formed by nucleons situated in closed shells, in particular the influence of these factors on the β-decay of heavy nuclei.

However, this question is still far from sufficiently clarified and will not be considered below.

The state of the theory of β-decay several years ago, summarized in well-known textbooks (Groshev and Shapiro¹, Fermi², Blatt and Weisskopf³) and review articles (Wu⁴), was characterized by the following:

1) After the experimental difficulties had been overcome, it was shown that the spectrum (the energy distribution of β-particles) of allowed transitions is in excellent agreement with Fermi’s theory. The distribution of the β-decay energy between the electron and the neutrino is determined by the statistical weight of the corresponding states. This result made it possible completely to reject the assumption of Konopinski and Uhlenbeck⁵, according to which the matrix element of β-decay depends on the momenta of the particles produced, i.e., on derivatives of the wave functions. It was thus shown that the expression for the matrix element determining the probability of the β-process contains, in the form of a product, the wave functions themselves of the four particles participating in the β-process (proton, neutron, neutrino, electron or positron), and not their derivatives.

2) The matrix element of the interaction (the interaction-energy density) must be invariant under a Lorentz transformation—this is an obvious requirement of the theory of relativity. From the four wave functions of particles with spin \(1/2\), an invariant can be constructed in several ways. For calculations of the probability of the β-process it is convenient to compose the invariant from terms written as the product of some quantity depending only on the two wave functions of the nucleons (index \(N\)) and the corresponding quantity depending only on the wave functions of the light particles (index \(L\)).

It is known that an invariant can be constructed in five ways: as the product of two scalars:

\[ S = S_N \cdot S_L; \]

as the scalar product of two four-dimensional vectors:

\[ V = (V_N \cdot V_L) = \sum_{i=1}^{4} V_{iN}\cdot V_{iL} = \]

\[ = V_{0N}\cdot V_{0L} - V_{1N}\cdot V_{1L} - V_{2N}\cdot V_{2L} - V_{3N}\cdot V_{3L}; \]

from two tensors:

\[ T = \frac{1}{2}\sum_{i,k=1}^{4} T_{ikN}\cdot T_{ikL}, \]

where \(T_{ik}\) is an antisymmetric tensor \((T_{ik}=-T_{ki})\); from two axial vectors:

\[ A=\sum_{i=1}^{4} A_{iN}\cdot A_{iL}, \]

where \(A_N\) and \(A_L\) are axial vectors (pseudovectors); and from two pseudoscalars:

\[ P=P_N\cdot P_L, \]

where \(P_N\) and \(P_L\) are pseudoscalars. Indices 1, 2, 3 refer to the spatial axes, and 4 to the time axis.

For definiteness, let us agree to express quantities with the index \(L\) in terms of the wave functions of the electron \(e^{-}\); \(\Psi_e\) describes the creation of \(e^{-}\) and the annihilation of \(e^{+}\), \(\Psi_e^{*}\) the annihilation of \(e^{-}\) and the creation of \(e^{+}\). If in the expressions \(S_L, V_L\), etc., one substitutes the wave functions of \(e^{+}\) instead of the wave functions of \(e^{-}\), then their character (scalar, vector, ...) is preserved, but the signs of some coefficients \(s, v, \ldots\) change. In the most general form, the problem of determining the elementary interaction was long ago formulated as the problem of determining five numerical coefficients: \(s, v, t, a, p\), with which five different invariants enter into the most general expression compatible with the theory of relativity:

\[ H=sS+vV+tT+aA+pP+(\text{Herm. conj.}). \]

The matrix element of the \(\beta\)-process is written as

\[ M=\int H\,d\tau . \]

Initially (in 1934), by analogy with the electromagnetic interaction, into which the 4-vector of charge density and current density enters, Fermi\({}^{6}\) assumed that \(H=vV\). This concrete assumption was not justified, which, of course, does not diminish the importance of Fermi’s remarkable work.

At the previous stage, summarized in the books and articles mentioned, it was shown that in the expression \(H\), in any case, one of the coefficients \(t\) or \(a\) is nonzero. In addition, a brilliant confirmation of the very structure of the expressions for \(H\) and \(M\) was obtained: among the forbidden spectra, cases were found in which \(M\) is equal to zero in the approximation in which the wave functions of the electron and neutrino are considered constant within the nucleus. In these cases (the so-called unique spectra, see \({}^{4}\)) the value of \(M\) depends on the derivatives \(\dfrac{\partial \Psi_e}{\partial x}\), \(\dfrac{\partial \Psi_\nu}{\partial x}\), i.e., on the momenta of the light particles \(p_e, p_\nu\), despite the fact that the momenta

do not enter into the elementary law (into the expression \(H\)). The dependence of \(M\) on \(p_e\) and \(p_\nu\) for these forbidden transitions has determined a certain form of the electron spectrum (differing from the form of an allowed spectrum), as well as the preferential emission of electrons in a definite direction in the decay of polarized nuclei and the correlation of the directions of \(\beta\)- and \(\gamma\)-rays in \(\beta\)-decay with the formation of an excited nucleus and the subsequent emission of a \(\gamma\)-quantum.

All these subtle effects have been observed experimentally. However, the elucidation of the concrete form of \(H\) has for the most part been carried out in recent years.

At first glance the determination of the five numerical constants \(s, v, t, a, p\) does not present great fundamental interest and resembles the work of a reference-book compiler. In reality, with these constants there is connected not only the quantitative side of the matter—the lifetimes of radioactive nuclei and the selection rules determining the probabilities of various transitions. Depending on the values of these constants, such a fundamental question as that of the charge independence of \(\beta\)-decay is decided in different ways, namely: whether the \(\beta\)-decay of a neutron with formation of an electron is exactly similar to the \(\beta\)-decay of a proton with formation of a positron and, further, whether one may extend to light particles the concepts of isotopic invariance and isotopic spin, so fruitful in the field of nuclear forces.

The second question, whose solution depends on the values of the constants, is the question of the possibility of representing \(\beta\)-decay as a sequence of two processes of a more customary type (more similar to the emission of a quantum), with the creation and subsequent decay of some hypothetical particle with integral spin. Attempts to construct such a theory were begun by Yukawa\(^7\) and Wentzel\(^8\) in 1936. Yukawa’s assumptions concerning \(\beta\)-decay proved to be incorrect; nevertheless, they played an enormous role in the discovery and investigation of mesons. The question of intermediate particles in \(\beta\)-decay remains unresolved up to the present time and continues to attract attention. Therefore, in recent years great efforts have been directed toward clarifying the values of the constants. Work of recent years is also characterized by special attention to allowed transitions and light nuclei—in this region theoretical predictions are the most reliable.

II. SELECTION RULES FOR ALLOWED TRANSITIONS

Neglecting the motion of the nucleons \(\left(\dfrac{v}{c}\ll 1\right)\), we obtain that in the expression for \(H\) one may neglect \((V_{1,2,3})_N\), while

\[ V_{0N}=S_N=\Psi_N^{*}\Psi_P \]

is a scalar with respect to rotation of the spatial axes;

in exactly the same way the components \((T_{14,24,34})_N\), \((T_{41,42,43})_N\), and \(A_{4N}\) may be neglected, while the remaining components form an axial vector (in the narrow, nonrelativistic sense—not with respect to a Lorentz transformation, but with respect to a spatial rotation), namely the spin vector:

\[ T_{12N}=A_{3N}=\Psi_N^*\sigma_3\Psi_p=-\Psi_N^*\sigma_z\Psi_p . \]

Finally, in the approximation \(\dfrac{v}{c}\ll 1\), \(P_N\) is also negligibly small.

The invariants \(S\) and \(V\) give the selection rule \(\Delta l=0\), “no” (\(l\) is the spin of the nucleus): it is obvious that the integral (matrix element) of a scalar can be different from zero only between two states with the same angular dependence and the same parity—the Fermi selection rules (“no” means that the parity does not change). The invariants \(T\) and \(A\) give the selection rules \(\Delta l=\pm 1,0\), “no,” with the important additional condition that a transition between two states with \(l=0\) is forbidden—the Gamow–Teller selection rules (abbreviated G.-T.).

One may say\(^3\) that the Fermi selection rules mean emission of an electron and a neutrino in a singlet state (with moment equal to zero), whereas the G.-T. selection rules mean emission of \(e^-+\nu\) in a triplet state with moment \(l=1\).

The decay \(\mathrm{Li}^6(l=1)\to \mathrm{He}^6(l=0)\) satisfies the G.-T. selection rules.*) Hence it was long ago concluded that \(t\) or \(a\) is different from zero.

The first indications that \(0\to 0\) transitions are also possible date to 1949, when Sherr\(^9\) investigated the positron decay \(\mathrm{C}^{10}\to \mathrm{B}^{10}\) and \(\mathrm{O}^{14}\to \mathrm{N}^{14}\). However, complete certainty has been reached only recently in the case of \(\mathrm{O}^{14}\). The spin of \(\mathrm{O}^{14}\) is equal to zero by the general rule applying to even-even nuclei. In the ground state \(\mathrm{N}^{14}\) has spin 1. However, both in the case of the electron decay \(\mathrm{C}^{14}\to \mathrm{N}^{14}\) and in the case of the positron decay \(\mathrm{O}^{14}\to \mathrm{N}^{14}\), the transition to the ground state of \(\mathrm{N}^{14}\) proceeds very slowly. In the case \(\mathrm{C}^{14}\to \mathrm{N}^{14}\), the so-called reduced lifetime \(f\cdot t=4\cdot 10^9\) sec is approximately \(10^6\) times greater than the times corresponding to allowed transitions. In the case of the transformation \(\mathrm{O}^{14}\to \mathrm{N}^{14}\), the decay energy is sufficient for formation of the excited state \((\mathrm{O}^{14}=\mathrm{N}^{14*}+e^+ +\nu+1.8\,\text{MeV})\) with subsequent emission of a \(\gamma\)-quantum \((\mathrm{N}^{14*}=\mathrm{N}^{14}+\gamma+2.2\,\text{MeV})\); owing to competition from decay to \(\mathrm{N}^{14*}\), the decay \(\mathrm{O}^{14}\to \mathrm{N}^{14}\) practically does not occur and has still not been reliably established.**) The small probability

*) The spin of \(\mathrm{Li}^6\) has been measured; the spin of \(\mathrm{He}^6\) has not been measured, but the rule that the spin of even-even nuclei is equal to zero has no exceptions in all the cases investigated.

