- Email: [email protected]
Collinear processes in the quark model
ANNALS
OF
44, 35-56
PHYSICS:
Collinear
(1967)
Processes
in the
Quark
Model
Yu. M. MALYUTA Physics
A quark
Institute
model
of the
of hadrons
SU(3) Relations are found of meson and baryon ential cross sections
x
Ukrainian
Academy
is proposed,
SU(3)
x
...
based
X SU(3)
of Sciences,
Kiev,
on the collinear (Zj + 1
U.S.S.R.
group
cofactors).
within the framework of this model for the partial widths resonances, as well as relations for the total and differof a number of elastic and inelastic scattering processes. I.
INTRODUCTION
In a number of recent researches (I)-( 4) many predictions have been made concerning collinear processeswithin the framework of collinear groups SU( 6), and SU(3) X SU(3). Unfortunately, some of these predictions have been in gross disagreement with the experimental data (5). The disagreement is, apparently, due to the violation of collinear symmetries by unitarity (6). In the hope of obtaining better predictions Lipkin proposed a new approach to the description of collinear processesbased on an “independent quark model” (7). By means of this approach it became possible to obtain relations which are not predicted by collinear symmetries, but which are in good agreement with experiment. Thus, the new approach proclaimed the idea of “symmetry without symmetry” ( 7). In the present paper we propose a quark model of hadrons based on the collinear group SU(3) x SU(3) x . . * X SU( 3) (2j + 1 cofactors), (1) where each cofactor correspondsto a definite helicity state.’ For the description of collinear processeswithin the framework of group (I), the impulse approximation (IO), ( 11) , well known in nuclear physics, is applied. In the quark language the impulse approximation is formulated as follows: the hadron-hadron scattering amplitude is equal to the sum of the amplitudes for scattering of each quark in the incident hadron by each quark in the target hadron. This approximation actually permits us to go beyond the framework of group (1) and leads to the relations that follow from “symmetry without symmetry”. 1 The group (1) is the subgroup of U[3(2j formation along the z-direction. The group for the description of hadron shell structure
+ l)] which commutes with a Lorentz U[3(2j + l)] was proposed in papers in the jj coupling scheme. 35
trans(8), (9)
36
MALYUTA
TABLE su (3) 8
I Multiplicity
Y, I 1, FS
Ull&
U21z3
-1, $8
II. HADRON
U3Ql U&l
DECAYS
In this section we shall obtain the branching ratios for different decay modesof mesons and baryons. Let us consider the two-particle partial decay of some hadron Hk+HiHj, (2) which can proceed through several isotopic channels. The partial width of the resonancein the LJ state is given by (12) :
(3)
COLLINEAR
PROCESSES
TABLE E’, I
IN
THE
QUARK
MODEL
37
II Multiplicity
1, N
where M is the massof the resonance,Pouti s the magnitude of the spatial momentum of the produced particles in the rest system of the resonance, and (Hk 1HiHi) is the vertex part for the decay (2). To find the relations between the vertex parts (Hk 1HiHi) we apply impulse
35
YALYUl’A
FIG.
approximation.
We restrict
1
ourselves to an investigation
of the decays
COLLINEAR
PROCESSES
IN
FIG.
