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Progress of Theoretical Physics, Vol. 44, No. 5, November 1970
On the Quark Counting Additivity Sum Rules
Masanori MATSUDA
Department of Physics, Faculty of General Education
Hiroshima University, Hiroshima
(Received May 11, 1970)
It is suggested that the inelastic amplitudes which generate the imaginary parts of the
elastic scattering amplitudes have an additivity property, instead of the quark counting additivity assumed for the elastic scattering amplitudes. It is shown that the resulting sum
rules are consistent with the experimental data. The present approach gives consistent results on the observed correlation between the total cross sections and the slopes of elastic
cross sections.
§ 1.
Introduction
Several authors 1)"'3) have proposed that the hadron-hadron scattering amplitudes may be expressed as the sum of the amplitudes with the index of quarkquark and quark-antiquark scattering, i.e. in the scattering A+ B~ A'+ B', the
amplitude is written as
(1)
where the summation extends on all possible combination of i and j which are
the quark indices associated with different hadrons respectively. This idea has
first been tested for the imaginary parts of the forward scattering amplitudes. 1)
It gives sum rules for the total cross sections for hadronhadron scattering, which
are in good agreement with experiment. Later the counting hypothesis has been
extended to inelastic processes and non-forward elastic scattering. 2),a)
At high energy the forward elastic scattering is dominated by the diffraction scattering. The diffraction scattering results from the absorption of the incident waves into the inelastic channels. In this case the responsible hadronhadron interactions are inelastic and the imaginary parts of the elastic amplitudes
are quadratic in these inelastic interactions. Therefore, the imaginary parts may
be regarded as being of a secondary nature. The real parts of the elastic amplitude are more directly related with the hadron interactions. Hence we may
take the viewpoint that some kind of regularity like additivity, if it exists reflecting the structure of the hadron interactions, might appear in the real part.
In the following we develope such an idea and discuss a counting rule for
the inelastic amplitudes. We show that sum rules may exist for the "coupling
constants" of the inelastic amplitudes. As a byproduct of this approach, we
On the Ouark Counting Additivity Sum Rules
1301
have an explanation of an existing certain correlation between total cross sections and slopes of elastic cross sections.
§ 2.
Quark counting 'sum rules for inelastic interactions
In the following discussion we neglect the spins of particles. To be consistent with the standpoint that the imaginary parts of the scattering amplitudes
are of the secondary character, we take the K-matrix formalism where all the
imaginary parts are generated as on-energy-shell rescattering effects and consider
that the real K-matrix elements are the primary quantities to which some kind
of additivity counting rule is applied.
To clarify our assertion, we take a two-channel model for diffractive elastic
scattering 4)' 5) where an inelastic process is introduced as the one expressing the
totality of inelastic effects. We consider a two-channel reaction
A+B~A+B
X
A+B~A+B,
where A and B are hadrons and A and B are hypothetical isobars which consist of the same quark components as A and B, respectively. To simplify the
kinematics, we assume that A and B have the same masses as A and B, respectively.*)
The S-matrix is given by
S= l+iK
1-iK'
KAB,AB
K=
-[ KAn,An
(2)
K~~·~~J.
KAB,AB
In order to give the dominance of diffraction scattering we take KAB,AB
= K A.i3,A.i3 = 0. For K An,A.i3 we assume that its partial wave projection is given by 5l
L)
3
Q (3) (1
(
inel= g
f AB,
L
AB \Zp 2
l
+L)
Zp
2
'
(3)
where p is the c.m.s. momentum, A the "mass parameter" characterizing inelastic
interaction range and UAn the "coupling constant". Here Ot< 3)(z)=(d 3/dZ 3)Qt(Z),
and QL (Z) is the Legendre function of the second kind.
This choice for the inelastic amplitudes does not restrict the validity of the conclusion of present analysis, as we confine our discussion only to the small momentum transfer phenomena.
*l Some interesting results followed from the cases mA~mA. and mn~mjj are discussed in reference 4).
1302
M. Matsuda
Now the partial wave S-matrix for the elastic A+ B scattering Is given by
inel)2
1 - (fAB,l
1 + (fAB,l
inel)2
SABl
(4)
•
In addition to Eq. ( 4) we also examine the eikonal approximation version
SzAB
= exp [- 2 (f}~~D 2 ].
(4')
In the following analysis we take A2 common to all channels.
By assuming a quark counting rule as in reference 1), we have
+ t (gJ(+p- gK+?~) -t- t (gK-p- gK-n),
g pp -t- g pp = 2 (gn:-p -t- gn:•p) - t(gK-p + gK+p),
gK-p + gK+p = t (gn:·p + gn:+p + gK·n + gK+?~),
gpp + g PP =! (gn:·p -t- gn:+p)
(5a)
(5b)
(5c)
and other relations corresponding to those given in reference 1). The left- and
right-hand sides of Eqs. (Sa)'"'-' (5c) determined from the experimental data of
total cross sections 6> for the case A2 = 0.4 (Ge V/ are given in Fig. 1. *> The value
A2 = 0.4 (Ge V) 2 is chosen so as to reproduce the forward slopes of the elastic
differential cross sections.
