Production Rates and Distincti~e Components of Single-Particle

1280

Progress of Theoretical Physics, Vol. 48, No. 4, October 1972

Production Rates and Distincti~e Components of Single-Particle Distributions*)

--Inclusive Reactions and Urbaryon Rearrangement Diagrams. Ill--

Kisei KINOSHITA and Hujio NODA

Department of Physics, Kyushu University, Fukuoka

(Received March 22, 1972)

The production mechanism of a particle in multi-particle production is distinguished according to whether it is composed of urbaryons coming partly from incident particles, or is composed only of ones newly created in the collision. The former mechanism, which is different for a particle and its antiparticle, gives constant multiplicity and rapid limiting behaviour in the fragmentation region. In the case of pp collisions, the limiting energy for this mechanism is somewhat above 10 Ge VI c. The latter mechanism gives the same increasing multiplicity and distribution for a particle and its antiparticle. The scaling energy in the latter mechanism including x-0 is quite different for rr, K and p, indicating strong mass dependence. The scaling energies in the reactions pp-7rr-X, K-X and pX are around 30 GeVIc, a few hundred GeVIc and 1 TeVIc, respectively. It is conjectured that the average multiplicity differences such as nn•-nn-, riK•-nK- become energy independent at energies somewhat above 10 Ge VIc for pp and K+p collisions.

§ l. Introduction

Recently, the single-particle distributions for pp --7 A0 X and K1 ° X at 13 ""'28 Ge VIc were reported with the interesting result that the weighted K1 ° distribution increases with energy, in contrast to the A0 distribution which already shows limiting behaviour.1l This difference should not be attributed to the one between baryons and mesons, since it is to be reminded that the pp--7-rrX shows limiting behaviour almost above 10 Ge VI c.2l In this paper, we shall solve this puzzle from the viewpoint of distinguishing the production mechanism according to the flow of urbaryon lines, which has been emphasized in previous papers.8l•'l Further, we investigate the relation between production rates and single-particle distributions based on the interrelation between exclusive and inclusive reactions through the reduction rule of urbaryon lines.'l

In § 2, basic concepts and main conclusions are summarized. In § 3, the distinction of production mechanisms are discussed in connection with the strange particle productions in pp collisions. In § 4, multi-pion production is studied. In § 5, the situation above 30 Ge VIc are discussed by utilizing Serpukhov data, ISR data and cosmic ray results. In § 6, numerical results are summarized

*l This work was presented at the meeting on "Scattering Theories and Strong Interaction" held at the Institute for Nuclear Study, on Feb. 14"-'16, 1972.

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Production Rates and Distinctive Components 1281

and related problems are discussed.

§ 2. Production mechanism, multiplicity and single-particle distribution

For convenience of later phenomenological studies, we summarize the general relations among production mechanism, multiplicity and single-particle distribution, and the outline of.our approximations in (i) and (ii). Also, the main conclusions of this work are briefly summarized in (iii).

(i) Production mechanism and multiplicity

In a reaction a+ b~c + d 1 + · · · + dN, the production mechanism of the particle c can be classified into the following three categories, according to the connectedness of urbaryon lines in the produced particle and initial hadrons :3l; 4)

Category I: All of the lines in c originate from all of the lines in either a or b.

Categary II: Part of the lines in a and/or b flows into c. Category III: None of the lines in a or b flow into c. The particle c

is made of urbaryon lines which are newly created in the collision.

For each exclusive reaction r specified by an urbaryon rearrangement diagram (abbreviated to URD), we denote the multiplicities of particle c belonging to these Categories by nc1(r), ncn(r) and ncm(r). The multiplicity of the particle c in the reaction r, nc (r), is written as the sum of three components as nc (r) =n/(r) +ncn(r) +ncm(r). The total multiplicity n(r) in the reaction r is also written as

(1)

where

(2)

By disr.egarding the cross terms of exclusive URD amplitudes in the sense of random phase approximation discussed previously,4l we can write the total inelastic cross section (J tot, inei ( = (J tot- (J ei) as

(J tot, inel = L: (J r ' r

where Gr denotes the channel cross section of the reaction r. tiplicities (denoted by upper bar) can be written as

n- 1, n. m _ "" n 1, n. m (r) (J /<J . C -~ C T tot,Inel• r

Since n1 (r) == 0, 1 or 2 in inelastic reactions, we have

n}= I:; n 1(r)Gr/Gtot,ine1<2 • r

(3)

The average mul-

(4)

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1282 K. Kinoshita and H. Noda

Similarly, we have

nii<(sum of minimal urbaryon numbers in a and b).