THE
QUARK
MODEL
39
2
for which the relative widths have been experimentally checked (13). Using Tables I and II, it is not difficult to construct diagrams contributing to the vertex parts (Hk 1HiHi) in the impulse approximation. These diagrams are pre-
40
MALTUTA
sented in Figs. 1-3, for meson decays,’ and in Figs. 4-10, for baryon decays.2 Denoting the topologically equivalent diagrams by the same parameters and applying formula (3), it is possible to compute the relative widths of the resonances. The results of the computations for meson resonances are given in Table IV, and for baryon resonances in Table V (the symbol I’0 denotes the relative width, uncorrected for the phase-space ( 2PoUt) 2L+1/M2). The results which are in good or satisfactory agreement with the experimental data (1s) are marked with asterisks. 2 Since the hadrons involved in the processes under consideration belong of different groups (l), it is necessary to refer to the common subgroup of SU(3) x @‘J. (where J, is the spin component in the direction of motion; parameter). Then the basic quark triplets of the group (1) may be written in
to multiplets these groups cp is the real the form
where (u1u2u3) is a unitary triplet and a,, is a helicity singlet. The graphic notations quarks are given in Table III (where B is the baryon number, Y is the hypercharge, is the isospin). The v-lines, demonstrating conservation of helicity, are omitted drawings. The numerals near the diagrams mark the number of equivalent diagrams.
of the and I in the
COLLINEAR
PROCESSES
III.
ELASTIC
IN
THE
QUARK
MODEL
41
SCATTERING
Let us apply impulse approximation to obtain the Johnson-Treiman relations (14) and someother relations for elastic scattering crosssectionswhich agree very well with the experimental data. We shall consider the following forward-scattering reactions: K+p + K+p,
K+n + K+n,
K-p + K-p,
K-n -+ K-n,
42
MALYUTA
Trip 4 a+p,
r-p
* r-p,
PP 4 PPY
pn. -+ pn,
PP 4 15P,
pn4pn.
(4)
Using Tables I and II, it is not difikult to construct diagrams contributing to the amplitudes of elastic scattering (4). These diagrams are presented in Figs. 11-15 (see footnote 2). We can now express all the scattering amplitudes (4) in terms of three parameters :
COLLINEAR
PROCESSES
IN
FIG.
THE
QUARK
6
= 6a + 3b, (K+n 1K+n)
= 6a + 3b,
{K-p
1 K-p)
= 6a + 3b + 2c,
(K-n
1K-n)
= 6a + 3b + c,
{r’p
[ ?r+p) = 6u + c,
(r-p
1x-p)
= 6u + 25
MODEL
43
44
MALYUTA
FIG.
7
(mJI PP>= 9% (P I P> = 9% (FPI l?P)= 9u + (@L] pn>= 9a +
56, 4c,
COLLINEAR
PROCESSES
IN
FIG.
where parameters
(a is the scattering vertices
THE
MODEL
45
8
a denote the diagrams with
contribution),
QUARK
parameters
vertices
a + b denote the diagrams with
MALYUTA
Fra.9
(b is the correction
due to unitary
symmetry
breaking),
and parameters
a + c
COLLINEAR
PROCESSES
IN
THE
QUARK
MODEL
47
FIQ. 10
(c is the annihilation contribution).
According to the optical theorem, a similar
4s
MALYUTA
TABLE QlMk
R
1
TABLE Decay
III
LJ
I
IV ro
1* 1* 1* 1* 12 3* 2* 1* 1*
TABLE Decay
LJ
D 312 DSlZ F 512 F 5/z P 512 F5/2 P 3/2 P 3/2 D 3/2 0312 h/2 P 312 0512 D 512 0512 D5/2 P612 D 312 D 312 & 12
Notation
r
rexp
1 1 1 0.03 1 0.29 0.56 1 0.14
30 38 60 40 91 5.5 3.6 50 50
V I-0
1 1 1 1 1 0 2 1 1 4 2 2 1 4 4 2 2 4 3 3
r
1* 1.94* 1* 0.64 0.22* 0* 1 2.13 1* 3.30* 3.08 0.35 1* 2.97 0.39* 2.43 0.11* 1 0.99 1.24
relp
29 56 75 9 15 1 10 90 15 30 5 30 60 3 10 16 10 65 5 25
COLLINEAR
PROCESSES
K+p -
K+p
K+~L-
K+n
IN
THE
QUARK
MODEL
49
dependence on the parameters will hold for total cross sections. Excluding parameters a, b, c, we obtain the following relations for the total cross sections:3 >$u( K+p)
- a(Kp)]
u(K+p) u(m) 42%)
- &P)
= u(K+n)
- u(Kn)
(5a)
= &rip)
- u(7r-p),
(5b)
= u(K+n),
(6)
= dP)>
(7)
= Men)
- dpn)l
(8a)
3 ReMions (5a, b) are the Johnson-Treiman relations (14). Relations (6) and (7) were obtained by Lipkin (7) within the framework of an independent quark model. Relations @a, b) were obtained by Freund (15) from universality and 8(E) invariance.
