STRANGE PARTICLES

(Scheme of Isotopic Multiplets)

L. Okun

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
2. Masses and decay schemes of \(K\)-mesons . . . . . . . . . . . . . . . . . . . . . . . . 536
3. Masses and decay schemes of hyperons . . . . . . . . . . . . . . . . . . . . . . . . . 537
4. Principal features of strange particles . . . . . . . . . . . . . . . . . . . . . . . . . 538
5. Types of interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
6. Isotopic spin of \(\pi\)-mesons and nucleons . . . . . . . . . . . . . . . . . . . . . . . 541
7. Isotopic spin of \(K\)-mesons and hyperons . . . . . . . . . . . . . . . . . . . . . . . 541
8. Strangeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
9. Strong interactions. \(\Delta S = 0\) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
   Production of strange particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
   Scattering and absorption of strange particles . . . . . . . . . . . . . . . . . . . . 546
   Hypernuclei and \(K\)-nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
10. Strong interactions. \(\Delta T = 0\) . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
11. Electromagnetic interaction. \(\Delta S = 0\) . . . . . . . . . . . . . . . . . . . . . . . 550
12. Decays of strange particles. \(\Delta S = \pm 1\) . . . . . . . . . . . . . . . . . . . . . 551
13. Decays of strange particles. \(\Delta T = \pm \frac{1}{2}\) . . . . . . . . . . . . . . . . 552
14. Other possible particles in the scheme of isotopic multiplets . . . . . . . . . . . . 554
15. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
    Appendix I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
    Appendix II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
    References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

1. INTRODUCTION

Experimental investigations and theoretical analysis of the properties of heavy mesons and hyperons have recently led to a reconsideration of physical concepts that had seemed quite firmly established.

The study of the reactions of production and decay of these “strange” particles has forced an expansion of the range of applicability of the concept of isotopic spin. The classification of elementary particles that arose on this basis—the scheme of isotopic multiplets \(^{1-6}\)—has, in the almost three years of its existence, not encountered a single experimental fact that would contradict it.

Another group of questions that arose in the study of various decays of \(K\)-mesons (and in particular the decays \(K^+ \to 3\pi\) and \(K^+ \to 2\pi\)) is connected with the concept of parity. The abandonment of the law of conservation of spatial parity in weak interactions, to which the experimental data compel us, entails a reconsideration of such concepts as time and charge parity. The possibilities that arise thereby are attracting close attention from both theorists and experimentalists and are now the object of intensive research.

In this review we shall consider questions connected with the isotopic spin of strange particles.

2. MASSES AND DECAY SCHEMES OF \(K\)-MESONS

Let us briefly recall the principal properties of “strange” particles\(^*\). Strange particles are heavy mesons (\(K\)-mesons) and hyperons. The name “\(K\)-mesons” refers to mesons whose mass is greater than the mass of the \(\pi\)-meson. Recently, mesons with mass \(966\,m\) have usually been called \(K\)-mesons. We shall also adhere to this terminology.

Positive, negative, and neutral \(K\)-mesons are known. \(K\)-mesons are unstable. The decays of \(K^+\)-mesons have been studied in the greatest detail. The following modes of decay of \(K^+\)-mesons have been established:

\[ \begin{aligned} K_{\pi2}^{+}(\theta^{+},\,\chi^{+})\qquad &K^{+}\to \pi^{+}+\pi^{0}+214\ \text{Mev} &&25\% \\[4pt] K_{\pi3}^{+}(\tau^{+})\qquad &K^{+}\to 2\pi^{+}+\pi^{-}+75\ \text{Mev} &\left.\begin{array}{l}\\[-6pt]\\\end{array}\right\} &8\%\\[-2pt] K_{\pi3}^{+}(\tau^{+\prime})\qquad &K^{+}\to 2\pi^{0}+\pi^{+}+84\ \text{Mev} && \\[4pt] K_{\mu2}^{+}\qquad &K^{+}\to \mu^{+}+\nu+388\ \text{Mev} &&60\% \\[4pt] K_{\mu3}^{+}(\chi^{+})\qquad &K^{+}\to \mu^{+}+\nu+\pi^{0}+253\ \text{Mev} &\left.\begin{array}{l}\\[-6pt]\\\end{array}\right\} &7\%\\[-2pt] K_{e3}^{+}(K_{\beta}^{+})\qquad &K^{+}\to e^{+}+\nu+\pi^{0}+359\ \text{Mev} && \end{aligned} \]

The numbers in percent indicate the relative abundances of the decays. In parentheses are given the designations of mesons that were introduced at a time when it was still assumed that a separate type of \(K\)-meson corresponded to each of the decays. Now, apparently, it should be considered that all these decays belong to one and the same \(K\)-meson. This is indicated by the fact that the masses of mesons decaying in different ways, with an accuracy up to \(2m\),\({}^{10}\) are equal to one and the same value (\(966\,m\)), and their lifetimes coincide within the limits of experimental errors \((\tau\sim 1.2\cdot 10^{-8}\ \text{sec}^{11-12})\). In addition, the abundances of the various \(K\)-decays do not depend on the method of production of the \(K\)-mesons and do not change as a result of the scattering of \(K\)-mesons by nuclei,\({}^{13-14}\) which is also an argument in favor of the assumption that all the decays listed above are decays of one and the same particle.

The decays of negative \(K\)-mesons have been studied considerably less well than the decays of \(K^+\)-mesons. This is explained by the fact that \(K^-\)-mesons are absorbed by nuclei of photographic emulsions in a time shorter than the decay time. \(K_{\pi2}^{-}\), \(K_{\pi3}^{-}\), and \(K_{e3}^{-}\)-decays have been observed.\({}^{9}\) The lifetime of \(K^-\)-mesons, within the limits of experimental error, coincides with the lifetime of \(K^+\)-mesons. The mass of \(K^-\)-mesons is equal to the mass of \(K^+\)-mesons.

The existence of two types of neutral \(K\)-mesons has been established: \(K_{1}^{0}\) and \(K_{2}^{0}\).

The most studied is the \(K_{1}^{0}\)-meson, which decays according to the scheme

\[ K_{1}^{0}\to \pi^{+}+\pi^{-}. \]

Another mode of decay of the \(K_{1}^{0}\)-meson is the decay\({}^{15}\)

\[ K_{1}^{0}\to \pi^{0}+\pi^{0}. \]

\(^*\) Detailed reviews of the experimental data relating to strange particles were published in \({}^{7-9}\).

The lifetime of the \(K_1^0\)-meson is \(1—2\cdot 10^{-10}\) sec (approximately 100 times less than the lifetime of the \(K^+\)-meson). The mass of the \(K_1^0\)-meson is equal to \(965\,m\).

The \(K_2^0\)-meson, whose existence was predicted by Gell-Mann and Pais \(^{16-17}\), was discovered experimentally quite recently \(^{(18)}\). The lifetime of this meson has been determined only very roughly; it lies between \(10^{-6}\) sec and \(10^{-9}\) sec. The observed decays of the \(K_2^0\)-meson apparently correspond to the schemes:

\[ K_2^0 \to \mu^\pm + \nu + \pi^\mp, \]

\[ K_2^0 \to e^\pm + \nu + \pi^\mp. \]

3. MASSES AND DECAY SCHEMES OF HYPERONS

Hyperons—particles heavier than the neutron but lighter than the deuteron—are denoted by the letter \(Y\). All known hyperons are unstable. The lifetime of all hyperons (with the exception of the \(\Sigma^0\)-hyperon) is of the order of \(10^{-10}\) sec. In all known reactions of hyperon formation, the rest mass of the hyperon necessarily contained within it the rest mass of the nucleon participating in the reaction: the hyperon arose, as it were, as a result of excitation of the nucleon. On the other hand, nucleons always arise in the decay of hyperons. This permits one to speak of hyperons as peculiar excited nucleons \(^{19}\).

The lightest hyperon, \(\Lambda^0\), has a mass of \(2182\,m\) and decays according to the scheme*)

\[ \Lambda^0 \to p + \pi^- + 37\ \text{Mev}. \]

Another mode of decay of the \(\Lambda^0\)-hyperon has also been observed \(^{11}\):

\[ \Lambda^0 \to n + \pi^0 + 41\ \text{Mev}. \]

The lifetime of the \(\Lambda^0\) hyperon is \(\tau \sim 3\cdot 10^{-10}\) sec.

The existence of three \(\Sigma\)-hyperons (\(\Sigma^+\), \(\Sigma^-\), and \(\Sigma^0\)) has also been established. Two types of \(\Sigma^+\)-hyperon decays have been observed:

\[ \Sigma^+ \to n + \pi^+ + 110\ \text{Mev}, \]

\[ \Sigma^+ \to p + \pi^0 + 116\ \text{Mev}, \]

the probabilities of these decays proving to be approximately equal.

The lifetime of the \(\Sigma^+\)-hyperon is \(\tau \sim 1\cdot 10^{-10}\) sec. The lifetime of the \(\Sigma^-\)-hyperon, decaying according to the scheme

\[ \Sigma^- \to n + \pi^- + 118\ \text{Mev}, \]

is equal to \(\tau \sim 2\cdot 10^{-10}\) sec.

It is interesting that not only the lifetimes differ, but also the energies released in the decays of the \(\Sigma^+\)- and \(\Sigma^-\)-hyperons (110 Mev and 118 Mev), and consequently also the masses of these hyperons. According to the latest measurements \(m_{\Sigma^+}=2327\,m\), \(m_{\Sigma^-}=2342\,m\), so that \(m_{\Sigma^-}-m_{\Sigma^+}\simeq 15\,m\).

The neutral \(\Sigma\)-hyperon (\(\Sigma^0\)) was discovered after its existence had been predicted by Gell-Mann \(^{1}\) on the basis of the isotopic-multiplet scheme. The \(\Sigma^0\)-hyperon decays according to the scheme:

\[ \begin{aligned} \Sigma^0 &\to \Lambda^0 + \gamma \\ &\ \ \ \downarrow p + \pi^- \end{aligned} \]

*) \(p\) and \(n\) denote, respectively, the proton and neutron, \(N\)—a nucleon.

The lifetime of the \(\Sigma^0\)-hyperon has not been measured; it should be expected to be close to \(10^{-20}\) sec. The mass of the \(\Sigma^0\)-hyperon is less than the mass of the \(\Sigma^-\)-hyperon, but greater than the mass of the \(\Sigma^+\)-hyperon.