**) According to\(^ {10}\), transitions to the ground level have a probability equal to \(3\%\) of the probability of decay to the excited level, which gives \(\ln ft=6.7\); according to\(^ {11}\), this probability is less than \(0.3\%\).

of the decay of \(O^{14}\) and \(C^{14}\) to the ground level of \(N^{14}\) is an astonishing fact, since by the G.-T. selection rules this decay is allowed \((\Delta I = 1)\); however, this cannot shake confidence in the existence of matrix elements that give the G.-T. rules. Obviously, the matrix element \(\sigma\) in this case is small (of order \(10^{-3}\) instead of a quantity of order 1) owing to an accidental almost complete compensation of terms with different signs; in the shell model, when definite states and definite orbital and total angular momenta \((l\) and \(j)\) are assigned to individual nucleons, such a compensation should not occur, but there are indications in the literature\(^{12}\) that tensor forces between nucleons can cause the appearance of terms of different sign and a sharp decrease of the matrix element.

The question of the transition to the ground level of \(N^{14}\), important from the point of view of explaining the experimental picture, is not essential for what follows; it is sufficient to know the properties of the state \(N^{14*}\) formed in the decay of \(O^{14}\). Owing to the charge independence of nuclear forces, the nucleus \(N^{14}\) must have a level similar to the ground states of \(C^{14}\) and \(O^{14}\) (belonging to the same value of the isotopic spin \(T = 1\)). Such a level is precisely the state \(N^{14*}\) formed in the \(\beta^+\)-decay of \(O^{14}\).

To prove this, let us note that the energy differences of the three nuclei—\(C^{14}\), \(N^{14*}\), and \(O^{14}\)—correspond exactly to the change of the Coulomb energy when a neutron is replaced by a proton; on this see also Zel’dzer’s article\(^{13}\), pp. 486 and 493. Since \(O^{14}\) and \(C^{14}\) have \(I = 0\), \(N^{14*}\) must also have spin \(I = 0\). (We note that, thanks to the long lifetime of \(C^{14}\), the equality to zero of its spin has been verified experimentally.)

Thus it has now been proved that the transition

\[ O^{14} \to N^{14*} \]

is a \(0 \to 0\) transition, which would be forbidden by the G.-T. rules. The existence of this transition with a small reduced lifetime proves that, in the general expression for the interaction, there are also terms that give the Fermi selection rules, so that \(s\) or \(v\) are not equal to zero, but have the same order of magnitude as \(t\) or \(a\).

Consequently, in the general expression for \(\hat H\) at least two terms are nonzero and of the same order.

Other recently discovered\(^{14,15,16}\) cases of \(0 \to 0\) transitions are the decays \(K^{38}\), \(Cl^{34}\), and \(Al^{26*}\), respectively, into \(Ar^{38}\), \(S^{34}\), and \(Mg^{26}\) with emission of \(e^+\).

In Fig. 1 we give the decay schemes of \(Al^{26}\) and \(Cl^{34}\).

A very interesting general property of odd-odd nuclei (\(Cl^{34}\), \(Al^{26}\)) turns out to be the presence of close levels with strongly differing spins. A detailed analysis made it possible to identify \(0 \to 0\) transitions in these cases. The radioactivity of \(Al^{26}\) in the ground state, corresponding to a large spin \((I = 5)\), is distinguished by a large period—\(10^6\) years \((\lg ft = 14)\)—and was reliably esta-

was established only very recently \(^{36}\); this discovery makes it possible conveniently to apply the method of tagged atoms to aluminum, which could not previously be done with the short-lived (6 sec.) \(Al^{26*}\).

Fig. 1.

The reduced time \(ft\) for \(Cl^{34}\) and \(Al^{26}\) is very close to \(ft\) for \(O^{14}\) \((3275 \pm 75)\); the study of these nuclei confirms the conclusions drawn from the decay of \(O^{14}\).

III. PROPERTIES OF VARIOUS INVARIANTS AND THE ANALOGY WITH THE ELECTROMAGNETIC INTERACTION

The concepts of a scalar, a polar and an axial vector in ordinary three-dimensional space are quite intuitive and require no explanation. However, the five basic quantities listed above and referring to 4-dimensional space are less familiar, especially when they refer to the field of particles. It may therefore not be superfluous to construct the corresponding quantities using the electromagnetic field as an example. In the latter case, an example of a scalar is the quantity \(\mathscr{E}^{2} - \mathscr{H}^{2}\), which is independent of rotation and motion of the axes and does not change sign under inversion of the spatial axes. Another example of a scalar is the square of the 4-vector of the potential \(\Phi_{0}^{2} - \Phi_{1,2,3}^{2}\). The classical example of a 4-vector is the vector-potential \(\Phi\) (its fourth component is the electrostatic potential, \(\Phi_{0} = \varphi\)). The electric and magnetic fields in the theory of relativity, as is well known, combine into an antisymmetric tensor:

\[ F_{ik} = \begin{pmatrix} 0 & \mathscr{H}_{3} & -\mathscr{H}_{2} & -i\mathscr{E}_{1} \\ -\mathscr{H}_{3} & 0 & \mathscr{H}_{1} & -i\mathscr{E}_{2} \\ \mathscr{H}_{2} & -\mathscr{H}_{1} & 0 & -i\mathscr{E}_{3} \\ i\mathscr{E}_{1} & i\mathscr{E}_{2} & i\mathscr{E}_{3} & 0 \end{pmatrix}. \]

The axial vector is in reality a tensor of the third rank, whose components are characterized by three indices \(A_{ikl}\); however, since this tensor is antisymmetric with respect to the interchange of each pair of the three indices, there are only four mutually distinct quantities \(A_{ikl}\), for which all three indices \(i, k, l\) are different. Then \(A_{ikl}\) may be numbered by the missing fourth index \(A_{ikl}=\pm A_m\), where \(i, k, l, m\) are the complete set of four indices corresponding to the four coordinates of the theory of relativity, and the sign depends on the order in which the indices \(i, k, l\) are arranged. Such a quantity in electromagnetic theory can be formed from the field tensor and the vector potential:

\[ \pm A_m=A_{ikl}=F_{ik}\Phi_l+F_{kl}\Phi_i+F_{li}\Phi_k. \]

An example of a pseudoscalar is the scalar product \((\mathcal{E}\cdot\mathcal{H})\). The Hamiltonian of the interaction of a charged particle with the electromagnetic field is described by the product of the 4-vector of current and the vector potential \(\Phi\).

If the charged particle has spin \(1/2\) and is described quantum-mechanically, then the Hamiltonian of its interaction is written in the form

\[ e(V_0\Phi_0' - V_1\Phi_1 - V_2\Phi_2 - V_3\Phi_3), \]

where \(V_0=\Psi^*\Psi\), \(V_i=\Psi^*\alpha_i\Psi\), and \(\alpha_i\) are the well-known Dirac matrices \(^{85,86}\).

Fermi \(^{6}\) chose the form of the \(\beta\)-interaction by analogy with this expression: from the coordinates of the heavy particles he constructed a 4-vector with components \(\Psi_N^*\Psi_P,\ \Psi_N^*\alpha\Psi_P\) (\(N\)—neutron, \(P\)—proton) and multiplied it by the same expression (also a 4-vector), constructed from the electron and neutrino functions and describing the “4-potential” of the field of light particles.

At low velocities the electrostatic interaction is the principal one; by analogy, in the \(\beta\)-interaction at low nucleon velocities the principal term is the product of four components: \(\Psi_N^*\Psi_P\Psi^*\Psi_{e--}\). An uncharged particle at rest possessing a magnetic moment \(\mathbf{M}\) (for example, the neutron) interacts with a magnetic field with energy

\[ \mathbf{M}\mathcal{H}=\mu(\boldsymbol{\sigma}\mathcal{H})=\mu(\Psi_N^*\boldsymbol{\sigma}\Psi_N)\mathcal{H}). \]

The magnetic moment is directed along the spin vector \(\boldsymbol{\sigma}\) (\(\mathbf{M}=\mu\boldsymbol{\sigma}\)); in the last expression the spin vector is considered as an operator acting on the neutron wave functions. A term proportional to the electric field is absent in the interaction of a particle at rest with the electromagnetic field, since such a term would describe the electric dipole moment of the particle; however, as is known, elementary particles cannot have a dipole

of a moment (a particle with spin \(1/2\) has no electrostatic moments at all). With respect to a Lorentz transformation, \(\mathscr H\) is part of a tensor of the second rank; correspondingly, in a relativistic generalization (for the case of a moving neutron) the interaction of a particle with an anomalous magnetic moment and a field is described as the product of two tensors: the tensor \(\Psi^* \beta \gamma_i \gamma_k \Psi^*\)*) and the tensor of the electromagnetic field. The tensor interaction in the theory of \(\beta\)-decay is analogous to the interaction of an anomalous magnetic moment with a field, with the difference that the “field” itself in the case of \(\beta\)-decay is described by a tensor composed of the electron and neutrino functions.

Along the way we have convinced ourselves that, at small velocities of heavy particles, the tensor variant gives the operator \(\sigma\). Thus, the linear combination \(aV + tT\) in the theory of \(\beta\)-decay is similar to the interaction of a particle possessing charge and an anomalous magnetic moment with an electromagnetic field. In the quantum theory of electromagnetic radiation the expressions \(eV_{kP}\Phi_k\) and \(\mu T_{ikP}F_{ik}\) describe the emission of one quantum by a particle with charge \(e\) and anomalous magnetic moment \(\mu\) (here the index \(P\) characterizes the charged particle, for example a proton; the indices \(i,k\), over which summation is performed, are associated with the coordinate axes).

To the invariants \(S\), \(A\), and \(P\) of the \(\beta\)-decay tensor one may also set in correspondence the appropriate expressions for interaction with the electromagnetic field. For this purpose \(S_P\), \(A_P\), and \(P_P\) must be multiplied respectively by a scalar, an axial vector, and a pseudoscalar composed of \(\Phi_i\) and \(F_{ik}\), as shown above. A characteristic feature of the three quantities indicated is that they are quadratic with respect to the fields and potentials. In the quantum theory of radiation the corresponding expressions, for example

\[ S_P \cdot (\mathscr E^2 - \mathscr H^2) = \Psi_P^* \beta \Psi_P \cdot (\mathscr E^2 - \mathscr H^2) \]

or

\[ S_P \cdot \Phi^2 = \Psi_P^* \beta \Psi_P \cdot (\Phi_0^2 - \Phi_1^2 - \Phi_2^2 - \Phi_3^2), \]

describe the simultaneous emission of two quanta by a particle possessing charge and magnetic moment; these expressions must enter the interaction Hamiltonian with coefficients \(\mu^2\) and \(e^2\).

It is immediately clear from this that if charge conjugation is performed, i.e. if one passes from particles to antiparticles, the expressions \(V_P\) and \(T_P\) change sign, while the expressions \(S_P\), \(A_P\), and \(P_P\) do not change sign. Five quantities, each composed of a pair of wave functions and differing in their properties with respect to transformations of the system

*) \(\beta, \gamma_i\) are Dirac matrices in the Pauli representation \(^{86}\).

coordinates, are divided into two groups (vector and tensor on the one hand, scalar, axial vector, and pseudoscalar on the other), differing in their properties with respect to the transition from particles to antiparticles.