50
MALYUTA
K-p e
K-p
K-n-+
K-n
FIG.
12
= 5[a( F-p) = 5b(PP)
- a(a+p)] - dl?n)l.
These relations are very well satisfied experimentally 18 BeV/c kinetic energy of the incident hadron. IV.
INELASTIC
(8b)
(16)
(SC) from
6 BeV/c
to
SCATTERING
Let us apply impulse approximation to obtain the relations for inelastic scattering differential cross sections in the forward and backward direction. We restrict ourselves to an investigation of the reactions K+p -+ K*‘A++,
K-p
4
Eon,
K-p
K-p
4
a*X*-,
-+ n--z*+,
COLLINEAR
PROCESSES
IN
THE
QUARK
MODEL
51
e x
Tr+p-Tr+p
2
*s-+-.-“ssr-,
FIG.
13
K-p * a+~-,
K-p + K”Eo,
K-p + K+Z-,
K-p + K*+e-,
r+p + K*+Z+,
(9)
n--p + K+Z-,
for which the crosssections da/dQ( 0”) and da/dQ( 180”) have been experimentally checked (5). Using Tables I and II, it is not difficult to construct diagrams contributing to the amplitudes of inelastic scattering (9). These diagrams are presented in Fig. 16 and 17 (see footnote 2). Denoting the topologically nonequivalent diagrams by different parameters, we obtain the following expressionsfor the scattering amplitudes (9) : (K+p 1K*‘A++)
= &u,
(K-p 1Z?“n) = 2b,
52
MALYUTA
PP'PP
P"-P"
FIQ.
(K-p
(K-p (K-p
1 .R-x*+)
=
a,
14
(K-p
I7r%*-)
= 2c,
I?r+r)
= 2c,
(K-p
1K”Eo) = 2c,
1K+K)
= 2c,
W-P
I K *+E-)
(7r’p 1K*+z+)
= b,
(n-p I K+;r;-)
= 25 = 2c
(a is the exchange scattering contribution, b is the annihilation contribution, and c is the superposit,ion of exchange scattering and annihilation contributions). Excluding parameters a, b, c, we obtain the following relations for the differential
COLLINEAR
cross sections in the forward $
(Ktp
[ K*‘A++)
PROCESSES
IN
and backward = 3g
THE
QUARK
MODEL
53
direction!
(K-p 1=-z*+),
4 The quantities denoted in (10) by the symbols &/dQ are equal to the differential cross sections with an accuracy up to the factors (P,,,/SPi,) exp [- (PM + UB)I t I], where P,,t/sPi, is the phase-space correction (12), and exp (-aH 1 t 1) is the structure factor which is the Fourier transform of the quark wavefunction in the hadron (IO), (11) (Pi, and P,,t are the magnitudes of the spatial momenta of the incoming and outgoing particles in the center-ofmass system, s is the square of the total energy, ( t / is the square of the momentum transfer).
54
MALYUTA
u-p --+Tr+~*-
K-P- lT+tFIG.
g
(I$(
Bz)
= 4%
(&I
16
K*+z+),
I’
(10)
= g (K-p 1E-+x-) = !!.g (K-p ( jy*+z-) = g (K-p ( KoEO) = g (7rp 1 &-f-z->.
