is considered for pseudoscalar, vector and scalar mesons: (7r, r/, K, ~ ' ) , (p, to, K*, ~b) and (t~, S*, K~ (1429) f0( 1581 ) ), respectively. Additionally, the isosinglet scalar meson tr is taken into account which represents the I = 0, J e = 0 + part of the correlated and uncorrelated 27r exchanges. A part of the tr exchanges between the A N and A A effectively represents the 27r exchange contributions from the A N - ~ N and A A - , ~ channel couplings, since the channel couplings are not explicitly treated here. This model is rather simple, compared with the theoretical models in recent years which make attempts at a microscopic description. We aim to build an effective model to be useful for studies of hypernuclei and YN interactions. We allow that the scalar and vector meson exchanges in our model involve effectively correlated and uncorrelated two octet-meson exchanges. The uncertainties arising from this treatment are removed in combined analyses with hypertriton and double A hypernuclei. As a result, our model is consistent with the data on A N scattering, hypertriton and double A hypernuclei of Ane, AABe l0 13 B. It is remarkable in comparison with other theoretical models that and aa the data on these double A hypernuclei are reproduced. With this model we demonstrate the A N as well as A A scattering phase shifts. Furthermore, the AA total cross section is predicted. No A A bound state is found. An advantage of the simplicity of this model is in application to hypernuclei with baryon number greater than three. Actually, it is quite remarkable that an application of our A N potential to four body systems has very recently provided a useful result about the binding energies of ~4He and A4H [9]. Furthermore, to check the predictive power of the present model in a future experiment, we display a result of potentials and phase shifts for the ~ N scattering. In Section 2 an overview of the N N interaction is presented. In Section 3 the potentials for N N , A N and A A systems are formulated. In Section 4 nonet OBEP parameters for N N , A N and A A systems are described. In Section 5 numerical results and a discussion on A N scattering, hypertriton, AA scattering, double A hypernuclei and potential structures are presented. In Section 6 ~ N OBEP and phase shifts are displayed. Section 7 is devoted to conclusions.
I A part of this paper was published as a brief letter [8].
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
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2. An overview of the N N interactions Traditionally, the two-nucleon interaction has been elucidated from outside of the force range. In a meson theoretical approach the two-nucleon interaction at approximately r ~> 1 fm has treated theoretically, while that at r < 1 fm phenomenologically. This approach is nowadays very refined. The one-boson-exchange potential (OBEP) describes the repulsive core at short distance by the heavy-meson-exchange contributions with the meson-baryon form factors. The contributions produce both static and velocity-dependent repulsion [7]. The mesonbaryon form factors are phenomenological. Recently, however, the pion-baryon form factor has been substantially understood [ I0]. On the other hand, it has been attempted to explain the repulsive core in terms of the quark degree of freedom based on the model QCD. This picture predicts the strong attraction generating the H dibaryon. However, this has not been found so far in spite of a great deal of experimental effort. Furthermore, there is the problem that no complete theory exists which unites the meson theory with the quark degree of freedom. Even the pion mass has not yet been explained in terms of QCD. The two-nucleon interaction at long distance r > 2 fm is believed to be well established. Recently, however, it has been clarified that the pion coupling constant is ambiguous. At the Blaubeuren conference the value of the pion coupling constant was controversially reported to be in the range 14.62-13.47 [11 ]. Therefore, we may conclude that the theory for the two-nucleon interaction is not at all well established in the full range: from the outside through the inside. It is dangerous if anyone should believe that the two-nucleon potential is now established and a certain potential should be uniquely qualified as the best. At the present stage a variety of approaches to the two-nucleon interaction should be attempted. So far one-boson-exchange potentials have been used for a model of the two-nucleon interaction. Among them we adopt the Ueda-Riewe-Green OBEP (OBEP-URG) in this paper [7]. This potential has nice features as follows. The model is constructed in the r space by involving the one-boson-exchange contributions from the mesons of 7r, ~r, p, w, r/, ~ and S* with the form factors at the meson-nucleon vertex. This potential describes very well the two-nucleon interaction below 350 MeV of incident proton laboratory energy. It is nice that, once the model is made by fixing the nine parameters in fitting the two-nucleon data, the model can be extended to hyperon and nucleon as well as hyperon and hyperon systems by SU(3) transformation without additional parameters such as the hard or soft core radius. This is an advantage of the OBEP-URG over some other models where the radius is adjusted as an intrinsic parameter to a hyperon and nucleon or a hyperon and hyperon system. Furthermore, it is nice that the model gives the short range interactions by the heavy mesons with form factors. This produces the effective repulsion due primarily to o3, incorporated by the velocity-dependent repulsion which is automatically generated in the model.
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K. Tominaga et a l . / N u c l e a r Physics A 642 (1998) 4 8 3 - 5 0 5
3. Equations and one-boson-exchange potentials 3.1. OBEP-URG for N N systems
The OBEP-URG involves the ingredients of pseudoscalar mesons 7r, r/, vector mesons w, p, and scalar mesons o-, S* and ~ with the vertex form factors [7]. The parameters of the potential are employed from model II of the OBEP-URG. The potential function is regularized with the vertex form factor in momentum space A2 F ( k 2) - k2 + A 2 ,
(1)
where k is the momentum transfer and A is a parameter. The regularized Yukawa function is Y(r)
=
( A 2 "~21 { \ a 2 - IzzJ -r e - I z r
( -- e -Ar
1+
A2-br2 ) }
2-----~-r
.
(2)
The total potential takes the form
Vtot(r, ~72)
= V(r)
-
~
1
[~2~b(r) ÷ q~(r)V2],
(3)
where the reduced mass Mr is given by (4)
Mr = I M u ,
and the V in Eq. (3) operates on all functions to the right. Furthermore, V ( r ) in Eq. (3) is given by V ( r ) = Vc(r) + V~(r)o'l • 0-z + VLs(r)L. S + Vr(r)S~2,
(5)
where L . S and $12 are given by L.S=IL2
. (oq ÷ 0"2) ,
S12 = 3 (0"1 " r)r2(o'2- r) - 0-1 "0-2.
(6)
Let us describe our treatment of the V z term in Eq. (3) for the spin-uncoupled case, for example. The Schr6dinger equation is written as
V2
p2
- 2M----~O+ Vtot(r, V 2 ) ,/// = ~ r
,t~,
(7)
where p is the magnitude of the center of mass momentum. The derivative part of the potential in Eq. (3) is eliminated by a transformation of the radial part of the wave function
u(r)~ ff ue(r)-
lv/i--T-~( ~ ,
(8)
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Table 1 The isospin factors m u l t i p l i e d by the potentials
I = 1, S = O
Potential 7r, p, d
~7, rlt,to, qS, S*, fo, o-
N N ~ N N ( I = O) N N --* N N ( I = 1) AN~ A N ( I = 1)
-3
AA ---+ A A ( I = O) ~ N --* ~ N ( I = O)
0 --3
~N --. ~N(1 = l)
I
I=O,S=O
1= ½, S = ± I
K,K*,K¢~
1 0
where g is the orbital angular momentum and
~9(r) = ~-~ U~(rr) yeo(O,¢ ) . The equation is rewritten for the radial part of the new wave function u~ff(r) as
d 2 ueffrr ) dr 2 g ~
g ( g + 1) utr)e" ,eff t' { - 2 M r V e f f ( r , p 2)
+pZ}u(r)~
ff =
0,
(9)
r2
where v~ff(r, p2) is the effective potential for an angular momentum eigenstate a defined by
V~(r) veff(r'p2) - 1 + ¢ ( r )
1 (~b,(r) /2 p2 ~b(r) 8Mr , ~ + ¢1( r ) + 2M----~1¢ (+r - - - - ~ "
(10)
The V,~(r) on the right-hand side of Eq. (10) is the original potential V ( r ) of Eq. (5) for the eigenstate or. The spin-coupled cases are treated similarly. The interaction Hamiltonian densities are assumed except for the isospin factors as follows. (a) For the N N pseudoscalar meson vertex
Hvs = igps~Ys~Pq~.
( 11 )
(b) For the N N scalar meson vertex H, = g.,.~C,q~.
(12)
(c) For the N N vector meson vertex
H,, = ig,,~y~Abfb ~ + ~f~, - - ~ 9-o ' u ~ ( O~(b~ - O~OSu ) ,
(13)
o'~,~
(14)
with =
[y~,,7~]/2i.
.Ad in Eq. (13) is a scaling mass. This is taken as the nucleon mass .A//= Mjv
(15)
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488
Table 2 The mass and the form factor parameters Kind
Mass (MeV)
A (MeV)
~t S K~ S* fo p K*
138.7 493.6 548.7 957.5 970.0 1429.0 993.0 1581.0 759.1 891.8 782.8 1019.4 484.3
2012.00 1129.47 1129.47 1129.47 1129.47 1629.00 1129.47 1798.30 1000.00 1129.47 1129.47 1129.47 1129.47
N A
938.5 1115.6
.A4
938.5
K
Table 3 The SU(3) parameters. Three sets of parameters (sets 2, A and B) are considered for vector mesons. Each set shares the parameter values for pseudoscalar and scalar mesons in common. The ps and sc represent the coupling of pseudoscalar and scalar mesons, respectively. The vc-g and vc-f represent the vector and tensor couplings of the vector mesons, respectively Set 2
0
gl/v/-~ gs/v/-4-~
ps
sc
-10.41 0.4338 3.7736 0.5211
55.59 1.4715 1.6248 0.8225
for all the c o u p l i n g s o f N N - ,
Set A
Set B
vc-g
vc-f
vc-g
vc-f
vc-g
vc-f
24.84 2.8783 0.8426 0.9347
24.84 0.0 4.1625 0.2500
26.17 2.8466 0.8426 0.9689
26.17 0.0 4.1625 0.2500
25.62 2.8599 0.8426 0.9548
25.62 0.0 4.1625 0.2500
N A - a n d A A - m e s o n in c o m m o n . Hv in Eq. ( 1 3 ) is
r e w r i t t e n as
Hv = iGv~y#¢d?~ + i F v { ~ ( 0 ~ )
- (0~)~O}~b~,
(16)
where
Gv = g,, + -MN - ~ fj.v ,
F~, = f v / 2 . A 4 ,
(17)
T h e n w e m a k e a p p r o x i m a t i o n ( A 1 ) as follows: -
( A 1) M a k e t h e e x p a n s i o n o f the p o t e n t i a l in t e r m s o f p / M ~ a n d take the t e r m s o f the o r d e r ( p / M u ) 2, w h e r e p is the c e n t e r o f m a s s m o m e n t u m in C M S . All the o t h e r h i g h e r o r d e r t e r m s are i g n o r e d . T h e V ( k ) in m o m e n t u m s p a c e is F o u r i e r - t r a n s f o r m e d into V ( r ) in r s p a c e b y
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
V(r) =
V(k)eik'r(27r)3.
489
(18)
Each component of the OBEP for the pseudoscalar, scalar and vector meson exchanges are
g2 [
~cl(r)=~
ZSI2
Y(2)O'l • o'2 +
Vpsd(r) -_ ~gps
-Y+2
47r
~
1y(2)
(19)
1 2M 2 ( ~ 2 Y + Y ~ 7 2 ) +
l+--~-~,,j
42r2
Y(')L . S ] ,
(20)
~ (vzY+Yv2) Mu fl,~ 2
2
× A1--~2ZS12+ ~g~ 3 + 4 - -,~M- u~ ft ' ) Y ( I ~ L . S .
(21)
In these equations
(
) (
)
e-mr
Y=\m~_m2 j \m~_m2 j Z O l i T
,
(22)
i=1,3 1
y(l~
1 dY
_ 2M~ r dr' (
y(2) =
1
2M 2 (vZY},
m2 "~ { 2m2 2 "~i~13(1~5 + -mi m_~_~)e -mir , - + cei \m 3-m lj.= r r
Z=\m~_m~/
(23) (24)
with eel= 1
ce2-
m2 - m~ m2_m2 ~
m22 - m~ ~ 3 - m32_m22 ~
(25)
where ml is a meson mass tz and the limit is taken as m2 ~ A and m3 ~ A. This result is given by Eq. (2) for Y in Eq. (22).
3.2. AN OBEP The Schr6dinger equation for the AN system is the same as those given by Eqs. ( 1 )-(10) except that the reduced mass is now given by Mr = . MNMA MA"
(26)
MN ÷
The Hamiltonian densities for the AA- and AN-meson vertexes are assumed to be the same as Eqs. ( 11 )-(17) except for the isospin factors and the first equation in Eq. (17). This is replaced by
G,. = g,, +
MI + M 2
-2---M J'''
(27)
K. Tominagaet al./Nuclear PhysicsA 642 (1998) 483-505
490
where (Mz, M2) = (MA, MA) and (Ma, MN) for AA- and AN-meson vertexes, respectively. We made approximation (A1) as described in the last subsection. Furthermore, we make approximation (A2) as follows. (A2) The retardation effect in the meson propagators with S 4= 0 is approximately taken into account in the denominator as follows: -
k 2 ÷ ~2 ~ k 2 + ~2,
where ~2 = i22 _
(28)
( M A -- M N ) 2.
