RELATIVISTIC TRANSFORMATIONS AND CONSERVATION LAWS OF ENERGY–MOMENTUM. APPLICATION TO SOME PROBLEMS IN COSMIC RAY PHYSICS

I. L. Rozental

INTRODUCTION

At present, theory is not capable of giving any unambiguous description of the mutual transformations of elementary particles. It therefore seems very valuable to develop methods based on unquestionably valid conservation laws and helping to reveal the characteristics of particles and their decay schemes. For experimentalists working in this direction, the possibility of identifying particles is of primary importance. In this respect, too, the consistent application of conservation laws can be of great assistance. The purpose of the present article is to set forth data on the application of the laws of conservation of energy–momentum to the study of particle decay*). Naturally, the article does not set out, in any systematic way, the experimental data on elementary particles currently available. These data, drawn mainly from a number of reviews (see, for example, \(^{1,2}\)), are used only to illustrate the general methods.

The successful application of the laws of conservation of energy and momentum to the problem of interest to us is based to a considerable extent on certain features of the coordinate system associated with the center of gravity of the particles, which determine the phenomenon under consideration (the \(c\)-system). The first, essentially trivial, characteristic of this system is the equation \(\sum \mathbf{p}=0\) (\(\mathbf{p}\) is the momentum of the \(i\)-th parti-

* We do not touch upon the application of other conservation laws (parity, spin, isotopic spin, etc.) to the solution of problems connected with elementary particles. A description of this very promising approach is a completely independent problem, partially solved in the review \(^{3}\). [See also work \(^{19}\).]

tions in the \(u\)-system). The other characteristic is associated with certain symmetry properties of the processes considered in the \(u\)-system. Thus, usually in a decay in the \(u\)-system (which is associated with the decaying particle) the phenomena are characterized by spherical symmetry in the sense that their averaged characteristics are identical with respect to any direction. A somewhat more complicated kind of symmetry occurs in collisions. The presence of a distinguished direction (the relative direction of motion of the particles) leads to axial symmetry*). The same type of symmetry may also occur in a decay accompanied by polarization effects. In this case the distinguished direction may be, for example, the spin direction of the primary particle.

The features indicated determine the advantages of considering various phenomena in the \(u\)-system, from which one can then pass to the laboratory coordinate system (\(l\)-system) by means of relativistic transformations. Derivations of the basic formulae of the theory of relativity can be found in any course on the theory of relativity (for example, L. Landau and E. Lifshitz\({}^{4}\)).

Notation Used

\(\bar E_0,\ \bar p_0\) — respectively the energy and momentum of the primary particle in the \(u\)-system;

\(E_0,\ p_0\) — the energy and momentum of the primary particle in the \(l\)-system;

\(\bar E,\ \bar p,\ \bar\vartheta\) — the energy, momentum, and polar angle of the direction of motion of some secondary particle (relative to the direction of motion of the primary particle) in the \(u\)-system;

\(\bar E_1,\ \bar p_1,\ \bar\vartheta_1,\ \bar E_2,\ \bar p_2,\ \bar\vartheta_2\) — the energies, momenta, and polar angles of definite secondary particles (which we denote by the indices 1, 2, ...) in the \(u\)-system;

\(E,\ p,\ \vartheta;\ E_1,\ p_1,\ \vartheta_1;\ E_2,\ p_2,\ \vartheta_2\) — the same quantities in the \(l\)-system;

\(M,\ \mu,\ \mu_1,\ \mu_2\) — the masses of the primary and secondary particles;

\(\bar\beta_0,\ \bar\beta,\ \bar\beta_1\) — the velocities of the primary and secondary particles in the \(u\)-system;

\(\beta_0,\ \beta,\ \beta_1\) — the velocities of the primary and secondary particles in the \(l\)-system;

\(V\) — the velocity of the \(u\)-system relative to the \(l\)-system**);

* ) The kinematic treatment of decay and collision processes has many features in common. Therefore, although the study of collisions is not included in our problem, we shall carry out the calculations in a form allowing them to be applied to the solution of this problem.

**) The speed of light is taken as the unit of velocity.

$$ \gamma=\frac{1}{\sqrt{1-V^{2}}}; $$

$\overline{N}(\overline{p}, \cos \overline{\vartheta})\,d\overline{p}\,d\cos \overline{\vartheta}\,d\overline{\varphi}$ is the number of secondary particles possessing momentum lying in the interval $\overline{p}, \overline{p}+d\overline{p}$ and emitted in the direction with azimuthal and polar angles lying, respectively, between $\overline{\varphi}$ and $\overline{\varphi}+d\overline{\varphi}$, and $\arccos \overline{\vartheta}$ and $\operatorname{arc}(\cos \overline{\vartheta}+d\cos \overline{\vartheta})$.

Since in what follows problems characterized by axial symmetry are analyzed, the function $\overline{N}(\overline{p}, \cos \overline{\vartheta})$ does not depend on the azimuthal angle $\overline{\varphi}$; $N(p,\cos\vartheta)$ is defined analogously to the function $\overline{N}(\overline{p}, \cos\overline{\vartheta})$.

1. GENERAL SCHEME OF KINEMATIC TRANSFORMATIONS FROM THE $u$-SYSTEM TO THE $l$-SYSTEM

A peculiarity of the relativistic transformations of the angles $\vartheta$, at which the secondary particles move*), from one coordinate system to another is their independence of the dynamical properties of the particles (for example, their masses).

