Fortschritte der Physik / Progress of Physics: Band 27, Heft 7 1979 [Reprint 2021 ed.] 9783112522721, 9783112522714

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT I) E K DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCHLUIIN. V. LÖSCHE, K. KITSCHL UND K. HOMPE

H E F T 7 • 1979 • B A N D 27

A K A D E M I E .

V E R L A G EVP 1 0 , - M 31728

.

B E R L I N

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ISSN 0 0 1 5 - 8 2 0 8 Portschritte der Physik 27, 313—354 (1979)

Hydrodynamical Theory of High-Energy Particles Interaction E . I . DAIBOG, I . L . ROSENTAL, JTT. A . TARASOV

Institute of Nuclear Physics, Moscow State University, Moscow,

USSR

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

Introduction Fermi Statistical Theory Pomeranchuk Statistical Model Physical Foundations of the Hydrodynamical Theory On the Dissipative Terms and Equation of State Hydrodynamic Expansion Distribution over Pseudorapidity rj Multiple Process Characteristics Based on Equation of State On Leading Particles in Multiple Processes Composition of Secondary Particles The Scale Invariance in the Hydrodynamical Theory On the Accuracy in Calculations of Characteristics of the Multiple Processes Comparison of the Hydrodynamical Theory with Experimental Data on pp Collisions. . . Hydrodynamical Model of Interaction of Nucleons with Composite Nuclei The Hydrodynamical Model of Interaction of High-Energy Photons and Leptons with Hadrons and Nuclei The Hydrodynamical Interpretation of Annihilation Processes Direct Production of Leptons . The Hydrodynamical Model and Equation of State of Hadron Matter On Some Fundamental Aspects of the Hydrodynamical Theory Relativistic Invariant Formulation of the Hydrodynamical Model Hydrodynamical and Thermodynamical Models . Hydrodynamical and Multiperipheral Parton Models

Literature

313 314 315 316 317 319 322 323 324 325 327 327 328 330 336 337 339 340 345 347 348 351 352

1. Introduction Production numerous particles in hadron-hadron collisions (multiple processes) is a privileg of strong interactions. Two features of multiple processes (the great number of particles in the finite state and strong interactions) cause the specific character of this phenomenon. The complication of its description is associated with the doubts of up-todate field theory application to the strong interactions. On the other hand participation of numerous particles in the process makes us believe that a quasi-classical approach can be successfully applied. Hence, it is expedient to describe multiple processes based on the quantum and classical concepts. 26

Zeitscbrilt, Fortschritte der Physik", Heft 7

314

E . I . DAIBOG, I . L . ROSENTAL, JTJ. A . TABASOV

Here an analogy with the nucleous models is relevant for example, the shell model is a quantum approach, while the gaseous model is a classical approach. However, there are differences in this analogy, while a nucleus is the totality of real particles (nucleons) that are the virtual particles (quarks, partons) that are significant for multiple processes. Nevertheless the most important aspect of such an analogy, possibility of quantum and classical approaches to describe both phenomena, nucleons and multiple processes, remains. It is believed that a nucleous is a typical example of nuclear matter; the spatial-time region corresponding to high-energy interacting particles is filled with a new type of matter-hadronic matter. The multiple process theory started with the prediction of the phenomenon just before it was experimentally confirmed. H E I S E N B E B G [2] making an attempt of explaining the showers which have long been observed in cosmic rays used a ^-interaction version. However (and it is essential) that just in early papers [i, 2]. Heisenberg pointed to the possible statistical description, that is caused by the high coupling value. Further the concepts manifested itself in paper FF.RMT [

g is the number of the particle internal degrees of freedom Kl is the Bessel function of an imaginary argument. Assuming Tj = fi — m, we obtain =

For p i

fi, we obtain

{TI)'-1

Vl +(PiM*)Pl.

(32)

dN = [34] 7

) I n refs. [46, 47], a c o m p a r i s o n w a s carried o u t for t h e t h e r m o d y n a m i c a l model. B u t t h e conclusions of b o t h m o d e l s for t h e distribution dN/dp ± are similar. 27

Zeitschrift „Fortschritte der Physik", Heft 7

330

E . I . DAIBOQ, I . L . ROSENTAL, JRR. A . TABASOV

Fig. 7 shows the dependence N9(E0)S). Up to the energies E0 < 100 GeV, the calculation with the exact formulas of the statistical theory (the dashed line); at the energy E0 > 100 GeV, the calculation in accordance with the hydrodynamical theory [4S\ (see, also [50]). Fig. 8 is a comparison of the theoretical [28] and experimental [5i] distributions dNjdr] at relatively small values r] (see also [44]).

I/

2k

I

Nr

ii t Apt

16

S

10 20

SO 102

103

10 GeV

Fig. 7. Dependence of the average multiplicity [45]

It is noteworthy that at the age of relatively short life of the strong interaction models almost all fundamental conclusions of the hydrodynamical theory have been drawn many years ago. Only the production of big accelerators permitted us to compare the conclusions of the model and the experimental data thoroughly.

F i g . 8 . The theoretical [28] and experimental 152] distributions

a) 1/7 =

30.8 GeV

b)

jT =

63.4 GeV

14. Hydrodynamical Model of Interaction of Nucleons with Composite Nuclei Multiple processes in high-energy hadron collisions are widely studied [54], but investigations of hadron-nucleus collisions revealed a number of interesting peculiarities. The nucleus is an object with a relatively great spatial extension. Therefore, a study of 8 ) Good agreement between the calculations of the distributions in y and x and the experimental data was also obtained (see, for example [53]).