On the other hand, nm and n~ are unlimited, as far as energetically possible.

(ii) Single-particle distribution and multiplicity

According to the previous work based on the structure of exclusive reactions and the reduction rule of urbaryon lines, the above Categories are related to the types of the URD for the single-particle distributions as follows:'>

a)

)(! X~ a c b a c b

O®P (!a) DrzH(la)

~I a c b

Hd® P(Ila)

~ a c b

przp

a c b HrzP(Ila)

a b c HrzH(Ilb)

uu nn a b c HdrzH(llb)

b)

FH~ a c b a c b

)I{

~~~~ a c b HrzH(Ilab)

Fig. 1. The URD for baryon (antibaryon)<a>+baryon<b>-7baryon<•>+anything(X) as the forward a+c+b-7a+c+b amplitude. (Suffix d implies double annihilations-creations of urbaryon lines.) a) Typical diagrams for baryon<a>+baryon<b>-7baryon<•>+X. b) Typical diagrams for antibaryon<a>+baryon<b>-7baryon<•>+X, not appearing in a).

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Production Rates and Distinctive Components

Category I: Category II:

Category III:

D@P(Ia), DQ9P(Ib), DQ9H(Ia), DQ9H(Ib). HQ9P(Ila), HQ9P(Ilb), HQ9H(Ila), HQ9H(Ilb), HQ9H(Ilab). PQ9P, )1{.*>

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where the types belonging to Categories I and II are further specified by the connectedness of urbaryon lines in c with a and b, denoted by Ia, lib, Ilab and so on. In Fig. 1, we illustrate these diagrams for baryon (antibaryon)<a> + baryon<b> ~baryon<•> +anything.

Throughout this paper, we shall neglect the types H®H and )1{, since these contributions decrease with s-1/ 3 and are quite unimportant at tens of GeV /c. Effect of them around 10 Ge V / c will be discussed in separated papers. On the other hand, we should retain the D®H type, because it remains asymptotically due to the singular behaviour at x:::::::1, although it does not show limiting behaviour at fixed x.8> In other words, drJ(M,/, s)/dM:1 due to the D®H type becomes energy indpendent as s~large. Within these approximations, the multiplicity belonging to each category and single-particle distribution for a+ b~c +anything (X) are related as follows:

n 1 = J ( drJ D®P(Ja) + drJ D®H(Ia) ) dq/rJ . + (Ib) c dq dq tot, mel , (5)

·• 'fi 11 = J (d(J H®P(IIa) + d(J H®P(nb)) dq/rJ .

c d d tot, mel , . . q q (6)

- m f drJ P®P d / n. = q (Jtot,inel. dq

(7)

It is to be noted here that the non-vanishing and s-independent behaviour of Wq

X (drJ/dq)=p(x,qJ.3,s) at x:::::::O gives logs dependence of the multiplicity.5> Therefore, such behaviour is admittable only for the P®P type. On the other hand, the limiting behaviour of p(x, qJ.3, s) at the fragmentation region is related to constant multiplicity and it is allowed for other types of contributions.

The above expressions, Eqs. (5) '"'"'(7), can be derived from Eq. ( 4). For example, let us consider n.m. Equation (4) can be rewritten as

- m ~ f drJr d (ill)/rJ n. = ~ dq (III) q tot, inel , (8)

since

where q (III) is the momentum of c belonging to Category III. Through the

*l The notation >K is suitable compared with H®P(III) adopted in Ref. 4), since its physical implication is quite different from H®P(Ila or lib), in spite of topological similarity.

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reduction rule of urbaryon lines,4l we have

:E d(f _ d!J P®P + d!J )!{ •

r dq(ill) dq dq (10)

By disregarding the second term on the right-hand side of Eq. (10), we obtain Eq. (7). Similarly, we can obtain Eqs. (5) and (6).

(iii) Asymptotic energies for multiplicities and single-particle distributions We shall summarize here the main conclusions deduced from later phenome

nological studies centered "on pp collisions. (1) Without respect to the kind .of particles, 'iien attains constant asymptotic

