403 Progress of Theoretical Physics, Vol. 45, No. 2, February 1971

Hadron Reactions and Urharyon Rearrangement Akizo KOBAYASHI, Takeo MATSUOKA, Kansuke NINOMIYA and Shoji SAW ADA

Department of Physics, Nagoya University, Nagoya (Received August 8, 1970)

crI

(phase space volume) cc s-2rnn

with r"'-'0.5, where s is the square of total energy in c.m.s. and nR is the least possible number of rearranged sakatons. On the basis of the analyses of the two-body reactions, we make a conjecture that a sakaton rearrangement brings a factor [(pi +pJ)2] -r

to the rearrangement amplitude where Pi and p1 are the four-momenta of hadrons.

§ I.

Introduction

From analyses 1) of the two-body and quasi-two-body processes in terms of rearrangement of sakatons (fundamental entities of the triplet models), it was found that the energy dependence of the cross sections are closely related to the number of rearranged sakatons nR as

(1) where s is the square of total energy in the center of mass frame. The experimental energy dependence of various two-body processes can be reproduced with r::::::::0.3~0.5. It has also been shown that the characteristic features of the inelastic twobody and quasi-two-body processes (appearance of forw~rd and/or backward peaks, dips, etc.) can be related with the type of rearrangement of sakatons. 2) In this paper, we examine the multiple production processes from the same viewpoint of the rearrangement of sakatons; It is shown in § 2 that Eq. (1) is well satisfied by the experimental data on the multiple production processes. In § 3, we make a conjecture on the energy dependent factors of sakaton rearrangement amplitude. Some remarks are given in § 4. In the Appendix the type of sakaton rearrangement for multiple production processes are summarized.

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

An aspect of the multiple production process is discussed from a viewpoint that reactions of hadrons are caused by rearrangement of the sakatons, the constituents of hadrons. It is shown that the energy dependence of experimental cross sections for multiple production processes as well as two-body reactions are approximately given by

A. Kobayashi, T. Matsuoka, K. Ninomiya and S. Sawada

404

§ 2. The energy dependence of the multiple production amplitudes

Fig. 1. Rearrangements for the strong reactions of disconnected type expressed by Tr(MMMM) ·Tr(MMMM) and Tr(MMM)Tr(MMM)Tr(MM) which are forbidden by the Iizuka-Okubo rule but not by rule (A).

Tr (MM· · ·M) Tr (MM· · ·M) · · · where M's denote the nonet mesons. The selection rule (A) forbids the interactions containing a factor Tr (M) but it allows the interactions expressed as Tr (MM) · Tr (MM) which are attributed to the elastic and the quasi-elastic processes in the case of two-body scatterings. (B) The selection rule PiP1 = ( -llf-Ji holds for each connected subsystem in the disconnected rearrangement diagrams, where Pi (P1 ) and Ji (J1 ) are the total parity and total angular momentum of the initial (final) subsystem as shown in Fig. 2. Selection rule (B) is a generalization of the rule specifying the elastic and quasi-elastic scatterings 4) which proceed without rearrangement of sakatons. Typical sakaton rearrangement diagrams for the multiple production processes are shown in Fig. 3 by taking MB~ MMMB process*) as examples. The diagrams shown in Fig. 3a are disconnected types, and those shown in Fig. 3b are connected types. The D 2-type rearrangement for the process such as o- -me. Fig. 2. Rule (B). *)

We denote .a non-exotic meson and baryon by M and B, respctively.

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

Multi-body reactions of hadrons as well as two-body scatterings are considered to be caused by various types of rearrangement of sakatons which are expressed by rearrangement diagrams. We summarized these rearrangement diagrams in the Appendix. The rearrangement diagrams are divided broadly into two categories, the connected and the disconnected. First. we discuss assumptions mainly concerning the disconnected types of sakaton rearrangements which will play important roles in the high energy multiple production processes. (A) Creation of sakaton pair is strongly suppressed when both of the created sakaton and antisakaton become constituents of a hadron. This selection rule is· the same as the Okubo-Iizuka rule jn the case of threemeson interactions. 3) Iizuka suggested also the suppression of the strong interactions of such disconnected type as shown in Fig. 1. These are expressed by