Hydrodynamical Theory of High-Energy Particles Interaction

331

h-A interactions is useful for clarifying a space-time picture of the particle-production process, and also the resonance-nucleon interactions and, maybe, even the interaction of nearly-free quarks. One should differ between the processes developing in the nucleus in the longitudinal direction (i.e., in the direction of primary particle motion) and in the transverse direction 100 GeV some characteristics of the multiple processes become energy-independent and reach the asymptotic regime. It should be stressed that the above-mentioned A-dependence of the characteristics is in contradiction to the model of intranuclear cascades based on the assumption that cascades are the greatest part of the quasifree particle interactions. This fact is an indication to the collective character of multiple processes in nuclei. Let us summarize briefly the experimental data on hA-collisions noting that much information has been already collected in numerous reviews and monographs (see, for example [55—59]). Here we confine ourselves by stating only some experimental facts in the energy region of E0 > 20 GeV which, to our opinion illustrate well the basic features of interactions. Multiplicity of Secondaries For classification we use here the following characteristics: Ns the multiplicity of shower particles (/? > 0.7); Nh the number of strongly-ionizing particles (/? < 0.7): Nh = Ng + Nb, where Ng and Nb are the numbers of grey and black tracks. An important characteristics of hA-collisions is the relation: ^

_ {NS)pA {Ns)pp

^ggj

The values of multiplicities in pp and pA-collisions correspond here to the same energy Eq . Figures 9 a, b show [60] the dependence RA (E0 ) for collisions of protons and pions with different nuclei — from C to Pb. It is seen from the figures that RA (E0 ) ~ const, for E 0 > 70 GeV. The value of RA is dependent of the number of (v), which is determined in the following way •200 GeV n-

1,0

2,0

?

3,0

U,0

A 50 GeV p + • 100 GeV p+ a200GeVp+

Fig. 10. a) Variation of B A with (v>, b) The data with indication of error at (v) = 3,5 and the line R A = f(v) + l)/2 are shown (I: Experiment; II. Ra= (? + l)/2)

Hydrodynamical Theory of High-Energy Particles Interaction

333

It should be noted now that the dispersion D = (Ns)2 — (Ns2) for hA collisions behaves itself like the dispersion in hp collisions: YD = a(Ns) + b. According to the da.ta in ref. [62] for pA collisions apA = 0.64 i 0.02 and for pp collisions app ~ 0.6. The quantity (iV3)/]/U for pp and pA-collisions is close to 2 and practically independent of E0 \6S\. Angular Distributions of Secondary Hadrons Analysis of the angular distributions of shower particles in the energy region E0 nc, the process is more complex — a simple wave running to the right will reach the shock one and will be reflected, i.e. the first reflected wave appears [29]. The solutions to the hydrodynamical equations for the potential %{r¡, y) for both the cases have been obtained recently in refs. [21, 66, 67] where a study was also made of the value of RA for different nuclei and at various values of Cs. The above-described asymmetry of secondaries developing in pA-collisions is apparently observed experimentally (figures 11, 12). This problem is investigated in ref. [21]. It is

336

E . I . D A I B O G , I . L . R O S E N T A I , J r . A . TAEASOV

shown t h a t the asymmetry can be more significant in the V-system if n ~ nc, (see also § 17). Now, let us t u r n to two phenomena discovered in hA-collisions — the cumulative effect and a strong dependence of the invariant cross sections E(d3aldpa) on the atomic number. The cumulative effect (see, for example, [6 2 GeV it was demonstrated t h a t a A* (ft ~ 1,1; see for example, [69]). One circumstance makes these effects similar: the question is of relatively rare events so it should be expected t h a t they are fluctuation events. When one tries to explain rare events deviating greatly from mean values in terms of a model one has to make, as a rule, new assumptions which are not found in the basic postulates of this model. For example, the cumulative effect is explained qualitatively by fluctuations in the collective motion of the fluid forefront elements. Setting the parameters of these fluctuations one can explain the observed peculiarities of the cumulative effect. B u t such quantitative estimates, to our opinion, can't be of great cognitive value. Similarly, a strong ¿4-dependence of the cross sections in the region of great px, is readily explained by rescattering effects during the virtual hadron phase or during transmission of secondaries through the nuclear matter. 15. The Hydrodynamical Model of Interaction of High-Energy Photons and Leptons with Hadrons and Nuclei This section deals with interactions of leptons (1) and photons (y) with nuclei Y+ A-*h + x

(61)

1 + A -> 1 + h + x

(62)

x is the system of undetectable hadrons. The basis for the possibility of applying the hydrodynamical approach is the fact t h a t the cross-section of inelastic interaction with nuclei increases slower than A. Approximately ain ~ (see, for example [55])10). This means t h a t for some time a photon is a "quasihadron" [70, 71]. Really, the cross section of yp-interaction: ojv ~ 10~28 cm af£. So if the effect of photon-hadron transformation were absent one would expect cr;„ P

+« +

x

(68)

+

X

(69)

P + P->5T + X.

(70)

The relation (68) is conditionally an inclusive formation of pions in pp-collision. (69) is the pion formation in pp-collisions; (70), in annihilation. One should expect on the basis of general postulates of the hydrodynamical model that the pion spectrum in reaction (69) will be more rigid than in reaction (68) (see § 9), and in reaction (70) more rigid than in (69). Recently, the statistical-hydrodynamical interpretation of (e+e~) annihilation into hadrons has attracted much attention [7-3—76]. Observations in the range of Ys : (wp> ~ 1:0.1:0.01. 3. The mean multiplicity of secondaries increases greatly with energy.

338

E . I . D A I B O G , I . L . R O S E N T A I , JTR. A . T A R A S O V

4. The angular distribution in the C-system is roughly isotropic. 5. Even if the total cross-section changes with energy E0 , it changes weakly and is nearly equal to 20 nb. All, the data are not readily explained from the standpoint of field theories (see [75]). But the mentioned experimental results (except, of course, the problem of the crosssection exceeding the limits of the model) can be explained in terms of the statisticalhydrodynamical theory. In this case, however, an important problem of the choice of the initial volume arises again. Since colliding are the point particles (leptons) it is natural that an initial state is spherically symmetric which explains for the absence of anisotropy. Here we observe a new interesting peculiarity: in rapid particle interactions the initial state was specified by an axial symmetry and in case of annihilation a spherical symmetry exists. However, both the values of the initial volume and its energy dependence11) are important here. It is assumed in a simple variant [73, 74\ that the initial volume is Ea — independent and equal to ~1/]M3. Using (4) and (5), it is easy to obtain for this case (N)

~S 3l*V 1l i.