The consideration of electromagnetic fields and potentials used above is, of course, only a more or less convincing pedagogical device. In reality the properties of quantities with respect to charge conjugation are proved with complete rigor directly from the properties of spinors and Dirac matrices, as was done by Tolhoek and Groot[^17] in 1950. In doing so one uses the properties of the charge-conjugation matrices \(C\), as shown in the appendix to the paper by Zel’dovich, Luk’yanov, and Smorodinskii[^18] *).

Tolhoek and Groot made the natural, at first sight, assumption that in \(\beta\)-decay either one group of interactions (the linear combination \(vV + tT\)) or the other (\(sS + aA + pP\)) must be realized ). Physically this requirement is formulated as the condition of complete symmetry of the interaction with respect to replacing one pair of particles by antiparticles, i.e., complete symmetry between neutron decay with emission of an electron and proton decay with emission of a positron *).

In the following paragraphs we shall see that experiment refutes the assumption of Tolhoek and Groot: the true interaction is a linear combination that includes representatives of both groups (\(sS + tT\) and, possibly, \(+ pP\)). We shall also see in what subtle and difficult-to-observe effects the arising asymmetry between \(\beta^-\)- and \(\beta^+\)-processes manifests itself—neither in the energy spectrum nor in the lifetimes of radioactive nuclei, as is well known, is there any symmetry.

It is necessary to make two cautions concerning the use of the preceding arguments. The groups of invariants \(V, T\) and \(S, A, P\) differ with respect to the transition from particles to antiparticles in one pair of particles, for example in the light particles. With respect to the transition from particles to antiparticles carried out for all (two heavy and two light) particles simultaneously, all five invariants are the same: they all do not change sign, and any linear combination of them is transformed into itself and is admissible. Mathematically this is obvious, since each invariant (for exam-

) For corrections see UFN 55*, issue 3, p. 467, 1955.

**) In Tolhoek and Groot’s work such a notation was chosen (without secondary quantization), in which \(S, A\), and \(P\) change sign, while \(V\) and \(T\) do not change sign; this is immaterial for what follows.

***) What is meant is the symmetry of \(\beta^-\)- and \(\beta^+\)-decay at equal energies of these processes, i.e., the symmetry of matrix elements. Of course, there is no symmetry in \(\beta^-\)- and \(\beta^+\)-processes in the pion: the difference between the neutron and proton masses and the Coulomb energy of the proton play the main role here.

…for example, \(S\), is the product of two factors referring to two pairs of particles \(S_N \cdot S_L\), as a result of which multiplying each of the factors by \((+1)\) or by \((-1)\) does not change the product. Physically this corresponds to the obvious exact equivalence, for example, of the decay of an antineutron into an antiproton and a positron and the decay of a neutron into a proton and an electron. The exact equivalence of particles and antiparticles lies at the very foundation of the modern theory and is not subject to any doubt. The difference between the invariants and the considerations of Tolhoek and Groot concerned the comparison of the processes \(N = P + e^- + \nu\) and \(P = N + e^+ + \nu\). The absence of symmetry of the latter two processes violates not the exact law of equivalence of particles and antiparticles, but only the approximate law of symmetry of the properties of the neutron and proton. This approximate symmetry of \(N\) and \(P\) is well justified in nuclear forces, evidently is not fulfilled in the interaction with the electromagnetic field, and, as is seen from experiment, is not fulfilled in the case of the \(\beta\)-interaction, which could not have been foreseen in advance.

The second caution concerns the analogy between the \(\beta\)-interaction and the electromagnetic interaction, more precisely the analogy between the emission of a pair of light particles \(\beta + \nu\) and the emission of a quantum. This analogy must not be carried too far; in particular, it must not be used to derive selection rules. In the electromagnetic interaction the particle does not change, its charge is conserved; closely connected with this is the well-known transversality condition for the field of the quantum (the vector potential, and also \(\mathcal{E}\) and \(\mathcal{H}\) of the quantum, are perpendicular to the direction of propagation), whence it follows, in particular, that emission of a single quantum is impossible in \(0 \to 0\) transitions.

In the vector interaction in the theory of \(\beta\)-decay the fields of the light particles and the four-dimensional vector constructed from them are not subject to the transversality condition; therefore a \(0 \to 0\) transition is entirely possible.

IV. STATE OF THE LIGHT PARTICLES FORMED IN \(\beta\)-DECAY

The expression for the interaction causing \(\beta\)-decay was presented above in the form of a sum of terms, each of which in turn is the product of two factors: a quantity depending on the state of the nucleons, for example \(S_N\) or \(V_{LN}\), etc., and a quantity depending on the state of the light particles \(S_L\) or \(S_{LL}\), etc. The selection rules concerning the state of the nucleus were derived from consideration of the matrix elements of quantities with the index \(N\), depending on the state of the nucleons. It was found that for allowed transitions the selection rules do not make it possible to distinguish \(S\) and \(V\), which in the nonrelativistic limit give identical expressions. Exactly the same is the case with the invariants \(T\) and \(A\).

Consideration of quantities with indices \(L\), referring to light particles, should reveal such properties of the process as the distribution of electrons and neutrinos by energy (spectrum), by directions of flight, and by spin directions.

Since the velocities of the light particles are by no means small compared with the speed of light \(c\), here one cannot use the nonrelativistic approximation, and on the basis of comparison of calculations with experiment one succeeds in distinguishing \(S\), \(V\), \(T\), and \(A\).

Let us consider allowed transitions in light nuclei, neglecting the influence of the Coulomb field.

Applying the usual technique of calculation with Dirac matrices, set forth, for example, in Heitler’s book \(^{19}\), it is easy to obtain the following well-known results:

1) After averaging over all directions of the spin and over the directions of the momenta of the electron and neutrino, the mean value of the matrix element for each of the four variants \((S, V, T, A)\), taken separately, does not depend on the energy of the electron or on the energy of the neutrino. Consequently, in each variant the form of the spectrum is entirely determined by the statistical weight of the different distributions of energy between the electron and the neutrino; each variant leads to the Fermi spectrum *). Therefore the allowed form of the Fermi spectrum does not make it possible to determine which of the four variants are realized in actuality.

2) In the presence of a linear combination of two invariants giving identical selection rules, i.e. giving the electron and neutrino in the same (triplet or singlet) states, the form of the spectrum changes \(^{20}\). The mean value of the square of the matrix element turns out to be equal to

\[ \left(s^2+v^2 \pm 2s\cdot v\,\frac{m_e c^2}{E_e}\right)|\langle 1\rangle|^2 + \left(t^2+a^2 \pm 2ta\,\frac{m_e c^2}{E_e}\right)|\langle \sigma\rangle|^2, \]

where \(\langle 1\rangle\) and \(\langle \sigma\rangle\) are matrix elements, taken over the nucleon wave functions, corresponding to the selection rules of Fermi and G.-T. The signs \(\pm\) refer respectively to electrons and positrons. Since \(S\) and \(A\), on the one hand, and \(V\) and \(T\), on the other, are transformed differently in the transition from electrons to positrons, it is not surprising that, when, for example, \(S\) and \(V\) are simultaneously realized, one obtains a formula asymmetric with respect to \(e^+\) and \(e^-\).

Experiment is in full agreement with the assumption \(sv=at=0\). The accuracy with which the Fermi spectrum of allowed transitions is realized experimentally \(^{21}\) limits these quantities from above:

\[ \varphi_{\Phi}=\frac{|s\cdot v|}{s^2+v^2}<0.1;\qquad \varphi_{\mathrm{G.-T.}}=\frac{|at|}{a^2+t^2}<0.1. \]

*) We do not consider the fifth variant—the pseudoscalar \(P\)—since it does not give allowed transitions.

Estimates of the accuracy with which allowed spectra agree with the Fermi spectrum are rather subjective, and other authors give, for the upper limit, values that differ considerably from 0.1:

\[ \varphi < 0.02^{22};\quad \varphi < 0.04^{23};\quad \varphi < 0.20^{24}. \]

Especially sensitive to the presence of terms \(\mp \dfrac{m_e c^2}{E_e}\) is the ratio of the probabilities of \(K\)-capture and positron decay of one and the same nucleus with formation of the same final nucleus.

The experiment\(^{25}\) consists in determining the ratio of the number of positrons to the total number of excited nuclei \(\mathrm{Ne}^{22*}\); the latter was determined from the yield of \(\gamma\)-quanta in the reaction

\[ \mathrm{Na}^{22} + e^- = \mathrm{Ne}^{22*} + \gamma, \]

or

\[ \mathrm{Na}^{22} = \mathrm{Ne}^{22*} + e^+ + \nu;\quad \mathrm{Ne}^{22*} = \mathrm{Ne}^{22} + \gamma, \]

the direct \(\beta\)-transition to the ground state

\[ \mathrm{Na}^{22} \to \mathrm{Ne}^{22} \]

amounts to only 0.06% of the decays and plays no role.

The spin of \(\mathrm{Na}^{22}\) was measured and found to be 3; the spin of \(\mathrm{Ne}^{22*}\) is apparently 2 (a transition according to the selection rules \(\Gamma\)-\(\mathrm{T}\)); the ratio of the probabilities of \(K\)-capture to \(\beta^+\)-emission, \(0.110 \pm 0.006\), is in excellent agreement with the theoretical value 0.1135 for \(at=0\); the authors give

\[ \varphi_{\mathrm{r.-t.}} \simeq \frac{|a|}{|t|} = -0.01 \pm 0.02. \]

Other measurements of the ratio of the probabilities of \(K\)-capture and \(\beta^+\)-decay, confirming the results of Sherr and Miller\(^{25}\), are given in works\(^{37,38}\). A refinement of the calculation carried out in\(^{39}\) led to a further decrease in the probable value of \(\varphi_{\mathrm{r.-t.}}\).

3) Averaged over the direction and spin of the neutrino and over the spin of the electron, the probability of electron emission does not depend on the direction of emission of the electron for any polarization of the initial nucleus and of the nucleus formed after the decay. In the case of polarized initial nuclei, \(\beta\)-particles are emitted isotropically; if, in the decay of an unpolarized nucleus, an excited nucleus is formed, then there is no correlation between the direction of the \(\beta\)-particle and the direction of the \(\gamma\)-quantum emitted after the decay. All this applies only to allowed transitions.

4) The decay probability, averaged over the spins of the neutrino and \(\beta\)-particle, depends on the angle \(\theta\) between their directions of emission, and in different ways in the different variants. This circumstance was first examined in detail by L. A. Sliv\(^{27}\). The Fermi form of the spectrum (see above, item 1) is obtained only after averaging over all \(\theta\). The direction of emission of the neutrino can be determined from the momentum of the electron and the momentum of the nucleus formed in the decay. Thus, from the correlation between the directions of emission of the neutrino and the \(\beta\)-particle (the \(\beta-\nu\) correlation), one can establish the law of interaction\(^{27,3,87}\).