Each component of the AN potential is derived as follows. For the exchange of scalar mesons o-, S* and K~
V~s*(r) = ~glg2 { - Y +
q- 8M-~ ) V2Y - (4M--~ + 4MA 2 )
x ( v Z Y + Y V 2)+
Vx~ (r) = glg2 ~
~
+~
r dr
{-Y + ,
,
4MNMA V 2 Y 1
J'
(29)
+ 8 ,7)
+ 4MNM-----~ 1 dY L S'~
× ( v Z Y + Y V 2 ) + 2MNMA r ~rr " J"
(30)
For the exchange of pseudoscalar mesons r/and K
gig2 VmK(r) = ~
{ 12MNMA 1 V2yo'l-o'2 + - - Z S' 1 2 4MNMA
}•
(31)
For the exchange of vector meson w
V~o(r)=(G1 - 2G2 - 2G3 + 4G4)Y
+{
(8__~u2
xG2
+( +
+
__1
+ 4MNMA +
4MN z 1
(
1
1
2MN 2
1 4h~N 2
1
2MNMA
1 )G4}V2 Y
2MN --------~ + MNM-----~A 2MA 2 1 1 1 2MNM------~AGl + MNMA G2 + --G3MNMa
-I- 2~N2
(
G1 +
2MNM-~ + 4
x(V2y+ yv2) +
+
8_~A2)
1
MNMA
l
MNMA
+ MNMA +
21~A2 G2
-}-M~)G 3 2
C4) G~
1 )
4MA 2
491
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505 1 M N2
4-
G4
1 M A2
1
r -~r " 1
-~ - Gl V2YO.I"o'2 6MNMA
(32)
- G1ZSI2, 4MNMA
where glg2 (
G,=--~-
MA f l )
(
1+-~--~-l
gig2 (
MN f2~ I +-~--72 / ,
MA fl ) MN f2
G =-fi7
7
M 2g2'
1 4- M u f 2 ~
glg2 MA fl
03- 4rr .Ad 2gl
--~-~2//
glg2 MA f l MN f2 G4 = - - - -
4rr .A4 2gl .A// 2g2" For the exchange of vector meson K* VK. (r) = (G'1 - G~2- G~ + G~4)Y +
1 _ _ l + G1 { (8__.M~N2 1 G'2 + 4MNMa 8~'I1_2) ' 4MzvMA 1 , ( 1 1 )G~}V2 Y
4MNM~A G3 + {
(MN 4- MA) 2
1 , ( 1 2MNMa Gl -
4(1
8a,IN2
3
t
4-
4-
where
1 G1
__1 4+ 2MNMA
' _ _ l 2MNMA G3
_ _1 G,4 } 1-dYL's 2MNMA . r dr
(
1
1
1 )G~} ( V 2y + Y V 2) 8MA2
8MN '''''~ 4- 4MNM~A
2MNMA
1 )GtlV2yo.l.O. 2
12MNMa + 8zVl (MA MN) 2 ( G', C 2 G; 4- G~ ) ~ Y tz 2 8/~N 2
-
+
__3 ~A2), 4MNMA + 8 G2
I___~)G , 3
8a/TN2 4MNMA 4- 8 ( 2 1 (MN 4- MA 2
4MNMA
1 1
+ 4MNMA
-
-
4- ~
2
- -1 4MNMA
G'I ZSj 2
_ _ V2y 4MNMA
,) (V2Y4- YV2)
1
!dYL.s
2MNMA r dr
J' (33)
K. Tominagaet al./Nuclear PhysicsA 642 (1998) 483-505
492
Gtl
gig2 ( M A + M N f , ) 1+ 2.M
= ~
(I+MA+MNf2) 2.M g2
gl
g,g2 ( MA+MNf,) G~ = ~ 1+ 2A,~ gl ,
'
MNWMAf2 2./91 g2'
glg2MN+Mafl (1+MA+MNf~2) 2M gl 2A4
G3 = 4~r
'
G'4 = glgz fl f2 ( MN + MA) z 47r gl g2 (2-M) 2
For the exchange of the mesons with S 4:0 the following exchange operator must be multiplied: P = -PxPa= _(_)L(_)S+I,
(34)
where Px is the space exchange operator and P,~ is the spin exchange operator. For l E, 10, 3E and 30 states, P takes the eigenvalue as follows: P= ~ +
/
(IE, 30),
--
(lo,
(35)
3E).
The total one-boson-exchange potential is given by the sum of the velocity-dependent and -independent terms as in Eq. (3), where Mr is given by Eq. (26) and & is given by (36)
= ~,~ + ~s. + P~OK¢+ ~o~ + P~K*,
where
(37)
~Pms* = 4Mr (4-~N2 + 4--~AZ) Y' ~OK~= 4Mr
(
1
__ 1
+ 4MAMN +
~p,,,=4Mr 2M-~MAGI --G2MNMA q~x* =4Mr +
2MuMA G1 +
(1
81('1N2
(
+
8~tN 2
4MNMA +
2 (MN + MA) 2 (Ma - MN) 2
~
+ 4MNMA + 8
3+
4MlvMA +
G4 Y,
(39)
G'z
G3
1 1 ~8MN + 4MNMA
__l ×
(38)
Y'
,_j)},.
1 ) 8/~A 2
, G4
(40)
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
493
3.3. AA OBEP The AA OBEP is given by Eqs. ( 1 9 ) - ( 2 5 ) , where Mu is replaced by MA: MN --+ MA.
(41)
3.4. lsospin factors The isospin factors in Table 1 are multiplied by the potentials. These factors are due to the SU(3) invariant Hamiltonians presented in Appendix A.
4. A nonet OBEP model for NN, A N and A A systems In this paper an SU(3) nonet meson scheme is adopted for pseudoscalar, vector and scalar mesons: (1r, Tl, K,~Tt), (p,~o,K*,~b) and (6, S*,K~(1429) fo(1581) ), respectively. Additionally, the isosinglet scalar meson o- is taken into account which represents the I = 0, j e = 0 + part of the correlated and uncorrelated 27r exchange contributions as well as a part of the iterated contributions from the AN-Y,N and AA-,~2f channel couplings. For the scalar mesons, K~ and fo, the masses are assumed to be 1429 and 1581 MeV, respectively, since there is experimental evidence for these values. The form factor masses are taken as the same as those for the OBEP-URG, apart from those of K~ (1429) and f0 ( 1581 ) which do not appear in OBEP-URG. For these scalar mesons, K~ and f0, the ratios of the form factor mass A to the meson mass m are fixed as A ( f o ) / m ( f o ) = A(K~)/m(K~) = A(S*)/m(S*). The masses and the values of A are summarized in Table 2. The SU(3) parameters are the following four for each set of nonet mesons: the coupling constants of the octet and singlet mesons, g8 and gl, respectively, the parameter ce for the F and D couplings, a = F/(F + D), and the mixing parameter 0 between the octet and singlet mesons. Let us now show how to derive the coupling constants for the nonet mesons. Model II by Ueda, Riewe and Green is employed for this purpose [7]. First, for pseudoscalar mesons we use the coupling constants determined in the NN analysis: gmv~, gNN~ and gNNn'. The mixing parameter 0 for the pseudoscalar mesons is determined by the formulatan 20 = (M 2 _ mn) 2 / (m,7, 2 - M8z), where the isosinglet octet-meson mass M8 is given by the Gellmann-Okubo mass formula M~ = ( 1 / 3 ) ( 4 m ~ - m2) and mn and m,, indicate the observed masses of r/ and r/~, respectively. Now the SU(3) relation provides the formula which determines the SU(3) parameters, gs, gs and or, as follows: gNN~r = g8, gNNn=COSO[---~(4ot--1)g81
-- sin 0gl,
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
494
gNN~,=sinO[---~/=3(4a--1)g8]+cosOg,.
(42)
Then we obtain all the other coupling constants by virtue of the SU(3) invariance. We show only the parts related to the couplings with NN, AN and AA. The remaining parts are shown in Appendix A 1
gNAK
=
v/~g8(1 + 2 a ) ,
gAAn=CosO
- - - ~ ( 1 -- a)g8
gAAn,=sinO[----~3(1--a)g8 ]
-- sin0gl, + COS0g|.
(43)
For the vector and scalar mesons the coupling constants are treated similarly as the pseudoscalar mesons except for the mixing parameters. For those mesons the mixing parameters are made adjustable for the following reasons. Let us consider the effect of the correlated and uncorrelated exchange of the ~- and p mesons which has totally I = 0 and jR = 1-, namely the same quantum number as w. More generally, we may consider the effects of the correlated and uncorrelated exchanges of the two octet mesons which have totally the quantum number jR = 1- They produce additional SU(3) components by: 8 x 8 = 1 + 8 s + 8 a + 10+ 10+27. The effect produces a shift of the mixing parameter of the vector mesons 0v from the value determined by the mass formula, Ov = 38.56 °, to some other value. Also for the scalar mesons, we make the mixing parameter of the scalar meson Os adjustable for the same reason as for the vector meson case. Behind this treatment we interpret the mesons as involving the ingredients of not only (qcl), but also (qglqgl). The latter ingredient is one of the origins producing a shift of the mixing parameter from the value of the mass formula. Concerning the coupling constants of o- with N and A, gNN~ and gAA,~, respectively, we allow them to be independent. The gNN,r is determined from the NN analysis. We now have three adjustable parameters: Or, Os and gAA,~.The three parameters are determined by fitting the Ap total cross sections [ 12] at PL = 150-400 MeV/c and the binding energy of the hypertriton and the separation energy of the double A hypernuclei 10 of AABe. The values of the SU(3) parameters and the coupling constants thus determined are summarized in Tables 3 and 4, respectively. We find that gAAc~is a little larger than gNN,r: gAAo-----1.88 and gNN~ = 1.78. This is interpreted as follows, gAA~ involves the effect of the I = O, jR = 0 + part of the uncorrelated 2~" exchange due to AN-~N or AA-NN channel coupling, while gNN~ does not involve the corresponding 2~" exchange, namely double iteration of the one-pion-exchange.
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
495
Table 4 Baryon-baryon-meson coupling constants. The coupling constants of sets A and B are shown for vector mesons. Each set shares the parameter values for pseudoscalar and scalar mesons in common. The vc-g and vc-f represent the vector and tensor couplings of the vector mesons, respectively. The values for g / x / - ~ and f / x/'4~ are shown Set A ps
sc
vc-g
Set B vc-f
vc-g
vc-f
NNTr 3.7736 [71 N N t 3 1 . 6 2 4 8 [71NNp 0.8426 17l 4.1625 [71 0.8426 171 4.1625 171 NNr/ 2.4021 [71NNS* 2.6038 [7] NNw 3.1718 171 0 171 3.1718 171 0 171 NN~' 0 [7] NNfo 0 [7] NNq~ 0 171 0 17] 0 [71 0 171 NAK -4.4493 NAK~ -2.4812 NAK* -1.4292 -3.6048 -1.4155 -3.6048 AA~7 - 1.9741 AAS* 0.5568 AAw 2.5414 - 1.5899 2.5597 - 1.5587 AA~ / 0.8036 AAfo --1.4022 AAq5 --1.2827 --3.2353 --1.2763 -3.2504 NNo" 1.7776 [71 AAtr 1.8787
5. Numerical results and discussion
5.1. AN scattering With these parameter values all the experimental values for Ap scattering cross sections which are presently available are reproduced by the potential as is shown in Fig. 1. Other observables and phase parameters are displayed in Figs. 2-4. In the low energy limit we find a considerable difference of our result from that of the Nijmegen soft-core model [1] for o-(3S1): our value of 358 mb is compared with Nijmegen's value of 160 mb, with a similar result for o'(ISo): our value of 231 mb is compared with Nijmegen's value of 190 mb (see Table 5). About the AN spin-spin interaction the splitting energy of the JP = 0 ÷ and 1+ states for the 4 H and 4He hypernuclei gives the best experimental information. Hiyama et al.
Elab(MeV)
Elab(MeV) 50 .
.
.
.
i
,
'
'
•
100
150
!
i
. . . .
50 ,
100 ,
,
,
,
,
150 . . . .
600
400
2O0
+ i
0
|
i
200
400
Plab(MeV/c)
I
600
200
i
I
400
P,ab(MeV/e)
i
,
600
Fig. 1. The cross sections (left) and the ratios of the forward to backward differential cross sections (right) for AN scattering. The theoretical result with parameter set B is compared with the data.
496
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
0.2
30
- - Plab=300MeV/c ...... Plab=400MeV/c
0.15
/ ...."x\ ...
20 0. l
- - Plab= 100MeV/c . . . . ptab=400MeV/c
lO
0.05 0
....
" V ' 7 - - - 7 " , ,'-r-'~. ' - r " 7 i , 50 100 150 0(deg)
0
0
,
0
50
100 O(deg)
150
Fig. 2. The differential cross sections at P£ = 100 and 400 MeV/c (left) and polarizations at PL = 300 and 400 MeV/c (right) for A N scattering. The theoretical result with parameter set B is used.
10
'
'
'
'
l
'
'
'
'
l
'
,
,
.
w
.
30
20
11
tpt -10 ,
.
,
.
I
0
50
,
,
,
,
I
.
I
I
I
100
I
Elab(MeV)
~
~
...... 3po . . . . 3p2 i
I
150
3pt
i
i
l
l
0
l
l
l
50
l
l
l
l
l
l
l
100
150
Elab(MeV)
Fig. 3. The S-wave phase shifts d(1So) and d(3Sl) (left) and the P-wave phase shifts d ( I p l ) , d(3po), d(3pl) and d(3p2) (right) for A N scattering. Parameter set B is used.
