137
Progress of Theoretical Physics, Vol. 15, No. 2, February 1956
Intermediate Coupling Meson Theory of Nuclear Forces, II
Y uk.ihisa NOGAMI and Hiroichi HASEGAWA*
Department of P-hysics, University of Osaka Prefecture, Sakai
and
Department of Physics, Osaka City University, Osaka*
(Rece1ved December 1, 1955)
In· the Part I of the same title a method has been proposed to construct the nuclear potent'a[
by means of the mtermediate coupling theory, where the charged scalar theory has been taken as an
example. In th1s paper, the method 1s applied to the realistiC case, symmetrical pseudo-scalar theory
w1th pseudo-vector coupling, and the static nuclear potential is obtamed up to the order of e-2r,
where x = p.r.
The results are found to depend on the cut-off procedure which is inevitably mtroduced to avoid
the divergence troubles. A cut-off factor should be chosen so that effects of the so-called contact
interaction do not extend onto the exterior region x:<: 1.0. We take a cut-oT factor of gaussian type
as a sUitable one. Our potential does not essentia1ly differ from the FST-potential and the BWpotential w1th cut-off. After the renormalization of the coupling constant, the potential of the order
of e-z comc1des With the conventional second order perturbation potential m the e,:terior region,
irrespectively of an assumed "structure" of the clothed nucleon. The potential of the ord"r of e-2z,
however, depends essentia1ly on the cut-off factor.
§ I. Introduction
In the interaction of nucleons with the meson field, the inertia of the self field is
large and its reactive effects play an important role. It is expected that these circumstances
are well exhibited by the intermediate coupling theory. In this, theory, meson configurations
are classified into bound -and unbound ones. The mesons in the bound configuration, the
so-called zero-mesons, are strongly bound to a nucleon. The mesons in the unbound configuration are called s-mesons. For the one nucleon problem, the intermediate coupling
theory has been applied extensively and we are now well acquainted with this, as far as the
one nucleon problem is concerned. However, its generalization to the two nucleon problem
is somewhat meandering, since it is not an easy task to solve the eigenvalue problem for
the cloud around two nucleons in ·a straightforward way.
Recently, one of us (H.H.) has proposed a method of construction of the nuclear
potential by means of the intermediate coupling theory taking the charged scalar theory as
an example1 >. The essential' point of this method is the transformation (1.3.1), in virtue
of which the two nucleon problem turns to be solved exploiting the knowledge about the
o::1e nucleon problem. The single clothed nucleon states are taken as a standard, and the
potential is obtained as the effect3 of change induced on the cloud in a course of approach
of two nucleons from infinitely separated positions.
Y. Nogami and H. Hase~awa
138
In this p<tper the metho:l is applied to the r.ealistic case where the nucleons interact
with the pseudo-scalar meson field through the pseudo-vector coupling. Effects of nucleon
recoil are neglect~. Moreover the following simplifying assumptions are employed: (i) effects
of the excited or isobar s~tes of the single clothed nucleon other than four states: (I,])=
(l/2, 1/2) 1., excited, (1/2, 3/2), (3 /2, 1/2), (3 /2, 3 /2) can be neglected, and (ii)
possibility of a transition of a nucleon from the ground state to any one of the excited
states accompanied by emission of the s·meson can be neglect~. These two assumptions
are called "one level approximation" hereafter. Res!=ficting the number of mesons exchanged
between two nucleons we evaluate the potential up to- the order of e- 2 x.
In actual evaluations, the divergence difficulties arise. Therefore, at the present stage,
we have to employ a cut-off pro:edure. A form of the cut-off factor F (k) should be
chosen so that the so-called contact interaction terms do not give appreciable effects on the
potential in the exterior region x ;G 1.0. We use the cut-off factor of gaussian type
F(k) =exp( -lCj2a2)
(1·1)
with an appropriate cut-off momentum a.
In § 2 a formal derivation of the potential is presented. In § 3 renorm:tlization of
th! coupling constant and the cut-off procedure are discussed. In § 4 the results are sUmmarized. Appendix contains brief reviews on the single clothed nucleon and details of the
calculations.