\-IT
+p-
K”+ t+
FIG. 17 TABLE
VI
:Process
K+p --) K*OA++ K-P ---) ,-i-f*+ K-p --) rr+iz*K-P --f ?r+zK-p --* K+r K-P + K*QK-p --f KOF
X
2.26 3 2.24 3 2.24 3 2.24 3 2.24 3 2.24 3 3
450 1360 80 37 9 3.5 8 2.5
f f f f f f
60 140 10 10 4 1.8
f
1
0.6 f
0.2
0.5 f 2.5 f
0.3 0.6
55
14 3.5 20 19 22 7 5.2 5.3 2.5
f6 f 1.8 f4 f1.4 f 1.3 f 0.6
?4 % 1 1 1 1 1 1 1 1 1 1 1
56
YALY
A comparison of Table VI, where predictions (10) factors indicated
UTA
relations ( 10) with the experimental data (5) is presented in X is the inverse strength factor that normalizes the data to (the comparison was carried out with an accuracy up to the in footnote 4). ACKNOWLEDGMENT
The author wishes to thank Professor N. N. Bogoliubov RECEIVED:
November
for valuable
discussions.
18, 1966 REFERENCES
J. J. COYNE, S. MESHKOV, D. HORN, M. KUGLER, AND Rev. Letters 16, 373 (1965). 2. D. V. VOLKOV, Nuovo Pimento 40,281 (1965). 3. J. KUPSCH, Nuovo Cimento 40, 287 (1965). 4. H. RUEGG AND D. V. VOLKOV, Nuovo Cimento 43, 84 (1966). 6. J. D. JACKSON, Phys. Rev. Letters 16, 990 (1965). 6. A. SALAM, in “Proceedings of 1965 the Oxford International Conference 1.
J. D. CARTER,
H. J. LIPKIN,
Phys.
on Elementary Particles.” Rutherford High Energy Laboratory, Oxford, England, 1965. 7. H. J. LIPKIN, Lecture Notes, Yalta International School, Yalta, U.S.S.R., April 1966. 8. M. M. MILLER, Phys. Rev. Letters 14,416 (1965). 9. Yu. M. MALYUTA, Nucl. Phys. 74, 625 (1965).
10. G. F. CHEW AND G. C. WICK, Phys. Rev. 85, 636 (1952). 11. G. F. CHEW AND M. L. GOLDBERGER, Phys. Rev. 87,778 (1952). la. K. NISHIJIMA, “Fundamental Particles.” Benjamin, New York, 13. A. H. ROSENFELD, Lecture Notes, Yalta International School,
1964. Yalta, U.S.S.R.,
April
1966. 14. K. JOHNSON AND S. B. TREIMAN, Phys. Rev. Letters 14, 189 (1965). 15. P. G. 0. FREUND, Phys. Rev. Letters 16, 929 (1965). 16. W. GALBRAITH, E. JENKINS, T. KYCIA, B. LEONTIC, R. PHILLIPS, RUBINSTEIN, Phys. Rev. 138, B913 (1965).
A. READ,
AND
R.
OF
44, 35-56
PHYSICS:
Collinear
(1967)
Processes
in the
Quark
Model
Yu. M. MALYUTA Physics
A quark
Institute
model
of the
of hadrons
SU(3) Relations are found of meson and baryon ential cross sections
x
Ukrainian
Academy
is proposed,
SU(3)
x
...
based
X SU(3)
of Sciences,
Kiev,
on the collinear (Zj + 1
U.S.S.R.
group
cofactors).
within the framework of this model for the partial widths resonances, as well as relations for the total and differof a number of elastic and inelastic scattering processes. I.
