STATISTICAL THEORY OF MULTIPLE PARTICLE PRODUCTION*)

S. Z. Belenky, V. M. Maksimenko,
A. I. Nikishov, I. L. Rozental

1. INTRODUCTION

It has recently become clear that in the collision of two nuclear-active particles of high energy, as a rule, several mesons are produced. A consistent interpretation of this phenomenon is still lacking, and it is difficult to expect substantial progress in this direction in the near future, since the quantitative side of meson theory is unsatisfactory, despite numerous attempts to improve it. This circumstance is connected with the fact that the modern formalism describes multiple processes in the form of a series in powers of the quantities \(g^2/\hbar c\). For mesons \(g^2/\hbar c \sim 1\), and this series either converges poorly or does not converge at all. Finally, multiple production is a many-particle problem. However, even in the case of comparatively simple and well-known laws of interaction of two particles, as, for example, in the case of electromagnetic interaction, this problem can be solved only approximately.

As a result of such a combination of circumstances, all the proposed theories of multiple processes \(^{1-5}\) are based on more or less arbitrary assumptions, the validity of which is tested by comparison with experimental data.

In the present article we shall dwell in detail on the statistical theory of multiple processes proposed by Fermi. The choice of this theory is due to the fact that, unlike others \(^{1,2,4}\), it can also be applied at comparatively moderate energies of \(1\text{–}5\) Bev, where there are numerous experimental data**).

The basis of this theory is the assumption that statistical equilibrium is established in a small volume between the various degrees of freedom of the system. The value of this idea lies in its generality. Indeed, since multiple production of mesons is a process in which many particles participate, it is natural to think that some aspects of the phenomenon have a statistical character. This is also supported by the fact that other theories of multiple processes, despite their quite different approaches, contain to a greater or lesser extent elements of statistics \(^{5,6}\).

*) The present work was written after the death of S. Z. Belenky and is a development of a report presented by the authors at the Conference on High-Energy Particle Physics (Moscow, 1956).

**) When the number of secondary particles \(n \gg 1\), the purely statistical approach often becomes unsuitable. In this case the expansion of the particles begins to play an essential role; to describe it L. D. Landau proposed using hydrodynamics. The hydrodynamical theory gives a qualitatively good description of experimental data on interactions at very high energies.

The assertion concerning the generality of the statistical approach can be given a simple mathematical form. It is known that the probability \(S_n\) of the transition of a system from state \(A\) to state \(B\), with the formation of \(n\) particles, is equal to*)

\[ S_n = 2\pi |H_{AB}|^2 \rho_{E_0,n}, \tag{1,1} \]

where \(H_{AB}\) is the matrix element corresponding to the transition from state \(A\) to state \(B\), and \(\rho_{E_0,n}\) is the density of levels in the final state, if the total energy of the system is equal to \(E_0\). The function \(\rho_{E_0,n}\) of \(n\) has a maximum at some \(n_{\max}\), and its sharpness increases with increasing \(n\). Therefore one may expect that, if \(H_{AB}\) is a sufficiently smooth function, the influence of the matrix element will be small, and the principal contours of the processes will be determined by the statistical factor.

It is obvious, however, that such an approach, which neglects the role of the matrix element, is approximate. One of the principal tasks of the present article is precisely to clarify the role of the statistical factor, i.e., the degree of approximation of the entire approach.

Let us make several general remarks concerning formula (1,1). The statistical factor \(\rho_{E_0,n}\) must, generally speaking, take into account all conservation laws: those of energy, momentum, angular momentum, electric charge, the difference between the numbers of nucleons and antinucleons, strangeness, isotopic spin. We shall take into account all the laws listed, except the conservation law for angular momentum. Such neglect may be interpreted as follows. Let our system have a classical character; then estimates made by Fermi\(^3\) show that taking account of the conservation law for angular momentum changes the statistical weight very little. From the point of view of quantum mechanics, neglect of this law means that the final state is classified by momenta; naturally, with such a classification we must take into account degeneracy with respect to ordinary spin and not take into account orbital angular momentum, which does not commute with momentum.

Let us now find the expression for the matrix element corresponding to the establishment of statistical equilibrium. To this end we write \(\rho_{E_0,n}\) in the form

\[ \rho_{E_0,n}= \frac{\Omega^{n-1}}{(2\pi^3)^{\,n-1}}\, f_{T,S,\ldots}\, \frac{dQ_n(E_0)}{dE_0}, \tag{1,2} \]

where \(\Omega\) is a normalization volume.

\(Q_n(E_0)\) is the volume occupied by the system in momentum space, with allowance for conservation of energy and momentum, and \(f_{T,S,\ldots}\) are factors taking into account the ordinary spins of the particles, identity, conservation of isotopic spin, etc. The exponent \(n-1\) (and not \(n\)) is a consequence of the law of conservation of momentum, which reduces the number of degrees of freedom by 3. Such a notation is, of course, possible when conservation of orbital angular momentum is neglected.

If the probability of realization of state \(B\) is determined by statistical equilibrium under the condition that the system is enclosed in a volume \(V\), then

\[ S_n= \frac{V^{n-1}}{(2\pi^3)^{\,n-1}}\, f_{T,S,\ldots}\, \frac{dQ_n(E_0)}{dE_0}, \tag{1,3} \]

*) Here and below we use a system of units in which

\[ \hbar = c = M = 1 \]

(\(M\) is the nucleon mass).

Comparing (1,1), (1,2), and (1,3), we obtain:

\[ |H_{AB}|^2=\left|\frac{V}{\Omega}\right|^{n-1}. \tag{1,4} \]

This expression has a simple physical meaning: the square of the matrix element is equal to the probability of finding all particles of the system in the volume \(V\).

2. INTERACTION BETWEEN PARTICLES. CHOICE OF VOLUME

The assumption that statistical equilibrium is established within a short interval of time is, at first glance, internally contradictory. Indeed, on the one hand, this requires a very strong interaction between the particles; on the other hand, formula (1,3) describes the statistics of noninteracting particles. The rigorous resolution of this contradiction lies in the calculation of the matrix element \(H_{AB}\). However, as already mentioned, it is at present impossible to carry this out. Therefore our task is to estimate the effective matrix element in the spirit of formula (1,3). For this purpose let us analyze the physical meaning of the two parameters entering (1,3)—the number of particles \(n\) and the volume \(V\). Our starting point is the assumption that at some instant of time, not necessarily coinciding with the beginning of the collision, there are particles in the volume \(V\) (which is not equal to the initial volume), the distances between which are equal to the radius of interaction; in this case the statistics of an ideal gas may be applied to the description of the system.

In his fundamental work Fermi\(^3\) assumed that the volume \(V\) is a Lorentz-contracted volume with linear dimensions equal to the range of action of nuclear forces \(R \sim \frac{1}{\mu}\) (\(\mu\) is the mass of the \(\pi\)-meson), i.e.

\[ V=V_0 \frac{2}{E_0}, \tag{2,1} \]

where \(V_0=\frac{4\pi}{3}\left(\frac{1}{\mu}\right)^3\) is the effective volume without allowance for Lorentz contraction. Obviously, such a definition is possible if the number of produced particles is close to unity or if the cross section of their interaction is negligibly small in comparison with \(\left(\frac{1}{\mu}\right)^2\). At present there are not sufficiently well-founded data on the cross section \(\sigma_{\pi-\pi}\) for \(\pi-\pi\) interactions. Rather, comparison of the conclusions of the statistical-hydrodynamical conception with experimental data makes it possible to draw some conclusion about the magnitude of this cross section\(^5,{}^7\).

We shall assume, together with Fermi, that the volume \(V\) is determined by relation (2,1). It is necessary, however, to emphasize the ambiguity of such a choice\(^*\).

Let us now turn to the question of the number of particles. The number \(n\) corresponds to the moment at which the interaction ends; however, it is by no means necessary that this number be equal to the observed number. It is possible that, owing to a strong resonant interaction, the particles formed complexes (quasiparticles), whose lifetime is greater than \(R\). The introduction of such complexes in the absence of additional experimental data would be arbitrary (and therefore devoid of significant value). However

\(^*\) If the cross section \(\sigma_{\pi-\pi}\sim (1/\mu)^2\), then allowance for the interaction of secondary particles in the spirit of the statistical conception should\(^7\) be made by changing the magnitude of the volume \(V\).

experiments on the scattering of \(\pi\)-mesons by nucleons prove the existence of a strong maximum for the scattering cross section at a \(\pi\)-meson energy \(\sim 200\) Mev, which indicates a strong resonant interaction of the \(\pi\)-meson and the nucleon,\(^{8}\) which can be described\(^{9-10}\) with the aid of an intermediate, so-called isobaric state, corresponding to a particle with ordinary and isotopic spins equal to \(3/2\). Therefore one may assume\(^{11}\) that in the collision process such quasiparticles are formed with ordinary and isotopic spins equal to \(3/2\)*). The mass of such a quasiparticle was taken, in accordance with\(^{9}\), to be \(1.32\). It is natural, in the spirit of the entire statistical concept, to assume that the probability of their formation is determined by the statistical weight. After the interaction is over, each isobar decays into a \(\pi\)-meson and a nucleon. The possibility of taking account of the resonant interaction with the aid of such a model can be justified from another point of view.\(^{14}\) Let us first consider an example due to L. D. Landau.\(^{14}\) We shall regard the particles in the zeroth approximation as noninteracting, and in the first approximation take into account only the pair interaction. Then the additional term in the sum of states, due to the interaction, can be written in the form\(^{15}\)

\[ Z = Z_1 + Z_2, \tag{2.2} \]

where \(Z_1=\sum e^{-\varepsilon_n/kT}\) (\(\varepsilon_n\) is the energy value in state \(n\)) is the sum of states over the discrete spectrum, while \(Z_2\) is the sum of states over the continuous spectrum with the pair interaction taken into account. This term is equal to

\[ Z_2 = \frac{1}{\pi}\sum_l \int_0^\infty (2l+1)\frac{d\delta_l}{dp} e^{-p^2/mkT}\,dp, \tag{2.3} \]

where \(\delta_l\) is the scattering phase with angular momentum \(l\), and \(p\) is the momentum. If the scattering has a resonant (in the limit, delta-function) character at some value of \(l\), then

\[ Z_2 = \frac{2l+1}{\pi} e^{-p_l^2/mkT}. \tag{2.4} \]

Thus, a term corresponding as it were to a state with energy \(p_l^2/m\) is added to the statistical sum.