Hadron Reactions and Urbaryon Rearrangement

405

uu uu u u O-O-type

0 1 -type

0 1 -type

0 2 -type

a. Disconnected type

n

H-type

Z-type

b. Cormected

w- type

type

Fig. 3. Some examples of a. the disconnected types and b. the H-types, Z-types and W-types of rearrangements for MB~MMMB.

son+ baryon~three o--mesons +baryon are forbidden by rule (B). It was found that in two-body processes in which the scattering without rearrangement corresponds to the Pomeron exchange dual to the background, while the scattering with rearrangement corresponds to the ordinary Regge pole exchange dual to the resonances. On the same footing it is suggested that the disconnected types would be related to the Pomeron exchange in multi-Regge pole model and may appear as the diffractive dissociation (D 1) and double diffraction dissociation (D2) or as one fire ball (Dh D-D) and two fire ball (D2) ~ith correlations of connected subsystem. The possible least numbers nR for various multiple production processes are given in Table I. Here we introduce the reduced total cross sections which are defined by (J = (J /(phase space volume), where (J is the total cross section for the multiple production process. Assuming the s-dependence of the reduced total cross sections as

(2a) the exponents n obtained from experimental cross sections 5) are summarized in Table I, PL being the incident momentum. As is seen from Table I, the value of n can be closely related to the least number of rearranged sakatons nR by

. (2b)

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

uuu

406

A. Kobayashi, T. Matsuoka, K. Ninomiya and S. Sawada

with r"--'0.57. Hitherto when we count the nR, we allow the rearrangement whose diagram is separated into three or more disconnected parts, that is, the rearrangement which produces hadrons in the final state connected to neither of two initial hadrons (e.g. the D-D type, D-D-D type, etc. shown in Fig. 3). If these types of Table I. The least possible numbers nR for various multiple production processes and the exponents n from experimental cross sections. The value of n is larger by one than that of reference 10) due to the flux factor Pcm -js. The experimental data used here are given in references 10) and 5). process

-">nn+nn+p-">pn+no -">n2n+ K+p-">pK0n+ -">pK+no K~p-">pK 0 npp~ppno

-">pnn+ n-p-">pn+2nn+p~p2n+n-

K+p-">pK+n+n-">pK0n+no ~nK0 2n+

PP-">ppn+nPP-">Ppn+nn-p~pn+2n-n°

-">n2n+2nn+p-">p2n+ n-no -">n3n + nK+p~pK+n+n-n°

-">pK02n+ n-">nK+2n+nK-p~pK-n+n-n°

pp-">ppn+n-n° -">pn2n + nn-p-">p2n+3nn+p-">p3n+2nK+p-">pK+2n+2n-">pK02n+n-n° pp-">pp2n +2nPP-"> pp2n+2n-

type

Range of PLab(GeV /c)

n

0.7-25 0.6-16 0.9-11.5 0.9-11.5 1.1-12.7 0.9-3.5 1.1-6.0 2.8-19 4-28.5

2.5±0.05 2.6±0.1 2.9±0.05 2.25±0.1 3.6±0.05 2.6±0.2 3.1±0.1 2.6±0.15 2.6±0.1

2 2 2 2 3 2 3 2 2

1.9-25 1.1-18.5 1.9-12.7 1.9-12.7 1.9-12.7 2-25 2.7-7

3.15±0.05 3.4±0.05 4.0±0.05 4.05±0.1 4.15±0.15 4.05±0.1 4.4±0.1

2(3) 2(3) 2(3) 3 4 2(3) 2(3)

1.9-25 1.6-16 1.8-18.5 1.8-16 1.6-5 1.6-10 1.9-5 3-12.7 2.2-10 2.2-10

4.75±0.05 4.65±0.05 5.15±0.05 4.0±0.15 4.8±0.3 5.35±0.2 4.5±0.3 4.7±0.3 5.2±0.1 5.6±0.15

4 4 4 4 4 5 4 4 4 4

H D D D D

2.7-16 2.7-8.5 3-8.3 2.7-8.3 5.5-28.5 5.7-7

5.25±0.1 5.3±0.25 6.25±0.2 5.75±0.15 6.6±0.2 5.1±1.4

4(5) 4(5) 4(5)

D-D(D) D-D(D) D-D(D)

5 4(5) 4(5)

D D-D(D) D-D(D)

