Inelastic Interactions of Particles at High Energies. II. (Energy and angular distributions of secondary particles)

Portschritte der Physik 15, 435-494 (1967)

Inelastic Interactions of Particles at High Energies. 11.

(Energy and angular distributions of secondary particles)

V. S. BARASHENKOV, V. M. MALTSEV

Joint Institute for Nu.clear Research, Laboratory of Theoretical Physics, Dubna

1. Introduction

The review paper [ I ] was devoted to a detailed study of the multiplicity and the composition of secondary particles produced in inelastic interactions of high energy particles. I n the present paper which is a continuation of the review [ I ] we are dealing with the properties of secondary particles related to their energy, momentum and angular distributions. As in ref. [ I ] we shall restrict ourselves to the energy region T 2 I GeV (T is the kinetic energy of the primary particle in the laboratory system). However data concerning the antinucleon annihilation will be considered for all energies starting with the threshold T = 0. Our aim is, as before, to summarize end analyze the available experimental data independently of any preconceived theoretical predictions. We hope that the conclusions drawn in this way may serve as a basis for construction of new theories. Some preliminary results were given in preprint [2]. The present paper is a complete and essentially revised version. We take an opportunity to thank K. D. Tolstov for many discussions which were very useful during the work. The notations used below are the same as in ref. [ I ] .

2. Energy and Momentum of Produced Particles

The distribution of produced particles over kinetic energy, their distribution over transversal momenta, the fraction of the energy spent for the formation of new particles are the quan$ities essentially characterizing the inelastic interaction. By analysing them we may obtain an important information on the intrinsic structure of colliding particles and on the interaction mechanism a t high energies. Further we shall use the center-of-mass system. This will allow one to eliminate non-essential effects of a pure kinematic origin. In particular, in doing, so we weaken the dependence of momentum and energy distributions upon angle of particle emission. In some cases, e.g. in a simultaneous production of a large number of pions the angular dependence can be neglected at all within the experi-

29 Zeitschrift ,,Fortschritte der Physik", Heft 7

436 V. 8. BARASHENKOV, V. M. ~ L T S E V

mental errors. While in the laboratory system, as is seen e.g. Fig. 1, even a small change of the angle leads to the number of produced particles being changed by a factor of 10. The higher is the energy of colliding particles, the greater is this

change. It is clear that the choice of the coordinate system does not affect the trans- % 177 verse particle momenta.

10-2

5 10 15 plGeV/cl

2.1. Mean P a r t i c l e Energy

At present there is already a great amount of experimental data on the energies and the momenta of particles produced in inelastic interactions. The first thing that is noteworthy is a weak dependence of the mean energy of produced particles on their kind and on the kind of colliding particles in all cases when the average multiplicity ?i iS large enough (2 3 f 4). Of decisive importance is in these cases only the energy

for formation of new particles. (Ec b the total energy of colliding particles in their c.m.s., M I and M , are the masses of these particles if the latter are reckoned among

Fig. 1. The number of protons and charged pions produced a t different angles in the laboratory system in the interaction of 20 GeV protons with the Be (0 = 9

nuclei 8.

The continuous curves are protons, dashed ones - pions, p is the produced particle momentum

:ind20") and ~1 (0 = 4.75 and 130) Ec = Ec - ( M I + H,) which can be spent

secondary ones). This is well seen from Tables 1-4 where the mean values

are calculated by the experimental energy spectra

Here aGj(9-)/a.Y is the differential cross section for production of particles of a certain kind in the j-th channel of inelastic reaction, nj is the number of these particles; here and everywhere below 7 is the kinetic and E = .Y + M the total energy of the particle in the c.m.s.). It may be said that in inelastic interactions of particles a t high energy (and in the case of the - N annihilation a t any energy) each degree of freedom has, on the average, the same kinetic energy. From this point of view the energy distribution of produced particles is a more convenient characteristic of inelastic interactions than the distribution of these particles over momenta which essentially depend on the particle masses, For instance, the average momentum of protons produced in x- - p collisions a t T = 7 GeV is nearly twice as large as the momentum of produced pions :

Fp = 0.9 GeV/c; pn = 0.5 GeV/c [39]

Inelastic Interactions of Particles a t High Energies I1 437

Table 1 Mean kinetic energy of nucleon after inelastic interaction

3 [MeV]

Interaction T[GeV] Method Protons Neutrons

- P - P x - P

P-P 0.81 0.925 0.97 1.5 2 2.75 3.5 4.2 6.2 8.7 9 9

18.9

1.7

9

8.5 f 1

-500

3 0.871 0.96 1 1 1.15 1.3 1.37 1.37 1.5 1.72 3.86 4.5 4.5 4.7 6.8 7 7.5 f 0,5

17

P- -" -1

(1.0 f 2.2)

N - N 6.2

27

(102 +- 103)

x- - N 1.37 4.5 4.5 7 7.3 9.89

H-DC [a, 51 Em [GI H-DC [7] H-DC [5] H-BC [8, 91 H-DC [5] Em [lo1 Em [111 Em [I21 Em ~ 3 1 Em ~ 1 4 1 Em [151 Em 1161 H-DC [I71 H-DC [18]

Em ~ 4 1 Em [lo1 Em [201 Em [211 IVICC (C12) [221

H-BC [23]

H-BC [25] Em,H-DC [261 H-BC [271

D-DC [291 H-DC [301 D-DC [311 H-DC [321 H-DC [331 H-BC 134, 351

Em [371 H-DC [38] Em [391

P-BC [41]

D-DC [31]

H-BC [24]

Em [281

Em 1361

Em [401

Em [421

Em r361

Em [a01 Em [371

Em [431 Em [441

61 f 7 84 & 9 78 f 4

192 f 23 166 f 3 243 f 50 350 f 36 310 & 44*) 510 f 64 602 & 87 582 f 45 430 & 50 620 f 37 76 f 5

115 f 11

540 f 100 596 f 90 535 f 180 320 & 65 582 f 96+)

36 & 4**) 135 f 2 81 f 5

90 f 9

149 f 13

115 & 14 480 f 12 390 i 49 581 f 49 288 38 372 f 30 386 f 64 750 f 10 837 & 268

252 * 28 575 & 97 393 i 36 439 * 107 756 f 144

59 f 6 85 5 8 84 f 5

172 5 40

44 f 6 109 f 18

90 f 5 74 f 8*) 90 f 6

151 f 25*) 116 f 8 132 f 19*) 113 f 18*) 126 & 17*) 185 f 54 530 & 17

407 f 66

113 f 13*)

*) The given value is mean for proton and neutron. **) +) The contribution of the leading nucleon (see below) is not included, therefore the value is strongly underestimated.

29*

is st,rictly speaking average for all heavy particles produced in inelastic interaction.

438 V. S. BARASHENKOV, V. M. MALTSEV

Table 2 Mean kinetic energy of pions after inelastic interaction

[MeV]

Interaction T [GeV] Method Ti+ x- (no)*)

P - P 0.81 0.925 0.97 0.97 1.5 2.0 2.7 2.75 4.2 6.2 8.7 9 9 9

18.9 24 27

P--n -1 0.5 + 1,5) 1.7

9 27

(1.0 + 2,2)

N - N 6.2 8.5 f 1

12.5 (10 +- 15) 24 25.8 27 50

20 + 150 (10 + 100)

100 100 100 100 200 200 300 (200 + 400) 500

500 (102 +- 103)

103 103 4.103 2.104

H-DC [41

H-DC [71

H-BC ~ $ 9 1 Em ~461 H-DC [51 Em ~411

Em ~ 3 1 Em ~141 Em 1151

H-BC [a81

H-DC ~171

Em [GI

H-BC [451 H-DC [51

Em [121

Em [a71 Em [161

Em [@I

H-DC [I81

Em [a71 Em [a91

Em [ZOI Em ~191

Em ~501

Em ~511 Em 1521

Em ~501

Em ~501

MCC ~561 Em [2841

Em [ell

CC(C12) [541 IC, MCC (LiH) [55]

MCC (C12) [ZZ]

MCC(CI2) [57] IC, MCC (LiH) [55]

MCC (C12) [ZZ]

IC, MCC (LiH) [55] MCC (P) [ZZ]

Em [591 Em [GO1

Em ~581

102 & 6 115 & 13 148 f 9

134 f 14 172 & 3

166 f 37

(173 f 36) 156 f 4 (172 * 9)

149 f 52 106 f 15**)

145 f 16**) 236 f 20**) 300 & loo**) 326 f 45**) 378 f 61**) 392 f 18**) 263 f 16**)

320 f 50 380 + 2

126 f 10

(144 & 4)

71 f 10

208 & 12**)

290 * 16**) 340 f 50**) 290 f 45**) 300 f 30 250 & 40**)

(330 5 35) 340 & 60**) 371 5 40**) 286 f 47**)

290 f 60 300**) 310 5 50**) 330 f 50**) 278 & 29**) 410 & 230**) 270**) 320**)

351 f 38**)

240**) 230 f 80**) 300**) 850**) 850**)


T a b l e 2 (continued)

[MeV]

Interaction T [GeV] Method TC+ n- (no)*)

r, -~ p -0 N O 0.05 0.08

0.15 0.47 0.47 2.44

(0 + 0.22)

H-BC H-BC H-BC P -BC

Em P-BC P-BC H-BC

369 f 16 363 f 17 (371 & 16) 246 f 18**) 240 f 12**)

239 -+ 19 262 f 21 (216 f 110)

206 & 20**) 318 f 49**) 278 & 15**) 321 f 15**)

17 - I1 0.08 P-BC ~ 3 1 362 f 24**) (0 + 0,23)

j? - Av N O N O N O 0.05 (0 + 0.1) 0.08 0.14 (0 + 0.23) 0.14 (0 t 0.23) 0.17 (0 f 0.25)

5 - p 0.9 x- - p 0.871

0.96 1 1 1.3 1.37 1.37 1.5 1.72 3.86 4.5 4.7 6.65 6.8 7.5 f 0,5 7.5 7.5 16 f 3

x- - n 0.6 0.82 0.9 4.5

x- - N 1.37

Em Em Em D-BC

Em Em

Em

Em

H-BC H-BC H-BC H-DC, Em

D-DC H-DC D-DC

H-BC

H-DC H-DC H-BC Em H-DC P-BC Em P-BC Em P-BC

H-BC H-BC

Em D-DC

H-BC

H-BC

239 f 12**) 251 f lo**) 197 f 21+) 231 f 17**)

208 & 28**) 206 f 20+)

227 f 25')

250 f 9**)

270 f 26**) 137 f 6 115 f 4 233 f 15 202 f 10 (204 f 16) 252 f 39 193 f 23 (262 5 38) 237 f 18 190 f 5 (246 f 24) 276 5 19 300 f 16 (233 f 22)

292 & 31+)

330 f 50+) 192 & 19 296 f 37 378 & 10 471 f 10 (522 f 27)

426 5 31 472 46 (669 & 106)

358 5 40**)

398 f 27

555 f 34 422 & 38**)

362 f 40 336 f 41 450 f 40**)

460 f 10 550 f 10 590 & 30

159 f 6+) (177 f 7) +) 163 f 9 178 f 11

(203 f 13) (233 & 17)

427 f 41**) 289 f 35++)

440 V. S. BARASHENKOV. V. M. MALTSEV

Tab le 2 (continued)

Interaction T [GeV] Method

6.65 P-BC [80] 365 5 22 430 f 24 6.8 E m [391 290 f 80**) 7 Em, P-BC [81] 390 f 30 390 & 30 (340 f 60) 7.5 Em [821 365 f 27**) 7.3 Em [431 306 f 31**) 9.89 Em [a41 332 f 25**)

*) The energy **) The given value is mean for positive and negative pions +) Obtained from data for x+ - p interaction by the isotopic invariance condition ++) The given value is mean for positive, negative and neutral pions

for neutral pions is given in brackets

Table 3 Mean kinetic energy of neutral K-mesons produced in inelastic interaction

Interactions T[GeV] Method [MeV]

P - P 9 E m [as] 426 f 240 N - N 50

(10 +- 100) MCC [83] 700 f 250 ~ ~~ ~~~ ~ ~

P - P -0 H-BC [84] 170 f 14 0,47 P-BC [65] 169 f 27

x- - p 5,86 P-BC [85] 355 f 26 6,65 P-BC [86] 314 5 32 7 3 f 03 P-BC [87] 366 f 13 7 3 f 0,5 P-BC [88] 311 f 33 10,86 P-BC [85] 458 f 77 16 H-BC [89] 390 -J= 30 17,86 P-BC [85] 500 f 100

Table 4 Mean kinetic energy of hyperons produced in inelastic interactions

Interactions T [GeV] Method c+ c A

N - AT - 50 MCC [83] 794 f 271*) 593 f 63 (10 + 1000)

x- - p 5.86 P-BC [85] 502 f 91 6.65 P-BC [86] 358 f 40 7.5 P-BC [88] 353 5 88 355 & 63 340 -j= 38 7.5 f 0,5 P-BC [90] 401 & 24

17.86 P-BC [85] 575 f 133 16 H-BC [89, 911 656 f 116 557 105 550 f 50

*) The given value is mean for Z+ and Z- hyperons.