6
,
,
•
•
i
,
•
,
•
,
,
,
,
•
,
/
7" 4
,
°`
7 "
'
.
•
•
•
I
'
,
,
•
!
,
i
I
t
,
,
,
I
'
4
2
7" 3D
7..7
~ 2
/
0
0
"
~ .
0
' ,
~
' .
~ ,
"
' i
50
~ t
-
' ,
2
~ ,
,
I
100
Elab(MeV)
|
,
•
,
I
,
150
Fig. 4. The D-wave phase shifts d(3Dl), d(3D2) and ~3 (right) for A N scattering. Parameter set B is used.
-2
t
0
,
,
,
I
50
t
i
i
100
Elab(MeV)
i
|
,
I
150
d(3D3) (left) and the mixing parameters El, 62 and
497
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505 i
30
.
.
.
.
i
.
,
•
.
i
•
/""""--.~
~20
--....
!
,
,
i
'
'
'
'
i
i
I
I
,
I
.
,
•
,
I
.
.
.
.
I
.
.
.
.
I
,
I
,
30
I-~ ""-~....
I
,
. . . . 3S l
! lO
10
0
0
50 100 Ei.b(MeV)
150
"
" I
0
.
.
.
50 100 Elab(MeV)
150
Fig. 5. The AN phase shifts d(ISo) and d(3Sl) in the case of hypertriton unbound (left) and overbinding with binding energy 2.43 MeV (right).
have reported that our A N potential gives about 0.7 MeV for the splitting energy: about 7/10 of the experimental value, 1.04 MeV, is reproduced. This is compared with the fact that existing standard potentials have so far provided much worse results [9].
5.2. Hypertriton Furthermore, we find that the binding energy for the hypertriton is in fair agreement with the experimental value as shown in Table 5: our theoretical value B(~tH) = 2.35 MeV due to the Faddeev formalism [ 13] is compared with the experimental value of 2.355 ± 0.05 MeV [ 14] (see Table 5). The calculation of the hypertriton binding energy is performed for the 1S0 and 3S1-3DI states of both the A N and N N subsystems. For the N N system the Paris potential is employed. The contribution from the partial waves higher than these is ignored. This contribution is thought to be less than about 0.05 MeV. The value of d(tS0) is found to be larger than that of d(3Sl) as shown in Fig. 2. If the value of d(lS0) is much larger than that of d(3S1 ) as shown in Fig. 5, while the Ap total cross sections are reproduced equally well, then the hypertriton is overbinding. On the other hand, if the value of d(ISo) is smaller than that of d(3Sl) as shown in Fig. 5, while the Ap total cross sections are reproduced equally well, then the hypertriton is unbound. It should be noted that the contributions of the AN l So and 3S1 states to the hypertriton binding energy are remarkably different from those to the AN scattering cross section. In the former the ANISo and 3S1 interactions in the A N N system work both in the A rearrangement three-body process and in the A N interaction with a remaining nucleon acting as a spectator, while in the latter the A rearrangement process does not exist. Therefore, in order to determine the relative strength of the ANISo and 3S1 contributions, it is significant to perform an analysis simultaneously of both the hypertriton binding energy and the A N scattering cross section.
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
498
Table 5 Binding energy of hypertriton, scattering lengths, effective ranges and cross sections at EL = 0. The numerical values of the scattering parameters for the Nijmegen soft core model [ 1] were derived by the present authors
Experiment Theory (set 2) Theory (set A) Theory (set B) Nijmegen SC
B(~IH) (MeV)
a(IS0) (fm)
r(IS0) (fm)
a(3Si) (fm)
r(3SI) (fm)
o-(Is0) (mb)
o-(3Si) (mb)
2.355 :]: 0.05 2.32 2.38 2.35 2.36 [13]
-2.65 -2.76 -2.71 -2.48
3.24 3.19 3.21 3.03
-1.80 -2.064 - 1.95 -1.32
3.71 3.46 3.56 3.26
221 239 231 190
305 400 358 160
Ejab(MeV) 10
20
i
i
30 40 50 60 i
"
i
"
i
•
!
i
i
800 _
_
.
3p
_
1
..... 3po .... 3p2
4
600
IS 0
_
400 c~
200 0
,
0
,
,
,
i
,
100
,
,
,
!
200
,
,
Plab(MeV/e)
,
i
I
i
300
i
i
0
i
|
20
i
~
Elab(MeV)
/
40
i
60
Fig. 6. The cross sections (left) and the ]So- and 3p-wave phase shifts (fight) for AA scattering. Parameter set B is used
5.3. A A scattering The cross section for the A A scattering is predicted as shown in Fig. 6. We find no A A b o u n d state. The d ( I S o ) , d ( 3 e 0 ) , d ( 3 p ] ) and d ( a P 2 ) for A A scattering are shown in Fig. 6. The A A d ( l S 0 ) is approximately similar to the A N d ( I S 0 ) at EL < 50 MeV. However, the A A d ( I S 0 ) is a little more attractive than the A N d(1S0). A l t h o u g h A A - _ ~ N c h a n n e l c o u p l i n g is not explicit in this calculation, we note that the contribution o f the uncorrelated exchange of K K due to the channel c o u p l i n g is represented effectively by the S* and fo exchange with the m i x i n g parameter Os adjusted.
5.4. Double A hypernuclei The difference ABAA b e t w e e n the two-A separation energy of double A hypernuclei BAA(AA Z ) and twice the o n e - A separation energy o f single A hypernuclei BA(aA--IZ)
ABAA = BAA( AA Z ) -- 2BA(Aa-1Z)
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
499
Table 6 The difference ABAA of the double A hypemuclei 6aHe, ~(liBe and AA J3 B, and the scattering lengths a(IS0) and the effective ranges r(ISo) for AA scattering Parameter set Set 2 Set A Set B Experiment
6aHe (MeV)
A IOABe ( MeV )
li~tB ( MeV )
a(ISo) (fin)
r(ISo)
3.43 3.72 3.60 4.7 [16]
3.95 4.28 4.14 4.3 [17]
4.70 5.12 4.94 4.9 [18]
-3.09 -3.64 -3.40
2.89 2.73 2.79
(fm)
is apparently sensitive to the AA potential. The ABAA'S for 6AHe, l ° B e and ]~4B are calculated by use of the present AA potential in a c o r e + A + A model [ 15]. The resulting theoretical values are in reasonable agreement with experimental data [ 16-18] as shown in Table 6. In the calculation the core-A potential is used which is made by folding the YNG-NS potential [ 19]. This is made from the Nijmegen soft-core potential to represent the effective A N interaction in the A nuclear matter. Its parameter is adjusted to reproduce the binding energy of the ground state of the core-A system.
5.5. Potential structure The structures of the A N potential are presented in this subsection. The total effective potential defined by Eqs. (3), (5) and (10) is divided into the two parts: one works for the singlet even (1E) and triplet odd (30) orbital-angular-momentum states with P = +1 in Eq. (34) and the other for the singlet odd (10) and triplet even (3E) orbital-angular-momentum states with P = - 1 . Furthermore, according to Eq. (5), each part is divided into the central, spin-spin, tensor, LS and V,~ parts as follows:
veff(r, p2) = v c e f f ( r ) + V J f ( r ) t r l . o ' z + V ~ eft . ( r ) S I 2 + V Left s ( r ) L ' S + V c b ( r , p 2 ), (44) where 1
[ v~e"(r)' k J f ( r ) ' v~rff(r)' V~Lsff(r)] - 1 + ~b(r~ [Vc(r), Vc~(r), Vr(r), VLs(r) ] (45) and
V4~(r'p2) -
1 (~,(r) /2 p2 ~b(r) 8Mr \ /+ 1~b(r) + 2M~r 1 + (b(r)"
(46)
Then the potential working for an angular momentum state a
veff(r,p 2) = [ Wcff(r) + veff(r)trl .o'2 + V-;ff(r)Sl2 +V~Lffs(r)L • S + Vo(r,p 2) l,~
(47)
is divided into the sum of each meson contributions rce~" and V~, where i identifies each meson, as follows:
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
500 100
100
50
50
0
/'° veff(r,p=0)
-50 -100
....
' .... 0.5
0
t .... 1 r(fm)
, .... 1.5
-50
2
-100
/ Ko
../,,. 0
0.5
//
e,itr)
X .......... 1 r(fm)
1.5
2
Fig. 7. The effective potential vfff(r, p = 0) and its meson components V~a~.f(r)for the AN iSo state.
" I ..........
100
"V " V o
.
.
.
.
.
.
.
.
.
.
50
"--4
-50 -100
/"
F --50 ~" / f f / 0
veff(r,p=0)
.
.
.
.
'
0
.
.
.
.
'
0.5
1
....
r(fm)
100 f . J . . . . . . 0 0.5
t , , ~ ,
1.5
O /" //"
/'/
" ,treffcv.t va,it~ j
/," . . . . . . . . . . 1 1.5 r(fm)
Fig. 8. The effective potential vfff(r, p = 0) and its meson components v2~f~(r) for the AA 1S0 state. V2a~(r ) + V¢(r, p2),
Vaeff ( r , p 2) = Z
(48)
i where eft
V~, i
1
(r) - 1 + d?(r)
Va,i(r).
(49)
O n the other hand, we have the relation
d~(r) = ~-~ fbi(r), i
Vc(r) = Z
Vc,i(r),
etc.
(50)
i
The structure o f the A A potential is treated similarly. The potentials Vfff(r, p = 0) and ff their m e s o n c o m p o n e n t s V~a,i(r) for the AN, AA and N N 1So states are shown for c o m p a r i s o n in Figs. 7, 8 and 9, respectively. Let us c o m p a r e the potential structure a m o n g the AN, AA and N N potentials. The potentials w o r k i n g in the 1So state have a f u n d a m e n t a l structure in c o m m o n : the large cancellation b e t w e e n the o- attraction and the to repulsion and the balance results in attraction at m e d i u m range ( R = 0 . 7 - 1 . 5 fm) and repulsion at short distance ( R < 0.6 f m ) . The strengths o f the attraction are in the relations: [V(NN)[ >>
K. Tominagaet al./Nuclear PhysicsA 642 (1998) 483-505 lOOi
lOG \'
50
'I\ ....
\'
o
-50
-50 ,
,
,
,
l
•
,
0.5
.
,
i
•
1
r(rm)
,
.
•
i
•
' ....
-
/
0
....
50
IS°
o
-100
501
,
i
-100
,
1.5
/ ,
0
/ ,
/; ,
!
./ /
, ,/~,
0.5
/"
.Y"
/ .
I
.
•
.
*
1 rffm)
Fig. 9. The effective potential v~aff(r,p = 0) and its meson components v~f~(r) for the
I
t
i
i
t
1.5
NN tso state.
Table 7 _~-_~[-meson coupling constants (set B) ps ~ ~_m~7 ~7 t
0.1593 -4.2978 1.2303
IV(AA)] >~ IV(AN)I. The
sc ~,~ _w~s* ~f0 ~o-
relation
vc-g
1.0480 - 1.2155 -2.6161 1.7776
~p ~oJ ~¢
IV(AA)] >~ IV(AN) I
0.7664 1.9667 -2.5129
vc-f -2.0813 - 1.5587 -3.2504
comes primarily from the
relations gUNo,(= 3.17) > gAA~o(=2.56) and gNN~(= 1.78) < gAAo, = (1.88). Furthermore, the relation IV(NN)I >> ]V(AA) I comes from two sources: (1) the different S* attractions due to gNNS,(= 2.6) > gAAS.(= 0.56); and (2) or and p contribute to the attraction at R = 0 . 8 - 1 . 0 fm in NN, while they do not in AN or AA.
6. ---.N OBEP and ~ N phase shifts Let us examine the implication of the present model for the ~ N interaction. The coupling effects of I = 0 ~ N with the AA and 2 2 channels and I = 1 ~ N with the A 2 channel are only effectively involved as mentioned before. The parameters are determined by g.~-=~ = - g 8 ( 1 - 2 a ) ,
g-=-=n = cosO
gz~n,=sinO[---~3(l+2a)g8 ]
1
----~_ (1 + 2ot)g 8 - sinOgl,
+ cos Ogl.
(51)
Additionally, we have an u n k n o w n parameter, g,~==. This is assumed to be g,~_=s = gowN. Except for this, set B parameters in Table 3 are used in this subsection. E N OBE potentials are given by Eqs. ( 2 6 ) - ( 4 0 ) , where MA is replaced by M z
MA ~ M~.
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
502 10
!
•
,
,
,
i
•
,
•
20
'
'
10
i
,
!
I=O
.
l
,
i
,
.
/-/
-
5
0
--'So ...... 3 e ...... Ol
-5
!
,
0
-
t
t
0
--tPI
-10
"\-..' -\. .
-
_ . . _ 3r-~ . . . . x./l
%, ~\~
I
t
t
a
...... 3po
t
50
--'--3PI
100
i
-20
I
. . . . 3p2 i
20
EI~b(MeV)
Fig. 10. The I = 0 ISo, I0
!
,
,
.
3ej and
3 S 1, E l , 3 0 1 ,
,
!
.
.
.