value, at energies somewhat above 10 GeV /c. Corresponding to this result, PH®P(x, q/, s) shows early limiting hehaviour in the fragmentation region, leaving some transcendental feature around x::::::::O.

(2) The differences of particle and antiparticle multiplicities such as fi;..- fi;.become energy independent at energies somewhat above 10 GeVjc. This is a consequence of (1), since we have

fie- fie= fie n- ficn::::::::const , (11)

from

fiem = ficm = (increasing with energy). (12)

(3) The scaling energy for the P&;;P type is sensitive to the mass of particle c. As energy increases, scaling behaviour is approached from below and from larger side of I xl. The log s. rise of the multiplicity ficm is given by the scaling behaviour at I xl ::::::::0, which is attained around 30 Ge V / c, a few hundred· Ge V / c and 1 Te V / c for 7r, K and p, respectively. ···

Numerical values of fien and fiem are summarized in the last section.

§ 3. Distinction between Categories II and III

In order to recognize the distinction between Categories II and III, it .is instructive to investigate strange particle productions in pp collisions. In Fig. 2, we show the energy dependences of the weighted distributions (J)qd!J / dq for pp~A0X and pp~K1°X at 13"'28 GeVjc.1l We see that the distribution for pp~A0X is almost energy independent in every x-interval, while that for pp~ K 1°X increases with energy. Since the antibaryon production, which belongs to Category III, in these energy intervals is quite negligible, we can attribute the A0 production to Category II. Furthermore, the single-particle distribution of A0

may be dominated by the H&;;P type which shows limiting behaviour. On the other hand, both Categories Hand I1I contribute to K 1° production;

dominated by the H&;;P and Ptg;P types, respectively, in the single-particle distribution. Therefore, the different behaviour of A0 -and K 1° production can be

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understood as the different nature of Categories II and ill, i.e., the different approach :to limiting h,ehaviour for the H®P and P®P types.

From these observations, we may conjecture that "the single-particle distribution due to Category II attains limiting behaviour of the H®P type at e~ergies somewhat above 10 GeVjc without respect .~o kinds of particles, while the scaling behaviour of the P®P type for the K-production is not yet saturated at--ene_rgies below 30GeVjc.

'''''''l:l ' f !

I I f t

r! t t

.,

,f"<-.41

10r<x<-.2~

o~--~------~----~

I I I I I I

20 30

J'"'o I J I

10

_(a) P,_ {GeV/c)

u .1..8s.. .:!!!:.... ( 10-a)

I .fitr1 dlxl X n 60 I 1 I f

p Ro p O<IXI<.1 I HeP

40

I .1 <IX1<.2 1 f I 20

I i I .2<1X1<.6 I I

0 p KO(KO) p 10 20 30 pep

(b) PL (GeVic) (c)

Fig. 2. a) Energy dependence of A inclusive 'cross_ section in four selected intervals of .x, given in Ref. 1).

b) Energy dependence of K1° inclusive' cross section in selected intervals of l.xl, given in Ref. I).

c) the URD for p+p--'>KO(l{O) +X.

The above conjecture is consistent with the energy dependence of the semiinclusive cross sections O"(pp~A°K+anything) and O"(PP--'>KK+anything), shown in Fig. 3. Since the strange particle production is small by one order of magnitude compared with the total inelastic cross s~ction, we can disregard the doublepair production of strange particles and obtain the average multiplicities by

'fi.t='fi.tn-::::::.O"(pp---'>A°KX)/O"tot,ine1-::::::.0.04,

'fiK=nxm-::::::.()"(PP---'>KKX)/O"tot,ine1,

(13)

(14)

with the result that nxm is almost linearly increasing in PL above 20 GeV jc. Our conjecture implies that the single-particle distributions of other hyperons

