Muons

(formerly μ-mesons), unstable elementary particles with spin 1/2, a lifetime of 2.2 × 10 ^-6 sec, and a mass approximately 207 times greater than that of the electron. There exist positively charged (μ⁺) and negatively charged (μ-) muons, which are particle and antiparticle with respect to one another. The muons belong to the lepton class—that is, they participate in electromagnetic and weak interactions but not in strong interactions.

Discovery and sources. Muons were first detected in 1936 in cosmic rays by the American physicists C. Anderson and S. Neddermeyer. At first an attempt was made to identify muons with a particle that, according to the hypothesis of the Japanese physicist H. Yukawa, would be the carrier of nuclear forces. However, although such a particle should interact intensively with atomic nuclei, experimental data showed that muons interact weakly with matter. This “paradox” was resolved in 1947 with the discovery of the pion, which has the properties of the particle predicted by Yukawa and decays into a muon and a neutrino.

The main source of muons in cosmic rays and high-energy particle accelerators is the decay of pions and kaons, which are produced intensively during collisions of strongly interacting particles (hadrons)—for example, protons (p) with nuclei:

(1, a) π⁺(K⁺) → μ⁺ + ν_μ

(l, b) π^-(k^-) → μ^- π + ν̄_μ

(here ν_μ and ν̄_μ are a muon neutrino and a muon antineutrino). Other sources of muons—the production of μ⁺μ- pairs by high-energy photons (γ) and electromagnetic decays of mesons of the type ρ → μ⁺ + μ ^- (so-called lepton decays of hyperons, such as Λ⁰ → ρ + μ + ν_μ)—usually play a much smaller role.

In cosmic rays at sea level, muons are the main component (about 80 percent) of the particles in cosmic radiation. Muon beams with an intensity of 10⁵–10⁶ particles per second are produced in modern high-energy particle accelerators.

The spin of a ν_μ that arises during decays (l,a) is oriented opposite the direction of its momentum, and the spin of a ν_μf from decays (l,b) is oriented in the direction of its momentum. From this it follows, on the basis of the laws of conservation of momentum and angular momentum, that the spin of a μ⁺ that is produced upon decay of resting π ⁺ or K ⁺ is directed opposite its momentum, whereas that of a μ^– is in the direction of the momentum (see Figure 1). Therefore, muons are partially or completely polarized in the direction of the momentum (μ -) or opposite to it (μ ⁺), depending on the kinematic conditions of the production of muons and on the energy spectrum of the pions and kaons.

Figure 1. Production of П^+- and μ^- muons during decays of and μ^--mesons at rest. The momenta ρv_+- and ρ_μ (ρ_v̄μ and ρ_μ p_μ-, respectively) of the decay particles r_μ and μ⁺ (v̄_μ and mu) are equal in magnitude and opposite in direction. The boldface arrows indicate the direction of the spins (polarization) of the particles s_vμ and s_μ (s_v̄μ and s_μ-).

Muon-muon interaction. Weak interactions of muons cause their decay in the following manner:

(2,a) μ⁺ → e⁺ + v_e + v̄_μ

(2,b) μ^- → e^- + v̄_e + v_μ

(where e⁺, e^-, v_e and v̄_e are a positron, an electron, an electron neutrino, and an electron antineutrino, respectively); these decays determine the “lifetime” of a muon in a vacuum. A μ^- “lives” a shorter time in matter: upon coming to a halt in matter, it is attracted by a positively charged nucleus and forms a “muon atom,” or μ-mesonic atom—a system that consists of an atomic nucleus, a μ^-, and an electron shell. In mesonic atoms the process of the capture of a μ^- by a nucleus may take place through weak interaction:

μ + _ZA → _Z-1 B + v_μ

(where Z is the charge of the nucleus). This process is analogous to the K-capture of electrons by the nucleus and reduces to the elementary interaction

μ^- + ρ → n + v_μ

(where η is a neutron). The probability of capture of a μ^- by the nucleus increases as Z⁴ for light elements and becomes equal to the probability of decay of a μ^- when Z ≈ 10. In heavy elements, the lifetime of μ ^- that are being stopped is determined primarily by the probability of their capture by nuclei and is 20–30 times shorter than their lifetime in a vacuum.

In a weak interaction, because of the violation of spatial parity, positrons emerge mainly in the direction of the spin of the μ ⁺ in decay (2,a), whereas the electrons in decay (2,b) are emitted mainly in the direction opposite the spin of the μ^-. Therefore, the directions of the spins of μ^- or μ⁺ may be determined by studying the asymmetry of the emission of electrons or positrons in such decays.

Modern experimental data show that muons participate in all known interactions in precisely the same way as the electron or positron, differing only in mass. This phenomenon is called μ-e universality. At the same time, the muon and electron differ in a certain inner quantum number, and this difference also exists for the neutrinos ν_μ and v_e that correspond to them. Proof of this is the fact that a neutrino that is produced together with a muon (for example, during the decay π⁺ → μ⁺ + ν_μ does not result in the production of an electron upon colliding with nucleons, as well as the fact that the neutrino-free decays μ^± → e^± + γ and μ^± — → 2e^± + e⁺ are not observed. A possible explanation of the difference between the muon and the electron is the hypothesis that μ^- and v_μ differ from e^- and v_e in the lepton charge (number) l: for e^- and v_e, l = +1, and for μ^- and v_e, l = +1; for their antiparticles l has opposite sign (the latter decays will then be prohibited by the law of conservation of the lepton number). The existence of μ^-e universality raises for the theory of elementary particles an important and as yet unresolved problem: since, according to contemporary theory, the mass of particles has a field origin—that is, is determined by the interactions in which a particle participates—it is not clear why the electron and the muon, which have totally identical interactions, are so different in mass. Hypotheses to the effect that muons have “anomalous” interactions (that is, interactions that the electron does not have) have been advanced, but such interactions have not been experimentally observed. On the other hand, the difference between the masses of the muon and electron may be related to the internal structure of leptons; as yet, however, even the approach to be taken to this problem is unclear. Thus, the existence of muons is one of the most interesting puzzles in nature, and the answer to the question may entail discoveries of fundamental importance.

