Multiple Processes

Multiple Processes


the production of a large number of secondary, strongly interacting particles (hadrons) in one event of the collision of particles at high energy. Multiple processes are characteristic of hadron collisions, but in rare cases they are also observed during collisions of other particles if their energy is sufficient to produce several hadrons (for example, in electron collisions in colliding-beam accelerators). In collisions of hadrons with an energy greater than a few giga electron volts (GeV), multiple processes predominate over processes of single production of mesons and elastic scattering of particles.

Multiple processes were first observed in cosmic rays, but their thorough study became possible after the development of high-energy charged-particle accelerators. Certain empirical laws governing multiple processes have been ascertained as a result of studies of the interaction of cosmic-ray particles with an energy of up to 106–107 GeV in a laboratory coordinate system and of accelerator-generated particles with an energy up to the order of 103 GeV (colliding beams).

The lightest hadrons—pions, which account for 70–80 percent of the secondary particles—are produced with the highest probability in multiple processes. Kaons and hyperons (˜ 10–20 percent) and nucleon-antinucleon pairs (of the order of a few percent) also account for a significant share. Many of these particles arise from the decay of resonances.

At high energies, the probability of a collision accompanied by a multiple process (the effective cross section of multiple processes) is almost independent of the energy of the colliding particles (it changes by no more than a few tens of percent upon a change in the collision energy by a factor of 104). The approximate constancy of the multiple-process cross section led to the use of the “black-ball” model to describe the processes of a hadron collision. According to the model, during each convergence of high-energy hadrons to distances less than the range of nuclear forces, an inelastic process of the multiple production of particles takes place; here the elastic scattering is primarily diffractive in nature (the diffraction of the de Broglie waves of the particles by the “black ball”). The model played an important role in the development of the theory of strong interactions (in particular, in establishing Pomeranchuk’s theorem of the equality of the effective interaction cross section of particles and antiparticles at extremely high energies). On the other hand, according to quantum field theory a slow growth of the multiple-process cross section is possible with an increase in energy E but may not take place faster than ln2E (Froissard’s theorem).

The number of particles produced in different collisions of hadrons of a specific energy varies sharply and in some cases is quite large (Figure 1). The average number of secondary particles <n> (average multiplicity) slowly increases with the energy E and is virtually independent of the type of hadrons colliding (Figure 1). Given the existing accuracy of measurement, the dependence of < n > on energy is described equally well by logarithmic functions or power functions (of the type Ev, where v < 1). This makes it difficult to choose among the various theoretical models of multiple processes, which predict different types of this dependence. The average multiplicity is much less than the maximum possible number of secondary particles, which is determined by the condition that the entire energy of the collision in the center-of-inertia system of colliding particles becomes the rest mass of the secondary particles. Thus, up to 70 pions could be produced in a collision of 70-GeV protons (from the Serpukhovo accelerator) with target protons. However, in reality the average multiplicity of charged particles at this energy is five to six particles. This means that only a small part of the collision energy goes to create the rest mass of the secondary particles—that is, the energy is expended primarily on imparting high kinetic energy (great momentum) to most of the generated particles. At the same time, a characteristic empirical principle of multiple processes is the fact that the transverse (to the collision axis) components p1 of the momenta of the secondary particles are usually small. The average value of p1 is approximately 0.3–0.4 GeV/c and is virtually constant in a very broad range of energies. Therefore, the secondary particles emerge in streams that are sharply directed and that taper in the direction of motion of the colliding particles with increasing energy (forward and backward in a center-of-inertia system; in the direction of motion of the incident particle in a laboratory system).

Figure 1. Average number of secondary charged particles navg as a function of the kinetic energy 0 of colliding particles in a center-of-inertia system. Different symbols are used to designate results that pertain to the scattering of π± mesons, K± mesons, and protons by nucleons.

