461
Progress of Theoretical Physics. Vol. 41, No.2, February 1969
One-Boson-Exchange Modes and Two- and Three-]Pion
Exchange in Nucleon-Nucleon Scattering. 11*)
Susumu FURUICHI, Hiroshi SUEMITSU,* Wataro WATARI**
and Minoru YONEZA W A*
Department of Physics, Rikkyo University, Tokyo
*Department of Phys'ics, Hiroshima University, Hiroshima
** Research Institute of Atomic Energy, Osaka City University, Osaka
. (Received August 26, 1968)
A detailed comparison is made between the theoretical two-pian-exchange amplitudes and
the experimental values for the left-hand-cut contribution in nucleon-nucleon scatte:ring for both
the T= 1 and T=O nucleon-nucleon states in order to confirm the conclusions of the previous
analysis where only the T=l states are analyzed. The conclusion obtained in the previous
analysis was that the uncorrelated two-pian-exchange amplitudes can explain the T= 1
nucleon-nucleon scattering together with the effects of the one-pion, one~p-meson and one-C!)meson exchanges and without effects of scalar mesons, if the squared momentum. of the
exchanged pions is cut off at tmax"-'9,u2. It is shown in this paper that this conclusion is
also valid for the T=O nucleon-nucleon states. Some positive evidence is found for the fact
that the two-pian-continuum contributions are favourable over the discrete (scalar boson) poletype contributions. Implication of the small cutoff momentum, tmax'"'-'9,u2, is discussed.
§ 1.
Introduction
In a previous paper!) (hereafter we refer to this paper as I), in terms of
the partial wave dispersion relation, we have analyzed the experimental values
for the left-hand-cut contributions 2) for the T= 1 nucleon-nucleon states by considering the one-, two- and three-pion-exchange effects.
There we have used the experimental vaules for the left-hand-cut contributions estimated according to Kantor's prescriptionS) and the theoretical two-pion
exchange amplitudes 4) calculated from the CGLN amplitudes. 5 ) The three-pion
exchange effects have been treated phenomenologically.6) The purpose of this
paper is to present a more detailed analysis ,of the problem focusing mainly on
the' two-pion-exchange contributions, taking both the T = 0 and T = 1 nucleonnucleon states.
In addition to this we also examine the one-boson-exchange modeF) by the
dispersion relation In connection with the K-matrix method how the unfavourable
features of the K-matrix method such as the p-meson coupling constants,
the S-wave problem, etc. will be improved by the dispersion theoretic
approach.
*) This report is a part of the results of the research project "Strong interaction ~t around
1 GeV" organized by the Research Institute for Fundamental Physics, Kyoto University, Kyoto, during
1967'"'-'1968.
'..
I
I
I
I
I
i
IPl
3D3!
3D2
PI
3
Dl
,
state
''.
1.9113
1.4184
1.0624
0.6838
0.6015
0.5116
0.4131
16
10
9
8
7
49
25
16
10
9
8
7
:25
2.8529
2.0139
1.4474
0.8875
0.7720
, 0.6487
0.5167
0.5104
0.4247
0.3405
0.2288
0.2017
0.1710
0.13(35
0.2182
0.1902
0.1597
0.1144
0.1024 '
0.0885
0.0721
49
7
8
9'
0.5468
0.4874
0.4089
0.2872
0.2551