In the general case the pair interaction can be taken into account analogously. It can be shown that the statistical weight \(S_{n+1}\) of a system of \(n\) \(\pi\)-mesons and one nucleon with the pair interaction taken into account is equal to

\[ S_n(M)=S_{0,n}(M)+\frac{2l+1}{\pi}\int S_{0,n-1}(M+\mu+E_1)\frac{\partial\delta_l,T}{\partial E_1}\,dE_1, \tag{2.5} \]

where \(T\) is the isotopic spin of the system, \(S_0\) is the statistical weight without allowance for the interaction, and \(S_{0,n-1}(M+\mu+E_1)\) is the statistical weight for a system of \(n-1\) mesons and one “particle” with mass \(M+\mu+E_1\) (\(E_1\) is the kinetic energy of the nucleon and meson in the system of their center of gravity). If the scattering phase has a resonant character for some \(l\), then the second term of formula (2.5) is equal to:

\[ (2l+1)S_{0,n-1}(M_1), \]

where \(M_1=M+\mu+E_1\).

*) The first assumption about the significant role of isobaric states in processes connected with the production of mesons was put forward by Belinfante\(^{12}\) and Peaslee.\(^{13}\) As for the very idea of isobaric states, it was put forward comparatively long ago in connection with the well-known difficulties of meson theory.\(^{58,59}\)

Thus, this term corresponds precisely to the isobar with mass \(M_1\), i.e., the resonance interaction can be described with the aid of quasiparticles possessing mass \(M_1\).

3. Isotopic spin, identity, distribution over charge states

The factor \(f_{T,S,\ldots}\) in the formula for the statistical weight can be divided into three factors: \(g_S\), \(\dfrac{1}{m!n!}\), and \(p_n(T)\)

\[ f_{T,S,\ldots}=g_S\frac{1}{m!n!}\,p_n(T). \tag{3,1} \]

The factor \(g_S\) takes into account the spin of the particles, the factor \(\dfrac{1}{m!n!}\) takes into account the identity of the available \(n\) \(\pi\)-mesons and \(m\) nucleons, and finally the function \(p_n(T)\) arises from the requirement of conservation of isotopic spin in a collision. Fermi \(^{3,16}\) did not take into account the ordinary spin of the particles. It is easy to see that taking it into account changes nothing, if one confines oneself to energies at which the probability of antinucleon production is negligibly small. Indeed, the spin of \(\pi\)-mesons is zero, while the number of nucleons in the collision does not change. Consequently, the multiplier \(g_S=m(2S+1)\), where \(m\) is the number of nucleons and \(S\) their spin, is common to all reactions and may be omitted. A different situation obtains when nucleons may be in isobaric states. In this case we shall assume that \(g_S=1\) for processes without participation of isobars and \(g_S=2k\) for processes with \(k\) isobars (since isobaric states with \(S=3/2\) are, in ordinary spin, expressed twice as many as nucleons).

Correspondingly, the multiplier \(\dfrac{1}{m!n!}\) is equal to \(\dfrac{1}{2!n!}\) for a process whose products are two nucleons (or two isobars) and \(n\) \(\pi\)-mesons, and is equal to \(\dfrac{1}{1!n!}\) for a process as a result of which there appear \(n\) mesons, one nucleon, and one isobar.

We proceed to the consideration of the isotopic statistical factor \(p_n(T)\). By definition, the multiplier \(p_n(T)\) is equal to the number of different isotopic states with given \(T\) and \(T_3\), in which the products of the reaction under consideration may be found. The values \(p_n(T)\) may be found by the rules of vector addition of angular momenta. For a system consisting of \(n\) mesons and \(m\) nucleons, we have \(^{17}\)

\[ p_n(T)\equiv p_{m,n}(T)=(2T+1)\sum_i \frac{(-1)^{i+n}}{2i+m+1} \binom{n}{i} \binom{2i+m+1}{i+\frac{1}{2}m-T}. \tag{3,2} \]

If among the \(m\) nucleons one is in an isobaric state, then \(p_n(T)\) is expressed through the function \(p_{m,n}(T)\) as follows:

\[ p_{N',\,m-1,\,n}(T)=p_{m,\,n+1}(T)-p_{m,\,n}(T). \tag{3,3} \]

The index \(N'\) in \(p_{N',\,m-1,\,n}(T)\) indicates the presence of one isobar. Accordingly, for two nucleons in isobaric states we obtain the following expression for \(p_n(T)\), through the functions defined by formulas (3,2) and (3,3):

\[ p_{N'N',\,m-2,\,n}(T)=p_{m,\,n+2}(T)-2p_{N',\,m-1,\,n}(T)-p_{m,\,n}(T). \tag{3,4} \]

The numerical values of the function \(p_{m,n}(T)\) for various values of \(T\), \(m\), and \(n\) are given in \({}^{17}\).

In some cases the system of colliding particles does not have a definite isotopic spin. Thus, in the collision of a \(\pi^{-}\)-meson with a proton, the system has, with probability \(a=1/3\), isotopic spin \(T=3/2\), and with probability \(b=2/3\), isotopic spin \(T=1/2\).* In this case it is necessary to calculate the probabilities for both values of \(T\) separately, to normalize the probabilities obtained for \(T=T_1\) and \(T=T_2\) in the same way (for example, so that their sum for each \(T\) is equal to 100), and, finally, to add the probabilities obtained with weights \(a\) and \(b\) for the cases \(T=T_1\) and \(T=T_2\), respectively. The result, however, changes little if one simply assumes that in such cases the factor taking account of conservation of isotopic spin has the form \(p_n=ap_n(T_1)+bp_n(T_2)\),

\[ f_{T,S,\ldots}=g_S\frac{1}{m!\,n!}\{ap_n(T_1)+bp_n(T_2)\}. \]

Table I gives the values of \(f_{T,S,\ldots}\) for a number of cases. Let us now turn to the question of the distribution of reaction products over charge states—

Table I

Values of the coefficients \(f_{T,S,\ldots}\), taking account of isotopic spin, identity, and ordinary spin

\(N\)—nucleon, \(N'\)—isobar, \(NN'3\)—a state with one nucleon, one isobar, and three \(\pi\)-mesons, etc.

states. The probabilities of charge states for various processes are given in Tables II—VI. The probabilities are normalized so that their sum for the given process is equal to 1. As before, we characterize the process by the number of isobars among the reaction products and by the number of mesons. For brevity, only the probabilities of charge states taking account of the decay of isobars are given. As

* In the general case the quantities \(a\) and \(b\) are determined by the isotopic spins of the colliding particles.

Table II

Charge distributions of the products of \(\pi^- — p\)-collisions.
For \(\pi^+ — p\)-collisions one must replace the signs of the meson charges by the opposite ones and replace \(p \to n,\ n \to p\).

Table III

Charge distributions of the products of \(\pi^- — n\)-collisions.
For \(\pi^+ — p\)-collisions one must replace the signs of the meson charges by the opposite ones and replace \(n \to p,\ p \to n\).

Table IV

Charge distribution of the products of n–p collisions

Table V

Charge distribution of the products of \(p-p\) collisions

Table VI

Charge distribution of the products of annihilation of nucleons (\(n\) or \(\tilde p\)) with antinucleons (\(\tilde n\) or \(p\)). The distribution for \(p\tilde n\) is obtained from the distribution for \(n\tilde p\) by replacing the sign of the meson charge by the opposite\(*\)

\(*\) In a recently published work \(^{23}\), an analytic expression was obtained for the probabilities of states with one neutral meson.

usually, we assume that the decay probabilities of the isobars are such that

\[ N^{++}\to (p0), \]

\[ N^{+}\to {2\over 3}(p0)+{1\over 3}(n+), \]

\[ N^{0}\to {1\over 3}(p-)+{2\over 3}(n0), \]

\[ N^{-}\to (n-) \]

(\(N^{++}\)-state with \(t_3=3/2\), \(N^{+}\)-state with \(t_3=1/2\)), i.e., the isobar \(N^{+}\) decays with probability \(2/3\) with emission of a \(\pi^0\)-meson and with probability \(1/3\) with emission of a \(\pi^+\)-meson, etc.

For each process the tables also give the values \(p_n(T)\). If the probabilities of the charge states are multiplied by the corresponding \(p_n(T)\), we obtain the isotopic statistical weights of the charge states. Usually, however, there is no need to use the latter. Calculations of charge distributions according to the statistical theory for various cases have been carried out in papers \(^{16,18-22}\). Details concerning the calculations are given in Appendix I. Possible deviations from the charge distributions determined by the statistical theory are discussed in Appendix II.

4. PHASE VOLUME

Obtaining a general formula for the magnitude of the volume \(Q_n(E_0)\) in the \(3(n-1)\)-dimensional phase space occupied by a system of \(n\) particles, taking into account the laws of conservation of energy and momentum, presents considerable difficulties. The expressions known in statistical physics are completely unsuitable, since in deriving them the condition \(\ln n \gg 1\) is assumed (Stirling’s formula is used), which is not at all satisfied in the cases of interest to us. Moreover, usually only the law of conservation of energy is taken into account, while momentum conservation is neglected*.

In his fundamental paper \(^{3}\), Fermi calculated the quantity \(Q_n(E_0)\), taking exactly into account only conservation of energy (momentum conservation was approximately taken into account in his formulas by replacing \(n\) by \((n-1)\)) for two limiting cases: the nonrelativistic case, when the total energy \(E\) of a particle is represented by the relation \(E=M^2+p^2/2M\) (\(p\) is the momentum of the particle, \(M\) its mass), and the ultrarelativistic case, when \(E=p\).

He also gave expressions for \(Q_n(E_0)\) for the “mixed” case, when some particles may be regarded as nonrelativistic and the others as ultrarelativistic; in this case the law of momentum conservation is taken into account approximately only for the nonrelativistic (heavy) particles. It is clear, however, that, on the one hand, the range of applicability of these formulas is very limited; on the other hand, the magnitude of the error due to the approximate nature of the calculations is not defined. Therefore, for quantitative comparisons of theoretical and experimental results, exact calculations of \(W_n(E_0)\) are necessary.