(71)

For a more general equation of state (40) ; J . + N + hadrons ans also pp-interactions and annihilation. A study of the degree of the universal character of inclusive spectra and the composition of the produced hadrons seems to be essential. (The universal character is taken here as independence of the characteristics of the colliding particle type at a given value of S). 17. Direct Production of Leptons Spectra of y-quanta and (11) pairs produced in pp-collisions with E0 > 200 GeV and P i > 1 — 1.5 GeV/c, practically repeat the pion spectra. This circumstance suggests an idea of a common mechanism of production, namely, of "the evaporation" of particles at an early stage of the hydrodynamical system. A physical picture of the phenomenon is represented as follows [78—79]. At first, when the energy of colliding particles is emitted in a Lorentz-compressed volume, the hadron vacuum gets excited. Then fluctuations of the charged currents (collision or annihilation of charged hadrons) lead, with a probability (x = 1/137 is the fine structure constant) to production of y-quanta and a 2 - 11-pairs. These fluctuations are developing during the entire stage of the hydrodynamical extension and the spectrum of y-quanta and 11-pairs is determined by summing the contributions of all elements at all intermediate values of temperature T T 0 ^ T A characteristic quantity possessing the dimension of time or length is 1 ¡T. So, the number and energy of y-quanta in the inherent fluid element dV' during the time dt '[7 1014 the part of energy lost to the electromagnetic component increases as compared with the region of more lower energies by 1.5 times. The above-mentioned effect of equilibrium of the y- and hadron degrees of freedom can be a possible explanation to this phenomenon. Unfortunately, one can speak here only about qualitative description since being unaware of the constant in eqs. (79, 80) one finds it difficult to give a sound value of the energy E0 at which the equilibrium considered can be set in. Another possible explanation of the effect in the framework of hydrodynamical theory is the circumstance that E0 ~ 1 0 u eY corresponds to the temperature T0 ~ M at which as was mentioned (see § 9), one observes an increase of the coefficient of inelasticity and consequently, an increase in the multiplicity (N). Here one should differ two different variants of what occurs at EQ & 1014 eV: a) a simultaneous increase of i f and {Ny)l(N„), b) an increase of only K. In the first case one should involve considerations stated in this section; in the second one, criticality of the value T0 ~ M. 18. The Hydrodynamical Model and Equation of State of Hadron Matter The characteristics of multiple processes calculated in the framework of hydrodynamical model are dependent of the equation of state (40) and specifically of the parameter a = c»2. That is why, comparison of predictions of the hydrodynamical theory with experimental data allows us, in principle, to approach a definition of the equation of state of superdense (q > 10 ls g cm - 3 ) hadron matter. This question is significantly beyond the scope of high-energy physics and is, for example, of great interest to cosmology in analysing of early stages of the Universe expansion. The question of the maximum (E 0 00, e or q - » 00) value of parameter a (or cs) is undoubtedly a matter of principle for high-energy physics. One can suppose that with increasing density q the sound velocity tends to one of the following values a 1, or a 1/3, a - > 0. The first possibility corresponds to the extremely rigorous equation of state and the second one to the asymptotic freedom. Discrimination of the value a = c s 2 = 1/3 is detected by the three-dimensionality of space (see § 5). For an ideal relativistic gas all directions are equally possible and p 4= 0. That is why p — s/3. In other words, if the potential energy at E0 - > 00 (or q —00) remains lower than the kinetic one, a — 1/3. In the modern language, this relation is apparently valid, for example, if the asymptotic freedom model is true. The same relation should be retained for all interactions which are

Hydrodynamical Theory of High-Energy Particles Interaction

341

characterized by a small parameter permitting us to use the perturbation theory. For example, the relation a = 1/3 is valid for a gas consisting of photons and relativistic electrons if the characteristic energy of the process is not unreally great (E meellix). No wonder, that all attempts to calculate the equation of state according to the perturbation theory led invariably to the relation a = 1/3 (see, for example [81, 82]). A maximally strong attraction due, for example, to resonance interaction (see § 18) leads to the fact that the pressure associated with the kinetic energy turns to zero which corresponds to a = 0. This picture naturally leads to some value of the maximum temperature T, the total energy is lost to the resonance production. There is no energy to an increase of T. I t should be noted that the limiting inclusion of resonances (i.e., the assumption that a number of resonances break at a mass m x and the spectrum itself is of the power shape) decreases the value of a, but a =)= 0. For example, the inclusion of resonances up to mx ~ 1.7 GeV leads to the value a ~ 0.2 [33, 34]. And vice versa, a maximum strong repulsion can lead to the value a ~ 1. This model is built [S3] in the classical limit when the concentration n oo. I t is clear, however, that this limit is not valid for analyzing the matter consisting of real hadrons since their length is ~ 10~13 cm (in the proper coordinate system) and therefore the hadron concentration n < 1039 particles/cm 3 . But if a decay of nucleons into quarks is assumed, it is obtained on some additional assumptions, that a -> 1/3 at Q > 1017 g cm - 3 [84]. It should be noted that if "three-dimensionality" of the space is violated, i.e. the system consists of relativistic particles moving linearly, it is natural that a -> 1. But for this case arises a question on the strength making relativistic particles move linearly (see § 21). Thus, the theory fails to give an unambiguous answer to the question of the extremely strict equation of state of the hadron state, though the value of a 1/3 (at E0 -> oo) seems to be rather suggestive. Let us turn now to the possibility of using the analysis of collisions of high-energy particles to define the parameter and consider firstly papers onppandthenonpA-collisions. Let us turn to the dependence (N(E 0 )). I t was noted earlier that the hydrodynamical theory can pretend to be describing the experiment at E0 > 100 GeV. At the energies E0 100 GeV, one should fit the results of predictions of the hydrodynamical theory onto the exact calculations of the statistical weights. Similar calculations [10] lead to the fact that with decreasing E0 the power exponent /? of the dependence (N) ~ E M the parameter /? reaches its limit 1/2 (see fig. 7). Therefore, if the dependence (N(E0)) is calculated on the basis of the statistical-hydrodynamical theory, it isn't at varience with the value* of a = 1/3 within the measured energy region. In ref. [S5] the experimental and theoretical distributions dNjdr] at different values of the parameter a are compared. Though the authors come to the conclusion that the shapes of curves are in agreement at values of a somewhat smaller than 1/3 the accuracy of calculations is, apparently, overstated. In ref, [28], the shape of distributions dN/drj and dNjdy in the energy region E0 ~ 400 to 1000 GeV is studied. At a point rj = 0 the distribution dNjdr) becomes somewhat wider and 1 ower than the Gaussian one and a small minimum can even appear with decreasing {p L ). Appearence of this minimum is due to limitation of the quantity (j) L ) (see for details § 7). In this paper the theoretical and experimental distributions dNjdr] are in good agreement at a = 1/3. The distributions in rapidity at fixed values of p L were compared in ref. [53], The comparison is also indicative of good agreement with the hydrodynamical theory at a = 1/3. In this theory a comparison of the mean multiplicity and the inclusive spectra overaged over leading particles with the experiment at E0 < 1000 GeV led to rough agreement at a = 0.2.

