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ANOMALOUS SCATTERING OF $\beta$-RAYS AND THE HYPOTHESIS OF “STICKING TOGETHER” OF ELEMENTARY PARTICLES
Some time ago, a suggestion was made concerning the possibility of a uniform interpretation of various multiple processes ($\beta$-decay, double $\beta$-decay, multiple production of mesons, etc.) as first-order processes.
As is known, in the case of ordinary $\beta$-decay the interaction energy of nucleons with the electron-neutrino field is taken in the form
$$ U_\beta = g_\beta \{ \psi^* A \varphi + \kappa \cdot c \}, \tag{1} $$
where $A$ is a matrix operator acting on the wave functions of the electron $\psi$ and the neutrino $\varphi$, the admissible forms of which are determined by the invariant properties of spinor functions. $g_\beta \equiv g_F \simeq 10^{-48}\ \text{erg}\cdot\text{cm}^3$ is the Fermi constant.
In the theory of double $\beta$-decay, treated as a first-order process, the interaction with the bielectronic field may be taken in an analogous form[^1]
$$ U_{\beta\beta}=g_{\beta\beta}\{ \bar{\psi} A \psi+\ldots \}; \tag{2} $$
In the case of multiple production of mesons, the interaction of nucleons with a field of $n$ mesons may be taken in the form[^2]
$$ U_n = g_1 \varphi + g_2 \varphi^2 + \ldots + g_n \varphi^n, \tag{3} $$
(or in closed form, for example, in the form \(U=ge^{\alpha\varphi}\) or \(U=g\varphi/(1+a\varphi^{2})\), etc.). Here \(\varphi\) is the wave function of one meson (for simplicity assumed to be a scalar and neutral one), \(g_n\) are the corresponding coupling constants.
After establishing the form of the interaction of nucleons with the field, we can calculate in the usual way the probability of \(\beta\)-decay, \(\beta\beta\)-decay, or the production of \(1,2,\ldots,n\) mesons. The coupling constant \(g_{\beta\beta}\) of double \(\beta\)-decay and the coupling constants with the meson field are for the time being determined empirically, as in the case of ordinary \(\beta\)-decay.
Thus one obtains a neutrinoless description of \(\beta\beta\)-decay, leading to probabilities that agree with experiment approximately as well as in the case of treating \(\beta\beta\)-decay as a second-order effect with the emission and absorption of a neutrino in an intermediate state.
In the case of multiple production of mesons in the collision of two high-energy nucleons (\(\sim 10^{12}\)—\(10^{13}\) ev), results are obtained that are close to the statistical variant of Fermi’s theory \(^{3}\) of this process, in which, however, there is no interaction energy, but only phase statistical factors are postulated. For example, for an exponential dependence of the interaction energy the mean number of mesons will increase with energy according to the law \(\bar n\sim \varepsilon^{2/3}\), whereas Fermi’s theory leads to the dependence \(\bar n\sim \varepsilon^{1/2}\) (in the center-of-inertia system).
The forms of the interaction energy listed above suggest the hypothesis that, instead of the emission of several particles, the emission of one composite particle may take place, for example of “fused” electrons and a neutrino, or of mesons. Such a hypothesis is quite natural if one takes into account the existence, under certain conditions, of decay processes of the type \(\tau\to\mu_{\pm}+\nu\), \(\tau_{\pm}\to\pi_{\pm}+ \pi_{+}+\pi_{-}\), etc., which are inverse to the process of “fusion.”
This hypothesis, asserting that fusion occurs with some probability, in particular under conditions of multiple production, is a development of the well-known theory of de Broglie \(^{4}\), in which the equation of motion, for example, of vector particles is obtained as a result of a special “fusion” of the equations of motion of two spinor particles. In this case the wave functions formed by “fusion” of a particle will be obtained as products of wave functions, for example \(\psi^{n}\to\Phi_{n}\).
In other words, it is possible that at least some of the heavier mesons (\(\tau\)-mesons) are fused \(\pi\)-mesons.
In an analogous way one may also assume possible “fusion” in double \(\beta\)-decay with the formation of “bielectrons,” and likewise in ordinary \(\beta\)-decay with the formation of an “electrino” (\(\varepsilon\)) as a result of the “fusion” of an electron and a neutrino*). The “electrino” must have spin 0 or 1 and, consequently, obey Bose statistics.
On the basis of these assumptions, \(\beta\)-decay should be regarded as a complex process of transformation of nucleons of two types, in the first of which the “electrino” participates as a particle actually emitted by the nucleons and subsequently decaying into an electron and a neutrino with a decay time \(\tau\sim10^{-10}\) sec. (for the magnitude of \(\tau\), see below):
\[ \begin{Bmatrix} p\\ n \end{Bmatrix} \to \begin{Bmatrix} n\\ p \end{Bmatrix} +\varepsilon_{\pm},\qquad \varepsilon_{\pm}\to e_{\pm}+\nu, \]
where the energy spectrum of the “electrino” will be monochromatic. In another part of the transformations an electron and a neutrino may still be emitted as before, while the “electrino” will participate in the process only
*) The theoretical possibility of fusion of electrons and neutrinos and, consequently, the existence of the “electrino” was pointed out by A. Sokolov \(^{5}\).
virtually. True, the spectrum of electrons formed in the decay of \(\varepsilon\) differs significantly from the ordinary \(\beta\)-spectrum. It will be cut off both on the side of small and on the side of large energies. Taking into account the slowing down of \(\varepsilon\) will lead to a smearing of the electron spectrum, which will already resemble the ordinary \(\beta\)-spectrum\(^6\).