also exhibit limiting behaviour of the H®P type. We obtain the total multiplicity of hyperons as ny · 0.1 from the data 0" (PP---'> YK +anything) = 3.1 ± 0.3 mb at 44.5 Ge V / c;6l This value of _ ny will persist for higher energies as far as

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::0 g b

iiif. ii,. ~ , f cr(pp-KK • any.) / ~08 y , f cr(pp-AK • any.) VI 2

.06 / '

!/! ?r .04 ~

~

o~--~lo~--~--~2~0----~--~3~0-

A. (GeV/c)

Fig. 3. Semi.inclusive cross setions for pp-7AK plus .anything and pp-7KK plus anything. Unshaded data are due to Ref. 1) , and shaded ones are explicitly referred to. Multiplicity scale for A and K is shown on the right·hand side of ver· tical axis. The straight line L: denotes an inter·

·polation with the 70GeV/c result·given by Eq. (21).

o-,01.

3 ~ §

2 ti (A) 1 0

3 Oi.ii _,f .. .a2 f. ,,,,_t-'--

bE 1 , ,. (B)

_j-! O"yoiC i lr-t-§...f- :--- t "::'----~

~ ~) I I

·+;:--~ c-~-~c-,<g.' 10 24 P,,.,.,GeV/c

Fig. 4. Semi·inclusive cross sections for strange. particle productions in re-p col. lisions, compiled· in Ref. 9).

antibaryon production is negligible, Concerning lower energy side, it is reported that 6 (pp~ YKX) = 1.6 ± 0.2 mb · at 8 Ge VIc/>

=1.23±0.2mb at 10GeVIc,8>

which imply that just 10 Ge VIc is somewhat insufficient for the asymptotic energy of Category II.

Our picture is consistent with the strange particle production cross section in the n-p collision below 25 Ge VI c,9> shown in Fig. 4. The gross feature can be summarized as follows :

6(n-p~YKX)-:::::.1 mb,

(J (n-p~ KKX) -:::::.2 mb +increasing part.

We obtain ny-:::::.0.05, which is just the half value of the pp case, indicating the target fragmentation belonging to Category II. The larger constant part of the KK production cross section is attributed to the fragmentation of the projectile pion, which is an aspect different from pp collisions.

§ 4. Multiple-pion production

The multiplicities of nucleons and pions in pp collisions at 19 Ge VIc were

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reported as10l

(15)

where strange particle productions were disregarded, since they contribute about

0.1 as a whole to the multiplicity. These values are related to three Categories

as follows :

(16)

From our conjecture introduced in the preceding section, it follows that:

(a) The values np=l.4 and n,.=0.6 are energy independent at energies some

what above 10 GeV jc, as far as the antibaryon production is negligible, and

({3) n ... - n .. -= 0.6, independent of energy, irrespective of the anti-baryon produc-

tion.**> The experimental result for pp~pX at 30 GeV jc gives np= 1.4,11> consistent with

our scheme. Further, the scaling behaviour of this reaction observed in an ISR

experimene2> indicates the energy independence of np in Te V region (a very

small increase of np due to Category III will be discussed in the next section).

Also, the constant inelasticity of proton observed in cosmic ray experiments18l is

consistent with limiting behaviour with the form p (x, q J.2, s) ocx.8>

Similarly, we have the energy independent difference of K± multiplicities m

pp collisions with value

nK·-nK-~ 0.5(J(pp~YKX) ~0.05' (J tot, inel

where the equal probability assumption of K+ and K 0 is used. Increase of the

K multiplicity at higher energies is due to Category III, and the above arguments

remain unaltered. In treating the single-particle distributions for pp~rc± X, it is very important

to distinguish the types H®P and Ptg)P, corresponding to Categories II and

ill. On the other hand, Bali et al.6> parametrized the average multiplicities as

n ... =c± log s+d±, and computed c± and d± from the single-particle distributions

without considering this distinction. The result with c+=/=c- obviously contradicts

the strange particle production data.14l At asymptotic energies where the P®P