D D D D H D H D D D-D(D) D-D(D) D-D(D) D H D-D(D) D-D(D) D D D D D

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

n-P-">Pn-nO

I

Hadron Reactions and Urbaryon Rearrangement Table I. process

I Range of

rc-p~p2rc+3rc-rc 0 ~n3rc+3rc-

rc+p~p3rc+2rc-rc 0 ~n4rc+2rc-

K+p~pK+2rc+2rc-rc 0 pp~pp2rc+2rc- rc 0 ~pn3rc+2rc-

(continued)

PLab (Ge V /c)

I

n

5.5-16 5.5-16 3.4-8.5 3.5-8 3.5-8.3 5.5-28.5 5.5-28.5

6.6±0.2 6.8±0.2 6.8±0.3 5.5±0.3 7.2±0.4 6.95±0.1 7.25±0.2

10-16

7.3±0.9

I 6(7)

3

PP~2rc+2rcPP~2rc+2rc-rc 0

PP~3rc+3rcPP~3rc+3rc-rc 0

0.8-5.5 1.5-12 1.5-12 1.5-7 1.5-7

""'-'5 ""'-'5.5 ""'-'6 ""'-'6.5

5 6 7 8 9 10

8 17.1

0 2.0±0.15 2.6

0 2 3

""'-'4

PP~4rc+4rcPP~4rc+ 4rc-rc0

quasi -elastic rc+p~p+p

rc+p~prc+

6 6 6 6 6 6 6

(backward)

4

D D D D D D D

I D-D(D)

H H H H H H H H D

H X, Z

n

1 8

@

elastic and quasi-elastic

o n±p- p±p •

rr.+p -pre+ (backward)

!:>.

pp-mrr multiple production processes

6

4

2

2

3

4

5

6

7

nR-

Fig. 4. Correlation between the energy dependence of cross sections of various high energy reactions and the least numbers of rearranged sakatons nR·

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

PP~rc+rc-

PP~rc+rc-rc 0

407

A. Kobayashi, T. Matsuoka, K. Ninomiya and S. Sawada

408

rearrangement are forbidden, then the possible least numbers of nR for the multiple production processes are given by those in the parentheses in Table I. In terms of nR redefined here the experimental energy dependence of the cross sections are reproduced more satisfactorily by (2). The correlation of experimental n and the nR is shown in Fig. 4.

§ 3. Energy dependent factor of the rearrangement amplitudes

l

(3) Each factor [(Pi+ p1i]-r corresponds to one-sakaton rearrangement from a hadron with momentum Pi to that with p1 shown in Fig. 5. The factors may be considered to come from the overlap integrals of the wave functions or the form factors of hadrons as the composite systems of sakatons. 6 ) Factorization (3) reproduces both the energy dependence (1) and the phase factors 1 and e 2i7tr of rearrangement amplitudes T x(s, t, u) and T H(s, t, u) in the small Jtl region. This is seen, for example, for the case of the MB---7MB process:

c

d

a

b

pl~l~~(" I

I

I

I

I I

I

I

r---------' pl

D-type

u

n

forward

H-type

X-type

Z-type

backward

~b

Fig. 5. A rearrangement in a multiple production .process a+b---7!+2+ ···+N

Fig. 6. Rearrangement diagrams for the ME-7MB processes (forward and backward).

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

In the previous analysis of the two-body scatterings, it was found that the spinnon-flip X- and H-type rearrangement amplitudes T x(s, t, u) and T H(s, t, u) have phase factors 1 and e 2i7tr, respectively, in the forward region. 1) From this fact and results (1), we propose a possible factor which is contained in the rearrangement amplitudes TnR (s, t, u) in the small Jta 1 and small JtbNI region in the following way:

409

Hadron Reactions and Urbaryon Rearrangement

(4a) (4b) In the MB---7 MB processes, there are the Z-type and D-type rearrangement contributions in addition to the X- and H-types (Fig. 6). For the Z-type rearrangement in the small lti-region we have

Tz(s, t, u) oc [ (Pa + PaYJ-r[ (Po+ PcYJ-r [ (Pa + PoYJ-r [ (Pc + Pd) 2] - 2r (4c)

= (-u)-2r(-s)-2r __ ~e2i"'rs-4r 8->large

T x(s, t, u) oc [ (Pa + Pc) 2lr [(Po+ Pa) 2]- 2r = ( - u)-sr ~s-sr,

(5a)

T H(s, t, u) oc [ (Pa + PcYlr [ (Pa + PoYJ-r [ (Pc + Pd) 2]-r [(Po+ Pd) 2]- 2r

(5b)

= (-t)-sr(-s)-2r~e2i"'rs-5r, 8->large

T z(s, t, u) oc [(Po+ Pa) 2]-r [ (Pa + Po) 2]-r [ (Pc + PaYJ-r = ( _ t)-r ( _

s

)-2r ~e2i1rr s-sr

(5c)

8->large

and

TD (s, t, u) oc [ (Pa + PcY]- 2r [(Po+ PaYJ-sr = ( - t)- 5r~S-

5

r.