The square of the matrix element, averaged over the spins of the electron and neutrino, as well as over the spins of the nucleus before and after the process,

depends only on the angle \(\theta\):

\[ M^2 = 1 + \alpha \frac{v_e}{c}\cos\theta, \]

where \(v_e\) is the electron velocity, and the values of the coefficient \(\alpha\) are given in the table:

	\(S\)	\(V\)	\(T\)	\(A\)
\(\alpha\)	\(-1\)	\(+1\)	\(+\dfrac{1}{3}\)	\(-\dfrac{1}{3}\)

In 1953, two groups of investigators independently carried out measurements of the \(\beta-\nu\) correlation in the decay \(\mathrm{He}^6 \to \mathrm{Li}^6\). Owing to the small mass of the nucleus and the large decay energy \((3.5\ \mathrm{MeV})\), the \(\mathrm{Li}^6\) ions acquire a rather large energy, up to \(200\ \mathrm{eV}\).

In the first work \(^{28}\), the number of coincidences between the registration of an energetic \(\beta\)-electron and the registration of the recoil nucleus by counters was measured as a function of the angle between the directions of motion of the electron and the recoil nucleus; the spectrum of electrons emitted at an angle of \(180^\circ\) to the direction of the recoil nucleus was also measured.

In the second work \(^{29}\), the spectrum of the recoil nuclei was measured. The measurement was made by the time of flight of the recoil nucleus from the place of decay to the counter.

In both works the tensor law of interaction was definitely established; in conjunction with the condition \(at=0\) (Sec. 2), it follows from this that \(t \ne 0,\ a=0\). The matrix elements \(S\) and \(V\) in this process are equal to zero (see § 2), so that these experiments give no information about \(S\) and \(V\).

For the sake of fairness it should be noted that such a conclusion had been drawn earlier by Langer and Moffat \(^{30}\) (see also \(^{3}\), pp. 583–584) in the study of forbidden transitions. However, at the present time, after the experimental difficulties of measuring \(\beta-\nu\) correlations have been overcome, the data obtained from the decay of \(\mathrm{He}^6\) should be regarded as much more convincing and reliable.

In 1954, investigations \(^{31,32}\) of the \(\beta-\nu\) correlation were carried out in the decay

\[ \mathrm{Ne}^{19} = \mathrm{F}^{19} + e^+ + \nu. \]

The spin of \(\mathrm{F}^{19}\) has been measured and found to be \(1/2\). From the value of the comparative lifetime \(ft=1600\), the transition belongs to the same category of allowed transitions as the transitions \(\mathrm{H}^3 \to \mathrm{He}^3\), \(\mathrm{N}^{13} \to \mathrm{C}^{13}\), etc.; thus there is every reason to suppose that \(\mathrm{Ne}^{19}\ (9n+10p)\) is in a state completely analogous to the ground state of \(\mathrm{F}^{19}\ (10n+9p)\).

In a β-transition between mirror nuclei with the same, but nonzero, spin, two decay variants take part at once: both the one that gives the Gamow–Teller selection rules (we already know that this is the tensor variant) and one of the two variants (\(S\) or \(V\)) that give the Fermi selection rules.

The correlation between the directions of the positron and the neutrino is determined by the expression \(1+\dfrac{\alpha v_e}{c}\cos\theta\), where, in the case of the interaction

\(sS+tT\)

\[ \alpha= \frac{\dfrac{1}{3}t^2|\langle\sigma\rangle|^2-s^2|\langle 1\rangle|^2} {t^2|\langle\sigma\rangle|^2+s^2|\langle 1\rangle|^2}, \]

\[ \frac{1}{3}>\alpha>-1, \]

and, in the case of the interaction \(vV+tT\),

\[ \alpha= \frac{\dfrac{1}{3}t^2|\langle\sigma\rangle|^2+v^2|\langle 1\rangle|^2} {t^2|\langle\sigma\rangle|^2+v^2|\langle 1\rangle|^2}, \]

\[ 1>\alpha>\frac{1}{3}. \]

The first experiments of Hamilton and Alford\(^{31}\) led to the value \(\alpha=-0.8\pm0.4\). As is evident from the error estimate made by the authors themselves, the accuracy of these experiments is insufficient. Later there appeared a paper by Maxson, Allen, and Jentschke\(^{32}\), in which the value \(\alpha=-0.21\pm0.08\) was found; such a value indicates with certainty that a linear combination of the \(S\) and \(T\) invariants is realized: \(v=0,\ s\ne0\)*).

*) We note that the object of investigation (\(\mathrm{Ne}^{19}\)) was chosen not very successfully, since in this case two variants take part in the transition at once. The clearest result and the greatest difference between \(S\) and \(V\) (\(\alpha=-1\) or \(\alpha=+1\)) should be obtained in the investigation of \(0\to0\) transitions, for example \(\mathrm{Cl}^{34}\to\mathrm{S}^{34}\). In the case of \(\mathrm{O}^{14}\to\mathrm{N}^{14}\), measurement of the recoil is complicated\(^{33}\) by the fact that, in the transition \(\mathrm{N}^{14*}\to\mathrm{N}^{14}+\gamma\), the momentum of the nucleus is additionally changed. From Maxson’s data it follows that in the case of \(\mathrm{Ne}^{19}\)

\[ K=\frac{t^2|\langle\sigma\rangle|^2}{s^2|\langle 1\rangle|^2}\approx1.5\pm0.4. \]

In \(0\to0\) transitions \(K\) is equal to zero. Although not zero, values of \(K\) (of order 0.5) smaller than in \(\mathrm{Ne}^{19}\) may be expected\(^{34}\) in the decay of nuclei with one nucleon in the \(p_{1/2}\) state (\(\mathrm{N}^{13}\to\mathrm{C}^{13}\), \(\mathrm{O}^{15}\to\mathrm{N}^{15}\)). The advantage of \(\mathrm{Ne}^{19}\) is that this atom does not form molecules; rupture of the diatomic molecules \(\mathrm{O}_2\), \(\mathrm{N}_2\), \(\mathrm{Cl}_2\) in the decay of one of the nuclei would substantially change the momentum of the recoil nucleus. This difficulty, however, can be overcome by using hydrogen compounds \(\mathrm{H}_2\mathrm{O}\), \(\mathrm{H}_3\mathrm{N}\), HCl. In view of the importance of the question, one should recommend to experimenters that the correlation also be determined in other nuclei besides \(\mathrm{Ne}^{19}\).

Thus a linear combination \(H=sS+tT\) of two invariants is realized (possibly also with the participation of a \(p\cdot P\) term*), which transform differently in the transition from \(\beta^-\) to \(\beta^+\) decay. In this case as well, analysis of the forbidden spectrum\(^{21}\) had already indicated that the combination \(H=sS\) agrees better with experiment than \(vV+tT\). The combination \(sS+tT\) is such that, in the allowed spectrum of \(\beta^-\) and \(\beta^+\) decays, terms

\[ \mp \frac{mc^2}{E_e} \]

do not appear; the decay probability, averaged over the spins of the \(\beta^-\) or \(\beta^+\) particles and the neutrino, shows no asymmetry with respect to \(\beta^-\) and \(\beta^+\). In what, then, does this asymmetry manifest itself?

Let us consider the decay of a polarized nucleon. For simplicity we shall speak of a free nucleon or of one nucleon in the state \(S_{1/2}\) (above a closed shell), with its spin angular momentum directed, for example, upward. We shall follow the direction of the spin of the nucleon formed as a result of \(\beta\)-decay, and the direction of the spin of the charged particle formed; we shall assume the velocity of the \(\beta^\pm\) particle to be small, choosing those cases in which the neutrino has carried off almost all the decay energy. In the \(sS+tT\) variant, apparently, \(s\) and \(t\) have opposite signs when the expressions for \(S_L\) and \(T_L\) are written in terms of the electron wave functions.

The sign of \(s/t\) has to be judged from the analysis of forbidden transitions (see\(^{51,54}\)**). As to the absolute value of \(|s|\) and \(|t|\), of one

*) As regards the participation of the \(pP\) term in the expression \(H\), it is difficult to obtain reliable information, because \(P\) plays a role only in forbidden transitions. It has been suggested that the \(\beta\)-decay of RaE \((83p+127n)\to \mathrm{Pb}^{210}\) is a transition with a change of parity\(^{40}\); in this case the nuclear matrix element \(P_N\) would be comparable with the (forbidden) matrix element of the \(T_N\)-interaction; from the decay spectrum a conclusion was drawn about the presence of a \(P\)-variant; the matrix element \(P_N\) over the nucleon wave functions and the coefficient \(p\) were refined in works\(^{41,42,43,44,45}\). Essential in these calculations is the accounting for the form of the electron wave function in the nucleus, first carried out in the work of L. A. Sliv\(^{35}\). The assumption of odd parity of the nuclear wave function of RaE is based on the shell theory; the assumption of spin equality is much less reliable. L. Pokrovskii showed that RaE has spin 1, so that the argument of the above-mentioned works falls away, and the lifetime and spectrum are explained by a combination of \(S\) and \(T\). On the other hand, from the fact that the \(\beta\)-decay of a \(\pi\)-meson into \(e+\nu\) is absent, one may conclude that the coefficient \(p\) must be sufficiently small\(^{46-50}\). However, the calculation is connected with an estimate of divergent integrals and cannot be regarded as wholly reliable and convincing.

**) In paper\(^{54}\) there are given upper and lower bounds for the ratio \(y=s/t\) in the expression \(H=t(T+yS)\) (Table 1 on p. 1204); the upper bound for \(y\) is equal (for several different decay reactions) to \(0.35;\ 0.25;\ 0.5;\ 0.7;\ 0.5\); the lower bound is \(0.7;\ -0.65;\ -0.7;\ -0.65\) (the value \(-0.5\) for Re\(^{188}\) is excluded by the note on p. 1208). It will be shown below that

\[ |y|=\frac{1}{\sqrt{R}}\approx 0.75\pm 0.05, \]

which is not compatible with a positive sign of \(y\).

order—see below, as well as Gerhardt’s works1. In this case the calculation leads to the following conclusions (Zel’dovich2)*.

In the decay \(p \to n + e^+ + \nu\), if the \(p\) is polarized (\(p\uparrow\) means that the spin is directed upward), positrons are produced predominantly polarized in the same way as the proton, i.e. \(e^+\uparrow\), whereas neutrons are produced with both polarizations: \(n\uparrow\) and \(n\downarrow\). The principal reactions are:

\[ p\uparrow = e^+\uparrow + \begin{cases} n\uparrow + \nu\downarrow,\\ n\downarrow + \nu\uparrow. \end{cases} \]

In the decay \(n \to p + e^- + \nu\), if the \(n\) is polarized (\(n\uparrow\)), the electrons are nevertheless produced unpolarized (or only weakly polarized); their spin is connected mainly with the spin of the proton that is formed. The principal reactions are:

\[ n\uparrow = \nu\uparrow + \begin{cases} p\uparrow + e^-\downarrow,\\ p\downarrow + e^-\uparrow. \end{cases} \]

In both cases, in \(\beta^+\)- and \(\beta^-\)-decay the spin of the charged particles \((e^+, e^-)\) is connected precisely with the spin of the proton, and not with the spin of the neutron, which is a manifestation of the asymmetry of \(\beta\)-decay with respect to the neutron and the proton.