IP 1
I
i
I
i
40 60 Ehb(MeV)
I
i
80
100
phase parameters for _~N scattering with set B parameters. I0
.
i
i
__
l= 1
Ipi
....... 3p0
i
!
-----3p1 ....
3p2 o-
--~
0
ISo
----- El ..... 3S1 . . . . 3D, -
-5
I
0
-
i
i
i
i
~
0 I
.
.
,
i
•
50 El~b(MeV)
Fig. II. The 1 = l ISo, 3Sh
. / " ._~fZ...............................
"-.
100
El, 3D1,
3pj a n d
0
I
20
i
i
40 60 Ehb(MeV)
i
80
100
phase parameters for --~N scattering with set B parameters.
'Pl
Table 8 The real part of the scattering lengths and the effective ranges for ~ N scattering with set B parameters
Scattering length (fm) Effective range (frn)
l=OlSo
1= 1 Is o
1=03si
I = 1 3s1
-5.79 x 10- I 8.5
-2.02 × 10-' 3.3 × 10
-3.52 × 10- t 1.74 × 10
-4.84 × 10- I 1.06 × 10
The phase parameters due to the ~ N potential are shown in Figs. 10 and 11. Furthermore, the low e n e r g y parameters are shown in Table 8. As o b s e r v e d f r o m Figs. 10 and l l , the I = 0, 1 ISo, 1 = 0, 1 3Sl and I = 0 3p1 phase shifts represent relatively small attractions for EL < 50 M e V in c o m p a r i s o n with the A N and A A phase shifts. T h e fact that g ~ = ~ = 0.16 is very small c o m p a r e d with gNNzr = 3.77 makes a large difference b e t w e e n ~ N and N N attractions. T h e ~ N potentials g e n e r a t i n g the
1S 0
phase shifts are displayed in Figs. 12 and 13.
In these figures the effective potential with g = s ~ = gNN,~ = 1.78 is displayed together with that with g ~ , ~ = gAA~r = 1.88. The differences b e t w e e n them w o u l d indicate the degree o f inherent ambiguity.
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
100
'
'
•
•
i
~
50
•
,
•
i
,
,
•
•
f~
i
,
,
•
100
,
....
X,~.\ .............
5O
I=0[~-]lSo
o
o
g _-.-o.=gNN~ ...... g£-o.=gAAo
i g0
-
-100
.
.
.
,
.
0
,
,
0.5
,
,
I
,
,
,
1 r(fm)
,
i
,
,
,
/:{
-50
veff(r,p=0)
-
503
/ ,i/ i
-100
,
1.5
,
,
0
,.,,:S
-:~-
.
",,
/,L,.
"" " ............/ ./o p / ,
,/'
i
0.5
,
on" v~,#-)
,
i
i
1 r(fm)
I
i
i
i
,
1.5
Fig. 12. The effective potentials V2~n(r,p = O) with g--=-,~= gnu,, = 1.78 and with gs-=,~ = g,la,~ = 1.88 are shown (left) by solid and broken curves, respectively, and its meson components for the -=N 1 = 0 I So state in the former c~se (right). 100
....
, ....
, ....
- - g "2-°'=gNN° ...... g==-o'=gAA~
50
, ....
100 F "
I'~
\'
50
I=1 ISo
' ' V
..........
• I*.~.>_..L
l ~,Nn
I--~
o
i
-50
. ~f(r,p=O)
. . .11. . . . .
0
0.5
. , ......... 1 1.5
~./
~.t / /,/
/
-100
f//o
-50 -100
,
0
r(fm)
,,/
. ~rr, ,
vc~.ikr)
:/
dil . . . . .
0.5
I.
, ....
1 r(fm)
, . . . ,
1.5
2
Fig. 13. The effective potentials V~,eft(r,p = 0) and its meson components for the ~N I = I ISn state. See caption of Fig. 12. 7. C o n c l u s i o n
In s u m m a r y ,
AN
and
AA
O B E P are presented in the nonet m e s o n s c h e m e with
effective scalar and v e c t o r m e s o n s incorporating the correlated and uncorrelated two octet mesons. T h e presently available data [ 12] for
Ap
cross sections and hypertriton are
reproduced. T h e
A N l S o phase shift is found to be m o r e attractive than 3&. Furthermore, 13 B are reasonably reproduced. the data for the separation e n e r g i e s o f 6aHe, ~ ° B e and AA
N o b o u n d state exists in the to
AN
I S o.
The
AA
AA
system. T h e
AA
1So
phase shift is a p p r o x i m a t e l y similar
total cross section b e l o w EL = 47 M e V as well as the ~ N
phase
shifts are predicted.
Acknowledgements
The authors w o u l d like to thank Y. Y a m a m o t o for valuable c o m m e n t s on this paper and E. H i y a m a for i n f o r m a t i o n o f their result prior to its publication. This w o r k was
504
K. Tominaga et a l . / N u c l e a r Physics A 642 (1998) 483-505
supported, in part, by a Grant-in-Aid for Scientific Research (No. 9225202) from the Ministry of Education, Science and Culture. Computer calculations were financially supported by RCNP, Osaka University, and carried out at the computer centers of RCNP, Kyoto, Osaka and Ehime Universities.
Appendix A The interaction Hamiltonian of the octet baryons N, A, 2 and ~ coupled with the octet mesons 7"r, r18 and K is given by Ho = glvN~r( N~ ~'Nl ) • zr + g ~ r ( N~ ~'N2) • ~r + gA Z~r( A t ~ + $ t A) • ~r +igzz~ ( ~ t × ~ ) . ~r + g~N~8 ( N~ N1 ) 718 + gzz,78 ( Nt2 N2) ~78 +gAA~s( At A)rl8 + g_~_=,Ts( ~ • ~)r/S + gNAK{ ( N~ K) A + At ( Kt Nl ) } +g_=Ax{ ( N ~ K c ) a + a t ( KtcN2) } + gN.~K{.,~"f " ( Kt~'Nj) + ( N ~ ' K ) . .,~} +gzzx{~t(Ktc~'N~) + ( N ~ ' K c ) • ~},
(A.1)
where N~ =
,
N2 =
~_
,
K =
Ko
,
Kc =
-K-
"
The SU(3) invariance of the interaction Hamiltonian describes the coupling constants by two parameters, gp and at,, as follows [20]: goner = gp, ga.~r = 2V~gp(
gz_=~r = - - g p 1 -- O l p ) ,
gNNrt8 = ~1 V~gp ( 4ap m 1), g-~zn8 = }v/3gp ( 1 -- ap), g n a t = -- ½'¢/3gp ( 1 + 2ap ), gN2"K = g p ( 1 - 2ap),
g~r
(1 -- 2ap ),
(A.3)
= 2gpap,
g-=-=u8 = - ½v/3gp ( 1 + 2ap ), gAArl8 = -- 2 v'3gp ( 1 -- ap ),
g~AK =
g~r
(A.4)
½v'~gp (4ap - I ),
(A.5)
= -ge,
where a t, is the ratio F / ( F + D). The interaction Hamiltonian of the octet baryons N, A, 2? and ~ coupled with the singlet meson rh is given by Hs = gNum ( N~ N1)7/1 ~- g-~m (N~N2)'ql -q- gAArh ( At A)rh + gzzm ( ~ t
.
,~)T]I" (A.6)
The SU(3) invariance requires that gNNm = gAA~h = gzz,7, = g-=-=m = gl.
(A.7)
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
505
In the nonet model we allow mixing between the neutral octet and singlet mesons, n8 and 71, respectively, with a mixing angle 0e In) = - sin 0e]nl) + COS0p[n8),
(A.8)
IT') = cos 0pin1 ) + sin 0pins ).
(A.9)
References I 1 I M.M. Nagels, T.A. Rijken and J.J. de Swart, Ann. Phys. 79 (1973) 338; P.M.M. Maessen, Th.A. Rijken and J.J. de Swart, Phys. Rev. C 40 (1989) 2226; V. Stoks, A fine-tuned version of the soft-core potential, unpublished. 121 A. Reuber, K. Holinde and J. Speth, Czech. J. Phys. 42 (1992) 1115. [31 Y. Fujiwara, C. Nakamoto and Y. Suzuki, Phys. Rev. C 54 (1996) 2180. 141 S. Sawada, T. Ueda, W. Watari and M. Yonezawa, Prog. Theor. Phys. 28 (1962) 991; 32 (1964) 380; for a review, see also S. Ogawa, S. Sawada, T. Ueda, W. Watari and M. Yonezawa, Prog. Theor. Phys. Suppl. 39 (1967) 140. 151 R. Bryan and B.L. Scott, Phys. Rev. 135 (1964) B 434; 177 (1969) 1435; see also extensive references in the first reference. 161 A.E.S. Green and T. Sawada, Rev. Mod. Phys. 39 (1967) 594. 17] T. Ueda and A.E.S. Green, Phys. Rev. 174 (1968) 1304; T. Ueda, EE. Riewe and A.E.S. Green, Phys. Rev. C 17 (1978) 1763. t8] T. Ueda, K. Tominaga, Y. Yamaguchi, N. Kijima, D. Okamoto, K. Miyagawa and T. Yamada, Prog. Theor. Phys. 99 (1998) 891. 19] E. Hiyama, M. Kamimura, T. Motoba, T. Yamada and Y. Yamamoto, Proc. Int. Conf. on Hypernuclear and Strange Particle Physics, BNL, USA, Oct. 13-18, 1997, Nucl. Phys. A (to be published). I10] T. Ueda, Phys. Rev. Lett. 68 (1992) 142. l 11 ] T.E.O. Ericson et al., Phys. Rev. Lett. 75 (1995) 1046; R. Arndt, Proc. Sixth Int. Symp. on Meson-Nucleon Physics and the Structure of the Nucleon, Blaubeuren, Germany, July, 1995; R. Timmermans, Proc. Sixth Int. Symp. on Meson-Nucleon Physics and the Structure of the Nucleon, Blaubeuren, Germany, July, 1995; B. Loiseau, Proc. Sixth Int. Symp. on Meson-Nucleon Physics and the Structure of the Nucleon, Blaubeuren, Germany, July, 1995. 1121 G. Alexander et al., Phys. Rev. 173 (1968) 1452; 13. Sechi-Zorn et al., Phys. Rev. 175 (1968) 1735; J.A. Kadyk et al., Nucl. Phys. B 27 (1971) 13. [131 K. Miyagawa, H. Kamada, W.Gl~Skle and V. Stoks, Phys. Rev. C 51 (1995) 2905; K. Miyagawa and W. Gl6kle, Phys. Rev. C 48 (1993) 2576. 1141 M. Juric et al., Nucl. Phys. B 52 (1973) 1. [151 Y. Yamamoto, H. Takaki and K. Ikeda, Prog. Theor. Phys. 86 (1991) 867; T. Yamada and K, Ikeda, Phys. Rev. C 56 (1997) 3216. 1161 D.J. Prowse, Phys. Rev. Lett. 17 (1966) 782. [171 M. Danysz et al., Phys. Rev. Lett. 11 (1963) 20; Nucl. Phys. 49 (1963) 121; R.H. Dalitz, D.H, Davis, P.H. Fowler, A. Montwill, J. Pniewski and J.A. Zakrzewski, Proc. R. Soc. London A 426 (1989) 1. 1181 S. Aoki et al., Prog. Theor. Phys. 85 (1991) 1287; C.B. Dover, D.J. Milleuer, A. Gal and D.H. Davis, Phys. Rev. C 44 (1991) 1905. 1191 Y. Yamamoto, T. Motoba, H. Himeno, K. lkeda and S. Nagata, Prog. Theor. Phys. Suppl. 117 (1994) 361. 1201 J.J. de Swart, Rev. Mod. Phys. 35 (1963) 916.
