On Parity Nonconservation in $\beta$-Decay*

I. S. Shapiro

§ 1. Introduction

Experimental data obtained recently on new particles ($K$ mesons, hyperons) once again bring the problem of $\beta$-decay to the forefront among the most important problems of nuclear physics, elementary-particle physics, and, apparently, physics in general.

To understand how events that would seem to have occurred in a region far removed from $\beta$-decay lead to such a substantial increase of interest in this phenomenon, one should recall that the interactions known so far in nuclear physics—the nuclear interaction of nucleons, the electromagnetic interaction, and the interaction responsible for $\beta$-decay (the $\beta$-interaction)—are characterized by dimensionless constants that differ very greatly in magnitude: the $\beta$-interaction constant is approximately $10^{12}$ times smaller than the nuclear constant and $10^{11}$ times smaller than the electromagnetic one (see Table Ia).

Table Ia

In such a situation it would not be surprising if the constants of the interactions responsible for the decays of other elementary particles had values differing by an order of magnitude from those indicated in Table Ia. This, however, is not the case. It was observed long ago that the constants of the interactions responsible for the decays of $\pi$- and $\mu$-mesons agree in order of magnitude with the constant of the $\beta$-interaction. Investigation of the properties of the new particles—$K$ mesons and hyperons—showed that the interactions causing the decays of these particles are also characterized by constants that agree in order of magnitude with the constant of the $\beta$-interaction (see Table Ib)**. This circumstance (nontri-

* Revised and supplemented report at the VII Annual Conference on Nuclear Spectroscopy (January 1957).

* The calculations whose results are given in Table Ib were performed by E. I. Dolinskii, using the latest experimental data. Similar calculations were carried out earlier by other authors (see Ya. B. Zel’dovich, UFN 59*, 377 (1956)). We note that in some cases of decays, in order to obtain dimensionless constants, besides the universal constants $\hbar$, $c$, the particle masses were also used.

...nontrivial, if one bears in mind the enormous gap in the values of the constants listed in Table Ia, compels one, at least for heuristic purposes, to assume that the decays of all particles indicated in Table Ib are caused by interactions having something essentially in common with the β-interaction. Following Gell-Mann^1, these interactions are commonly called weak, in contrast to the strong (nuclear) and electromagnetic interactions.

Table Ib

If the point of view expressed is correct, then from the properties of \(K\)-mesons and hyperons there may follow definite consequences, amenable to experimental verification, for the processes of β-decay of nucleons.

§ 2. The “\(\tau-\theta\) problem”

The most remarkable property of \(K\)-mesons is the equality (within experimental errors) of their masses and lifetimes. Especially well established (to an accuracy of up to \(0.2\%\)) is the closeness of the masses of the so-called \(\tau\)- and \(\theta\)-mesons—particles decaying respectively into three and into two \(\pi\)-mesons (see Table II). The lifetimes of these particles also coincide within the limits of experimental error (10–20%). On the basis of what has been said, it is natural to think that here we are dealing with one and the same particle, decaying by different paths that compete with one another.

Table II

In this connection, the determination of the spins and parities of the \(\tau\)- and \(\theta\)-mesons becomes of great interest; these, of course, must be identical if both particles are identical. Information about the spin and parity of the \(\tau\)-meson can be obtained by studying the energy spectrum or angular distribution of the \(\pi\)-mesons produced in the decay of the \(\tau\)-meson and comparing the experimental data obtained with the results of theoretical calculations for various combinations of spin and parity.

At present the best-studied decay is that of the \(\tau^+\)-meson into 3 charged \(\pi\)-mesons:

\[ \tau^+ \to 2\pi^+ + \pi^- . \]

In Fig. 1, borrowed from the work of Shapiro, Dolinskii, and Mikhailova\(^{2}\), a comparison is shown between the experimental data and the theoretical results

Fig. 1.

for the energy spectrum of the \(\pi^-\)-mesons produced in the decay of the \(\tau^+\)-meson. The experimental data represented by the histogram include 492 cases of \(\tau^+\)-meson decay. The theoretical curves were obtained on the basis of fairly general phenomenological (although perhaps not quite rigorous) considerations, an analysis of which it would be inappropriate to enter into here (see \(^{2}\)). As is evident from the figure, one can hardly doubt that the theoretical curve closest to the experimental data is that corresponding to spin and parity \(0^-\) or \(2^-\) (the latter being less probable). Other authors arrive at the same conclusion (see the literature in \(^{2}\)). On the other hand, as is not difficult to show, for the spin and parity of \(\theta\)-particles decaying into two \(\pi\)-mesons (let us recall that the spin of the \(\pi\)-mesons is equal to 0), only the combinations indicated in Table III\(^*\) are possible.

\(^*\) Strictly speaking, for charged particles the combinations \(1^-\), \(3^-\), etc., are also possible. Since, however, neutral \(\theta\)-particles, which are in all probability isobaric analogues of charged \(\theta\)-particles, decay into two \(\pi^0\)-mesons (i.e., into two identical bosons), odd spin values should be excluded. For the questions considered here, however, this is immaterial.