Inelastic Interactions of Particles at High Energies I1

while their average kinetic energies are practically the same :

441

The assumption that each degree of freedom has, on the average, the same amount of kinetic energy in the secondary particle system underlies the Fermi statistical theory of multiple particle production and various hydrodynamical generaliza- tions. However, the equality of the kinetic energies of produced particles is very rough. So, in spite of large measurement errors, the kinetic energy of secondary protons in N - N collision is noticeably higher than that of pions; on the contrary the average kinetic energy of pions produced in TC - N collisions essentially exceed that in N - N collisions. The measurements of the energy spent by a primary particle for the new particle production were performed by many authors and showed that in high energy region this energy is, as a rule, less than half the amount of the secondary particle energy. (See 0 2.2). Consequently, among secondary particles there must be such whose kinetic energy considerably exceeds the average kinetic energy of the remaining particles. This effect becomes more and more essential as the energy of the incident particle increases, since the average kinetic energy of secondary particles .F increases rather rapidly with increasing T. The study of the energy spectra of secondary particles showed that only one is, as a rule, a “leading” particle essentially different in energy. (A similar conclusion is obtained in cosmic-ray experiments, in particular, in analysing the mean free paths of shower particles, for the details see refs. [92, 931). I n the case of N - N interactions a leading particle is the nucleon. This is just the reason for that the average kinetic energy of secondary nucleons exceeds that of pions. After having excluded the “leading” nucleon the kinetic energies of pions and second nucleon are close. The fact that the average kinetic energy of secondary pions in TC - N collisions exceeds that of pions produced in N - N collisions may be explained by that in the case of x - N interactions the leading particle may be both nucleon and pion. However, it should be noted that these conclusions are as yet of a qualitative character since the accuracy of the available experimental data is not good yet and the energy of the leading particle can be determined only approximately. Only the fact that the leading particle exists is well established; this implies that a complete energy spread does not occur, i.e. inelastic collisions a t very high energies may not be regarded purely statistically (e.g. by means of the hydrodynamical model). The fact that one very fast particle essentially differs in energy as compared to the other is in good qualitative agreement with the model of peripheral interactions. The inelastic interaction characteristics which are not related directly to the leading particle sufficiently clear show the statistical features. This is just the reason for the success in explaining some important aspects in inelastic interactions reached earlier in various statistical theories. It should be stressed that the abovementioned weak dependence of the energy .T on the kind of produced and colliding particles is observed only in the c.m.s. In going over to the laboratory system this dependence becomes very noticeably. For example, the average kinetic energies of no and x- mesons produced in inelastic


X- - p interactions at T = 6.65 GeV in the 1ab.sys. are twice as high as those in the c.m.s. [74]

- - ,Yn- = (1.55&0.08) GeV, F x o = (0 .76j0.12) GeV.

Formally it is due to the fact that in the c.m.s. the angular distributions of particles of various kinds strongly differ from each other (see chapter 11). More- over, under the relativistic transformation to the laboratory system the difference in the particle masses is very essential (comp. proton and pion spectra. Fig. 1) . With increasing energy of colliding particles the average energy of produced particles increases slowly. As

- ,T = 2 ; E, N T'ls; N T'I4

n

then it may be expected that the degree of increase of .F is about the same as for the average multiplicity E. This conclusion i s valid also if account is taken of the fact that the main fraction of the energy is taken away by the leading particle, since the energy of this particle is .Fe = O L E , (a 2: 0.5 + 0.7) see 5 2.2. and therefore ,F = F, - OLE,/E N T'I4. Fig. 2 gives the experimental data on the energies of pions produced in N - N collisions a t an energy up to Y - lo4 GeV. It is seen that the energy dependence

1- i I - L - - I I a 'v 100 150 200 &,LGeVl

0,4

a 2

I

I I I I I I I I I 2 3 4 5 &, LGel

O L I

Fig. 2. The mean kinetic energy of pions produced in inelastic interactions. By 0 , a, and A we mark the values related to p - p, p - n, N - N and 3 - N interactions. The continuous curve denotes the dependence y ( T ) = 0.1 + 0.1 T'ii, the dashed line are the average value at low energies where the dependence p ( T ) is stronger than T1/4

Inelastic Interactions of Particles at High Energies I1 443

.F ( T ) can be approximated by the function

F ( T ) N 0.1 + 0.1 TI/*. (3)

Essential deviations are only in the energy region of several GeV. The experimental values of the nucleon kinetic energy and of other heavy particles which are available only up to several dozens of GeV can be also approximated by the function (3). However measurement, errors are for the time being very large therefore some other methods of approximation are also possible. In any case we may assert that the mean kinetic energy of produced particles does not increase faster than T’!4. It is interesting to note that the position of the peak in the spectrum, i.e. the value of the most likely (peak) kinetic energy of produced particles remains practically constant 3, = (2 + 3) mnc2 in a large energy interval from several GeV to hundreds of thousand GeV (see Fig. 3). The increase of is mainly due to a relative extension of the area of the high energy part (“tail”) of the spectrum [92, 941. For a fixed colliding particle energy this part of the spectrum ,F > .Tp is described by the power function

wl-ci4

W ( . Y ) N .F-=, (. > 0) . (4) 1 5 10 50 &/m,

pig. 3. The energy spectra (non-normalized) of pions produced in inelastic collisions of very high- energy nucleons [92, 941. E = T + mn is the total c.m.s. pion energy. T is the kinetic energy of a primary nucleon in the laboratory system

This follows from the shape of the spectrum of photon-electron cascade produced by cosmic particles in the atmosphere and a dense matter (see below). In the laboratory system the values of 9 and ,Yp increase as Tala and T’I. respectively, in any case not slower than T‘Ia does. The dependence of the mean particle energy on the number of star-prongs was investigated in numerous paper. I n x - N interactions particles belonging to few- prong event are, on the average, more hard than those belonging to many-prong events. For example, the average momentum of XI mesons of two-prong events produced in inelastic x- - p interactions a t T = 6.8 GeV is 620 60 MeV/e and those of four- and six-prong events are 520 f 50 and 460 + 60 MeV/c [39]. It is interesting to note that in x- - p interactions negative pions are produced in two-prong events always much more hard than positive one. So, for T = 16 GeV the mean energy of negative pions in two-prong events is (1.05 f 0.1) GeV and that of positive pions only (0.83f0.1) GeV; in four-prong events the mean energies of positive and negative pions are practically the same :

- .Fn- = (0.51 & 0.05) GeV; y=,+ = (0.52 f 0.05) GeV [77].


This result points to a different mechanism of negative and positive pion production in few-prong x - N interactions. The average energy (momentum) of secondary particles produced in N - N collisions also decreases with increasing multiplicity. So, the mean energy of positive pions in stars with the number of prongs n = 2,4, 6, 8 produced in p - p collisions a t T = 24 GeV is 1210, 740, 630 and 470 MeV respectively [as], with increasing n the proton energy increase. A quite different conclusion about the dependence of F ( n ) was obtained in refs. [ l a , 961: the average momentum of nf mesons and protons produced in N - N collisions a t T = 9 GeV is independent there of the number of prongs: ?jn = = 325 & 30, 370 f 50 MeV/c and jjjP = 1152 & 90, 1028 f 80 MeV/c for stars with n = 2 + 4 and n = 5 + 7 [ l a ] respectively. This result may be under- stood only if we assume that in the transition to events with larger number of prongs the energy of secondary neutral particles (or their number) strongly decreases. The latter seems to be very surprising especially if we take into account that according to the data of many experimental works a t T >> 1 GeV the energy of secondary pions with charge of different signs little differ from one another. Noticeable changes are only observed in the case of n: - N interactions for one- and two- prong events. Apparently, the measurements in refs. [ l a , 961 are not precise, moreover in a later paper [47] in p - p interactions a noticeable dependence cT (n) is seen (in p - n interactions the values of ,yn are as before independent of the number of prongs).

2.2. Ine l a s t i c i ty Coefficient

The mean energy 7 is closely connected with the problem of the magnitude of the energy consumed for production of new particles in inelastic interaction, what, in turn, closely connected with the inelastic interaction mechanism. So, the “head-on” collisions with relatively small impact parameters described by the Fermi statistical theory are characterized by a noticeably larger inelasticity than “peripheral collisions” where the main fraction of the energy is, as a rule, carried away by the only particle. The investigation of the energy losses for new particle production is especially important in the cosmic energy region. In some cases the measurement of these losses is the only means of obtaining information about the produced particle energy. The energy transferred to new particles may be conveniently characterized by the inelasticity coefficient K which is defined as the ratio of the total energy of all newly produced particles to that of colliding particles. It is obvious that the inelasticity coefficient determined in such a way cannot assume larger than unity 1). This valne corresponds to the annihilation processes

l) In some works the inelasticity coefficient is defined as the ratio of the total energy of all new particles to the total kinetic energy of colliding particles, or to its part which may be spent for the new particle production. However such a definition is not quite convenient since in the case of annihilation processes the coefficient may take arbitrarily large values near the threshold. Besides, even for N - N interactions K will be dependent on the chosen frame of reference (cf. (6)). In the high-energy region when T > Mi different definitions of the inelasticity coefficient are, in their essence, the same.


L c T

Fig. 4. The distribution of the inelasticity coefficient in AT - N interactions at T - 200 f 400 GrV!55]

Fig. 5. The distribution of the inelasticity coeffieieiit in n - N interactions at T = 7 GrV [401

of colliding particles ; in these cases in the final state there are no particles of those kinds which were in the initial state: all secondary particles are here newly ori- ginated. The inelasticity of all other types of interactions is less than unity, in this case K < 1 even when the maximum possible energy E , = W, - ( M I + M,) is transferred to new particles. As far as the energy transferred to the new particles turns out to be somewhat different for each act of inelastic interaction (Figs 4, 5, 6) then it is convenient to introduce the mean inelasticity coefficient

where N ( k ) is the number of inelastic interactions with the definite value of the inelasticity coefficient. Owing to the symmetry of the initial system particles produced in N - N collisions are scattered in the c.m.s. almost symmetrically with respect to the plane 0 = $2 perpendicular to the velocity vectors of the primaries. I n this case the energy losses consumed for new particle production in the laboratory system and in the c.m.s. are connected by a simple relation: d EL = y,d E, and the corresponding inelasticity coefficients KL and K , turn out

n = 2

n=3 T

40 -

n = 4

40

20 40LJ++-!5 0 L b

012 44 46 ' OI8 K: Fig. 6. The distribution of tho inelasticity coeffi-

cient in events with different number of prongs n x - N interactions a t T = 7 GPY I # O l . . X is the number of int(wctions

4-46 V. S. BARASHENKOV, V. M. MALTSEV

to be identical2)

In doing so, we have taken into account that the relativistic transformation factor

where EL and E, are the total energies of colliding particles in the laboratory system and in the c.m.s. respectively. In other cases, e.g. for 7~ - N interactions the coefficients KL and K , differ. In some papers the inelasticity coefficient is considered in the so-called ‘‘mirror’’ frame of reference where the incident particle and the target particle exchange places [55, 821. For N - N interactions the mean inelasticity coefficient determined in such a way is

while in other cases K* $; KL ; but every where a larger value of the inelasticity coefficient in the laboratory system corresponds, on the average to its larger values in the mirror system and in the c.m.s. (In particular, for inelastic x - N interactions with K* 5 0.5 and K* > 0.5 the corresponding mean values of K, are about 0.3 and 0.7). It should be emphasized that the analysis of the “mirror coefficient” K* gives nothing new as compared to K . However this coefficient can be only expressed in terms of the energies of colliding particles and the recoil nucleon energy

K* = KL = K ,

Here T* and F* is the nucleon kinetic energy in the “mirror” frame of reference before and after collision, E i is the total energy of colliding particles in this frame of references. Therefore when the energy of other secondary particles (e.g. of xO-meson) is unknown nevertheless the determination of the coefficient K* enables one to obtain information on the inelasticity of the interaction. Besides the total energy losses A E it is very interesting to consider separately what fraction of the energy of primaries is spent for production of particles of a definite kind. Such losses may be characterized by the inelasticity coefficients

and so on.

2, We stress that the equality holds on the average only; for some events the angular distribution of particles may be strongly asymmetrical. If we are interested only in the charged or neutral particles then system p + n cannot be considered as symmetrical, even on the average since the angular distributions of produced charged particles are in this case asymmetrical (see 3.1). Note that from eqs. (6) it immediately follows the connection between the average energies of pions in the laboratory system and in the c.m.s.

Z, == Ez (E,/EL) (6‘) -

For this is should be recalled that A E, = En&; A EL = %,EL, where B = 7 + m,, Zz is the average number of produced pions. Eq. (6’) is very useful in practice.


Starting from the energy dependence of the average number of produced particles E and their energy 7 considered in the previous section it may be expected that in the very high cosmic energy region the inelasticity coefficient

En (.F + m,) n, Fn K, = - N-

EC Eo

will be constant or a quantity very weakly dependent on the energy, and the total inelasticity coefficient

must not, in any case, decrease with increasing colliding particle energy. The values of the inelasticity coefficients in the acceleration energy range can be obtained by summing directly the produced particle energies given in Tables 1-4 (It should bear in mind that the mean value of K determined by eq. (5) can be only approximately replaced by the product of 7 and E ) . As to the cosmic energy range, the energy distribution of produced particles will be, as a rule, either unknown or badly known. Therefore to determine the inelasticity coefficient it is necessary to use different assumptions and indirect methods. The available values of the coefficient K, are given in Table 5. From this table it is seen that a t energies T 5 30 GeV the fraction of the total energy E, spent for pion production in N - N collisions is about 30-35 per cent. In the case of pion-nucleon collisions a t energies exceeding several GeV the values of the inelasticity coefficient in Table 5 are twice as large as those for K in N - N interactions. This is apparently due to that values of K are calculated neglecting the leading pion in x - N collisions, i.e. under the assumption that the energies of all the secondary pions are, on the average, the same. It is interesting to investigate this problem experimentally. It should be expected that in one part of inelastic x - N-interactions a high- energy nucleon will be observed while in other - a high-energy pion. Knowing a relative number of these two types of interactions some important conclusions on the intrinsic structure of the pion may be drawn [114]. Since the cross section for strange particle and antinucleon production in the region of accelerative energies is small enough (see refs. [115, 1161) then the energy consumed for production of such kind particles is insignificant as compared to d Ec and the coefficients K and K, practically coincide. In the slow antinucleon annihilation the total energy of primaries is consumed for production of new particles, mainly of pions. However, a t T > 1 GeV non- annihilation processes play a noticeable role. It may be expected that energy increase the contribution of these processes will increase and a t T > 1 GeV the inelasticity coefficient in w - N interactions will be approximately the same as for N - N and x - N interactions. At energies T > 30 GeV the inelasticity coefficient is known only for nucleon- nucleon collisions. The experimental data are there very inaccurate and strongly differ from one another. The bad accuracy of the inelasticity coefficient in cosmic ray experiments is mainly due to the difficulty of an exact determination of the primary particle


Table 5 Inelastic coefficient Kn (the c.m.8.)