0.2177
0.1741
0.2320
0.2121
0.1838
0.1356
0.1220
0.1058
0.0862
49
25
16
10
0.9272
0.6760
0.4717
0.2565
0.2119
0.1647
0.1157
1.2783
1.1303
0.9420'
0.6433
0.5631
0.4692
0.3599
95 MeV
0.6032
0.4550
0.3280
0.1859
0.1551
0.1219
0.0868
0.4941
0.4471
0.3830
0.2733
0.2422
0.2051
0.1607
50 MeV
49
25
16
10
9
8
7
Tlab
3.5000
2.3700
1.6496
0.9770
0.8436
0.7032
0.5553
0.8090
0.6454
0.4980
0.3190
0.2780
0.2328
0.1832
0.8709
0.7602
0.6224
0.4227
0.3723
0.3145
0.2484
1.1601
0.8231
0.5607
0.2965
0.2433
0.1878
0.1309
2.2008
1.9100
1.5599
1.0344
0.89S0
0.7408
0·5605
142 MeV
4.1157
2.6560
1.7859
1.02l6
0.8759
0.7248
0.5681
1.2016
0.9095
0.6720
0.4096
0.3530
0.2921
0.2270
1.2997
1.1086
0.8857
0.5833
0.5099
0.4270
0.3335
1.3909
0.9578
0.6367
0.3279
,0.2675
0.2052
0.1420
3.6030
3.0631
2.4517
1.5822
1.3636
1.1146
0.8329
210 MeV
7
8
9
10
16
25
49
7
8
9
10
16
. 25
49 (p.2)
(~4.4834)
7.4787
( -2.7149)
6.0845
I (-2.3462)
4.7311
( -1.9416)
3.4405
( -1.5155)
,
54.1852
( -8.4158)
28.2795
( -6.1764)
16.0634
59.9676
. (-10.1632)
32.7773
( -7.7394)
19.3066
( '-:5.7972)
9.3572
( -3.6351)
7.6760
( -3.1612)
6.0166
( -2.6335)
4.4017
( -2.0626)
50 MeV
48.9628
( -13.6382)
24.7469
( -9.7090)
13.6877
( -6.8591)
6.1773
( -4.0163)
4.9898
( -3.4409)
3.8491
( -2.8236)
2.7744
( -2.1816)
53.8160
( -16.3148)
28.4484
( -12.0683)
16.2966
( -8.8072)
7.6464
( -5.3459)
6.2263
( -4.6109)
4.8407
( -3.8094)
3.5101
( -2.9542)
95 MeV
44.7564
( -17.8446)
22.0561
( -12.3998)
'11.9623
( -8.5845)
5.2836
( -4.9100)
4.2476
( -4.1831)
3.2596
( -3.4131)
2.3362
( -2.6198)
48.9327
( -21.1981)
25.1954
( -15.3213)
14.1387
( -10.9651)
6.4859
( -6.5064)
5.2554
( -5.5818)
4.0641
( -4.5860)
2.9303
( -3.5340)
,142MeV
40.0481 '
( -22.5529)
19.1980
( -15.2579)
10.2036
( -10.3432)
4.4129
( -5.7807)
3.5317
( -4.8990)
2.6970
( ~3.9757)
1.9228
( -3.0332)
43.5342
( -26.5966)
21.7803
( -18.7364)
11.9647
( -13.1391)
5.3684
( -7.6239)
4.3295
( -6.5077)
3.3314
( -5.3187)
2.3894
( -4.0749)
210 MeV
reference 2), therefore, it is different from Furuichi's h4 ) by a
factor i(L- L'), where L and Ii are the. angular momenta in the
initial and final states respectively.
a) Here the partial wave amplitude h is defined by Eq. (1) of
ISO
3S1
state
Tlab
Table 1. ,The two-pion exchange amplitudes for the T=O nucleon-nucleon states and the ISO state (g".2/4rr= 14.4, p.; pion mass).
The numerical figures.in the parentheses for the 3S1 and ISO states are the once subtracted values, h 27r (Tlab) -h2",(0).a)
~
::::l
~
~
~
t:<
~
~
';:::.<'
~
~
::::l
~
"".
"i
~
'No
~
~
~
~
u~
~
'No
N.
~
C'\l
~
C/.)
~
;::s-.
:-,.
~.
~
~
~
tv
m
One-Bason-Exchange Modes and Two- and Three-Pion Exchange
§ 2.
463
Kantor amplitudes and two-pion-exchange amplitudes
In this paper we at~empt to determine the values of various parameters
(coupling constants of bosons) so that they give best fit to the experimental
data of the 3pJ, IP!, HDJ, ID2 and PI amplitudes by minimizing X2 as was done in
1. In the present analysis we take the coupling constants gi as parameters rather
than the squared coupling constants gi 2 so that gi 2 >0 are always' guaranteed.