This asymmetry is maximal, under the assumptions made, in the case in which \(|s| = |t|\). Then, in the limit of a small velocity of the charged particle, only decays with the above-stated relation of spin directions would occur. For \(|s| < |t|\) these decays predominate.

It is unlikely that this very subtle effect will be observed directly in the near future. However, after the values of the constants \(s\) and \(t\) and the equality to zero of the remaining constants have been established by experiments that are simpler to carry out, the existence of polarization effects is established by the theory unambiguously. Thought polarization experiments play an important role in attempts to penetrate more deeply into the internal mechanism of beta decay.

V. NUMERICAL VALUES OF THE INTERACTION CONSTANTS

After clarifying exactly which invariants play a role in \(\beta\)-decay, it remains to determine the numerical values of the two constants \(s\) and \(t\). Both of them are real, as follows from the general conditions of the reality of the Hamiltonian and the symmetry of the theory with respect to past and future3. In the literature these constants

are often denoted by \(s=g_F\) and \(t=g_T\), respectively, because \(S\) gives the Fermi selection rules, and \(T\) the Gamow–Teller selection rules. Their dimensions are \(\mathrm{erg}\cdot\mathrm{cm}^3\) (in paper \(^{18}\), p. 363 contains a misprint).

Of special interest is the quantity \(R=t^2/s^2\), which characterizes the ratio of the decay probabilities according to the Gamow–Teller and Fermi rules; above, in considering polarization experiments, we obtained the first hint of the special properties of the case \(R=1\); in the last paragraph these properties will be revealed more fully.

To determine \(s^2\) and \(t^2\) one uses the well-known expression for the probability of allowed \(\beta\)-decay:

\[ w=\frac{1}{2\pi^3}\frac{m_e^5 c^4}{\hbar^7}\, f(\eta_0,Z)\left(s^2|\langle 1\rangle|^2+t^2|\langle\sigma\rangle|^2\right). \]

The half-life is expressed in terms of the probability as \(t_{1/2}=\dfrac{\ln 2}{w}\); \(f(\eta_0,Z)\) is a known function of the maximum momentum \(\eta=p_{\max}/m_e\) and of the charge \(Z\) of the daughter nucleus. For \(Z=0\), \(f(\eta_0,0)\) is proportional to the statistical weight of the electron–neutrino system; \(f(\eta_0,Z)\) takes into account the influence of the Coulomb field of the nucleus. To determine \(s^2\) and \(t^2\) from the measured lifetimes and decay energies, it remains to find the nuclear matrix elements.

The first works dealt with mirror nuclei with odd \(A\). Considering an odd nucleon as being in the field of central forces of a spherically symmetric even-even residue (in the case of nuclei with \(A=4n+1\)*), it was possible, from the parity and total spin of the nucleus, to determine the state (orbital angular momentum \(l\)) of the odd nucleon; its total angular momentum \(j\) coincides with the angular momentum of the whole nucleus. Thus the angular and spin dependences of the wave function were completely determined. In the transformation of mirror nuclei, the radial functions of the neutron and proton in the nuclei before and after the transformation coincide when Coulomb forces are neglected, and the corresponding integral is determined from the normalization conditions. Consideration of the angular and spin functions leads to the following values of the matrix elements, summed over all directions of the total spin of the final state:

Transition	\(s_{1/2}\to s_{1/2}\)	\(p_{1/2}\to p_{1/2}\)	\(p_{3/2}\to p_{3/2}\)	\(d_{3/2}\to d_{3/2}\)	\(d_{5/2}\to d_{5/2}\)
\(|\langle 1\rangle|^2\)	1	1	1	1	1
\(|\langle\sigma\rangle|^2\)	3	\(1/3\)	\(5/3\)	\(3/5\)	\(7/5\)

\[ \text{*} \]

*) In the case of nuclei with \(A=4n-1\), one considered a “hole”—the missing nucleon in the field of the residue.

Comparing the probabilities of decay of nuclei with different ratios \(\dfrac{|\langle \sigma\rangle|^2}{|\langle 1\rangle|^2}\), one can in principle determine each of the quantities \(s^2\) and \(t^2\) separately. Such calculations were carried out by a number of authors \(^{56,57}\); in paper \(^{57}\) the conclusion was drawn that \(R=1\pm0.20\).

At present it has become clear with certainty that this result is incorrect\(^*\). The authors mentioned grossly overestimated the accuracy with which the assumption of the shell theory concerning the motion of a nucleon in the constant field of the core is fulfilled; the motion of the nucleon perturbs the core to a large extent. The shell theory does indeed make it possible to predict the largest component of the wave function, determining those quantities which change discretely: the parity of the nucleus, the total angular momentum of the nucleus. However, in calculating the matrix elements of \(\beta\)-processes (as also in calculating the magnetic moments of nuclei), other components of the wave function, describing other possibilities for realizing the same total spin and parity of the nucleus as a result of a different distribution of the nucleons among single-particle levels, exert a noticeable influence. The spin-orbit and spin-spin interactions of the nucleons “entangle” and mix these states.

Let us turn to a rigorous theory, in which no assumptions are made about the properties of nuclear forces (in particular, about their central character and potential character), apart from their charge independence. In developing such a theory consistently, one must assume that in \(\beta^-\)-decay any neutron of the initial nucleus can be transformed into any proton of the final nucleus, so that the exact operator of \(\beta\)-transformation, according to the Fermi selection rules, written explicitly, is the sum of operators

\[ \sum_{n=1}^{A}(\tau_+)_{n}, \]

for positron decay correspondingly we obtain

\[ \sum_{n=1}^{A}(\tau_-)_{n}; \]

the sum is taken over all \(A\) nucleons (\(A\) is the atomic weight). We use here the notation of the theory of isotopic spin: the operator \((\tau_+)_n=(\tau_x+i\tau_y)_n\) transforms the \(n\)-th nucleon into a proton if it was a neutron, and gives zero if the \(n\)-th nucleon was a proton in the initial nucleus; the operator \(\tau_-\) transforms a proton into a neutron and gives zero for a neutron. Thanks to this definition, the summation may be extended over all nucleons of the nucleus. The sums \(\sum \tau_+\) and \(\sum \tau_-\) may be represented as \(T_+\)- and \(T_-\)-operators of rotation in the space of isotopic spin, changing its projection \(T_Z\) (i.e. the charge of the nucleus) without changing its absolute value. Thus, under the assumption of charge

\(^*\) Trigg’s result \(^{56}\), \(R=2\), although obtained by unconvincing calculations, is closer to reality.

independently of nuclear forces, Fermi’s selection rules include the selection rule for isotopic spin: \(\Delta T=0\). For \(\beta\)-transitions satisfying this condition one obtains

\[ |\langle 1\rangle|^2=T(T+1)-T_{Z\,\text{initial}}T_{Z\,\text{final}}. \]

In this case the wave function of the nucleus may differ arbitrarily from a simple product of one-particle wave functions of the individual nucleons. Equality of \(T\) in the initial and final states (\(\Delta T=0\)) ensures the similarity of the wave functions of the initial and final nuclei*).

For \(T=1/2\), \(T_{\text{initial}}\) and \(T_{\text{final}}=\pm 1/2\), we obtain \(|\langle 1\rangle^2|=1\). Consequently, in the \(\beta\)-transformation of one of a pair of “mirror” nuclei into the other, for example \(N_{13}\) (seven protons, six neutrons) into \(C_{13}\) (six protons, seven neutrons), the matrix element is exactly equal to unity, precisely as if only one excess nucleon participated in the \(\beta\)-transformation.

According to elementary representations, in the example \(N_{13}\to C_{13}\) six neutrons and six protons in both nuclei are in identical states, and the \(\beta\)-transformation of the inner six protons is forbidden by the Pauli principle, since under \(\beta\)-transformation an inner proton would find itself in a state already occupied by one of the six inner neutrons; only one outer (odd) nucleon participates in the process. In the theory of isotopic spin, the fulfillment of the Pauli principle is ensured automatically, and the elementary result \(|\langle 1\rangle^2|=1\) for one odd nucleon in mirror nuclei finds its confirmation and proves to be exact. The mixing, mentioned above, of various possible states of closed shells and of the outer nucleon by nuclear forces does not impair the exactness of the equality \(|\langle 1\rangle^2|=1\).

The triad of mirror nuclei \(C_{14}\), \(N^*_{14}\), \(O_{14}\) has isotopic spin \(T=1\), \(T_Z=1,0,-1\). According to the formula given, for the process \(O_{14}\to N^*_{14}\), \(|\langle 1\rangle^2|=2\). In elementary representations one can speak of two nucleons above a closed shell: \(O_{14}=(6\text{ neutrons}+6\text{ protons})+2\text{ protons}\), \(N_{14}=(6\text{ neutrons}+6\text{ protons})+\text{neutron}+\text{proton}\).

Let us recall again that the exactness of the equality \(|\langle 1\rangle^2|=2\) in this case corresponds to the exactness of the assumption of charge independence of nuclear forces, and not to the considerably poorer accuracy of the shell theory and the assumption of independent nucleons.

Thus, the known cases of \(0\to 0\)-transitions, where \(|\langle \sigma\rangle|^2=0\), namely \(O_{14}\to N^*_{14}\) and \(Cl_{34}\to S_{34}\), of nuclei with isotopic spin \(T=1\), have made it possible to determine with great accuracy\({}^{11}\)

\[ s=q_F=1.37\cdot 10^{-49}\ \text{erg}\cdot\text{cm}^3. \]

*) Coulomb corrections in the wave functions of the nuclei in the present case apparently do not exceed \(1\text{–}2\%\) \({}^{5}\).