Interactions T [GeV] Method K n

P - P 2.7 3.5 6.2 8.7 9 9 9

14 18.9 27 30

Em [as] Em [ lo] Em [IZ] Em [I31 Em [97] Em 1141 Em [15] E m [98] Em [I61 Em [49] C [991

0.20 f 0.04 0.13 f 0.03 0.25 f 0.03 0.30 f 0.02 0.29 f 0.05 0.32 f 0.02 0.33 f 0.08 0.32 f 0.06 0.25 & 0.06 0.28 f 0.03 0.40

P--" 9 Em 1971 0.23 f 0.03 25.8 Em [52] 0.29 f 0.06 27 Em [49] 0.30 f 0.04

N - N 3 6.2 8.7 9

15

20 24 25.8 27 40 65 70

100 100 100 150 (10 + 300) 160 200 200 250 300 400 400 750 103 103 1.2 . 103

2.108

2.5 103 ( 5 . 102 + 5

IC, c [I001 E m [I91 Em [I011 Em [ZO]

MCC (C12) [22,102]

Em [51] Em [52] Em [21] MCC (C12) [22, 1021 IC [I031 MCC(C12) [22, 1023 IC [I031 MCC(Cl2) [ZZ, 1023 MCC [I041 Em [50]

IC [I031 MCC(C12) [ Z Z , 1021 Em [I051 IC [I031 MCC (Lik) [55] MCC(C12) [22, 1021 1C [I031 IC [I031 MCC [I041 Em [58] MCC(C12) [22, 1021

Em [I061

Em [93]

IC, c [I001

104)

0.16 f 0.04 0.23 f 0.03 0.33 f 0.09 0.22 f 0.05

+ 0.02 0'52 - 0.41 0.27 f 0.03 0.31 f 0.05 0.29 & 0.05*) 0.45 5 0.1 0.35 f 0.14 0.47 f 0.05 0.27 -j= 0.09 0.39 f 0.05 0.22 f 0.07 0.4 0.6

0.43 f 0.08 0.19 f 0.07 0.31 & 0.05 0.41 0.08 0.3 0.18 f 0.09 0.36 f 0.08 0.34 f 0.14 0.1 0.15 0.06 f 0.04

+ 0.25 - 0.30

0,25


Table 5 (continued)

449

Interactions T [GeV] Method K ,

2.8.103 Em [I071 0.3 + 0.07 0'4 - 0.013

5.103 Em [I041 0.2 104 Em [I091 0.22 f 0.05 3 . lo3 + lo5 Em [ I I O ] 0.3 105 Em [I041 0.2 3.105 Em [I101 0.25

2 . 10' + 6 . 10' IC [I081

~~ ~

P - P N O H-BC 1841 0.99 f 0.005 0.47 H-BC [65] 0.985 f 0.005 0.92 H-BC [I121 0.98 f 0.005 1.26 H-BC [I121 0.98 f 0.006

x- -. p 0.96 1 .o 1.3 1.37 1.5 1.72 4.7 6.65 6.8 7.0 7.5 9.86

16

H-BC [73] H-BC [27] D-DC [29] H-DC [31] H-DC [32] H-DC [33] H-DC [38] P-BC [74] Em [39] Em, P-BC [81] Em [76, 821 Em [44] H-BC [77]

0.21 f 0.04 0.25 f 0.06 0.23 f 0.04 0.29 f 0.06 0.26 & 0.06 0.29 & 0.06 0.43 f 0.06 0.56 f 0.1 0.42 f 0.07 0.49 f 0.08 0.51 f 0.11 0.57 f 0.05 0.56 f 0.06

*) The value is calculated by pion energy in c.m.s. If the pion energy in the laboratory system is taken as a basis E, = 2.2 GeV then K = 0.35. It should benoted that the average energies of pions given in ref. [52] E~ = 0.48 & 0.06 and E ~ , = 2.2 f 0.2 GeV do not satisfy eq. (6').

energy. Most methods of determination are based on assumptions which are not always fulfilled (see below). The part of the data given in Table 5 has been obtained from the analysis of the interaction of cosmic particles with Light nuclei. Since the number of internuclear collisions only slightly exceeds in this case the unity [117, 1191 and the inelasticity coefficient weakly depends on the energy then the obtained in this way values of K, must be close to the corresponding values for N - N interactions. The most precise data for light nuclei (for carbon and the middle nucleus of air) are given in Fig. 7. As is seen, in a large energy interval from 10 GeV to T - N lo7 GeV the inelasticity coefficient of inelastic nucleon-nucleus interaction is 40 + 50 per cent. The inelasticity coefficient of inelastic nucleon-nucleon interaction must be somewhat smaller, however it should be expected that it will remain about constant (If use is made of the internuclear cascade model then this coefficient is 30-40 per cent). The situation is worse as to the analysis of the interaction in emulsion where there is a large admixture of heavy nuclei. It is difficult to distinguish between


nucleon-nucleon and nucleon-nucleus stars in emulsion even in experiments on accelerators where a rich statistics of detected events allows one to use quite a number of selection rules [73, 117, 1241. This is especially difficult to do for cosmic rays. The admixture of the interactions with heavy nuclei is a source of additional errors in the inelasticity coefficient, this especially affects the determination of the primary particle energy. The very different values of K , in Table 5 are also due to that many papers contain the values of the inelasticity coefficient obtained from the processing of only a few stars, while its magnitude strongly changes from event to event (Figs. 4-6). Besides, large errors may also be d ueto the usually employed assumption that the energy consumed for charged pion production is twice as high as the neutral pion energy. Such a distribution of the energy is average only; in some stars there may be noticeablp f uctuations.

1-

46

02

I I I 1 I - - -

I - - + - f i I

-

4 4 - f f -

- - - - -

I I I I I

This leads to that most values of K , in the region of energies higher than the accelerative ones should be considered only as a rough estimate. It is interesting that in spite of large fluctuations in the values of K, obtained from the analysis of the cosmic particle interactions in diffusion chamber and in emulsion decreases on the average rather rapidly with increasing T. For example, according to the data of the emulsion work [125]

K, T-(0.3+0.4)

and to ref. [22] the K , decreases even more rapidly: from 0.4 at T = lo2 GeV to 0.1 at T = lo3 GeV. The conclusion about the rapid decrease of the multiplicity coefficient K , with energy has been first obtained in the works of the Bristol Group from the comparison of the energy spectrum of primary cosmic particles with that of gamma quanta generated by the latter [93, 104, 1261. The exponent of the integral spectrum of gamma rays (i.e. the slope of the spectral curve in the logarithmic scale, cf. Fig. 18 in 2.4) turned out to be about twice as small as the exponent of the cosmic particle spectrum. Since the gamma ray energy is defined by the energy of the decayed neutral pion then this may occur only when K , 3 Kn0 - -T-'/* The fact that K decreases with increasing T was also noted in refs. [127 -- 1301.

Indastic Interactions of Particles a t High Energies I1 451

It is difficult to bring all these results in agreement with the energy dependence of the mean multiplicity and the average energy of pions produced in inelastic interactions. So, e.g. to explain the result of the Bristol Group it should be assumed that ?i and .y remain constant or one of the values decreases with increasing T (so that the product 7i. However the analysis shows that in all cases when a strong dependence K,(T) has been obtained in measuring in emulsion or in diffusion chamber, the energy of the primary particle was determined in such a way that it might be noticeably overestimated. In particular, as MURZIN 11311 has recently showed, the steeply

- const).

falling power energy spectrum of the primary cosmic particles gives no possibility to measure accurately even the average energy of these particles if corrections depending on the exponent of their energy spectrum and on the mean square error in measuring the energy of separate particles are not taken into account. This is well seen from Fig. 8 where are plotted the values of the coefficient obtained in ref. [129] by the usual photoemulsion method, the primary particle energy being determined by the Castagnoli formula. The theoretical curve for K ( T ) given in the same figure is obtained by the Castagnoli formula but neglecting the above corrections, although the underlying “true” coefficient K was assumed to be constant and equal to unity3). An additional source of errors is the Casta- gnoli’s assumption that in the c.m.s. the secondary particles are scattered symmetrically with respect to the plane 0 = nl2. As was already noted the latter holds only for a large number of events, therefore the Castagnoli formula applied to separate events can lead to essential errors (Table6).

K 10

1

( 2 7

0,o I 1 10 102 103

TZGeVl Fig. 8. The points are the experimental values of

Kfrorn ref. [129]. I t is noteworthy that at T cc 100 GeV most measurements lend to a meaningless result K > 1. The continuous cnrve is the dependence K ( T ) into which the “true” inelasticity coef0cient K = const = 1 transforms, if corrections for the shape of the energy spectrum of cosnlic particles and mean square errors of measurement of the energy of separate particles are neglected. The dashed line is the limiting value of the calculated curve a t T - m: If (T) --f 0.024 &rue = 0.024 [ I311

As to the results of refs. [93, i04, 1261, the measurements recently made by BARADSEY, GRIGOROV et al. [134, 1351 with the aid of an ionization calorimeter have shown that the exponent of the gamma-ray integral spectrum little differs from that of the primary particle spectrum. The underground experiments with high-energy p-mesons show also that the exponents of the energy spectra of cosmic particles and of generated secondaries little differ from each other [136].

3, In ref. [I311 it was also shown that neglecting the corrections for the shape of the energy spectrum of cosmic particles we are led to wrong conclusions that the mean multiplicity of produced particles is approximately constant a t very high energies. This just underlies the conclusion of MALHOTRA [I321 on “saturation” of the processes of multiple production of particles a t T > lo2 + lo3 GeV (see page 380 in ref. [ I ] ) . YULDASHEV [I331 has analysed in detail different methods of measurement of K , and pointed t o a systematic overestimation of the cosmic particle energy.

50 Zeitschrift ,,Fortschrittc der Physik“, Heft 7

452 V. S. BARASHENKOV, V. M. MALTSET

Table 6 The primary particle energy determined by different methods [57]

Events with symmetrical tracks trical tracks

Events with non-symme-

Number of event 1 2 3 4 5 6 7 8 ~~~

Measurement in ionization 220 250 260 210 100 300 60 280 colorimeter (T[GeV])

Measurement by the Castagnoli 200 290 340 160 260 50 280 50 formula (T[GeV])

A strange difference of the results of their measurements from the data of other works was noted by the authors [93,126]. I n a later work [92] it was indicated that the dependence K , ( T ) is considerably weaker: A K, /K, 40 per cent in the energy interval from 2 - lo3 to 5 - 106 GeV. Within the accuracy of the experiment a so insignificant change of K , practically implies the energy dependence of the inelasticity coefficient. This is also indicated in a subsequent paper of this group

The constancy or a very weak-dependence of K , on the energy follows also from the most accurate experiments made with the aid of an ionization calorimeter. The energy of the primary was in this case determined by summing up the energy of the electron-photon cascade generated by this particle in the absorber thik layer [55, 103, 133, 1341. The inelasticity coefficient obtained in such a way remains approximately the same for cosmic-rays as for accelerative energies T = 10 + 30 GeV. The same result is obtained by analysing experimental data on the altitude dependence of the nucleon energy spectrum in the atmosphere (see, e.g. [133]). As the inelasticity coefficient KK for K-meson production in the region of cosmic- ray energy only fragmentary data are available, which however show that in this case KK < K , as for the accelerative energies [92]. The energy carried away by hyperons a t T 5 30 GeV is also only a small fraction of the total energy of colliding particles. At very high energies the number of produced hyperons is, as usual, relatively small (not more than 20 per cent of the total number of produced particles, see [I, 115, 1161) and the energy carried by them cannot become noticeable. Thus, it may be asserted that in a wide energy range from several GeV to 7' N lo7 GeV the inelasticity coefficients KK, KY, K r are far smaller than K , and the total one is about 30-40 per cent. As was already noted, the conclusion about the approximate constancy of the inelasticity coefficient a t high energies follows from the comparison of available experimental data on the energy dependence of the mean multiplicity of secondary particles Z ( T ) and their average energy y ( T ) . However the fact that the coefficient K is constant is more definitely established than the shape of y ( T ) (Tabeles 2 and 5), therefore it is better to take as a basis the experimental data on E(T) and K ( T ) and compare them in order to get a sufficiently reliable conclusion that a t T > 1 GeV . y ( T ) - T'I4 and the main fraction of the energy

[ l l O ] .

(1 - K ) E , zz (65 +- 75 yo) E , N T'I.


is concentrated only on two secondary particles which are of the same kind as the colliding particles (more exactly on one of them, see 9 2.1.). Sometimes one says that after inelastic interaction the incident particle keeps the main fraction of its original energy although, strictly speaking, one may imply only the primary and secondary particles of the same kind. From Figs. 5 and 6 it is seen that in the distributions of inelastic x - N interactions over the inelasticity coefficient there appear two separate maxima ccrrespond- ing to different multiplicity. However, it is not clear as yet to what extent this conclusion is true. Maybe the minimum of the curve in the distributions K (n) in Figs. 5 and 6 is due to the insufficiently correct identification of secondary protons. This problem will be considered in the subsequent section in discussing the structure of the momentum spectra of secondary particles. Accelerator's measurements performed by different authors show that irrespecti- vely of whether the structure in the distributions K(n) exists, in fact, or not, the inelasticity coefficient in 'ic - N as well as in N - N collisions increases, on the average, with increasing number of produced particles. To interactions with larger inelasticity there corresponds, as a rule, larger multiplicity of secondary particles. Small inelasticity coefficients are mainly observed in few-prong stars, large ones both in few-and many-prong stars. At the same time different authors have obtained different quantitative data on the increase of the inelasticity coefficient with increasing multiplicity n. So, in refs. [ l a , 9/31 for N - N collisions at T = 9 GeV one obtained a so strong dependence K (n) that the mean energy of produced particles -7 was practically independent of their number. The same result for p - n interactions at 9 GeV was obtained in i-ef. [47]. I n other papers the increase of the inelasticity coefficient is much weaker. The dependence of the inelasticity coefficient on the multiplicity of produced particles was also investigated in cosmic-ray experiments. I n refs. [103, 1331 for interactions with the number of produced particles n > 9 a t T = lo2 + lo3 GeV one got KE = 0.19 & 0.02. Wit,hin the experimental errors. This value does not differ from K$ = 0.17f0.02 which is got for interactions with smaller multiplicity. In subsequent papers (see, e.g. [137]) more noticeable increase of K is noted in the transition to events with larger number of prongs. Here there is apparently no essential difference from the region of acceleration energies.