type satisfies the scaling behaviour, the average multiplicities should be para-

*> Charge independence requires this value within 1.4±0.1, from more accurate values of 'iin•.

**> Charge conservation constraint for pp collisions is written as

2=~ Q0 (n0 -n;;) =~ Q~Cn.II-n;;II) +nPI, • •

where summation runs over positively charged particles, Our conjecture implies energy independ

ence of each difference 'iic-'iic,

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metrized as

(17) Not the ratio but the difference of n± distributions is useftill, since the PQ!;;P type is common to n± and

(18)

In the fragmentation region, i.e. I xi ~0, the n± spectra show limiting behaviour with n+ larger than n-.'AJ This feature is due to the HQ!;;P dominance, and it will survive for higher energies. However, in the central region characterized by I xi ::::::0, the HQ!;;P type still contributes at lower energies giving n+ excess over n-. There, the PQ!;;P type contribution domi~~tes ~s ~nergy incre~~es, and the HQ!;;P type gradually diminishes, resulting in eqwil production rates of n±. The scaling energy of the PQ!;;P type may be taken to be about 30 Ge VIc, where the increase in charged multiplicity seems to attain logs behaviour.15l The·· scaling behaviour of the PQ!;;P type is most slowly attained at I xi ::::::0, and the multiplicity for Category III is very sensitive to the behaviour at !xi::::::O.

§ 5. , Situations above 30 Ge:V I c

The reaction pp~K1°X is far from scaling behaviour below 30 GeV lc, as discussed in § 3, contrary to pp~n± X which qualitatively attain scaling behaviour including !xi ::::::0 at energies around 30 Ge VI c. This may be due to the difference of the asymptotic energy of the PQ!;;P type contribution, reflecting the sensitivity of the production dynamics to the mass of the produced particle belonging to Category III. Therefore, it is very interesting to investigate the K production at energies a.bove 30 Ge VI c. ' First, we investigate the Serpukhov data.lB),lT) At 70 qev I c, the x-In- ratio of the spectrum from AI target was measured at small fixed angle as a. function of momentum. The ratio reaches maximum value (8.5 ± 0.3) X 10-2 at the top intensity of the n- spectrum.17l Therefore, we obtain nK-In.,-::::::0.1. Since the most of the negative charged components are n- and K-, the absolute value of nK- can be evaluated from the average charged multiplicity nch as follows:

(19)

We have nch = 5.1 at 70 Ge VIc, from the interpolation of the acqelerator 'data up to 30 Ge VIc and the cosmic ray data by Jones et aP8l compiled by W roblewski.15l Thus we obtain

nK-= 0.12, n,.-=1.4, that is,

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Comparing this value with the data below 30 GeVIc shown in Fig. 3, we see that

the almost linear increase of nxm still continues around 70 Ge VI c. Also, the

K± In± ratio at 70 Ge VIc 'can be· evaluated as follows:

nx.+nxn ... +n .. -

2nx-+ (nx.-nx-) ~0.24+0.05~o.o8 . (20) 2n .. -+ (n ... - n .. -) 2.8 + 0.6

This value is yet far fr.om the cosmic ray data, K±ln±:::::::0.25±0.08.19> If we ex

trapolate the increase of nx by parametrizing

nx= (Pi(GeVIc) -5.0) x 3.7 x 10~8, (21)

which passes 0.24: at 70 Ge VIc, the crossing energy with the asymptotic form5>

nx= 0.89 log (~J x 0.2 (22)

is about 300 Ge vIc, where E is the inciden~ energy in the laboratory frame and

MN'js the nucleon mass. The asymptotic energy will become higher, if we adopt

10-' further moderate approach than

R \ J.

----:~ " '\ --~~ : Kl£ ......... ~ ..... +. . ..

.... . . .. ''\ ' +\

\ "-'.... .., . '":"'-·........... ·~ ·' \

........... "' ... ........ p!TC

.... g .... 0 ' .. ' ' ......

•• £=70 GeV ' .. 52 GeV .. 43 GrN

• 35GrN + 20GeV