(5d)

Owing to the energy dependent factor and phase factor in (4a) and (4b) there are correspondence between the amplitudes T x(s, t, u) ± T H(s, t, u) and the Regge amplitudes TRegge with strongly degenerate trajectories a (t) and residues {3 (t) in the forward regions if we take - 2r + 1 =a (t):

T x(s, t"-'0, u) ± T H(s, t"-'0, u) oc (1 ± e 2i"'r) s- 2r ~T

. (r = ± 1) Regge

=

{3 (t) (1 ± e-i,.a(t)) , ( ) sm na t

sa(t)-1.

(6)

where r is the signature of the Regge trajectory. On the other hand, as is shown by (5), in backward meson-baryon scattering, the sakaton rearrangement amplitudes do not correspond to the conventional Regge-pole amplitudes with baryon trajectories. The factorization (3) satisfies the requirements on the phase difference between s-u crossed amplitudes imposed by the generalized Pomeranchuk theorem. 7)

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

from (3). In the region, this type will give very small contribution. In the forward region the D-type rearrangement has neither energy-dependent factors nor phase factors. This type of rearrangement should not be treated on an equal footing with the other types because it occurs without rearrangement and corresponds to the diffraction scattering as mentioned in § 2. In the backward meson-baryon scattering where u = - (Pa + Pc) 2 = - (Po+ pdy rvO, we have

410

A. Kobayashi, T. Matsuoka, K. Ninomiya and S. Sawada The theorem states that the amplitude of the process a+ b-7c+ d, F(s, t) = g (t) sa

Hadron Reactions and Urharyon Rearrangement Akizo KOBAYASHI, Takeo MATSUOKA, Kansuke NINOMIYA and Shoji SAW ADA

Department of Physics, Nagoya University, Nagoya (Received August 8, 1970)

crI

(phase space volume) cc s-2rnn

with r"'-'0.5, where s is the square of total energy in c.m.s. and nR is the least possible number of rearranged sakatons. On the basis of the analyses of the two-body reactions, we make a conjecture that a sakaton rearrangement brings a factor [(pi +pJ)2] -r

to the rearrangement amplitude where Pi and p1 are the four-momenta of hadrons.

§ I.

Introduction

From analyses 1) of the two-body and quasi-two-body processes in terms of rearrangement of sakatons (fundamental entities of the triplet models), it was found that the energy dependence of the cross sections are closely related to the number of rearranged sakatons nR as

(1) where s is the square of total energy in the center of mass frame. The experimental energy dependence of various two-body processes can be reproduced with r::::::::0.3~0.5. It has also been shown that the characteristic features of the inelastic twobody and quasi-two-body processes (appearance of forw~rd and/or backward peaks, dips, etc.) can be related with the type of rearrangement of sakatons. 2) In this paper, we examine the multiple production processes from the same viewpoint of the rearrangement of sakatons; It is shown in § 2 that Eq. (1) is well satisfied by the experimental data on the multiple production processes. In § 3, we make a conjecture on the energy dependent factors of sakaton rearrangement amplitude. Some remarks are given in § 4. In the Appendix the type of sakaton rearrangement for multiple production processes are summarized.

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

An aspect of the multiple production process is discussed from a viewpoint that reactions of hadrons are caused by rearrangement of the sakatons, the constituents of hadrons. It is shown that the energy dependence of experimental cross sections for multiple production processes as well as two-body reactions are approximately given by

A. Kobayashi, T. Matsuoka, K. Ninomiya and S. Sawada

404

§ 2. The energy dependence of the multiple production amplitudes

Fig. 1. Rearrangements for the strong reactions of disconnected type expressed by Tr(MMMM) ·Tr(MMMM) and Tr(MMM)Tr(MMM)Tr(MM) which are forbidden by the Iizuka-Okubo rule but not by rule (A).