Let us now note that processes of multiple production can be described by means of nonlinear terms in the interaction of nucleons with the electron-neutrino or meson fields. The presence of such nonlinear terms of interaction is, to some extent, equivalent to the inclusion of nonlinearities in the very equations for a free field of one or another particle, for example of the type first proposed by D. Ivanenko\(^7\),
\[ D\psi+\lambda\cdot \psi\psi^{*}\psi=0, \]
or, in the case of a real field,
\[ D\psi+\lambda\psi^{3}=0, \]
where \(D\) is the operator of the relativistic Dirac equation, \(\lambda\) is a constant. Analogous nonlinear additions to the equations of meson fields lead to a weakening of the interactions between nucleons carried by this field, apparently in general qualitative agreement with the empirical data.
Without dwelling further on questions of the theory of multiple production and particle disintegration, let us turn to a brief account of the anomalies of \(\beta\)-particles and of the possible connection of these anomalies with the existence of some of the predicted particles (“electrino”), as was recently once again indicated in the work of D. V. Skobeltsyn\(^8\).
The anomaly of \(\beta\)-particles was pointed out long ago. Studying the tracks of \(\beta\)-particles in Wilson’s chamber in connection with the verification of the theory of pair formation, Skobeltsyn and Stepanova in 1934\(^ {7,9}\), and then other researchers, came to the conclusion that positrons are produced by \(\beta\)-particles with a cross section \(10^{4}\) times greater than the theoretical value, if they arose as a component of an electron-positron pair produced by a \(\beta\)-particle or a \(\gamma\)-quantum emitted by the same source. However, the impossibility of identifying these positive particles with positrons was indicated by the absence of annihilation radiation. This anomaly was subsequently connected with the so-called anomalous scattering of \(\beta\)-particles, consisting in the fact that near the source the scattering through angles close to \(90^\circ\) exceeds ordinary Coulomb scattering (for example, according to Mott’s theory) by 30–40 times, and also in the fact that \(\beta\)-particles often experience sharp losses of energy, not explained by ordinary bremsstrahlung. At the same time the anomalous scattering was observed only at distances of 10–20 cm from the source (a \(\beta\)-radioactive element) in the absence of significant magnetic fields. Let us note that up to the present time the results of some investigators are in fairly good agreement with Mott’s theory, whereas others point to the anomalies described above. There arises the need to take more accurately into account the conditions of the experiments.*)
*) Recently, Gretsinger and Ribe\(^ {13}\) carried out experiments in a Wilson chamber with the aim of clarifying the nature of the “positive” tracks arising near a \(\beta\)-source (\(P^{32}\)), the identification of which with positrons is impossible. Special attention in these experiments was paid to excluding other possibilities for the appearance of “positive” tracks (for example, from electrons going toward the source, or electrons having positive curvature as a result of strong multiple scattering). With the aid of a stereoscopic analysis of tracks in the Wilson chamber and a suitable arrangement of the source, the fraction of such tracks was reduced to a few percent. The ratio of the fraction of positive tracks to the fraction of electron tracks was found by them to be \(3\cdot 10^{-4}\), which is in general smaller than was reported in other investigations with \(P^{32}\).
A number of attempts were made to explain anomalous scattering (the supposition of a nonelectric interaction of β-particles with nuclei, the hypothesis of neutron emission in the collision of β-particles with a nucleus, the supposition that nuclear rotation is excited in the collision of β-particles with nuclei, etc.). However, all these explanations proved unsatisfactory. A completely different solution of the problem—the supposition that new particles are present in β-radiation—was put forward by A. F. Ioffe (1938)\(^{10}\) and other investigators, who ascribed to the new particles masses both greater than and less than the electron mass. Analyzing his experiments, Thibaud (1946)\(^{12}\) arrives at the supposition of the existence in β-radiation of superlight particles (with mass \(\sim 10^{-11}m_0\)), having a small charge (\(\sim 10^{-4}e\)), but possessing a relatively large magnetic moment. Subsequent work led to masses and charges of these hypothetical particles equal to several \(m_0\) and to the electron charge, respectively.
Independently of experiments with β-radiation, some investigators of cosmic rays\(^{11}\) came to the conclusion that in cosmic radiation there are new particles with mass \(\sim(2 \div 10)m_0\) and lifetime \(10^{-9}\div 10^{-10}\) sec., which were called λ-mesons.