The situation reflected in Table III confronts us with a dilemma:

1) either the $\tau$- and $\theta$-mesons are different particles, and we are dealing with a case of degeneration in which, besides equality of the particle masses, their lifetimes are equal or very close;

2) or the $\tau$- and $\theta$-mesons are identical particles, so that we are dealing with a single particle in whose decay, however, the law of parity conservation is not fulfilled*). Although the first possibility, being less radical,

Table III

seems more acceptable, the facts, as will be seen from what follows, speak against it. We shall discuss here only the second possibility, which leads to a number of consequences essential for $\beta$-decay and which can be subjected to experimental verification.

§ 3. PHENOMENA IN THE $\beta$-DECAY OF NUCLEI THAT DEPEND ON NONCONSERVATION OF PARITY

If parity is not conserved in the decay of $K$-mesons, then, in view of the generality noted above of all weak interactions, one should expect that nonconservation of parity will also occur in the $\beta$-decay of nuclei. Let us note, however, that nuclear states will continue to be characterized by a quite definite parity, since weak interactions play a negligibly small role in the formation of nuclear states. Nonconservation of parity in the processes of $\beta$-decay thus reduces to the fact that the parity of the radiation will not be definite. In other words, the wave function of the radiation (electron, neutrino) will be neither even nor odd: the square of the modulus of the wave function of the radiation will change under mirror reflection of space. This circumstance is explained in Table IV, where the primed quantities refer to the coordinate system obtained from the given one by mirror reflection, and the symbols $\mathbf{p}$ and $\boldsymbol{\sigma}$ denote the momenta and spins of the particles participating in the $\beta$-process under consideration. Noting that the quantity $|f(\mathbf{p}, \boldsymbol{\sigma})|^2$ is proportional to the probability of the $\beta$-process, we arrive at a simple way of finding effects in which nonconservation of parity in $\beta$-decay may manifest itself. Indeed, since in the case of nonconservation of parity the quantity $|f(\mathbf{p}, \boldsymbol{\sigma})|^2$ changes under mirror reflection, it can be represented in the form of a sum of terms even ($F_+$) and odd ($F_-$) with respect to mirror reflection, depending on the quantities measured in the experiment $|\mathbf{p}|$, $\mathbf{p}$, $\boldsymbol{\sigma}$. Thus, a necessary condition for observing an effect depending on nonconservation of parity is such an arrangement of the experiment in which, from the measured quantities, an odd quantity $F_-(\mathbf{p}, \boldsymbol{\sigma})$ (a pseudoscalar)**) can be constructed.

*) As far as can be judged, this hypothesis was first put forward by Feynman in a discussion at the 6th Rochester Conference on High-Energy Physics$^3$. The first detailed discussion of the question belongs to Lee and Yang$^4$.

**) An even quantity $F_+(\mathbf{p}, \boldsymbol{\sigma})$ (a scalar) can always be composed; a constant or any even power of a pseudoscalar are even quantities.

It should be emphasized here that the possibility of constructing a pseudoscalar from the measured quantities is only a necessary, but not a sufficient, condition for the dependence of the effect on the conservation or nonconservation of parity (there may

Table IV

\[ \psi_{\mathrm{em}}=f(\mathbf p,\boldsymbol\sigma)\times(\text{scattered wave}) \]

be cases in which the coefficient of the constructed pseudoscalar turns out to be zero).

On the basis of what has been set forth, it is not difficult to see that nonconservation of parity cannot affect the phenomena that have chiefly been studied up to now in nuclear β decay: investigations of the shape of the β spectrum, of the angular electron—neutrino correlation, and of β—γ angular correlations (see Table V). In the first and last cases there are in general no quantities from which a pseudoscalar could be constructed. In the case of the electron—neutrino correlation, formally, one can construct the pseudoscalar \(\mathbf p_N[\mathbf p_\beta,\mathbf p_\nu]\) from the momenta of the recoil nucleus \((\mathbf p_N)\), the electron \((\mathbf p_\beta)\), and the neutrino \((\mathbf p_\nu)\); however, because of the coplanarity of all three vectors, this quantity vanishes identically.

Table VI indicates the principal phenomena in which nonconservation of parity may manifest itself. These include:

a) the angular distribution of electrons in the β decay of polarized nuclei (nuclear spin \(\boldsymbol\sigma_N\)), which, in the case of nonconservation of parity, will be anisotropic even for allowed transitions (see Table VI)*;

* It should be especially emphasized that in this case what is required is not the so-called ordering of nuclei (arising, for example, at low temperatures in crystalline structures as a result of quadrupole interactions), but precisely polarization of the nuclei, in which the mean value of the projection of the spin of the nuclei onto some direction in space (for example, the direction of an external magnetic field) is different from zero.