§ 2. Derivation of the potential
(1) The Hamiltonian
The positions of two nucleons are denoted by x 1 and x 2, respectively.
interaction Hamiltonian is
H
•ut
= __=-!____
_fJ_"' -(il ~<il f dk F (k)_ u. (k). ~il··rx·, + c c
V2tra p. 7 '<>
J V2K«tl'<> '
••
ut
Then the
(2 ·1)
Here, the suffix z( = 1, 2) refers to the nucleons, the suffices a ( = 1, 2, 3) and l ( = 1, 2, 3)
refer to the meson charge and the spatial component, respectively. The dummy suffices a
and l are to be automatically summed up. The direction of r = x 1 - x~ is chosen as the
third coordinate axis. For other notations, refer to Appendix A. The wave functions of
zero-mesons are taken to have the following form :
1
r.pJ(k)
i
q F(k)k e-•k""+e-•kx•
= v: vz·ii·'-"; "hK
1
.v 2
'
1
i
g F(k)k e-ikxl_e-ik:x•
r.pHk) =~----=.-~ - ~--~--.
v;a .v2n·' p. .V2K
.v 2
(2 ·2)
(l=1, 2, 3)
The normalization factors Vi and 1-- ~ are .functions of internucleon distance r. For verylarge r, both are nearly equal to V, the normalization factor in the case of one nucleon
Intermediate Coupling Meson Theory of Nuclear Forces, II
139
problem [ cf; (A· 4)]. While, when the two nucleons approach closely, VJ and Vi' tend
to v'TV and zero, respectively.
We expand the meson field ¢rs(k) into the complete orthonormal set involving rp;(k)
and rpr (k) as its six members :
(2. 3)
¢! (k)
=A~frp!* (k) +A~trp'l* (k)
+a: (k).
A~f and A~! or A~ 1 and A~ 1 are the creation or the annihilation operators of the symmetric
and the antisymmetric zero-mesons. The commutation relations are
(2 ·4)
others=O.
a! (k) and a" (k) are the operators for s-mesons.
In order to express all the quantities in the language of each single nucleon, we
perform the canonical transformation (I · 3 · 1) :
A~V = (A~l
+ A~l) I v'T,
A~1/*= CA~t+A':J) 1
vz,
A~~)= (A~~- A:l) I
A~~>*=
v'T'
(2 ·5)
CA!t-A:nl -vz.
A~~>* and AW are the creation and the annihilation operators of the zero-mesons which
extefid around the nucleon 1, and so on.
The total Hamiltonian H is rearranged as follows :
H=lfo+H~/s 1 +H~;>+Ho~+H.,
(2·6)
lfo= (Koo+UJ ~Q(il + Uu~Qjil
'
'
+ U2 ~A£it* A~tj + U2~~ (3A~~~* A'"3 (il -A~~>* A~?)
'
i
+ w~ A~'l* A~P + w~~ (3A~iJ* A~{ -A~l* A;JJ)
i ... J
i"eJ
(2. 7)
(2 ·8)
H;.=
GKVoo~f dkK[v.rpt(k) a·Y> +v (3rpt(k) o-,\1> -rp1(k) o-ji>) J r~tla .. (k) e
67r
....
JJ
1
1,.. 00 J
+c.c.,
(2 ·9)
H.=.\ dkKa! (k) a.. (k),
(2 ·10)
140
Y. Nogami and H. Hasegawa
with
Here the conventional notations
(2 ·11)
are used, of which the diagonal elements vanish, and
1
!J<ilV-<'>~.
t - 3 [A<il*A<ilaa
CJ:l
"a. "-'ac;> (A«>*
.a:l +A<a.a>) J•
(2. 12)
Coefficients U's, W's{' u's and v's, the functions of x= pr, are defined by
u,=- (1/2n 2) fdkF(k)2 cos k3r·P/K2 ,
u1= - (1/21l'
2)
JdkF(k)
2
cos
kor· (3k
2
3
-k2)
/2~,
(2·13)
v,=- (1/4rr) \' dkF(k) 2 cos k3r·P/K3 ,
J
and
(2·14)
Other functions are unnecessary for later calculations and are not given here.
u's etc., of course, depend on the cut-off factor F(k). In the case of 110 cut-off, we
have
Ue(x) =(1 +~+~)e-x +2(a' (x)- a(x)),
X
~
X
'
(2 ·15)
X
V8 (x) =K0(x) - K 1(x) jx,
v1 (x) =K2 (x).
Kn (x)'s are the modified Bessel functions of the second kind.
Terms involving the deltatype functions are the m-called contact intera::tion. Even in the case of cut-off, u's and v's
can be approximated by the no cut-off functions (2 ·15) for large x, and U's ar~ of the
order of e- 2"'. The expressions (2 · 7) ..._ (2 · 9) are obtained after the omission of terms
higher than e-~x for large x.