Tr (MM· · ·M) Tr (MM· · ·M) · · · where M's denote the nonet mesons. The selection rule (A) forbids the interactions containing a factor Tr (M) but it allows the interactions expressed as Tr (MM) · Tr (MM) which are attributed to the elastic and the quasi-elastic processes in the case of two-body scatterings. (B) The selection rule PiP1 = ( -llf-Ji holds for each connected subsystem in the disconnected rearrangement diagrams, where Pi (P1 ) and Ji (J1 ) are the total parity and total angular momentum of the initial (final) subsystem as shown in Fig. 2. Selection rule (B) is a generalization of the rule specifying the elastic and quasi-elastic scatterings 4) which proceed without rearrangement of sakatons. Typical sakaton rearrangement diagrams for the multiple production processes are shown in Fig. 3 by taking MB~ MMMB process*) as examples. The diagrams shown in Fig. 3a are disconnected types, and those shown in Fig. 3b are connected types. The D 2-type rearrangement for the process such as o- -me. Fig. 2. Rule (B). *)

We denote .a non-exotic meson and baryon by M and B, respctively.

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

Multi-body reactions of hadrons as well as two-body scatterings are considered to be caused by various types of rearrangement of sakatons which are expressed by rearrangement diagrams. We summarized these rearrangement diagrams in the Appendix. The rearrangement diagrams are divided broadly into two categories, the connected and the disconnected. First. we discuss assumptions mainly concerning the disconnected types of sakaton rearrangements which will play important roles in the high energy multiple production processes. (A) Creation of sakaton pair is strongly suppressed when both of the created sakaton and antisakaton become constituents of a hadron. This selection rule is· the same as the Okubo-Iizuka rule jn the case of threemeson interactions. 3) Iizuka suggested also the suppression of the strong interactions of such disconnected type as shown in Fig. 1. These are expressed by

Hadron Reactions and Urbaryon Rearrangement

405

uu uu u u O-O-type

0 1 -type

0 1 -type

0 2 -type

a. Disconnected type

n

H-type

Z-type

b. Cormected

w- type

type

Fig. 3. Some examples of a. the disconnected types and b. the H-types, Z-types and W-types of rearrangements for MB~MMMB.

son+ baryon~three o--mesons +baryon are forbidden by rule (B). It was found that in two-body processes in which the scattering without rearrangement corresponds to the Pomeron exchange dual to the background, while the scattering with rearrangement corresponds to the ordinary Regge pole exchange dual to the resonances. On the same footing it is suggested that the disconnected types would be related to the Pomeron exchange in multi-Regge pole model and may appear as the diffractive dissociation (D 1) and double diffraction dissociation (D2) or as one fire ball (Dh D-D) and two fire ball (D2) ~ith correlations of connected subsystem. The possible least numbers nR for various multiple production processes are given in Table I. Here we introduce the reduced total cross sections which are defined by (J = (J /(phase space volume), where (J is the total cross section for the multiple production process. Assuming the s-dependence of the reduced total cross sections as

(2a) the exponents n obtained from experimental cross sections 5) are summarized in Table I, PL being the incident momentum. As is seen from Table I, the value of n can be closely related to the least number of rearranged sakatons nR by

. (2b)

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

uuu

406

A. Kobayashi, T. Matsuoka, K. Ninomiya and S. Sawada

with r"--'0.57. Hitherto when we count the nR, we allow the rearrangement whose diagram is separated into three or more disconnected parts, that is, the rearrangement which produces hadrons in the final state connected to neither of two initial hadrons (e.g. the D-D type, D-D-D type, etc. shown in Fig. 3). If these types of Table I. The least possible numbers nR for various multiple production processes and the exponents n from experimental cross sections. The value of n is larger by one than that of reference 10) due to the flux factor Pcm -js. The experimental data used here are given in references 10) and 5). process