The heaviest of the firmly established hyperons is the cascade hyperon \(\Xi^-\) (xi minus). The mass of the \(\Xi^-\)-hyperon is \(2586\,m\). The decay of the cascade hyperon proceeds according to the scheme

\[ \begin{gathered} \Xi^- \to \Lambda^0 + \pi^- + 67\ \text{MeV},\\ \downarrow\\ p+\pi^-+37\ \text{MeV}. \end{gathered} \]

The direct decay

\[ \Xi^- \to n+\pi^- \]

has not been observed.

The lifetime of the cascade hyperon has not been established accurately and is of the order \(10^{-10}\) sec.

Decays of hyperons with the formation of \(\mu\)-mesons and electrons have not yet been observed.

4. BASIC FEATURES OF STRANGE PARTICLES

\(K\)-mesons and hyperons, like \(\pi\)-mesons and nucleons, are strongly interacting particles. They are copiously produced in collisions of \(\pi\)-mesons and nucleons with energies of the order of several Bev.

Thus, at an energy of \(\sim 1.5\) Bev the cross section of the reaction

\[ \pi^- + p \to K^0 + \Lambda^0 \]

is equal\({}^{20}\) to \(1\) mbarn, which amounts to as much as \(3\%\) of the total cross section for the collision of \(\pi\)-mesons with protons at these energies. They have large scattering cross sections on nuclei and nucleons. Metastable compounds of strange particles with nucleons (the so-called \(\Lambda\)-nuclei) are known. The existence of \(\Lambda\)-nuclei indicates that the interaction between a \(\Lambda\)-hyperon and a nucleon is comparable with the interaction between two nucleons.

At the same time, the production and interaction of \(K\)-mesons and hyperons with nucleons have a number of characteristic features.

One of the principal features of strange particles is their associated production, an example of which is the well-known reaction \(\pi^- + p \to \Lambda^0 + K^0\). Reactions of single production of both \(K\)-mesons and hyperons are forbidden. Thus, despite specially undertaken searches, the reaction

\[ N+N\to \Lambda+N, \]

has not been found, although its threshold is considerably lower than the threshold of the repeatedly observed reaction

\[ N+N\to K+\Lambda+N. \]

However, not every associated production of strange particles is allowed: for example, reactions of pair production of hyperons

\[ N+N\to \Lambda+\Lambda(\Sigma) \]

have not been found and are forbidden.

No less surprising, at first sight, is the fact that both the production and the interaction with nucleons of \(K^+\)- and \(K^-\)-mesons are completely different.

At energies of the order of \(1\)–\(2\) Bev, \(K^-\)-mesons are produced approximately two orders of magnitude less often than \(K^+\)-mesons.

In collisions with nucleons and nuclei, \(K^+\)-mesons can either scatter or undergo charge exchange. Along with scattering and charge exchange, \(K^-\)-mesons can be absorbed by nucleons, thereby forming hyperons, for example

\[ K^- + p \to \Sigma^+ + \pi^- . \]

The selection rules arising in reactions involving strange particles are closely connected with the selection rules that determine the metastability of strange particles.

The long lifetime of strange particles in decays into \(\pi\)-mesons and nucleons is in striking contrast with the short time in which they manage to be formed in collisions of \(\pi\)-mesons and protons. The decay times of strange particles, as we saw above, are of order \(10^{-8}\)—\(10^{-10}\) sec. The time of their formation can be determined if one estimates, say, the collision time of a fast \(\pi\)-meson with a nucleon, during which a \(\Lambda\)-particle has a high probability of being formed. This time is of order \(\dfrac{\hbar}{\mu c^2} \sim 10^{-23}\) sec, which is approximately \(10^{13}\) times shorter than the decay time.

Attempts were made \(^{21-23}\) to explain the slowness of the decay of strange particles by their large spin. It was assumed, for example, that the spin of the \(\Lambda\)-hyperon is \(S \geq 11/2\). In this case the comparatively slow \(\pi^-\)-meson and proton arising in the decay with a large orbital angular momentum must spend approximately \(10^{-10}\) sec in order to overcome the centrifugal barrier due to this momentum.

Indeed, the wave function of a \(\pi\)-meson with angular momentum \(l\), emitted in the decay of a \(\Lambda^0\)-hyperon with momentum \(k\), at distances \(r \ll 1/k\) has the form

\[ \frac{1}{\sqrt{v\tau_0}} \cdot \frac{(2l-1)!!}{r(kr)^l}, \]

where \(v = k/m\), and \(\tau_0\) is the lifetime of the \(\Lambda^0\)-hyperon. We determine this time by matching the solution for \(r > r_0\) with the solution for \(r < r_0\), where \(r_0\) is the “size” of the \(\Lambda\)-hyperon,

\[ \frac{1}{\sqrt{v\tau_0}} \cdot \frac{(2l-1)!!}{r_0(kr_0)^l} \sim \frac{1}{r_0^{3/2}}, \]

whence

\[ \tau_0 \sim \frac{r_0}{v} \cdot \frac{[(2l-1)!!]^2}{(kr_0)^{2l}} . \]

For a \(\pi\)-meson energy \(\varepsilon \sim 37\,M_e v\), \(k = 2\sqrt{\mu\varepsilon}\), \(r_0 \sim \dfrac{1}{\sqrt{M\mu}}\), and \(l=5\), we obtain \(\tau_0 \sim 10^{-10}\) sec, which agrees well with the known lifetime of the \(\Lambda\)-hyperon.

However, as was shown in a number of papers \(^{24-27}\), such a model of the \(\Lambda^0\)-hyperon is incompatible with the very fact of observation of \(\Lambda\)-nuclei. \(\Lambda\)-nuclei are nuclear fragments containing a \(\Lambda^0\)-particle; their lifetime is comparable with the lifetime of a free \(\Lambda^0\)-particle (\(\tau > 10^{-12}\) sec). Two types of decay of \(\Lambda\)-nuclei are possible: mesonic and nonmesonic. In the first case the \(\pi\)-meson formed in the decay of the \(\Lambda^0\)-hyperon leaves the nucleus; in the second it is absorbed by some nucleon in the nucleus. This latter process is analogous to the phenomenon of internal conversion in atoms: the role of the \(\gamma\)-quantum is played here by the \(\pi\)-meson, and the role of the electron by the nucleon. The momentum of the nucleon that has absorbed the \(\pi\)-meson, \(p \sim \sqrt{M\mu} \gg k\), and therefore the centrifugal barrier for the nucleon is much more “transparent” than for the \(\pi\)-meson, and the effective lifetime of the \(\Lambda\)-hyperon in the nucleus due to nonmesonic decay is reduced by the factor \((p/k)^{2l}\). For \(l=5\) this factor is \(\sim 10^7\), i.e. the lifetime of a \(\Lambda^0\)-hyperon with spin \(11/2\) in a nucleus would be

are \(10^5\)—\(10^7\) times smaller than the experimentally observed times, and this means that \(\Lambda\)-nuclei would not be observed at all. Calculation shows that smaller values of the spin of the \(\Lambda^0\)-hyperon \((9/2,\ 7/2,\ 5/2)\) would also lead to an anomalously large probability of nonmesonic decay of \(\Lambda\)-nuclei\(^*\).

We shall not dwell on other attempts to explain the metastability of strange particles and the features characterizing their production and interaction with nucleons, but shall turn to the scheme of isotopic multiplets. Within this scheme, created by Gell-Mann \(^{1-3}\) and Nishijima \(^{4-6}\), the most surprising properties of strange particles were consistently explained and a number of predictions were made, which were subsequently brilliantly confirmed experimentally.

At the basis of the scheme of isotopic multiplets lies the concept of isotopic spin and the classification of interactions according to their magnitude and according to their isotopic properties.

5. TYPES OF INTERACTIONS

Interactions between elementary particles naturally fall into three classes.

1. Strong interactions between nucleons, \(\pi\)-mesons, and strange particles.

These interactions are responsible for the production of \(\pi\)-mesons and strange particles, for their scattering and absorption by nuclei. These interactions account for the forces acting between nucleons in nuclei. The dimensionless coupling constant\(^ {**}\) for strong interactions is \(g^2 \gtrsim 1\). Processes due to strong interactions are characterized by time intervals \(\sim 10^{-23}\) sec, as we saw in the example of the production of the \(\Lambda^0\)-hyperon.

1. Electromagnetic interaction between all charged particles and quanta of the electromagnetic field.

The electromagnetic interaction is responsible for the decays \(\Sigma^0 \to \Lambda^0 + \gamma\), \(\pi^0 \to 2\gamma\); it also accounts for the forces acting between charged particles. The coupling constant of the electromagnetic interaction is \(e^2 = \dfrac{1}{137}\). Processes due to the electromagnetic interaction, if there are no additional prohibitions, are approximately 100 times slower than processes due to strong interactions.

1. Weak interactions account for \(\beta\)-decay, the decays \(\pi \to \mu + \nu\), \(\mu \to e + 2\nu\), the absorption of \(\mu^-\)-mesons by nuclei, and the decays of strange particles. The dimensionless constant\(^ {***}\) of these interactions is \(f^2 \sim 10^{-13}\)—\(10^{-14}\). Correspondingly, the characteristic times for them are of order \(10^{-10}\) sec. The long lifetime of the neutron (13 min) is due to the fact that the energy \(\varepsilon\) released in \(\beta\)-decay is very small \((\varepsilon/\mu \sim 5 \cdot 10^{-3})\), while the probability of decay is proportional to \((\varepsilon/\mu)^5\). It is interesting to note that interactions with constants larger than \(f^2\), but smaller than \(e^2\), are unknown.

\(^*\) We note that in work \(^{28}\) there is an incorrect statement that a large spin of the \(\Lambda^0\)-particle would lead to an anomalously small nonmesonic decay of \(\Lambda\)-nuclei.

\(^ {**}\) Here and below \(\hbar = c = 1\).

\(^ {***}\) In various slow processes the interaction constants have different dimensions and, correspondingly, different magnitudes. Thus, the decay constant of \(\Lambda \to p + \pi^-\) is dimensionless and is approximately \(f_\Lambda^2 \sim 10^{-13}\). The \(\beta\)-decay constant has dimension \(f_\beta^2 \sim 10^{-24} m^4\), where \(m\) is the electron mass. If, however, the mass of the \(\pi\)-meson \(\mu\) is taken as unity, then \(f_\beta^2 \sim 10^{-14}\). Such a choice of the unit of mass corresponds to the system of units \(\hbar = c = \mu = 1\) and corresponds to a unit length \(\sim 10^{-13}\) cm. In these units the coupling constants of all weak interactions are approximately equal to one another.