In a number of works, refinements were made to the formulas obtained by Fermi. Thus, Yang and Christian \(^{24}\) and Block \(^{25}\) calculated the statistical weights numerically with the aid of electronic machines. However, the difficulties encountered along this path are so great that calculations could be carried out for a number of particles not exceeding \(4\)--\(5\). In papers \(^{26,27}\) an expression is given for the statistical weight of three particles, under the condition that one of the particles has a prescribed momentum \(p\). General expressions for arbitrary \(n\) for the two limiting—ultrarelativistic and nonrelativistic—cases, with exact allowance for both the law of conservation of energy and the law of conservation

* Apparently, both approximations are equivalent in the sense that for \(n\gg 1\) the influence of the law of conservation of momentum can be neglected.

of momentum, are obtained comparatively simply[^28-29]. Finally, Milburn in [18] gives formulas for \(W_n(E_0)\), when the mass of one particle is taken into account exactly, while the remaining \(n\) particles \((n \le 3)\) are considered ultrarelativistic.

We shall describe[^30] a general method for computing \(Q_n(E_0)\); the general expressions obtained with its aid include, as special cases, the formulas mentioned above. The phase volume \(Q_n(E_0)\) of a system of particles for which the conditions

\[ \sum_{i=1}^{n}\sqrt{p_i^2+\mu_i^2}\leq E_0 \quad \text{and} \quad \sum_{i=1}^{n}\mathbf p_i=0 \; *) \]

are fulfilled (we consider the \(C\)-system; for a generalization of the formula to the case of nonzero total momentum, see below in § 5) is, by definition, the following quantity:

\[ Q_n(E_0)= \underbrace{\int\ldots\int}_{3n} \delta\!\left(\sum_{i=1}^{n}\mathbf p_i\right) U\!\left(E_0-\sum_{i=1}^{n}\sqrt{p_i^2+\mu_i^2}\right) \prod_{i=1}^{n}d\mathbf p_i, \tag{4,1} \]

where \(\delta(z)\) is the delta-function, and \(U(z)\) is the Heaviside step function, equal to unity for positive argument and to zero for negative argument. Using the identity \(dU(z)/dz=\delta(z)\), we obtain the following formula:

\[ W_n(E_0)=\frac{dQ_n(E_0)}{dE_0}= \underbrace{\int\ldots\int}_{3n} \delta\!\left(\sum_{i=1}^{n}\mathbf p_i\right) \delta\!\left(E_0-\sum_{i=1}^{n}\sqrt{p_i^2+\mu_i^2}\right) \prod_{i=1}^{n}d\mathbf p_i, \tag{4,2} \]

which can be transformed to the following form (see Appendix III):

\[ \begin{aligned} W_n(E_0)=& \left(\frac{\pi}{2}\right)^{n-1}E_0^{3n-4} \frac{\pi^n\left(\prod_{i=1}^{n}\nu_i^2\right)}{8\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{e^{i(x+y)}(x+y)^n(x-y)^2}{(xy)^n} \\ &\times \prod_{i=1}^{n}H_2^{(2)}\!\left(2\nu_i\sqrt{xy}\right)\,dx\,dy, \end{aligned} \tag{4,3} \]

where \(H_2^{(2)}(z)\) is the Hankel function of the second kind, \(\nu_i=\dfrac{\mu_i}{E_0}\). It proves possible to obtain the value of the integral appearing in (4,3) in the form of a power series in the quantities \(\nu_i^2\) and \(\nu_i^2\ln \dfrac{1}{\nu_i}\). The general method for computing the terms of this series is set forth in Appendix III; here we shall give the values of several first terms, writing all expressions for \(W_n(E_0)\) in the form

\[ W_n(E_0)=\left(\frac{\pi}{2}\right)^{n-1}E_0^{3n-4}\sum_{i=1}^{\infty}P_i(\nu). \tag{4,4} \]

*) \(\mathbf p_i,\ \mu_i\) are the momentum and mass of the \(i\)-th particle.

a) All particles have the same mass \(\mu\) \((\nu_i=\nu)\):

\[ \begin{gathered} P_1(\nu)=D_n^{(0)},\qquad P_2(\nu)=C_n^1D_n^{(1)}\nu^2,\\ P_3(\nu)=C_n^1D_n^{(2)}\nu^4\ln\frac1\nu,\\ P_4(\nu)=\left\{\left(\frac34 C_n^1+C_n^2\right)D_n^{(2)}+\frac12 C_n^1F_n^{(2)}\right\}\nu^4,\\ P_5(\nu)=\left(-\frac13 C_n^1+2C_n^2\right)D_n^{(3)}\nu^6\ln\frac1\nu,\\ P_6(\nu)=\left\{\left(-\frac{17}{36}C_n^1+\frac32 C_n^2+C_n^3\right)D_n^{(3)} +\left(-\frac16 C_n^1+C_n^2\right)F_n^{(3)}\right\}\nu^6,\\ P_7(\nu)=C_n^2D_n^{(4)}\nu^8\ln^2\frac1\nu,\\ P_8(\nu)=\left\{\left(\frac1{24}C_n^1+\frac1{12}C_n^2+3C_n^3\right)D_n^{(4)} +C_n^2F_n^{(4)}\right\}\nu^8\ln\frac1\nu, \end{gathered} \tag{4,5} \]

and so on\(^*\).

Here the following notation has been introduced:

\[ \begin{gathered} D_n^{(A)}=(-1)^A \frac{(4n-2A-4)!}{(3n-2A-4)!(2n-A-1)!(2n-A-2)!},\\ F_n^{(A)}=D_n^{(A)} \left\{2(4n-2A-4)!\,\alpha(4n-2A-3)-2(3n-2A-4)!\times\right.\\ \left.\times\alpha(3n-2A-3)-(2n-A-1)!\,\alpha(2n-A)-(2n-A-2)!\times \alpha(2n-A-1)\right\}, \end{gathered} \tag{4,6} \]

\[ \alpha(z)= \begin{cases} 0, & z=1,\\[4pt] \displaystyle \sum_{m=1}^{z-1}\frac1m, & z=2,3,4,\ldots,\\[8pt] \displaystyle (-1)^z+1\,|z|!, & z=0,-1,-2,\ldots . \end{cases} \]

\(C_n^k\) are binomial coefficients.

b) All particles have different masses\(^ {**}\):

\[ \begin{gathered} P_2(\nu_1,\ldots,\nu_n)=D_n^{(1)}\sum_{i=1}^n \nu_i^2,\qquad P_3(\nu_1,\ldots,\nu_n)=D_n^{(2)}\sum_{i=1}^n \nu_i^4\ln\frac1{\nu_i},\\ P_4(\nu_1,\ldots,\nu_n)= D_n^{(2)}\left\{\frac34\sum_{i=1}^n\nu_i^4+\frac14\sum_{i,j=1}^{n\,\prime}\nu_i^2\nu_j^2\right\} +\frac12 F_n^{(2)}\sum_{i=1}^n\nu_i^4,\\ P_5(\nu_1,\ldots,\nu_n)=\\ = D_n^{(3)}\left\{-\frac13\sum_{i=1}^n\nu_i^6\ln\frac1{\nu_i} +\frac12\sum_{i,j=1}^{n\,\prime}\nu_i^2\nu_j^2 \left(\nu_i^2\ln\frac1{\nu_i}+\nu_j^2\ln\frac1{\nu_j}\right)\right\}. \end{gathered} \tag{4,7} \]

\[ \text{\(^*\) We computed 12 terms of the series; however, we do not give them all because of the cumbersome nature of the corresponding expressions.} \]

\[ \text{\(^ {**}\) The value of \(P_1(\nu_1,\ldots,\nu_n)\) does not depend on \(\nu_i\) and is the same in all cases.} \]

A prime on the summation sign means that, in summing, the term with \(i=j\) is omitted.

The generalization of the remaining terms of the series to this case presents no difficulty; however, owing to the cumbersome nature of the expressions, we shall not write them out here. For \(\nu \to 0\) the particles should be regarded as ultrarelativistic; in this case formula (4.4) has the form

\[ W_n(E_0)_{\mathrm{u.r.}} = \left(\frac{\pi}{2}\right)^{n-1} D_n^{(0)} E_0^{3n-4} = \left(\frac{\pi}{2}\right)^{n-1} \frac{(4n-4)!}{(3n-4)!(2n-1)!(2n-2)!}\, E_0^{3n-4}. \tag{4,8} \]

Derivation of formula (4.8) in \(^{28,29}\).

Expression (4.4) gives the exact solution of the problem; however, for large \(\nu\) this series converges poorly, and even the use of 10–15 of its terms does not make it possible to compute \(W_n(E_0)\) for values of the kinetic energy

\[ T_0 \sim \sum_{i=1}^{n} \mu_i . \]

Therefore it is expedient to obtain more convenient formulas for this region of energies; in particular, one may use a “mixed” formula, but one must take exact account in it of the law of conservation of momentum, since, apparently, neglect of it is a source of considerable errors. Such calculations were also carried out in work \(^{30}\).

Expression (4.2) for the “mixed” case has the following form (\(m\) “heavy” particles of mass \(M\), \(n\) relativistic particles):

\[ W_{m,n}(E_0)= \left(\frac{1}{2\pi}\right)^4 \int_{-\infty}^{\infty} e^{i(E_0-mM)\tau_1}\,d\tau_1 \int_{-\infty}^{\infty} \iiint d\tau_2\,d\tau_3\,d\tau_4 \times \]

\[ {}\times \left[ \int_{-\infty}^{\infty} e^{-i[p^2/2M+(\boldsymbol{\rho}\boldsymbol{\tau})]}\,dp \right]^m \left[ \int_{-\infty}^{\infty} e^{i[\tau_1 p+(\boldsymbol{\rho})]}\,dp \right]^n . \tag{4,9} \]

By means of cumbersome calculations we obtain the following expression:

\[ W_{m,n}(E_0)= \frac{ 2\pi^{\,n+\frac{3}{2}(m-1)} (mM)^{\,3n+\frac{3}{2}m-4} M^{\frac{3}{2}m} }{ (2n-1)! }\,S . \tag{4,10} \]

\(S\) is defined differently, depending on the parity of the number \(m\)
\[ \left( \alpha=\frac{2(E_0-mM)}{mM} \right): \]

a) \(m\) even

\[ S= \frac{1}{\sqrt{\pi}} \left\{ \operatorname{arctg}\sqrt{\alpha} \sum_{k=1}^{3n+\frac{3}{2}m-3} p_{3n+\frac{3}{2}m-3-k}\, \alpha^{\,3n+\frac{3}{2}m-3-k} - \right. \]

\[ \left. {}-\sqrt{\alpha} \sum_{l=0}^{3n+\frac{3}{2}m-5} q_l \alpha^l \right\}, \tag{4,11} \]

where

\[ \left. \begin{gathered} p_{3n+\frac{3}{2}m-3-k} = \frac{(-1)^{k-1}(2k-1)!\left(2n-\frac{3}{2}-k\right)!} {[(k-1)!]^2\left(3n+\frac{3m}{2}-3-k\right)!\,2^{2k-3}\sqrt{\pi}}, \\[6pt] q_l=\sum_{i=0}^{l}(-1)^i\frac{p_{l-i}}{2i+1}, \end{gathered} \right\} \tag{4,12} \]

b) \(m\) is odd

\[ S=(1+\alpha)^{\,n+\frac{3m}{2}-3} \sum_{k=0}^{2n-1} a_k(1+\alpha)^k + \sum_{r=0}^{n+\frac{3m}{2}-3} b_r\alpha^r, \tag{4,13} \]

where

\[ \left. \begin{gathered} a_k=(-1)^{k-1} C_{2n-1}^{k} \frac{\left(k-\frac{3}{2}\right)!} {\left(k+\frac{3m}{2}-3\right)!}, \\[8pt] b_r=\sum_{k=0}^{2n-1}(-1)^k C_{2n-1}^{k} \frac{\left(k-\frac{3}{2}\right)!} {\left(k+\frac{3m}{2}-r-3\right)!\,r!}. \end{gathered} \right\} \tag{4,14} \]

In the case where the “heavy” particles have different masses, in the expressions given above the substitution

\[ mM \to \sum_{i=1}^{m} M_i;\qquad M^{\frac{3m}{2}} \to \prod_{i=1}^{m} M_i^{3/2}. \tag{4,15} \]

must be made.