Table V

Allowed transitions I.
Effects independent of parity conservation

b) longitudinal polarization of the electrons, i.e. the preferential direction of the electron spin \(\boldsymbol\sigma_e\) along or opposite to the direction of the electron momentum \(\mathbf p_\beta\). The magnitude of the polarization
\(P=\dfrac{n_\downarrow-n_\uparrow}{n_\downarrow+n_\uparrow}\), where \(n_\uparrow\) and \(n_\downarrow\) are the numbers of electrons with spins parallel and antiparallel to \(\mathbf p_\beta\), is equal to zero in the case of allowed transitions if parity is conserved, and may be different from zero if parity conservation does not take place;

c) angular \(\beta-\gamma\) correlation with selection of quanta with a specified (right or left) circular polarization (we note that, from the quantum point of view, selection of circular polarization is equivalent to specifying the projection of the photon spin \(\boldsymbol\sigma_\gamma\) on the direction of its propagation). If parity is conserved, the correlation function \(W_{\beta\gamma}(\vartheta)=\mathrm{const}\), i.e. isotropy will occur. If parity is not conserved, the correlation function will have the form indicated in Table VI*).

*) Regarding this effect, it should be noted that measurement of the circular polarization of \(\gamma\)-quanta, although associated with certain difficulties, is nevertheless entirely feasible, as is shown, for example, by work\(^5\), in which the circular polarization of \(\gamma\)-radiation from the polarized nuclei \(N^{180}\) was measured and thus the sign of the magnetic moment of one of the excited states of this nucleus was determined. As an effect sensitive to the circular polarization of \(\gamma\)-radiation, one uses the strong dependence, noted by Zeldovich\(^6\) and Fano\(^7\), of the Compton scattering cross section back on polarized electrons on the sign of the circular polarization of the quantum (magnetized iron or another ferromagnet may serve as the scatterer with polarized electrons).

Table VI

Allowed transitions II.
Effects dependent on conservation of parity

§ 4. FORMAL THEORY

For the calculation of the quantities \(a\), \(P\), \(B\) (see Table VI), by measuring which one can establish the facts of nonconservation of parity in \(\beta\)-interactions, one should use the phenomenological theory of \(\beta\)-decay, which in the case of nonconservation of parity must be formally modified as shown in Table VII. This modification amounts to the fact that we must now require invariance of the Hamiltonian (energy density) of the \(\beta\)-interaction \(H_\beta\) only with respect to Lorentz transformations and rotations of space, but not mirror reflections, as was required in the theory providing for conservation of parity.

The Hamiltonian of the \(\beta\)-interaction \(H_\beta\), as is known, is the product of two quantities \((p,n)\) and \((e,\nu)\), one of which \(((p,n))\) is composed of the wave functions of nucleons, the other \(((e,\nu))\) of the wave functions of light particles. In the theory providing for conservation of parity, equality of the transformation properties of both quantities with respect to all transformations is required—Lorentz transformations, rotations, and reflections of space. Thus, if the quantity \((p,n)\) is a scalar (the scalar variant of the theory of \(\beta\)-decay),

I. S. SHAPIRO

then the quantity \((e,\nu)\) must also be a scalar if \((p,n)\) is a vector, a tensor, etc.; then the quantity \((e,\nu)\) must correspondingly also be a vector, a tensor, etc. Under this condition the scalar product \((p,n)(e,\nu)\), and consequently also

\[ H_\beta=g(p,n)(e,\nu) \]

where \(g\) is the interaction constant, will be invariant with respect to all the transformations listed above.

In the case of nonconservation of parity, the transformation properties of the quantity \((p,n)\) and of the expression constructed from the wave functions of the electron and neutrino must be the same with respect to rotations of space and Lorentz transformations, but must differ under mirror reflections of space. Thus, for example, if \((p,n)\) is a scalar, then the expression constructed from the wave functions of the light particles must be a sum of a scalar and a pseudoscalar, etc., as is shown in Table VII. The Hamiltonian \(H_\beta\) constructed in this

Table VII

Formal theory

way will change under mirror reflections of space, but will be invariant with respect to rotations and Lorentz transformations. It is essential that, in the case under consideration, each variant of the theory of \(\beta\)-decay will, generally speaking, be characterized not by one but by two constants \(g\) and \(g'\) (see Table VII). As is now known, in reality there is a superposition of the scalar and tensor variants (for a remark concerning a possible admixture of the axial-vector variant, see § 6). For this reason, in the new theory four constants will appear: two constants of the scalar variant \((g_S, g'_S)\) and two constants of the tensor variant \((g_T, g'_T)\). Using the formal theory constructed according to the indicated principles, one can calculate the quantities of interest to us \(a\), \(P\), \(B\) (see Table V). For a superposition of the scalar and tensor variants and neglecting the Coulomb field of the nucleus, the results of such calculations, carried out by V. V. Turovtsev, are given in Tables VIII, IX, X*). Here \(I,m\) denote the spin of the nucleus and its projection on the direction of the \(z\)-axis, while \(a\) and \(b\) are matrix elements of the nuclei, independent of \(m\), for allowed \(\beta\)-transitions,

\[ a=\int 1,\qquad b=\frac{1}{m}\int \sigma_z . \]

*) In the work of Lee and Yang cited above\({}^{4}\), the coefficient \(a\) was calculated for a superposition of the tensor and axial-vector variants.

As is seen from the tables, the quantities \(\alpha, P, B\) are proportional to the real part of the product of the constants \(gg'\) and also depend on the ratio of the moduli of these constants.