The experimental values for the left-hand-cut contributions, i.e. the Kantor amplitudes, which are used in the present analysis, have' been tabulated in references 1) and 2) .*) The theoretical two-pion-exchange amplitudes are summarized
in Table 1. In this Table we give the amplitudes for the T = 0 states and the
ISO state. For the amplitudes in other states, please refer to I and for the contributions from the deuteron, see the appendix of reference 2).
§ 3.
Analysis of Kantor amplitudes by the one-pion,
two-pion and one-hoson-exch,ange amplitudes
In the present analysis we consider only the contributions from vector mesons
and scalar mesons in addition to the pion (one-pion and twoMpion exchanges).
Other types of mesons which may' relate to the three-pion exchange are not
considered, since these do hot improve the fit with experiment remarkably as
was seen in 1.
For vector and scalar mesons we consider both possibilities of 1=0 and
1=1. We, however, assume that their masses are iso-spin independent. The
vector meson mas,ses are taken to be 750 Me V which is close to those of the P
meson and the (j) meson.
The coupling constants of these mesons are determined from the Kantor
amplitudes of the above mentioned nine states at three energies, T 1ab = 95, 142
and 210 MeV and four (T= 1) states at 50 MeV. Therefore, the expected value
of the chi square, X2, is 31 minus the number of free parameters and the standard
deviation is AX2 = ../2 xX2~7.
'
(A) Before we give a detailed analysis of the two-pion exchange which will be
given in the next subsection, let us see in this subsection how the consideration
of the T=O nucleon-nucleon data influences the previous conclusions which have
been obtained from the analysis of the data for the T= 1 states. We also see
the difference of the K-matrix method and the dispersion theoretic: method. In'
*) By taking the new phase shifts solutions at 50 MeV (N. Hoshizaki, Prog. Theor. Phys. 38
(1967), 1203) which were obtained from the more precise experiments recently done at INS, University of Tokyo, the T=l nucleon-nucleon amplitudes haye been re-calculated in referenc,e 1).
These T=l new phase shifts at 50 MeV may naturally affect the T'=O phase shifts at 50 MeV which
have not been obtained yet. Therefore, we do not involve the T=O amplitudes at 50 MeV in our
discussion. More strictly speaking, the change in the T=O phase shifts at 50 MeV will also affect
the Kantor amplitudes at other energies, but this effect is small.
464
S. Furuichi, H. Suemitsu, W. Watari and M. Yonezawa
Table II. Comparison of the results of. the present analysis which consider both the T=l and 0
nucleon-nucleon states with the previous ones!) from the T= 1 states and also those of the
K-matrix method. (ms=450 MeV)
- - - - - - - - - - - - 1 - - - - 1 - - - - - --- ----- --------------- ----------- --- ----- ---- -- --- - - - - - - - - - - 1 - - - - - - - - -
G//41C
f//41C
fr,/gp
21.92
3.24
1.66
12.09
5.97
-1.73
- - - - - - - - - - - - -- -
gs2/41C(I= 0)
2.57
- - - - - ---------- - - - - - - - - - - - - - - - - - - - - -
3.35
i
0.35
----- -----
_g's2/41C (I= 1)
0.00
0.00
0.06
~-~-------.-----
Gr}/41C+ G//41C
G.,f.,/41C+ Gpfr,/41C
f.,2/41C+ fr,2/41C
gs2/41C+ g's2/41C
30.20
11.00
4.04
2.57
17.48
2.60
1.69
14.37
2.11
1.64
30.13
10.97
4.34
2.86
32.09
12.92
6.95
3.35
24.52
8.00
2.72
0.41
------- ---'----
---------1
0.49
0.01
---- -------·-··--···------i---~·--------
24.63
,8.13
1.96
-0.41
28.30
9.91
3.53
0.50
-
86.0
28:92
lO.37
2.78
-0.48
-.~---
54.4
60.5
Table II, we compare the .results of I with those obtained from the analysis
which also involves T= 0 states for some typical cases. As was assumed in the
previous analysis, here we take 450 Me V for the scalar meson masses.