6. ISOTOPIC SPIN OF $\pi$-MESONS AND NUCLEONS

It is well known that the strong interactions of $\pi$-mesons and nucleons possess the property of isotopic invariance. In the absence of electromagnetic and weak interactions, the proton and the neutron are strictly definite isotopic states of one and the same particle—the nucleon—whose isotopic spin is equal to $T=1/2$. Here the projection of the isotopic spin $T_3=+1/2$ corresponds to the proton, while the projection $T_3=-1/2$ corresponds to the neutron. The relation between the charge $Q$ and the projection $T_3$ is expressed for nucleons by the formula $Q=T_3+1/2$.

Similarly, the $\pi^+$-, $\pi^-$-, and $\pi^0$-mesons are regarded as three isotopic states of one particle—the $\pi$-meson—described by an isotopic vector $(T=1)$. The projections of this vector $T_3=\pm 1$ correspond to the $\pi^\pm$-meson, and the projection $T_3=0$ corresponds to the $\pi^0$-meson. The charge of the $\pi$-mesons is related to $T_3$ by the relation $Q=T_3$.

Thus, particles are regarded as components of isotopic multiplets, the number of components in a multiplet being equal to $2T+1$. Isotopic invariance of strong interactions requires that the total isotopic spin of the system be conserved in strong interactions. The isotopic invariance of the strong interactions of $\pi$-mesons and nucleons is at present confirmed by the entire body of experimental data concerning the scattering of $\pi$-mesons and nucleons by nucleons and the levels of light nuclei (see, on this point, the reviews $^{29\text{--}34}$).

Electromagnetic interactions, which depend on charges, violate the conservation of isotopic spin $T$. In this case the degeneracy in isotopic spin is lifted and a difference arises in the masses of the $\pi^\pm$- and $\pi^0$-mesons, of the proton and the neutron.

It is essential, however, that, while causing changes of isotopic spin $(\Delta T=0,\pm 1)$, the electromagnetic interaction conserves $T_3$. This property of the electromagnetic interaction, called by Gell-Mann the principle of minimal interaction, rests on the assumption that photons have no interactions other than the ordinary interaction with the charges and currents of real and virtual particles. In this case the interaction Lagrangian depends only on $Q$, and consequently only on $T_3$, and commutes with $T_3$. The principle of minimal interaction would be violated if, for example, the interaction of the anomalous magnetic moment of the nucleon with the electromagnetic field could not be reduced to the interaction of the electromagnetic field with the charges and currents of particles arising as a result of the virtual dissociation of the nucleon.

Weak interactions not only violate the conservation of $T$, but, unlike electromagnetic interaction, also violate the conservation of $T_3$. With respect to the decays of strongly interacting particles into leptons ($\mu$-meson, electron, neutrino), it would be more correct to say that in these decays the conservation of $T_3$ is not violated, but is bypassed. This is connected with the fact that the concept of isotopic spin cannot be extended to leptons. At least, attempts made until recently to generalize the concept of isotopic spin to leptons have not been successful.

7. ISOTOPIC SPIN OF $K$-MESONS AND HYPERONS

From the fact that the strong interactions of $\pi$-mesons and nucleons are isotopically invariant, and from the fact that $K$-mesons and hyperons interact strongly with $\pi$-mesons and nucleons, it follows that the strong interactions of $K$-mesons and hyperons among themselves and with nucleons and $\pi$-mesons must also be isotopically invariant. Otherwise virtual $K$-mesons and hyperons would violate the isotopic invariance of the interaction of $\pi$-mesons with nucleons and of nucleons with nucleons. This reasoning

cannot be considered absolutely convincing, since the conservation of isotopic spin in the interactions of \(\pi\)-mesons and nucleons has not been tested with very high accuracy. Nevertheless, the extension of the concept of isotopic spin to \(K\)-mesons and hyperons now appears not only fully justified, but also necessary.

The problem of including strange particles within the framework of the isotopic-spin scheme was solved most successfully in the works of Gell-Mann \(^{1-3}\) and Nishijima \(^{4-6}\), whose basic ideas we shall set forth here.

In discussing the possible values of the isotopic spin of strange particles, it should first of all be emphasized that elementary particles with charge exceeding one elementary charge \(e\) have so far not been observed.

We shall assume that there are no doubly and multiply charged elementary particles. We shall show that in this case the isotopic spin of an elementary particle cannot exceed unity. Indeed, the magnitude of the isotopic spin \(T\) determines the number of particles in a given isotopic multiplet, equal to \(2T+1\).

\(T=0\), an isotopic singlet, one particle.

\(T=\frac{1}{2}\), an isotopic doublet, two particles. One corresponds to the projection \(+\frac{1}{2}\), the other to \(-\frac{1}{2}\).

\(T=1\), an isotopic triplet, three particles, corresponding to the three projections \(T_3=\pm 1, 0\).

An isotopic spin \(\frac{3}{2}\) would correspond to four particles. Particles belonging to one isotopic multiplet and corresponding to different \(T_3\) must have different charges \(Q\), and an increase of \(T_3\) by one corresponds to an increase of \(Q\) by one. Obviously, in the quartet corresponding to spin \(\frac{3}{2}\) there must be at least one doubly charged particle. Thus the possible isotopic spins of singly charged particles are \(T=0,\frac{1}{2},1\).

As we have already said, the difference in the masses of particles belonging to a given isotopic multiplet is connected with the difference in the electromagnetic (but not strong!) interactions of these particles. Consequently, the masses of such particles must differ by an amount smaller than the mass of the \(\pi\)-meson by at least 1–2 orders of magnitude. Thus, the masses of the neutron and proton differ by approximately \(2m\), and the masses of the \(\pi^\pm\)- and \(\pi^0\)-mesons differ by \(8m\). The same must also apply to strange particles. Let us consider from this point of view the isotopic multiplets of strange particles. The \(\Lambda^0\)-hyperon has no charged analogues. It cannot be assumed that this hyperon belongs to one multiplet with the \(\Sigma\)-hyperons, since the latter are heavier than the \(\Lambda^0\)-hyperon by approximately \(160m\). Consequently, the \(\Lambda^0\)-hyperon is an isotopic singlet, and the isotopic spin of the \(\Lambda^0\)-hyperon is \(T=0\).

In the multiplet of \(\Sigma\)-hyperons in 1953, when the scheme of isotopic multiplets was being created, only the \(\Sigma^+\)- and \(\Sigma^-\)-hyperons were known. Since their charges differ by two units, there had to be one more particle in this multiplet (the \(\Sigma^0\)-hyperon), and from the condition that elementary particles be singly charged it immediately followed that the \(\Sigma\)-hyperons form an isotopic triplet. The \(\Sigma^0\)-hyperon thus predicted \(^{1}\) was soon discovered experimentally \(^{35-36}\). Recent experiments \(^{37}\) have definitively confirmed the existence of this particle.

The choice of the isotopic spin of the cascade hyperon \(\Xi^-\) is not as obvious as the choice in the case of the \(\Lambda^0\)- and \(\Sigma^0\)-hyperons. This is connected primarily with the fact that the experimental material concerning the \(\Xi^-\)-hyperon is much poorer. Suffice it to say that in accelerators these particles have so far not been observed. The existence of the \(\Xi^-\)-hyperon has been firmly established; the \(\Xi^+\)-hyperon has not been found. Thus, one must choose between \(T=0\)

and \(T = 1/2\). In the first case the \(\Xi\)-multiplet would consist of one \(\Xi^-\)-hyperon, in the second—of \(\Xi^-\)- and \(\Xi^0\)-hyperons. The second possibility was adopted, since it makes it easy to coordinate all the data relating to the production and decay of the \(\Xi\)-hyperon.

The isotopic spin of the \(K\)-mesons was determined on the basis of the firmly established fact of the existence of the reaction

\[ \pi^- + p \to K^0 + \Lambda^0 . \]

Since the total projection \(T_3\) of the \(\pi^- + p\) system is equal to \(-1/2\), and for the \(\Lambda^0\)-hyperon \(T=0\), the isotopic spin of the \(K\)-meson must be half-integral. From the condition of unit charge it follows that for the \(K\)-meson \(T=1/2\). This leads to the fact that \(K^+\)- and \(K^-\)-mesons cannot be components of one multiplet, but belong to different doublets. In this case one doublet combines \(K^+\) and \(K^0\), while the corresponding antiparticle doublet contains \(K^-\) and \(\overline{K^0}\) (anti-\(K^0\)). It is essential here that there are two types of neutral \(K^0\)-mesons: \(K^0\) and \(\overline{K^0}\).

From the fact that the \(K^0\)-meson and the \(\overline{K^0}\)-meson are different particles, Gell-Mann and Pais\(^{16}\) in 1954 concluded that, alongside the then known \(K_1^0\)-meson, there must exist a \(K_2^0\)-meson (see above). They indicated that the lifetime of the \(K_2^0\)-meson must be longer than the lifetime of the \(K_1^0\)-meson and that the \(K_2^0\)-meson must have other modes of decay. In 1956 the \(K_2^0\)-meson was discovered experimentally\(^{18}\), and the theoretical prediction was brilliantly confirmed. The ideas pertaining to this are set forth in detail in review\(^{38}\); here we shall note only that \(K_1^0\) and \(K_2^0\) do not coincide with \(K^0\) and \(\overline{K^0}\), but are their combinations:

\[ K_1^0 = \frac{1}{\sqrt{2}}\left(K^0 + \overline{K^0}\right), \]

\[ K_2^0 = \frac{1}{\sqrt{2}}\left(K^0 - \overline{K^0}\right). \]

Proceeding from the fact that in the decay of a free particle charge parity is conserved, Gell-Mann and Pais regarded \(K_1^0\) as a charge-parity-even particle, and \(K_2^0\) as a charge-parity-odd particle. If in the decay of \(K\)-mesons parity is not conserved, then the argument of Gell-Mann and Pais connected with charge parity must also be changed. This question will be considered in detail in the second part of our review.

The classification of strange particles by isotopic spins given above is in itself not very convincing. It will become considerably more reliable after certain consequences of this classification, relating both to the strong and to the weak interactions of strange particles, have been considered. However, for convenience in the further arguments, we must explicitly introduce the quantum number “strangeness,” which has already been introduced implicitly.

8. Strangeness

We have already said that the charge of a particle \(Q\) is connected with the projection of the isotopic spin \(T_3\). Thus, for \(\pi\)-mesons \(Q = T_3\); for nucleons \(Q = T_3 + 1/2\). Combining these formulas, one may write:

\[ Q = T_3 + n/2, \]

where \(n\) is the number of nucleons contained in the particle minus the number of antinucleons.