342

E . I . DAIBOG, I . L . ROSENTAL, JTJ. A . TABASOV

In ref. [S7] the inclusive distributions for pp-collisions in the energy region 200 sS EL ^ 1500 GeV were studied according to the refined hydrodynamical model. A set of differential equations describing three-dimensional motion was numerically solved without averaging over the radial transverse coordinate r. The authors [57] come to the conclusion that better agreement of the distributions in rapidity and the transverse momenta is obtained at a = 0.16, i.e. the corresponding spectrum of masses of the resonances is of the form m" at x ~ 2. To all appearance, it is reasonable to put here several questions: 1. What values of a = c / can be fitted to a very weak energy-dependence of {N)l Apparently, for the value of a = 0.16 this dependence will prove to be excessively strong. 2. What phase of the collision process can these values of the parameter a correspond to? At the initial stage at high energies the hadron matter density can exceed the nuclear one by hundreds of times and so to what extent one can speak of the resonance spectrum? The third phase of decay which determines the composition of secondaries is in good agreement with the value of a — 1/3. That is why one can't exclude the possibility of changing the velocity of sound in decay in the hadron cloud cooling. 3. What phase exerts the greates influence upon the distributions in rapidity and other characteristics? In other words, to what extent do these characteristics yield information on the properties of the hadron matter in a superdense state? To all appearence, a study of the equations of the hydrodynamical theory at a = c / = const, excessively simplifies the solution of these questions. Now, let us go over to thepA-collisions. Most of the papers dedicated to this problem are published in the last 3—4 years. Side by side with explanation of experimental data, these papers study the questions of the equation of hadron matter state. A new possibility is, apparently, an investigation of the dependence of the relation RA on .4 and the parameter a. If the tube length doesn't exceed the value of ne (60) the entropy of separate sections of the nucleon-type system after the shock-wave transmission should be calculated to determine RA. In the F-system the matter of these sections is at rest. Then (81)

If n > ne, the progressive wave can overtake the shock wave and the calculation becomes more involved since the matter behind the shpck wave will not be at rest. This can be for great tubes and small values of Cs. The entropy (multiplicity) and the quantity RA for different values of Cs were calculated in refs. [66, 67]. The value of RA for n> n3 diminishes weakly as compared with (81). In ref. [66], a study was made of the pA-collision for nuclei of the elliptic form, the parameters of this ellipse which is stretched along the inicident beam being chosen from conditions of conformity to the nuclei with a diffuse boundary (see below). In this approximation the most close agreement of the value of RA with experiment is achieved for a a 0.13 which corresponds to the mass spectrum of resonances m3. The question of adequacy of this value to a slow increase of (N a ) with energy arises here as well. The author [66] presents no data on the value of the mean tube (ri) in nuclei of the form he considers. Another paper [67] used the model of the nucleus with the density determined by the Woods-Saxon formula in order to calculate RA [&S]. eir) = eo i + exp -

(82)

Hydrodynamical Theory of High-Energy Particles Interaction

343

Here d is the diffuseness parameter. The parameters b and d are different for various nuclei but they can be represented rather well by the same formula in a wide region of values of A. The quantity g0 is obtained from the normalization conditions oo Aat f

o

e

( r ) r

2

d r =

(83)

A.

Agreement with experiment in ref. [67] is achieved, to all appearence, for the value of a 0.3. The author doesn't present the values of the mean tube (ri) either. It should be noted that the pA-collision cross-section is very weakly dependent of the shape of the nuclear density — an increase in the transparency of nuclei with the diffusive boundary is cancellated by an increase in their geometric dimensions. For example, for heavy elements the inelastic scattering is well approximated by the geometric one: nr02Ail3 where r0 a 1,26 • 10~13 cm, i.e. coincides with the cross section of collisions on nuclei with a sharp boundary of the radius R = «v41/3. For spherical nuclei with an abrupt boundary the mean tube value (n) is readily obtained. The mean path length of a nucleon in the nucleus is R ( i ) = j

2(B2

-

=

T

s

=

T

0 where R is the nucleus radius. Thus =

= f

(84)

Let us present the values of {ri) for various nuclei: Table 1 Element

A1

Cu

Ag

Pb

(n)