For calculating the lifetime of β-active nuclei it is convenient to use the quantity

\[ A_F^{-1}=\frac{g_F^2 m_e^5 c^4}{2\pi^3\hbar^7\ln 2} =\frac{1}{6550}\ \mathrm{sec}^{-1}\ (\pm 2.3\%), \]

\[ t_{1/2}\cdot f(\eta_0)=A_F\bigl(|\langle 1\rangle|^2+R|\langle\sigma\rangle|^2\bigr)^{-1}. \]

The quantity \(|\langle\sigma\rangle|^2\) (more precisely, the sum

\[ \sum_{x,y,z}\left|\sum_n \sigma_{nx,y,z}(\tau_x+i\tau_y)_n\right|^2 \]

over all nucleons) is not an integral of motion. In a nucleus only the total angular momentum \(J\) has a definite value; its separation into orbital and spin parts and, still more, the assignment of a definite orbital angular momentum to individual nucleons cannot be carried out exactly. The only “nucleus” for which \(|\langle\sigma\rangle|^2\) is known exactly is the free neutron \((|\langle\sigma\rangle|^2=3)\). Using this value and the neutron lifetime measured by Robson\({}^{60}\), \(t_{1/2}=12.8\pm2.5\) min, Gerhart obtained \(R=1.37\pm0.40\). In the case of the neutron, owing to the difficulty of the experiment, the lifetime has been measured with an accuracy considerably lower than for other nuclei. The accuracy proved insufficient to assert with confidence that \(R>1\). A reasonable compromise is needed between experimental accuracy and rigor of calculation.

Blatt\({}^{61}\) gave a special analysis of the question of the value of \(|\langle\sigma\rangle|^2\) for the transition \(\mathrm{H}^3\to\mathrm{He}^3\) and came to the conclusion that, even under very artificial assumptions, \(|\langle\sigma\rangle|^2<3.01\); a larger value is practically quite improbable. The energy and decay time (18.1 kev, 12.4 yr) are known with sufficient accuracy. Taking \(|\langle\sigma\rangle|^2<3\), one obtains\({}^{11,62}\) \(R>1.77\). An indirect confirmation of the value \(|\langle\sigma\rangle|^2=3\) in the case \(\mathrm{H}^3-\mathrm{He}^3\) is the closeness of the magnetic moments of \(\mathrm{H}^3\) and the proton (2.979 and 2.793 nuclear magnetons), as well as the closeness of the magnetic moments of \(\mathrm{He}^3\) and the neutron (−2.128 and −1.914 nuc. mag.), in accordance with the representation of \(\mathrm{H}^3\) and \(\mathrm{He}^3\) as an α-particle with a “hole” in the \(S_{1/2}\) state.

It may be concluded that

\[ R=\frac{t^2}{s^2}\simeq 1.75\pm0.15, \]

so that \(R\) is certainly greater than unity. Apparently, the relation between the lifetimes of the neutron and of other nuclei does not agree with theoretical predictions. The discrepancy reaches \(\sim 20\text{–}30\%\), which lies at the limit of the experimental accuracy attained. A more precise measurement of the neutron lifetime is highly desirable.

The basic premise of a rigorous theory of β-decay is the assumption that the elementary interaction of each nucleon located in the nucleus with the electron-neutrino field does not differ from the interaction of a free nucleon. In calculating the probability of β-decay it is necessary to sum the contribution depending ...

from the possibility of transforming any neutron into any proton (“additivity” of the β-interaction). The nuclear forces and the binding of nucleons in the nucleus, according to this assumption, affect the β-decay of nucleons only indirectly, by determining the energy and the form of the wave functions of the transforming nucleons.

However, the form and the numerical constants of those operators whose application to the wave functions of nucleons in the nucleus gives the probability of β-decay are assumed to be the same for free and bound nucleons.

This assumption appears plausible theoretically, since the distance between nucleons in the nucleus (\(\sim 2 \cdot 10^{-13}\)) considerably exceeds that radius of the nucleon within which some of its internal structure is manifested,* \(\sim 4 \cdot 10^{-14}\). The binding energy of nucleons in the nucleus, \(20\)--\(30\) \(M_{\beta}\), is also small in comparison with the energy of the virtual mesons surrounding the nucleon (see the following chapter).

An indirect experimental confirmation of the assumption of the additivity of the β-interaction is the good agreement with theory of the ratio of the decay probabilities of two different nuclei \(O_{14}\) and \(Cl_{34}\), for which the nuclear matrix elements are known. A careful investigation of neutron decay, including measurement of the correlation between the directions of emission of the electron and the neutrino, would make it possible directly to compare both constants characterizing the β-interaction (\(s\) and \(t\)) of a free neutron and of a nucleon in the nucleus.

VI. MESON CORRECTIONS IN THE ELEMENTARY β-INTERACTION

Let us suppose that, after overcoming all difficulties, the quantities \(s\) and \(t\) have been measured exactly; according to the basic hypothesis of the additivity of the β-interaction in the nucleus, the quantities \(s\) and \(t\) pertain to free nucleons.

From the point of view of modern field theory, the “free nucleon” is not an elementary particle with spin \(1/2\) obeying the Dirac equation. Owing to the strong interaction inherent in nucleons with \(\pi\)-mesons, an isolated nucleon in vacuum gives rise to \(\pi\)-mesons around itself. By the law of conservation of energy, without external action these \(\pi\)-mesons cannot move away from the nucleon that produced them, i.e. they cannot manifest themselves as free \(\pi\)-mesons. The \(\pi\)-mesons produced in vacuum by an isolated nucleon must always follow the nucleon that produced them, now arising, now being annihilated; thus we are speaking of virtual mesons. It is customary to speak briefly of a free nucleon as of a nucleon

* This radius is determined from the interaction of the neutron with the electron and from the deviations in the scattering of electrons by the proton from Coulomb scattering by a point charge.

in the meson “coat.” The particle located inside this “coat” of virtual mesons—the “bare” nucleon—obeys the Dirac equation (the question of whether the nucleon has other “clothes,” apart from the \(\pi\)-meson “coat,” remains open at present). In any external interaction, not only the properties of the “bare” nucleon are manifested, but also the influence of the “coat.”

The very idea of mesons and of the strong interaction of nucleons with mesons arose from the analysis of nuclear forces between nucleons. The presence of a meson field, forming a “coat” around an isolated nucleon, is manifested in the anomaly of the magnetic moments of the neutron and the proton.

Returning to the theory of \(\beta\)-decay, we must state that the constants \(s\) and \(t\) refer to the nucleon in its “coat.” A natural question arises: what can be said about the constants of the \(\beta\)-interaction of the “bare” nucleon? How does the “coat” affect the \(\beta\)-interaction?

It is possible that, only because of mathematical difficulties, there does not at present exist a consistent, logically closed relativistic theory of the interaction of \(\pi\)-mesons with nucleons. The mathematical difficulties do not even allow one to compare with experiment the existing simplest assumptions about the initial equations of such a theory, so that it is still unclear whether such new physical ideas are needed. Under these conditions, for orientation it is expedient, following Chu \(^{64}\), Finkelstein and Moszkowski \(^{65}\), to take the simplest variant of the theory of the meson “coat,” one that lays no claim either to rigor or to relativistic invariance of formulation.

The mass of the nucleon is taken to be very large in comparison with the mass of the meson; when a nucleon emits a meson, we neglect the velocity and kinetic energy acquired by the nucleon as a result of recoil. The emission of mesons with momentum not exceeding some \(p_m\) is considered; the possibility is not taken into account that the “coat” consists of more than one meson, nor is the possibility of formation of nucleon–antinucleon pairs.

The interaction of a nucleon with a meson is taken in a form corresponding to the limiting case of a resting (but spin-possessing) nucleon:

\[ H=f\left(\Psi_N^{*}\sigma\Psi_N\right)\mathbf{k}\varphi, \]

i.e. as the scalar product of the vector (operator) spin of the nucleon and the momentum \(\mathbf{k}\) of the meson (\(\varphi\) is the wave function of the meson).

The authors of the work consider only the emission of a neutral \(\pi^{0}\)-meson; thus, for example, in the case of neutron decay, in addition to the direct process

\[ \mathrm{n}=\mathrm{p}+e^{-}+\nu, \]

a process is considered with the same initial and final states, but with intermediate emission of a \(\pi^0\)-meson:

\[ \mathrm{n}=\mathrm{n}+\pi^0=\mathrm{p}+e^-+\nu+\pi^0=\mathrm{p}+e^-+\nu. \]

It is assumed that emission of \(\pi^+\)-mesons by a neutron is impossible, and after emission of a \(\pi^-\)-meson \((\mathrm{n}=\mathrm{p}+\pi^-)\) \(\beta^-\)-decay would be impossible.

According to perturbation theory, the matrix element of the transition through two intermediate states with a \(\pi^0\)-meson in the intermediate state can be written in the form

\[ M'_n=-\frac{1}{\varepsilon_n^2} (\mathrm{n}|\Gamma'|\mathrm{n},\pi^0_n) (\mathrm{n},\pi^0_n|O|\mathrm{p},e^-,\nu,\pi^0_n)\times \]

\[ \times(\mathrm{p},e^-,\nu,\pi^0_n|\Gamma'|\mathrm{p},e^-,\nu). \]

In this expression \(\Gamma\) is the operator of meson creation, \(O\) is the operator of the \(\beta\)-process, the index \(n\) characterizes the momentum of the virtual meson, and \(\varepsilon_n\) is the energy of the virtual meson, equal to

\[ \varepsilon_n=\sqrt{m_\pi^2c^4+p_n^2c^2}. \]

Since \(\Gamma\) does not act on light particles, and \(O\) does not act on \(\pi^0\)-mesons, this expression may be written more simply:

\[ M'_n=-\frac{1}{\varepsilon_n^2} (\mathrm{n}|\Gamma'|\mathrm{n},\pi^0_n)(\mathrm{n}|O_\Pi|\mathrm{p})(\mathrm{p}|\Gamma'|\mathrm{p},\pi^0_n)(\Psi_0|O_L|e^-\nu). \]

The full operator of the \(\beta\)-transformation \(O\) is represented here as the product \(O=O_N\cdot O_L\) of operators acting on nucleons and light particles (cf. in § 1 \(S=S_N S_L\), etc.). The symbol \((\Psi_0|O_L|e^-,\nu)\) means that from the vacuum \((\Psi_0)\) two particles, \(e^-,\nu\), arise under the action of the operator \(O_L\). If there are different paths (the direct one—the matrix element \(M_0\)—and with emission of a virtual \(\pi^0\)-meson—the matrix element \(M_n\)) for carrying out one and the same transition, the matrix elements \(M_0\) and \(M_n\) are added; the matrix element with meson corrections taken into account has the form

\[ M'=M_0+\sum_n M_n. \]

Summation over \(n\) in fact represents integration in the phase space of the meson momentum vector \(\mathbf{k}\) over the whole volume \(|\mathbf{k}|<k_{\max}\).