The study of multiple processes is essential for determining the structure of hadrons and constructing a theory of strong interactions. Of particular importance in this regard are the conformities to principle that have been established in the study of a special class of multiple processes, called inclusive processes, in which events that result in the production of a specific particle

c, regardless of what other particles (X) accompany the production of the particle c and in what quantity, are selected from among a large number of multiple processes that take place upon collisions of a and b hadrons. In 1967, A. A. Logunov, who established on the basis of quantum field theory the limiting laws of the growth of the cross section of limiting processes with increased energy (laws that are analogous to Froissard’s theorem), pointed out the importance of studying inclusive processes. A unique scaling law in the microcosm—called scale invariance, or scaling—has been detected by experimental study of inclusive processes in the Serpukhovo accelerator (1968) and by comparison of the data with the results of experiments conducted at lower energies. Scale invariance consists in the fact that at different collision energies the probability of production of an “inclusive” c particle with a specified value of longitudinal momentum PL (the projection of momentum on the direction of motion of the colliding particles) is a universal function of the variable x = PL/Pmax, where pmax is the maximum possible value (at the given energy) of the longitudinal momentum of the c particle (Figure 2). Thus, the longitudinal momenta of secondary particles increase in proportion to the collision energy. Indications of the existence of this type of dependence were previously obtained in the study of cosmic rays. This dependence stemmed from the fact that the energy spectrum of the second component of cosmic rays almost exactly duplicates the form of the energy spectrum of the primary component (G. T. Zatsepin and co-workers). Scale invariance has profound physical import. In 1969, R. Feynman proposed an explanation of scale invariance, based on model representations of the composite structure of hadrons. (The American physicist K. Wilson pointed out the possibility of this principle in 1963.)

Figure 2. Graph illustrating the scale invariance in the inclusive process p 4- p + p → π + X (where p is a proton, π is a negative pion, and X is the aggregate of the other hadrons produced in the reaction). Dependence of the quantity (2/π)xdσ/dx, which is proportional to the differential cross section for the production of a π — meson dσ/dx, on x = pL/pmax is plotted. Experimental data for different collision energies fit, to the accuracy of the measurement errors, into a universal dependence on x. Different symbols designate data that pertain to different collision energies (momenta) in a laboratory system; the points at 1,500, 1,100, 500, and 24 GeV/c were obtained in experiments on colliding beams, accelerator at CERN, and those at 70 GeV/c were obtained in a Soviet-French experiment at Serpukhovo.

Experimental data have shown that scale invariance is observed upon collisions not only of elementary particles but also of atomic nuclei at relativistic energies.

Because of the lack of a complete and consistent theory of strong interactions, various theoretical models are used to explain the empirical regularities that are observed in multiple processes. It is assumed in statistical hydrodynamic models— developed in the works of W. Heisenberg, E. Fermi, L. D. Landau (1949–53), and others—that during the brief duration of a collision the statistical equilibrium among the particles formed as a result of the collision has time to become established for strongly interacting particles. This makes it possible to calculate many characteristics of multiple processes, in particular the average multiplicity, which should increase with energy according to the exponential law Ev with an exponent v < 1 (in the Fermi-Landau theory, v = 1/4). It is assumed in another class of models (the Italian physicists D. Amati, S. Fubini, and A. Stangellini; the Soviet physicists E. L. Feinberg and D. S. Chernavskii) that the production of secondary particles takes place in “peripheral” or “multiperipheral” interactions of hadrons that occur as a result of the interchange of a virtual pion or another particle. The idea that the strong interaction is accomplished at high energies through the exchange of a particular state—the “reggeon,” which is a sort of stream of particles with a momentum that varies monotonically from particle to particle —has been used extensively since the end of the 1960’s for theoretical analysis of multiple processes. These concepts (which were developed in particular by such Soviet physicists as V. N. Gribov and K. A. Ter-Martirosian) make possible quantitative explanation of many regularities in multiple processes. According to the “multiperipheral” models and the “reggeon” model, the average multiplicity should increase in proportion to the logarithm of the collision energy.


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Feinberg, E. L. “Mnozhestvennaia generatsiia adronov i statisticheskaia teoriia.” Uspekhi fizicheskikh nauk, 1971, vol. 104, fasc. 4, p. 539.
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