If (4,10) is expanded in a series in powers of \(\alpha\) and only the first term is retained, then we obtain the expression given by Fermi\(^3\) (and also in the review\(^ {14}\)):

\[ W_{m,n}(E_0)_{\text{Fermi}} = \frac{ 2\pi^{\,n+\frac{3}{2}(m-1)} (mM)^{3n+\frac{3}{2}m-4} M^{\frac{3}{2}} m_a^{\,3n+\frac{3m}{2}-\frac{5}{2}} } { \left(3n+\frac{3m}{2}-\frac{5}{2}\right)! }. \tag{4,16} \]

This expression differs substantially from (4,10) (Fig. 1). For \(n=0\), formula (4,10) becomes

\[ W_m(E_0)= \frac{ (2M\pi)^{\frac{3}{2}(m-1)} }{ m^{3/2}\left(\frac{3m}{2}-\frac{5}{2}\right)! } T^{\frac{3m}{2}-\frac{5}{2}} \tag{4,17} \]

—the formula for the case when all particles are nonrelativistic. For a geometric interpretation of the last formula, it is useful to give its simple derivation.

The equations \(\sum_{i=1}^{m} p_i^2/2M = E_0 - mM\) and \(\sum_{i=1}^{m} p_i = 0\) determine a certain second-order surface in a \(3(m-1)\)-dimensional phase space; it is easy to show that this surface is an ellipsoid whose

Fig. 1. The ratio
\[ R_{2,n}(E_0)=\frac{W_{2,n}(E_0)}{W_{2,n}(E)_{\mathrm{u.r.}}} \]
of statistical weights to their expression obtained when rest masses are neglected, for the following systems: two nucleons and four mesons \((R_{24})\), and two nucleons and five mesons \((R_{25})\). The statistical weights were calculated by formulas (4.4) and (4.18). The same ratios, calculated by Fermi’s formula (4.16), are shown by the dotted lines. Along the abscissa is plotted the energy in the \(C\)-system in units of \(Mc^2\).

three semiaxes are equal to
\[ a_1=\sqrt{2M(E_0-mM)/m}, \]
and the remaining \(3m-6\) semiaxes are equal to
\[ a_2=\sqrt{2M(E_0-mM)}. \]
Using the formula for the volume of a \(3(m-1)\)-dimensional ellipsoid
\[ V_{3(m-1)}=\frac{\pi^{3(m-1)/2}}{[3(m-1)/2]!}\prod_{i=1}^{3(m-1)} a_i, \]
we obtain \(Q_m(E_0)\); after differentiation with respect to \(E_0\) we arrive at (4.17).

Fig. 2. The ratio \(R_{2,n}(E_0)\) for systems consisting of two nucleons and \(n\) mesons. The initial portions of the curves were calculated by formula (4.18); the portions at high energies by formula (4.4). The interpolation regions between the two formulas are marked by dotted lines.

We next give, graphically, the dependence of \(W_{m,n}(E_0)\) for the most interesting case of two nucleons and \(n\) \(\pi\)-mesons (Fig. 2).

![Figure 3 graph: dependence of \(\bar n(E_0)\) on \(\gamma_c\) and \(E_k\), with curves labeled 1 and 2.]

Fig. 3. Dependence of \(\bar n(E_0)\), the most probable number of mesons produced in a \(p\)—\(p\) interaction, for the chosen variant: curve 1—the total number of \(\pi\)-mesons, curve 2—the number of charged \(\pi\)-mesons. Plotted along the abscissa are: \(E_k\)—the kinetic energy of the nucleon moving in the \(L\)-system and \(\gamma_c = 1/\sqrt{1 - V^2}\), where \(V\) is the velocity of the \(C\)-system relative to the \(L\)-system.

![Figure 4 graph: relative probability \(S(\%)\) versus number \(n\) of produced \(\pi\)-mesons.]

Fig. 4. Relative probability, in %, of the production of \(n\) \(\pi\)-mesons in the collision of two nucleons with \(E_k = 10\) Bev. The calculation was carried out for the chosen variant.

The initial portions of the curves were computed from formulas (4.10), which for this special case have the form \((M=1)\):

\[ \left. \begin{aligned} W_{2,1}(E_0) &= 8\pi^2\left\{(\alpha^2+6\alpha+5)\operatorname{arctg}\sqrt{\alpha} -\left(\frac{13}{3}\alpha+5\right)\sqrt{\alpha}\right\},\\ W_{2,2}(E_0) &= \frac{4\pi^3}{3}\left\{\left(\frac{1}{10}\alpha^5-\frac{1}{2}\alpha^4+5\alpha^3+35\alpha^2+\frac{105}{2}\alpha+\frac{231}{10}\right)\operatorname{arctg}\sqrt{\alpha}\right.\\ &\qquad\left. -\left(-\frac{1}{10}\alpha^4+\frac{56}{105}\alpha^3+\frac{553}{25}\alpha^2+\frac{224}{5}\alpha+\frac{231}{10}\right)\sqrt{\alpha}\right\},\\ W_{2,3}(E_0) &= \frac{\pi^4}{30}\left\{\left(\frac{1}{24}\alpha^8-\frac{1}{7}\alpha^7+\frac{1}{2}\alpha^6-\frac{7}{3}\alpha^5+\frac{105}{4}\alpha^4+231\alpha^3+\right.\right.\\ &\qquad\left. +\frac{1001}{2}\alpha^2+429\alpha+\frac{7293}{56}\right)\operatorname{arctg}\sqrt{\alpha} +\frac{1}{56}\left(\frac{7}{3}\alpha^7-\frac{79}{7}\alpha^6+\right.\\ &\qquad\left.\left. +\frac{467}{15}\alpha^5-\frac{2129}{15}\alpha^4-\frac{772409}{105}\alpha^3-\frac{107393}{5}\alpha^2-21593\alpha-7293\right)\sqrt{\alpha}\right\}, \end{aligned} \right\} \tag{4.18} \]

and so on.

The remaining part of the curves was computed by means of the series (4.4). For convenience, the graph plots along the ordinate axis the quantities

\[ W_{2,n}(E_0)/W_{2,n}(E_0)_{\mathrm{u.r.}}=R_{2,n}(E_0), \]

where \(W_{2,n}(E_0)_{\mathrm{u.r.}}\) corresponds to ultrarelativistic particles (formula (4.8)).

In conclusion, we give the dependence \(\bar n(E_0)\) of the most probable number of produced \(\pi\)-mesons on energy (Fig. 3) and the relative probabilities of production of different numbers of mesons in the collision of nucleons with energy \(10\) Bev (Fig. 4).

5. DISTRIBUTION OF SECONDARY PARTICLES BY MOMENTA

Up to now we have been interested in the distribution with respect to the number of particles. In this section we shall consider the momentum distribution of secondary particles. Let us find the probability \(\omega_n(E_0,p)\,dp\) that, in a system consisting of \(n\) particles, one of them with mass \(\mu\) carries momentum (in the \(C\)-system) lying in the interval \((p,p+dp)\)*). Using the fact that the volume occupied by this particle in phase space is equal to \(4\pi p^2dp\), and that the momentum of the whole system is equal to zero, it is easy to obtain that

\[ \omega_n(E_0,p)\,dp = 4\pi p^2 W_{n-1}\left(E_0-\sqrt{p^2+\mu^2},p\right)\,dp. \tag{5.1} \]

In order to compute \(\omega_n(E_0,p)\), it is necessary to generalize the expressions obtained earlier (see § 4 and Appendix III) for the quantities \(W_n(E)\), in which the total momentum of the system was assumed equal to zero, to the case when \(p\ne0\). Let \(W_n(E,p)\) be the required quantity. Let us consider the general case \(^{31**}\). The generalization of formula (4.4), which represents the exact solution of the problem when \(p=0\), is carried out as follows:

a) the quantity \(\nu_i=\mu_i/E\) must be replaced by

\[ \frac{\mu_i}{\sqrt{E^2-p^2}}, \]

* We recall that \(E_0\) is the total energy of the colliding particles in the \(C\)-system.