(i)
hex=n+ V + S
The first example is the case hex = n + V + S. If we sum up the coupling
constants of the 1= 0 and 1= 1 mesons in the pn~sent results, the obtained values
are in good agreement with the previous ones, as is shown in Table II.
In this Table we also show the coupling constants obtained by the K-matrix
method from the same experimental data which were used to evaluate the Kantor
_amplitudes. Comparison of the numerical figures in the second and fourth rows
of, this Table indicates that larger O)-meson coupling constant and smaller p-meson
coupling constant are _obtained in reference 7) compared with the dispersion
relation approach, though the ·1=1 scalar meson contribution is unlikely in
both cases. - The jp/gp ratio, which is negative in the K-matrix method, turns
to be positive and now is consistent. with the information from the electromagnetic form factor of nucleon. *)
*) It is, however, noted that the negative value is not characteristic of the· K-matrix method.
In fact we obtain a positive value for small ms and g.",2/41C, as shown in the recent analysis (M.Kikugawa, W. Watari and M. Yonezawa, Prog. Theor. Phys. Suppl. Extra Number (1968), 160).
One-Bason-Exchange Modes and Two- and Three-Pion Exchange
For the S-wavestates, we use the once
s,ubtracted dispersion relation to estimate
the Kantor amplitudes in order to secure
the convergence of the dispersion integrals. We have, therefore, to com pare
the opce subtracted value of the lefthand-cut contribution with the expenmental data. *), The results are given in
Fig. 1. The agreement with experiment
is improved over the case of the K-matrix
method, though 110t satisfactory.
Generally speaking, the coupling
constants determined from the K-matrix
method seem to tend larger than those
by the dispersion theoretic one. This
is due to'the fact that only the on-energyshell rescattering effect IS involved III
the ](-matrix method.
(ii)
h ex - h27r = TC
-5.
I
x
I
x
I
x
I
-10.
x
+ V +S
The next two exam pIes are of the case
hex - h27r = rc + V + S. In the corresponding
cases in I of these exam pIes, the three
vector coupling constants G V 2 = G(} + G/,
Gvfv=G.,f.,+ Gpfp and f/=f.,2+ fp2 do
not satisfy the relation
200 MeV
100
o
465
I
70.
x
50.
Com parison of the previous results
with the present ones show that the viola-,
tion of this inequality is due to the small
fv 2j 4rc in the previous results. The pre- 30.
vious values,' however, are very close to
the present values. We have the largest
difference for fv 2j 4rc. Even in this case
10.
the prevIOUS value IS smaller than the
present value only by 20 %.
O'~~--I~O-O--~~--~O~O--MeV
This analysis, combined with the
Fig. 1. Comparison of the theoretical S-wave
results of the previous analysis, may be
amplitudes of the case n+ V +S with the
enough to indicate that the TC + V + S
experimental data.
*)For this problem see the discussion (p. 278) of S. Furuichi, Prog. Theor. Phys. Supp!. No.
39 (1967), 190.
466
S. Furu£ch£, 1-1. Suemitsu, IV. Watari and M. Yonezawa
(or 2rr) can well approximate nucleon-nucleon scattering and, therefore, the previous
analysis which considers only the T= 1 nucleon-nucleon states gives essentially
correct answers.
The best value for tmax may be at around 9/i. If we examine the values of
tmax more closely, a rather small tmax as ::;8/i was favoured in the previous
analysis. The consideration of the T= 0 nucleon-nucleon states slightly increases
this value and we have tmax?,:-9fJ,2 in the present analysis which may somewhat
slacken the impression of. the too small tmax of the previous analysis.
.
(B) In the following we make a detailed investigation of the case (ii). We
have shown in I that if the squared two-pion momentum t = - (k 1 + k 2) 2 is cut off
at t max r'-J9fJ,2, then the two-pion amplitudes assume reasonable magnitude and the
1=0 scalar meson can be regarded as a phenomenological substitute for the twopion-exchange amplitudes. The main purpose of the present paper is to confirm
this conclusion by performing more refined analysis by considering the T=O
nucleon~nucleon states as well as the T= 1 states.