2.3. Momentum Spec t r a

The features of the inelastic interactions we are concerned with in this section are similarly manifested both in momentum and energy spectra. Therefore it is quite sufficient t.0 consider only one of these cases. We shall consider the momentum spectrum since this type spectrum is treated in most experimental papers4).

4, We can pass to the energy spectrum by a simple relation:

where M is the mass of producedparticles, W ( p ) is their momentum spectrum. It is worthnot- ing that there is no simple connection between the mean quantities Fand j5, therefore when an experimental work gives only the mean momentum, the mean energy remains unknown.

30*

452 V. S. BARASKENKOV, V. M. MALTSEV

I n contrast to the mean energy the distributions of particles over momenta turn out to essentially different depending on the kind or particles and the type of inelastic interaction. Fig. 9 gives the momentum spectrum of protons in inelastic x- - p interactions at T = 3.86 GeV. Peaks are clearly seen which go beyond the statistical errors (about 2500 protons were analysed in ref. [138]).

I 7 - T= 9/86 GeV

Fig. 9. The momentum spectra of protons produced in inelastic n- - N interactions a t different energirc. The histograms for T = 3.86 and 9.86 GeV are taken from refs. [ I N , 1401 obtained by summing up eslw rimental data of refs. 1138, 1391 and 1141, 1431. Events with the number of prongs PZ 2 6 whose contribution is 10 per cent are neglected in the histogram a t T = 3.86 GeV (see Table 8, ref. 111). The continuous histograms for T = 7 and 16 GeV are taken from refs. [40 ] and 1421 respectively. The dashed line is the histogram from ref. [ I 4 4 ] . As is seen the position of the maxima on the histograms from refs. [40. 1441 is different

The statistics of events detected a t high energies is much more poor therefore only very rough details of the structure are revealed in the experimental distribution W (13). So, some recent measurements have shown that in the spectra of nucleon and hyperons produced in x-N interactions a t T > 1 GeV two separate peaks are observed (Figs. 9, 10). These peaks are especially clearly seen of the spectra for stars with different number of prongs are separated. From Figs. 11 and 12 it is seen that the spectrum structure is demonstrated both in few- and many-prong events. However, the high-energy peak is more clearly seen in few-prong events. Generally speaking, evidence for the structure of the momentum spectra has been obtained for a long time already (e.g. ref. [38] dated 1957). However the accuracy of measurements was not good, anomalies observed in spectra may always be accounted for by possible statistical errors. But the matter is not only in the accuracy of measurements. The analysis made in refs. [40, 42, 87, 901 has shown that the peaks in the baryon spectra are related to


absolutely different types of inelastic interactions. These types of interactions are characterized by, in addition to different multiplicity, different inelasticity coefficients as well as by different transverse momenta and angular asymmetries of produced particles (as we shall see below). The four-momentum transfer to the

Fig. 10. The momentum spectra of strange particles produced in X- - p interactions a t T = 7 GeV [871. The hatched area of the spectrum corresponds to four-momenta t.ransferred to a A-hyperon d 2 0.7 OeV/c

20 40L 0

4 4 0,s .1,2

n'5

L 46 2,0 p LGeV/cl

Fig. 11. The inonientnm distribution of protons in x- ~ p events with different number of pronm 1 1 :

I' = i GeV, [401

456 V. S. BARASHENHOV, V. M. M ~ ~ T S E V

baryon turns out to be essentially different too. From Fig. 10 and 12 e.g. it is well seen that in collisions corresponding to low-energy peak in the spectrum the A- hyperon receives a four-momentum transfer

A = f ( p A - PP)P (PA - PP)” which is several times larger than that in collisions corresponding to the high-energy peak and small multiplicity of produced particles. In total distributions over A

20 .I.;;- T

15

10

5

70 t n=2 I

50

30

I0

n = 4,6 I :I 10

0 0,4 48 $2 $6 pTGeV/cl Fig. 12. The nionirntum distribution of A-hyprrons

produced in ~ - - p interactions a t G = 7 GeV depending on the number of prongs in the star [a71 The hatched area corresponds to four-momenta transferred to a A-hyperon A 2 0.7 GeV/e

two peaks are clearly seen (see, e.g. Figs. 13 and 14). There are some indications that inelastic interactions with small A are also characterized by small energy and three- momentum transfers [go]. By comparing the corresponding areas of the A-hyperon momentum spectrum in Fig. 10 it may concluded that a t T = 7 GeV interactions with small momentum transfer (the shaded part of the spectrum) is about one-third of the total number of inelastic x - N interactions. The same estimate is ob- tainedfor x-NinteractionsatT= 16 GeV. The ratio of the areas of the high- and low-energy maxima in the proton spectra (refs. [40,42]) issomewhatlarger. However experiments lead to a rather uncertain situations for proton spectra, data of various authors are often con- tradictory. So, in ref. [39] there is no structure in the proton momentum spectrum a t all; the distribution of protons obtained in ref. [144] for the same energy T = 7 GeV has two separate maxima but these maxima are just in places where there are minima in the distribution W ( p ) (refs. [40, 421 (see Fig. 9)). In ref. [42] the second maximum is clearly seen while in refs. [ I41 -1431 indicating to the structure of W ( p ) the momentum distribution p of protons at T = I0 GeV has not twoseparatemaxima for largep (Fig. 9). It is not yet clear what is the reason for these contradictions. In refs. [13Y, 1401 it is supposed that the existence of the two verv vronounced

“ I

peaks in the proton spectrum is mainly due to insufficiently correct analysis of the nature of second protons with intermediate values of the momenta. The mean number of protons per act of inelastic x- - N interaction Zp = 0.27 determined in ref. [40] is, in fact, much smaller than their average number Ep = 0.44 f 0.04 ob-


tained in ref. [39] where “two-hump” structure of the proton spectrum W ( p ) was not observed. The systematic loss of a fraction of protons in the intermdiate momentum region could also explain the minimum of the curve in the inelasticity coefficient distri-

Big. 13. The distribution of the recoil protons in high-energy inelastic x- - N interactions depending on the four-momentum transfers d 1421

bution K ( n ) and in the spectrum W ( d ) . All these problems are to be thoroughly investiga- ted5). As to the structure of baryon spectra in inelastic N - N interactions, the situation is even less definite than in the case of x - N interactions. Quite a series of papers is known which are devoted to experimental investigation of inelastic p -p and p -n interactions a t T > 1 GeV (see, e.g. r12, 14-16] and so on) and in which no significant particularities in the distributions of protons over momenta was not indicated; there is Only Onemaximum in thehistogram W ( p ) . On the other hand, the Alma-Ata group used the emulsion method and quite clearly obser-

Fig. 14. The distribution of A-hyperons in inelastic x--p interactions a t T = 16 GeV over the four-momen- turn transfers A [ S O ]

5, If the structure of baryon spectra is analysed by means of the model of central and peripheral interactions then additional maxima in these spectra may be explained by the contribution of peripheral processes with resonant many-particle intermediate states (i.e. by processes with the exchange of resonons @, /, K* and so on in the intermediate states). To explain the hyperon spectrum structure in Figs. 10, 12, 14 it is sufficient to take into account the resonon K* with mass M = 890MeV [l45, 1461. The proton spectra in Figs. 9, 11, 13 a t T = 7 GeV may be accounted for by assuming the existence of a boson resonon with mass M = 2 GeV [147]. With decreasing energy T the peak in the spectrum W ( p ) corresponding to the definite value of M is displaced to the left, to the region of smaller p . In particular, the structure of the proton spectrum which is due to resonons with M 5 1 GeV and corresponding at T = 7 GeV to the region p > 1.6 GeV/c where the measurement errors are very large is revealed in the middle part of the spectrum a t lower energies. This is just the reason for the presence of a large number of peaks in the histogram Fig. 9 a t T = 4 GeV. On the contrary, with increasing T peaks due to resonances peripheral processes must go into the region of large momenta.

458

ved at T = 9 and I9 GeV two separate maxima. The distributions depending on the four-momentum transfer (Fig. 15 and 16) have a similar shape. The mini-

mum in the distribution of protons over I = IF' - j5 j in p - p-collisions a t


u 0 Q4 0,8 1,2 TCGeVl

T= 19 GeV 0,20 P-P

0 Q4 Q8 t 2 1,6 2,O TCGeVl

T = 19 GeV ?$ 0,3 P- n

0 0,4 0,8 1,2 1,6 TCGeVl 0 0,4 0,8 12 1,6 2/0 TCGeVl

Fig. 15. The distribution of protons after inelastic p - p and p - n collisions over their kinetic energy T (t,he c.m.s.) [42] . In the case of p - n interactions the continuous histograms are related to protons emitted in the backward semisphere, the dashed ones to protons emitted to the forward semisphere The given data are obtained from the analysis of two-prong p - p and three-prong p - n interactions

A N

J= 9 GeV (51 T=19GeV P-P 0/4 P-P

-

a2L (GeV/c) '1 A ~ C ( G ~ V / C ) ~ I Fig. 16. The distribution of protons from inelastic two-prong p - p and three-prong p - n interactions over the

four-momentum transfer A [ re ]


T = 9 GeV is also indicated in ref. [go]. Nothing definite can be said, as yet, about the reliability of these results. Some further experiments are needed. Up to this point of our presentation we are concerned only with the baryon spectra. The momentum distributions of produced pions have the clearly expressed structure when colliding particle energy is about 1 GeV. I n a number of papers it was shown that this structure is related to secondary interactions of produced particles (soe, e.g. refs. [27, 148, 2491). At present there are indications concerning the structure of the pion spectra a t higher energies too. This follows, in particular, from the experimental data on x- -p interactions at T = 7.5 and 16GeV obtained in refs. [74, 76, 82, 1501 (fig. 17). As in the case of proton spectra the structure is more clearly expressed in few-prong events. On the other hand, in ref. [39] (T = 6.8 GeV) andin other papers no anomalies in the pion spectra were observed. This is also the case in investigating the spectra of pions produced in N - N interactions (refs. 8,9 , 97). This problem is also unsolved. Asingle peak is detected in the momentum distributions of K-mesons produced in high energy particle collisions ; within the experimental errors no structure is here revealed. Though, the K-meson spectra are presently known very in- accurately. A single peak is observed also in the momentum spectra of particles produced in the antinucleon annihilation. (refs. [62, 65, 2511 and others). The measurement errors are here not too large. The momenta of pions produced together with strange particles have been investigated in detail in ref. [87]. The main part of the spectrum of such pions is very similar to the spectra of pions from reactions not involving strange particles. A noticeable difference is seen only in the range of large momenta : for accompanying pions this part of the spectrum is essentiallycut off.

3/0 t d n= 1,2

I T

44

d n-’5

Fig. 17. The momentum distribution of rc* ilicsons produced in inelastic rc - N interaction5 a t T = 7.5 GeV depending on the nunihcr of prongs [ R Z ]

This is due to a relatively large fraction of the energy consumed for heavy particle production. No structure was observed in the pions spectra. Thus, in spite of the fact that a t present there is already a large amount of experimental data on the momentum distributions of produced particles in inelastic


interactions, many important problems remain still unsolved. On the basis of the above experimental material it may be apparently asserted that the momentum distributions of particles have remarkable structure not only for energies of several GeV but also for higher ones. The structure of momenta spectra is more clearly seen in the case of interactions with small multiplicity of secondary particles and more strongly displayed for baryons than for x - and K-mesons. How- ever, any definite conclusions may not be drawn yet. In this connection is should be stressed that a t present it is much more important to reduce statistical and systematical errors for a fixed energy than perform several, but less accurate experiments for different energies.

2.4. Energy Spec t r a of E l e c t r o n - P h o t o n Cascades

Especial attention should be focussed on the spectra of gamma-rays, electrons and positrons generated in matter by fast cosmic particles. The analysis of these sepctra gives important information on the probabilities of hyperon production a t very high energies. In some papers these spectra are considered in connection with a possible change in the mechanism of inelastic interactions at ultrarelativistic energies. In an emulsion exposed at high altitude one detects photon-electron cascades produced by gamma quanta, electrons and positrons in higher layers of the atmosphere as well as in the nuclear interactions occuring directly in emulsion and in the layers of a heavy matter introduced in it. If the only source of gamma quanta is the decay of neutral pions produced in nuclear interactions and the bremsstrahlung of secondary electrons and positrons then the spectra of cascades corresponding to interactions in the atmosphere and in solid matter must be of similar nature. As is seen from Fig. 18 this is well fulfilled experimentally up to energies T,,,, e

At higher energies in some papers (e.g. refs. [92, 104, 110, 152, 1531) it was established that the spectrum generated by gamma rays, electrons and positrons from the atmosphere falls faster than that generated by particles from stars in emulsion. Although this results ist not a t present completely definite (there are papers, e.g. [ l54] , where rather smooth change of N,,,, a t Tease 2 lo3 GeV was observed), it is however interesting to clear up reasons for the change of the energy dependence of the number of electron-photon cascades in the atmosphere at very high energies In ref. [155] the radiative decay of high-energy xo mesons produced in hyperon decays in the atmosphere was indicated as a possible reason for such a change. In the laboratory system the energy of decayed mesons is proporbional to that of primary cosmic particles and the energy of neutral pions produced directly from inelastic collisions is proportional to TI/. (see section 2.1). Hence, for T, - 5 . lo3

(2 -+ 5) . lo3 GeV6)

6) We recall that the lifetime of the neutral pion is t - s; therefore the relativistic retardation is not long and in both cases neutral pions decay immediately after they have been produced. If the amount of the matter per 1 om2 in the emulsion stock and in higher layer of the atmosphere is the same, both types of spectra should not differ from each other at all. The difference between spectra a and b (Fig. 18) at T,,,, N (2 + 5 ) . lo3 GeV is due to that the thickness of the emulsion stock used in refs. [92,104,152] was 1.3 of the mean free path and that of the atmosphere over the stock was about three mean free paths. The number of cascades formed in the atmosphere is, in this case, larger than that formed by emulsion stock particles and their spectrum decreases faster with increasing T,,,,.