INTRODUCTION
In a number of recent researches (I)-( 4) many predictions have been made concerning collinear processeswithin the framework of collinear groups SU( 6), and SU(3) X SU(3). Unfortunately, some of these predictions have been in gross disagreement with the experimental data (5). The disagreement is, apparently, due to the violation of collinear symmetries by unitarity (6). In the hope of obtaining better predictions Lipkin proposed a new approach to the description of collinear processesbased on an “independent quark model” (7). By means of this approach it became possible to obtain relations which are not predicted by collinear symmetries, but which are in good agreement with experiment. Thus, the new approach proclaimed the idea of “symmetry without symmetry” ( 7). In the present paper we propose a quark model of hadrons based on the collinear group SU(3) x SU(3) x . . * X SU( 3) (2j + 1 cofactors), (1) where each cofactor correspondsto a definite helicity state.’ For the description of collinear processeswithin the framework of group (I), the impulse approximation (IO), ( 11) , well known in nuclear physics, is applied. In the quark language the impulse approximation is formulated as follows: the hadron-hadron scattering amplitude is equal to the sum of the amplitudes for scattering of each quark in the incident hadron by each quark in the target hadron. This approximation actually permits us to go beyond the framework of group (1) and leads to the relations that follow from “symmetry without symmetry”. 1 The group (1) is the subgroup of U[3(2j formation along the z-direction. The group for the description of hadron shell structure
+ l)] which commutes with a Lorentz U[3(2j + l)] was proposed in papers in the jj coupling scheme. 35
trans(8), (9)
36
MALYUTA
TABLE su (3) 8
I Multiplicity
Y, I 1, FS
Ull&
U21z3
-1, $8
II. HADRON
U3Ql U&l
DECAYS
In this section we shall obtain the branching ratios for different decay modesof mesons and baryons. Let us consider the two-particle partial decay of some hadron Hk+HiHj, (2) which can proceed through several isotopic channels. The partial width of the resonancein the LJ state is given by (12) :
(3)
COLLINEAR
PROCESSES
TABLE E’, I
IN
THE
QUARK
MODEL
37
II Multiplicity
1, N
where M is the massof the resonance,Pouti s the magnitude of the spatial momentum of the produced particles in the rest system of the resonance, and (Hk 1HiHi) is the vertex part for the decay (2). To find the relations between the vertex parts (Hk 1HiHi) we apply impulse
35
YALYUl’A
FIG.
approximation.
We restrict
1
ourselves to an investigation
of the decays
COLLINEAR
PROCESSES
IN
FIG.
THE
QUARK
MODEL
39
2
for which the relative widths have been experimentally checked (13). Using Tables I and II, it is not difficult to construct diagrams contributing to the vertex parts (Hk 1HiHi) in the impulse approximation. These diagrams are pre-
40
MALTUTA
sented in Figs. 1-3, for meson decays,’ and in Figs. 4-10, for baryon decays.2 Denoting the topologically equivalent diagrams by the same parameters and applying formula (3), it is possible to compute the relative widths of the resonances. The results of the computations for meson resonances are given in Table IV, and for baryon resonances in Table V (the symbol I’0 denotes the relative width, uncorrected for the phase-space ( 2PoUt) 2L+1/M2). The results which are in good or satisfactory agreement with the experimental data (1s) are marked with asterisks. 2 Since the hadrons involved in the processes under consideration belong of different groups (l), it is necessary to refer to the common subgroup of SU(3) x @‘J. (where J, is the spin component in the direction of motion; parameter). Then the basic quark triplets of the group (1) may be written in
to multiplets these groups cp is the real the form
where (u1u2u3) is a unitary triplet and a,, is a helicity singlet. The graphic notations quarks are given in Table III (where B is the baryon number, Y is the hypercharge, is the isospin). The v-lines, demonstrating conservation of helicity, are omitted drawings. The numerals near the diagrams mark the number of equivalent diagrams.
of the and I in the
COLLINEAR
PROCESSES
III.