For \(\pi\)-mesons, \(n=0\); for the proton and neutron, \(n=1\); for the antiproton and antineutron, \(n=-1\). For the deuteron, for example, \(T_3=0,\ n=2,\ Q=1\).

It is easy to see that this formula cannot be extended to strange particles by understanding \(n\) as the number of baryons) minus the number of antibaryons. Indeed, for the \(\Lambda^0\)-hyperon \(T_3=0,\ n=1,\) while \(Q=0\), and the difference \(Q-T_3-\dfrac{n}{2}\) is not equal to zero, as it is for nucleons and \(\pi\)-mesons. It is precisely this difference that determines \(S\)—the “strangeness” of the particle*)

\[ S=2\left(Q-T_3-\frac{n}{2}\right). \]

For ordinary particles (\(\pi\) and \(N\)), \(S=0\). For the \(\Lambda^0\)-hyperon,

\[ S=-2\times \frac{1}{2}=-1. \]

For \(\Sigma\)-hyperons the strangeness is also

\[ S=-1, \]

since for them \(Q=T_3\).

For the \(\Xi^-\)-hyperon \(Q=-1,\ T_3=-\dfrac{1}{2},\ n=+1\), and therefore \(S=-2\). Since, when \(T_3\) changes by one unit, \(Q\) also changes in the given multiplet by one unit, it is obvious that the value of \(S\) is the same for all particles of the given multiplet.

For \(K^+,\ K^0\)-mesons, according to the formula \(S=+1\); for \(K^-\) and \(\overline{K}{}^0\)-mesons, \(S=-1\).

Thus, the isotopic multiplets of strange particles turn out to be, as it were, shifted relative to the multiplets of ordinary particles. The scheme shown in Fig. 1 results***).

Fig. 1.

When passing from a particle to an antiparticle, strangeness changes sign. Strangeness is not a new quantum number, since it is uniquely determined by the quantum numbers \(Q,\ T_3\), and \(n\). However, introducing \(S\) is convenient, since it makes it possible to take into account simultaneously the conservation laws for \(Q\) and \(T_3\).

Below we shall consider the strong interactions of strange particles from the standpoint of the conservation of strangeness \(S\) and isotopic spin \(T\). Selection rules with respect to isotopic spin for weak interactions will also be considered:

\[ \Delta S=\pm 1\left(\Delta T_3=\mp \frac{1}{2}\right)\quad \text{and}\quad \Delta T=\pm \frac{1}{2}. \]

9. STRONG INTERACTIONS. \(\Delta S=0\)

Production of strange particles. Let us consider the features characterizing the production and interaction of strange particles with nucleons. The strong interactions responsible for these processes must proceed with conservation of isotopic spin and, consequently, of strangeness as well. We shall first dwell on the consequences of strangeness conservation in strong interactions, since this conservation law is more general (it is not violated even when electromagnetic interaction is taken into account) and since the most interesting prohibitions are connected precisely with it.

) Baryons is the general name for nucleons and hyperons.
) The term “strangeness” was introduced by Gell-Mann. The same quantity was called the \(\eta\)-charge by Nishijima.
**) Figs. 1–3 and Table I are borrowed, with some changes, from work \(^{39}\).

Since the strangeness of \(\pi\)-mesons and nucleons is equal to zero, strange particles cannot be produced in collisions of \(\pi\)-mesons and nucleons singly. At a minimum, two particles must be produced simultaneously (such that their total strangeness is equal to zero). Thus, the conservation of strangeness gives an explanation of the well-known fact of the associated production of strange particles. It is easy to verify that in such reactions as

\[ \pi^- + p \to \Lambda^0 + K^0, \]

\[ \pi^- + p \to \Sigma^- + K^+, \]

\[ p + p \to \Lambda^0 + K^+ + p \]

strangeness is conserved.

However, not every associated production of strange particles turns out to be allowed. In particular, as is easy to verify, strangeness is not conserved in reactions of pair production of hyperons:

\[ n + n \to \Lambda^0 + \Lambda^0, \]

\[ p + n \to \Lambda^0 + \Sigma^+. \]

Specially designed experiments showed that these reactions are indeed forbidden \(^{40}\). The cascade hyperon \(\Xi^-\), having strangeness \(-2\), must be produced with two \(K\)-mesons (\(K^+\) or \(K^0\)), the strangeness of each of which is \(+1\). An example of such a reaction is the observed reaction of production in nuclear interaction

\[ \Xi^- + K^0 + K^0\,^{41}. \]

Since \(K^+\)- and \(K^-\)-mesons have different strangenesses, the reactions of their production are completely different. \(K^+\)-mesons are produced together with \(\Lambda\)- and \(\Sigma\)-hyperons:

\[ \pi^+ + n \to K^+ + \Lambda^0, \]

\[ p + p \to K^+ + \Sigma^+ + n. \]

\(K^-\)-mesons can be produced only together with \(K^+\) (or \(K^0\))-mesons, for example:

\[ \pi^- + p \to K^+ + K^- + n. \]

(We are speaking of energies at which antihyperons still cannot be produced.) Since the formation of a \(\Lambda\)-particle requires considerably less energy than the formation of a \(K\)-meson, the thresholds of the reactions producing \(K^-\)-mesons exceed the thresholds for producing \(K^+\)-mesons (see Table I). This leads to the fact that at energies \(\sim 1\text{--}2\ \mathrm{Bev}\) \(K^-\)-mesons are produced about two orders of magnitude less often than \(K^+\)-mesons.

The difference between \(K^+\)- and \(K^-\)-mesons is especially clearly seen in the example of the reactions

\[ \pi^- + p \to K^+ + \Sigma^- \]

and

\[ \pi^- + p \to K^- + \Sigma^+. \]

The first of these reactions is allowed by strangeness; the second is forbidden. Experiment confirms this conclusion \(^{42}\).

Table I gives the thresholds for the production of strange particles in reactions proceeding with conservation of strangeness. The energies of the \(\pi\)-mesons and nucleons are given in the laboratory coordinate system. It was assumed that the energy of nucleons in the target nuclei is equal to \(25\ \mathrm{Mev}\).

Table I

Thresholds for the formation of strange particles in BeV

Particles formed	π-meson—nucleon	π-meson—nucleus	Nucleon—nucleon	Nucleon—nucleus
\(K\)	0.75	0.59	1.57	1.11
\(\overline{K}\)	1.34	1.08	2.48	1.82
\(\Lambda\)	0.75	0.59	1.57	1.11
\(\Sigma\)	0.90	0.69	1.80	1.31
\(\Xi\)	2.25	1.73	3.75	2.86

Scattering and absorption of strange particles. \(K^+\)- and \(K^-\)-mesons behave differently in collisions with nucleons. A \(K^+\)-meson, upon collision with a nucleon, can either be scattered,

\[ K^+ + p \to K^+ + p, \]

\[ K^+ + n \to K^+ + n, \]

or undergo charge exchange:

\[ K^+ + n \to K^0 + p. \]

There are no other reactions at not very high \(K\)-meson energies. This is due to the fact that there is no system with strangeness equal to \(+1\) and with an energy smaller than the energy of the \(K+N\) system. The interaction cross section of \(K^+\)-mesons with protons at energies from 30 MeV to 100 MeV does not exceed 30 mbarn.

The system \(K^- + N\) does not have the smallest energy among all systems with strangeness \(-1\). The same strangeness is possessed by the systems \(\Lambda+\pi\) and \(\Sigma+\pi\), whose proper energy is substantially smaller. As a result, in collisions of a \(K^-\) (\(\overline{K^0}\))-meson with a nucleon, along with scattering and charge-exchange reactions, processes of absorption of \(K^-\)-mesons proceed with greater probability:

\[ K^- + p \to \Lambda^0 + \pi^0, \qquad K^- + n \to \Lambda^0 + \pi^-, \]

\[ \Sigma^0 + \pi^0, \qquad \Sigma^0 + \pi^-, \]

\[ \Sigma^+ + \pi^-, \qquad \Sigma^- + \pi^0. \]

\[ \Sigma^- + \pi^+. \]

The total interaction cross section of \(K^-\)-mesons with protons is of the order of 100–200 mbarn at energies of the order of 30 MeV.

In collisions with nucleons of \(\Lambda\)-hyperons, elastic scattering is possible and, if the energy of the \(\Lambda\)-particle is sufficiently large, the formation of \(\Sigma\)-hyperons:

\[ \Lambda^0 + p \to \Sigma^+ + n, \]

\[ \Lambda^0 + p \to \Sigma^0 + p. \]

etc.

Nuclear collisions of \(\Sigma\)-hyperons, as experiments have shown, are accompanied by charge exchange

\[ \Sigma^-+p\to\Sigma^0+n \]

and the formation of \(\Lambda^0\)-hyperons

\[ \Sigma^-+p\to\Lambda^0+n. \]

The cross section for the interaction of \(\Sigma^-\)-hyperons with protons at low energies (\(\sim 10\) MeV) is hundreds of mbarn. Without violating strangeness conservation, \(\Xi\)-hyperons in collisions with nucleons and nuclei can be absorbed as a result of the reaction

\[ \Xi^-+p\to\Lambda^0+\Lambda^0, \]

or, if their kinetic energy is sufficient, as a result of the reactions:

\[ \Xi^-+p\to\Lambda^0+\Sigma^0, \]

\[ \Xi^-+n\to\Lambda^0+\Sigma^-. \]

The allowed reactions of production and absorption of strange particles are shown in Fig. 2.

Hypernuclei and \(K\)-nuclei. It is known that \(\Lambda^0\)-hyperons can form stable compounds with nucleons, the so-called \(\Lambda\)-nuclei. The lifetime of such \(\Lambda\)-nuclei (\(\gtrsim 10^{-12}\) sec) is determined by the lifetime of the \(\Lambda^0\)-hyperon. The metastability of \(\Lambda\)-nuclei is quite understandable if one takes into account that the \(\Lambda^0\)-hyperon has the lowest excitation energy of all particles with strangeness \(-1\).

As a result of the interaction of \(\Sigma\)-hyperons with nucleons, the formation of \(\Lambda^0\)-hyperons is allowed by strangeness and is energetically favorable.