So far we have fixed the masses of the scalar meson as· 450 Me V and the
pion-nucleon coupling constant gn:2/4rr as 14.4. In the following analysis we treat
these quantities also as adjustable parameters.
First let us search for the best values for these parameters in the case
hex = 1C + V + S. The obtained results are shown in Fig. 2 and the obtained best
values are gn:2/41C=14 and ms=390 MeV. Therefore we take two values of ms,
ms=450 MeV and 390 MeV and try the case hex-h2n:(gn:2/4rr,t~ax) =rr+ V+S.
Since our interest is ina critical case where the two-pion-exchange amplitudes takes over the role of the. scalar meson, and hence, gs2/4rrr'-...'0, the results
should be rather insensitive to the scalar
meson mass and these special choices of
ms will not impair the. validity ot the
obtained conclusion.
85
The results of the x2-minimum research analysis are given in Figs. 3 and 4.
Of other results, we would like to
note that the case hex = h 2n: + rr + V + (S)
gives better . fit than the· case hex =1C
80
+ V + S, if we choose tmaxr'-...'9fJ,2 (X~in"-'50
for the former a~d X~in~80 for the latter).
This would be an evidence for the physical reality of the two-pion-continuum
effects. In order to see that this improve75 360 380 400· 420 440 ms
menf comes from the continuum nature
(MeV)
of the L<l part rather than the higher
Fig. 2. The X2-curves vs. the scalar meson
angular
momentum pion-pion states with
mass for variolfs values of the pion-nuL>2, we have separated the pion-pion
cleon coupling constant gn:2/4rr.
One-Eason-Exchange Modes and Two- and Three-Pion Exchange
80
m s =450 MeV
X2
ms
80
x"
70
==
467
390MeV
70
60
60
50
50
12.5
13.0
13.5
14D
14.5
Fig. 3a. The X2-curves vs. the pion-nucleon coupling constant g,,2/4rc for the various values
of tmax. ms=450 M~V.
9si;'47T
ms=450MeV
- - [=0
-------- [= 1
1.5
13.0
12.5
9~/477'
13.5
14.0
14.5
g~/477'
Fig; 3b. The X2-curves vs"the pion-nucleoncoupIing constant g",.2/4rc the varidus values of
tmax. ms = 390 MeV.
2
gs}
ms= 390MeV
~7T
1.5
1=01= 1
1.0
1.0
0.5
0.5
tmax::::
?
t~
~
...t---::aJi"IO ___ ~--------I-__~
tmax=:__:~"Y-~x_~____
13.0
13.5
14.0
14.5
9~/477'
Fig. 4a. . The scalar meson coupling constants
gs2/4rc and g's2/4rc vs. the pion-nucleon coupling constant [/,,2/4rc for various values tmax.
ms=450MeV.
.
i mcx=812.5
t mcx=?
13.0
13.5
t
14.0 . 2 14.5
9"0/477'
Fig. 4b. The scalar meson coupling constants
gs2/4rc and g's2/4rc vs. the pion-nucleon cou·
pIing constant g,,2/4rc for various values tmax.
ms=390MeV.
states with L>2 from the two-pion-exchange amplitudes and repeated the analysis
for the remainder (L<l). The results show that the improvement is mainly
due to the continuum nature of the L<l part. Mindr improvement is also made
by the parts with L>2 which are necessarily involved in the two-pion exchange.
We also plot the scalar coupling constants Ys2/4n (1 = 0) and Ys'2/4n (1 = 1) vs. ,
y,,2/4n in Fig. 4. When the two-pi on-exchange amplitudes are taken into account,
sometimes the coupling constant Ys'2/4n retains finite values. The maximum
values which we have encountered with is 0.09 ± 0.09. From the physical point
of view, this small coupling constant Can be regarded as zero.