Inelastic Interactions of Particles a t High Energies I1 46 1

GeV neutral pions to be produced it is necessary that in both cases the energy of cosmic nucleons would be about lo4 and lo7 GeV respectively. However, the number of primary particles fast decreases with increasing energy. If the number of hyperons produced in inelastic collisions at ultraviolet energies T 2 1 0 4 GeV is large

Fig. 18a. The number of gamma-quanta, electrons and positrons with an energy T 2 T,.,, co

-V0&8e = 1 N ( T ) d T T'dSe

registrated in the emulsion stock irradiated a t the altitude h = 11 km (in units cm-* ster-' 5-') [92, 104, 1521. The spectrum generated by gamma quanta, electrons and positrons produced in the atmosphere is shown. By o , and x show the results of measurements in different stocks (with the plates of C, Pb, W); lahoratory system

Fig. 18b. The same as in Fig. 18a spectrum, generated by particles produced in iiuclear interactions in emulsion

462 V. S. RARASHENKOV, V. M. MALTSEV

enough. The main part of photon-electron cascades with T,,,, > ( 2 -+ 5 ) . lo3 GeV will be of a decay origin. Next, if the hyperon energy TY is not too high then their decay in emulsion stock in the atmosphere has equal probability. However when the energy TY becomes high Ty > TY* so that the relativistic retardation is important, the main part of hyperons “perish” in inelastic interactions before they have time to decay. The number of high-energy cascades then noticeably decrease 7 . The critical energy TY* is determined by the relation:

where zy and LY are lifetime and the mean free path of the hyperon in the atmosphere (LY in units “mass/cm2”) ; h and p are t,he altitude and the atmosphere pressure a t the observation point p > Ly. At the sea level T ; - lo4 GeV and a t the altitude h - 11 km at which the emulsion was exposed [110,152] TI’ 5 . lo4 GeV. The corresponding energy of gamma-quanta produced by decayed no mesons is lo3 and 5 lo3 GeV what well agrees with experimental data of Fig. 18. Basing on this idea and using the available experimental information on inelastic interactions of particles a t very high energies in ref. [110] one obtained the following ratio of integral fluxes of gamma-quanta which are generated by directly produced neutral pions and by those from hyperon decay:

for T,,, = 5 . 102, 5 . lo3 and 5 . lo4 GeV respectively. PY is the hyperon production probability in inelastic interaction of a cosmic particle with the nucleus of the air. If no more than one hyperon is produced per act inelastic of interaction a t T 2 2 lo5 GeV, even for PY = I (i.e. UY = bin) the number of cascades of the hyperon origin is sufficient to explain an experimentally observed sharp decrease in the spectrum of N,,,, . However a t very high energies the multiple hyperon production is quite possible (we recall that a t T - lo4 GeV the average number of produced heavy particles is 7iT = 0.2 ?i - 6; Fig. 6, ref. [ I ] ) . In this case the contribution of electron- photon cascades of the hyperon origin may become predominant. Although such an explanation is, of course, very ingenious and attractive however a t present it does not seem to be very convincing. In some papers the difference of the energy spectra of photon-electron cascades in the atmosphere and in dense matter is regarde das an indication to an essentially different mechanism of the inelastic interactions of particles a t T 2 lo5 GeV (e.g. refs. [92, 110, 1561). In particular, the difference of the spectra in Figs. 18a and 18b may be explained under the assumption that in inelastic interactions of ultrahigh-energy particles one or two neutral pions are produced with large pro-

’) Note that for excited (resonant) states of baryons (e.g. N* + N + x ) such an effect should be absent since their lifetime is excessively small and the relativistic retardation as for neutral pions, is practically of no importance.


bability which receive the main fraction of the primary particle energy. Then the laboratory lifetime of these pions would be very long. Only a small fraction of the energy due to slow particle decay would be transferred to the proton-electron cascades over the emulsion stock. However such an explanation seems to be hardly more convincing than that for hyperon decay. In any case, we would not like to make any conjections on new unknown mechanisms of interactions where customary principles my be udes.

2.5. Par t i c l e D i s t r ibu t ion over Transverse Momentum

Typical examples of the distribution of particles produced in inelastic interactions over the transverse momentum p

are given in Figs. 19 and 20. Tables 7-9 and Fig. 21 give the average values

F l = J P l W @ l ) dP1. (12)

0 0,4 0,8 0 44 0,8 0 0,4 0,8 1,25[GeV/cl

Pig. 19. The transverse momentum distribution of protons, A-hyperons, nt and K-mesons for inelastic x- _- g interactions at difft-rent energies [29, 8U, 891

Fig. 20. The transverse momentuni distribution of particles produced in inelastic N - N collisions at T = 500 I;?\ IZZJ The continuous histogram are heavy particles, the dashed one x* mesons

464 V. S. BARASRENKOV, V. M. MALTSEV

Besides the given values we note also the mean transverse momentum of cascade 3-hyperons. We known only the two values of this momentum: (13,)s = 318 &

35 MeV/c, obtained in ref. [I841 from the analysis of interaction of 7 GeV x-- mesons with the carbon nuclei a t T 7 GeV and 0,)s = 580 f 60 MeV/c obtained in ref. [159] for x- - p interactions at T = 9.86 GeV.

10 GeV 1 16GeV i 23,6 Gev

0 / 2 L - L L 1 L - L 1 . - . L - - - r ? K p A Z z T K p A Z n K p A 1

Fig. 22. The dependence of the mean transverse momentum on the kind of produced particles. The values of p , in units GeV/c

From the above experimental data it is seen that the transverse momenta of produced particles very weakly depend on the energy and the type of interaction. This dependence is much weaker than for the mean energy and the total momenta of particles. One can see a very slow (roughly speaking, logarithmic) increase of the mean values of p, only in a very wide energy interval. Yet, this increase cannot be considered as reliably established since at cosmic energies T > 10 GeV the errors of measurement are very large. At acceleration energies the mean transverse momenta of various kind particles somewhat differ from one another within the experimental errors and amount, on the average, to about 350 MeV/c.


The values of PI appear to increase with the particle mass. This is seen, e.g. from data in Fig. 22. But this problem is not entirely solved yet; in a number of papers one obtained that the calues of the proton transverse momenta are smaller than the pion ones. At cosmic energies the situations is not quite clear too. The mean value of jil for pions at T = 102 + lo6 GeV is about 400 + 500 MeV/c ; pions with the transverse

Table 7 Mean transverse momentum of nucleons after inelastic interaction

Interaction T [GeV] Method ii 1 [MeV/cI*

P - P 0.81 3.5 4.2 9 9

14 18.9 24 27

H-DC [a] Em [I01 Em [I571 Em [I51

Em [98] Em [I61

Em [21]

3 m [971

H- BC [48]

320 f 46 232 f 21 265 f 31 437 f 52 320 f 30 372 f 37n 217 f 31 270 f 40 326 & 29

(306 f 44)

p - n 27 Em [21] 281 f 26

N - N 6.2 Em [I91 9 Em [I41

500 MCC ( C12) [22] (10 .+ 103)

n- - p 1.3 P-BC [29]

4.5 Em [37]

4.7 H-DC [38] 6.65 P-BC [74] 6.8 Em [39]

7.3 Em [43] 9.86 Em [44]

7.2 H-BC [I581

9.86 H-BC [I591 11.26 H-BC [I601 16 f 3 M-BC [77] 16.86 PF-BC [I611

338 f 56 372 f 25 310 f 44**)

330 28 + 43 248 - 33 346 f 18 313 * 41 370 f 40 330 f 26 330 & 60 382 & 69 429 f 50 411 f 34 420 & 40 300 & 22 (490 f 35)

x- - n 6.8 Em [39] 346 & 33

x - i v 7 Em [39] 370 f 40 7.3 Em [43] 330 f 60 7.5 Em [82] 388 f 41+)

-360++) ~- *) Values given in brackets are related to neutrons, all the remaining ones to protons. **) Strictly speaking, this value is related to all heavy particles. +) For the inelasticity coefficient in the mirror “coordinate system” K* 5 0.5. ++) Under the condition that K* > 0.5. A ) The contribution of p - n events with n 5 5 is also included.


Table 8 Mean transverse momentum of pions after inelastic interaction

Interac- tion T [GeV] Method

p - p 0.81 3.5 4.2 9 9 14 18.9 24 27

H-DC [a] Em [I01 Em [I571 Em [I51 Em [97] Em [98] Em [I61

Em [49] H-BC [48]

200 f 30 111 f 10 142 f 12 314 f 37 210 * 20 344 f 32**) 154 f 13

180 f 17 345 * 15 (330 f 20)

p - 11 27 Em [49] 185 f 17

N - - N 6.2 9

24 25.8 27 50 (19 + 100) - 100 - 140 (10 + 500) 150 (10 + 300) 200 200 250 300 500

500 (102 -+ 103)

2.103 2.5.103 2.8.103 3.8.103

5.103

(103 + 104) 4.103 8.4.103 2.3.104 4.5.104 5.104

2.105 3.3.105

8 . lo4 2.105

Em [I91 Em [I41 Em [51] Em [52] Em [ZI] MCC ( ~ 1 2 7 ) [mi CC [I631 Em [I641

Em [50]

Em [I051 MCC(LiH) [57] Em [I661 Em [I091 CC [I671

MCC(C12) [22] Em [I061 Em [I091 Em [I071 Em [I681

Em [I677

Em [59] Em [I681 Em [I091 Em [I621 Em [I701 Em [I711 Em [I721 Em [ I l l ] Em [I731

225 f 33 245 f 30

240 f 30 312 & 25 310 f 230

220 f 16 270 f 50

280 & 30

240 f 160 N 350 370 f 80 334 f 72 310 f 230

308 f 23 300 * 50 494 f 90 N 300

316 f 54

682 f 150

450 f 74

[398 f 251

[365 -& 301

[ 370 t 3 r407 f 321

[415 * 801 [344 5 1041 “4201 [ U O f 981 [490 f 831 [667 f 1771

Inelastic Interactions of Particles a t High Energies I1

Table 8 (continued)

467

Interac- tion T [GeV] Method lj 1 [MeVIcI

X - - p 1.3 P-BC [29] 385 f 30 [471 f 401 [236 f 201

230 + l4 - 18 4.5 Em [37]

4.7 H-DC [38] [506 & 171 5.96 PF-BC [Ira]*) 293 & 19 (315 f 16) 6.65 P-BC [80] 363 f 14 (362 f 14)

(340 f 68) 6.8 Em [39] 335 & 30 7.2 H-BC [I581 332 & 28 7.3 Em [43] 270 5 20 7.5 Em [76] 250 5 30 9.86 H-BC [I591 320 10 [300 5 201

11.26 H-BC [I601 339 * 15 16 & 3 H-BC [77] 365 f 10 (370 & 10) 16.86 PF-BC [761]+) 407 f 17 (414 & 16)

[412 f 211 17.86 P-BC [I501 [397 f 361 17.96 PF-kC [164]+) 357 * 28 (365 f 21)

x- - n 4.5 Em [37]

x - N 4.5 6.65 6.8 7.3 7.5

7.5 7.5 9.86

140 (10 + 500) 104

Em [I751 P-BC [80] Em [39] Em [43] Em [I761

P-BC [I771 Em [82] Em [44] Em [I641 Em [I641

[120]++)

290 50 337 f 17 (337 16) 310 f 20 270 f 20 256 18n) 324 f 32nn) 290 f 20 286 f 18 318 & 16 270 f 50

[260 * 1001

2 300

*) The given value is related to the mean momentum of charged X* mesons; in round and square brackets are indicated the values only for negative and neutral pions respectively. **) See footnote A ) to Table 7. +) Measurements are performed in propan-freon bubble chamber. ++) Obtained from the analysis of wide atmosphere shower and is, strictly speaking, related to the mean momentum of all nuclear-active particles in jet of the shower. A) For the inelasticity coefficient in the “mirror” system of coordinates K* 2 0.5. A) Under the condition that K* > 0.5.

momenta of the order of several GeV occur only occasionally. This value is by half order smaller than the mean transverse momentum of heavy particles obtained from the analysis of interactions of secondary neutral particles in the central part of showers which are due to cosmic rays [92, 931. On the other hand, in

31 Zeitschrift ,,Fortschritte der Physik”, Heft 7

468 V. S. BARASRENKOV, V. M. MALTSEV

Table 9 Mean transverse momenta of produced strange particles

F l [MeVlcI Interac- tions T[GeV] Method Ko,Ko A CO c+ P - P 14 Em [98] 312 .f 45*)

23.6 0.6 H-BC [I791 373 f 33 396 & 50 510 f 70 950 f 100 24 H-BC [I801 380 f 40 360 80**)

N - N 5 MCC [I811 540 & 110 450 f 60 640 f 190f) 9 Em [I821 327 f 14++) . -

f 160 + 140++) + 140++) 60 MCC [I831 530 - 520 - 60 520 - 60

~~

x - - p 7 P-BC [87] 384% 11 383& 12 7.5 P-BC [SS] 393 f 35 388 & 35 587 f 53 559 f 85 9.86 H-BC [I591 370 f 20 4 7 0 i 2 0 530 % 30+)

11.26 H-BC [I601 376 f 42a) 417 f 56A) 16 H-BC [89, I791 410 5 30 460 f 40 650 f 80 650 & 90 16 H-BC [I801 380 & 50 410 & 7 0 A A )

*) See footnote A ) to Table 7. **) The mean value for A and Co hyperons. +) The given value is mean for C+ and C- hyperons. ++) The mean value for hyperons A, Co, C+, C-. a) The value is only related to four-prong events. A A ) The given value is obtained from the analysis of interaction of protons with emulsion nuclei and is mean for C+ and C- hyperons.

ref. [22] one has obtained approximately equal values of ji, for pions and heavy particles produced in N - N collisions a t T = lo2 f lo3 GeV. In the foregoing sections one has already said that the experimental data of some authors point to the existence of the two types of inelastic interactions which are characterized by different momentum spectra, different multiplicity and so on. These two types can be observed in particle distributions over the transverse momenta too. So, the average value of the transverse momentum of A-hyperons produced in 5c- - p interactions a t T = 7 GeV is 420 25 and 295 5 14 MeV/c for collisions with large and small momentum transfers respectively. ( A > 700 and A < 700MeV/c; of. 2.3). This considerably differs from the mean value = = 383 & 12 MeV/c [87]. If the inelastic interactions are divided into two groups according to the energy spent for production of new particles then in a group characterized by large values of this energy the transverse momenta of particles turn out to be, on the average, noticeably larger than in the other groups. This is especially clearly seen for pions (Table 8 and Figs. 23 and 24). At very high energies the contribution of interactions with large p I somewhat increases, especially for heavy particles. However we may conclude about the existence of some structure in the distributions W(ji , ) with even larger care than in the case of the spectra W ( p ) and W ( A ) considered previously. The two groups of interactions with mean values of the transverse momentum ji, E 200 t 300 MeV/c, p L E 500 -+ 800 MeV/c may


I I I 0 - 2 %

31*


be singled out from the experimental distributions W ( p ) but within the measurement errors this distribution is not very reliable, This is well seen, e.g. from the distributions W ( F L ) for stars with different number of prongs (Figs. 23 and24). I n contrast to the spectra of W ( p ) and W ( A ) the shape of the transverse momentum distributions in few- and many-prong events is, within the experimental errors: the same.