The reactions

\[ \begin{aligned} \Sigma^-+p&\to\Lambda^0+n,\\ \Sigma^+ + n&\to\Lambda^0+p,\\ \Sigma^0+n&\to\Lambda^0+n \end{aligned} \]

lead to the fact that the existence of \(\Sigma\)-nuclei is, generally speaking, impossible. If, however, a \(\Sigma\)-nucleus consisted of a \(\Sigma^-\) (\(\Sigma^+\))-particle and one or several neutrons (protons), then the transition \(\Sigma\leftrightarrow\Lambda\) would be forbidden by charge conservation and such a nucleus would be stable. Similar \(\Sigma\)-nuclei apparently have not been observed.

\(\Xi\)-nuclei also, generally speaking, are impossible because of the reactions

\[ \Xi^-+p\to\Lambda^0+\Lambda^0, \]

\[ \Xi^0+n\to\Lambda^0+\Lambda^0. \]

However, a system consisting of a \(\Xi^-\) (\(\Xi^0\))-hyperon and one or several neutrons (protons) would be stable.

Fig. 2.

The long-term existence (on a nuclear time scale) of \(K^-\) (\(\overline{K^0}\))-mesons in nuclear matter is impossible, since the formation of hyperons is energetically favorable and allowed by strangeness.

A system consisting of a \(K^+\) (\(K^0\))-meson and nucleons (a \(K\)-nucleus) would be stable on the nuclear time scale, since inelastic reactions of \(K\)-mesons

with nuclei are forbidden by strangeness. The lifetime of such a system would be determined by the lifetime of the \(K_1^0\)-meson. The probabilities of mesonic and nonmesonic decays of \(K\)-nuclei have been calculated in \(^{43}\). In order for the formation of \(K\)-nuclei to be possible, it is necessary that there exist a sufficiently strong attraction between the nucleon and the \(K\)-meson. An analysis of the interference between Coulomb and nuclear scattering of \(K\)-mesons shows \(^{44}\) that the \(K\)-meson is attracted to the nucleon. The intensity of this attraction, judging from the magnitude of the scattering cross section of \(K\)-mesons, is such that the existence of \(K\)-nuclei is apparently possible \(^{45}\).

10. STRONG INTERACTIONS. \(\Delta T = 0\)

Up to now we have considered only the consequences of the conservation, in strong interactions, of strangeness (or, what is the same thing, \(T_3\)). These consequences are in brilliant agreement with experiment. A stronger requirement is the conservation not only of \(T_3\), but also of \(T\)—the requirement of isotopic invariance of the strong interactions of strange particles, considered in papers \(^{1,5,6,46-53}\).

Using conservation of isotopic spin, one can relate the cross sections of different reactions that differ only in the charge states of the particles participating in them. Thus, for example, for the reactions producing a \(\Lambda^0\)-hyperon

\[ \begin{aligned} 1)&\quad \pi^- + p \to \Lambda^0 + K^0,\\ 2)&\quad \pi^+ + n \to \Lambda^0 + K^+ \end{aligned} \]

the relation \(\sigma_1=\sigma_2\) is valid. This equality already follows from the requirement of isotopic symmetry, which is less stringent than the requirement of isotopic invariance. (Isotopic symmetry requires that the cross sections of reactions, one of which can be obtained from the other by replacing all particles by particles with the opposite sign of \(T_3\), be equal.)

The relations following from isotopic invariance are especially simple for reactions with a deuteron, whose isotopic spin is equal to zero. Thus, for the reactions

\[ \begin{aligned} 1)&\quad K^- + d \to \Sigma^- + p,\\ 2)&\quad K^- + d \to \Sigma^0 + n \end{aligned} \]

the relation \(\sigma_1=2\sigma_2\) holds.

This and other more complicated relations can be obtained simply by means of the formulas contained in Appendices I and II. A number of isotopic relations are easily derived by means of Shmushkevich’s method \(^{54,55}\). Examples of such relations, obtained by Shmushkevich’s method, are contained in \(^{47}\).

Experimental verification of isotopic relations will make it possible to determine whether the strong interactions of strange particles are indeed isotopically invariant.

If the interactions of \(\Lambda^0\)-hyperons with nucleons are isotopically invariant, then \(\Lambda\)-nuclei, analogously to ordinary light nuclei, should form isotopic multiplets \(^{51}\). At the same time, since the isotopic spin of the \(\Lambda\)-particle is equal to zero, its strong interactions with the proton and the neutron must, already by virtue of isotopic symmetry, be equal.

Let us consider the simplest \(\Lambda\)-nuclei. The systems \(\Lambda p\) and \(\Lambda n\) have not been observed. Apparently the interaction of a \(\Lambda^0\)-particle with a nucleon is not as strong as the interaction between nucleons. In addition, the radius of this interaction must be approximately twice as small as the radius of the interaction between two nucleons. This is explained \(^{50}\) by the fact that the \(\Lambda N\)-forces are due, on the one hand, to exchange of \(K\)-mesons, whose mass is of the order of \(3.5m_\pi\), and, on the other hand, to pairs of \(\pi\)-mesons. Exchange of single \(\pi\)-mesons between a \(\Lambda^0\)-particle and a nucleon is forbidden. This circumstance is connected with the fact that the isotopic spin of the \(\Lambda^0\)-particle is zero, while the isotopic spin of the \(\pi\)-meson is equal to

unit. Therefore the process \(\Lambda \rightleftarrows \Lambda+\pi\) is forbidden if isotopic spin is conserved, and only the transitions \(\Lambda \to \Sigma+\pi\) and \(\Sigma \to \Lambda+\pi\) are possible.

The isotopic spin of \(\mathrm{H}_3^*\) (Λ-tritium) is zero, since the systems \(nn\Lambda\) and \(pp\Lambda\) \((\mathrm{He}_3^*)\) have not been observed.

The Λ-nuclei \(\mathrm{H}_4^*\) and \(\mathrm{He}_4^*\) form an isotopic doublet. It is interesting that the existence of \(\mathrm{H}_4^*\) was predicted by Dalitz\({}^{51}\) on the basis of isotopic symmetry and the fact of the existence of \(\mathrm{He}_4^*\). The binding energies of these nuclei are equal to within Coulomb corrections.

The hypothesis of isotopic invariance of the strong interactions of strange particles makes it possible to analyze these interactions with the aid of amplitudes with a given value of \(T\). Thus experiments\({}^{37}\) on the capture of \(K^-\) in hydrogen give, for the probabilities of the reactions

\[ \begin{aligned} 1)\quad &K^-+p\to \Sigma^-+\pi^+,\\ 2)\quad &K^-+p\to \Sigma^+ + \pi^-,\\ 3)\quad &K^-+p\to \Sigma^0+\pi^0 \end{aligned} \]

the ratio \(\omega_1:\omega_2:\omega_3=2:1:1\).

Using formula (3) of Appendix II, we obtain for these probabilities

\[ \omega_1=\left|-\frac{f_0}{\sqrt{6}}+\frac{f_1}{2}\right|^2, \]

\[ \omega_2=\left|-\frac{f_0}{\sqrt{6}}-\frac{f_1}{2}\right|^2, \]

\[ \omega_3=\left|-\frac{f_0}{\sqrt{6}}\right|^2, \]

Here \(f_0\) and \(f_1\) are complex amplitudes with isotopic spin \(T=0\) and \(T=1\), respectively. Writing \(f_0=\rho_0 e^{i\varphi_0}\), \(f_1=\rho_1 e^{i\varphi_1}\), one can determine from the ratio \(\omega_1:\omega_2:\omega_3\) the values of \(\rho_0/\rho_1\) and \((\varphi_0-\varphi_1)\); they are:

\[ \frac{\rho_0}{\rho_1}=\sqrt{3}, \qquad \cos(\varphi_0-\varphi_1)=\frac{\sqrt{2}}{4}. \]

This conclusion does not agree with the results of work\({}^{52}\), whose author, on the basis of an analysis of data on the capture of \(K^-\)-mesons in nuclei, came to the conclusion that the reactions

\[ K^-+N\to \Sigma+\pi \]

proceed mainly through the channel with \(T=1\).

Another example: data on the scattering of \(K^+\)-mesons with energies \(\lesssim 100\) MeV in photoemulsions\({}^{53,56,57}\) indicate that the scattering cross section on a free proton is \(20\text{--}28\) mbarn, while the scattering cross section per nucleon in a nucleus is \(6\text{--}9\) mbarn. To explain this discrepancy, a hypothesis was proposed\({}^{53}\) according to which the scattering of \(K^+\)-mesons occurs through the channel with \(T=1\). In this case the elastic scattering \(K^+ + p \to K^+ + p\) will have a cross section 4 times larger than the scattering \(K^+ + n \to K^+ + n\) (see Appendix II).

Another possible explanation of the small scattering cross section of \(K^+\)-mesons on nuclei is given in\({}^{58}\). This explanation is based on the fact that, at comparatively low energies of \(K^+\)-mesons (\(\lesssim 50\) MeV), inelastic reactions in

collisions of a \(K\)-meson with a nucleus (and, in particular, the charge-exchange reaction \(K^{+}+n\to K^{0}+p\)) must be suppressed. This is due to the Pauli principle, which restricts the number of states available to the proton produced in charge exchange. The decrease of the cross section due to the Pauli principle has been calculated on the basis of the Fermi-gas model in \(^{59}\). This decrease becomes insignificant as the energy of the \(K\)-meson increases.

In conclusion, we emphasize that the isotopic invariance of the strong interactions of strange particles is as yet only a hypothesis. Experimental verification of this hypothesis is of fundamental importance.

11. ELECTROMAGNETIC INTERACTION. \(\Delta S=0\)

The fact that the electromagnetic interaction conserves strangeness (the principle of minimal interaction) means that the production of strange particles by \(\gamma\)-quanta (photoproduction) must necessarily be associated. Pair photoproduction of hyperons must be forbidden; for example, the reaction

\[ \gamma+d\to \Lambda+\Sigma^{+} \]

and other reactions violating conservation of strangeness. The electromagnetic interaction also cannot cause decays of strange particles, with the exception of one case, when strangeness does not change in the decay. This exception is the decay of the \(\Sigma^{0}\)-hyperon:

\[ \Sigma^{0}\to \Lambda^{0}+\gamma. \]

For both the \(\Sigma^{0}\)-hyperon and the \(\Lambda^{0}\)-hyperon \(T_{3}=0\), so that in this decay \(\Delta T_{3}=\Delta S=0\). As regards isotopic spin, the \(\Sigma^{0}\)-hyperon has \(T=1\), while the \(\Lambda^{0}\)-hyperon has \(T=0\), and \(\Delta T=1\), in accordance with the selection rules for the electromagnetic interaction. It should be expected that the lifetime of the \(\Sigma^{0}\)-hyperon is approximately \(10^{-20}\) sec.