(2) The potential
The potential is obtained as the x-dependent eigenvalue of the total Hamiltonian I-1.
Let the potentials of the order of e-"" and e- 2"" for large x be denoted by V(e-x) and V',
141
Intermediate Coupling Meson Theory of Nuclear Forces, II
V(e--") is given by the diagonal element of the last row of (2·7):
respectively.
V(e--") =diagonal element of
3f1 (
g2 )c-(l>-<~>)[u (O"<l>(j"cz>) +us ]
4 ;r
'"' '"'
·'
t
t
e
J2 ,
(2 ·16)
where
(2·17)
It should be noted that 't' and (J are the isospin and the spin operators of a bare nucleon.
Similarly to the case of charged scalar theory, we must introduce the isospin operator T
and the spin operator S of a clothed nucleon. For instance, ( r1 JTC2 J) is equal to 1 for
the charge triplet ~tate of two clothed nucleons, and - 3 for the charge singlet state.
(S'1 JSC2J) is equal to 1 for the spin triplet state of two clothed nucleons, and -3 for
the spin ~inglet state.
Moreover we introduce the operator
(2 ·18)
corresponding to the operator sl2'
Since the diagonal element of -r~>..-~z> · O"l >O"j 2> is T~1 >T~2 > · Sjl>Sfl · (P0 -P"'-P~+Pa:) 2 ,
as is shown in Appendix A, (2 ·16) becomes
V(e--")
= J:l.{ g2 \(P0 -Pa-Pt+PaY (T~ >y~~>)f u,(Se()Sf>) +ue~ 12].
3
4;;-
(2 ·19)
The factJr (P0 -Pa-P~+Pa:) 2 corresponds to the factor (1-Ptl) 2 in the charged scalar
theory. V(e-•) can be interpreted as the potential due to an exchange of one zero-meson
between two nucleons.
V' consists of two parts, V.lis and V, 80 • Vdis comes from the effect of distortion of
the zero-meson cloud of a nucleon due to the existence of the other nucleon nearby.
And we obtain, from the diagonal element of the first and the second row of (2 · 7) :
(2·20)
K0ofdo is the energy of the ground state and n0 the mean number of zero-mesons in the
cloud of the ground state nucleon.
V1so is the potential energy due to the effect of the excitation of the clothed nucleons
to the isobar levels and the effect of the s-meson emission. The method of its evaluation
is the same as in I, and is given in Appendix B. It sl;10uld be noted that the s-meson
exchange plays no role.
§ 3.
Coupling constant renormaliz!ltion and cut-off procedure
The potential up to the order of e- 2-" is expressed in terms of the functions u., u,
v, and v1• The shape of these functions depends strongly on an assumed form of the
cut-off factor in contrast to the case of the charged scalar theory. These circumstances
142
Y. Nogami and H. Hasegawa
are due to the appearance of the "contact interaction!' As is seen from (2 ·15), if the
cut·off procedure is not applied, the contact interaction term becomes zero el!!cept an infinitely
steep peak at x=O, and can be disregarded since it has no physical effects. However,
when the cut-off procedure is applied, the peak is spread out. But the effects of the spread
of this peak must be suppressed in the region of large x.
For this purpose, the cut-off factor must be a function which smoothly decreases with
increasing k. We employ a cut-off factor of gaussian type (1 · 1) as a suitable one. The
cut-off momentum a is now to be determined.
Using (1·1) and (2·13), we have
l/a"{ 1 -
u, ( x) -- -e-"'
-·e
x
1
~
-Vr:
.p>,(ax
7 - _ - 1 )}
2
a ·
(3 ·1)
with
Therefore V(e-"') is, for large x,
V(e-"')
= .f!_(JL)(P0 -Pa-P,+Pa.) 2ella'
3
47r
(3 ·2)
In this region, V' is very small compared with V(e-"') and the resultartt potential is
predominatingly determined by (3 · 2). According to the analysis of Otsuki and Tamagaki4 ),
the effective coupling constant
(3 ·3)
should be renormalized to 0.08.
Table 1
v
(Po- .. ·) 2
g2/4rr
I
I
I
I
0.2
0.4
0.6
0.633
0.341
0.224
0.114
0.221
0.343
3.04
4.11
I
a
!