The agreement of the present model with experiment is as good as the
original sum rules, 1> and the present model is equally acceptable as the old one
as far as the judgement of the acceptability is based on the fit to experiment.
2.5
(the eikonal approximation model)
A
2.0
A
A
A
ii
<il
•
A
A
A
0
a
0
+
0
co
t:J,<(
A
01
0
•
2.0
A
0
•
A
A
0
0
•
•
•
•
(the K-matrix model)
1.5
+
)(
+
)(
+
X
+
+
+
+
l(
l(
l(
)<
X
1.0L----1L5_ _ _2L0----::"=----::':::----::'!::------:-"::-25
30
35
40
s (GeV) 2
Fig. 1. Validity of relations (Sa), (5b) and (5c).
We have taken A2=0.4(GeV) 2.
•: !i'pp+!i'P:P• 0: i(g,.+p+!l,.-p) +Hux+p-!7K+n) +Hux-p-Ux-n), D.: 2(g,.+P+g"-P)
-HgK+p+Ux-p), X: UK+p+UK-p• +: t(g,.+p+!l,.-p+!lx+n+!i'K-n).
These values were calculated from the experimental data in reference 6). The results of the eikonal model for Eq. (5c) are almost the same with those of the Kmatrix model.
*> Here we have assumed that all
UAB
have the same sign (positive for convenience).
On the Ouark Counting Additivity Sum Rules
1303
Here let us make some comments on the analyses by Imachi, Matsuoka,
Ninomiya and Sawada. 3 ) They have taken the viewpoint that the counting rule
should be assumed for the "interaction" not for the amplitude. As the "interaction", they have taken the imaginary potentials W (r) and discussed the elastic
and total cross sections in the eikonal approximation. Their W (r) is related
to the Fourier transformation of SKAn,A.iJ (p, p') dp' KAn,A.iJ (p', p"), and therefore
their additivity rule is not the same as the present one.
In their analysis, they have introduced a phenomenological parameter r; as
a factor which would account for the difference of wave functions between mesons and baryons and have multiplied r; (r;- 1) to the baryon-baryon (meson-meson)
potentials determined from the additivity. Such a phenomenological parameter
is used neither in the original additivity hypothesis nor in the present approach.
In our opinion, an interesting feature of the quark counting sum rule is the
prediction of the relation between meson-baryon scattering and baryon-baryon
(anti-baryon) scattering. Introduction of this parameter reduces this interesting
feature of the model.
In the symmetrical limit, the original counting rule g1ves
(6)
which roughly expresses the present experimental situation.
In the present model, the counting gives
(7)
The prediction for the total cross sections depends on the details of K AB,A.s,
but the choice for parameters to reproduce the slopes of the elastic differential
cross sections gives sum rules consistent with experiment as has been seen in
Fig. 1. [For example, if we assume O",.N=24 mb, then from Eq. (7) we have
O"NN=O"N.N=45mb (44mb) for A2 =0.4(GeV) 2 in the K-matrix model (the eikonal
approximation model).] In the lowest order approximation in which only the g2
terms are retained, we have
(JNN = ( ~
2
(j1tlV'
r.
(lowest)
(8)
In the analysis of Imachi et al., the Born approximation gives
(JNN
3
-=-r;.
0",.1v
2
(lowest)
(9)
With a suitable choice for the imaginary potentials, they have obtained a fit to
the experimental data with r; = 1.43. This value is interestingly compared with
the extra ! in Eq. (8) which, however, is due to an entirely different physical
source.
As for the difference between the present model and the usual quark count-
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M. Matsuda
ing model, it will be interesting to examine the meson-meson total cross sections.
In the symmetrical limit, we have
(JMB
3
(j'MM
2
(the usual additivity rule 1))
(10)
(the present model),
since the damping of the cross section due to the rescattering becomes weaker
as gAB decreases and the lowest term gives a good approximation. Similarly,
the corresponding value is about iftl in the model of Imachi et al.
§ 3.
The forward slopes of elastic scattering
In the previous section we have stated that the value of the parameter A2
was chosen so as to reproduce the forward slopes of elastic scattering. In this
section we give the slopes obtained.