2

2.66

3.16

3.95

A comparison of data from table 1 and ref. [60] shows that the values of (n} are close to the values of (v} [59] exceeding the latter but slightly. In order to calculate RA for different nuclei one should average over the tube length RA being calculated with (81) for n < nc and RA being represented by sufficiently more complicated formulas for n> nc [21, 66, 67]. But calculations show that for all nuclei the value of RA slightly differs from that for the mean tube: (85) this difference decreases with increasing parameter a. Now compare the value of RA (85) with the experimental data in Table 1 (for the proton energy E0 = 200 GeV). Table 2 Element

A1

Ra = ((») + l)/2 1.5 RAFIK p 1.52

Cu

Ag

Pb

1.83 1.84

2.08 2.11

2.48 2.53

344

E . I . DAIBOG, I . L . KOSENTAL, J U . A . TAEASOV

It follows from data in Table 2 that the values (85) fit well the experimental data — even to a better degree than the values of l/2((v) + 1) [60]. Agreement with the experiment is good at a 1/3. Thus, the experimental data for the relation R A are, to all appearance, in good agreement with the tube model for spherical nuclei with an abrupt boundary. Inclusion of other possible forms of the nuclear density will change the results weakly, to all appearance. A possibility of determining the velocity of sound Cs was suggested in ref. [21] which studies the asymmetry of the angular distribution of secondaries in collisions between nucleons and the nuclei of emulsion and hydrogen. The asymmetry, i.e. the value of shift of the distribution maxima for pp and pA collisions is dependent of the velocity of sound and can be observed even for (n) < nc. Comparison of the experimental and theoretical values of the shifts at the energy of E0 — 200 GeV gives the values of a ~ 1/3. We should note here an important circumstance. One of the authors (Ju. T.) calculated the distributions dN/dtj in pA collisions for the case when the velocity of sound Cs was temperature-dependent and changed smoothly from the initial value of CJ* at the first stage of decay to the final value of CJ l/|/3. Calculations for two values of C3° = 0.7 and Cs° = 0.5 indicated that the value of maximum shift is, mainly, dependent of the initial values and not of the final values of C3. Consequently, this value, in principle, can specify the equation of state of the hadron cloud in the initial strongly-compressed state. These calculations are advisable for other characteristics of decay as well. One of the recent papers [90] gave the calculations of the asymmetry of the energy distribution of secondaries in pA-collisions. Calculations at a — 1/3 and a = 1/7.5 gave great differences in the width and character of the distributions, for example, for hydrogen and tungsten. Experimental data are not yet obtained. A question of a possible form of the equation of state of a strongly-condensed hadron matter was considered in refs. [21, 91]. These papers analyzed the dependence of the parameter a assumed in the form (86) Then, the mean multiplicity N = B0 In

(87)

From agreement with the experimental data one can obtain the value of K ~ 0.8. Substitution of this value into (86) at E0 = 200 GeV yields a ~ 1.3. This value is close to the values obtained from calculations of the distribution asymmetry [2I\. At E0 =70 GeV the corresponding value of Cs from (86) will be a ~ 0.25. The comparison with experimental distributions in pseudo-rapidities for the emulsion and hydrogen at the proton energy 70 GeV [92], doesn't contradict to the value of a 0.3. Of course, one should note that with a further energy decrease the applicability of the hydrodynamical theory fails to be well-founded. If the dependence (87) is used for the energy E0 = 1000 GeV, the corresponding value a ~ 0.5. It can be shown that the corresponding shift of maxima in this case should increase by more than 20%. Thus, we sum up the analysis carried out in this section: 1. At present, one can not draw final conclusions on the quantity C3, i.e. on the form of the equation of state of the hadron matter at superhigh densities. To this end, further experimental and theoretical studies are required.

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345

2. It seems to be trustworthy that at E0 ~ 200 GeV available now the velocity of sound in hadron matter is close to the value of C3 ~ 0.6 o r a ~ 1/3. 3. So, an alternative exists — the velocity of sound has already reached the asymptotic value Cs = 1/V3 i.e. the hadron system satisfies the requirement of the asymptotic freedom. In this case such a characteristic as, for example, the peak shift in the pA collisions will not rise with further increase in Ea. Another possibility is: the value a ~ 1/3 for Ea ~ 200 GeV is intermediate and continues to rise with increasing energy. The question which of the two possibilities is actually fulfilled is undoubtedly of interest.

19. On Some Fundamental Aspects of the Hydrodynamical Theory The hydrodynamical theory is built in terms of quasiclassical presentations. In this connection two questions arise: first, if there is an analogy with the LagrangeHamilton formalism and, second, what are the limits of application of the classical approach. The first question which has been already discussed in the early works on the multiplicity processes (see, e.g., ref. [2]). has a simple qualitative answer: fluid just as the field is a system with an infinite number of freedom degrees expanding in the space-time. However, in a particular attempt to follow the connection between the two presentations this connection cannot be made unambiguous since the classical description uses notions (entropy, temperature etc.) quite different from those used by the field theory (field amplitude). Nevertheless, as the energy density, or Lagrangian, is expressed in terms of the classical and quantum notions, there is a hope to find successfully the analogy between the two approaches. This problem was stated in ref. [9S\ where the scalar fields were investigated. The idea of the further consideration of the Lagrangian was division into two terms L = L0 + Liat

(88)

where L0 = —1/2 ( 1 real particles (see § 3). A possible way of getting out of this difficulty is to make an assumption that the volume