The clearest conclusions from a large body of experimental material were drawn by Skobeltsyn\(^{8}\), who states the conviction that there exist in β-radiation particles with mass equal to several electron rest masses \((3\pm0.3)m_0\), which spontaneously decay in flight (like \(V\)-particles in cosmic radiation) with the emission of neutral particles (for example, neutrinos) and electrons.
The lifetime of the hypothetical particles, estimated from the path length, turns out to be of order \(10^{-10}\) sec. The presence in β-radiation of 5 to 10 percent of such particles leads, according to Skobeltsyn, to qualitative and quantitative agreement with the data of a whole series of experiments. The supposition of an “electrino” makes it possible to explain the anomaly of β-particles. As a result of the decay of the “electrino” (in collision with nuclei or spontaneously) into an electron and a neutrino, the decay electrons can constitute appreciable angles with the direction of motion of the “electrino” (anomalous scattering and abrupt energy losses). In a magnetic field such electrons also lead to the appearance of tracks with positive curvature (“positive” tracks).
Skobeltsyn analyzes in detail the angular distribution of the energy losses of β-particles arising as a result of the in-flight decay of unstable hypothetical particles, which it is reasonable to call the “electrino,” emphasizing thereby the connection with the hypothesis stated above.
Knowing the empirical value of the lifetime of the “electrino” with respect to decay into an electron and a neutrino and calculating, according to the decay scheme, the decay probability, one can estimate the corresponding coupling constant. The lifetime of \(\varepsilon\), according to Skobeltsyn’s experiments (\(\sim 10^{-10}\) sec.), corresponds to a coupling constant of the “electrino” with the electron-neutrino field of order
\[ g'_{\varepsilon}\sim 10^{-11}\;(\mathrm{erg}\cdot\mathrm{cm})^{1/2}. \]
Further, assuming that the particle actually or virtually emitted is only the “electrino,” one can find the coupling constant of the \(\varepsilon\)-field with nucleons \(g_{\varepsilon}\) from the relation
\[ g_F=\frac{4\pi g'_{\varepsilon}g_{\varepsilon}}{\chi_{\varepsilon}^{2}}, \]
where \(g_F\) is the Fermi constant, \(\chi_{\varepsilon}=\dfrac{m_{\varepsilon}c}{\hbar}\) (\(m_{\varepsilon}\) is the rest mass of the “electrino”), and consequently also calculate the probability of emission of the “electrino” by nucleons. As an estimate, the constant \(g_{\varepsilon}\) turns out to be of order
\[ 10^{-17}\;(\mathrm{erg}\cdot\mathrm{cm})^{1/2}. \]
If one takes for the “electrino” the value of the rest mass following from Skobel’tsyn’s analysis, equal to several electron rest masses ($\sim 3m_0$), then it becomes clear that not all $\beta$-emitters will emit real “electrinos.”
A direct experimental verification of the existence of new particles and confirmation of the theoretical hypothesis mentioned above is undoubtedly of primary interest both for refining the interpretation of $\beta$-decay and for the general theory of elementary particles, in connection with the proposed recently nonlinear theory of multiple processes and the nonlinear generalization of the equations of quantum field theory.
S. L.
CITED LITERATURE
- D. Ivanenko and N. Kolesnikov, DAN 81, 771 (1951); see also: R. Winter, Phys. Rev. 83, 1070 (1951).
- D. Ivanenko and V. Lebedev, DAN 80, 352 (1951); see also: R. Glauber, Phys. Rev. 84, 315 (1951), where, somewhat later, the same results were obtained for weak interaction with a nonlinear coupling to the field.
- E. Fermi, Phys. Rev. 81, 683 (1951); Progr. Theor. Phys. 5, 570 (1950).
- L. De Broglie, Théorie générale des particules à spin (Méthode de fusion), Paris, 1943.
- A. Sokolov, DAN 21, 35 (1938).
- D. Ivanenko and N. Kolesnikov, DAN 87, 923 (1952).
- D. Ivanenko, Sow. Phys. 13, 141 (1938). D. Ivanenko and A. Brodsky, DAN 84, 682 (1952); see also L. Schiff, Phys. Rev. 84, 1 (1951).
- D. V. Skobel’tsyn, in the collection In Memory of S. I. Vavilov, Publishing House of the Academy of Sciences of the USSR, Moscow, 1952.
- D. Skobel’tsyn, Izv. AN SSSR, ser. fiz., No. 1–2, 75 (1938); D. Skobel’tsyn, DAN 21, 435 (1938); Stepanova, ibid., p. 91.
- Izv. AN SSSR, ser. fiz., 1938, p. 99.
- See, for example, Broadbent and Jassby, Proc. Roy. Soc. 192, 364 (1948), Freter, Phys. Rev. 73, 41 (1948), and others.
- J. Thibaud, Compt. Rend. 223, 984 (1946).
- G. Groetzinger and F. L. Ribe, Phys. Rev. 87, 1003 (1952).