Table VIII

Allowed transitions III

\[ \begin{gathered} \left. \begin{array}{c} \alpha\\ P\\ B \end{array} \right\} \sim \frac{v}{c}\, \frac{\operatorname{Re}(g,g'^{*})}{|g|^{2}+|g'|^{2}} \end{gathered} \]

Table IX

Allowed transitions IV

Table X

Allowed transitions V

It follows from this that, generally speaking, even in the case of nonconservation of parity the effects considered above may be absent if

one of the constants, for example \(g\), is real, and the other (\(g'\)) is purely imaginary. Thus, a purely formal theory alone is insufficient for clarifying the question of experimentally observed phenomena associated with nonconservation of parity. A purely formal conception is also unsatisfactory for the following reasons. The starting point of such a theory, as we have seen, is the construction of an interaction Hamiltonian \(H_\beta\) that is not invariant with respect to mirror reflection of the coordinate system. Strictly speaking, such an approach is physically meaningless, since the energy density of the interaction must be invariant under any transformation of reference frames. Indeed, otherwise some threshold reaction—say, the splitting of a nucleus by a \(\gamma\)-quantum—could occur, for example, in a right-handed coordinate system and be impossible in a mirror, left-handed coordinate system; this is obviously meaningless, since the splitting or non-splitting of a nucleus is an objective fact independent of the reference frame.

For the reasons set forth, further investigation of the question requires discussion of hypotheses concerning the causes of parity nonconservation. At the same time, such a modification of modern theoretical concepts is necessary under which parity nonconservation would be compatible with the invariance of the energy of the \(\beta\)-interaction with respect to mirror reflection of space.

§ 5. WHAT DOES PARITY NONCONSERVATION MEAN?

Two kinds of causes of parity nonconservation are conceivable:

1. Parity nonconservation may be a consequence of an internal asymmetry of particles with respect to right and left. This point of view, already contained in the first paper of Lee and Yang\(^4\), was most distinctly formulated for the first time by Landau\(^9\) and was also developed (in another version, see below) by Ioffe\(^ {10}\).

2. Parity nonconservation may be due to the properties of space, to its structure at small lengths. Some considerations on this matter will be given at the end of the present article.

Let us first consider the first possibility in more detail.

The concept of parity is inseparably connected with the assumption that a particle is an object symmetric with respect to “right” and “left.” For this reason, under mirror reflection of space (in which, as is known, “right” is replaced by “left”) the particle will, as it were, transform into itself; in other words, the particle will in no way differ from its mirror image. In this case, which alone was considered in the old theory, the wave function of the particle will change only insignificantly under mirror reflection: according to the general principles of quantum mechanics, the wave functions of two physical systems (in the present case, a particle and its mirror image) that do not differ from each other in any way can differ only by a numerical factor whose modulus is unity\(^*\). Thus, if \(\psi\) is the wave function of the particle, then under reflection

\[ \psi \to \xi \psi, \tag{1} \]

where

\[ |\xi| = 1. \]

\(^*\) For simplicity we speak here of particles with zero spin. In the presence of spin, the wave function of the particle will, under reflection, be multiplied by a certain numerical matrix, which, however, does not change the matter. For transformations of wave functions of particles with spin under reflections, see\(^ {11}\).

ON NONCONSERVATION OF PARITY IN \(\beta\)-DECAY

Since a twice repeated reflection is, on the one hand, the identity transformation, and, on the other hand, under it

\[ \psi \to \xi^2 \psi, \]

it follows that

\[ \xi^2 = 1, \]

or

\[ \xi = \pm 1. \]

The number \(\xi\) is called, as is well known, parity. All this is true only so long as, under mirror reflection, a particle goes over into itself, i.e., so long as we consider the particle to be an object symmetric with respect to “right” and “left.” If this is not so, then the arguments leading to the concept of parity can no longer apply. Indeed, suppose that a particle possesses a certain internal asymmetry with respect to right and left, analogous, for example, to a molecule of tartaric acid, which rotates the plane of polarization of light to the right or to the left. For definiteness, we shall say that in this case a certain right- or left-handed screw is associated with the particle. Since, upon reflection in a mirror, a right-handed screw goes over into a left-handed one, under mirror reflection of space the particle will no longer go over into itself, that is, it will not be identical with its mirror image. For this reason the wave function of the particle is no longer obliged to transform according to the law (1); it must change in a more essential way. Since, as we have seen, the concept of parity is a consequence of the transformation (1), in this case the concept of parity is automatically eliminated: no parity whatever can be assigned to a particle internally asymmetric with respect to “right” and “left.” It is clear that, by virtue of the elimination of the very concept of parity, the question of the law of conservation of parity also disappears.