4.84
Intermediate Coupling Meson Theory of Nu=lear Forces, II
143
+
The value of V determines the " structure " of the meson clould and ( P0 - Pa-P, Pa,) 2
depends only on V, but neither on g nor on a. Then noting that there is the relation
(A· 5) among g, V and a, we can determine the values of g and a for given V. The
values of the parameters determined in this manner are listed in Table 1. As to the
structure of the meson cloud of the single clothed nucleon, we use Takeda's results5>.
The functions u, u1, v, and v, are shown in Figs. 1 and 2. The effects of the
contact interaction are spread almost to x""" 1.0. In the interior region x;S 1.0, the sign
of the central potenti~l of V(e-"") changes.
•
/
f
I
I
I
I
-I
I
I
I
I
I
I
I
I
-2
I
I
I
I
- - - - cut·off
--------no
cut-off
- - - - - no cut-off
X
0~----------~-----------L--o.s
1.0
1.5
Fig. l.
u 8 (x) and v 8 (x) with a=4.1l,u
and with no cut·olf.
§ 4.
Fig. 2.
Ut(x) and v 1 (x) with a=4.1l,u
and with no cut·olf.
Results and discussions
The results are shown in Figs. 3 and 4. Qualitative tendency of the potential is
tabulated in Table 2. There is no serious discrepancy among our results and the FSTpotentiafl and the BW-potential3l. As to V', v;11• is repulsive and the central part of v;.o
is always ·a~ractive, and the central part of the resultant V1 is attractive for all states.
While the tensor part of V1 changes its sign with states and regions.
144
Y. Nogami and H. Hasegawa
(p.)
Moreover, V.so mainly consists of
the effects of the isobaric transitions of
nucleons. The effects ~f s-meson turn to
be very small. Therefore the potential up
to the order of e- 2~, i.e, V(e-x) V1
0.5
+
is almost determined only by the effects
of zero-mesons. This fact suggests that
the zero-mesons assumed in the intermediate coupling theory give a fairly good
picture of the real meson cloud around
the nucleons in the problem of the low
energy nuclear forces.
Table 2.
-0.5
Fig. 3.
The potential m even state.
V=0.4, a=4.ll,u.
ev.en
state
(p.)
0.5
odd
state
Tendency of V 1 •
x;:Sl.o
x::2:LO
singlet
-
-
triplet
central
-
-
tensor
+
+
singlet
-
-
triplet
central
-
-
tensor
-
+
_ .... _ _tensor
The effects of the cut-off procedure
appear in two ways : (i) effect on shape
of the functions u's, etc. and (ii) effect
"
on the coefficients by which the functions u's etc. are to be multiplied. The
effect (i) is estimated from the spread
of
the contact term. Using the gaussian
Stnglet [ 1n stnglet state ]
V' <V(e-•)
cut-off with the cut-off momentum
-0.51
a;;:;4.0p, this effect is suppressed to be
small enough in the exterior region
Fig. 4. The potential in odd state. V=0.4, a=4.ll,u.
where
1.0. The effect (ii) is due
to the change of the energy levels of excited states, etc. For example, K 00 obtained by
the straight cut-off [ cf. (A· 2) which is usually employed in the one nucleon problem,
is smaller than that obtained by the gaussian cut-off with common cut-off momentum.
Consequently, the differences between energy levels are small and the effect of the isobaric
transitions becomes larger in the case of the straight cut-off _than in the case of the gaussian
cut-off.
In Fig. 5 the results of the straight cut-off and of the gaussian cut-off are shown
for the triplet odd state. Here, of course, if we apply the straight cut-off to (2 · 13),
-
1.5
x>
.~
J,
Intermediate Coupling Meson Theory of Nuclear Forces, II
145
the functions u's etc. oscillate inconveniently for x"<:l.O. Therefore we have used u's et·.
obtained by the straight cut-off only in the interior region x$1.0, and connected them
smoothly to the functions obtained without cut-off (2 · 15) in the exterior region.
The effect (i) is characteristic to
tellSor
the two nucleon problem, while the effect
1.5
(ii) appears also in the one nucleon
0
-----0.5
problem, e.g. meson scattering by nucleon.
.-'
The effect (ii) on V(e-x) disappears after
/
/,,~;'
the "coupling constant renormalization ".