Table 10 Mean transverse momentum of protons for events with different number of prongs n

F 1 lMeVic1 Interac- tion T [GeV] Method n = 2 n = 4 n = 6

P - P 9 Em [47] 368 f 36 439 f 37 549 f 71*) 9 Em [I851 390 f 30 430 f 40 430 f 80*)

14 Em 1981 358 f 24**) 394 f 32+)

x- - p 4.7 H-DC 1381 388 f 65 323 f 54++) 6.8 Em 1391 380 f 70 410 f 80 7 Em [81] 300 f 60n) 410 f 80 7.2 H-BC [I521 290 f 30 350 f 40 370 & 90 9.86 H-BC [I431 420 f 20

10.1 H-BC 11861 429 f l l a n ) 11.26 H-BC [I601 411 f 34.)

n = 3 n = 5 n = 7

p - n 9 Em [47] 355 f 43 441 f 45 543 f 75 . .) - 9 Em [I871 317 f 25 -

*) Mean value for n = 6 and 8. **) Mean value for n = 2 and 4. +) This value is related to all interactions with n 2 5. ++) Mean value for n = 4 and 6. A ) There is an admixture one-prong stars of x- - n interactions.

v) Only for events with strange particles. ..) Mean value for n = 7 and 9.

A) Mean value for protons and neutrons.

At the same time it i s worth noting the dependence of the distributions W ( p l ) on the number of particles produced in inelastic interaction. From Tables 10 and 11 it is well seen that in the transition to events with larger number of prongs the mean transverse momentum of protons increases and that of produced pions remains almost constant. This point out that the production of particles in events with small and large number of prongs is due to different space regions. Let us consider this important problem in more detail.

2.6. Spaoe Dimensions of t h e I n t e r a c t i o n Region

Using the uncertainty relations


Tabel le 1 1 Mean transverse momentum of charged pions in stars with different number of prongs TL

47 1

P 1 [MeV/cI Interac- T [GeV] Method tion n = 2 n = 4 n = 6 n = 8

~~~ ~

P - P 9 Em [47] 414f72 260528 355f45*) 14 Em ~981 372 f 20**) 295 1 15+) 24 H-BC [&?I 360f30++) 340*20 355320 355&40

X- - p 3.36 4.7 5.96 6.8 7.2 7.5 9.86 9.86 9.86

10.1 11.26 16 17.96

H-BC [I881 H-DC [38] PF-BC [I741 Em [391 H-Be [I581 Em ~761 H-BC [I431 Em [441 H-BC [I891 H-BC [I861 H-BC [IGO] H-BC [77] PF-BC [I741

335142 362 5 33 414 f 31*) 310 f 400 400 f 70 230 f 50

407 & 62

415f40 349 f 32,)

360f 40++)

291 f 15 360 f 40 330 f 30 320 f 20 270 f 40 250 f 40

364 f 36

348 $ 5 A ) 339* 1 5 A A ) 360f20 365520 350f30 366 f 21

360 f 30" A)

400 & loo*)

250 5 50*) 300.6 f 3.9

270 f 20* * )

n = 3 n = 5 n = 7 n = 9

p - n 9 Em [47] 252f33 291539 9 Em [I871 212112

22 8 & 250)

~~~ ~~

r r - n 7 Em [81] 260 f 30 310f 50+) 7.5 Em [76] 240&40 250f50 9.86 Em [44] 305 34 270 f 22 +)

A) Mean value for TC* and xo mesons. * * ) The value is related to x0 mesons from stars with n = 6 and 8. A A ) See footnote v) to Table 20. 0 ) Mean values for n = 7 and 9. *) Mean values for n = 6 and 8. **) Mean values for n = 2 and 4. +) See footnote +) to Table 10. ++) The value is related to x- mesons. a) Measurements are performed in proton-freon bubble chamber.

*) See footnote a) to Table 10. A) Mean value for n = 4 and 6.

where

is the transverse momentum dispersion, one can estimate average dimensions of the space region Q = (?:)'/a in which the inelastic interaction proceeds. Since the experimental distributions W ( p , ) are known still with rather large errors then (Ap;) in eq. (13) can be replaced by the mean value of PL (e.g. for protons in


x- - p interactions at T = 7 GeV the difference @;)‘/a - p , = 40 & 90 MeV [ I S S ] ) in this case

1 Pl

p s r .

We should bear in mind that the quantity p determined in such a way cannot be considered as an impact parameter since this quantity noticeably depends on the interactions of particles produced in the initial act of an inealstic collision. From the data given in Tables 7-11 it follows that the main part of inelastic interactions occurs in the region p cm. This region is about of the same magnitude as the nuclear radius of particles obtained from the analysis of experiments on elastic scattering [190] and within the experimental errors is independent of the kind of colliding particles. With increasing energy T the mean dimension of the region of inelastic interaction appears to decrease slowly (cf. Fig. 21). However this conclusion should be treated very carefully. The experimental data given in Fig. 22 indicate that the production of heavy particles is connected with smaller space regions than the pion production, though this conclusion is not very reliable too. When the multiplicity of produced particles increases the space region of pion production remains practically unchanged while the space size of p characterizing the proton production noticeably decreases. The latter may be considered as one more indication to the existence of the two types of inelastic interactions : interactions with small multiplicity of secondary particles formed a t relatively far distances between colliding primaries (peripheral interactions) and interactions with a large number of produced particles which proceed for relatively small distances between colliding particles (“centralinteractions”). The fact that the mean transverse momentum of pions is practically independent of the multiplicity is in agreement with that these pions are formed as a result of collision of a meson from the peripheral “cloud” of one primary with the central part (“kernel”) of another. The dimensions of the region in which the “crystallization” of mesons occurs is determined mainly by the radius of action of nuclear forces (Model of central and peripheral interactions is presented in more detail in refs. [191, 1921). As was already noted in section 2.3 the mechanism of production of positive and negative particles appears to be essentially different. However, the transverse momentum of these pions turns out to identical with good accuracy. For example, a t T E 16 GeV

(0.5 -+ 0.8)

= 420 f 40, 360 5 20, 350 & 20, 420 & 40 MeV/c

(j5j)x- = 410 f 40, 360 & 20, 380 f 20, 320 30MeV/c,

respectively for stars with the number of prongs n = 2, 4, 6, 8. This also shows that, in spite of Werent mechanism, the dimensions of the space regions in which positive and negative pions are produced, are approximately the same. Thus, in spite of the fact that the picture of the primary act can be strongly shaded by secondary interactions of produced particles when they fly apart, it may be asserted on the basis of the uncertainty relations that the above two types of inelastic interactions are due to different “degree of peripherality”.


To estimate numerically the “degree of peripherality” a more careful analysis is needed in combination with concrete theoretical models as well as more detailed experimental information. Inserting the known experimental data into the uncertainty relation for the longitudinal component of the momentum p,,

and into eq. (13) it can be seen that in the longitudinal direction the interaction region is reduced about yc times (yo = 1/1/1 - v21c2 is the Lorentz coefficient for the c.m.s.) :

(<)’/q($)l/* = 272i$)1q((3)’/2 yc .

12 ) l / S / ( S )l’21exp

According to the data of ref. [165] for T = 4, 7 and 16 GeV x- - p interactions the ratio

(Ychheor

is 1.15 f 0.11, 1.11 f 0.17 and 0.85 f 0.2 respectively. It should be stressed that this result is not so trivial as it may seem a t first sight since it is related to the c.m.8. where the interaction region has zero velocity. As is seen, the bulk of secondaries is formed when the relative motion of colliding primaries is essential. This result is an experimental foundation of a phenomenological use of the Lorentz contraction coefficient L/yc in various statistical models of inelastic interactions.

3. Angular Distributions of Produced Particles

In the laboratory system particles produced in inelastic interactions a t high energies are emitted mainly a t small angles to the direction of motion of the primary. With increasing energy the angular distribution is concentrated in the region of

. .-.____ ..__ - . _ -..__

Fig. 25. Inelastic collision of the proton a t T - 3 . 108 GeV with the emulsion nucleon (1061. The internal and external cones are clearly seen in the angular distribution of tracks

ever smaller angles. Very often in the angular distribution two cones are peaked forward, one is narrow (internal) and the other more diffusive (external) (Fig. 25). All these features are of a purely kinematic character. In particular, the narrow and diffusive cones are formed while the relativistic contraction of the angles of particles emitted in the c.m.s. into the forward and backward semi-sphere. I n what follows, just as in the case of the momentum distributions, we shall use the center-of-mass system.


3.1. Anisot ropy a n d Asymmet ry of Angular D i s t r ibu t ions

Figs. 26 and 27 give characteristic examples of the angular distributions of particles produced in N - N and x - N interactions a t high energies

I do(@) W ( 0 ) = - __

b in a 0 *

T = 9 G e V P-P

T +-1

0 I 45 0 -0,s - I

T= 9 GeV P - n

T - I000 GeV N - N

45 :“:c-s 0 -0,s cos 0 O I 45

Big. 26. The angular distributions of particles produced in inelastic collisions of nucleons. At T = 9 GeV the continuous histogramm are protons, the dashed ones xi mesons [471; histograms a t T - 300 and 1000 GeV are related to all secondary charged particles [ZZ, 551

Fig. 27. The angular distribution of particles in inelastic x- - p interactions a t T = 7 GeV [40, 871


(Here and below the angle 0 is determined with respect to the direction of the velocity vector of a particle which is incident in the laboratory system.) As is seen in all cases particles are emitted anisotropically in p - p interactions, owing to the symmetry of the initial system, the angular distributions of particles are symmetrical with respect to the angle 0 = 4 2 ; in other cases a strong asymmetry is observed. The nucleons attempts always to concerve their initial direction of motion after collision. The same can be said about the angular distributions of A and X-hyperons: the overwhelming majority of these particles are emitted in the direction in which the nucleon was moving before collision. Bt present little is known about the production of E-hyperons. However, in a few cases investigated these hyperons are presumably emitted in the direction of motion of the primary nucleon. For example, in the case of K- - p interactions at T = 2.24 GeV the number of E- hyperons emitted into the same semi-sphere as the primary proton exceeds about twice the number of E--hyperons emitted into the other semi-sphere [193]. There are experimental indications that an analogous situation takes place for x - N interactions too [184]. Thus, in all cases the main part of produced baryons is distributed in comparatively narrow solid angles round the directions of the velocity vectors of the primary nucleons. The angular distributions of x and K-mesons produced in inelastic N - N and x - N interactions are more isotropic and symmetric than the angular distributions of baryons. In pion-nucleon interactions most mesons are emitted into the semi-sphere which is inverse to that into which baryons are emitted. In this case the mean angular distributions negative of pions in x- - p and x- - n interactions are noticeably more asymmetric than the angular distributions of positive pions (Fig. 27) ; on the contrary, in interactions positive pions are distributed more symmetrically. It may be said that in inelastic x - N interactions the produced pions also try to conserve the direction of motion and the sign of the primary pion charge. The angular distributions of pions and kaons produced in the annihilation of slow antinucleons do not differ, within the experimental errors, from the isotopic ones. The asymmetry of the particle emission becomes noticeable only a t T > 1 GeV, when the non-annihilation inelastic processes are already essential. In this case the behaviour of the angular distributions is, in outline, the same as for N - N interactions. In particular, the produced nucleons and antinucleons try to conserve the direction of motion of the corrresponding primary particles. Por example, in jj - p interactions at T r 0.92 GeV the number of protons and neutrons emitted in the direction of motion of the primary proton are respectively (14 * 6) and (2.7 5 2.7) times larger than in the inverse direction. On the contrary, most antinucleons are emitted into the same semi-sphere as the primary antiproton: the number of produced antiprotons and antineutrons emitted in the direction is respectively (15 f 13) and (3.4 4) times larger than that in the inverse direction [194]. Antihyperons created while annihilating are emitted mainly in the same direction as antinucleons (see, e.g. ref. [67]) . It is very important to stress that all conclusions about the direction of the emission of particles are true only on the average for a large number of inelastic interactions. However, considerable deviations can be observed in separate acts of inelastic interaction. For example, in stars formed in inelastic p - p interactions

476 V. S. BARASIIENKOV, V. M. MALTSEV

an asymmetrical emission of particles is observed with a large probability (> 50 per cent) (Fig. 28). The number of stars with “left” and “right” asymmetry is approximately the same. Therefore, the angular distribution of produced particles turns, out on the average, to be symmetrical. Deviations from the mean angular distributions is more often observed in few- prong stars. However, the probability of such deviations is so large that they cannot be accounted for by simple statistical fluctuations.