Landau\(^9\) put forward a hypothesis in which the internal “right-left” asymmetry of particles is associated with charge. This means, speaking pictorially, that if a particle with positive charge (for example, a proton) is characterized by a right-handed screw, then the antiparticle, i.e., the particle with negative charge (for example, an antiproton), is characterized by a left-handed screw. Since the mirror reflection of a right-handed screw is a left-handed screw, the mirror reflection of a particle must be an antiparticle. In this variant of the theory it turns out to be possible to achieve invariance of the Hamiltonian of the \(\beta\)-interaction \(H_\beta\) with respect to transformations of mirror reflection of space\(^*\), if the interaction constants entering the Hamiltonian \((g_S, g'_S, g_T, g'_T\), etc.) are either real, or all purely imaginary. As is seen from Table VIII, in this variant of the theory the quantities \(\alpha\), \(P\), \(B\) will be nonzero and, consequently, all the effects considered above (see Table VI) associated with nonconservation of parity will occur.

In the variant under consideration, the symmetry of the theory with respect to charge conjugation is lost, i.e., with respect to replacement of all particles by antiparticles

\(^*\) This is achieved, in accordance with what was said above, by changing the transformation law of the wave functions: because of the internal right-left asymmetry of the particle associated with charge, under space inversion, instead of the transformation law (1), there is the transformation

\[ \psi_{\text{part}} \xrightarrow{\,r \to -r\,} \xi \cdot \psi_{\text{antipart}} \]

(where \(\xi\), in the general case, is a number or a matrix), which Landau calls “combined inversion.”

(without reflection of space). Indeed, let us imagine, for example, a system consisting of light waves and positively charged particles. Since a certain “right-left” asymmetry is now inherent in a definite sign of the charge, for example a “right-handed screw,” such a system of charges can rotate the plane of polarization of light to the right, whereas the charge-conjugate system, consisting of negative charges, will rotate the plane of polarization to the left. It is obvious that in Landau’s conception the concept of parity can be introduced only for absolutely neutral particles, i.e., for particles which have no antiparticles or, in other words, for which the particle and antiparticle are identical. In this (and only in this) case the particle will be identical with its mirror image and, consequently, for the wave function of a neutral particle \(\psi^0\) we shall again have a transformation of the type (1):

\[ \psi^0 \to \xi_k \psi^0,\qquad \xi_k=\pm 1. \tag{2} \]

The quantity \(\xi_k\), however, will not be equal to the parity \(\xi\) in the traditional sense of this word. Indeed, the operation of reflection in Landau’s version consists, if one uses the terminology of the old theory, of reflection and charge conjugation. With the first of these operations in the old theory there is associated the ordinary parity \(\xi\); with the second, in the case of absolutely neutral particles, the so-called charge parity \(\xi_c\)\(^*\). In view of what has been said, \(\xi_k\) is equal to the product of \(\xi\) and \(\xi_c\),

\[ \xi_k=\xi\xi_c. \tag{3} \]

Landau proposed to call the quantity \(\xi_k\) the combined parity.

Examples of absolutely neutral particles may be the quanta of the electromagnetic field, \(\pi^0\)-mesons, and \(K^0\)-mesons. In contrast to this, neutrons and, as will be seen from what follows, neutrinos are not absolutely neutral particles.

The combined parity of the potentials of the electromagnetic field is equal to \(+1\) \((\xi=-1,\ \xi_c=-1)\), that of \(\pi^0\)-mesons is \(-1\) \((\xi=-1,\ \xi_c=+1)\); as for \(K^0\)-mesons, as has recently been established, there exist two sorts of \(K^0\) mesons with \(\xi_k=\pm 1\). The combined parity of an isolated absolutely neutral system will be conserved. The difference from the old theory consists in the fact that in the latter the parity \((\xi)\) and the charge parity \((\xi_c)\) are separately conserved, whereas in Landau’s version only the product of these quantities is conserved. We have seen that the conception proposed by Landau leads to a number of experimentally testable consequences, indicated in Tables VI, VIII, IX, X, since in this variant \(\operatorname{Re} gg'^* \ne 0\). From the considerations developed so far, however, it is not yet possible to draw any conclusions about the magnitudes of \(g'/g\), on which the quantitative aspect of the matter depends. Using Tables VIII, IX, X, it is not difficult to notice that all effects attain their maximum magnitude at \(g=g'\). In particular, the magnitude of the longitudinal polarization of \(\beta\)-particles (electron or neutrino) \(P\) then becomes simply equal to \(\pm v/c\), where \(v\) is the velocity of the particle, \(c\) the velocity of light in vacuum. If one assumes that the mass of the neutrino is identically equal to zero, i.e. \(v_\nu=c\), then we obtain that all neutrinos in \(\beta\)-decay will be completely longitudinally polarized (along or opposite to the direction of motion), depending on whether the signs of \(g\) and \(g'\) coincide or differ. A remarkable feature of the variant of the theory under consideration is the possibility noted by Landau\(^9\)

\(^*\) If a particle is absolutely neutral, then under charge conjugation its wave function will transform according to the law \(\psi^0 \to \xi_c \psi^0\), where \(\xi_c=\pm 1\) is the charge parity (for more detail see \({}^{11}\)).

and, independently, somewhat later, Lee and Yang\(^{12}\), the possibility of assuming that all neutrinos in the world, irrespective of the manner of their generation, are completely longitudinally polarized (“longitudinal neutrinos,” in Landau’s terminology). From this assumption it follows automatically that the neutrino mass must be identically equal to zero. This circumstance makes the variant of the theory proposed by Landau especially attractive and entails a number of additional consequences.