ELASTIC
IN
THE
QUARK
MODEL
41
SCATTERING
Let us apply impulse approximation to obtain the Johnson-Treiman relations (14) and someother relations for elastic scattering crosssectionswhich agree very well with the experimental data. We shall consider the following forward-scattering reactions: K+p + K+p,
K+n + K+n,
K-p + K-p,
K-n -+ K-n,
42
MALYUTA
Trip 4 a+p,
r-p
* r-p,
PP 4 PPY
pn. -+ pn,
PP 4 15P,
pn4pn.
(4)
Using Tables I and II, it is not difikult to construct diagrams contributing to the amplitudes of elastic scattering (4). These diagrams are presented in Figs. 11-15 (see footnote 2). We can now express all the scattering amplitudes (4) in terms of three parameters :
COLLINEAR
PROCESSES
IN
FIG.
THE
QUARK
6
= 6a + 3b,
{K-p
1 K-p)
= 6a + 3b + 2c,
(K-n
1K-n)
= 6a + 3b + c,
{r’p
[ ?r+p) = 6u + c,
(r-p
1x-p)
= 6u + 25
MODEL
43
44
MALYUTA
FIG.
7
(mJI PP>= 9% (P I P> = 9% (FPI l?P)= 9u + (@L] pn>= 9a +
56, 4c,
COLLINEAR
PROCESSES
IN
FIG.
where parameters
(a is the scattering vertices
THE
MODEL
45
8
a denote the diagrams with
contribution),
QUARK
parameters
vertices
a + b denote the diagrams with
MALYUTA
Fra.9
(b is the correction
due to unitary
symmetry
breaking),
and parameters
a + c
COLLINEAR
PROCESSES
IN
THE
QUARK
MODEL
47
FIQ. 10
(c is the annihilation contribution).
According to the optical theorem, a similar
4s
MALYUTA
TABLE QlMk
R
1
TABLE Decay
III
LJ
I
IV ro
1* 1* 1* 1* 12 3* 2* 1* 1*
TABLE Decay
LJ
D 312 DSlZ F 512 F 5/z P 512 F5/2 P 3/2 P 3/2 D 3/2 0312 h/2 P 312 0512 D 512 0512 D5/2 P612 D 312 D 312 & 12
Notation
r
rexp
1 1 1 0.03 1 0.29 0.56 1 0.14
30 38 60 40 91 5.5 3.6 50 50
V I-0
1 1 1 1 1 0 2 1 1 4 2 2 1 4 4 2 2 4 3 3
r
1* 1.94* 1* 0.64 0.22* 0* 1 2.13 1* 3.30* 3.08 0.35 1* 2.97 0.39* 2.43 0.11* 1 0.99 1.24
relp
29 56 75 9 15 1 10 90 15 30 5 30 60 3 10 16 10 65 5 25
COLLINEAR
PROCESSES
K+p -
K+p
K+~L-
K+n
IN
THE
QUARK
MODEL
49
dependence on the parameters will hold for total cross sections. Excluding parameters a, b, c, we obtain the following relations for the total cross sections:3 >$u( K+p)
- a(Kp)]
u(K+p) u(m) 42%)
- &P)
= u(K+n)
- u(Kn)
(5a)
= &rip)
- u(7r-p),
(5b)
= u(K+n),
(6)
= dP)>
(7)
= Men)
- dpn)l
(8a)
3 ReMions (5a, b) are the Johnson-Treiman relations (14). Relations (6) and (7) were obtained by Lipkin (7) within the framework of an independent quark model. Relations @a, b) were obtained by Freund (15) from universality and 8(E) invariance.
50
MALYUTA
K-p e
K-p
K-n-+
K-n
FIG.
12
= 5[a( F-p) = 5b(PP)
- a(a+p)] - dl?n)l.
These relations are very well satisfied experimentally 18 BeV/c kinetic energy of the incident hadron. IV.