19.2 GeV

0.4 PIPma•

\

"' \ +' \

\ \ \ \ \ +. ., ' \ \ '

+\

0.8

\ \ \ \ \

t\ 1.0

Fig. 5. The production ratios of K- and p with respect to n- in proton-alumi~ium collisions between E=20....,70 GeV, plotted as a function of P/Pmax• where P is the secondary laboratory momentum and P mu: the maximam momentum for the heavier particle,·

Eq. (10). Anyway,· the scaling energy of the P&;;P type including x:::::::O · should be greater than 300

Gevlc . Next, we consider the ap

proach to the scaling for PP-" pX in the fragmentation region. We can approximate

PIPmax:::::::x,

where P is the absolute value of the laboratory momentum of the secondary particle in the forward direction and Pmu is its kinematical maximum, as far as P is nearly equal to its longitudinal component. Therefore, the ratio of the K~ In~ spectrum shown in Fig.

5 can be taken as the ratio of the p(x, qJ.2, s) in x2;0.4, i.e., in the projectile fragmentation region. Since the limiting behaviour of 77:~ is already attained at energies above 30 Ge VIc and such large x, the energy dependence of the. ratio can be taken as that of

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K- distribution. From Fig. 5, we see that the limiting behaviour of K- is attained from larger side of x, and 70 Ge V / c is almost sufficient for limiting behaviour at x2':0.4. This observation supplements the previous argument that the scaling behaviour in the region including x=O is not yet attained at 70 GeVjc.

On the other hand, the pjn- spectrum shown in Fig. 5 indicates that: · (1) The p distribution also attains limiting behaviour from below, from

the larger side of x, with x?0.7 at 70 GeV /c. (2) In the region x$0.5, 70 Ge V / c is yet insufficient for limiting behaviour. It should be reminded here that pp~ K-X and pp~ pX are uniquely classi

fied into the P®P type. Therefore, we may conclude that the scaling form of the P®P type is approached from below gradually with energy from the larger side of JxJ, and the approach to scaling behaviour is different for n-, K- and p. The scaling behaviour in the whole region including x.::::::::O may be attained at around 30 Ge V / c for n and somewhat above 300 GeV jc for K-. As for p, the energy dependence of the p/n- spectrum even for x.::::::::0.4, in contrast to K- /n-,

0.1

0.01

0 .... E CIJ u .... g_

500 1100 1500 GeV/c

• •

indicates that its scaling en-ergy may be much higher 0·1 0·2 x 0·3 0.4

0.1

0.0

than that of K-. Fig. 6. The particle ratio p/7-c- at q.L=0.4GeV/c at ISR Now,theCERN-ISRdata energies and 24GeV/c, given in Ref. 20).

for negative particle production has become available.20> As shown in Fig. 6, the spectrum of the pjn- ratio at q .L = 0.4 Ge V / c and x.::::::::0.07 "'0.2 increases with factor about 7 from 24 Ge V / c to ISR energies. This increase is greater than the factor about 3 at x.::::::::0.4 from 19 Ge V / c21l to 70 Ge V / c.16> Within the ISR energies, the spectra at 1100 and 1500 Ge V / c overlap well within errors and the one at 500 Ge V / c seems to be slightly lower. Therefore, we may consider that 1 Te V / c is almost sufficient for the scaling behaviour for pp~ pX, described by the P®P type.

Combining these results, we conclude that the saturation energy of the scaling distribution for the P(g)P type, which gives the logs dependence of ncm, is sensitive to the mass of the produced particle c.· It is interesting that the ratio mp2 : m1r2 : m,..2.::::::::1: 0.28: 0.02 seems to be roughly proportional to the ratio of the scaling energies in the pp collision. This point may be an important clue to un-

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Production Rates and Distinctive Components

derstand the dynamics of multi-particle productions in the central region.