-">nn+nn+p-">pn+no -">n2n+ K+p-">pK0n+ -">pK+no K~p-">pK 0 npp~ppno

-">pnn+ n-p-">pn+2nn+p~p2n+n-

K+p-">pK+n+n-">pK0n+no ~nK0 2n+

PP-">ppn+nPP-">Ppn+nn-p~pn+2n-n°

-">n2n+2nn+p-">p2n+ n-no -">n3n + nK+p~pK+n+n-n°

-">pK02n+ n-">nK+2n+nK-p~pK-n+n-n°

pp-">ppn+n-n° -">pn2n + nn-p-">p2n+3nn+p-">p3n+2nK+p-">pK+2n+2n-">pK02n+n-n° pp-">pp2n +2nPP-"> pp2n+2n-

type

Range of PLab(GeV /c)

n

0.7-25 0.6-16 0.9-11.5 0.9-11.5 1.1-12.7 0.9-3.5 1.1-6.0 2.8-19 4-28.5

2.5±0.05 2.6±0.1 2.9±0.05 2.25±0.1 3.6±0.05 2.6±0.2 3.1±0.1 2.6±0.15 2.6±0.1

2 2 2 2 3 2 3 2 2

1.9-25 1.1-18.5 1.9-12.7 1.9-12.7 1.9-12.7 2-25 2.7-7

3.15±0.05 3.4±0.05 4.0±0.05 4.05±0.1 4.15±0.15 4.05±0.1 4.4±0.1

2(3) 2(3) 2(3) 3 4 2(3) 2(3)

1.9-25 1.6-16 1.8-18.5 1.8-16 1.6-5 1.6-10 1.9-5 3-12.7 2.2-10 2.2-10

4.75±0.05 4.65±0.05 5.15±0.05 4.0±0.15 4.8±0.3 5.35±0.2 4.5±0.3 4.7±0.3 5.2±0.1 5.6±0.15

4 4 4 4 4 5 4 4 4 4

H D D D D

2.7-16 2.7-8.5 3-8.3 2.7-8.3 5.5-28.5 5.7-7

5.25±0.1 5.3±0.25 6.25±0.2 5.75±0.15 6.6±0.2 5.1±1.4

4(5) 4(5) 4(5)

D-D(D) D-D(D) D-D(D)

5 4(5) 4(5)

D D-D(D) D-D(D)

D D D D H D H D D D-D(D) D-D(D) D-D(D) D H D-D(D) D-D(D) D D D D D

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

n-P-">Pn-nO

I

Hadron Reactions and Urbaryon Rearrangement Table I. process

I Range of

rc-p~p2rc+3rc-rc 0 ~n3rc+3rc-

rc+p~p3rc+2rc-rc 0 ~n4rc+2rc-

K+p~pK+2rc+2rc-rc 0 pp~pp2rc+2rc- rc 0 ~pn3rc+2rc-

(continued)

PLab (Ge V /c)

I

n

5.5-16 5.5-16 3.4-8.5 3.5-8 3.5-8.3 5.5-28.5 5.5-28.5

6.6±0.2 6.8±0.2 6.8±0.3 5.5±0.3 7.2±0.4 6.95±0.1 7.25±0.2

10-16

7.3±0.9

I 6(7)

3

PP~2rc+2rcPP~2rc+2rc-rc 0

PP~3rc+3rcPP~3rc+3rc-rc 0

0.8-5.5 1.5-12 1.5-12 1.5-7 1.5-7

""'-'5 ""'-'5.5 ""'-'6 ""'-'6.5

5 6 7 8 9 10

8 17.1

0 2.0±0.15 2.6

0 2 3

""'-'4

PP~4rc+4rcPP~4rc+ 4rc-rc0

quasi -elastic rc+p~p+p

rc+p~prc+

6 6 6 6 6 6 6

(backward)

4

D D D D D D D

I D-D(D)

H H H H H H H H D

H X, Z

n

1 8

@

elastic and quasi-elastic

o n±p- p±p •

rr.+p -pre+ (backward)

!:>.

pp-mrr multiple production processes

6

4

2

2

3

4

5

6

7

nR-

Fig. 4. Correlation between the energy dependence of cross sections of various high energy reactions and the least numbers of rearranged sakatons nR·

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

PP~rc+rc-

PP~rc+rc-rc 0

407

A. Kobayashi, T. Matsuoka, K. Ninomiya and S. Sawada

408

rearrangement are forbidden, then the possible least numbers of nR for the multiple production processes are given by those in the parentheses in Table I. In terms of nR redefined here the experimental energy dependence of the cross sections are reproduced more satisfactorily by (2). The correlation of experimental n and the nR is shown in Fig. 4.