The production and decay of \(\Sigma^{0}\)-hyperons were detected in the reactions

\[ \pi^{-}+p\to \Sigma^{0}+K^{0} \]
\[ \downarrow \]
\[ \Lambda^{0}+\gamma \]
\[ \downarrow \]
\[ p+\pi^{-} \]

and

\[ K^{-}+p\to \Sigma^{0}+\pi^{0} \]
\[ \downarrow \]
\[ \Lambda^{0}+\gamma \]
\[ \downarrow \]
\[ p+\pi^{-} \]

from the nonconservation in these reactions of the energy–momentum balance (the \(\gamma\)-quantum produced in the decay \(\Sigma^{0}\to\Lambda^{0}+\gamma\) is not registered in the chambers in which these reactions were observed).

Differences in the masses of strange particles belonging to a given isotopic multiplet, and in particular of the \(\Sigma^{+}\)- and \(\Sigma^{-}\)-hyperons, are due to the electromagnetic interaction. At first sight it seems that the inequality of the masses of the \(\Sigma^{+}\)- and \(\Sigma^{-}\)-hyperons (\(\Delta m=15\,m\)) contradicts the principle of charge symmetry,

finding its expression in the fact that the masses of the electron and positron, of the proton and antiproton, and of the \(\pi^+\)-meson and \(\pi^-\)-meson are equal to one another. In this connection it is appropriate to emphasize that the \(\Sigma^+\)- and \(\Sigma^-\)-hyperons are not a particle and an antiparticle, as is the case for \(e^+\) and \(e^-\), \(p\) and \(\tilde p\), \(\pi^+\) and \(\pi^-\), and the principle of charge symmetry is not applicable to them.

Qualitatively, one can understand the origin of the mass difference of the \(\Sigma^+\)- and \(\Sigma^-\)-hyperons if one considers the virtual chains of strong interactions that contribute to the masses of these hyperons:

\[ \Sigma \to N + \overline K \to \Sigma, \]

\[ \Sigma \to \Xi + K \to \Sigma. \]

In the absence of electromagnetic interaction, the contributions of these processes to the masses of the \(\Sigma^+\)- and \(\Sigma^-\)-hyperons are equal. If, however, the electric charges of the real and virtual particles are “switched on,” differences arise between the chains for the \(\Sigma^+\)-hyperon and the \(\Sigma^-\)-hyperon:

\[ \Sigma^+ \to p + \overline K^0, \qquad \Sigma^+ \to \Xi^0 + K^+, \]

\[ \Sigma^- \to n + K^-, \qquad \Sigma^- \to \Xi^- + K^0, \]

and the masses of these particles will no longer be equal to one another; as shown in \(^{60}\), the difference in the electromagnetic structure of the \(\Sigma^+\)- and \(\Sigma^-\)-hyperons should lead to the fact that these particles will also have different magnetic moments.

12. DECAYS OF STRANGE PARTICLES. \(\Delta S = \pm 1\)

Of the decays of strange particles we shall consider here only decays into strongly interacting particles (\(\pi\)-mesons and nucleons). We shall not consider leptonic decays \((K_{\mu2}, K_{\mu3}, K_{e3})\), since the concept of isotopic spin is not applicable to leptons.

Within the framework of the scheme of isotopic multiplets, the metastability of strange particles is explained in a natural way. Since the strangeness of strange particles is different from zero, while the strangeness of \(\pi\)-mesons and nucleons is equal to zero, decays of strange particles into \(\pi\)-mesons and nucleons due to strong and electromagnetic interactions, which conserve strangeness, are forbidden, and strange particles decay by means of weak interactions. It is easy to see that for all particles, except the cascade hyperon, the strangeness in all known decays into \(\pi\)-mesons and nucleons must change by one unit (Fig. 3). For the \(\Xi^-\)-hyperon, if for it one chooses \(T = \frac12\), the change of strangeness in the decay \(\Xi^- \to \Lambda^0 + \pi^-\) is also equal to \(\Delta S = 1\). If one assumes that the unobserved direct decay \(\Xi^- \to n + \pi^-\) is forbidden, then one may conclude that weak interactions always change \(S\) by one unit. In this case the only mode of decay for the \(\Xi^0\)-hyperon is the decay

\[ \Xi^0 \to \Lambda^0 + \pi^0, \]

which makes observation of the \(\Xi^0\)-hyperon very difficult. It is usually assumed that interactions changing strangeness by 2 should be approximately \(10^{13}\) times weaker than the known decay interactions.

Fig. 3.

Since the conservation laws of electric charge and baryon number are absolute, it is clear from the definition of strangeness that a change of strangeness in a decay by $\Delta S=\pm 1$ leads to a change of the third projection of the isotopic spin by $\Delta T_3=\mp \frac12$.

13. DECAYS OF STRANGE PARTICLES. $\Delta T=\pm \frac12$

If one assumes that in decays of strange particles not only $\Delta T_3=\frac12$, but also $\Delta T=\frac12$, then a number of interesting consequences arise, $^{61-70}$ some of which are in good agreement with experimental data.

Let us consider the decay $K^+ \to \pi^+ + \pi^0$. The two $\pi$-mesons formed as a result of this decay can, generally speaking, have $T=0, 1$, and $2$. However, the value $T=0$ is excluded by the fact that for this system $T_3=1$. If the $K^+$-meson has spin equal to zero, then the two $\pi$-mesons arise in an $S$-state, and consequently their wave function must be symmetric in the isotopic variables. This excludes the value $T=1$. If we now assume that in the decay $\Delta T=\frac12$ and recall that for the $K$-meson $T=\frac12$, then the state with $T=2$ is also excluded, and the decay $K^+ \to \pi^+ + \pi^0$ proves to be forbidden. It is easy to see that for the decay $K_1^0 \to \pi^+ + \pi^-$ there is no similar prohibition. This conclusion is in qualitative agreement with the known difference in lifetimes for the decays $K^+ \to \pi^+ + \pi^0$ $(\tau \sim 10^{-8}\ \mathrm{sec})$ and $K_1^0 \to \pi^+ + \pi^-$ $(\tau \sim 10^{-10}\ \mathrm{sec})$.

Of course, if the selection rule $\Delta T=\frac12$ were absolutely strict, the lifetime of the $K^+$-meson would be arbitrarily large. In order to obtain the observed lifetime of the $K^+$-meson, it is necessary to assume that the decay amplitude contains a small admixture with $\Delta T=\frac32$ (or $\Delta T=\frac52$). The magnitude of this admixture must be $\sim 10\%$ of the amplitude with $\Delta T=\frac12$.

Taking into account all that was said above about the decay $K^+ \to 2\pi$, it is easy to see that in the decay $K^0 \to 2\pi$, the $\pi$-mesons arise in a state with $T=0$. This immediately gives the ratio between the probabilities of the alternative decays:

\[ 1)\quad K_1^0 \to \pi^+ + \pi^- \]

and

\[ 2)\quad K^0 \to \pi^0 + \pi^0, \]

namely $w_1/w_2=2$ (see Appendix I). Experimentally this ratio has been determined very roughly and, in order of magnitude, is close to unity.

The hypothesis that in the decay $K^+ \to 3\pi$ the change $\Delta T=\frac12$ leads to the fact that the three $\pi$-mesons arise in this decay in a state with $T=1$.

This makes it possible to relate the probabilities of the decays

\[ 1)\quad K^+ \to 2\pi^+ + \pi^- \]

and

\[ 2)\quad K^+ \to 2\pi^0 + \pi^+ . \]

It turns out, $^{69,71,72}$ that $w_2=0.25w_1$, if all $\pi$-mesons fly out in an $S$-state. Taking into account the difference of the masses of $\pi^+$ and $\pi^0$ proves substantial in calculating the phase volumes because of the smallness of the energy released and leads $^{61}$ to the relation $w_2=0.325w_1$. This relation agrees well with the experimental data.

For the decays of the $\Lambda^0$-hyperon

\[ 1)\quad \Lambda^0 \to p + \pi^- \]

and

\[ 2)\quad \Lambda^0 \to n + \pi^0 \]

under the assumption \(\Delta T=\frac12\), one obtains \(w_1/w_2=2\). This ratio apparently does not agree with the experimental data \(^{15,42}\), which give for it a value \(\sim 0.5 \div 1\).

Analogous isotopic relations can also be derived for decays of \(\Lambda\)-nuclei \(^{63,66}\). Thus, for the decay probabilities of \(\Lambda\)-tritium, whose isotopic spin is zero, one obtains:

\[ w(H_3^* \to \pi^-+2p+n)+w(H_3^* \to \pi^+ +3n)= \]

\[ =2w(H_3^* \to \pi^0+2n+p). \]

Interactions for which \(\Delta T=\frac12\) may be regarded as isotopically invariant if one introduces, in a purely formal way, a “particle” \(s\) carrying “strangeness” (for the \(s\)-“particle” the name “spurion” was proposed). The strangeness of the “spurion” is \(+1\), and its charge is zero. Thus defined, the “spurion” must have \(T=\frac12\) and \(T_3=-\frac12\).

With the help of the “spurion,” decays of strange particles can be represented in the form

\[ s+\Sigma \to N+\pi, \qquad s+\Lambda \to N+\pi, \]

\[ s+\Xi \to \Lambda+\pi, \qquad K \to s+2\pi \]

and, in determining isotopic relations, one can use the formulas contained in Appendices I and II.

It should be noted that even a small admixture of an interaction with \(\Delta T=\frac32\) strongly changes the relation between decays. In particular, this applies to the decays of the \(\Lambda^0\)-hyperon \(^{70}\).