Nuclear Physics A 642 (1998) 483-505
A one-boson-exchange potential for AN, AA and systems and hypernuclei K. Tominaga a, T. Ueda a, M. Yamaguchi a, N. Kijima a, D. Okamoto a, K. Miyagawa b, T. Y a m a d a c a Faculty of Science, Ehime University, Matsuyama, Ehime 790-8577, Japan h Faculty of Science, Okayama University of Science, Okayama 700-0005, Japan c Laboratory of Physics, Kanto Gakuin University, Yokohama 236-8501, Japan
Received l 1 May 1998; revised 26 August 1998; accepted 9 September 1998
Abstract N N OBEP due to Ueda, Riewe and Green is extended to A N and AA systems in the framework of the nonet mesons and the SU(3) invariance. The data for Ap cross sections and hypertriton are reproduced. The A N ~So phase shift is found to be more attractive than the 3S~. Furthermore, the data for the separation energies of 6aHe, la°aBeand AA t3 B a r e reasonably reproduced. No bound state exists for the AA system. The AA ~So phase shift is similar to the A N I So. The AA total cross section is predicted. ~ N phase shifts are also predicted. (~ 1998 Elsevier Science B.V. PACS: 13.75.Ev; 21.80.+a; 13.75.Cs; 27.20.+n Keywords: A - N OBEP; A - A OBEP; ~ - N OBEP; Hypertriton; Double A hypernuclei
1. I n t r o d u c t i o n In recent years various theoretical approaches to hyperon-nucleon and h y p e r o n hyperon interactions have been presented. The Nijmegen group has presented OBE models with soft or hard cores [ 1 ]. The Jiilich group has presented an OBE model combined with the two-meson-exchange contribution due to the isobar degree of freedom [2]. The N i i g a t a - K y o t o group has proposed a hybrid model where pseudoscalar and scalar meson exchanges are taken into account in hybrid with the quark exchange for short distance [ 3 ] . However, none of these attempts is fully consistent with the experimental facts concerning the A N scattering and the single and double A hypernuclei. Historically, the one-boson-exchange potential ( O B E P ) for the N N interactions has been advanced by several theoretical groups independently since 1962, including one of 0375-9474/98/$ - see front matter (~) 1998 Elsevier Science B.V. All rights reserved. PII S0375-9474(98) 00485-0
484
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
the present authors (T.U.) and Bryan [4-7]. A realistic N N OBEP was first proposed by Ueda and Green [7]. This potential involves 7r, r/, p, to, t3, S* and o- exchanges with the vertex form factors which are introduced to represent the nucleon structure phenomenologically and describe the scattering data below EL ~< 330 MeV as well as the deuteron data. The vertex form factor regularizes the Yukawa potential. This model employs no additional phenomenological core. The short-range repulsion due to the to and due to the velocity-dependent terms from the vector and scalar mesons provides sufficient and necessary repulsion for a realistic potential. In this paper we extend the potential to A N and A A systems, l SU(3) invariance was first applied to hyperon-nucleon systems by the Nijmegen group [ 1 ]. Here, an SU(3) nonet meson scheme is considered for pseudoscalar, vector and scalar mesons: (7r, r/, K, ~ ' ) , (p, to, K*, ~b) and (t~, S*, K~ (1429) f0( 1581 ) ), respectively. Additionally, the isosinglet scalar meson tr is taken into account which represents the I = 0, J e = 0 + part of the correlated and uncorrelated 27r exchanges. A part of the tr exchanges between the A N and A A effectively represents the 27r exchange contributions from the A N - ~ N and A A - , ~ channel couplings, since the channel couplings are not explicitly treated here. This model is rather simple, compared with the theoretical models in recent years which make attempts at a microscopic description. We aim to build an effective model to be useful for studies of hypernuclei and YN interactions. We allow that the scalar and vector meson exchanges in our model involve effectively correlated and uncorrelated two octet-meson exchanges. The uncertainties arising from this treatment are removed in combined analyses with hypertriton and double A hypernuclei. As a result, our model is consistent with the data on A N scattering, hypertriton and double A hypernuclei of Ane, AABe l0 13 B. It is remarkable in comparison with other theoretical models that and aa the data on these double A hypernuclei are reproduced. With this model we demonstrate the A N as well as A A scattering phase shifts. Furthermore, the AA total cross section is predicted. No A A bound state is found. An advantage of the simplicity of this model is in application to hypernuclei with baryon number greater than three. Actually, it is quite remarkable that an application of our A N potential to four body systems has very recently provided a useful result about the binding energies of ~4He and A4H [9]. Furthermore, to check the predictive power of the present model in a future experiment, we display a result of potentials and phase shifts for the ~ N scattering. In Section 2 an overview of the N N interaction is presented. In Section 3 the potentials for N N , A N and A A systems are formulated. In Section 4 nonet OBEP parameters for N N , A N and A A systems are described. In Section 5 numerical results and a discussion on A N scattering, hypertriton, AA scattering, double A hypernuclei and potential structures are presented. In Section 6 ~ N OBEP and phase shifts are displayed. Section 7 is devoted to conclusions.
I A part of this paper was published as a brief letter [8].
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
485
2. An overview of the N N interactions Traditionally, the two-nucleon interaction has been elucidated from outside of the force range. In a meson theoretical approach the two-nucleon interaction at approximately r ~> 1 fm has treated theoretically, while that at r < 1 fm phenomenologically. This approach is nowadays very refined. The one-boson-exchange potential (OBEP) describes the repulsive core at short distance by the heavy-meson-exchange contributions with the meson-baryon form factors. The contributions produce both static and velocity-dependent repulsion [7]. The mesonbaryon form factors are phenomenological. Recently, however, the pion-baryon form factor has been substantially understood [ I0]. On the other hand, it has been attempted to explain the repulsive core in terms of the quark degree of freedom based on the model QCD. This picture predicts the strong attraction generating the H dibaryon. However, this has not been found so far in spite of a great deal of experimental effort. Furthermore, there is the problem that no complete theory exists which unites the meson theory with the quark degree of freedom. Even the pion mass has not yet been explained in terms of QCD. The two-nucleon interaction at long distance r > 2 fm is believed to be well established. Recently, however, it has been clarified that the pion coupling constant is ambiguous. At the Blaubeuren conference the value of the pion coupling constant was controversially reported to be in the range 14.62-13.47 [11 ]. Therefore, we may conclude that the theory for the two-nucleon interaction is not at all well established in the full range: from the outside through the inside. It is dangerous if anyone should believe that the two-nucleon potential is now established and a certain potential should be uniquely qualified as the best. At the present stage a variety of approaches to the two-nucleon interaction should be attempted. So far one-boson-exchange potentials have been used for a model of the two-nucleon interaction. Among them we adopt the Ueda-Riewe-Green OBEP (OBEP-URG) in this paper [7]. This potential has nice features as follows. The model is constructed in the r space by involving the one-boson-exchange contributions from the mesons of 7r, ~r, p, w, r/, ~ and S* with the form factors at the meson-nucleon vertex. This potential describes very well the two-nucleon interaction below 350 MeV of incident proton laboratory energy. It is nice that, once the model is made by fixing the nine parameters in fitting the two-nucleon data, the model can be extended to hyperon and nucleon as well as hyperon and hyperon systems by SU(3) transformation without additional parameters such as the hard or soft core radius. This is an advantage of the OBEP-URG over some other models where the radius is adjusted as an intrinsic parameter to a hyperon and nucleon or a hyperon and hyperon system. Furthermore, it is nice that the model gives the short range interactions by the heavy mesons with form factors. This produces the effective repulsion due primarily to o3, incorporated by the velocity-dependent repulsion which is automatically generated in the model.
486
K. Tominaga et a l . / N u c l e a r Physics A 642 (1998) 4 8 3 - 5 0 5
3. Equations and one-boson-exchange potentials 3.1. OBEP-URG for N N systems
The OBEP-URG involves the ingredients of pseudoscalar mesons 7r, r/, vector mesons w, p, and scalar mesons o-, S* and ~ with the vertex form factors [7]. The parameters of the potential are employed from model II of the OBEP-URG. The potential function is regularized with the vertex form factor in momentum space A2 F ( k 2) - k2 + A 2 ,
(1)
where k is the momentum transfer and A is a parameter. The regularized Yukawa function is Y(r)
=
( A 2 "~21 { \ a 2 - IzzJ -r e - I z r
( -- e -Ar
1+
A2-br2 ) }
2-----~-r
.
(2)
The total potential takes the form
Vtot(r, ~72)
= V(r)
-
~
1
[~2~b(r) ÷ q~(r)V2],
(3)
where the reduced mass Mr is given by (4)
Mr = I M u ,
and the V in Eq. (3) operates on all functions to the right. Furthermore, V ( r ) in Eq. (3) is given by V ( r ) = Vc(r) + V~(r)o'l • 0-z + VLs(r)L. S + Vr(r)S~2,
(5)
where L . S and $12 are given by L.S=IL2
. (oq ÷ 0"2) ,
S12 = 3 (0"1 " r)r2(o'2- r) - 0-1 "0-2.
(6)
Let us describe our treatment of the V z term in Eq. (3) for the spin-uncoupled case, for example. The Schr6dinger equation is written as
V2
p2
- 2M----~O+ Vtot(r, V 2 ) ,/// = ~ r
,t~,
(7)
where p is the magnitude of the center of mass momentum. The derivative part of the potential in Eq. (3) is eliminated by a transformation of the radial part of the wave function
u(r)~ ff ue(r)-
lv/i--T-~( ~ ,
(8)
K. Tominaga et a l . / N u c l e a r Physics A 642 (1998) 483-505
487
Table 1 The isospin factors m u l t i p l i e d by the potentials
I = 1, S = O
Potential 7r, p, d
~7, rlt,to, qS, S*, fo, o-
N N ~ N N ( I = O) N N --* N N ( I = 1) AN~ A N ( I = 1)
-3
AA ---+ A A ( I = O) ~ N --* ~ N ( I = O)
0 --3
~N --. ~N(1 = l)
I
I=O,S=O
1= ½, S = ± I
K,K*,K¢~
1 0
where g is the orbital angular momentum and
~9(r) = ~-~ U~(rr) yeo(O,¢ ) . The equation is rewritten for the radial part of the new wave function u~ff(r) as
d 2 ueffrr ) dr 2 g ~
g ( g + 1) utr)e" ,eff t' { - 2 M r V e f f ( r , p 2)
+pZ}u(r)~
ff =
0,
(9)
r2
where v~ff(r, p2) is the effective potential for an angular momentum eigenstate a defined by
V~(r) veff(r'p2) - 1 + ¢ ( r )
1 (~b,(r) /2 p2 ~b(r) 8Mr , ~ + ¢1( r ) + 2M----~1¢ (+r - - - - ~ "
(10)
The V,~(r) on the right-hand side of Eq. (10) is the original potential V ( r ) of Eq. (5) for the eigenstate or. The spin-coupled cases are treated similarly. The interaction Hamiltonian densities are assumed except for the isospin factors as follows. (a) For the N N pseudoscalar meson vertex
Hvs = igps~Ys~Pq~.
( 11 )
(b) For the N N scalar meson vertex H, = g.,.~C,q~.
(12)
(c) For the N N vector meson vertex
H,, = ig,,~y~Abfb ~ + ~f~, - - ~ 9-o ' u ~ ( O~(b~ - O~OSu ) ,
(13)
o'~,~
(14)
with =
[y~,,7~]/2i.
.Ad in Eq. (13) is a scaling mass. This is taken as the nucleon mass .A//= Mjv
(15)
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
488
Table 2 The mass and the form factor parameters Kind
Mass (MeV)
A (MeV)
~t S K~ S* fo p K*
138.7 493.6 548.7 957.5 970.0 1429.0 993.0 1581.0 759.1 891.8 782.8 1019.4 484.3
2012.00 1129.47 1129.47 1129.47 1129.47 1629.00 1129.47 1798.30 1000.00 1129.47 1129.47 1129.47 1129.47
N A
938.5 1115.6
.A4
938.5
K
Table 3 The SU(3) parameters. Three sets of parameters (sets 2, A and B) are considered for vector mesons. Each set shares the parameter values for pseudoscalar and scalar mesons in common. The ps and sc represent the coupling of pseudoscalar and scalar mesons, respectively. The vc-g and vc-f represent the vector and tensor couplings of the vector mesons, respectively Set 2
0
gl/v/-~ gs/v/-4-~
ps
sc
-10.41 0.4338 3.7736 0.5211
55.59 1.4715 1.6248 0.8225
for all the c o u p l i n g s o f N N - ,
Set A
Set B
vc-g
vc-f
vc-g
vc-f
vc-g
vc-f
24.84 2.8783 0.8426 0.9347
24.84 0.0 4.1625 0.2500
26.17 2.8466 0.8426 0.9689
26.17 0.0 4.1625 0.2500
25.62 2.8599 0.8426 0.9548
25.62 0.0 4.1625 0.2500
N A - a n d A A - m e s o n in c o m m o n . Hv in Eq. ( 1 3 ) is
r e w r i t t e n as
Hv = iGv~y#¢d?~ + i F v { ~ ( 0 ~ )
- (0~)~O}~b~,
(16)
where
Gv = g,, + -MN - ~ fj.v ,
F~, = f v / 2 . A 4 ,
(17)
T h e n w e m a k e a p p r o x i m a t i o n ( A 1 ) as follows: -
( A 1) M a k e t h e e x p a n s i o n o f the p o t e n t i a l in t e r m s o f p / M ~ a n d take the t e r m s o f the o r d e r ( p / M u ) 2, w h e r e p is the c e n t e r o f m a s s m o m e n t u m in C M S . All the o t h e r h i g h e r o r d e r t e r m s are i g n o r e d . T h e V ( k ) in m o m e n t u m s p a c e is F o u r i e r - t r a n s f o r m e d into V ( r ) in r s p a c e b y
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
V(r) =
V(k)eik'r(27r)3.
489
(18)
Each component of the OBEP for the pseudoscalar, scalar and vector meson exchanges are
g2 [
~cl(r)=~
ZSI2
Y(2)O'l • o'2 +
Vpsd(r) -_ ~gps
-Y+2
47r
~
1y(2)
(19)
1 2M 2 ( ~ 2 Y + Y ~ 7 2 ) +
l+--~-~,,j
42r2
Y(')L . S ] ,
(20)
~ (vzY+Yv2) Mu fl,~ 2
2
× A1--~2ZS12+ ~g~ 3 + 4 - -,~M- u~ ft ' ) Y ( I ~ L . S .