**) The calculation of the quantity \(W_n(E,p)\) for the ultrarelativistic case was carried out in \(^{29}\).

b) the quantities \(D_n^{(A)}\), \(F_n^{(A)}\) are computed by the formulas \((p/E=k)\):

\[ D_n^{(A)}=\frac{(1-k^2)^{n-2}}{2k}(-1)^A \sum_{r=0}^{n} C_n^r \left\{ \frac{(1+k)^{\,n-r+1}(1-k)^r}{(2n-r-A-1)!(n+r-A-2)!} -\right. \]
\[ \left. -\frac{(1+k)^{\,n-r}(1-k)^{r+1}}{(2n-r-A-2)!(n+r-A-1)!} \right\}, \tag{5,2} \]

\[ F_n^{(A)}=\frac{(1-k^2)^{n-2}}{2k}(-1)^A \sum_{r=0}^{n} C_n^r \left\{(1+k)^{\,n-r}(1-k)^{r+1}\times\right. \]
\[ \times\left[ \frac{a(n+r-A)}{(2n-r-A-2)!} +\frac{a(2n-r-A-1)}{(n+r-A-1)!} \right] -(1+k)^{\,n-r+1}(1-k)^r\times \]
\[ \left. \times\left[ \frac{a(n+r-A-1)}{(2n-r-A-1)!} +\frac{a(2n-r-A)}{(n+r-A-2)!} \right]\right\}. \tag{5,3} \]

In the expressions obtained, when computing the momentum distribution it is necessary, according to (5,1), to put

\[ E=E_0-\sqrt{p^2+\mu^2}. \]

We next consider the mixed case (\(n\) ultrarelativistic particles, \(m\) nonrelativistic ones); here the formula is

\[ W_{n,m}(E,p)= \frac{2^{3n}\pi^n(2\pi)^3(m-1)!/2\,M^{3m/2}}{(mM)^{3/2}} \left(T-\frac{p^2}{2mM}\right)^{3n+\frac{3m}{2}-\frac{5}{2}} \times \]
\[ \times \sum_{k=0}^{\infty} C_{k+2n-1}^{k}(-1)^k(2k+1)!! \left(\frac{T-\frac{p^2}{2mM}}{mM}\right)^k \times \]
\[ \times \left\{ \sum_{r=0}^{k} \frac{C_k^r}{(2r+1)!!} \frac{\left(\frac{2p^2}{2mMT-p^2}\right)^r} {\left(p+3n+\frac{3m}{2}-\frac{5}{2}-r\right)!} \right\}. \tag{5,4} \]

It is easy to see that all the expressions given, for \(p=0\), pass into the corresponding formulas of § 4. Formula (5,5) has a simple form for \(n=0\) (all particles are considered nonrelativistic)\(^{29}\):

\[ W_m(E)= \frac{(2\pi M)^3(m-1)!/2}{m^{3/2}\left(\frac{3m}{2}-\frac{5}{2}\right)!} \left(T-\frac{p^2}{2mM}\right)^{\frac{3m}{2}-\frac{5}{2}}. \tag{5,5} \]

We shall also give the exact formula for a system consisting of three particles:

\[ \omega_3(E_0,p)\,dp =p^2\left(E_0-\sqrt{p^2+\mu^2}\right)^2 \left\{(1-k^2)^2(3-k^2)+\right. \]
\[ \left. +2k^2(1-k^2)(\nu_1^2+\nu_2^2) -(3+k^2)(\nu_1^2+\nu_2^2) \right\}\times \]
\[ \times \frac{\sqrt{[1-(\nu_1+\nu_2)^2][1-(\nu_1-\nu_2)^2]}} {(1-k^2)^2}\,dp, \tag{5,6} \]

where \(\nu_1\) and \(\nu_2\) characterize the masses of the other two particles.

For illustration of the formulas obtained, Fig. 5 gives the momentum distributions of \(\pi\)-mesons in the annihilation of a nucleon–antinucleon pair at rest *).

Fig. 5. Normalized momentum distributions of \(\pi\)-mesons in the annihilation of antinucleons at rest.

6. COMPARISON OF THEORETICAL AND EXPERIMENTAL DATA ON \((N—N)\)- AND \((\pi—N)\)-COLLISIONS AT ENERGIES \(1—5\) Bev **)

Let us first consider nucleon–nucleon collisions. For brevity, we shall call the calculation without taking “isobars” into account, i.e. without taking into account the meson–nucleon interaction in the state \((3/2,\ 3/2)\), variant \(A\) of the statistical theory, and the calculation taking this interaction into account—variant \(B\). The statistical weights of the various processes are given in Table VII.

Table VII

Statistical weights of processes in \(n—p\)- and \(p—p\)-collisions.
In variant \(A\), only the processes \(NN\), \(NN1\), \(NN2\), etc. are possible.

) The annihilation of antinucleons is considered in detail in § 7.
*) An analysis of collisions of nucleons with energy \(\gg 10\) Bev with heavy nuclei from the point of view of statistical theory was carried out by I. A. Ivanovskaya and D. S. Chernavskii \(^{56}\).

Neutron collisions with protons at an average energy of the incident particles \(E_k=1.7\) Bev were experimentally studied in work \(^{24}\). There were found 185 three-prong reactions and not a single 5-prong one that could be interpreted as \((pp+---)\). On this basis the authors believe that the probability of the production of three mesons may be neglected. Among the three-prong reactions, cases were singled out of the formation of one meson \((pp-)\) and two mesons \((pn+-)\) and \((pp-0)\). The ratio

\[ \frac{(pn+-)+(pp-0)}{(pp-)} \]

varies, within the experimental errors, from 3 to 4. The calculation according to variant \(A\) gives for this ratio the value 0.2, whereas according to variant \(B\) we obtain \(3.5^{11}\). In addition, with the aid of Tables IV and VII it is not difficult to see that the probability of formation of 5-prong stars for both variants, in accordance with experiment, is small.

If one tries, by changing the parameter \(R\), which determines the dimensions of the interaction region, to bring into agreement with experiment the ratio

\[ \frac{(pn+-)+(pp-0)}{(pp-)} \]

for variant \(A\), this will lead to a strong increase in the probability of production of three mesons, which again will sharply contradict experiment.

As for the ratio between \((pn+-)\) and \((pp-0)\), its experimental value is determined less accurately. The experimental value of the ratio

\[ \frac{(pn+-)}{(pp-0)} \]

is equal to 3; however, the value 7 and even 11 do not contradict experiment. According to variant \(A\) this ratio is equal to \(2.5—3\) and does not depend on the energy. For variant \(B\) the indicated ratio is equal to 2.5 at an energy below the \(NN'\) threshold (1 Bev); at energies \(1—1.4\) Bev it is equal to \(3—4\), and, finally, at energy \(E_k=1.7\) Bev

\[ \frac{(pn+-)}{(pp-0)}=6. \]

With a further increase in energy this ratio again decreases.

The neutron beam used in \(^{24}\) contained particles with energies \(1—2.3\) Bev. A considerable part of the neutron spectrum is situated near the threshold for the process \(N'N'\) or even below it. This circumstance does not permit a calculation of the distribution of the reaction products by momenta for variant \(B\), since near the threshold the theory is not applicable. With regard to the angular distribution of the reaction products it is necessary to make the following remark. In the experimental study of neutron collisions with protons, and also of \(\pi\)-meson collisions with nucleons, the following feature of the interactions was found: the incident particle has a tendency to preserve the direction of its motion after the collision. This feature cannot be explained on the basis of the assumption that statistical equilibrium is reached in the system (even if one takes into account the law of conservation of momentum). Thus, the angular distribution of the reaction products turns out to depend most essentially on the form of the matrix element and cannot be explained by statistical theory.

Let us consider \(p—p\)-collisions at 2.7 Bev. We compare the ratio of the number of cases of 4-prong reactions to the total number of inelastic collisions (the products of 4-prong reactions are 4 charged particles and any number of neutral ones). The experimental \(^{32,33}\) ratio

\[ \frac{\sigma_{(4\text{-prong})}}{\sigma_{\text{inel}}}=0.16. \]

The values calculated according to variants \(A\) and \(B\) are 0.11 and 0.19, respectively. The experimental ratio of the numbers of cases of production of one, two, and three mesons is equal to \(36:48:16\). The ratios calculated according to variants \(A\) and \(B\) are respectively \(68:28:4\) and \(36:52:12\).

For energy 5.3 Bev, a comparison of the calculated and experimental distribution of the products of \(p—p\)-collisions is given in Table VIII \(^{22}\).

Let us next dwell on the ratio

\[ \rho=\frac{\pi^{+}}{\pi^{-}} \]

—the number of particles born in collisions of \(\pi^{+}\)-mesons to the number of \(\pi^{-}\)-mesons. For \(p-p\) collisions at \(2\text{--}3\) Bev and \(p+{}_{4}\mathrm{Be}^{9}\) collisions at \(1\text{--}2.3\) Bev the calculated and experimental values are given in Table IX\({}^{34}\). It is assumed that in \(p+{}_{4}\mathrm{Be}^{9}\) collisions the incident proton interacts with one of the Be nucleons as with a free particle.

Table VIII

Distribution according to the number of beams of shower \(p-p\)-collision products at \(5.3\) Bev

Let us now consider the multiple production of mesons in collisions of \(\pi\)-mesons with nucleons\({}^{20,37}\). In the center-of-inertia system the colliding meson and nucleon move with different velocities. We shall assume that the volume of the interaction region, as also for nucleon–nucleon collisions,

Table IX

Experimental and theoretical values of \(\rho=\dfrac{\pi^{+}}{\pi^{-}}\)
in \(p-p\) and \(p+{}_{4}\mathrm{Be}^{9}\) collisions

is determined by the meson cloud of the nucleon, Lorentz-contracted owing to the motion of the nucleon. The statistical weights of various processes in \(\pi^{-}-p\) collisions for energies \(E_k=1.4\) and \(4.5\) Bev are given in Table X.

Table X

Statistical weights of the processes of \(\pi^{-}-p\) collisions at \(1.4\) and \(4.5\) Bev

*) In work\({}^{35}\), for variant \(A\) at energy \(E_k=2\) Bev, the value \(\rho=1.7\) is given. This is probably simply a misprint. From Tables IV–V it is easy to find that \(\rho=1.7\) only in the case when the probability of birth of one meson may be neglected in comparison with the probability of birth of two mesons (cf. also\({}^{36}\)). In the case under consideration, for variant \(A\), rather the opposite holds.

Using the distribution over charge states given in Table III, it is easy to find the probabilities of the various reactions. For an energy of 1.4 Bev the statistical weight of elastic scattering has a rather considerable value. The calculated value of elastic scattering must be compared with the so-called incoherent elastic scattering \(^{18}\). This scattering is due to the emission from the region of strong interaction, as a result of the reaction, of only one meson. In addition, according to quantum mechanics, any inelastic scattering must be accompanied by elastic diffraction scattering through small angles. In the experimental determination of the magnitude of incoherent scattering it is assumed that the latter appears as scattering through large angles.

Table XI

Distribution of charged products of \(\pi^- - p\) collisions at 1.4 Bev

Table XI gives a comparison with experiment of the distributions of charged products of \(\pi^- - p\)-collisions. In this table \((\pi^- + p)_{\mathrm{el}}\) is the sum of \((p-0)\) and \((p-00)\), while \((\pi^+ + \pi^-)\) is respectively the sum of \((n+-)\) and \((n+-0)\). The comparison with experiment for variant \(A\) was also carried out in Ref. \(^{18}\) *).