Let us consider, for example, the decay of \(\pi^\pm\)-mesons into a \(\mu\)-meson and a neutrino:

\[ \pi \to \mu + \nu . \tag{4} \]

If the neutrinos are completely longitudinally polarized, then, since the spin of the \(\pi\)-meson is equal to zero, the \(\mu\)-mesons will also be completely longitudinally polarized (the sign of the longitudinal polarization of the \(\mu\)-mesons will, evidently, be opposite to the sign of the longitudinal polarization of the neutrino). As is known, the \(\mu\)-meson formed in process (4) subsequently itself decays according to the scheme

\[ \mu \to e + \nu + \tilde{\nu} \tag{5} \]

or, possibly,

\[ \mu \to e + 2\nu, \tag{6} \]

where \(\tilde{\nu}\) denotes an antineutrino. Since here we are dealing with the decay of polarized \(\mu\)-mesons, the same phenomenon will be observed as in the \(\beta\)-decay of polarized nuclei: the angular distribution of the decay electrons will be anisotropic with respect to the direction of polarization of the \(\mu\)-meson, i.e. with respect to the direction of its motion before it came to rest. The angular distribution function for decay according to scheme (5) will have the form

\[ dW(\varepsilon,\vartheta)=2\varepsilon^2(3-2\varepsilon)(1+\alpha\cos\vartheta)\,d\varepsilon, \tag{7} \]

\[ \alpha=\frac{2ab}{a^2+b^2}\frac{2\varepsilon-1}{3-2\varepsilon}, \tag{8} \]

where \(\varepsilon\) is the ratio of the electron energy to the maximum possible energy, and \(a\) and \(b\) are unknown real constants entering into the interaction Hamiltonian\(*\).

Another most important consequence of the longitudinal-neutrino hypothesis is the nonidentity of the neutrino and the antineutrino, which in this version of the theory are different particles. This is easy to understand in the light of all that has been said above, since the “right-left” internal asymmetry of the particles in the case of a longitudinal neutrino receives an especially vivid expression: as the “right-handed screw” associated with the neutrino one may regard the neutrino spin, directed, for all neutrinos, along their direction of motion. Under mirror reflection of space, the right-handed screw will become a left-handed one, and we shall obtain a particle whose spin is directed opposite to its motion. This particle is

\(*\) The interaction Hamiltonian in this case will have the form \(H_\beta=(e,\mu)(\tilde{\nu},\nu)\). From the wave functions of longitudinal neutrinos and antineutrinos one can form only a vector combination, more precisely

\[ (\tilde{\nu},\nu)=(\text{vector})+(\text{pseudovector}). \]

Correspondingly, for \((e,\mu)\) we have:

\[ (e,\mu)=a\,(\text{vector})+b\,(\text{pseudovector}). \]

obviously, antineutrino, since in Landau’s variant the mirror image of a particle is its antiparticle. We see that the neutrino and the antineutrino are clearly distinguished in this case: whereas the longitudinal polarization of the neutrino is \(P_\nu=+1\), the longitudinal polarization of the antineutrino is \(P_\nu=-1\)*). Thus, in the hypothesis of the longitudinal neutrino, the neutrino and antineutrino are different particles. This conclusion is consistent with the present experimental data on the absence of double \(\beta\)-decay of \(\mathrm{Ca}^{48}\) with the emission of only two electrons (without neutrinos)—see \(^{13-15}\), and also with the negative result of searches for the process

\[ \bar{\nu}+\mathrm{Cl}^{37}\to \mathrm{A}^{37}+e^- \]

under the action of neutrinos emitted from a nuclear reactor (see \(^{15a}\)).

Interesting consequences of the longitudinal-neutrino hypothesis also follow for the form of the spectrum of electrons produced in the decay of \(\mu\)-mesons. Theoretically the form of the spectrum is determined by a single real parameter \(\rho\), introduced by Michel \(^{16}\) and depending on the variant of the theory. If the \(\mu\)-meson decays according to scheme (6), then under the longitudinal-neutrino hypothesis

\[ \rho=0. \tag{9} \]

If, however, scheme (5) takes place, then

\[ \rho=0.75. \]

The experimental data completely exclude the possibility (6) and speak in favor of (5) (\(\rho=0.64\pm0.10\) and \(\rho=0.57\pm0.14\); see \(^{17,18}\)).

We have considered one of the variants of the theory of asymmetric particles, in which the internal “right-left” asymmetry of particles is connected with charge. Ioffe \(^{10}\) considered another possibility, in which the screw asymmetry of particles was connected with the evolution of processes in time. According to this hypothesis, the mirror image of a particle at the instant \(t\) is its state at the instant \(-t\). Since, because of the presence of screw asymmetry, the mirror image of a particle is not identical with the particle itself, in Ioffe’s variant the theory becomes noninvariant with respect to time reflection). The requirement of invariance of the \(\beta\)-interaction Hamiltonian \(H_\beta\) with respect to mirror reflection of space*) leads in this theory to the condition

\[ \operatorname{Re} gg'^{*}=0, \]

as a result of which all the effects considered above that depend on nonconservation of parity disappear. Instead of them there arise other phenomena, for example, such as the correlation of the polarizations of the electron and the recoil nucleus in \(\beta\)-decay, etc., whose experimental investigation is considerably more complicated than the detection of the effects listed in Table VI.