Interesting conclusions can be drawn if one assumes that in decays of \(\Sigma\)-hyperons \(\Delta T=\frac12\) \(^{65,66,68}\). The decay amplitudes

\[ \Sigma^+ \to n+\pi^+ \qquad a_+, \]

\[ \Sigma^+ \to p+\pi_0 \qquad a_0, \]

\[ \Sigma^- \to n+\pi^- \qquad a_- \]

are then expressed through the amplitudes \(f_3\) and \(f_1\), corresponding respectively to \(T=\frac32\) and \(T=\frac12\) (see Appendix II):

Spin and parity \(\Sigma\)	\(\frac12+\)	\(\frac12-\)	\(\frac32+\)	\(\frac32-\)	All other values
\(\alpha_1-\alpha_3\)	\(\sim 0^\circ\)	\(25^\circ\)	\(40^\circ\)	\(\sim 0^\circ\)	\(\sim 0^\circ\)

\[ a_+=\sqrt{\frac13}\,f_3+\sqrt{\frac23}\,f_1, \]

\[ a_0=\sqrt{\frac23}\,f_3-\sqrt{\frac13}\,f_1, \]

\[ a_-=\sqrt{3}\,f_3. \]

Using the unitarity and symmetry condition for the \(S\)-matrix in the form \(SS^*=1\), and the smallness of the decay interaction, it is easy to show that

\[ f_3=i\rho_3 e^{i\alpha_3}, \qquad f_1=i\rho_1 e^{i\alpha_1}, \]

where \(\rho_3\) and \(\rho_1\) are real, and \(\alpha_3\) and \(\alpha_1\) are the phases of \(\pi\)-meson scattering on a nucleon in states with \(T=3/2\) and \(T=1/2\), respectively. The parity and orbital angular momentum of these states are determined by the spin and parity of the \(\Sigma\)-hyperon. The ratio of the probabilities of the decays \(\Sigma^+\to p+\pi^0\) and \(\Sigma^+\to n+\pi^+\) is then equal to \((w=|a|^2)\):

\[ X=\frac{w_0}{w_+}= \frac{2+2z^2-4z\cos(\alpha_1-\alpha_3)} {1+4z^2+4z\cos(\alpha_1-\alpha_3)}, \]

where

\[ z=\frac{\rho_1}{\rho_3\sqrt{2}}, \]

and the ratio of the lifetimes of the \(\Sigma^+\)- and \(\Sigma^-\)-hyperons is equal to:

\[ Y=\frac{w_-}{w_+ + w_0}=\frac{3}{1+2z^2}. \]

Eliminating \(z\) from these equations and substituting the experimentally determined values of \(X\) and \(Y\), one can determine the quantity \((\alpha_1-\alpha_3)\) and thereby the spin and parity of the \(\Sigma\)-hyperon could be determined. However, the latest experimental data\(^{37}\) are such \((X\simeq 1,\; Y\simeq 1/2)\) that the difference \(\alpha_1-\alpha_3\) calculated in this way turns out to be close to \(70^\circ\). Such a large phase difference cannot arise for any values of the spin and parity of the \(\Sigma\)-hyperon and indicates that the selection rule \(\Delta T=1/2\) does not hold in this case *).

14. OTHER POSSIBLE PARTICLES IN THE SCHEME OF ISOTOPIC MULTIPLETS

In conclusion let us consider the question of whether the scheme of strange particles considered above contains as yet unfilled “vacancies.” Going through all possible combinations of \(n\), \(T_3\), \(Q\), and \(S\), it is easy to see that, within the framework of ordinary particles, the existence of four more particles is possible: two hyperons and two mesons. The isotopic spins of these particles must be equal to zero.

From the relation \(Q=T_3+\dfrac{1}{2}+\dfrac{S}{2}\) for hyperons it follows that, for \(T_3=0\), a positively charged hyperon with \(Q=+1\) and strangeness \(S=+1\), and a negatively charged hyperon with \(Q=-1\) and strangeness \(S=-3\), are possible. Gell-Mann\(^{1}\) proposed calling the first of them \(Z^+\) (zeta plus), the second \(\Omega^-\) (omega minus). For mesons \(Q=T_3+S/2\), and two mesons with \(T_3=0\) and \(S=\pm2\) are possible. Gell-Mann called them \(\omega^\pm\). Here \(\omega^-\) is the antiparticle with respect to \(\omega^+\).

For metastability of the \(Z^+\)-hyperon it is necessary that its mass be less than the sum of the masses of the nucleon and the \(K\)-meson. In that case \(Z^+\) would decay into a nucleon and a \(\pi\)-meson. The presence of the \(Z^+\)-hyperon would lead, however, to more

*) If one considers the decay of the \(\Sigma\)-hyperon with nonconservation of parity, then the indicated contradiction does not arise and the selection rule \(\Delta T=1/2\) can be retained.

low threshold for the production of \(K^{-}\)-mesons in reactions of the type

\[ \pi^{-}+p \to Z^{+}+K^{-} \]

and to the possibility of pair production of hyperons in reactions of the type

\[ p+n \to Z^{+}+\Lambda^{0}. \]

For the metastability of the \(\Omega^{-}\)-hyperon, its mass must be less than the sum of the masses of the \(\Xi\)-hyperon and the \(K\)-meson. In this case the \(\Omega^{-}\)-hyperon will decay into a \(\Xi\)-hyperon and a \(\pi\)-meson (or also into a \(K\)-meson and a \(\Lambda\) (or \(\Sigma\))-hyperon, if there is enough energy). If, however, the mass of the \(\Omega^{-}\)-hyperon is less than the mass of the \(\Xi\)-hyperon, then the decay of the \(\Omega^{-}\)-hyperon will proceed by emission of leptons, since decay into a \(\Lambda\)-hyperon and a \(\pi\)-meson would correspond to \(\Delta S=2\), and decay into a nucleon and a \(\pi\)-meson to \(\Delta S=3\).

For the metastability of \(\omega\)-mesons their mass must be less than twice the mass of \(K\)-mesons.

If \(m_\omega > m_k\), then the \(\omega\)-meson will decay with \(\Delta S=1\) according to the scheme

\[ \omega^{\pm}\to K^{\pm}+\pi \;(\text{or } \gamma). \]

In addition to these particles, the existence of one more is possible, namely a meson with \(T=0\) and \(Q=0\). However, its metastability, if such a meson were to prove metastable, could not be explained by the conservation of strangeness, since the strangeness of such a meson is equal to zero.

15. CONCLUDING REMARKS

The scope of the review has not allowed us to dwell on a number of interesting directions connected with the theory of strange particles. We have in mind works devoted to the extension of the usual formalism of isotopic spin (the introduction of the so-called \(\omega\)-space\({}^{73}\), four-dimensional isotopic space\({}^{74,75}\), etc.), as well as works in which the classification of strange particles is carried out on the basis of various physical model concepts (the theory of the deformable form factor\({}^{77-80}\), the composite model of strange particles\({}^{81}\), etc.). A number of the works mentioned here, and a detailed exposition of questions belonging to this circle of ideas, are contained in the collection Systematics of Elementary Particles, edited by Yu. A. Yappa\({}^{82}\).

APPENDIX I

In this appendix formulas are given by means of which the product of wave functions with isotopic spins \(T_1\) and \(T_2\) and projections \(T_1^3\) and \(T_2^3\) is expressed in terms of the eigenfunctions of the operator of total isotopic spin \(T\) and projection \(T_3\). This corresponds to the expansion, in eigenfunctions of \(T\) and \(T_3\), of the wave function of two particles, one of which has isotopic spin \(T_1\) and projection \(T_1^3\), and the other—\(T_2\) and \(T_2^3\).

Let us introduce the following notation:

\[ \chi^{T_1T_2}_{T_1^3T_2^3} \]

is the wave function of two particles with isotopic spins \(T_1\) and \(T_2\) and projections \(T_1^3\) and \(T_2^3\).

In what follows the upper indices are omitted for brevity \(\bigl(\chi_{T_1^3T_2^3}\bigr)\).

\[ \varphi^T_{T_3} \]

is the wave function with isotopic spin \(T\) and projection \(T_3\). The relation between the functions \(\chi\) and \(\varphi\) is given by the Clebsch—Gordan coefficients

(see A. I. Akhiezer, V. B. Berestetskii, Quantum Electrodynamics, pp. 25 and 67):

1. \(T_1=\frac{1}{2},\ T_2=\frac{1}{2}\)

\[ \begin{aligned} 1)\quad &\chi_{\frac12\,\frac12}=\varphi^1_1,\\ 2)\quad &\chi_{\frac12\,-\frac12}=\frac{1}{\sqrt2}\varphi^1_0+\frac{1}{\sqrt2}\varphi^0_0,\\ 3)\quad &\chi_{-\frac12\,-\frac12}=\varphi^1_{-1},\\ 4)\quad &\chi_{-\frac12\,\frac12}=\frac{1}{\sqrt2}\varphi^1_0-\frac{1}{\sqrt2}\varphi^0_0. \end{aligned} \]

2. \(T_1=1,\ T_2=\frac{1}{2}\)

\[ \begin{aligned} 1)\quad &\chi_{1\,\frac12}=\varphi^{\frac32}_{\frac32},\\ 2)\quad &\chi_{0\,\frac12}=\sqrt{\frac{2}{3}}\,\varphi^{\frac32}_{\frac12}-\sqrt{\frac{1}{3}}\,\varphi^{\frac12}_{\frac12},\\ 3)\quad &\chi_{-1\,\frac12}=\sqrt{\frac{1}{3}}\,\varphi^{\frac32}_{-\frac12}-\sqrt{\frac{2}{3}}\,\varphi^{\frac12}_{-\frac12},\\ 4)\quad &\chi_{-1\,-\frac12}=\varphi^{\frac32}_{-\frac32},\\ 5)\quad &\chi_{0\,-\frac12}=\sqrt{\frac{2}{3}}\,\varphi^{\frac32}_{-\frac12}+\sqrt{\frac{1}{3}}\,\varphi^{\frac12}_{-\frac12},\\ 6)\quad &\chi_{1\,-\frac12}=\sqrt{\frac{1}{3}}\,\varphi^{\frac32}_{\frac12}+\sqrt{\frac{2}{3}}\,\varphi^{\frac12}_{\frac12}. \end{aligned} \]

3. \(T_1=\frac{3}{2},\ T_2=\frac{1}{2}\)

\[ \begin{aligned} 1)\quad &\chi_{\frac32\,\frac12}=\varphi^2_2,\\ 2)\quad &\chi_{\frac12\,\frac12}=\frac{\sqrt3}{2}\varphi^2_1-\frac{1}{2}\varphi^1_1,\\ 3)\quad &\chi_{-\frac12\,\frac12}=\frac{1}{\sqrt2}\varphi^2_0-\frac{1}{\sqrt2}\varphi^1_0,\\ 4)\quad &\chi_{-\frac32\,\frac12}=\frac{1}{2}\varphi^2_{-1}-\frac{\sqrt3}{2}\varphi^1_{-1},\\ 5)\quad &\chi_{-\frac32\,-\frac12}=\varphi^2_{-2},\\ 6)\quad &\chi_{-\frac12\,-\frac12}=\frac{\sqrt3}{2}\varphi^2_{-1}+\frac{1}{2}\varphi^1_{-1},\\ 7)\quad &\chi_{\frac12\,-\frac12}=\frac{1}{\sqrt2}\varphi^2_0+\frac{1}{\sqrt2}\varphi^1_0,\\ 8)\quad &\chi_{\frac32\,-\frac12}=\frac{1}{2}\varphi^2_1+\frac{\sqrt3}{2}\varphi^1_1. \end{aligned} \]