The question of the possible existeʘce of other leptons with a mass greater than that of the muon is also related to the problem of μ-e universality. If the interactions of “heavy” leptons were found to be the same as for μ and e, then some of their properties (particularly their lifetime and modes of decay) could be predicted theoretically. If such leptons exist and their mass is greater than 0.5 giga electron volt (GeV), then, by virtue of their properties, they may have gone unnoticed in most experiments. Therefore, special experiments, apparently with high-energy neutrinos (or photons), are required to search for “heavy” leptons.

Penetrating power. Since high-energy muons have no strong interactions, they are decelerated in matter only as a result of electromagnetic interactions with the electrons and nuclei of the substance. Up to energies of the order of 10ⁿ-10¹² eV, muons lose their energy mainly to ionization of the atoms of the medium, whereas at higher energies energy losses caused by the production of electron-positron pairs, the emission of gamma quanta of bremsstrahlung, and the splitting of atomic nuclei become essential. Since the mass of the muon is much less than that of the electron, energy losses of fast muons to bremsstrahlung and pair production are much less than the energy losses of fast electrons to bremsstrahlung or of gamma quanta to the production of e⁺e^- pairs. These factors account for the high penetrating power of muons as compared both to hadrons and to electrons and gamma quanta. As a result, the muons of cosmic rays not only penetrate the earth’s atmosphere easily but also descend to quite significant distances beneath the surface, depending on their energy. In subterranean experiments muons of cosmic rays with an energy of 10¹²-10¹³ eV have been recorded at a depth of several kilometers.

Muons that stop in matter. Slow muons may stop in matter upon losing their energy to the ionization of atoms. In most cases μ ⁺ adds an atomic electron, forming a system analogous to the hydrogen atom (a muonium). A muonium can enter into chemical reactions analogous to those of the hydrogen atom. Because of interaction with the magnetic moments of electrons of the substance, the μ⁺ (whose spin was originally directed opposite its entry into the substance) partially loses its polarization. This can be assessed from the change in the asymmetry of emission of positrons from decay (2,a). By studying the process of depolarization of μ⁺ in matter in the presence of external magnetic fields, it is possible to establish the chemical reactions into which the muonium enters and to determine the rate of the reactions. Muon chemistry, a new trend in the study of the properties of matter and chemical reactions using positive muons, has arisen in recent years.

As noted above, negative muons that stop in matter may form muon mesonic atoms. The Bohr radius of a muon mesonic atom is equal to h²/m_μ e²Z, where m and the mass and charge of the muon, Z is the charge of the nucleus, and h is Planck’s constant. This quantity is (m_μ/m_e)Z times less than the Bohr radius of the hydrogen atom (m_e is the mass of the electron). Therefore, the muon “orbits” that correspond to the lower energy levels of a mesonic atom are much closer to the nucleus than are the electron orbits. When Z ≈ 30–40 the dimensions of muon “orbits” are comparable to those of the nuclei, and the distribution of the electric charge in the nucleus strongly affects the energy of the lower state of the mesonic atom. Here the distance between the energy levels of mesonic atoms is m_μ/m_e ≈ 207 times greater than for the corresponding hydrogenlike atom (with a nuclear charge Z) and may be tens or hundreds of kilo electron volts, or, for heavy elements, a few megaelectron volts.

Muon mesonic atoms originally arise in excited states, and then, upon successively emitting gamma quanta or transferring energy to the atomic electrons, pass into the ground state. By measuring the energy of the gamma quanta emitted during transitions between the levels of mesonic atoms, it is possible to obtain information on the size of nuclei, the distribution of electric charge in the nucleus, and other characteristics of the nucleus.

The behavior of mesonic atoms of hydrogen and its isotopes (deuterium and tritium) is highly distinctive. The single positive charge of the nucleus in these mesonic atoms is completely screened by the charge of the negative muon. Therefore, such a system, whose dimensions are of the order of 2 × 10^-11 cm, behaves in matter like a slow neutron: it “freely” penetrates the electron shells of atoms and may draw close to other nuclei. This accounts for the possibility of a number of specific phenomena; in particular, mesonic atoms of hydrogen or deuterium may add yet another nucleus and form the mesonic molecules ρρμ, dpμ, or ddμ, which are analogous to the molecular ions of hydrogen, H₂⁺, HD⁺, or D₂⁺ (where d is a deuterium nucleus, or deuteron). The nuclei in such molecules, which are located at short distances from each other, may enter into the fusion reactions d + p → ³He + γ or d + Φ → ³He + n, d + d → T + p, which result in the release of energy (T is a tritium nucleus). After the event the μ^- is often freed of its bond to the nucleus and may then induce a new fusion reaction by successively forming a muon mesonic atom and a muon mesonic molecule. That is, it acts as a catalyst of nuclear reactions. However, the μ^- cannot practically be used to produce energy from nuclear fusion, since the number of nuclear reactions induced by a muon during its lifetime is very small.