We shall not dwell in greater detail on this variant of the theory of asymmetric particles, since the experimental data that have appeared very recently speak against it and do not contradict the variant considered by Landau and by Lee and Yang.

*) Of course, what is important here is the very fact of a difference in signs, and not the terminology. One could call the particle with \(P_\nu=-1\) the neutrino; then for the antineutrino one would have \(P_\nu=+1\).

**) It remains, however, invariant with respect to charge conjugation.

***) Of course, with the new transformation law for the wave functions

\[ \psi_{+t}\to \xi\psi_{-t}. \]

§ 6. Experimental Data

The experimental data mentioned above concern the angular distribution of electrons in the β-decay of polarized Co\(^ {60}\) nuclei \(^{19}\), and the angular distribution of electrons in the decay of μ-mesons relative to the direction of motion of the latter \(^{20}\). In both cases an asymmetry of the angular distribution “forward–backward” was found, which is evidence of nonconservation of parity.

The clearest data were obtained in the experiment with polarized Co\(^ {60}\) nuclei. The well-known decay scheme of Co\(^ {60}\) is shown in Fig. 2. The β-transition

Fig. 2.

Visible labels in the figure: Co\(^ {60}\); Ni\(^ {60}\); \(5.27\cdot 10^5\), \(0^+\); \(99.8\%\) \((\lg ft=7.46)\); \(\sim 10^{-3}\%\); \(\beta^- 1480\), \(0.15\%\) \((\lg ft=12.6)\); \(2504;\ 4^+\ 10^{-11}\) sec; \(2180;\ 2^+\); \(1332;\ 2^+\ 7.5\cdot 10^{-13}\) sec; \((\text{energy in keV})\); \(0\), \(0^+\).

in this case, according to a number of indications, is allowed and belongs to the type \(I \to I-1\) with \(I=5\). The polarization of Co\(^ {60}\) nuclei, obtained at low temperatures with the aid of a magnetic field by the Gorter–Rose method, amounted to about 60% (the degree of polarization was measured from the anisotropy of the γ-radiation of Ni\(^ {60}\) formed in the decay of Co\(^ {60}\)). The mean energy value of the electrons registered by the scintillation counter corresponded to the quantity \(\frac{v}{c}\), approximately equal to 0.6. As is not difficult to see from the formula given in Table IX, the maximum possible value of the coefficient (for \(g=g'\)) in this case is the quantity 0.36, which, within the experimental errors, agrees with the experimental value 0.4 measured from the “forward–backward” asymmetry. Thus, the experiments on the β-decay of Co\(^ {60}\) do not contradict the longitudinal-neutrino hypothesis. It should be noted that these experiments do not yet imply confirmation of Landau’s general conception (the mirror image of a particle is an antiparticle), since according to this hypothesis, as we noted above, it is required that all the constants of the β-interactions be simultaneously either real or purely imaginary. In the β-decay of Co\(^ {60}\) the scalar interaction, by virtue of the selection rules, plays no role, and essentially only the tensor interaction enters. For this reason it proves impossible to establish the phase difference of the constants of the scalar and tensor interactions from experiments with Co\(^ {60}\). Let us note that the pseudovector interaction, if it exists, must be characterized by a constant with a very small real part if the tensor constant is real (this follows from the form of the β-spectrum for allowed transitions, see \(^{8}\)), moreover \(\left|\frac{g_A}{g_T}\right|^2 \ll \frac{1}{3}\).

\(\{g_A\)—the constant of the pseudovector interaction}. As Lee and Yang note\(^{12}\), because of the action of the Coulomb field of the nucleus, the expression for \(\alpha\) may contain a term proportional to the imaginary part of the product of the tensor and pseudovector constants. The accuracy of the experiments described above is apparently still insufficient for clarifying the role of the pseudovector variant. We note that, because of the certainly different phases of the tensor and pseudovector constants, the latter must be equal to zero if Landau’s general conception is correct.

§ 7. CONCLUSION

It was indicated above that the conception of intrinsically asymmetric “screwlike” particles is not the only conceivable way of explaining nonconservation of parity. Moreover, it should be noted that, for all its attractiveness and ingenuity, it leaves unresolved the question of the reason for the difference between weak interactions, on the one hand, and strong (nuclear) and electromagnetic interactions, on the other, with respect to parity conservation. For these and certain other considerations it seems interesting to consider the question of parity nonconservation from the standpoint of a change in our ideas about the structure of space. If one proceeds from this point of view, then one may suppose that the structure of space is such that the transformation of mirror reflection in it is impossible. In other words, the concepts of “right” and “left” do not exist in such a space, just as, for example, the concepts of an inner and an outer normal to the surface of a Möbius strip do not exist. In the language of transformation theory this means that the group of transformations which carries the space into itself and leaves one point fixed is connected, i.e., is not split into two separate sets (rotations and reflections), as is the case in Euclidean space. We shall call such a space nonorientable or one-sided. Since in a one-sided space mirror reflections are impossible, the problem of parity nonconservation is solved in a trivial way: the concept of parity ceases to exist.