Therefore, if we suppress the effect (i)
I
I
I
in the exterior region, V (e-x) conincides
- - - - - stratght
I
I
with the conventional second order perturI
- - - gausswa
I
I
I
bation potential, irrespectively of the -0.5
I
f
form of the. cut-off factor, and also of (p.)
I
the " structure " of the clothed nucleon. Fig. 5. Comparison of potentials for tr1plet odd state
obtamed by gausstan cut-off and by stra1ght cut-off.
(Because the structure of the clothed
V=0.4, a=4:u,u.
nucleon is determined by V, and the
same effective coupling constant is obtained after renormalization for any value of V.)
Thus the potential V(e-x) is well established, as long as we understand that the coupling
constant is the renormalized one and the isospin and the spin operators are the operators
for the clothed nucleons. However, the total potential V(e-x) V' can by no means be
independent of the cut-off procedure. In other words, V' depends essentially on the structure
of the clothed nucleon including the isobar states, through the effect (ii).
The above results have been shown in Figs. only for V=0.4. It would be desirable
to choose larger cut-off momentum, e.g. a"'"'-nucleon mass, so that the appearance of the
effect (i) is confined in the more interior region. However, for such a large cut-off
momentum, .. V would become '""- 1.0. Then the one level approximation which simplifies
evaluations would not be applicable.
-------
,~
-
,,
+
The authors wish to express their thanks to Prof. M. Taketani and Dr. S. Machida
for their critical discussions and to Messrs. Y. Munakata, A. Komatsuzawa, S. Otsuki and
J. Iwadare for their discussions and interest in this work. One of the authors (Y.N.) is
also grateful to Prof. Z. Shirogane and Prof. T. Inoue for their continual encouragement.
Appendix A.
The single clothed nucleon
As a preparation, we briefly review the static problem ~f a single clothed nucleon5181 •
The nucleon is considered to be at rest at the origin and interacting with the symmetrical
pseudo-scalar meson field through the pseudo-vector coupling.
Then the Hamiltonian is*
··-----------
* The
natural unit (t=c=l) is used.
146
Y. Nogami and H. Hasegawa
f dkK¢! (k)¢Cl(k),
_ - i g _ f F(k)
H.,nt. /-
(A·1)
Hmeson=
3 -,CJCI"t
v
with
2rr 11
·
1
dk-~k1 ·nCJ(k)
+c.c.,
V2K
K= vfl+ti.
Here ¢! ( k) and ¢Cl ( k) are the creation and the annihilation operators of a meson with
momentum k. Suffix a ( = 1, 2, 3) refers to the charge of the meson, and l ( = 1, 2, 3)
to the spatial component. r Cl and CJ"1 are the usual isospin and spin operators, respectively.
f1 is the rest mass of meson, and g is the coupling constant.
F(k) is the cut-off factor. For instance, in the case of the straight cut-off,
F(k)={~
In the case of no cut-off, a ~ oo .
Now we split the ro..e>on field
for
{k<
k >a.
a
(A·2)
¢,. (k) into bound and unbound parts, by means of
¢Cl(k) =ACl19r(k) +aCl(k),
¢! (k)
=A~'f! (h)+ a! (k).
(A·3)
The "wave function" of the zero-meson 9 1 (k) is taken to have the form
(/)t(k) =J__ _,_jl__ F(k)kt'
'
V V2ica p. V2K
(/=1, 2, 3),
(A·4)
where V is the normalization constant :
~=-1-(JL)2J__[dkF(kW.
(2rr) 3 11
3
J ·2K
3
(A·5)
aCl ( k) , the so-called s-meson part, is orthogonal to 9[ ( k) . A!1 and Arl 1 are the creation and
the annihilation operators of zero-mesons, respectively. The commutation relations are
(A·6)
Using the expansions (A· 3) , the total Hamiltonian 1s rewritten in the form
f
H=K0o-fl+[A:1 dkKff[(k)aa.(k) +c.c.]
(A·7)
+ \ dkKa! (k) aCl (k),
J
with
Koo=
f
dkKI'ft(k)
12,
(independent of/),
(A·8)
Intermediate Coupling Meson Theory of Nuclear Forces, II
147
K0of2 is the Hamiltonian of the system composed of a bare nucleon and the zero-meson
cloud.
The eigenvalue equation for a clothed nucleon
(A·9)
has been already solved by several authorsr>Bl. The normalization constant V is the coupling
constant between the zero-mesons and the bare nucleons, and the structure of the clothed
nucleon is determined by the value of V.