(21)
In these equations
(
) (
)
e-mr
Y=\m~_m2 j \m~_m2 j Z O l i T
,
(22)
i=1,3 1
y(l~
1 dY
_ 2M~ r dr' (
y(2) =
1
2M 2 (vZY},
m2 "~ { 2m2 2 "~i~13(1~5 + -mi m_~_~)e -mir , - + cei \m 3-m lj.= r r
Z=\m~_m~/
(23) (24)
with eel= 1
ce2-
m2 - m~ m2_m2 ~
m22 - m~ ~ 3 - m32_m22 ~
(25)
where ml is a meson mass tz and the limit is taken as m2 ~ A and m3 ~ A. This result is given by Eq. (2) for Y in Eq. (22).
3.2. AN OBEP The Schr6dinger equation for the AN system is the same as those given by Eqs. ( 1 )-(10) except that the reduced mass is now given by Mr = . MNMA MA"
(26)
MN ÷
The Hamiltonian densities for the AA- and AN-meson vertexes are assumed to be the same as Eqs. ( 11 )-(17) except for the isospin factors and the first equation in Eq. (17). This is replaced by
G,. = g,, +
MI + M 2
-2---M J'''
(27)
K. Tominagaet al./Nuclear PhysicsA 642 (1998) 483-505
490
where (Mz, M2) = (MA, MA) and (Ma, MN) for AA- and AN-meson vertexes, respectively. We made approximation (A1) as described in the last subsection. Furthermore, we make approximation (A2) as follows. (A2) The retardation effect in the meson propagators with S 4= 0 is approximately taken into account in the denominator as follows: -
k 2 ÷ ~2 ~ k 2 + ~2,
where ~2 = i22 _
(28)
( M A -- M N ) 2.
Each component of the AN potential is derived as follows. For the exchange of scalar mesons o-, S* and K~
V~s*(r) = ~glg2 { - Y +
q- 8M-~ ) V2Y - (4M--~ + 4MA 2 )
x ( v Z Y + Y V 2)+
Vx~ (r) = glg2 ~
~
+~
r dr
{-Y + ,
,
4MNMA V 2 Y 1
J'
(29)
+ 8 ,7)
+ 4MNM-----~ 1 dY L S'~
× ( v Z Y + Y V 2 ) + 2MNMA r ~rr " J"
(30)
For the exchange of pseudoscalar mesons r/and K
gig2 VmK(r) = ~
{ 12MNMA 1 V2yo'l-o'2 + - - Z S' 1 2 4MNMA
}•
(31)
For the exchange of vector meson w
V~o(r)=(G1 - 2G2 - 2G3 + 4G4)Y
+{
(8__~u2
xG2
+( +
+
__1
+ 4MNMA +
4MN z 1
(
1
1
2MN 2
1 4h~N 2
1
2MNMA
1 )G4}V2 Y
2MN --------~ + MNM-----~A 2MA 2 1 1 1 2MNM------~AGl + MNMA G2 + --G3MNMa
-I- 2~N2
(
G1 +
2MNM-~ + 4
x(V2y+ yv2) +
+
8_~A2)
1
MNMA
l
MNMA
+ MNMA +
21~A2 G2
-}-M~)G 3 2
C4) G~
1 )
4MA 2
491
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505 1 M N2
4-
G4
1 M A2
1
r -~r " 1
-~ - Gl V2YO.I"o'2 6MNMA
(32)
- G1ZSI2, 4MNMA
where glg2 (
G,=--~-
MA f l )
(
1+-~--~-l
gig2 (
MN f2~ I +-~--72 / ,
MA fl ) MN f2
G =-fi7
7
M 2g2'
1 4- M u f 2 ~
glg2 MA fl
03- 4rr .Ad 2gl
--~-~2//
glg2 MA f l MN f2 G4 = - - - -
4rr .A4 2gl .A// 2g2" For the exchange of vector meson K* VK. (r) = (G'1 - G~2- G~ + G~4)Y +
1 _ _ l + G1 { (8__.M~N2 1 G'2 + 4MNMa 8~'I1_2) ' 4MzvMA 1 , ( 1 1 )G~}V2 Y
4MNM~A G3 + {
(MN 4- MA) 2
1 , ( 1 2MNMa Gl -
4(1
8a,IN2
3
t
4-
4-
where
1 G1
__1 4+ 2MNMA
' _ _ l 2MNMA G3
_ _1 G,4 } 1-dYL's 2MNMA . r dr
(
1
1
1 )G~} ( V 2y + Y V 2) 8MA2
8MN '''''~ 4- 4MNM~A
2MNMA
1 )GtlV2yo.l.O. 2
12MNMa + 8zVl (MA MN) 2 ( G', C 2 G; 4- G~ ) ~ Y tz 2 8/~N 2
-
+
__3 ~A2), 4MNMA + 8 G2
I___~)G , 3
8a/TN2 4MNMA 4- 8 ( 2 1 (MN 4- MA 2
4MNMA
1 1
+ 4MNMA
-
-
4- ~
2
- -1 4MNMA
G'I ZSj 2
_ _ V2y 4MNMA
,) (V2Y4- YV2)
1
!dYL.s
2MNMA r dr
J' (33)
K. Tominagaet al./Nuclear PhysicsA 642 (1998) 483-505
492
Gtl
gig2 ( M A + M N f , ) 1+ 2.M
= ~
(I+MA+MNf2) 2.M g2
gl
g,g2 ( MA+MNf,) G~ = ~ 1+ 2A,~ gl ,
'
MNWMAf2 2./91 g2'
glg2MN+Mafl (1+MA+MNf~2) 2M gl 2A4
G3 = 4~r
'
G'4 = glgz fl f2 ( MN + MA) z 47r gl g2 (2-M) 2
For the exchange of the mesons with S 4:0 the following exchange operator must be multiplied: P = -PxPa= _(_)L(_)S+I,
(34)
where Px is the space exchange operator and P,~ is the spin exchange operator. For l E, 10, 3E and 30 states, P takes the eigenvalue as follows: P= ~ +
/
(IE, 30),
--
(lo,
(35)
3E).
The total one-boson-exchange potential is given by the sum of the velocity-dependent and -independent terms as in Eq. (3), where Mr is given by Eq. (26) and & is given by (36)
= ~,~ + ~s. + P~OK¢+ ~o~ + P~K*,
where
(37)
~Pms* = 4Mr (4-~N2 + 4--~AZ) Y' ~OK~= 4Mr
(
1
__ 1
+ 4MAMN +
~p,,,=4Mr 2M-~MAGI --G2MNMA q~x* =4Mr +
2MuMA G1 +
(1
81('1N2
(
+
8~tN 2
4MNMA +
2 (MN + MA) 2 (Ma - MN) 2
~
+ 4MNMA + 8
3+
4MlvMA +
G4 Y,
(39)
G'z
G3
1 1 ~8MN + 4MNMA
__l ×
(38)
Y'
,_j)},.
1 ) 8/~A 2
, G4
(40)
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
493
3.3. AA OBEP The AA OBEP is given by Eqs. ( 1 9 ) - ( 2 5 ) , where Mu is replaced by MA: MN --+ MA.
(41)
3.4. lsospin factors The isospin factors in Table 1 are multiplied by the potentials. These factors are due to the SU(3) invariant Hamiltonians presented in Appendix A.
4. A nonet OBEP model for NN, A N and A A systems In this paper an SU(3) nonet meson scheme is adopted for pseudoscalar, vector and scalar mesons: (1r, Tl, K,~Tt), (p,~o,K*,~b) and (6, S*,K~(1429) fo(1581) ), respectively. Additionally, the isosinglet scalar meson o- is taken into account which represents the I = 0, j e = 0 + part of the correlated and uncorrelated 27r exchange contributions as well as a part of the iterated contributions from the AN-Y,N and AA-,~2f channel couplings. For the scalar mesons, K~ and fo, the masses are assumed to be 1429 and 1581 MeV, respectively, since there is experimental evidence for these values. The form factor masses are taken as the same as those for the OBEP-URG, apart from those of K~ (1429) and f0 ( 1581 ) which do not appear in OBEP-URG. For these scalar mesons, K~ and f0, the ratios of the form factor mass A to the meson mass m are fixed as A ( f o ) / m ( f o ) = A(K~)/m(K~) = A(S*)/m(S*). The masses and the values of A are summarized in Table 2. The SU(3) parameters are the following four for each set of nonet mesons: the coupling constants of the octet and singlet mesons, g8 and gl, respectively, the parameter ce for the F and D couplings, a = F/(F + D), and the mixing parameter 0 between the octet and singlet mesons. Let us now show how to derive the coupling constants for the nonet mesons. Model II by Ueda, Riewe and Green is employed for this purpose [7]. First, for pseudoscalar mesons we use the coupling constants determined in the NN analysis: gmv~, gNN~ and gNNn'. The mixing parameter 0 for the pseudoscalar mesons is determined by the formulatan 20 = (M 2 _ mn) 2 / (m,7, 2 - M8z), where the isosinglet octet-meson mass M8 is given by the Gellmann-Okubo mass formula M~ = ( 1 / 3 ) ( 4 m ~ - m2) and mn and m,, indicate the observed masses of r/ and r/~, respectively. Now the SU(3) relation provides the formula which determines the SU(3) parameters, gs, gs and or, as follows: gNN~r = g8, gNNn=COSO[---~(4ot--1)g81
-- sin 0gl,
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
494
gNN~,=sinO[---~/=3(4a--1)g8]+cosOg,.
(42)
Then we obtain all the other coupling constants by virtue of the SU(3) invariance. We show only the parts related to the couplings with NN, AN and AA. The remaining parts are shown in Appendix A 1
gNAK
=
v/~g8(1 + 2 a ) ,
gAAn=CosO
- - - ~ ( 1 -- a)g8
gAAn,=sinO[----~3(1--a)g8 ]
-- sin0gl, + COS0g|.
(43)
For the vector and scalar mesons the coupling constants are treated similarly as the pseudoscalar mesons except for the mixing parameters. For those mesons the mixing parameters are made adjustable for the following reasons. Let us consider the effect of the correlated and uncorrelated exchange of the ~- and p mesons which has totally I = 0 and jR = 1-, namely the same quantum number as w. More generally, we may consider the effects of the correlated and uncorrelated exchanges of the two octet mesons which have totally the quantum number jR = 1- They produce additional SU(3) components by: 8 x 8 = 1 + 8 s + 8 a + 10+ 10+27. The effect produces a shift of the mixing parameter of the vector mesons 0v from the value determined by the mass formula, Ov = 38.56 °, to some other value. Also for the scalar mesons, we make the mixing parameter of the scalar meson Os adjustable for the same reason as for the vector meson case. Behind this treatment we interpret the mesons as involving the ingredients of not only (qcl), but also (qglqgl). The latter ingredient is one of the origins producing a shift of the mixing parameter from the value of the mass formula. Concerning the coupling constants of o- with N and A, gNN~ and gAA,~, respectively, we allow them to be independent. The gNN,r is determined from the NN analysis. We now have three adjustable parameters: Or, Os and gAA,~.The three parameters are determined by fitting the Ap total cross sections [ 12] at PL = 150-400 MeV/c and the binding energy of the hypertriton and the separation energy of the double A hypernuclei 10 of AABe. The values of the SU(3) parameters and the coupling constants thus determined are summarized in Tables 3 and 4, respectively. We find that gAAc~is a little larger than gNN,r: gAAo-----1.88 and gNN~ = 1.78. This is interpreted as follows, gAA~ involves the effect of the I = O, jR = 0 + part of the uncorrelated 2~" exchange due to AN-~N or AA-NN channel coupling, while gNN~ does not involve the corresponding 2~" exchange, namely double iteration of the one-pion-exchange.
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
495
Table 4 Baryon-baryon-meson coupling constants. The coupling constants of sets A and B are shown for vector mesons. Each set shares the parameter values for pseudoscalar and scalar mesons in common. The vc-g and vc-f represent the vector and tensor couplings of the vector mesons, respectively. The values for g / x / - ~ and f / x/'4~ are shown Set A ps
sc
vc-g
Set B vc-f
vc-g
vc-f
NNTr 3.7736 [71 N N t 3 1 . 6 2 4 8 [71NNp 0.8426 17l 4.1625 [71 0.8426 171 4.1625 171 NNr/ 2.4021 [71NNS* 2.6038 [7] NNw 3.1718 171 0 171 3.1718 171 0 171 NN~' 0 [7] NNfo 0 [7] NNq~ 0 171 0 17] 0 [71 0 171 NAK -4.4493 NAK~ -2.4812 NAK* -1.4292 -3.6048 -1.4155 -3.6048 AA~7 - 1.9741 AAS* 0.5568 AAw 2.5414 - 1.5899 2.5597 - 1.5587 AA~ / 0.8036 AAfo --1.4022 AAq5 --1.2827 --3.2353 --1.2763 -3.2504 NNo" 1.7776 [71 AAtr 1.8787
5. Numerical results and discussion
5.1. AN scattering With these parameter values all the experimental values for Ap scattering cross sections which are presently available are reproduced by the potential as is shown in Fig. 1. Other observables and phase parameters are displayed in Figs. 2-4. In the low energy limit we find a considerable difference of our result from that of the Nijmegen soft-core model [1] for o-(3S1): our value of 358 mb is compared with Nijmegen's value of 160 mb, with a similar result for o'(ISo): our value of 231 mb is compared with Nijmegen's value of 190 mb (see Table 5). About the AN spin-spin interaction the splitting energy of the JP = 0 ÷ and 1+ states for the 4 H and 4He hypernuclei gives the best experimental information. Hiyama et al.