Hydrodynamical Theory of High-Energy Particles Interaction

351

where hadrons are enclosed corresponds to a certain finite state and, hence, has no constant value but increases proportionally to (,W)[105]. This model is close in this respect to the statistical theory with the expanding volume (§3). Spectrum (107) can be obtained in the dynamical models, e.g. in the dual resonance model (this approach is described in detail in ref. [106]) as the result of degeneration of the corresponding resonance levels, i.e. the number of resonance states with mass M. The dual resonance averaged statistically is considered as the fireball. From the thermodynamical point of view existence of two types of hadron systems then appears to be possible: 1) fire-balls-elements with the limiting temperature Tt and hadron systems for which the limiting temperature is lacking. It must be noted that such an approach introduces a very significant dynamical element; such a model should be rather considered as a dynamical one. The idea of an essential uncertainty of the boundary between the statistical and dynamical principles in microphysics runs all through this review, though. Note that straight forward introduction of the limiting temperature leads to existence of the elementary length [107} (109) Usually it is assumed that Tl ~ 160 MeV; then I ~ 10~13 cm, which is in disagreement with numerous experimental data on electromagnetic interactions, (e.x. measurements of the anomalous magnetic momenta of muons and electrons): It seems that the elementary length I ~ 10~13 cm is also lacking for strong interactions (measurement of scattering for the zero angle). Therefore, the only possibility to keep representation of the limiting temperature is to assume that the elementary length I ~ 10 -13 cm is not unique. For example, assuming that this length arises at the quark-parton level of the substance state. In conclusion the difference in the physical sense of temperatures T} (hydrodynamical theory) and Tt (thermodynamical model) should be emphasized. Finite temperature T{ characterizes change in the substance state similar to the phase transition. Superfluidity setting in at a given temperature can serve as its analog (see ref. [108]). Introduction of the limiting temperature T t corresponds to that of a new world constant. However, some numerical results of the two models (e.g. distribution dN\dpx) are insensitive to such "niceties".

22. Hydrodynamical and Multiperipheral Parton Models

In spite of superficial difference between the classical and quantum approaches to the multiple processes both these directions lead to similar conclusions [54], Solution of this seeming paradox is based on rather profound basis [109]. The main of them are: 1) quasi- one-dimensional motion (limitation p ±), which follows from the initial conditions in the hydrodynamical theory and is postulated in the multiperipheral model and 2) an equilibrium state due to strong interaction. This analogy can be seen in the classical and quantum languages. The hydrodynamical theory interprets the quasi-one-dimensional gas in the configuration space, while the multiperipheral model deals with the equilibrium gas in the rapidity space [110, 111]. The length of the cylinderical volume in this space is Y = In {8jM2), while the transverse dimension is pL ~ 300MeV. Motion of the gas is, therefore, strictly one dimensional. Therefore, if the particles are put in series by increasing energies of the seondary particles E1 1. Now the parton (quark) treatment of the multiperipheral model is accepted [54]. Therefore the black box elements shown in Fig. 13 can be identified with the partons. The thermodynamical model leading to the limiting temperature Tt (and a 0) shows a strong resonance interaction between the elements (partons). The hydrodynamical model corresponds to a moderate interaction between the elements not violating the perfectness of the " p a r t o n " gas (a ~ 1/3). The multiperipheralism is characterized by a specific interaction which makes (for S oo) the partons move strictly one dimensionally (a -> 1). The difference between the multiperipheralism and hydrodynamical theory can be also stated in the quantum terms [109]. The matrix element corresponding to transition (2 -> N) of particles for the statistical-hydrodynamical theory depends on the scalar product (PoPi) {Pq and pi are the four momentum of the primary and i-th secondary particles) and in the multiperipheral model — on the products (pipi-i). However it proves eventually t h a t the effect of the initial momentum P0 (because of one dimensionality) is lost and the difference between the two directions is obliterated.

Literature [2] W. HEISENBERG, Z. Phys. 101, 533 (1936). [2] W . HEISENBERG, Z. P h y s . 126, 569 (1949).

[3] E. FERMI, Progr. Theor. Phys. 5, 570 (1950). [4] I. JA. POMERANCHUK, Dokl. Abad. Nauk SSSR 73, 889 (1951). [5] L. D. LANDAU, IZV. Akad. Nauk SSSR, Ser. Fiz. 17, 51 (1973). [6] H . W . LEWIS, I . R . OPPENHEIMER, S. A . WOUTHUYSEN, P h y s . R e v . 73, 127 (1948).

[7] E. L. FEINBERG, D. S. CHERNAVSKU, Dokl. Akad. Nauk SSSR 81, 796 (1953). [8] D . AMATI, S. FUBINI, A . STANGELLINI, NUOVO C i m e n t o 26, 896 (1962).

[9] K. A. TER-MARTIROSIAN, Zh. eksper. teor. Fiz. 44, 341 (1963).

[20] S. Z. BELENKII, W . M. MAKSIMENKO, A. I . NIKISHOV, I . L . ROZENTAL, U s p e c h i fiz. N a u k 62, 1 (1957).

[22] E. L. FEINBERG, Uspechi fiz. Nauk 104, 539 (1971). [22] L. D. LANDAU, E. M. LIFSHITS, „Mekhanika sploshnixh' sred" Gostekhizdat 1953. [23] S. Z. BELENKII, L . D . LANDAU, U s p e c h i fiz. N a u k 6 6 , 309 (1955). 14

) Indeed, in [222] the viscosity coefficient is constant and is as low as required.

Hydrodynamical Theory of High-Energy Particles Interaction

353

[14] C. Iso, K. MORI, N. NAMIKI, Proc. Int. Confer, on Cosmic Rays Moscow, 1959, 1, 220 (1960). [15] E. L. WEINBERG, Trudy FIAN 19, 155 (1965). [16] I . L . ROZENTAL, D . S. CHERNAVSKII, U s p e c h i fiz. N a u k 5 2 , 185 (1954).

[27] I . L . ROZENTAL, Zh. eksper. teor. Fiz. 81, 278 (1956).