§ 6. Concluding remarks

(i) Numerical values of particle multiplicities

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Here we compile our estimates of particle multiplicities in pp collisions in our scheme.

The energy independent differences of particle and antiparticle multiplicities:

n,..-n,.-=0.6±0.05,

nx.-nx-=0.05±0.01,

n,-np=l.4±0.05,

nn-nr.=0.55±0.10'

n"-n..:c=0.04±0.01,

ny-ny=0.10±0.0l for PL:2:;13GeV/c.

These differences are due to the production mechanism belonging to Category II, except for proton to which Category I also contributes.

The scaling energies PL~ of the P@P types and corresponding asymptotic behaviours of multiplicities:

n: PL0-::::.30Ge V jc, n~-=n,.--n~-==-0.35log(E/ MN),

K: PL0:2:;300GeV/c, niJ-=nx-==-0.09log(E/MN).

p: PL0""1TeVjc, nPm=np-::::.O.Ollog(E/MN),

where we have used, as an input,p/n--:::::.3% atq.L=0.2GeV/c, 0.07<x<0.2 and ISR energies.20l

The values of multiplicites and their ratios at some typical energies are listed in Table I.

Table I. Particle multiplicities and ratios.

19 4.02 1.0 0.02 --..10-8 1.6 3.2 0.037

30 4.5 1.2 0.04 ? 1.5 2.2 0.043

70 5.1 1.4 0.12 0.03 1.4 1.4 0.083

1000 7.2 2.0 0.51 0.06 1.3 1.1 0.23

(ii) New experiments on multiplicities and single.particle distributions

In order to check our scheme on Category II, bubble-chamber work below 30 GeV /c will be useful. We predict the energy indepedence of n,..-n,.-, nx• -nx- and n,.(-:::::.0.7) for K+p collisions in the energy region PL>10 GeV /c.

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The situations for n±p collisions may not be so different, as far as non-scaling terms can be neglected. It goes without saying that the bubble chamber data in the Serpukhov region are waited for. Another important problem is the validity of the additivity relation

where na-+cn and nb_cn are due to the single-particle distributions represented by the H<Z;P(IIa) and HQ9P(IIb), respectively (see Eq. (2)). Also interesting are accurate measurements of the production ·cross. section of each hyperon in various collisions below 30 Ge VI c.

Concerning Category III and the corresponding single-particle distribution of the P®P type, the energy dependences of pp~K-X and pp~J5X in the central region, x:::::O, is very interesting. At this point, we expect future experiments at Serpukhov, NAL and CERN-ISR. If the production mechanism in the central region is characterized by the mass. ot the produced particle, the production rate of anti-hyperon may be of the same order of magnitude as that of ]5. This point should be carefully examined by the "big accelerators." Below 30 Ge VIc, the energy dependences of the single-particle distributions for K+p~K-Xand n+p~ K-X are of primary interest. We predict rapid increase of the spectrum with energy, especially at x::::O. These reactions are also interesting in the Serpukhov energy region.

(iii) Comments on other approaches

Recently, Tye and V eneziano22> argued that negative non-scaling contribution in the single-particle distribution should necessary, in order to balance with the positive non-scaling terms satisfying the consistency with flat .total cross section for (ab) exotic reactions. They concluded that, in our terminology, the contric bution of the H<Z;P(IIa) type in the region x>O should be accompain~ed by negative non-scaling contribution. The increase of the pp~K1°X spectrum with energy was adopted as a supporting evidence. The pitfall of their argument is the a priori assumption of early scaling of the P®P type. Our scheme is the opposite: early limiting behaviour of the H®P type and variant approach to scaling from below for the P®P type. The behaviour of pp~K-X and pp~J5X favours our scheme and the limiting behaviour of pp~ A0X can be consistently understood.