There are the following constants of motion : the total angular momentum ], its third
component J~ and similar quantities I, I 3 in the charge space. For the clothed nucleon in
the ground state, I= J= 1I 2. I 3 = 1I 2 state is the proton state of a clothed nucleon, and
I 3 = - 1/2 state is the neutron state. The probability that the isospin and the spin of a
bare nucleon in a clothed nucleon are the same as those of a clothed nucleon is denoted
by P0 ; the probability that the bare nucleon isospin is reversed, by P~ ; the probability
that the bare nucleon spin is reversed, by P0 ; and the probability that the bare nucleon
isospin and spin are both reversed, by P0 , . Of course
(A·10)
Denoting the probability of finding the total orbital angular momentum of the meson cloud
to be 1 by PH we have
(A·ll)
Hereafter the ground state is denoted by suffix 0, and the expectation value of any
operator c9 for the ground state by ( 0 I01 0) . Now we introduce the isospin operator
T and the spin operator S for a clothed nucleon ; for instance,
!(T1 -iT2 ) jproton state of a clothed nucleon)
=I neutron state of a clothed nucleon),
T 3 jproton (neutron) state of a clothed nucleon)
(A·12)
= 1 ( - l) Iproton (neutron) state of a clothed nucleon) .
Then the diagonal element of •a.rr1 is expressed in ~rms of Ta. S1, namely,
(A·13)
As to the excited or isobar states we consider only the following four states : (I, ])
= (!, !) lst excited> (~, ~) 1 .west> (~, !) lowest> G, ~)lowest• The level of the ~tate (~, ~)lowest
C;Jincides with that of(~, !)lowest· We denote the above states by suffix li{~O), and matrix
elements of any operator c9 by ( 0 Ic9jv), (vI c9jv') , etc.
Appendix B.
E v tluation of V1, 0
Here we calculate V,80 , the effects of isobaric transitions and emission and absorption
of s-mesons.
148
Y. Nogami and H. Hasegawa
( 1) .
Matrix elements
First we write down the necessary non-diagonal elements of H 0, If_~~) and H,.;,. Under
the assumption of the one level approximation, the third row of 1fo (2 · 7) is omitted.
Now, there is a well known formula
(z;JA.t/O) =VB,(z;Jr.. CT1 /0)
(B·1)
with
B,=!Jj(1-f2J
where K 00 !J, is the energy difference between the JJ-state and the ground state. Then the
non-diagonal element between the ground state and the excited state /ZJ1 ZJ2) is, (two nucleons
are in the JJ1- and JJ2-states, respectively)
( j)
1
li
2
IH 0 /00) -X
-
\lt\12
2)CTO)(J'(£) I00)
(z; l li 2 J-Cl)-(
"<' "a
l
l
(B·2)
with
(B·3)
("' . c ) +Gp.
Y llt\12 _GKoo
_ _ _ c111+c112 Vt
-Ue
67.
3
•
Further, the state is specified by the following quantities; (il>. j 1 ) and (i2, j 2) , the isospins
and the rpins of nucleons 1 and 2 respectively; i and j, the isospin :md the spin of the
two nucleon system; m, the third component of j. The state vector is represented by
Jv1 ZJ2 ; il> i2 , i; j 1, j 2, j, m). Especially the ground state is, using capital letters, represented
by /00; ~'~'I; ~' !, ], M) or simply /I;], M). Because of the charge mdependence,
there is no need to tpecify the third component of i. Then the matrix element (B · 2)
is expressed as follows :
(vlv2; iJ> i2, i; jl, j2, j, m /Ho/I; ], M)
~ [ x,.,,a,( -1 )•-·""W(1/2, 1;2, j, j,, J,
1)
+ Y ~, >"6 >"s (2]+ 1) (], 2 ; M, 0 li M) U (;:
X ( -1) ii-l+Y.ailamNW(1j2, 1/2,
X (vJI/rCl)CTCI>JJO) (v2J!r12JCTI2JJ!O).
i~> i 2
;
:~;
m
I, 1)
(B·4)
Here (], 2; M, 0/ j, M), W( ...... ) and U( ...... ) are the Clebsch-Gordan, Racah and
generalized Racah coefficients, respectively. (v[JrCT/JO) 1s the so-called physical part of the
matrix elemene>.
Next, the system composed of one s-meson and one nucleon is to be considered.