4. \(T_1=2,\ T_2=\frac{1}{2}\)

\[ \begin{aligned} 1)\quad &\chi_{2\,\frac12}=\varphi^{\frac52}_{\frac52},\\ 2)\quad &\chi_{1\,\frac12}=\frac{2}{\sqrt5}\varphi^{\frac52}_{\frac32}-\frac{1}{\sqrt5}\varphi^{\frac32}_{\frac32},\\ 3)\quad &\chi_{0\,\frac12}=\sqrt{\frac{3}{5}}\varphi^{\frac52}_{\frac12}-\sqrt{\frac{2}{5}}\varphi^{\frac32}_{\frac12},\\ 4)\quad &\chi_{-1\,\frac12}=\sqrt{\frac{2}{5}}\varphi^{\frac52}_{-\frac12}-\sqrt{\frac{3}{5}}\varphi^{\frac32}_{-\frac12},\\ 5)\quad &\chi_{-2\,\frac12}=\sqrt{\frac{1}{5}}\varphi^{\frac52}_{-\frac32}-\frac{2}{\sqrt5}\varphi^{\frac32}_{-\frac32},\\ 6)\quad &\chi_{-2\,-\frac12}=\varphi^{\frac52}_{-\frac52},\\ 7)\quad &\chi_{-1\,-\frac12}=\frac{2}{\sqrt5}\varphi^{\frac52}_{-\frac32}+\frac{1}{\sqrt5}\varphi^{\frac32}_{-\frac32},\\ 8)\quad &\chi_{0\,-\frac12}=\sqrt{\frac{3}{5}}\varphi^{\frac52}_{-\frac12}+\sqrt{\frac{2}{5}}\varphi^{\frac32}_{-\frac12},\\ 9)\quad &\chi_{1\,-\frac12}=\sqrt{\frac{2}{5}}\varphi^{\frac52}_{\frac12}+\sqrt{\frac{3}{5}}\varphi^{\frac32}_{\frac12},\\ 10)\quad &\chi_{2\,-\frac12}=\frac{1}{\sqrt5}\varphi^{\frac52}_{\frac32}+\frac{2}{\sqrt5}\varphi^{\frac32}_{\frac32}. \end{aligned} \]

5. \(T_1=1,\ T_2=1\)

\[ \begin{aligned} 1)\quad &\chi_{11}=\varphi^2_2,\\ 2)\quad &\chi_{01}=\frac{1}{\sqrt2}\varphi^2_1-\frac{1}{\sqrt2}\varphi^1_1,\\ 3)\quad &\chi_{-11}=\frac{1}{\sqrt6}\varphi^2_0-\frac{1}{\sqrt2}\varphi^1_0+\frac{1}{\sqrt3}\varphi^0_0,\\ 4)\quad &\chi_{1\,-1}=\frac{1}{\sqrt6}\varphi^2_0+\frac{1}{\sqrt2}\varphi^1_0+\frac{1}{\sqrt3}\varphi^0_0,\\ 5)\quad &\chi_{0\,-1}=\frac{1}{\sqrt2}\varphi^2_{-1}+\frac{1}{\sqrt2}\varphi^1_{-1},\\ 6)\quad &\chi_{-1\,-1}=\varphi^2_{-2},\\ 7)\quad &\chi_{10}=\frac{1}{\sqrt2}\varphi^2_1+\frac{1}{\sqrt2}\varphi^1_1,\\ 8)\quad &\chi_{00}=\sqrt{\frac{2}{3}}\,\varphi^2_0-\sqrt{\frac{1}{3}}\,\varphi^0_0,\\ 9)\quad &\chi_{-10}=\frac{1}{\sqrt2}\varphi^2_{-1}-\frac{1}{\sqrt2}\varphi^1_{-1}. \end{aligned} \]

STRANGE PARTICLES

As an example of the use of the formulas given in this appendix, let us consider the reactions \(K^-+d\to \Sigma+N\).

The isotopic wave function of a system consisting of \(\Sigma^-\)-particles \((T=1,\ T_3=-1)\) and a proton \((T=\tfrac12,\ T_3=\tfrac12)\), in the general case, is expressed in terms of isotopic wave functions with \(T=\tfrac32\) and \(T=\tfrac12\). From equality (2.3) it follows that

\[ \chi_{\Sigma^-p}=\sqrt{\frac{2}{3}}\,\varphi^{3/2}_{-1/2} +\sqrt{\frac{1}{3}}\,\varphi^{1/2}_{-1/2}. \]

Similarly, the function of the system \(\Sigma^0+n\) has the form

\[ \chi_{\Sigma^0 n}=\sqrt{\frac{2}{3}}\,\varphi^{3/2}_{-1/2} +\sqrt{\frac{1}{3}}\,\varphi^{1/2}_{-1/2}. \]

The isotopic spin of the deuteron is zero, that of the \(K^-\)-meson is \(\tfrac12\), and the total isotopic spin of the system \(K^-+d\) is \(\tfrac12\). We shall denote the wave function of this system by \(\psi^{1/2}\). If the interaction is isotopically invariant, then in the final state the isotopic spin is also equal to \(\tfrac12\), and \(\varphi^{3/2}=0\). In this case the amplitudes of the reactions \(K^-+d\to\Sigma^-+p\) and \(K^-+d\to\Sigma^0+n\) have the form \(-\sqrt{\frac{2}{3}}(\varphi^{1/2}\psi^{1/2})\) and \(\sqrt{\frac{1}{3}}(\varphi^{1/2}\psi^{1/2})\), and the corresponding cross sections are in the ratio \(2:1\).

APPENDIX II

In this appendix expressions are given for the amplitudes \(a\) of reactions of the type

\[ X^{T_1}_{T^3_1}+Y^{T_2}_{T^3_2}\to U^{T_3}_{T^3_3}+V^{T_4}_{T^3_4} \]

in terms of amplitudes with a given isotopic spin \(f^T\). Here \(X^{T_1}_{T^3_1}\) denotes a particle with isotopic spin \(T_1\) and projection of isotopic spin \(T^3_1\); the same applies to \(Y, U, V\). The reaction cross sections \(\sigma\) are related to the amplitudes \(a\) by the relation

\[ \sigma=|a|^2. \]

The relations given below are obtained simply with the help of isotopic functions (see Appendix I), if one assumes that the interaction is isotopically invariant and takes into account that the functions are orthonormal.

For brevity we shall omit the indices \(T\) in the expression for the amplitude \(a\), leaving only the indices \(T_3\): \(a(T^3_1T^3_2;\ T^3_3T^3_4)\). In the amplitudes \(f^T\), the indices indicating \(T_3\) are omitted. It is easy to show that, by virtue of isotopic symmetry,

\[ a(T^3_1T^3_2;\ T^3_3T^3_4) = a(-T^3_1-T^3_2;\ -T^3_3-T^3_4). \]

1. \(X^{1/2}+Y^{1/2}\to U^{1/2}+V^{1/2}\)

\[ a_1\left(\frac12\ \frac12;\ \frac12\ \frac12\right)=f^1, \]

\[ a_2\left(-\frac12+\frac12;\ \frac12+\frac12\right) =\frac12\left(f^1+f^0\right), \]

\[ a_3\left(-\frac12+\frac12;\ +\frac12-\frac12\right) =\frac12\left(f^1-f^0\right). \]

2. \(X^1+Y^{1/2}\to U^1+V^{1/2}\)

\[ a_1\left(1\frac12;\ 1\frac12\right)=f^{3/2}, \]

\[ a_2\left(0\frac12;\ 0\frac12\right) =\frac13\left(2f^{3/2}+f^{1/2}\right), \]

\[ a_3\left(0\frac12;\ 1-\frac12\right) =\frac{\sqrt2}{3}\left(f^{3/2}-f^{1/2}\right), \]

\[ a_4\left(-1\frac12;\ -1\frac12\right) =\frac13\left(f^{3/2}+2f^{1/2}\right), \]

\[ a_5\left(-1\frac12;\ 0-\frac12\right) =\frac{\sqrt2}{3}\left(f^{3/2}-f^{1/2}\right). \]

3. \(X^1+Y^1\to U^1+V^1\)

\[ a_1(11;\,11)=f^2, \]

\[ a_2(01;\,01)=\frac{1}{2}(f^2+f^1), \]

\[ a_3(01;\,10)=\frac{1}{2}(f^2-f^1), \]

\[ a_4(-11;\,-11)=\frac{1}{6}(f^2+3f^1+2f^0), \]

\[ a_5(-11;\,1{-}1)=\frac{1}{6}(f^2-3f^1+2f^0), \]

\[ a_6(-11;\,00)=\frac{1}{3}(f^2-f^0), \]

\[ a_7(00;\,00)=\frac{1}{3}(2f^2+f^0). \]

4. \(X^{1/2}+Y^{1/2}\to U^1+V^1\)

\[ a_1\left(\frac12\,\frac12;\,10\right)=\frac{f^1}{\sqrt2}, \]

\[ a_2\left(\frac12\,\frac12;\,01\right)=-\frac{f^1}{\sqrt2}, \]

\[ a_3\left(-\frac12\,\frac12;\,1{-}1\right)=-\frac{f^0}{\sqrt6}+\frac{f^1}{2}, \]

\[ a_4\left(-\frac12\,\frac12;\,-11\right)=-\frac{f^0}{\sqrt6}-\frac{f^1}{2}, \]

\[ a_5\left(-\frac12\,\frac12;\,00\right)=\frac{f^0}{\sqrt6}. \]

Expressions for \(a\) in terms of \(f\) make it possible to obtain relations between the probabilities of reactions differing in the charge states of the particles participating in them. Thus, relations (1) of this appendix make it possible to compare the cross sections of the interactions of \(K^+\)-mesons with protons and neutrons, if it is assumed that the channel with \(T=1\) is dominant: \(f^1\gg f^0\). In this case the amplitudes of the reactions \(K^+ + p\to K^+ + p\), \(K^+ + n\to K^+ + n\), \(K^+ + n\to K^0 + p\) are equal to \(f^1\), \(\frac12 f^1\), \(\frac12 f^1\), respectively, and the corresponding cross sections are in the ratio \(1:\frac14:\frac14\).

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