Berger and Krzywicki28> stressed the importance of exclusive reactions to understand the inclusive reactions. we quite agree with them . on this assertion. However, the considerations of these authors are rather limited to kinematical aspects only. In contrast, we have stressed the role of urbaryon lines to understand the interrelation between exclusive and inclusive reactions.'> Alorig this line of thought, we have analysed here various data distinguishing· the production mechanisms due to Categories II and III, and the single-particle distribu~ tions belonging to the H®P and P®P types.

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Concering average multiplicities, there are some questions on the two energy regimes discussed in Ref. 23). The first regime with PL;$100 Ge VIc was characterized by the 3-dimensional isotropic gass with noc power of energy. However Peyrou plots of pions at energies near 19 Ge VIc already reveal onedimensional character, due to universal ql.-cutof£.24) The second regime with PL ~500 GeV lc was characterized by one-dimensional relativistic gas with 'fi,-,_,log s. In our estimates, approach to logs behaviour is quite different for n, K and p. This feature may be an important check point for any dynamical model.

The authors would like to thank Professor M. Uehara for helpful discussions and encouragement.

References

1) E. L .. Berger, B. Oh and G. A. Smith, Phys. Rev. Letters 28 (1972), 322. 2) H. B~ggild, K.H. Hansen and M. Suk, Nuc. Plys. B27 (1971), 1.

B. Carazza and G. Marchesini, Phys. Letters 35B (1971), 436. 3) K. Kinoshita and H. Noda, Prog. Theor. Phys. 46 (1971), 1639, 1642; 47 (1972), 1300. 4) H. Noda and K. Kinoshita, Prog. Theor. Phys. 48 (1972), 877. 5) N. F. Bali, L. S. Brown, R. D. Peccei and A. Pignotti, Phys. Rev. Letters 25 (1970), 557. 6) J. Bartke et al., Nuovo Cim. 29 {1963), 8. 7) M. Firebaugh et al., Phys. Rev. 172 (1968), 1354. 8) S. M. Holmgren et al., Nuovo Cim. 51A (1967), 305. 9)· J. M. Waters et al., Nucl. Phys. B17 (1970), 445.

10) H. B~ggild et al., Nucl. Phys. B27 (1971), 285. 11) E. W. Anderson et al., Phys. Rev. Letters 19 (1967), 198. 12) L. G. Ratner et al., Phys. Rev. Letters. 27 (1971), 68. 13) Y. Fujimoto and S. Hayakawa, Handbuch der Physik 4612, 115 (Springer, 1967). 14) J. Arafune and H. Sugawara, INS Report-175 (1971). 15) A. K. Wroblewski, Rapportou~s talk at Kiev Conference, 1970. 16) Yu. B. Bushnin et al., Phys. Letters 29B (1969), 48.

F. Binon et al., paper 308 submitted to Lund Conference (1969). 17) Yu. M. Antipov et al., Phys. Letters 34B (1971), 164. 18) · L. W. Jones et al., Phys. Rev. Letters 25 (1970), 1679. 19) M. Koshiba et al., J. Phys. Soc. Japan 22 (1967), 1321. 20) A. Bertin et al., Phys. Letters 38B (1972), 260. 21) J. V. Allaby et al., CERN Report 70-12 (1970). 22) S. H. H. Tye and G. Veneziano, Phys. Letters 38B (1972), 30 .

. 23) E. L. Berger and A. Krzywicki, Phys. Letters 36B (1971), 380. 24) loc. cit. 10).

See also M. Namiki, I. Ohba and Yap Sue-Pin, Prog. Theor. Phys. 47 (1972), 1247. M. S. Chen and F. E. Paige, Phys. Letters 38B (1972), 249.

Note added in proof: The increase of the P®P type spectra with energy can be explained by the form3l wqckr/dq= (Mx2/s)PF(ql.2 +mc2), due to the energy dependence of Mx2/s :::::1-vx2+4(ql.2+mc2)/s for fixed x and ql.. Quantitative agreement is remarkable, as will be shown in the V-th paper of the present series.

In the IV-th paper (Prog. Theor. Phys. 48 (1972), No. 6b), the systematics of the inclusive phenomenology is ·discussed.

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Production Rates and Distincti~e Components of Single-Particle

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