[18] G. A. MILECHIN, I. L. ROZENTAL, Zh. eksper. teor. Fiz. 83, 197 (1957). [19] G. A. MILECHIN, Zh. eksper. teor. fiz. 85, 1185 (1958). [20] E. V. SHURYAK, Yadern. Fiz. 16, 395 (1972). [21] Y n . A . TABASOV, Y a d e r n . F i z . 8 6 , 7 7 0 (1977).

[22] Y u . N. OCHELKOV, O. F . PBILUZKII, I . L. ROZENTAL, V. Y. U s o v , „Relativistkaja Kinetika i

Gidrodinamika", M. Atomizdat (1978). [23] V . P . VOLGENSKAYA, L . I . SABICHEVA, IZV. A k a d . N a u k S S S R , S e r . F i z . , 8 0 , 1954 (1968).

[24] E. V. SHURYAK, Yadern. Fiz. 320, 549 (1974). [25] A ) M . I . GOBENSTEIN, V . P . SHELEST, G . M . ZINOVIEV, P h y s . L e t t e r s 6 0 B , 2 8 3 ( 1 9 7 6 ) ; B ) D . MORRISON, P r e p r i n t C E R N , 7 3 — 4 6 (1973).

[26] E. I. DAIBOG, I. L. ROZENTAL, Acta Phys. Hung., Suppl. 29, 3, 267 (1970). [27] E . I . DAIBOG, I . L . ROZENTAL, Y a d e r n . F i z . 14, 2 2 6 (1971). [28] O. CARRUTHERS, MINH DUONG-VAN, P h y s . R e v . D 8, 859 (1973).

[29] A. A. EMELIANOV, "Trudy FIAN" 29, 169 (1966). [30] S. A . GURVITZ, E . I . DAIBOG, I . L . ROZENTAL, Y a d e r n . F i z . 1 4 , 1263 (1971).

[31] S. Z. BELENKH, Dokl. Akad. Nauk SSSR 99, 523 (1954). [32] I . N . SISSAKYAN e t al., " T r u d y F I A N " 5 7 , 164 (1971).

[33] E . V. SHURYAK, Phys. Letters 42 B, 357 (1972).

[34] E. V. SHURYAK, Intern. Symposium on Deep Inelastic and Multiple Processes, Dubna (1973). [35] R. SINGER et al., Phys. Letters 60 B, 385 (1976). [36] F. C. WINKELMANN et al., Phys. Letters 56 B, 101 (1975). [37] N . ANGELOV e t al., Y a d e r n . F i z . 2 5 , 117 (1977).

[3S] G. JANCHO et al., Nuclear P h y s . B 1 2 4 , 1 (1977).

[39] H. KURK et al., Nuclear Phys. B128, 397 (1977). [40] K. BOCKMANN, Proc. Symp. on Mult. Prod. Processes and Incl. Reactions, Serpuchov, 1976, 38 (1977). [41] V. G. GBISHIN, Proc. XVIII Intern, Conf. on High Energy Phys., Tbilisi (1976) A 2 - 6 . [42] E. I. VOLKOV, T. N. KANAEEK, Yadern. Fiz. 26, 1130 (1977). [43] R. FEYNMAN, Phys. Rev. Letters 23, 1415 (1969). [44] E . I . DAIBOG, YU. P . NIKITIN, I . L . POZENTAL, Y a d e r n . F i z . 1 6 , 1315 (1972). [45] P . CARRUTHERS, MINH DUONG-VAN, P h y s . L e t t e r s 4 1 B 5 9 7 (1972). [46] R . HAGEDORN, NUOYO C i m e n t o 5 6 A , 1027 (1968).

[47] R. LISTIENNE, Proe. I l l Int. Conf. Multipart. Reactions, 629 Zakopane (1972). [48] British-Scandinavian Collabor., Phys. Letters 44 B, 527 (1973). [49] Annual Report, CERN (1972). [50] E . I . DAIBOG, YU. P . NIKITIN, I . L. ROZENTAL, IZV. A k a d . N a u k SSSR, Ser. Fiz., 37, 1396

(1973). [52] G. BABBELLINI, M. Bozzo et al., Phys. Letters 39 B, 294 (1972). [52] Pisa-Stony Brook Collabor., Proc. XVI Int. Conf. High Energy Phys., Batavia (1972). [53] F. COOPER, E. SCHONBERG, Phys. Rev. Letters 30, 880 (1973). [54] Yu. P. NIKITIN, I. L. ROZENTAL „Teorija mnojestvennych processov", Atomizdat 1976. [55] Yu. P. NIKITIN, I. L. ROZENTAL, F. I. SERGEEV, Uspechi fiz. Nauk 121, 4 (1977). [56] V. S. MURSIN, L. I. SARICHEVA, „Mnojestvennye processy pri vysokich energijach", Atomizdat (1974). [57] V. S. BARASHENKOV, V. D. TONEEV, „Vzaimodeistvye vysokoenergichnykh chastic i jader s jadrami", Atomizdat (1972). [5S] W. BUSZA, Invited talk X I I Recontre de Moriond March 1977, Haute Savoie-France. [59] W. BUSZA, Review of Experim. Data on hA Collision at High Energies. Proc. VII Int. Coll. on Multipart. Reactions, Tutzing, June (1976). [60] W. BUSZA et al., Proc. XVIII Int. Conf. of High Energy Phys., Tbilisi (1976). [61] P. R. VISHWANATH et al., Phys. Letters B 53, 479 (1979). [62] A. GUBTU et al., Phys. Letters B 50, 391 (1974). [63] M . I . ATANELISHVILI e t a l . , Z h . e k s p e r . t e o r . F i z . 1 8 , 4 9 0 ( 1 9 7 9 ) ; 1 9 , 4 0 5 (1974).

354

E . I . DAIBOG, I . L . ROSENTAL, J U . A. TABASOV

[64] W. BUSZA et al., Phys. Rev. Letters 84, 836 (1975). [65] I. L. ROZENTAL, Uspechi fiz. Nauk 116, 271 (1975).