With tmax = 10,u2 and y,,2/4n = 13.5 we obtain a solution which is considered to
be very close to the over-all minimum of X2 • This solution is given in Table III
and in Fig. 5. Figure 5a is concerned with those states of which scattering
468
S.Furuichi, H. Suemitsu, W. Watari and M. Yonezawa
1.5
o
x
100
200MeV
~--~~-+~~r-~~
-0.6
I
x
-1.0
I 1
-1.4
1.0
-1.8
0.5
0.4
x
0.0 1----r-----1Ft----t~--+-l-0.0 1---L--10l-0--L..-2-0+-01- MeV
x
-0.4
x
100
100
-0.8
-1.0
-1.2
200MeV
-0.2
1
I
-0.6
1
-1.4
x
-1.0
I
1
x
-1.4
x
x
-1.8
1.0
I
I
0.8
x
x
2.5
0.6
100
200 MeV
t
.
'02
2.0
1
!
1,5
I
1.0
(a )
x
x
x
100
200 MeV
Fig. 5. Comparison of the theoretical amplitudes of the case h2n' + n-+ V + S with the experimental
ones. (a) Those which are used to determine the coupling constants, (b) some of higher
angular momentum states and, (c) the S states.
One-Bason-Exchange Modes and Two- and Three-Pion Exchange
100
0.0
200 MeV
100
O.
469
:200 MeV
I
P2
-0.1
f
-0.2
i
f
!
-0.3
5.
0.3
0.2
x
I
0.1
0.0 L - - - - - L - _ - I - - _ - - - - L - _ - 4 - _
100
200MeV
0.0
-0.1
l
-10.
100
200MeV
I
x
I
-·0.2
0.2
0.1
70.
t
x
0.0 '----'~~-__ll---L-----+I100
200MeV
100
0.0
-0.1
x
200MeV
x
50.
30.
x
10.
-0.2
O. L - _ - L - _ - - 1 . ._ _.l....-_..:..L100
,~OO
MeV
-0.3
-0.4
(b )
(C)
Fig. 5. Comparison of the theoretical amplitudes of the case h2",+n+ V+S with the experimental
ones. (a) Those which are used to determine the coupling constants, (b) some of higher
angular momentum states and (c) the S states.
470
S. Furuichi, H. Suemitsu, W. "YVatari and M. Yonezawa
Table III. The coupling constants of the various mesons for tmax=10,u2, g".2j4n=13.5 (ms=450MeV)
in the case hex = h2n: + n + V + S.
gn:2/4n= 13.5,
Goo2/4n= 1O.68±1.46,
foo2/4n=0.59±0.16,
foo/ goo = 0.44,
Gp2 /4n = 13.30± 1.40,
.t;, 2/4n= 1.98±0.20,
gs2/4n=0.00±0.14,
g! s2/4n=0.05±0.05,
!phIp = 1.70,
X~'in =52.38.
amplitudes are used to determine the coupling constants. The fit of the theory
to the experiment is generally good except for the sD l and sD2 amplitudes. The
agreement with the experimental data is satisfactory ab'out these states and other
higher angular momentum states. We give the theoretical and experimental
amplitudes of the P2, sFJ and IFs in Fig. 5b as examples of higher angular
momentum states.
As noted before, in the calculation of the Kantor amplitudes for S states,
we have used the once subtracted form for the right-hand-cut contribution. This
means that if we use the present values for the Kantor amplitudes we should compare the once-subtracted values for the theoretical and experimental amplitudes or
we have to treat the scattering lengths as parameters. The once subtracted values
of the theoretical two-pion-exchange S-wave amplitudes are given in Table I in
the parentheses.
The results are given in Fig.5c. It shows that the agreement with the experimental data is satisfactory both for the sSl and ISO states. These results
imply that the S-wave amplitudes can be explained by the contributions from
the one-pion, two-pion (including p-meson) and CO-meson exchange to considerable
extent except for the scattering lengths. Conversely, the interactions which are
not involved in the present analysis are of such short range nature as to mainly
affect only on scattering lengths.
§ 4.