In the present model, we have a considerable suppression of the cross sections from those of the lowest order terms due to the on-energyX pp
13.0
18.9
a PP
shell-rescattering effect such as
o 7Cp
11.8
removmg the extra 23 factor Ill
• rtp
12.0
10.0
Eq. (8). This unitarity effect, on
A Kp
. . Kp
the other hand, prompts to increase
the slopes of the forward di:ffraction peak of th'e elastic cross sections. In the present model the
numerical values of the logarithmic slopes at t = 0 are approximately expressed in terms of the
total cross sections rJ T as
8.0
r
15.9
7.0
,,~
6.0
fe
11.9
20
2
+ 3.6 X 10- 2rJT(mb)
[ + 4.8 X 10- 6 T • A
4
10
1
A (Ge V 2)
A (Ge v- 2) rv3.08
30
40
50
60
O'T(mb)
Fig. 2. Correlation between a'T and A. The number
attached to each of the experimental data 7) is the
squared c.m.s. energy s. The solid curve shows
the theoretical values calculated by the K-matrix
model, Eq. (4), and the broken denotes the theoretical values by the eikonal approximation
model, Eq. (4 1 ) • Here A2 is taken as 0.4 (GeV) 2•
2
for Eq. (4)
for Eq. (5)]
(11)
for A2 rv0.3rv0.6 Ge V 2 and (J T<60
mb. In Fig. 2 we give the theoretical curves of A versus rJ T for
A 2 = 0.4 (Ge V)2 together with the
experimental data. 7 ) The general
feature of the experimental data
On the Ouark Counting Additivity Sum Rules
1305
Is consistent with the theoretical calculation. *l The correlation between A and
like (11) is the characteristic of any model which involves the rescattering corrections as in the present model and some of the cross over effects such as the
one observed between pp and pp elastic differential cross sections may be partly
reduced to this rescattering effects. 8l
(J r
§ 4. Some remarks
In this paper we have shown that if we introduce a fictitious channel A+ B
representing the totality of inelastic channels, then there exists the quark counting additivity sum rules which are consistent with the experimental data.
So far we have not referred to the relation of the individual inelastic channels with their totality A+ B. It will be quite natural to take
(KAB,AB)2 =
L: (KAB,ao)2,
(12)
ab
where KAB,ao is the K-matrix element for the transition from the initial state
A+ B to the real inelastic channel a+ b.
Obviously one simple way to obtain the additivity sum rule for KAB,AB would
be to assume that there is one-to-one correspondence among the inelastic channels for which the additivity rule hold with approximately satisfying KMB,mo / K BB,M
,....._,-f, etc. The dominance of the quasi-two-body reaction (without pair creation
or annihilation of quarks) is necessary in order that the simple counting rule is
effectively applied. The success of the present model might be considered as
giving a certain simplicity which exists in the complexity of all inelastic channels.
Acknowledgements
The author would like to express his sincere gratitude to Professor M.
Y onezawa for suggesting this subject and many valuable discussions. He is also
much obliged to Mr. M. Kawasaki and Dr. M. Kobayashi for their useful discussiOns. Thanks are also due to Professor K. Sakuma, Professor S. Ogawa
for their interest in this work.
References
1)
2)
3)
H. ]. Lipkin and F. Scheck, Phys. Rev. Letters 16 (1966), 71.
E. M. Levin and L. L. Frankfurt, Soviet Phys.-JETP Letters 2 (1965), 65.
D. Ito, Soryushiron Kenkyu (mimeographed circular in Japanese) 32 (1965), 217.
G. Alexander, H. J. Lipkin and F. Scheck, Phys. Rev. Letters 17 (1966), 412.
H. R. Rubinstein and H. Stern, Phys. Letters 21 (1966), 447.
M. Imachi, T. Matsuoka, K. Ninomiya and S. Sawada, Prog. Theor. Phys. 38 (1967), 1198;
40 (1968)' 353.
*l As for the experimental data of A, here we have taken the values from the fit with (da-j dt) /
(da/dt)t=o=exp(At+Bt2 ) for p+p, n±-p, K±-p and exp(At) for ]5-p. It is noted that the experimental values of slopes would vary depending on the assumed form of distribution.
1306
4)
5)
6)
7)
8)
M. Matsuda
T. Matsuoka, K. Ninomiya and S. Sawada, Prog. Theor. Phys. 41 (1969), 1533.
M. Kawasaki, M, Kobayashi and M. Yonezawa, Prog. Theor. Phys. 44 (1970), 557.
M. Kawasaki, Prog. Theor. Phys. 44 (1970) 1618.
M. Kawasaki, M. Kobayashi and M. Y onezawa, Lett. Nuovo Cim. 111 (1970), 655.
W. Galbraith, E. W. Jenkins, T. F. Kycia, B. A. Leontic, R. H. Phillips, A. L. Read and
R. Rubinstein, Phys. Rev. 138 (1965), B913.
K. ]. Foley, S. J. Lindenbaum, W. A. Love, S. Ozaki, J. ]. Russell and L. C. L. Yuan,
Phys. Rev. Letters 11 (1963), 425, 503.
K. J. Foley, R. S. Gilmore, S. J. Lindenbaum, W. A. Love, S. Ozaki, E. H. Willen, R.
Yamada and L. C. L. Yuan, Phys. Rev. Letters 15 (1965), 45.
K. J. Foley, R. S. Jones, S. J. Lindenbaum, W. A. Love, S. Ozaki, E. D. Platner, C. A.
Quarles and E. H. Willen, Phys. Rev. Letters 19 (1967), 330, 857.
See S. Sawada, Soryushiron Kenkyu (mimeographed circular in Japanese) 40 (1969), D89.