It is clear, however, that such a hypothesis can apply only to the structure of space at small lengths, since the Euclidean character of macroscopic regions of space is an experimental fact\(^*\). Since parity conservation takes place in nuclear and electromagnetic interactions, this means that the nonorientability of space (if it exists at all) must manifest itself only at distances small in comparison with the lengths essential for electromagnetic and nuclear interactions. For the latter, as is known, the characteristic lengths are of the order of the Compton wavelength of the \(\pi\)-meson, i.e., lengths of order \(10^{-13}\) cm. Thus the conception being developed may have meaning if it turns out that the processes caused by weak interactions occur at distances small compared with \(10^{-13}\) cm. Everything that we know about weak interactions does not contradict such an assumption. The only constant with the dimension of length that can be constructed from the Fermi constant \(g\) \((g \simeq 10^{-49}\,\text{erg}\,\text{cm}^3)\) and the universal constants \(\hbar\) and \(c\) is

\[ l_0=\sqrt{\frac{g}{\hbar c}}\simeq 6\cdot 10^{-17}\ \text{cm}, \]

at least \(10^3\) times smaller than the characteristic nuclear length. It is further noteworthy that the energy spectrum of \(\pi^-\)-mesons presented in Fig. 1,

\(^*\) One cannot assume that weak interactions condition a small non-Euclidean character of the entire space, since the effects of parity nonconservation are not small—the amplitudes of the even and odd parts of the radiation wave function are comparable in magnitude.

formed in the \(\tau\)-decay of the \(K\)-meson, agrees extremely accurately with the theoretical curve for spin and parity \(0^{-}\), calculated with account of \(\pi\)-mesons emitted with zero orbital angular momenta. Zero orbital angular momenta, with the mean de Broglie wavelength of the \(\pi\)-mesons of the order of \(10^{-13}\) cm, agree with the assumption that, in the \(\tau\)-decay of \(K\)-mesons caused by the weak interaction, the emission of \(\pi\)-mesons takes place from distances much smaller than \(10^{-13}\) cm.

In the concept under consideration, space is nonhomogeneous: it is nonorientable at small distances and orientable at distances large in comparison with a certain critical length \(l_{0}\). The structure of such a space is explained schematically in Fig. 3, where the regions of nonorientability are symbolically represented by circles.

Fig. 3.

In a nonhomogeneous space, however, the concept of a particle’s momentum cannot be strictly defined. When a particle moves in such a space there would occur, speaking pictorially, scattering of the particle by the inhomogeneities of space, so that there would be a certain uncertainty in the momentum, \(\Delta p/p\), depending on the ratio \(l_{0}/\lambda\), where \(\lambda\) is the wavelength of the particle. If, however, one assumes that the properties of space are not something prepared in advance, so that a particle enters this space “like a tenant into a ready-made apartment” (Weyl), but that, on the contrary, the properties of space are determined by the physical processes taking place in it (as, for example, is the case in the general theory of relativity), then the uncertainty in momentum noted above will manifest itself only in processes in which weak interactions are essential, in particular in the decay of elementary particles, but not in their free motion or in strong (nuclear) and electromagnetic interactions. As already indicated, this uncertainty must be some function of the ratio \(l_{0}/\lambda\), and for \(l_{0}/\lambda \ll 1\) it may be proportional to \(l_{0}/\lambda\). If, in accordance with what was said above, one assumes that \(l_{0}\) is of the order of \(10^{-16}\) cm, then in the decay of \(K\)-mesons \(\Delta p/p\) will be of the order of \(10^{-3}\), i.e. about \(0.1\%\).

Let us note that \(\beta\)-interactions turn out to be very strongly singular (for example, the \(\beta\)-interaction of nucleons \(\sim 1/r^{3}\) in the \(g^{2}\)-approximation, which follows already from the dimension of the constant \(g\)). Since it is not yet clear at precisely what distances the singular interactions should be cut off, it is not excluded that at sufficiently small distances “weak” interactions will cease to be weak and will prove to be the most essential ones for the structure of elementary particles.

As is clear from the foregoing, the fact of nonconservation of parity may lead to very far-reaching and unusual consequences. The development of physics has taught us, however, to treat habitual concepts that seem self-evident with caution. I think that a good conclusion to this article may be supplied by the words of Riemann, spoken almost 100 years ago in his famous lecture “On the hypotheses which lie at the foundations of geometry”: “The empirical concepts on which the determination of spatial metric relations is based—the concepts of a rigid body and of a ray of light—apparently lose all definiteness in the infinitely small. It is therefore quite conceivable that the metric relations of space in the infinitely small do not correspond to our geometrical assumptions; we should indeed have to accept this position if, with its help, the observed phenomena were more simply explained.”

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