It should also be noted that for both variants \(A\) and \(B\) about \(1/5\) of all reactions should give states containing only neutral products \((n0)\), \((n00)\), and \((n000)\).

A comparison with experiment of the distributions of charged products of \(\pi^- - p\)-collisions according to the number of prongs at 4.5 Bev is given in Table XII.

Since only one nucleon participates in meson–nucleon collisions, the difference between the results of the calculation for variants \(A\) and \(B\) turns out to be smaller than in the case of nucleon–nucleon collisions.

Table XII

Distribution of charged products of \(\pi^- - p\)-collisions by number of prongs at an energy of 4.5 Bev

Let us now turn to the consideration of the energy distribution of the reaction products. We shall assume that, in the system in which the isobar is at rest, its decay is isotropic. We neglect the width of the isobar level.

In the decay of an isobar in flight, the momentum spectrum of the mesons is given by the expression \(^{38}\)

\[ N(p)\,dp=\frac{p\,dp}{2\sqrt{r^2+p^2 p_c^2 V}}, \qquad \gamma\left|p_c-E_cV\right|\leq p\leq \gamma\left|p_c+E_cV\right|. \]

\[ \text{*) The probabilities of the charge states given in }^{18}\text{ differ somewhat} \]
from the corresponding values in Table XI. This is due to the fact that the radius of the interaction region (in contrast to that adopted by us) is chosen in \(^{18}\) to be

\[ R=\frac{2\pi\hbar}{Mc}=1.32\cdot 10^{-13}\ \text{cm}. \]

Here \(p_c = 0.23\ \mathrm{Bev}/c\) is the momentum of the meson in the decay of a resting isobar, \(V\) is the velocity of the isobar, \(\gamma = (1 - V^2)^{-1/2}\), \(E_c = (\mu^2 + p_c^2)^{1/2}\). In the case where the isobars have a velocity spectrum \(f(P_b)\) (for which it is necessary that among the reaction products, in addition to the isobar, there be at least two more particles), the spectrum of mesons formed in their decay is determined by the expression

\[ N(p)\,dp = \]

\[ = \frac{p\,dp}{2\sqrt{p^2+\mu^2}\,p_c}\,M_b \int_{P_{b\min}(p)}^{P_{b\max}} \frac{f(P_b)}{P_b}\,dP_b, \]

where

\[ P_{b\min}(p)=\frac{V_{b\min}M_b}{\sqrt{1-V_{b\min}^2}}, \]

\(V_b\) is the modulus of the root of the equation

\[ p=\frac{\left|p_c^2 \pm E_c V_b\right|}{\sqrt{1-V_b^2}} . \]

The index “\(b\)” refers to the isobar; \(M_b\), \(P_b\) are the mass and momentum of the isobar, etc. The integral depends on \(p\) through the function of the lower limit.

Fig. 6. Momentum distribution of mesons from various processes: 1 — spectrum of free mesons in the process \(NN\pi\); 2 — spectrum of mesons from isobar decay in the process \(NN\pi\); 3 — spectrum of mesons in the process \(NN'\); 4 — spectrum of mesons in the process \(NN'\); 5 — spectrum of free mesons in the process \(NN'\pi\); 6 — spectrum of mesons in the process \(NN\pi\).

Let us consider the energy distribution of mesons produced in collisions \(p + {}_{4}\mathrm{Be}^{9}\) at an incident-proton energy of \(2.3\ \mathrm{Bev}\). According to \(^{36,39}\), the meson spectrum in the reaction under consideration is approximately

Fig. 7. Energy distribution of \(\pi^{+}\)-mesons in \(p + {}_{4}\mathrm{Be}^{9}\) collisions at \(2.3\ \mathrm{Bev}\): 1 — experiment; 2 — calculation according to variant \(B\) of the statistical theory; 3 — calculation according to variant \(A\) of the statistical theory.

such as it should be if the incident proton interacts with one of the nucleons of the Be nucleus as with a free one. In Fig. 6 are shown

spectra of the various processes. The calculated and experimental spectra of \(\pi^+\)- and \(\pi^-\)-mesons are shown in Figs. 7 and 8. The experimental spectrum was obtained by transformation from the \(L\)-system, where the observation was made at an angle of \(32^\circ\) to the beam direction, to the \(C\)-system; in the latter system this spectrum is obtained in the angular interval \(106^\circ\)—\(74^\circ\). Of course, it would be more correct to carry out

Fig. 8. Energy distribution of \(\pi^-\)-mesons in \(p^- + \mathrm{Be}^9\) collisions at \(2.3\ \mathrm{BeV}\): 1—experiment; 2—calculation according to variant \(B\) of the statistical theory.

the comparison with the experimental spectrum in the \(C\)-system, integrated over the solid angle; however, such a spectrum is not yet known. As is seen from Fig. 7, the calculation according to variant \(B\) leads to satisfactory agreement with experiment.

We shall also give the momentum distributions of \(\pi\)-mesons and nucleons from the reactions
\[ \pi^- + p \to \begin{cases} p+\pi^-+\pi^0,\\ n+\pi^+ + \pi^- \end{cases} \]
at \(1.4\ \mathrm{BeV}\). The corresponding experimental

Fig. 9. Meson spectrum in the \(N2\pi\) process, averaged over the energy of the incident mesons. Dashed line—experiment; solid line—calculation according to the statistical theory (variant \(A\)).

data were obtained in work \(^{40}\). The momentum distributions of mesons, averaged over the energy of the incident particles, for variants \(A\) and \(B\) are shown in Figs. 9 and 10 (solid lines). The dashed lines in these figures show the experimental distributions. Similar results for the momentum distribution of protons are shown in Figs. 11—12.

Fig. 10. Meson distribution by momentum from reactions \((p-0)\) and \((n-+)\). Dashed line—experiment, solid line—calculation according to the statistical theory (variant B).

Fig. 11. Momentum distribution of nucleons in the process \(N2\pi\). Dashed line—experiment, solid line—calculation according to the statistical theory (variant A).

Fig. 12. Momentum distribution of nucleons in reactions \((p-0)\) and \((n+-)\). Dashed line—experiment, solid line—calculation according to the statistical theory (variant B).

Let us finally consider the production of \(K\)-mesons and hyperons within the framework of statistical theory. We shall assume that the ordinary spin of the \(K\)-mesons is zero, and that of the hyperons is \(1/2\). We take into account only those reactions in which strangeness is conserved (see, for example, \(^{41}\)). For such reactions the statistical weights are calculated in the usual way. Consider \(\pi^- - p\) collisions at \(1.4\) and \(4.5\) Bev. The statistical weights of processes without the production of unstable particles are given in Table X. Relative to them we give the statistical weights of processes with the production of unstable particles. At an energy of \(1.4\) Bev only the following reactions are energetically possible:

\[ \pi^- + p \to \Lambda^0 + \vartheta^0, \]

\[ \pi^- + p \to \Sigma + \vartheta . \]

Their statistical weights are 10 and 13, respectively. The ratio of the cross section for the production of unstable particles to the cross section for the production of \(\pi\)-mesons thus turns out to be \(0.23\) for variant \(B\) and \(0.4\) for variant \(A\). The experimental value \(\frac{\sigma(K)}{\sigma(\pi)}\) is much smaller, \(\sim 1/30\), and, apparently, does not change appreciably when the energy of the incident \(\pi\)-meson is increased to \(4.5\) Bev \(^{40,42}\). The ratio \(\frac{\sigma(K)}{\sigma(\pi)}\) calculated from statistical theory at \(4.5\) Bev is even somewhat larger than at an energy of \(1.4\) Bev. The statistical weights of the processes \((\Lambda^0 \vartheta \pi)\), \((\Sigma \vartheta \pi)\), \((\Xi \vartheta \vartheta)\), and \((N \vartheta \vartheta)\) are, respectively, 7, 12, 3, and 6. Some contribution to the cross section for the production of unstable particles should be made by reactions with 4 particles \((\Lambda^0 \vartheta \pi \pi)\), \((\Sigma \vartheta \pi \pi)\), etc. Thus \(\frac{\sigma(K)}{\sigma(\pi)}\) turns out to be \(\sim 0.3\).

If it is assumed that the volume of the region of strong interaction is smaller for unstable particles than for \(\pi\)-mesons, then \(\frac{\sigma(K)}{\sigma(\pi)}\) will decrease correspondingly. More detailed experimental data are needed, however, in order to determine whether such a volume can be chosen for unstable particles so as to obtain agreement with experiment at different energies.

7. APPLICATION OF STATISTICAL THEORY TO PROCESSES ASSOCIATED WITH THE ANNIHILATION OF ANTINUCLEONS

Recently it has been established \(^{43-45}\) that, in the annihilation of a nucleon–antinucleon pair, a star with several prongs is formed. It is known that the applicability of perturbation theory to processes associated with annihilation is highly doubtful; therefore it is expedient in this case as well to use statistical theory. A justification for such an approach is the release of a comparatively large (\(\geqslant 1.8\) Bev) energy in processes of this kind. As was mentioned earlier, already at such an energy statistical theory correctly predicts a number of features of particle collisions (see § 6). In the first calculations \(^{46,47}\) of particle multiplicities in annihilation, based on the application of statistical theory, approximate expressions for the statistical weights were used*).

*) In paper \(^{47}\) an error was made in calculating the statistical weights. In the system of units chosen by us, formula (1.3) has the form

\[ S_n(E_0) \simeq \left(5 \frac{V}{V_0}\right)^{n-1} f_{\tau,S,\ldots}\frac{dQ_n(E_0)}{dE_0} \]

(where \(V_0\) is defined by (2.1)), whereas in \(^{47}\) the factor 5 is absent.

Here we shall give the results of calculations by the exact formulas (4,5), taking, as before, \(R=1.4\cdot 10^{-13}\ \mathrm{cm}^{31}\). We shall consider further two variants: in the first of them (variant I) we shall assume that, in annihilation, only \(\pi\)-mesons are formed, and in the second (variant II), that both \(\pi\)-mesons and \(K\)-particles are formed.

Variant I:

\[ \widetilde N+N\to n\pi . \]

Table XIII

The calculated distribution of stars according to the number \(n\) of \(\pi\)-mesons in annihilation at rest is given in Table XIII, and in Table XIV—the distribution of stars according to the number of charged particles \(n_{\mathrm{ch}}\), obtained on the basis of work \(^{21}\) (see Appendix I).