Elab(MeV)
Elab(MeV) 50 .
.
.
.
i
,
'
'
•
100
150
!
i
. . . .
50 ,
100 ,
,
,
,
,
150 . . . .
600
400
2O0
+ i
0
|
i
200
400
Plab(MeV/c)
I
600
200
i
I
400
P,ab(MeV/e)
i
,
600
Fig. 1. The cross sections (left) and the ratios of the forward to backward differential cross sections (right) for AN scattering. The theoretical result with parameter set B is compared with the data.
496
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
0.2
30
- - Plab=300MeV/c ...... Plab=400MeV/c
0.15
/ ...."x\ ...
20 0. l
- - Plab= 100MeV/c . . . . ptab=400MeV/c
lO
0.05 0
....
" V ' 7 - - - 7 " , ,'-r-'~. ' - r " 7 i , 50 100 150 0(deg)
0
0
,
0
50
100 O(deg)
150
Fig. 2. The differential cross sections at P£ = 100 and 400 MeV/c (left) and polarizations at PL = 300 and 400 MeV/c (right) for A N scattering. The theoretical result with parameter set B is used.
10
'
'
'
'
l
'
'
'
'
l
'
,
,
.
w
.
30
20
11
tpt -10 ,
.
,
.
I
0
50
,
,
,
,
I
.
I
I
I
100
I
Elab(MeV)
~
~
...... 3po . . . . 3p2 i
I
150
3pt
i
i
l
l
0
l
l
l
50
l
l
l
l
l
l
l
100
150
Elab(MeV)
Fig. 3. The S-wave phase shifts d(1So) and d(3Sl) (left) and the P-wave phase shifts d ( I p l ) , d(3po), d(3pl) and d(3p2) (right) for A N scattering. Parameter set B is used.
6
,
,
•
•
i
,
•
,
•
,
,
,
,
•
,
/
7" 4
,
°`
7 "
'
.
•
•
•
I
'
,
,
•
!
,
i
I
t
,
,
,
I
'
4
2
7" 3D
7..7
~ 2
/
0
0
"
~ .
0
' ,
~
' .
~ ,
"
' i
50
~ t
-
' ,
2
~ ,
,
I
100
Elab(MeV)
|
,
•
,
I
,
150
Fig. 4. The D-wave phase shifts d(3Dl), d(3D2) and ~3 (right) for A N scattering. Parameter set B is used.
-2
t
0
,
,
,
I
50
t
i
i
100
Elab(MeV)
i
|
,
I
150
d(3D3) (left) and the mixing parameters El, 62 and
497
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505 i
30
.
.
.
.
i
.
,
•
.
i
•
/""""--.~
~20
--....
!
,
,
i
'
'
'
'
i
i
I
I
,
I
.
,
•
,
I
.
.
.
.
I
.
.
.
.
I
,
I
,
30
I-~ ""-~....
I
,
. . . . 3S l
! lO
10
0
0
50 100 Ei.b(MeV)
150
"
" I
0
.
.
.
50 100 Elab(MeV)
150
Fig. 5. The AN phase shifts d(ISo) and d(3Sl) in the case of hypertriton unbound (left) and overbinding with binding energy 2.43 MeV (right).
have reported that our A N potential gives about 0.7 MeV for the splitting energy: about 7/10 of the experimental value, 1.04 MeV, is reproduced. This is compared with the fact that existing standard potentials have so far provided much worse results [9].
5.2. Hypertriton Furthermore, we find that the binding energy for the hypertriton is in fair agreement with the experimental value as shown in Table 5: our theoretical value B(~tH) = 2.35 MeV due to the Faddeev formalism [ 13] is compared with the experimental value of 2.355 ± 0.05 MeV [ 14] (see Table 5). The calculation of the hypertriton binding energy is performed for the 1S0 and 3S1-3DI states of both the A N and N N subsystems. For the N N system the Paris potential is employed. The contribution from the partial waves higher than these is ignored. This contribution is thought to be less than about 0.05 MeV. The value of d(tS0) is found to be larger than that of d(3Sl) as shown in Fig. 2. If the value of d(lS0) is much larger than that of d(3S1 ) as shown in Fig. 5, while the Ap total cross sections are reproduced equally well, then the hypertriton is overbinding. On the other hand, if the value of d(ISo) is smaller than that of d(3Sl) as shown in Fig. 5, while the Ap total cross sections are reproduced equally well, then the hypertriton is unbound. It should be noted that the contributions of the AN l So and 3S1 states to the hypertriton binding energy are remarkably different from those to the AN scattering cross section. In the former the ANISo and 3S1 interactions in the A N N system work both in the A rearrangement three-body process and in the A N interaction with a remaining nucleon acting as a spectator, while in the latter the A rearrangement process does not exist. Therefore, in order to determine the relative strength of the ANISo and 3S1 contributions, it is significant to perform an analysis simultaneously of both the hypertriton binding energy and the A N scattering cross section.
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
498
Table 5 Binding energy of hypertriton, scattering lengths, effective ranges and cross sections at EL = 0. The numerical values of the scattering parameters for the Nijmegen soft core model [ 1] were derived by the present authors
Experiment Theory (set 2) Theory (set A) Theory (set B) Nijmegen SC
B(~IH) (MeV)
a(IS0) (fm)
r(IS0) (fm)
a(3Si) (fm)
r(3SI) (fm)
o-(Is0) (mb)
o-(3Si) (mb)
2.355 :]: 0.05 2.32 2.38 2.35 2.36 [13]
-2.65 -2.76 -2.71 -2.48
3.24 3.19 3.21 3.03
-1.80 -2.064 - 1.95 -1.32
3.71 3.46 3.56 3.26
221 239 231 190
305 400 358 160
Ejab(MeV) 10
20
i
i
30 40 50 60 i
"
i
"
i
•
!
i
i
800 _
_
.
3p
_
1
..... 3po .... 3p2
4
600
IS 0
_
400 c~
200 0
,
0
,
,
,
i
,
100
,
,
,
!
200
,
,
Plab(MeV/e)
,
i
I
i
300
i
i
0
i
|
20
i
~
Elab(MeV)
/
40
i
60
Fig. 6. The cross sections (left) and the ]So- and 3p-wave phase shifts (fight) for AA scattering. Parameter set B is used
5.3. A A scattering The cross section for the A A scattering is predicted as shown in Fig. 6. We find no A A b o u n d state. The d ( I S o ) , d ( 3 e 0 ) , d ( 3 p ] ) and d ( a P 2 ) for A A scattering are shown in Fig. 6. The A A d ( l S 0 ) is approximately similar to the A N d ( I S 0 ) at EL < 50 MeV. However, the A A d ( I S 0 ) is a little more attractive than the A N d(1S0). A l t h o u g h A A - _ ~ N c h a n n e l c o u p l i n g is not explicit in this calculation, we note that the contribution o f the uncorrelated exchange of K K due to the channel c o u p l i n g is represented effectively by the S* and fo exchange with the m i x i n g parameter Os adjusted.
5.4. Double A hypernuclei The difference ABAA b e t w e e n the two-A separation energy of double A hypernuclei BAA(AA Z ) and twice the o n e - A separation energy o f single A hypernuclei BA(aA--IZ)
ABAA = BAA( AA Z ) -- 2BA(Aa-1Z)
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
499
Table 6 The difference ABAA of the double A hypemuclei 6aHe, ~(liBe and AA J3 B, and the scattering lengths a(IS0) and the effective ranges r(ISo) for AA scattering Parameter set Set 2 Set A Set B Experiment
6aHe (MeV)
A IOABe ( MeV )
li~tB ( MeV )
a(ISo) (fin)
r(ISo)
3.43 3.72 3.60 4.7 [16]
3.95 4.28 4.14 4.3 [17]
4.70 5.12 4.94 4.9 [18]
-3.09 -3.64 -3.40
2.89 2.73 2.79
(fm)
is apparently sensitive to the AA potential. The ABAA'S for 6AHe, l ° B e and ]~4B are calculated by use of the present AA potential in a c o r e + A + A model [ 15]. The resulting theoretical values are in reasonable agreement with experimental data [ 16-18] as shown in Table 6. In the calculation the core-A potential is used which is made by folding the YNG-NS potential [ 19]. This is made from the Nijmegen soft-core potential to represent the effective A N interaction in the A nuclear matter. Its parameter is adjusted to reproduce the binding energy of the ground state of the core-A system.
5.5. Potential structure The structures of the A N potential are presented in this subsection. The total effective potential defined by Eqs. (3), (5) and (10) is divided into the two parts: one works for the singlet even (1E) and triplet odd (30) orbital-angular-momentum states with P = +1 in Eq. (34) and the other for the singlet odd (10) and triplet even (3E) orbital-angular-momentum states with P = - 1 . Furthermore, according to Eq. (5), each part is divided into the central, spin-spin, tensor, LS and V,~ parts as follows:
veff(r, p2) = v c e f f ( r ) + V J f ( r ) t r l . o ' z + V ~ eft . ( r ) S I 2 + V Left s ( r ) L ' S + V c b ( r , p 2 ), (44) where 1
[ v~e"(r)' k J f ( r ) ' v~rff(r)' V~Lsff(r)] - 1 + ~b(r~ [Vc(r), Vc~(r), Vr(r), VLs(r) ] (45) and
V4~(r'p2) -
1 (~,(r) /2 p2 ~b(r) 8Mr \ /+ 1~b(r) + 2M~r 1 + (b(r)"
(46)
Then the potential working for an angular momentum state a
veff(r,p 2) = [ Wcff(r) + veff(r)trl .o'2 + V-;ff(r)Sl2 +V~Lffs(r)L • S + Vo(r,p 2) l,~
(47)
is divided into the sum of each meson contributions rce~" and V~, where i identifies each meson, as follows:
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
500 100
100
50
50
0
/'° veff(r,p=0)
-50 -100
....
' .... 0.5
0
t .... 1 r(fm)
, .... 1.5
-50
2
-100
/ Ko
../,,. 0
0.5
//
e,itr)
X .......... 1 r(fm)
1.5
2
Fig. 7. The effective potential vfff(r, p = 0) and its meson components V~a~.f(r)for the AN iSo state.
" I ..........
100
"V " V o
.
.
.
.
.
.
.
.
.
.
50
"--4
-50 -100
/"
F --50 ~" / f f / 0
veff(r,p=0)
.
.
.
.
'
0
.
.
.
.
'
0.5
1
....
r(fm)
100 f . J . . . . . . 0 0.5
t , , ~ ,
1.5
O /" //"
/'/
" ,treffcv.t va,it~ j
/," . . . . . . . . . . 1 1.5 r(fm)
Fig. 8. The effective potential vfff(r, p = 0) and its meson components v2~f~(r) for the AA 1S0 state. V2a~(r ) + V¢(r, p2),
Vaeff ( r , p 2) = Z
(48)
i where eft
V~, i
1
(r) - 1 + d?(r)
Va,i(r).
(49)
O n the other hand, we have the relation
d~(r) = ~-~ fbi(r), i
Vc(r) = Z
Vc,i(r),
etc.
(50)
i
The structure o f the A A potential is treated similarly. The potentials Vfff(r, p = 0) and ff their m e s o n c o m p o n e n t s V~a,i(r) for the AN, AA and N N 1So states are shown for c o m p a r i s o n in Figs. 7, 8 and 9, respectively. Let us c o m p a r e the potential structure a m o n g the AN, AA and N N potentials. The potentials w o r k i n g in the 1So state have a f u n d a m e n t a l structure in c o m m o n : the large cancellation b e t w e e n the o- attraction and the to repulsion and the balance results in attraction at m e d i u m range ( R = 0 . 7 - 1 . 5 fm) and repulsion at short distance ( R < 0.6 f m ) . The strengths o f the attraction are in the relations: [V(NN)[ >>
K. Tominagaet al./Nuclear PhysicsA 642 (1998) 483-505 lOOi
lOG \'
50
'I\ ....
\'
o
-50
-50 ,
,
,
,
l
•
,
0.5
.
,
i
•
1
r(rm)
,
.
•
i
•
' ....
-
/
0
....
50
IS°
o
-100
501
,
i
-100
,
1.5
/ ,
0
/ ,
/; ,
!
./ /
, ,/~,
0.5
/"
.Y"
/ .
I
.
•
.
*
1 rffm)
Fig. 9. The effective potential v~aff(r,p = 0) and its meson components v~f~(r) for the
I
t
i
i
t
1.5
NN tso state.
Table 7 _~-_~[-meson coupling constants (set B) ps ~ ~_m~7 ~7 t
0.1593 -4.2978 1.2303
IV(AA)] >~ IV(AN)I. The
sc ~,~ _w~s* ~f0 ~o-
relation
vc-g
1.0480 - 1.2155 -2.6161 1.7776
~p ~oJ ~¢
IV(AA)] >~ IV(AN) I
0.7664 1.9667 -2.5129
vc-f -2.0813 - 1.5587 -3.2504
comes primarily from the
relations gUNo,(= 3.17) > gAA~o(=2.56) and gNN~(= 1.78) < gAAo, = (1.88). Furthermore, the relation IV(NN)I >> ]V(AA) I comes from two sources: (1) the different S* attractions due to gNNS,(= 2.6) > gAAS.(= 0.56); and (2) or and p contribute to the attraction at R = 0 . 8 - 1 . 0 fm in NN, while they do not in AN or AA.