If the meson correction to the scalar \(\beta\)-interaction is considered, then in the nonrelativistic approximation \(S_\Pi=1\), and the correction contains

\[ \Gamma S_\Pi \Gamma'\sim(\boldsymbol{\sigma}\mathbf{k})\,1\,(\boldsymbol{\sigma}\mathbf{k})=k^2. \]

If the tensor \(\beta\)-interaction \(T_{\parallel}=\sigma\) is considered, then from the anticommutation of the perpendicular components of the operator \(\sigma\) one obtains

\[ \Gamma T_{\parallel}\Gamma=(\sigma k)\sigma(\sigma k)=k^2(\sigma_k-\sigma_\perp), \]

where \(\sigma_k\) is the component of \(\sigma\) parallel to \(k\), and \(\sigma_\perp\) is the sum of the two components perpendicular to \(k\). On averaging over all directions of \(k\) we obtain:

\[ (\sigma k)\sigma(\sigma k)=-\frac{1}{3}k^2\sigma . \]

One may say that the virtual \(\pi^0\)-meson is emitted in a \(p\)-state with orbital angular momentum \(l=1\) by virtue of the flip of the nucleon spin, so that the \(\sigma\) of the nucleon in the intermediate state with a virtual \(\pi^0\)-meson has the sign opposite to the sign of \(\sigma\) for the “bare” nucleon. For the result it is very important that \((n|\Gamma|n,\pi^0)\) and \((p|\Gamma|p,\pi^0)\)—the matrix elements of the interaction of the neutron and proton with the \(\pi^0\)-meson—have different signs; the \(\pi^0\)-meson is one of the members of the isotopic triplet \(\pi^+,\pi^0,\pi^-\), and the expression for the Hamiltonian of the interaction of the \(\pi^0\)-meson with nucleons contains the matrix \(\tau_3\) in isotopic-spin space, which has different signs for the neutron and the proton. Such a structure of the expression is necessary in order that the theory with three mesons give charge independence of the nuclear forces. Recently the difference in the signs of the interaction with the proton and the neutron was tested directly in experiments on photoproduction of \(\pi^0\)-mesons on deuterons, where the contributions of the proton and neutron interfere with one another. The different signs of \(\Gamma\) for the proton and the neutron are essential in calculating the meson correction to the \(\beta\)-interaction. Finally, in \(^{66}\) the following expressions are given for the meson corrections (a prime denotes inclusion of the correction, the subscript 0 refers to the “bare” nucleon):

\[ s'=s_0(1-\varepsilon),\qquad t'=t_0\left(1+\frac{1}{3}\varepsilon\right),\qquad R_0=R'\left(\frac{1-\varepsilon}{1+\frac{1}{3}\varepsilon}\right)^2, \]

where

\[ \varepsilon=\frac{c^2}{m_\pi^2}f^2\int_0^{k_{\max}} \frac{k^3}{\left(\sqrt{m_\pi^2c^4+k^2c^2}\right)^3}\,k^2\,dk. \]

According to Chu \(^{64}\), from data on meson scattering, on the magnetic moment of the neutron, and on nuclear forces it follows that the best value is \(f^2=0.058,\ k_{\max}=5.51\,m_\pi c\) (\(k_{\max}/c\) turns out to be close to the nucleon mass, equal to \(6.5\,m_\pi\)).

In this case one obtains \(\varepsilon=0.237\) and

\[ R_0=(1.75\pm0.15)\cdot \left(\frac{1-0.237}{1+0.079}\right)^2 =0.98\pm0.08. \]

Of course, the accuracy of the meson calculation is much lower, but in any case one may state that the experimental data on β-decay do not contradict the assumption that \(R_0 = 1\), i.e. \(|s|=|t|\), for the interaction of a “bare” nucleon; the sign and order of magnitude of the experimentally measured deviation of the value \(R'\) from unity are in reasonable agreement with the idea of the influence of meson corrections on β-decay.

In connection with the calculations carried out in work \(^{65}\), several remarks should be made which, however, do not alter the conclusion indicated above.

1) In \(^{65}\) processes with β-decay of the \(\pi\)-meson are not considered. The decay \(\pi^\pm \to e^\pm + \nu\) would give a correction of order \(f\), but in fact it does not occur \(^{66}\).

The process \(\pi^\pm \to \pi^0 + e^\pm + \nu\), observation of which is very difficult, would give a correction of order \(f^2\), i.e. just the same as that considered above \(^{65}\). However, according to Zeldovich’s calculations \(^{50}\), the invariants \(S, T\), and \(P\) entering the β-interaction do not give such a decay (here \(S\) differs substantially from \(V\)). Thus, the neglect of meson decay in work \(^{65}\) is justified precisely for the expression of the β-interaction which is actually realized.

2) The expression for the matrix element \(M'_n\) is based on the following expression for the wave function \(\Psi'\) of the nucleon in the “fur coat”:

\[ \Psi' = \Psi_0 + \sum_n c_n \Psi_n, \]

where \(\Psi_0\) is the wave function of the “bare” nucleon, \(\Psi_n\) is the wave function of the system nucleon plus virtual meson with momentum \(\mathbf{k}_n\), and the coefficients \(c_n\) in first-order perturbation theory are equal to

\[ c_n = \frac{(n|\Gamma|n,\pi_n)}{E_n}. \]

The expression for \(\Psi'\) is correct to first order in \(c_n\), i.e. to first order in the coupling constant \(f\). But the meson correction to the β-interaction turned out to be of order \(f^2\). Consequently, in the expression \(\Psi'\) one should also have kept track of terms of order \(f^2\) (or \(c_n^2\)), which may give a correction of the same order in the final result. Taking into account terms of order \(f^2\), it turns out that in the expression \(\Psi'\) the normalization must be restored:

\[ \Psi' = \frac{\Psi_0 + \sum_n c_n \Psi} {\sqrt{\,1+\sum_n c_n^2\,}} . \]

Changing the normalization, one must take into account not only neutral but also charged mesons. In this case the corrected expressions obtained are:

\[ \frac{s'}{s_0}=\frac{1-\varepsilon}{1+3\varepsilon},\qquad \frac{t'}{t_0}=\frac{1+\frac{1}{3}\varepsilon}{1+3\varepsilon}. \]

The ratio \(R'/R_0\) is not changed in this case. The change in the absolute values \(s_0\) and \(t_0\) may be of interest in connection with comparison \(^{65}\) of these constants with constants describing other processes in which neutrinos are produced \((\mu^-+p=n+\nu,\ \mu^\pm=e^\pm+2\nu)\). Restricting ourselves to second order in \(f\), all corrections, including the normalization corrections, may be written in a very symmetric form:

\[ O'=O_0+\sum_n \Pi_n O_0\Pi_p-\frac{1}{2}\left(\sum_n \Pi_n\Pi_n\right)O_0-\frac{1}{2}O_0\left(\sum_n \Pi_p\Pi_p\right), \]

where

\[ \Pi_n=\frac{1}{E_n}\big[(n|\Gamma|n,\pi^0)+(n|\Gamma|p,\pi^-)\big], \]

\[ \Pi_p=\frac{1}{E_n}\big[(p,\pi^0|\Gamma|p)+(n,\pi^+|\Gamma|p)\big]. \]

The terms with charged mesons in the expressions \(\Pi\) identically give zero with the operator of the \(\beta\)-interaction \(O_0\) in the first correction term \(\Pi O_0\Pi\), but not in the two other terms, which arise from

Fig. 2.

the change of normalization. These latter have the form of the “free-end correction” known in quantum electrodynamics (Feynman \(^{67}\), Dyson \(^{68}\)). To the three correction terms of the formula one may associate three diagrams (Fig. 2).

The example considered shows that the meaning of the free-end corrections consists in a change of normalization; in particular, the coefficient \(1/2\) with which the contribution of the two end (normalization) diagrams enters is obtained quite transparently.

3) The calculation of meson corrections can be carried out in a relativistically invariant perturbation theory for the pseudoscalar coupling of a meson with a nucleon field.

For \(ISI\) and for the end-point corrections, divergent expressions are obtained. The formulation of the problem differs from the usual renormalization problem in quantum electrodynamics: there the quantity, for example, of the electric charge of the “bare” particle (electron, proton) \(e_0\) must be eliminated from the final formulas; the aim is to find corrections referring to the final values of the wave vector of the electromagnetic field (additional magnetic moment, Lamb shift of levels) for an experimentally specified charge \(e\), characterizing the interaction of the “particle in the vacuum” with a static field. In \(\beta\)-decay we are interested precisely in the ratio \(s'\), \(t'\) and \(s_0\), \(t_0\). Therefore the divergent expressions must not be renormalized, but cut off. The pseudoscalar theory is characterized, moreover, by a large coupling constant, which makes perturbation theory inapplicable. On the whole, nonrelativistic calculations \(^{65}\) by the method of \(^{63,64}\), with constants adjusted to the magnetic moment, appear more reasonable than invariant perturbation theory.

It will be necessary to return to the question of meson corrections in \(\beta\)-decay when, in the relativistically invariant theory of the interaction of nucleons and \(\pi\)-mesons, the difficulties connected with the large coupling constant and divergences have been overcome.

VII. ON INTERMEDIATE PARTICLES IN THE THEORY OF \(\beta\)-DECAY

In his famous work of 1935, Yukawa \(^{7}\), along with the assumption of the role of mesons in nuclear forces, proposed the following scheme of \(\beta\)-decay:

\[ \mathrm{n}=\mathrm{p}+\pi^-;\qquad \pi^-=\mathrm{e}^-+\nu, \]

in which the meson plays the role of an intermediate particle. Estimating the mass of the meson from the radius of nuclear forces and the coupling constant of the meson with nucleons from the magnitude of these forces, Yukawa, from the probability of \(\beta\)-decay, found the coupling of the meson with the electron-neutrino field and the probability of \(\beta\)-decay of a free meson.

It is well known that, after the first apparent confirmations, the picture became considerably more complicated; it turned out that the free mesons producing electrons and positrons in decay are \(\mu\)-mesons, weakly interacting with nucleons and having spin \(1/2\); the \(\pi\)-mesons, strongly interacting with nucleons, as it turned out \(^{65}\), do not decay by the reaction \(\pi^\pm=\mathrm{e}^\pm+\nu\); their decay proceeds according to the scheme \(^{70}\)

\[ \pi^\pm=\mu^\pm+\nu. \]

None of the mesons presently known and directly observed can be regarded as responsible for the \(\beta\)-process in the sense of Yukawa’s scheme \(^{7}\).

Let us note that a process analogous to the \(\beta\)-process (or, more precisely, to \(K\)-capture), namely the capture of \(\mu\)-mesons by nuclei according to the elementary reaction

\[ \mu^-+p=n+\nu, \]

apparently can be described by a Yukawa-type scheme with the virtual formation of a \(\pi\)-meson:

\[ \mu^-+p=\pi^-+\nu+p=n+\nu . \]

Bethe and Marshak \(^{71}\) (see also \(^{72,84}\)) note that the probabilities of this process are in reasonable agreement with the probability of \(\pi \to \mu\)-decay*).

These ideas should be further developed. From the latter scheme of \(\mu\)-capture one can find not only the total probability, but also draw definite conclusions concerning the form of the \(pn\mu\nu\)-interaction and the choice of the system of invariants entering into the general expression of type \(H\).