Table XIV

From the tables it is easy to obtain that the mean number of all particles per star is equal to 3.4, and the mean number of charged particles is 2.2 for \(\widetilde{\mathrm p}-\mathrm p\) (or \(\widetilde{\mathrm n}-\mathrm n\)) collisions and 2.4 for \(\widetilde{\mathrm p}-\mathrm n\) (or \(\widetilde{\mathrm n}-\mathrm p\)) collisions.

If annihilations occur in flight, the increase of the mean multiplicity as a function of the energy of the incident antinucleon is described with good accuracy by the formula

\[ \overline n \simeq 2.87E^{1/4}, \tag{7,1} \]

where \(E\) is the total energy in the \(L\)-system of the nucleon–antinucleon pair.

Variant II: along with the process \(N+\widetilde N\to n\pi\), the process

\[ \widetilde N+N\to 2K+n\pi \]

is possible. The distribution of the stars formed in annihilation over possible final states is given in Table XV.

Table XV

From this table it is seen that in approximately \(65\%\) of the cases a pair of \(K\)-mesons must be present in a star; moreover, the number of \(K\)-mesons amounts on average to about \(30\%\) of the number of all particles.

The theoretical calculations presented can be compared with the experimental data obtained by É. Serge et al.\(^{46}\). In the photoemulsions, approximately 30 annihilation stars were recorded with an average number of \(\pi\)-mesons \(\sim 5 \pm 1\) per star. It was not possible to obtain the exact number of \(K\)-particles formed in this process; only one pair of \(K\)-particles was definitely recorded, and two pairs were identified as \(K\)-particles only tentatively.

Consequently, the probability of the production of \(K\)-mesons in annihilation is only a few percent of the probability of the formation of \(\pi\)-mesons. This relation, as already mentioned, also holds as a result of particle collisions (see § 6). Therefore it should be concluded that the calculation in variant B gives a significantly overestimated number of \(K\)-mesons. This circumstance also indicates that, within the framework of the statistical theory, \(K\)-mesons must be treated differently from \(\pi\)-mesons.

APPENDIX 1

PROBABILITIES OF CHARGE STATES IN THE STATISTICAL THEORY

We shall briefly set forth a method for finding the charge distributions of the reaction products. Let
\[ [T, T_3]_{t_3^{(1)},\, t_3^{(2)},\, \ldots,\, t_3^{(n)}} \]
denote the eigenfunction of the isotopic-spin operator \(\hat T^2\) and its projection \(\hat T_3\), with eigenvalues \(T(T+1)\) and \(T_3\), respectively; \(t_3^{(i)}\) is the isotopic variable of the \(i\)-th particle of the system. Consider a system consisting of two subsystems with isotopic spins \(T'\) and \(T''\). Then, according to the rules for vector addition of angular momenta, the isotopic spin of the whole system can take the values
\[ T=T'+T'',\ T'+T''-1,\ \ldots,\ |T'-T''|. \]
Suppose that we know the probabilities of the charge states of the subsystems for arbitrary \((T', t'_3)\), \((T'', t''_3)\), and find the charge distribution of the system in the state \(T, T_3\). Since the charge distribution of a subsystem is determined by the value of the projection of the isotopic spin of the subsystem (for a fixed value of the isotopic spin itself), the problem reduces to determining the probabilities with which, in the state \(T, T_3\), the values of the projections \(t'_3, t''_3\) occur. These probabilities are determined by the squares of the corresponding Clebsch–Gordan coefficients in the expansion
\[ [T,T_3]=\sum_{T'_3}\left(T',T'',T'_3T''_3\mid T,T'',T_3\right)[T',T'_3][T'',T''_3]. \tag{I,1} \]

Formulas for calculating the Clebsch–Gordan coefficients
\[ (T',T'',T'_3,T''_3\mid T',T'',T,T_3) \]
are given in monographs\(^{48-49}\); numerical values for some cases can be found in \(^{13}\).

In the simplest case each subsystem consists of only one particle. Let these particles be \(\pi\)-mesons. We find the charge distribution of such a system in the state \(T=2,\ T_3=0\). In the particular case under consideration, expression (I,1) has the form
\[ [2,0]_{12}=\sqrt{\frac{1}{6}}\,[1,-1]_1[1,1]_2+ \sqrt{\frac{2}{3}}\,[1,0]_1[1,0]_2+ \sqrt{\frac{1}{6}}\,[1,1]_1[1,-1]_2. \tag{I,2} \]

Here
\[ [1,-1]_1\equiv[1,-1]_{t_3^{(i)}} \]
assumes the values \(1,0,-1\); \([1,-1]_{t_3^{(i)}}\) differs from zero only when
\[ t_3^{(i)}=-1. \]

From (I,2) it follows that, in the state under consideration, \(T=2,\ T_3=0\), \(\pi^{+}\)- and \(\pi^{-}\)-mesons occur with probability \(1/3\) (they correspond to the first and last terms on the right-hand side in (I,2)) and with probability \(2/3\) two \(\pi^0\)-mesons occur.

Let us denote this charge distribution as follows:

\[ (2,0)=\frac{2}{3}(00)+\frac{1}{3}(+-). \]

In the same way one finds the probabilities of charge distributions for other \(T\) and \(T_3\). As a result, the following charge distributions are obtained for a system of two mesons:

\[ \begin{array}{ll} (0,0)=\dfrac{2}{3}(+-)+\dfrac{1}{3}(0,0), & (2,2)=(++) \\[6pt] (1,1)=(+0), & (2,1)=(+0) \\[6pt] (1,0)=(+-), & (2,0)=\dfrac{1}{3}(+-)+\dfrac{2}{3}(0,0) \\[6pt] (1,-1)=(-0), & (2,-1)=(-0) \\[6pt] & (2,-2)=(--) \end{array} \tag{I,3} \]

Next one may consider a system consisting of two mesons (subsystem I) and one nucleon (subsystem II). Let us find the charge distribution in the state \(T=\frac{3}{2},\ T_3=-\frac{3}{2}\). Since the isotopic spin of the nucleon is equal to \(\frac{1}{2}\), the entire system can have \(T=\frac{3}{2}\), provided that the subsystem of two mesons has isotopic spin 1 or 2.

In the first case the isotopic function of the system is

\[ \left[\frac{3}{2},-\frac{3}{2}\right]_{123} = \left[\frac{1}{2},-\frac{1}{2}\right]_1[1,-1]_{23}, \]

and the corresponding charge distribution is

\[ \left(\frac{3}{2},-\frac{3}{2}\right)=(n-0), \]

while in the second case we correspondingly obtain

\[ \left[\frac{3}{2},-\frac{3}{2}\right]_{123} = \sqrt{\frac{4}{5}} \left[\frac{1}{2},\frac{1}{2}\right]_1[2,-2]_{23} - \sqrt{\frac{1}{5}} \left[\frac{1}{2},-\frac{1}{2}\right]_1[2,-1]_{23}, \]

\[ \left(\frac{3}{2},-\frac{3}{2}\right) = \frac{4}{5}(p--) + \frac{1}{5}(n-0). \]

Within the framework of the statistical theory these two distributions are equiprobable. The desired distribution is, therefore, the arithmetic mean

\[ \left(\frac{3}{2},-\frac{3}{2}\right) = \frac{6}{10}(n-0)+\frac{4}{10}(p--). \]

Analogous expressions are not difficult to write also for other values of \(T\) and \(T_3\). Continuing this process for systems with a larger number of particles, one can obtain the charge distributions for all processes of interest to us.

APPENDIX II

CHARGE DISTRIBUTION AND ISOTOPIC INVARIANCE

Up to now we have assumed that the various isotopic states are equally probable. If this assumption is not used, then the charge distribution will not be completely determined by specifying the isotopic spin of the system and the numbers of particles of different kinds in it. In particular, a dependence will appear on the behavior of the isotopic function under permutation of identical particles (cf. \(50\text{–}51\)).

Using first only isotopic invariance, one can trace what changes in the charge distribution may be expected if the assumption of equal probability of the admissible isotopic states in some cases proves to be incorrect (cf. also \(^{53}\)).

Consider a system consisting of two subsystems:

1) a subsystem of \(m\) nucleons (among which there may be nucleons in arbitrary states), and
2) a subsystem of \(n\) mesons. Restricting ourselves to the cases \(m \leq 2,\ n \leq 5\), we shall find the charge distributions of the subsystems; after this it is not difficult to obtain the charge distribution of the whole system, analogously to the way this was done in Appendix I. It is obvious that the charge distribution of a system containing no more than two particles is completely determined by specifying \(T\) and \(T_3\). Let us consider in detail\(^{50}\) the case \(n=3,\ T=1\) and \(T_3=1\).

Since a subsystem consisting of two mesons may have isotopic spin 0, 1, and 2, there are three isotopic functions of a system of three mesons describing states with \(T=1\). These functions may be written in the following form:

\[ [1,1]_{123}=\sqrt{\frac{1}{3}}\,[1,1]_1\{[1,1]_2[1,-1]_3-[1,0]_2[1,0]_3+[1,-1]_2[1,1]_3\}, \]

\[ [1,1]'_{123}=-\sqrt{\frac{1}{4}}\,[1,0]_1\{[1,1]_2[1,0]_3-[1,0]_2[1,1]_3\} \]
\[ +\sqrt{\frac{1}{4}}\,[1,1]_1\{[1,1]_2[1,-1]_3-[1,-1]_2[1,1]_3\}, \]

\[ [1,1]''_{123}=\sqrt{\frac{1}{10}}\,[1,1]_1 \left\{ \sqrt{\frac{1}{6}}\,[1,-1]_2[1,1]_3+ \sqrt{\frac{2}{3}}\,[1,0]_2[1,0]_3+ \right. \]
\[ \left. +\sqrt{\frac{1}{6}}\,[1,1]_2[1,-1]_3 \right\} -\sqrt{\frac{1}{10}}\,[1,0]_1 \left\{ \sqrt{\frac{1}{2}}\,[1,1]_2[1,0]_3+ \sqrt{\frac{1}{2}}\,[1,0]_2[1,1]_3 \right\} \]
\[ +\sqrt{\frac{3}{5}}\,[1,-1]_1[1,1]_2[1,1]_3. \]

Upon permutation of the isotopic variables the functions written above go over into linear combinations of themselves. The representation of the permutation group realized by these functions is reducible. The basis functions of the irreducible representations turn out to be the following:

\[ \begin{aligned} e_0&=c'\{[1,1]_1(x_2x_3)+[1,1]_2(x_1x_3)+[1,1]_3(x_1x_2)\},\\ e_1&=c\{2[1,1]_1(x_2x_3)-[1,1]_2(x_1x_3)-[1,1]_3(x_1x_2)\},\\ e_2&=c\{2[1,1]_2(x_1x_3)-[1,1]_1(x_2x_3)-[1,1]_3(x_1x_2)\}, \end{aligned} \tag{II,1} \]

where

\[ (x_ix_j)=\sqrt{\frac{1}{3}}\{[1,1]_i[1,-1]_j-[1,0]_i[1,0]_j+[1,-1]_i[1,1]_j\}, \]

\(c'\), \(c\) are normalization coefficients.