Conclusions and a few remarks
. In this paper we have repeated the same kind of analysis of I, considering
both the T= 1 and. 0 nucleon-nucleon states, and we have shown the followings:
(1) The previous results obtained from the analysis of the T= 1 states also
hold 'even if we consider the T= 0 nucleon-nucleon states. We can obtain the
two-pion-exchange amplitudes which well explain nucleon-nucleon scattering if
we take "t max rv9ji. In this case the 1=0 scalar meson, which is an essential
ingredient in the OBE model, is no longer necessary.
(2) For the OBE model, the treatment of the dispersion relation has shown
an improvement on the ratio of the p-meson coupling constant, fp/g p, which is'
negative in the K-matrix method and inconsistent with the electromagnetic form
factor.*) Also some improvement is observed about the S-wave states.
*)
For this problem, see the footnote on page 464.
One-Bason-Exchange Modes and Two- and Three-Pion Exchange
471
(3) The most notable fact in the present results is that the two-pion-exchange
amplitudes give better fit' to the experimental data th~m the one-scalar-meson
exchange.
(4) This is consistent with the analysis of the nucleon electromagnetic form
factors in which the two-pion-exchange effects with uncorrelated parts markedly
improve the fit with experiment. 8) These results indicate the favourable features
of the two-pion-exchange-continuum contributions and their reality.
(5)
When we taketmax"--'9ti, the I=L=Opart dominates the two-pionexchange amplitudes and the 1= L = 1 part becomes significantly reduced; as is
seen in the change of the coupling constants for the cases with and without
two-pion-exchange. This small amplitude for the 1= L = 1 part is quite natural
from the effective' mass distribution 4) of the pions in the uncorrelatedtwo-pion
exchange for small mass region.
(6) The cutoff parameter for the squared momentum of exchanged pions
tmax should be understood in connection with the pion-pion interaction, and other
short range interactions. In fact, it has been shown that the pion-pion interaction
which does not lead into a resonance, serves as 'a cutoff factor. 9) This value of
tmax is, therefore, considered in general to be state-dependent.
In this sense the
obtained parameter for the p meson may still subject to some change, as the
present t,max' may be considered determined mainly in respect of the I==' L = 0 state.
(7) It is interesting that the small tmax just corresponds to the low scalar
meson mass"-"400 Me V which was obtained from the OBE analysis of the Kantor
amplitudes, though the restrictions from the magnitude of the two-pion-exchange
amplitude is the main factor of the requirement for such a small tmax.
(8) This result implies that the force range of the contributions from the
1= L = 0 part or the scalar meson in nucleon-nucleon scattering is considerably
longer than that of the 1=1 part (the p-meson range) and the three-pion or the
(j) meson.
This may allow us to divide the interaction range into
I
range of (In) >range of (IS) >range of (1 V) ....
(9) The interpretation, of these ranges in terms of the two-pion exchange
gives with t max r-J9/i
range of (I=L=O or IS»(3,a)-I,
and, therefore, this range is very characteristic of the two-pion exchange. From
the mass distribution of the exchanged two' pion, it is seen4) that the L1 (1236)
resonance gives very important contribution to a small t region for the 1= L = 0
part. Considering the smallness of the contrib~tions from other higher resonances 10 )
in such small t region, the present results obtained for the L1 (1236) resonanGe
approximation may be valid to a considerable extent.
As the features of the interaction in the intermediate region become considerably cleared up now, .it is possible to attack the inner region of nuclear in-
472
S. Furuichi, H. Suemitsu, W. Watari and M. Yonezawa
teractions of the smaller range than that of (1 V). The cutoff tmax,,--,9/-L2 may be
interpreted as a possible representation of inner effects. Also, the consideration
of the inner region effects is essential to explain the S-wave data without sub. traction.
Acknowledgements
The authors would like to express their gratitude to Professor S. Sawada,
Dr. K. Watanabe and Dr. H. Kanada for valuable discussions. Some parts of
the numerical calculations were carried with HITAC 5020 at the Computer
Center, University of Tokyo.
References
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