6. ---.N OBEP and ~ N phase shifts Let us examine the implication of the present model for the ~ N interaction. The coupling effects of I = 0 ~ N with the AA and 2 2 channels and I = 1 ~ N with the A 2 channel are only effectively involved as mentioned before. The parameters are determined by g.~-=~ = - g 8 ( 1 - 2 a ) ,
g-=-=n = cosO
gz~n,=sinO[---~3(l+2a)g8 ]
1
----~_ (1 + 2ot)g 8 - sinOgl,
+ cos Ogl.
(51)
Additionally, we have an u n k n o w n parameter, g,~==. This is assumed to be g,~_=s = gowN. Except for this, set B parameters in Table 3 are used in this subsection. E N OBE potentials are given by Eqs. ( 2 6 ) - ( 4 0 ) , where MA is replaced by M z
MA ~ M~.
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
502 10
!
•
,
,
,
i
•
,
•
20
'
'
10
i
,
!
I=O
.
l
,
i
,
.
/-/
-
5
0
--'So ...... 3 e ...... Ol
-5
!
,
0
-
t
t
0
--tPI
-10
"\-..' -\. .
-
_ . . _ 3r-~ . . . . x./l
%, ~\~
I
t
t
a
...... 3po
t
50
--'--3PI
100
i
-20
I
. . . . 3p2 i
20
EI~b(MeV)
Fig. 10. The I = 0 ISo, I0
!
,
,
.
3ej and
3 S 1, E l , 3 0 1 ,
,
!
.
.
.
IP 1
I
i
I
i
40 60 Ehb(MeV)
I
i
80
100
phase parameters for _~N scattering with set B parameters. I0
.
i
i
__
l= 1
Ipi
....... 3p0
i
!
-----3p1 ....
3p2 o-
--~
0
ISo
----- El ..... 3S1 . . . . 3D, -
-5
I
0
-
i
i
i
i
~
0 I
.
.
,
i
•
50 El~b(MeV)
Fig. II. The 1 = l ISo, 3Sh
. / " ._~fZ...............................
"-.
100
El, 3D1,
3pj a n d
0
I
20
i
i
40 60 Ehb(MeV)
i
80
100
phase parameters for --~N scattering with set B parameters.
'Pl
Table 8 The real part of the scattering lengths and the effective ranges for ~ N scattering with set B parameters
Scattering length (fm) Effective range (frn)
l=OlSo
1= 1 Is o
1=03si
I = 1 3s1
-5.79 x 10- I 8.5
-2.02 × 10-' 3.3 × 10
-3.52 × 10- t 1.74 × 10
-4.84 × 10- I 1.06 × 10
The phase parameters due to the ~ N potential are shown in Figs. 10 and 11. Furthermore, the low e n e r g y parameters are shown in Table 8. As o b s e r v e d f r o m Figs. 10 and l l , the I = 0, 1 ISo, 1 = 0, 1 3Sl and I = 0 3p1 phase shifts represent relatively small attractions for EL < 50 M e V in c o m p a r i s o n with the A N and A A phase shifts. T h e fact that g ~ = ~ = 0.16 is very small c o m p a r e d with gNNzr = 3.77 makes a large difference b e t w e e n ~ N and N N attractions. T h e ~ N potentials g e n e r a t i n g the
1S 0
phase shifts are displayed in Figs. 12 and 13.
In these figures the effective potential with g = s ~ = gNN,~ = 1.78 is displayed together with that with g ~ , ~ = gAA~r = 1.88. The differences b e t w e e n them w o u l d indicate the degree o f inherent ambiguity.
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
100
'
'
•
•
i
~
50
•
,
•
i
,
,
•
•
f~
i
,
,
•
100
,
....
X,~.\ .............
5O
I=0[~-]lSo
o
o
g _-.-o.=gNN~ ...... g£-o.=gAAo
i g0
-
-100
.
.
.
,
.
0
,
,
0.5
,
,
I
,
,
,
1 r(fm)
,
i
,
,
,
/:{
-50
veff(r,p=0)
-
503
/ ,i/ i
-100
,
1.5
,
,
0
,.,,:S
-:~-
.
",,
/,L,.
"" " ............/ ./o p / ,
,/'
i
0.5
,
on" v~,#-)
,
i
i
1 r(fm)
I
i
i
i
,
1.5
Fig. 12. The effective potentials V2~n(r,p = O) with g--=-,~= gnu,, = 1.78 and with gs-=,~ = g,la,~ = 1.88 are shown (left) by solid and broken curves, respectively, and its meson components for the -=N 1 = 0 I So state in the former c~se (right). 100
....
, ....
, ....
- - g "2-°'=gNN° ...... g==-o'=gAA~
50
, ....
100 F "
I'~
\'
50
I=1 ISo
' ' V
..........
• I*.~.>_..L
l ~,Nn
I--~
o
i
-50
. ~f(r,p=O)
. . .11. . . . .
0
0.5
. , ......... 1 1.5
~./
~.t / /,/
/
-100
f//o
-50 -100
,
0
r(fm)
,,/
. ~rr, ,
vc~.ikr)
:/
dil . . . . .
0.5
I.
, ....
1 r(fm)
, . . . ,
1.5
2
Fig. 13. The effective potentials V~,eft(r,p = 0) and its meson components for the ~N I = I ISn state. See caption of Fig. 12. 7. C o n c l u s i o n
In s u m m a r y ,
AN
and
AA
O B E P are presented in the nonet m e s o n s c h e m e with
effective scalar and v e c t o r m e s o n s incorporating the correlated and uncorrelated two octet mesons. T h e presently available data [ 12] for
Ap
cross sections and hypertriton are
reproduced. T h e
A N l S o phase shift is found to be m o r e attractive than 3&. Furthermore, 13 B are reasonably reproduced. the data for the separation e n e r g i e s o f 6aHe, ~ ° B e and AA
N o b o u n d state exists in the to
AN
I S o.
The
AA
AA
system. T h e
AA
1So
phase shift is a p p r o x i m a t e l y similar
total cross section b e l o w EL = 47 M e V as well as the ~ N
phase
shifts are predicted.
Acknowledgements
The authors w o u l d like to thank Y. Y a m a m o t o for valuable c o m m e n t s on this paper and E. H i y a m a for i n f o r m a t i o n o f their result prior to its publication. This w o r k was
504
K. Tominaga et a l . / N u c l e a r Physics A 642 (1998) 483-505
supported, in part, by a Grant-in-Aid for Scientific Research (No. 9225202) from the Ministry of Education, Science and Culture. Computer calculations were financially supported by RCNP, Osaka University, and carried out at the computer centers of RCNP, Kyoto, Osaka and Ehime Universities.
Appendix A The interaction Hamiltonian of the octet baryons N, A, 2 and ~ coupled with the octet mesons 7"r, r18 and K is given by Ho = glvN~r( N~ ~'Nl ) • zr + g ~ r ( N~ ~'N2) • ~r + gA Z~r( A t ~ + $ t A) • ~r +igzz~ ( ~ t × ~ ) . ~r + g~N~8 ( N~ N1 ) 718 + gzz,78 ( Nt2 N2) ~78 +gAA~s( At A)rl8 + g_~_=,Ts( ~ • ~)r/S + gNAK{ ( N~ K) A + At ( Kt Nl ) } +g_=Ax{ ( N ~ K c ) a + a t ( KtcN2) } + gN.~K{.,~"f " ( Kt~'Nj) + ( N ~ ' K ) . .,~} +gzzx{~t(Ktc~'N~) + ( N ~ ' K c ) • ~},
(A.1)
where N~ =
,
N2 =
~_
,
K =
Ko
,
Kc =
-K-
"
The SU(3) invariance of the interaction Hamiltonian describes the coupling constants by two parameters, gp and at,, as follows [20]: goner = gp, ga.~r = 2V~gp(
gz_=~r = - - g p 1 -- O l p ) ,
gNNrt8 = ~1 V~gp ( 4ap m 1), g-~zn8 = }v/3gp ( 1 -- ap), g n a t = -- ½'¢/3gp ( 1 + 2ap ), gN2"K = g p ( 1 - 2ap),
g~r
(1 -- 2ap ),
(A.3)
= 2gpap,
g-=-=u8 = - ½v/3gp ( 1 + 2ap ), gAArl8 = -- 2 v'3gp ( 1 -- ap ),
g~AK =
g~r
(A.4)
½v'~gp (4ap - I ),
(A.5)
= -ge,
where a t, is the ratio F / ( F + D). The interaction Hamiltonian of the octet baryons N, A, 2? and ~ coupled with the singlet meson rh is given by Hs = gNum ( N~ N1)7/1 ~- g-~m (N~N2)'ql -q- gAArh ( At A)rh + gzzm ( ~ t
.
,~)T]I" (A.6)
The SU(3) invariance requires that gNNm = gAA~h = gzz,7, = g-=-=m = gl.
(A.7)
K. Tominaga et al./Nuclear Physics A 642 (1998) 483-505
505
In the nonet model we allow mixing between the neutral octet and singlet mesons, n8 and 71, respectively, with a mixing angle 0e In) = - sin 0e]nl) + COS0p[n8),
(A.8)
IT') = cos 0pin1 ) + sin 0pins ).
(A.9)
References I 1 I M.M. Nagels, T.A. Rijken and J.J. de Swart, Ann. Phys. 79 (1973) 338; P.M.M. Maessen, Th.A. Rijken and J.J. de Swart, Phys. Rev. C 40 (1989) 2226; V. Stoks, A fine-tuned version of the soft-core potential, unpublished. 121 A. Reuber, K. Holinde and J. Speth, Czech. J. Phys. 42 (1992) 1115. [31 Y. Fujiwara, C. Nakamoto and Y. Suzuki, Phys. Rev. C 54 (1996) 2180. 141 S. Sawada, T. Ueda, W. Watari and M. Yonezawa, Prog. Theor. Phys. 28 (1962) 991; 32 (1964) 380; for a review, see also S. Ogawa, S. Sawada, T. Ueda, W. Watari and M. Yonezawa, Prog. Theor. Phys. Suppl. 39 (1967) 140. 151 R. Bryan and B.L. Scott, Phys. Rev. 135 (1964) B 434; 177 (1969) 1435; see also extensive references in the first reference. 161 A.E.S. Green and T. Sawada, Rev. Mod. Phys. 39 (1967) 594. 17] T. Ueda and A.E.S. Green, Phys. Rev. 174 (1968) 1304; T. Ueda, EE. Riewe and A.E.S. Green, Phys. Rev. C 17 (1978) 1763. t8] T. Ueda, K. Tominaga, Y. Yamaguchi, N. Kijima, D. Okamoto, K. Miyagawa and T. Yamada, Prog. Theor. Phys. 99 (1998) 891. 19] E. Hiyama, M. Kamimura, T. Motoba, T. Yamada and Y. Yamamoto, Proc. Int. Conf. on Hypernuclear and Strange Particle Physics, BNL, USA, Oct. 13-18, 1997, Nucl. Phys. A (to be published). I10] T. Ueda, Phys. Rev. Lett. 68 (1992) 142. l 11 ] T.E.O. Ericson et al., Phys. Rev. Lett. 75 (1995) 1046; R. Arndt, Proc. Sixth Int. Symp. on Meson-Nucleon Physics and the Structure of the Nucleon, Blaubeuren, Germany, July, 1995; R. Timmermans, Proc. Sixth Int. Symp. on Meson-Nucleon Physics and the Structure of the Nucleon, Blaubeuren, Germany, July, 1995; B. Loiseau, Proc. Sixth Int. Symp. on Meson-Nucleon Physics and the Structure of the Nucleon, Blaubeuren, Germany, July, 1995. 1121 G. Alexander et al., Phys. Rev. 173 (1968) 1452; 13. Sechi-Zorn et al., Phys. Rev. 175 (1968) 1735; J.A. Kadyk et al., Nucl. Phys. B 27 (1971) 13. [131 K. Miyagawa, H. Kamada, W.Gl~Skle and V. Stoks, Phys. Rev. C 51 (1995) 2905; K. Miyagawa and W. Gl6kle, Phys. Rev. C 48 (1993) 2576. 1141 M. Juric et al., Nucl. Phys. B 52 (1973) 1. [151 Y. Yamamoto, H. Takaki and K. Ikeda, Prog. Theor. Phys. 86 (1991) 867; T. Yamada and K, Ikeda, Phys. Rev. C 56 (1997) 3216. 1161 D.J. Prowse, Phys. Rev. Lett. 17 (1966) 782. [171 M. Danysz et al., Phys. Rev. Lett. 11 (1963) 20; Nucl. Phys. 49 (1963) 121; R.H. Dalitz, D.H, Davis, P.H. Fowler, A. Montwill, J. Pniewski and J.A. Zakrzewski, Proc. R. Soc. London A 426 (1989) 1. 1181 S. Aoki et al., Prog. Theor. Phys. 85 (1991) 1287; C.B. Dover, D.J. Milleuer, A. Gal and D.H. Davis, Phys. Rev. C 44 (1991) 1905. 1191 Y. Yamamoto, T. Motoba, H. Himeno, K. lkeda and S. Nagata, Prog. Theor. Phys. Suppl. 117 (1994) 361. 1201 J.J. de Swart, Rev. Mod. Phys. 35 (1963) 916.