The function \(e_0\) goes over into itself under any permutation of coordinates, i.e. transforms according to the one-dimensional representation of the permutation group; the two functions \(e_1\) and \(e_2\) transform according to a two-dimensional representation, to which the Young scheme \(\yng(2,1)\) \(^{4*}\) corresponds. In particular, \(e_2\) is obtained from \(e_1\) by permuting the variables of the 1st and 2nd mesons.

The complete wave function must be symmetric with respect to the simultaneous permutation of any pair of meson coordinates (isotopic and spatial simultaneously). Let \(f_0(1,2,3)\) be a symmetric function of the spatial coordinates of the mesons, and let \(f_1(1,2,3)\) and \(f_2(1,2,3)\) be functions of the spatial coordinates forming a basis of the two-dimensional representation \(\yng(2,1)\) of the permutation group. Then, using the functions (II,1), one may compose two complete wave functions normalized to 1:

\[ F_1=f_0e_0, \tag{II,2} \]

\[ F_2=\sum_{i,k}^{2}a_{ik}f_ie_k=\sum_{k=1}^{2}a_ke_k. \tag{II,3} \]

STATISTICAL THEORY OF MULTIPLE PRODUCTION

The coefficients \(a_{ik}\) are chosen so that the function (II,3) transforms into itself under interchange of any pair of meson coordinates; the \(a_{ik}\) are thereby determined uniquely, and no other complete wave functions can be constructed from \(f_0, f_1, f_2\) and the functions (II,1). Indeed, from the standpoint of group theory, the construction of the symmetric complete wave function of a system of mesons consists in finding the basis function of the identity representation of the direct product of two representations, one of which is determined by the functions of the spatial variables of the mesons, the other by the functions of the isotopic variables. But it is known that the direct product of two different irreducible representations does not contain the identity representation, while the direct product of a representation by itself contains the identity representation, and moreover only once.\(^{48}\)

The charge distribution in the state (II,2) is equal to

\[ (1,1)\ \Box\Box\Box = \frac{4}{5}(++-)+\frac{1}{5}(+00), \tag{II,4} \]

and in the state (II,3)

\[ (1,1)\ \Box\Box = \frac{1}{2}(++-)+\frac{1}{2}(+00). \tag{II,5} \]

The charge distribution in the state (II,3) is determined by any of the functions \(e_k\), since \(e_1\) and \(e_2\), and an arbitrary linear combination of them, give one and the same charge distribution.

Every state of a system of three mesons with total spin \(T=1\) can be decomposed into the functions (II,2) and (II,3). Let the squares of the coefficients of the decomposition be equal to \(a\) and \(b\) (\(a+b=1\))\(^*\). Then the charge distribution in the state under consideration is equal to the sum of the charge distributions (II,4) and (II,5), taken with weights \(a\) and \(b\), respectively.

Suppose that the function of the spatial coordinates \(\psi(1,2,3)\) is such that all \(3!\) functions obtained from the original one by all possible permutations of the coordinates are linearly independent. These functions carry the regular representation of the permutation group, which is reducible and contains each irreducible representation as many times as its dimension. In the case under consideration the basis functions of the irreducible representations are as follows: one completely symmetric \(f_0\), two sets of two functions \(f_1, f_2\) and \(f'_1, f'_2\), each set forming the basis of the representation \(\Box\Box\), and, finally, one antisymmetric function. Thus, starting from a coordinate function of general type, one can construct one complete wave function of type (II,2) and two complete wave functions of type (II,3). Counting the states described by these functions as equally probable, we obtain the charge distribution of the statistical theory.

We have examined in detail the state with \(T=1\) of a system of three mesons. Analogous results may also be obtained for other values of \(T\) for systems of 3–5 mesons.

The isotopic functions of states with isotopic spin \(T\), forming the basis of some irreducible representation, again give one and the same charge distribution, just as any linear combination of them does. It is therefore unnecessary to use the explicit form of these functions. The charge distribution of the functions is given in Table XVI. The numbers above the Young diagrams indicate the dimension of the corresponding irreducible representation.

With the aid of Table XVI one can easily obtain the charge distributions of systems consisting of 1–2 nucleons and 3–4 mesons, whose isotopic functions form the basis of some irreducible representation of the permutation group.

A trivial example of deviation from the charge distribution of the statistical theory is provided by processes occurring near threshold. Thus, for the case
\(\pi^- -- n \to N+4\pi\), according to the statistical theory we have

\[ \sigma(p+---):\sigma(p---00):\sigma(n+---0):\sigma(n--000)=6{,}7:7{,}3:16{,}7:4{,}3. \]

If one meson produced in the decay is chosen, the ratio changes only slightly—to \(5:5:20:5\).

\(^*\) It is not difficult to verify (see, for example, in \(^{48}\) § 94) that the integral \(\int F_1F_2\,d\tau\) (the interference term) vanishes.

Table XVI

Charge distributions of the basis functions of irreducible representations of the permutation group. Distributions for functions with negative \(T_3\) are obtained from the corresponding positive ones by changing the sign of the meson charge to the opposite.

Continuation of Table XVI

However, at threshold, when all mesons emerge in an \(s\)-state, only the cross section \(\sigma\ \text{[[unclear: symbol]]}\) is possible. Then the isotopic function of the state is as follows:

\[ \left[\frac{3}{2},-\frac{3}{2}\right] = \sqrt{\frac{4}{5}}\,p[2,-2]\ \text{[[unclear: symbol]]} - \sqrt{\frac{1}{5}}\,n[2,-1]\ \text{[[unclear: symbol]]}. \]

Using the charge distributions in the states \((2,-2)\) and \((2,-1)\), given in Table XVI, we obtain for the state under consideration

\[ \sigma(p+---):\sigma(p---00):\sigma(n+---0):\sigma(n--000)=24:4:4:3. \]

It is also seen from Table XVI that there are no other states with a symmetric meson coordinate function. Thus, near threshold one should have

\[ \frac{\sigma(p+---)}{\sigma(n+---0)}=6, \]

whereas far from threshold, according to the statistical theory, this ratio is equal to \(0.4\).

APPENDIX III

DERIVATION OF THE GENERAL EXPRESSION FOR THE PHASE VOLUME OF A SYSTEM OF \(n\) ARBITRARY PARTICLES

Using the integral representation of the \(\delta\)-function, we write (4.2) in the form:

\[ W_n(E_0)= \left(\frac{1}{2\pi}\right)^4 \int_{-\infty}^{\infty} e^{-iE_0\tau_1}\,d\tau_1 \int_{-\infty}^{\infty}\int\int d\tau_2\,d\tau_3\,d\tau_4 \times \prod_{i=1}^{n} \int_{-\infty}^{\infty}\int\int e^{\,i\left[\tau_1\sqrt{p_k^2+\mu_k^2}+(\tau \mathbf p_k)\right]}\,d\mathbf p_k, \tag{III, 1} \]

where \(\tau=(\tau_2,\tau_3,\tau_4)\). Turning to the singular functions \(\Delta^{(1)}\), \(\Delta\) introduced by Dirac, \({}^{53}\) it can be shown that

\[ \int_{-\infty}^{\infty}\int\int e^{\,i\left[\tau_1\sqrt{p_k^2+\mu_k^2}+(\tau \mathbf p_k)\right]} \,d\mathbf p_k = \frac{(2\pi)^3}{i}\, \frac{d}{d\tau_1} \times \left[ \Delta^{(1)}(\mu_k,\tau_1^2-\tau^2) + i\Delta(\mu_k,\tau_1^2-\tau^2) \right]. \tag{III, 2} \]

Using further the representation of \(\Delta^{(1)}\), \(\Delta\) in terms of Hankel functions (see, for example, \({}^{54}\)) and making the change of variables
\[ \tau_1=\frac{x+y}{E_0},\qquad \tau=\frac{x-y}{E_0}, \]
we obtain (4.3)\(*\).

The path of integration in (4.3) lies on a straight line parallel to the real axis and tending to it from below. The main difficulty of the subsequent calculations lies in the fact that the Hankel function has at zero an essential singularity of logarithmic type. To overcome this difficulty, we use Jordan’s lemma (here this lemma is applicable, since the Hankel function \(H_2^{(2)}(z)\to z^{-1/2}\)) and transform the contour of integration as follows: its beginning is at the point \(i\infty-\delta\) \((\delta\to 0,\ \delta>0)\), then it goes parallel to the imaginary axis, passes around zero from below, and arrives at the point \(i\infty+\delta\). Thus, the integration is carried out along the banks of a cut along the imaginary axis. We next reduce the computation of (4.3) to the computation of a sum of residues. Substituting into (4.3) the expansion for the Hankel function raised to the \(n\)-th power, we choose from the expression obtained—

\(*\) Another derivation of formula (4.3) was proposed earlier in work \({}^{28}\).

terms giving a contribution to the given \(P_i(y)\), where the coefficients will be certain combinations of integrals of two principal types:

\[ I_1(k)=\int_{-\infty}^{\infty}\frac{e^{iz}\,dz}{z^k} \quad \text{and} \quad I_2(k,l)=\int_{-\infty}^{\infty}\frac{e^{iz}\ln^l z\,dz}{z^k}. \tag{III, 3} \]

The first integral, by the residue theorem, is equal to

\[ I_1(k)=2\pi i^k\Gamma(k). \tag{III, 4} \]

We transform the second integral as follows:

\[ I_2(k,l)=\frac{d^l}{dk^l}\int_{-\infty}^{\infty}\frac{e^{iz}\,dz}{z^k} =\frac{d^l}{dk^l}I_1(k) =2\pi\frac{d^l}{dk^l}\left(i^k\Gamma(k)\right). \tag{III, 5} \]

The last expression can be represented with the aid of logarithmic derivatives of gamma-functions, which in turn are expressed through the Riemann zeta-function (see, for example, \(^{55}\)). By means of rather laborious, although simple, algebraic transformations we obtain the expressions for the terms \(P_i(y)\) given in the text (see (4,5) and (4,7)).

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