[66] N. MASUDA, Phys. Rev. D 15, 1314 (1977). [67] B . ANDEBSSON, NUOVO Cimento 3 8 A , 329 (1977). [6S] A. M. BALDIN e t al., Y a d e r n . Fiz. 20, 1201 (1974).

[69] I. W. CRONIN et al., Phys. Rev. D 11, 3105 (1975). [70] V. N. GRIBOV, Zh. eksper. teor. Fiz. 67, 1278 (1969). [71] R. FEYNMAN, The Proton-Hadron Interaction, Ed. Benjamin (1972). [72] Y u . P . NIKIHN, I . L . ROZENTAL, Y a d e r n . Fiz. 24, 665 (1976).

[73] E. V. SHURYAK, Phys. Letters 84B, 509 (1978). [74] F . COOPER, G. FRY, E . SCHONBERG, P h y s . R e v . L e t t e r s 82, 862 (1974). [75] J . BJORKEN, B . L . IOFFE, Uspechi fiz. N a u k 116, 115 (1975).

[76] E. L. FEINBERG, Proc. V Int. Coll. Multipart. Reactions, Leipzig (1974). [77] a) G. HANSON et al., Phys. Rev. Letters 85. 1609 (1975); b) M. I . GORENSTEIN e t al., Y a d e r n . F i z . 26, 537 (1977).

[7S] E. L. FEINBERG, Izv. Akad. Nauk SSSR, Ser. Fiz., 26, 622 (1962). E. L. FEINBERG, IZV. Akad. Nauk SSSR, Ser. Fiz., 84, 1987 (1970). [79] I. N. SISSAKJAN, E. L. FEINBERG, D. S. CHERNAVSKII, Zh. eksper. teor. Fiz. 85, 356 (1968).

[SO] S. I. NiKOLSKn et al., Proc. Int, Conf. on Cosm. Rays Hobart 6, 2259 (1971). [52] A. I . AKHIEZER, S. V. PELETMINSKII, Zh. eksper. teor. Fiz. 38, 1829 (1960). [82] R . BOWERS e t al., P h y s . R e v . D 7, 296 (1973); ibid 2278, ibid 2289.

[53] YA. B. ZELDOVICH, Zh. eksper. teor. Fiz. 41, 1609 (1961). [84] J . COLLINS, M. PERRY, Phys. Rev. Letters 34, 1353 (1975). [55] E. SUHONEN et al., Phys. Rev. Letters 81, 1567 (1973). [56] O. V. Jmov, E. V. SHURYAK, Yadern. Fiz. 21, 54 (1975). [57] B. ANDERSSON et al., Nuclear Phys. B 1 1 2 , 413 (1976). [SS] D. SAXON, R . WOOD, P h y s . R e v . 9 5 , 577 (1954).

[S9] R. W. WILLIAMS, Rev. Mod. Phys. 36, 815 (1964). [90] N . MASUDA, R . WeiNBER, Phys. Letters 70 B , N I (1977).

[91] I. L. ROZENTAL, Yu. A. TARASOV, Proc. XV Int. Conf. on Cosm. Rays, Plovdiv, 7, 515 (1977). [92] K. D. TOLSTOV, Proc. XVIII Intern. Conf. on High Energy Phys., Tbilisi (1976). [93] G. A. MILEHIN, Proc. Int. Conf. on Cosm. Rays Moscow, 1959, 1, 223 (1960). [94] D. I. BLOKHINTSEV, Soviet Physics-JETP 32, 350 (1957).

[95] A. A. TYAPKIN, Physics of Elementary Particles and Atomic Nuclei, Moscow, Atomizdat 8, 544 (1977). [96] R. C. HWA, Phys. Rev. 10 D, 2260 (1974). [97] C. B . CHIÙ, E . C. SUDARSHAN, KUO-RSIAN WANG, P h y s . R e v . 12 D , 902 (1975). [9S] E . M. ILGENFRITZ, J . KRIPFGANZ, H . J . MÒHRING, F o r t s c h r . P h y s . 2 6 , 123 (1977).

[99] R. WOLF, Fortschr. Phys. 24, 529 (1976). [J00] R. HAGEDORN, Suppl. Nuovo Cimento 8, 147 (1965); R. HAGEDORN, J . RANFT, ibid 6, 169 (1968).

[101] H. GROTE, R. HAGEDORN, J . RANFT, Particle Spectra, Ed. CERN Geneve (1970).

[102] N. M. GERASIMOVA, D. S. CHERNAVSKII, Zh. eksper- teor. Fiz- 29, 372 (1955). [103] J . RANFT, G. RANFT, P h y s . L e t t e r s 3 2 B , 207 (1970); J . KRIPFGANZ, G. RANFT, J . RANFT,

Nuclear Phys. B 56, 205 (1973). [104] S. C. FRAUTSCHI, C. J . HAMER, NUOVO C i m e n t o 13 A , 645 (1973).

[105] M. I. GORENSTEIN et al., Zh. eksper. teor. Fiz., Pisma 76, 456 (1973). [106] V. P . SHELEST, G. M. SINOVJEV, V. A. MIRANSKII „Modeli s i l n o v z a i m o d e i s t v u j u s s c h i k h ele-

mentarnykh chastic", chast II, Atomizdat, Moskwa (1976). [107] D. A. KIRZHNITZ, V. YA. FAINBERG, Zh. eksper. teor. Fiz., 18, 590 (1973). [108] A. MANN, R . WEINER, N u o v o C i m e n t o 1 0 A , 625 (1972).

[¿00] I. L. ROZENTAL, IZV. Akad. Nauk SSSR, Ser. Fiz., 38, 939 (1974). [HO] M. BANDER, Phys. Rev. 6 D, 164 (1972). [111] R. C. ARNOLD, Preprint Argonne National Laboratory ANL/HEP 7241 (1972). [112] M. I. GORENSTEIN, Yadern. Fiz. 24, 1222 (1976).

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