From this, further, one can obtain selection rules for the capture of \(\mu\)-mesons by nuclei and the probability of formation, upon capture of a \(\mu\)-meson, of an excited nucleus in one or another definite state. With the present technique for obtaining \(\mu\)-mesons in accelerators and for scintillation studies of gamma spectra, these more refined conclusions of the theory are accessible to experimental verification.

Returning to \(\beta\)-decay, let us note that the hypothesis of an electrino (for references to the literature see \(^{73}\)), emitted from a \(\beta\)-active nucleus and decaying into \(e^-+\nu\), is outwardly analogous to Yukawa’s scheme; however, the assumptions of a small mass of the electrino and of the possibility of its real (not virtual) production lead to contradictions with experiment \(^{74}\).

What are the general features of theories with an intermediate particle? Considering a process involving four particles \(A\), \(B\), \(C\), and \(D\) with spins \(1/2\), for example the process

\[ A+B=C+D, \]

we write the interaction Hamiltonian in the form

\[ H'=g\,(A^*OB)(C^*OD). \]

Correspondingly the matrix element

\[ M=\int H'\,dx \]

*) In Zeldovich’s work \(^{50}\) the reverse attempt was made: to regard the decay \(\pi^\pm \to \mu^\pm+\nu\) as a consequence of vacuum polarization with the production of virtual \(\pi^+p\)-pairs by a \(\pi\)-meson, the \(pn\mu\nu\)-interaction being considered elementary.

or is the sum of such terms. In \(H'\) the quantities \(A, B, C, D\) are the wave functions of the corresponding particles; all functions are taken at one value of the coordinate \(x\), to which the value \(H'(x)\) of the energy density also refers.

In the theory with an intermediate particle \(\Phi\) of integer spin, the elementary interaction is written as follows:

\[ H' = e_1(A^*OB)\Phi + e_2(D^*OC)\Phi + (\text{Herm. conj.}). \]

The two terms describe two elementary reactions \(A=B+\Phi\) and \(C+\Phi=D\); the constants \(e_1\) and \(e_2\) now have the dimension of electric charge. Strictly speaking, the matrix element of the process of interest to us, in which \(\Phi\) appears only virtually, is described by a double integral:

\[ M \sim e_1 e_2 \iint dx_1 dx_2\,(A^*(x_1)OB(x_1))\Phi(x_1)\Phi^*(x_2)(C^*(x_2)OD(x_2)). \]

The larger the mass of the particle \(\Phi\), the shorter the distance it has time to traverse from the point of creation \(x_1\) to the point of disappearance \(x_2\); the product \(\Phi(x_1)\Phi^*(x_2)\) approaches the Dirac \(\delta\)-function, and the whole expression \(M\) approaches the expression \(M=\int H'\,dx\), where the role of \(g\) is played by the quantity

\[ g \approx \frac{e_1 e_2 \hbar^2}{m_\Phi^2 c^2} \]

(numerical factors are omitted; \(m_\Phi\) is considerably greater than the masses of all the particles \(A, B, C, D\)). The spin of the particle \(\Phi\) depends on what the operator \(O\) is.

Let us consider a Yukawa-type scheme, in which the initial nucleus emits a meson, being transformed into the final nucleus, while the meson decays into \(e+\nu\). It is clear that the Fermi selection rules, in particular the \(0 \to 0\) transition, require the emission of a meson with zero spin, decaying into \(e+\nu\) in a singlet state.

The G.-T. selection rules require a meson with spin 1.

The experimentally observed combination \(sS+tT\) for the description according to the Yukawa scheme would require the introduction of two hypothetical mesons with different spins (0 and 1), but with approximately equal values of \(e_1 e_2/m_\Phi^2\).

Such a scheme appears highly artificial; therefore the hypothesis of intermediate mesons \(^{7,76,82,83}\) has at present been abandoned. There exists, however, another possible type of intermediate particles, first proposed by Wentzel \(^{8}\) and subsequently considered in works by Japanese authors \(^{76-80}\) and by the author of the present article \(^{81}\). We shall not follow the historical order in the exposition, and shall consider this scheme as applied to the modern data on \(\beta\)-decay.

As was clarified above, the assumption that for the interaction of a “bare” nucleon \(t_0^2=-s_0^2\) (i.e. \(R_\nu=-1\)) is quite probable; it is also possible that \(|p_0|=|s_0|=|t_0|\) (at present one can neither reliably confirm nor refute the presence of a pseudoscalar interaction). Let us assume that for a “bare” nucleon the interaction is described by the combination

\[ H' = s_0(S-T+P) \]

with numerically equal coefficients at \(S\), \(T\), and \(P\).

This combination can be written in a simpler form if the wave functions of different particles are combined in another way*)

\[ H' = 2s_0(\bar S+\bar P), \]

where

\[ \bar S=(\Psi_p^*\beta\Psi_{e^+})(\Psi_\nu^*\beta\Psi_n) \]

in contrast to

\[ S=(\Psi_n^*\beta\Psi_p)(\Psi_\nu^*\beta\Psi_{e^-}); \]

\(\bar P\) is written analogously. Each factor in \(\bar S\) and \(\bar P\) contains one nucleon wave function and one wave function of a light particle. The possibility of such a notation is due to the symmetry properties of the expression \(H'\) noted by Finkelstein and Kaus \(^{52}\): this expression transforms into itself when the wave functions of the neutron and neutrino are interchanged. From the point of view of the theory of intermediate particles in \(\beta\)-decay, expressions of the type \(\bar O(\bar S,\bar P)\) correspond to the scheme

\[ p = L + e^+, \qquad L = n+\nu. \]

The intermediate particle \(L\), evidently, is heavier than the nucleon and, according to the modern classification, should be called not a meson but a hyperon; \(L\) possesses nuclear charge in the sense of \(^{75}\): \(L\) is formed from a nucleon and decays with the formation of a nucleon.

The two terms in the expression \(H'(\bar S\) and \(\bar P)\) correspond to two types of intermediate hyperons \(L\) and \(L'\), both hyperons having zero spin, the same mass, identical coupling constants \(e_1\) and \(e_2\) for the two reactions, and differing only in parity.

In the energy region usual for \(\beta\)-decay, the theory with intermediate hyperons leads to conclusions identical with the ordinary theory based on the expression

\[ H' = sS+tT+pP. \]

*) For the technique of passing from \(\bar O\) to \(O\) for any \(O\), see \(^{20}\).

Without taking meson corrections into account one obtains \(t=-s=p\); the experimentally observed \(|t|>|s|\) can then be explained by meson corrections; the conclusions concerning the ratio of the polarization of the nucleon and of the charged particles, considered above, are obtained directly. It is not difficult to convince oneself without calculations that the expression

\[ H' = 2s(\bar S+\bar P) \]

gives, for the polarization of electrons in the decay of polarized nucleons, precisely the conclusions that were obtained earlier\({}^{34}\) and described above for the case \(R=1\).

Let us consider the production of a nonrelativistic electron. In this case the term \(\bar P\) vanishes; only one term \(\bar S\), corresponding to the production of one intermediate isobar \(L\), plays a role.

When, in the reaction \(\mathrm{p}=L+e^{+}\), a particle \(L\) with zero spin is produced, it “forgets” the initial spin of the proton and the spin of the emitted positron; in the subsequent decay the spins of \(n\) and \(p\) turn out to be uncorrelated with one another. Likewise the spins of \(e^{+}\) and \(\nu\) are not correlated; the probabilities of forming a triplet and a singlet (of total \(e+\nu\) angular momentum equal to 1 or 0) must be in the ratio \(Q\), equal to the ratio of the statistical weights, i.e. \(3:1\). In the usual theory the expression \(H=sS+tT\) gives the ratio

\[ Q=t^{2}|\langle\sigma\rangle|^{2}/s^{2}|\langle1\rangle|^{2}=3R^{2}; \]

for a free nucleon we find

\[ Q=\frac{3t^{2}}{s^{2}}=3R^{2}. \]

Consequently, \(Q=3\) corresponds to \(R=1\).

Fig. 3.

The scheme of the isobars \(L\) and \(L'\) does not contradict the result \(R\ne1\), if the action of meson corrections is taken into account. Denoting the isobar line by a double stroke, let us draw the diagram of the \(\beta\)-process of a “bare” nucleon and the diagram describing the meson correction (Fig. 3).

In the second diagram, the emission and absorption of a \(\pi^0\)-meson create a correlation between the spin of the neutron before decay and the spin of the proton formed (the virtual \(\pi^0\)-meson possesses orbital angular momentum). In the first diagram the line \(L(L')\) does not give such a correlation. Owing to the meson correction, a partial correlation of the spins of \(e^+\), \(\nu\), corresponding to \(R \ne 1\), is also realized.

For allowed transitions one obtains the usual statistical form of the spectrum, with corrections of the type

\[ \left(1 \pm \frac{E_e}{(m_L-m_n)c^2}\right); \]

for \((m_L-m_n)c^2 \gg m_e c^2\) such corrections are not easy to detect. If from such corrections it were possible to determine \(m_L\) and to make the natural assumption \(e_1=e_2\), then the data on \(\beta\)-decay would make it possible to predict all the properties of the particles \(L\) and \(L'\).

These include, first of all, the lifetimes of the free particles \(L\) and \(L'\) (for \(m_L \simeq 2m_n\), \(t \sim 10^{-19} - 10^{-20}\) sec), as well as the energy dependence of the cross section for the resonant reaction

\[ e^- + p = L \]

with the subsequent

\[ L \begin{cases} \nearrow p+e^-\\ \searrow n+\nu \end{cases}. \]

With the very small expected lifetime of \(L\) and \(L'\), the formation of \(L\) and \(L'\) will manifest itself experimentally as resonant scattering of electrons and resonant capture of electrons at a certain large energy of order \((M_L-M_n)c^2\).

Let us recall once more that the possibility of such an interpretation of \(\beta\)-decay essentially requires the presence, in the linear combination of invariants, of a pseudoscalar \(P\) with coefficient \(p \simeq t \simeq s\) (to within meson corrections of order 1).

The absence of electric charge in the particles \(L\) and \(L'\) follows from the difference in the signs of \(s\) and \(t\) (if the signs were the same, \(L\) and \(L'\) would have to be positively charged). The ideas about intermediate particles give particular acute interest to the further experimental and theoretical investigation of the detailed regularities of \(\beta\)-processes. Among the nearest and most important tasks one may name the determination of the sign of the ratio \(s/t\) and the presence of a pseudoscalar interaction.

Thus, in principle, the fascinating possibility is not excluded that, by studying \(\beta\)-processes, one may reveal the existence and properties of new elementary particles.

I take this opportunity to express my gratitude to S. S. Gershtein, L. A. Sliv, and Ya. A. Smorodinskii for discussing the questions touched upon in the article and for valuable comments.

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1. * The question of the polarization of \(\beta\)-particles and its measurement is systematically set forth in \(^{53,26}\). ↩

2. \(^{34}\) ↩

3. \(^{17,20,55}\) ↩