I

Fig. 28. Symmetrical and antisymmetrical stars in thr c.nI.8. produced in p - p interactions in the Wilson chamber for T I 300 GeV [57]. The momenta of colliding protons are directed horizontally. The vectors are proportional to particles moments

We may think that stars with different character of asymmetry are created as a result of peripheral interactions of various types. From this viewpoint a separate experimental investigation of stars with ‘‘left’’ and “right” asymmetry is of

great interests). It is convenient to characterize quan- titatively the produced particle an- W l O ) gular asymmetry by the ratio of the numbers of particles emitted into the backward and forward semi-sphere n’/n‘, respectivelys). Whenidenticalparticlescollide Z/% = 1 the angular distribution can be conveniently characterized by the ratio of an angle which contains the quarter

0 JT of all secondary particles to the corresponding angle O?,, = 2/3n for the isotropic distribution :

3 4 3 d v P ‘ i r

respectively (17)

Fig. 20. The angular distributionin the identical particlc collision. The hatched areas correspond to thc

ted in the forward and backward semi-spherr angles in which there arc half of particles emit- A = - (@ + @”) 2 O?,,

8 , Within the framework of the presently available theoretical models (e.g. refs. [195--1971 stars with “left” asymmetry in the c.m.s. may be interpreted as a result of interaction of the peripheral region of an incident particle with the kernel of a target-particle; on the contrary, in the interaction of the kernel of an incident particle with the periphery of a target-particle stars with “right” asymmetry should be formed. Symmetrical stars may be interpreted as a result of collisions of the particle kernels, collisions of their (identical) peripheral shells or as a result of collision when two symmetrical interactions such as kernel-periphery occur simultaneously. 9, In some papers the angular asymmetry is characterized by the ratio d = (2 - t ) / ( s + Z). However, the quantity s/& is a more clear and simple characteristics. Obviously, ;/; = = (1 + d)/(l - A ) .

Inelastic Interactions of Particles a t High Energies I1

Table 12

inelastic interactions a t high energies (the c. m. 9.)

Asymmetry of angular distributions of protons and pions produced in

477

+ t nln Interac- tions T [GeV] Method P (n)* x+ x-

P - P P - - " 1 H-DC

(0.6 t 1.5) 1.6 H-DC (1 + 2.2) 9 Em 9 Em

14 Em 19 Em

1 1 [I71 1.02f0.2 (0.9750.2) 1.00*0.39 1.04f0.24

[I81 0.61 & 0.16 (1.41 f 0.53) 0.71 f 0.24 1.45 f0.35

[97]+ 2.4 f0.5.) [47] 1.78 f 0.5 [98] 0.40 & 0.08 [4Z] 0.61 f 0.060)

1.41 f0.15**) 1.63 f 0.47**) 0.96 f 0.23**)

P - P -0 0.47 0.92 0.92 0.92 0.92 2.44

p - 11 -0 -

Em P-BC H-BC H-BC H-BC H-BC H-BC Em

[681 1 1 1651 1 1

~ 9 8 1 1.24f0.15++) [I991 0.11 f0.02 (0.07&0.04) 0.84-&0.19 1.18f0.27 [I941 0.07 f 0.06 (0.37 5 0.37) 0.91 f0.43**)

~ 7 1 0.901 0.07**) [72] 0.05.fO.01 1.0 5 0.22

[@I 1 1

X - - p 0.96 1 1 1.15 1.3 1.5 1.72 4,5 4.5 4.7 5.26 7.2 7.5 f 0.5

1 6 f 3 17.96

H-BC [ZOO] 1.54 f 0.34 H-BC [27] 0.54 & 0.22 (0.98 & 0.28) 1.23 f 0.36 0.97 i 0.32 Em,H-DK[26] 0.81 & 0.03A) 1.33h0.04 Em [28] 0.22 f 0.01A) P-BC [29] 0.37f0.09 (0.35f0.07) 1.23f0.1 1.54iO.l H-DC [32] 0.29&O.lA) 1.62 * 0,58A)

Em [37] 0.19 i0.05 1.53 f 0.11**) Em [361 0 1.71 & 0.2**) H-DC [38] 0 (0.38 f 0.18) 1.41 f 0.26**) PF-BC [I741 0.29 f 0.52 2.66 f 0.67 H-BC [I381 0 1.35i0.12 1.51 f O . 1 1 P-BC [4I] 0 2.14 f 0.7 2.04 f 0.34 H-BC [77] 0 1.46f0.12 2.12j0.11 PF-BC [I741 4.1 j l . 8 4.2 i1.1

H-DC [33] 0.09fO.03 (0.44&0.2) 0.5930.22 1.9210.3

~~

K- - p 1.79 H-BC [I931 0.36 f 0.06

*) I n brackets data for neutrons **) The given values are mean for xf and x- mesons +) For prongs with n 2 3 ++) For prongs with K mesons pair production. The values are mean for X+ and x- mesons .) Only for protons with I)* 2 0.4 GeV/c V ) Only for events with n = 3 A) The given values are mean for p and n

) Only for channels with production of an additional pion

478 V. S. BARASHENKOV, V. 11. MALTSEV

T a b l e 1 3 Asymmetry of angular distributions of strange particles produced in inelastic interactions

at high energies (the 0.m.s.)

?i/& Interac- tion T [GeV] Method KO, Ko A z*

~~

P - P 1 1 1

- p 0.92 H-BC [I981 1.37 & 0.18*) 0.92 H-BC [I941 2.7 f 0.3 2.24 H-BC [201] 0 0

x- - p 0.829 H-BC [202] 0.871 H-BC [202] 0.895 H-BC [203] 0.9 H-BC [203] 0.9 H-BC [204] 0.96 H-BC [205] 1.035 H-BC [206] 1.75 H-BC [207] 5.86 P-BC [85] 2.1 50 .1 7.5f0.5 P-BC [75] 2 f 0.7 7.5f0.5 P-BC [87] 1.650.1 7.5 i0 .5 P-BC [208] 1 f0.5

10.86 P-BC [85] 1.36 f 0.09 15.86 H-BC [209] 2.21 & 0.03 16 H-BC [91] 17 P-BC [85] 1.4 f 0.1

*) This value is related to the charged K-mesons.

0.52 f 0.08 0.47 f 0.06 0.45 f 0.04 0.46 f 0.04 0.46 f 0.01 0.93 hO.18 0.59 f 0.07 0.46 f 0.02

0.14 hO.01 0.138 & 0.03 0.15 f 0.02 0.12 * 0.01 0.56 f 0.7

0.72 0.01

0.41 f 0.03 0.31 f 0.21

Table 14 Anisotropy of angular distributions of protons,

produced in inelastic p - p collisions a t high energies (the c.m.s)

T [GeV] Method A

0.81 H-DC [a] 0.49 2 H-BC [8,9,210] 0.49 2.85 H-BC [ell] 0.45 3.5 Em [I01 0.66 9 Em [97] 0.27

18.9 Em [I61 0.34

where and @f;l are the angles which contain half of particles emitted into the backward and forward semi-sphere respectively (Fig. 29). If we do not take into account the statistical fluctuations, then @:I, = Nilo). 10) Sometimes, instead of A, one uses the quantity I = Os/,/304, where Os/& is the angle which contain three-quarters of produced particles. Since 03,4 + Of,, = TC then within the accuracy of the statistical fluctuations

1 = [(n - Of[)/(319:,~)] = [(3 - 2A)/(6A)] and A = 3/[2 (1 - 3I)].


Table 15 Anisotropy of the angular distributions of charged x-mesons produced in inelastic p - p

collisions a t high energies (the c.m.s.)

T [GeV] Method ~

A

0.81 1.5 2.85 3.5 6.2 9 9 9

18.9 27

100 300 250

10 3.5.103

H-DC [a] H-DC [5] H-BC [ Z l l ] Em [lo1 Em r121 Em [I41

Em [971 Em [I61 Em [a91 MCC [$GI MCC(LiH) [55] Em [I661 Em [el21 Em [164]

Em [151

0.51 *) 0.58*) 0.79* 0.52 0.53 0.50 0.84 0.82 0.55 0.94**) 0.84**) 0.86**) 0.82**) 0.65**) 0.69**)

*) The given value is related to x+-mesons. **) The mean value is relativ to all charged secondary particles a t T > 10 GeV. The overwhelming majority of these particles (= 80 per cent) are TC* mesons.

Table 16

The dependence of the angular asymmetry of charged x-mesons produced in inelastic p - n and x- - n interactions on the number of prongs in star n = 1.3 . . . (the c.m.5.)

21% Interac- tion T [GeV] Method 1 3 5

p-71L 9 Em [97] 2.14 i 0.5 1.44 5 0.5*) p - 7 2 9 Em [42,187] 2.36 f 0.45

n- - n 9.86 Em [44] 2.20 & 0.75 1.06 0.24 n - - n 7 Em [39] 2.2 f 0.3**) 1.9 i 0 . 2 1.0 5 0 . 1

*) The given value is related to stars with the number of prongs n 2 5. **) There is a noticeable admixture of the two-prong stars from inelastic x- - p interactions.

The values of n'/% and A are given in Tables 12-17. The main part of data is related to the acceleration energy region. At higher energy it is difficult to distinguish between p - p, p - n and nucleon-nucleus interactions; therefore for cosmic energy one gives only the values of the anisotropy A which comparatively weakly depends on the type of interaction. From the tables it is seen that with increasing T the asymmetry of the particle emission rapidely increases. This is especially seen for x - N interactions. Even at energies of the order of several GeV the overwhelming majority of secondary nucleons and hyperons are emitted into the backward semi-sphere. Though it


T a b l e 17 The dependence of the angular asymmetry of particles produced in inelastic x- - p interac-

tions on the number of prongs in star n = 2,4, . . . (the c.m.5.)

ys Part-

T [GeV] Method icles 2 4 6 8

0.871 2.0

3.36

5.96 6.65 6.8 7.5 9.86 9.86

9.86 10.1

11.26

1 6 f 3 1 6 3 3 17.96

H-BC [24] H-BC [213]

H-BC [I881

PF-BC [I741 P-BC [74] Em [39]

H-BC [I431 H-BC [I891

Em [44]

P-BC [75]

H-BC [I861

H-BC [I601

H-BC [77] H-BC [77] PF-BC [I741

x i x+

xo

P n x+

x-

x- x- x-

x + - KO, KO X+

x- x- x* x+

xo P n A KO x+

x-

x- x-

1.74 f 0.04 5.74 f 0.35 4.7 f2.6 4.25 f 1.4 2.2 f0.3**) (2.6 f 0.5)++)

5.4 f3.2

0.88 f 0.46 10 f5.9 3.8 f l . 9

1.14 f 0.05 1.19 f0.14 1.18 f 0.16 0.76 & 0.07 0.72 & 0.06 0.54 f 0.05

2.14 i 0 . 6 2.17 30 .6 1.5 f0.2 1.7010.17

2.0 10 .6 1.11 f0.03 2.07 f0.03 1.83 -10.06 0.17 & 0.01 0.18 i0.03 0.24 f 0.14 0.88 f 0.46 1.34 1 0.27 2.86 & 0.6 1.75 -j= 0.5

1.82 f 1.1*)

1.05 f 0.15n 1.24 f 0.07 1.35 f 0.08

1.0 *0.1+)

1.4 &0.4*)

1.73 f0.34 1.72 +0,75 1.12 f0.23 1.26 f 0.46

*) The mean value for n = 6 and 8. **) There is a noticeable admixture of one-prong stars from x- - n interactions. +) For n 2 5 (there is a noticeable admixture of x- - n interactions). ++) This value is related to zero-prong stars (n = 0). A ) The mean value for n = 4 and 6.

should be taken into account that the errors indicated in the tables &d (n'/%) are in all the cases, purely statistical. Besides, there are also significant systematic errors which are, in particular, due to an inaccurate identification of produced particles. These errors especially strongly affect the baryon symmetry. For example, even a small change in the number of baryons emitted in x - N interactions into the forward semi-sphere leads to very large changes in the quantity n'ln'. Unfortunately, the determination of the magnitude of such errors depends on many purely subjective factors. As to the angular distributions of particles ih the case of p - p interactions, the proton emission becomes noticeably more anisotropic with increasing energy T ; while the anisotropy of the angular distributions of pions remains approximately constant over a large energy interval from several dozens of MeV to hundreds of

Inelastic Interactions of Particles at High Energies I1 48 1

GeV. (At the same time in some stars the quantity A can change by a factor of several hundreds.) In the foregoing paragraph it was established that the mean transverse momentum of produced particles

weakly depends on energy and the total momentump increases about a t T'I.. These results can be brought in agreement with the weak energy dependence of the pion angular distributions only if the energy of particles emitted a t small angles (and in the case of N - N interactions in the range of angles 0 - iz too) increases with increasing T far more rapidly than the mean value F . This conclusion well agrees with the results of direct measurements of the energies of particles produced a t different angles [15, 39, 106, 1071. The most fast particles are mainly produced in stars with large anisotropic and small number of prongs. In the region of small angles, of importance is the leading particle whose energy increases as T'la. We should consider in detail the angular distribution of particles in inelastic p -n collisions. In refs. [47,97, 2141 it was established that protons and charged pions in p - n collisions a t T = 9 GeV are mainly emitted into the forward semi- sphere, i.e. in the direction of motion of the primary proton. The same result is obtained in a recent paper of the Praha Group [20]. Hence, this results in important theoretical conclusions in the resonant character of peripheral interactions (e.g., ref. [196]). However in the works of the Alma-Ata Group [15, 421 as well as in ref. [98] opposite experimental result was obtained: protons were mainly emitted into the backward semi-pshere (i.e. the charge exchange reactions occur with a large probability), positive and negative pions were mainly emitted into the backward semisphere. The cause of different results is not clear yet, apparently it is due to systematic errors while identifying produced particles. In particular, in ref. [82] one indicates that the source of disagreement may be due to an unjustified use [47, 2141 of the assumption that the velocity of secondary particles in the c.m. s. equal to that of center-of-mass system relative to laboratory system. Since the problem of the angular symmetry in p - n interactions is very important for checking different peripheral interaction models, it is very interesting to continue experimental investigations along this line. As the measurements of many authors show, the angular distributions of produced particles very strongly depend on the multiplicity. Few-prong stars turn out as a rule to be much more anisotropic than many-prong stars. Among many-prong stars there is a large percentage of practically isotropic ones.This is well seen from Tables 16, 17. It is interesting that the isotropy in stars with strange particles comes when the values of n are smaller than in stars without strange particles. For example, at T'r 7 GeV the angular distribution of pions produced in x- - p interactions together with A-hyperons is practically isotropic already in four-prong stars [87]. In stars with a definite number of prongs n the anisotropy increases with increasing energy of colliding particles. In all the cases the angular distribution of heavy particles is more anisotropic than that of pions. In ref. [40], by the example of x - N interactions, a t T N 7 GeV one investigated the dependence of angular distributions of produced particles on the magnitude of the inelasticity coefficient. The results of measurement are given in Fig. 30.