We specify the state composed of one nucleon in the -ground state and one s-meson, by
the isospin j_, the spin j and the s-meson momentum k. Then the matrix element of Ho, 11 > is
Intermediate Coupling Meson Theory of Nuclear Forces, II
149
X Vc" 1 (v1[[r<1J<T<1l[Jo) (K-K00 ) r;(k) /-./ (2il + 1) (2j1+ 1),
(8· 5)
(8·6)
r;(kr=r;"f(k)r;1 (k)/3.
where
The matrix element of H 0f> is similar to (8 · 5) .
Finally, the matrix elements of H0~ e.g. (Os; v2 [H0',[00) can be obtained by the
following substitution in (8 · 4) .
GK00
Y \lt\2 --7>·--v,
(8· 7)
67<
(2)
Effect> of isobaric transitions and interactions with s-mesons
Since we are concerned with the potential up to the order of e- 2", the processes
containing the exchange of an s-meson between two nucleons are inhibited. As (Os, vf~fOO),
(v, Os[Ho:[oo) and (v1 v2 [H0 f00) are
Ho.'
•
Ho,•n
Ho
(0, 0) - - <•· 0) - - (0 '· 0) - - (0, 0)
Type I.
the quantities of the order of e-"'
for large x, the potential energy of
the order of e- 2" is obtained by a
(Os, ·~
~H'o..
Ho,"/
sort of perturbation method. There
Ho~m " '
/
are two types of transition schemes,
(0, 0)
(OJ, Os)
(0, 0) ~ (>, >2)
Type II.
just as in I, which are shown in
Fig. 6. ~su is the sum of the
(•, Os)
contributions of respective processes.
In the process of the type I,
Fig. 6. Transition schemes. v~(Os) indicates that a nucleon
makes transition from 11-state to the ground state
contribution V" 0 is given by (I · 6 · 9\
emitting one s-meson.
I.e.
I
H~ IH•·"/(
(8·8}
with abbreviations
(vOflfofOO) =a,
(OO[H;.:[os, 0) =~(K-K00 )VJ(k),
(Os, Oflfo~1l[vo) =i.(K-K00 )r;(k)
and
(8·9)
In the pro!:ess of the type II, we obtain contribution V" 1 " " ' using an appropriate
approximation, as follows :
Y. Nogami and H. Hasegawa
150
(B·10)
with abbreviations
(111 11 2 [~[00)
(oo[H0~[0s,
=a',
V2)
=~1
(OO[H0 ~[v~>
(K -K00 ) f{J(k),
(Os, V2 [H0~1 l[v 1 !.12 )
= (Os,
Os[H0~1 l[vl>
(v 1, Os[H0i21 [!.11 !.12 )
= (Os,
Os[~~ZJ
R1 = R(KoofJl +J.22R(O)),
Os)
=~2 (K -K00 )
f{J(k),
Os) =21 (K-K00 )f{J(k),
[Os2 ,
V2)
=A 2 (K-K 00 )f{J(k),
!J1=!J,,,
d' = - (J./R2+A 22 R1) /Koo(QJ +!J2) ·
It is easily seen that (B · 9) is a special case of (B · 10).
In V, 0 and V,,"' the terms containing a 2 are due to the isobari:c transitions, and the
terms containing ~2 to the em1ss10n and reabsorption of the s-mesons. The cross terms
are due to the interference of these two processes.
References
1)
2)
3)
4)
S)
6)
7)
H. Hasegawa, Prog. Theor. Phys. 13 (1955), 47.
Hereafter this paper will be cited as I, and the formula e.g. (3.1) In I as (1.3.1).
N. Fukuda, K. Sawada and M. Taketani, Prog. Theor. Phys. 12 (1954), 156.
K. Inoue, S. Mach1da, M. Taketam and T. Toyoda, Prog. Theor. Phys. 15 (1956), 122.
The1r potential IS Cited as FST-potential.
K. A. Brueckner and K. M. Watson, Phys. Rev. 92 (1953), 1023.
The1r potential IS Cited as BW-potential.
S. Otsuki and R. Tamagaki, Prog. Theor. Phys. 14 (1955), 52.
G. Takeda, Phys. Rev. 95 (1954), 1078.
D. Ito, Y. Miyamoto andY. Watanabe, Prog. Theor. Phys. 13 (1955), 594.
G. Racah, Phys. Rev. 62 (1942), 438.
A. Ar1ma, H. Horie and Y. Tanabe, Prog. Theor. Phys. 11 (1954), 143.