§ 3. Energy dependent factor of the rearrangement amplitudes

l

(3) Each factor [(Pi+ p1i]-r corresponds to one-sakaton rearrangement from a hadron with momentum Pi to that with p1 shown in Fig. 5. The factors may be considered to come from the overlap integrals of the wave functions or the form factors of hadrons as the composite systems of sakatons. 6 ) Factorization (3) reproduces both the energy dependence (1) and the phase factors 1 and e 2i7tr of rearrangement amplitudes T x(s, t, u) and T H(s, t, u) in the small Jtl region. This is seen, for example, for the case of the MB---7MB process:

c

d

a

b

pl~l~~(" I

I

I

I

I I

I

I

r---------' pl

D-type

u

n

forward

H-type

X-type

Z-type

backward

~b

Fig. 5. A rearrangement in a multiple production .process a+b---7!+2+ ···+N

Fig. 6. Rearrangement diagrams for the ME-7MB processes (forward and backward).

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

In the previous analysis of the two-body scatterings, it was found that the spinnon-flip X- and H-type rearrangement amplitudes T x(s, t, u) and T H(s, t, u) have phase factors 1 and e 2i7tr, respectively, in the forward region. 1) From this fact and results (1), we propose a possible factor which is contained in the rearrangement amplitudes TnR (s, t, u) in the small Jta 1 and small JtbNI region in the following way:

409

Hadron Reactions and Urbaryon Rearrangement

(4a) (4b) In the MB---7 MB processes, there are the Z-type and D-type rearrangement contributions in addition to the X- and H-types (Fig. 6). For the Z-type rearrangement in the small lti-region we have

Tz(s, t, u) oc [ (Pa + PaYJ-r[ (Po+ PcYJ-r [ (Pa + PoYJ-r [ (Pc + Pd) 2] - 2r (4c)

= (-u)-2r(-s)-2r __ ~e2i"'rs-4r 8->large

T x(s, t, u) oc [ (Pa + Pc) 2lr [(Po+ Pa) 2]- 2r = ( - u)-sr ~s-sr,

(5a)

T H(s, t, u) oc [ (Pa + PcYlr [ (Pa + PoYJ-r [ (Pc + Pd) 2]-r [(Po+ Pd) 2]- 2r

(5b)

= (-t)-sr(-s)-2r~e2i"'rs-5r, 8->large

T z(s, t, u) oc [(Po+ Pa) 2]-r [ (Pa + Po) 2]-r [ (Pc + PaYJ-r = ( _ t)-r ( _

s

)-2r ~e2i1rr s-sr

(5c)

8->large

and

TD (s, t, u) oc [ (Pa + PcY]- 2r [(Po+ PaYJ-sr = ( - t)- 5r~S-

5

r.

(5d)

Owing to the energy dependent factor and phase factor in (4a) and (4b) there are correspondence between the amplitudes T x(s, t, u) ± T H(s, t, u) and the Regge amplitudes TRegge with strongly degenerate trajectories a (t) and residues {3 (t) in the forward regions if we take - 2r + 1 =a (t):

T x(s, t"-'0, u) ± T H(s, t"-'0, u) oc (1 ± e 2i"'r) s- 2r ~T

. (r = ± 1) Regge

=

{3 (t) (1 ± e-i,.a(t)) , ( ) sm na t

sa(t)-1.

(6)

where r is the signature of the Regge trajectory. On the other hand, as is shown by (5), in backward meson-baryon scattering, the sakaton rearrangement amplitudes do not correspond to the conventional Regge-pole amplitudes with baryon trajectories. The factorization (3) satisfies the requirements on the phase difference between s-u crossed amplitudes imposed by the generalized Pomeranchuk theorem. 7)

Downloaded from http://ptp.oxfordjournals.org/ by guest on January 13, 2016

from (3). In the region, this type will give very small contribution. In the forward region the D-type rearrangement has neither energy-dependent factors nor phase factors. This type of rearrangement should not be treated on an equal footing with the other types because it occurs without rearrangement and corresponds to the diffraction scattering as mentioned in § 2. In the backward meson-baryon scattering where u = - (Pa + Pc) 2 = - (Po+ pdy rvO, we have

410

A. Kobayashi, T. Matsuoka, K. Ninomiya and S. Sawada The theorem states that the amplitude of the process a+ b-7c+ d, F(s, t) = g (t) sa