As is seen in events with smaller values of the inelasticity the anisotropy is much larger. All these results point again to the existence of the two types of inelastic interactions,

n’5 P

3

Fig. 30. The dependence of the angular distribution of protons and negative pions produced in inelastic x - N interactions at T = 7 GeV on the inelasticity coefficient [401. The continuous and dashed histograms are the distributions for K* 5 0.5 and K* 2 0.5 respectively. n is the number of prongs

3.2. Angular Distr ibut ions of Par t ic les i n theSuperhighEnergy Region. “Fire Balls”

To describe the angular distributions of produced particles in cosmic-ray experiments, instead of the probability W (@), one often use the function

N ( S @ ) @(@) = lg

N ( 2 n ) - N ( s @ )

where N (50) is the number of tracks in the angular interval (0, 0) in the laboratory system ll).

11) In experimental works the function @ is usually written in the form: @ = lg [ (P/ l - P)] where F = N ( S O ) / N ( ( n ) .

Inelastic Interactions of Particles at High Energies 11 483

If the angular distribution of particles in the c.m.s. is isotropic then it is not difficult to show that

(19) where yc is the factor of the relativistic transformation from the c.m.s. to the laboratory system [93,215,216). In this case, depending on x = lg tan @ the function @ ( O ) is represented as a straight line with the slope d@/dx = 2 which inter- sects the axis x at the point x = -1g yc (Fig. 31 a).

@(@) = 2 lg tan 0 + 2 lg yc

0 5c

I I

-{ 0 x -q 0 x Fig. 31. Different methods of description of the angular distribution of produced particles

-4 -3 -4 -3

k - I c - l I I '

T= 1,3. 105CeV T = 4,2 104Gev T = 9,s. ~ O ~ G ~ V

Fig. 33. Angular distributions which can be represented as a superposition of the angular distributions of partirles produced in the two fire balls dccay [215, 2181. T is the primary nucleon energy

In the anisotropic distribution of particles in the c.m.s., but W ( n / 2 ) + 0 , the f xct ion @ (2) is described by the curve which little differs from the straight line with the slope d @ldx < 2 ; if W (3212) = 0 then the curve representing @ (x) has clearly shown inflection in intersecting the axis x (Fig. 31 b and c). The upper and lower branches of the function describe particles emitted in the laboratory system in the narrow and diffusive cones respectively. In investigating events formed in emulsion by ultra-high-energy T 2 lo3 GeV cosmic particles it was established that the asymptotic slope of the upper and lower branched of the function @(x) are the same and close to two [217]. The examples of such angular distributions are given in Fig. 32. This means that there exist two systems of coordinates in one of which the angular distribution


484

of particles emitted into the narrow cone is close to the isotropic distribution while in the other about isotropic is the angular distribution of particles of the diffusive cone. The velocities of motion of these system of coordinates with respect to the general mass centre are close in magnitude and opposite in sign. As a rule, this result is pictorially presented as the emission in opposite directions of two isotropically decayed “fire balls” which essentially affect the shape of the angular distributions only a t very high energies. At lower energies and, in particular, in the acceleration energy region the velocities of emission of “fire balls” are not yet large and their angular distributions are overlapped (Figs. 31 b and c). Such a phenomenological model was first proposed by COCCONI [217] and the Po- lish Group [215, 216, 218-2201 and then was widely used. I n addition to the integral function @ (x) in papers devoted to the study of cosmic rays one often consider the function


The form of this function strongly depends on the magnitude of the asymmetry of angular distributions in the c.m.s. The isotropic angular distribution is described by the function f ( x ) which is well approximated by the Gaussian curve with dispersion c = 0.39. The deviations from the isotropiy lead to the increase of c. To the angular distributions which kinematically may be interpreted as the dispersion of two “fire balls” there correspond “two-hump” function f (x) with the same dispersion of “humps” just as for the case of the isotropic angular distribution (Fig. 31). Although the existence of the coordinate systems distinguished with respect to the ultra-high-energy distribution of particles is a reliably established fact, its physical interpretation is far from being unambiguous. The existence of the two distinguish coordinate systems and the two-hump character of the angular distributions f ( x ) with isotropic gaussian shape of both humps is not necessarily connected with the decay of two separate objects. In particular, the angular distribution of particles in elastic p - p scattering expressed in the variables x = lg tan 0 is of the same shape as those in inelastic interactions. From the purely kinematical point of view the former may be interpreted as a result of the decay of two “fire balls” although the formation of such objects is here excluded [221]. In ref. [221] it was shown that all available experimental data on two-maximum angular distributions are well accounted for if the conservation law of angular momentum is taken into account and it is assumed that all two-maxima distributions are due to peripheral collisions of primaries. No hypotheses on “fire balls” are in this case needed. The assumption on an essentially peripheral character of inelastic interactions with two maxima angular distribution [221] is proved by the experimental fact that the two-maxima distributions are mainly observed in events with not too large number of shower particles n, < 20, small number of black tracks nh 5 5 and large value of dispersion c > 0.6. Although the picture of formation of “fire balls” is ingenious and very attractive it cannot be regarded as an experimentally established fact and should be treated very carefully. On the other hand, a t present there are no serious arguments against this picture. A compromise situation can also take place when in some stars the two-maximum character of the distributions is due to the fire ball decay and in other peripherality of interaction.

Inelastic Interactions of Particles at High Energies 11

The solution of the problem depends, first of all, on experiment. Since the accuracy of cosmic-ray experiments is for the time being not large it is difficult to determine accurately the percentage of N - N interactions which contain two singled out frames of reference. According to the Polish Group's data, within the statistical fluctuations all the inelastic interactions with not too high multiplicity may belong to this class of u interactions [215, 2181. Following the experimental data of ref. [222] this effect is revealed in about 20 per cent of all inelastic interactions.

485

4. Quasi-Elastic Collision

The analysis of the interactions of ultrarelativistic 12) cosmic-ray nucleons has provided evidence that these particles often emerge from collisions against nucleons a t rest with little loss of energy. (quasi-elastic collisions). This fact sharply contradicts the commonly accepted description of N -. N interactions according to the Fermi statistical model. Therefore a t CERN an attempt has been made to study experimentally such interactions. A collimated beam of protons of well-defined energy (between 8 and 24 GeV) was scattered on a Be foil. The momentum spectra of protons scattered at small angles 0 I 60 m rad e were investigated. The most surprising result was that the spectra

v - possess a clearly cut structure. Each of them contains an elastic peak and one or two quasi-elastic peaks superposed on the continuous spectrum which is extended from highest momenta up to lowest ones and due to multiple production processes (Fig. 33). It should be stressed that quasi-elastic peaks appear only when in the spectra there is an elastic peak, the difference between the elastic peak and the highest quasi- elastic one being constant and equal to A p = I GeV/c. It depends neither on the scattering angle nor one the primary proton momentum. The correlation between quasi-elastic and elastic peaks allows to assume that the first is due to x0 meson exchange and arises as a result of the well-known resonances observed in x - N scattering [113, 168, 1781. This can be explained qualitatively as follows : The quasi- elastic peak is due to the diffraction dissociation of the target-nucleon bv an

& dp dR

400

200

50

25

0

T= 2,868 GeV

T= 13 GeV

11 I2 f3 pCGeV/cl Fig. 33. The momentum spectrum of protons in the

quasi-elastic p ~ p scattering

" incident ultrarelativistic proton into a nucleon and a pion occurs a t our energies, only when N 1 GeV energy is absorbed. The absorption processes accompanying the quasi-elastic scattering lead to elastic diffraction scattering whose peak is always present in proton momentum spectra. The fact that the height of the quasi-elastic peaks decreases with increasing momentum transfer in the same

la) The Lorentz factor is y > 1.

32*


way as the corresponding values for elastic scattering confirms the above hy- pothesis about the connection of these processes. The quasi-elastic scattering of x and K mesons on protons was not investigated experimentally. However, from the point of view of N - N scatt,ering some predictions can be made concerning the momentum spectra of x and K mesons under- going quasi-elastic scattering. The spectra of x and K-mesons scattered at amall angles must also have the structure and contain, in addition of diffraction peak, quasi-elastic peaks, however in the given case they are connected with the production or resonance x x and K x states. Consequently, their shift with respect to the elastic peak is no longer the same as for N - N scattering, but is defined by the known resonances in x x and K x scattering. We may hope that the quasi-elastic scattering will help to solve many still unclear problems of inealstic interactions with very small momenta transfers.

5. Conclusion

In ref. [ I ] and in the present review we have considered in detail experimental characteristics of inelastic interactions of particles a t T 2 1 GeV energy. The main conclusion is the existence of the two types of inelastic interactions which are characterized by different inelasticity coefficient and the four-momentum transfer ; the energy of produced particles and their multiplicity also turn out to be different for both types of interact ions. Inelastic interactions with small multiplicity are characterized by much larger anisotropy in the angular distributions. I n spite of large experimental errors both types inelastic interactions can be revealed in all the basic characteristics of inelastic interactions. The analysis of the particle distributions over their transverse momentum shows that both types interactions proceed in space regions of different radii, one of types may be compared with the “peripheral” interaction and the other with the “central” one. As to the number of “central” and “peripheral” collisions a t present we may only assert that the latter occur considerably more often than the former. This conclusions well agree with the theoretical predictions. We note with satisfaction that the model of central and peripheral collisions which was developed in our laboratory during the last ten years, now is being proved experimentally and becomes generally known. Among secondaries produced in inelastic collisions there is, as a rule, a “leading” particle which carries away 60-70 per cent of the total energy. I n a wide energy interval from several GeV to hundreds of thousand GeV the inelasticity coefficient remains approximately constant and is about 30-40 per cent. After se- paration of the leading particle in average distributions of stars over the multiplicity and the kinetic energy of produced particles purely statistical features are clearly revealed. To interprete them it is sufficient to consider phase volumes. In particular, the kinetic energy per one degree of freedom of the secondary particle system is with good accuracy independent of the kind of colliding and produced particles. In all cases only the energy which is spent for new particle production is to be taken into account. The mean number of produced particles E is proportional to T‘i4 their mean kinetic energy (excluding the leading particle) is also proportional to T‘I4 and the mean transverse momentum jil is less or equal to In T . Among secondaries about


20 per cent are particles with mass larger than that of the pion. Note one more important conclusion which follows from the analysis of the presently available experimental data. The mechanism of inelastic interactions at T > 105 GeV can be possibly changed. This conclusion is as yet unreliable but further experimental investigations in this direction are of great interest. Apparently, the main experimental problem is the improvement of the measurement accuracy. In this connection it is more important to investigate carefully the details of inelastic interactions only for several values of energies (e.g. x -- N interactions a t T = 7 and 16 GeV) and N - N interactions a t T = 9 and 25 GeV) than to perform many, but not accurate, measurements a t various energies. In conclusion we should characterize briefly the state of the inelastic interaction theory a t large energies. Various statistical theories were used to estimate the average characteristics of inelastic interactions. The only x N resonance N& was taken into account, nevertheless calculations were in good agreement with experiment. Without N& agreement was noticeably worse. However a t present a large amount of x N and x x resonances is available; the width of most resonances is by far larger than that of N&, the former seem to give an essential contribution to the statistical weights of partial channels. It is not clear why the earlier calculations gave good results. Apparently we should return to statistical calculations, what is especially important a t present in connection with the calculation of particle beams for super-power accelerators being under construction. For the time being we should note that there is no satisfactory statistical theory. To analyse inelastic interactions one often use the onemeson theory or equivalent Weizsaecker-Williams method. However in this case too, one should use the statistical theory in order to calculate interactions with large number of produced particles in the vertices of one-meson graphs. From this point of view it may be said that the one-meson theory differs from the earlier used purely statistical one only by that the interaction kinematics is more accurately taken into account. The difficulty becomes greater when a t high energies one cannot apply the main one-meson theory assumption that the vertex function is independent of the momentum transfer A and its module is porportional to the appropriate cross section observed. Thus, the situation in the inelastic interaction theory a t high energy is a t present extremely unsatisfactory.

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Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Arthur Losche, Prof. Dr. Rudolf Ritschl und Prof. Dr. Robert Eompe. Manuskripte siud zu richten an die Schriftleitung : Dr. Lutz Rothkirch, 11. Physikalisches Institut der Humboldt-Universitat Berlin, 104 Berlin, Hessische Str. 2. Verlag: Akademie-Verlag GmbH, 108 Berlin, Leipziger Str. 3-4, Fernruf 220441, Telex-Nr. 0112020, Postscheckkonto: Berlin 35021. Die Zeitschrift ,,Fort- schritte der Physik" erscheint monatlich; Bezugspreis cines Heftes 8, - (Sonderpreis fur die DDR 0,- MDN). Bestellnummer dieses Heftes: 1027/15/7. - Satz und Druck: VEB Druckhaus ,,Maxim Gorki", 74 Altenburg, Bez. Leipzig, Carl-von-Ossietzky-Str. 30-31. - Veroffentlicht linter der Lizenznummer 1324 des Presseamtes

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Inelastic Interactions of Particles at High Energies. II. (Energy and angular distributions of secondary particles)