INELASTIC
(8b)
(16)
(SC) from
6 BeV/c
to
SCATTERING
Let us apply impulse approximation to obtain the relations for inelastic scattering differential cross sections in the forward and backward direction. We restrict ourselves to an investigation of the reactions K+p -+ K*‘A++,
K-p
4
Eon,
K-p
K-p
4
a*X*-,
-+ n--z*+,
COLLINEAR
PROCESSES
IN
THE
QUARK
MODEL
51
e x
Tr+p-Tr+p
2
*s-+-.-“ssr-,
FIG.
13
K-p * a+~-,
K-p + K”Eo,
K-p + K+Z-,
K-p + K*+e-,
r+p + K*+Z+,
(9)
n--p + K+Z-,
for which the crosssections da/dQ( 0”) and da/dQ( 180”) have been experimentally checked (5). Using Tables I and II, it is not difficult to construct diagrams contributing to the amplitudes of inelastic scattering (9). These diagrams are presented in Fig. 16 and 17 (see footnote 2). Denoting the topologically nonequivalent diagrams by different parameters, we obtain the following expressionsfor the scattering amplitudes (9) : (K+p 1K*‘A++)
= &u,
(K-p 1Z?“n) = 2b,
52
MALYUTA
PP'PP
P"-P"
FIQ.
(K-p
(K-p (K-p
1 .R-x*+)
=
a,
14
(K-p
I7r%*-)
= 2c,
I?r+r)
= 2c,
(K-p
1K”Eo) = 2c,
1K+K)
= 2c,
W-P
I K *+E-)
(7r’p 1K*+z+)
= b,
(n-p I K+;r;-)
= 25 = 2c
(a is the exchange scattering contribution, b is the annihilation contribution, and c is the superposit,ion of exchange scattering and annihilation contributions). Excluding parameters a, b, c, we obtain the following relations for the differential
COLLINEAR
cross sections in the forward $
(Ktp
[ K*‘A++)
PROCESSES
IN
and backward = 3g
THE
QUARK
MODEL
53
direction!
(K-p 1=-z*+),
4 The quantities denoted in (10) by the symbols &/dQ are equal to the differential cross sections with an accuracy up to the factors (P,,,/SPi,) exp [- (PM + UB)I t I], where P,,t/sPi, is the phase-space correction (12), and exp (-aH 1 t 1) is the structure factor which is the Fourier transform of the quark wavefunction in the hadron (IO), (11) (Pi, and P,,t are the magnitudes of the spatial momenta of the incoming and outgoing particles in the center-ofmass system, s is the square of the total energy, ( t / is the square of the momentum transfer).
54
MALYUTA
u-p --+Tr+~*-
K-P- lT+tFIG.
g
(I$(
Bz)
= 4%
(&I
16
K*+z+),
I’
(10)
= g (K-p 1E-+x-) = !!.g (K-p ( jy*+z-) = g (K-p ( KoEO) = g (7rp 1 &-f-z->.
\-IT
+p-
K”+ t+
FIG. 17 TABLE
VI
:Process
K+p --) K*OA++ K-P ---) ,-i-f*+ K-p --) rr+iz*K-P --f ?r+zK-p --* K+r K-P + K*QK-p --f KOF
X
2.26 3 2.24 3 2.24 3 2.24 3 2.24 3 2.24 3 3
450 1360 80 37 9 3.5 8 2.5
f f f f f f
60 140 10 10 4 1.8
f
1
0.6 f
0.2
0.5 f 2.5 f
0.3 0.6
55
14 3.5 20 19 22 7 5.2 5.3 2.5
f6 f 1.8 f4 f1.4 f 1.3 f 0.6
?4 % 1 1 1 1 1 1 1 1 1 1 1
56
YALY
A comparison of Table VI, where predictions (10) factors indicated
UTA
relations ( 10) with the experimental data (5) is presented in X is the inverse strength factor that normalizes the data to (the comparison was carried out with an accuracy up to the in footnote 4). ACKNOWLEDGMENT
The author wishes to thank Professor N. N. Bogoliubov RECEIVED:
November
for valuable
discussions.
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