As is known (see, for example, $^{4}$),

$$ \operatorname{tg}\overline{\vartheta}=\frac{1}{\gamma}\frac{\sin\vartheta}{\cos\vartheta-\dfrac{V}{\beta}}, \tag{1} $$

and, conversely,

$$ \operatorname{tg}\vartheta=\frac{1}{\gamma}\frac{\sin\overline{\vartheta}}{\cos\overline{\vartheta}+\dfrac{V}{\overline{\beta}}}. \tag{2} $$

Using the relations

$$ \overline{\beta}=\frac{\overline{p}}{\overline{E}}\quad \text{and} \quad \beta=\frac{p}{E}, $$

equations (1) and (2) may be written in the following form:

$$ \operatorname{tg}\overline{\vartheta}=\frac{1}{\gamma}\frac{p\sin\vartheta}{p\cos\vartheta-VE}, \tag{1a} $$

$$ \operatorname{tg}\vartheta=\frac{1}{\gamma}\frac{\overline{p}\sin\overline{\vartheta}}{\overline{p}\cos\overline{\vartheta}+V\overline{E}}. \tag{2a} $$

Since

$$ \overline{p}\sin\overline{\vartheta}=p\sin\vartheta \tag{3} $$

*) Secondary are all particles after the act of collision or decay and those caused by them (i.e. newly produced particles, $\delta$-nucleons, as well as the primary particle).

and

\[ \bar p \cos \bar\vartheta=\gamma[p\cos\vartheta-VE]^{*}), \tag{4} \]

\[ p\cos\vartheta=\gamma[\bar p\cos\bar\vartheta+V\bar E], \tag{5} \]

then

\[ \bar p=\gamma\sqrt{[E-pV\cos\vartheta]^2-\frac{\mu^2}{\gamma^2}}, \tag{6} \]

\[ p=\gamma\sqrt{[\bar E+\bar pV\cos\bar\vartheta]^2-\frac{\mu^2}{\gamma^2}}. \tag{7} \]

In passing from one coordinate system to another, the distribution functions in momenta and angles transform as follows:

\[ \bar N(\bar p,\cos\bar\vartheta)\,d\bar p\,d\cos\bar\vartheta\,d\bar\varphi = N(p,\cos\vartheta)\,J\,dp\,d\cos\vartheta\,d\varphi, \tag{8} \]

where the Jacobian

\[ J= \frac{\partial\bar p}{\partial p}\frac{\partial\cos\bar\vartheta}{\partial\cos\vartheta} - \frac{\partial\bar p}{\partial\cos\vartheta}\frac{\partial\cos\bar\vartheta}{\partial p} \]

can be obtained, using (1a), (6), and the equality \(d\bar\varphi=d\varphi\):

\[ J=\frac{p^2\gamma}{\bar p^{\,2}E}(E-pV\cos\vartheta) = \frac{p^2}{\gamma E}\, \frac{(E-pV\cos\vartheta)} {(E-pV\cos\vartheta)^2-\dfrac{\mu^2}{\gamma^2}}. \tag{9} \]

In the polar coordinate system \((\bar p,\bar\vartheta)\), the region in which the quantities \(\bar p,\bar\vartheta\) are contained for each given process is bounded by the curves

\[ \bar p=\bar p_{\max}=\Phi_1(\bar\vartheta) \tag{10a} \]

and

\[ \bar p=\bar p_{\min}=\Phi_2(\bar\vartheta), \tag{10b} \]

where \(\Phi_1,\Phi_2\) are certain functions determined by the particular physical processes and conservation laws.

Thus, in decay into three particles, the momenta of the secondary particles may take any values from 0 to \(\bar p_{\max}\), determined by the quantities \(M,\mu_1,\mu_2,\mu_3\) (see equation (54)). For example, in the case

*) In paper 5, formula (4) was erroneously written in the form

\[ \bar p\cos\bar\vartheta=\pm\gamma[p\cos\vartheta-VE]. \]

The presence of two signs leads to ambiguity of the Lorentz transformation of momenta, which in itself already testifies against the presence of two signs.

of the decay of a \(\mu\)-meson into an electron and two neutrinos, the momentum is \(p_{\max}\sim \dfrac{M}{2}\). In decay into two particles, their momenta in the \(\mu\)-system can take strictly definite values, depending only on the values \(M,\mu_1,\mu_2\) (see (29)).

Substituting into (10) the values of the quantities \(\bar{\vartheta}\), \(\bar p\), determined in accordance with (1) and (6), we obtain the following equations describing the curves that bound regions in the coordinate system \(p,\vartheta\):

\[ \gamma \sqrt{(E-pV\cos\vartheta)^2-\frac{\mu^2}{\gamma^2}} = \Phi_1\left[ \operatorname{arctg}\frac{1}{\gamma} \left( \frac{p\sin\vartheta}{p\cos\vartheta-VE} \right) \right], \tag{11a} \]

\[ \gamma \sqrt{(E-pV\cos\vartheta)^2-\frac{\mu^2}{\gamma^2}} = \Phi_2\left[ \operatorname{arctg}\frac{1}{\gamma} \left( \frac{p\sin\vartheta}{p\cos\vartheta-VE} \right) \right]. \tag{11b} \]

Equations (11a) and (11b) are greatly simplified in practice, since in decay \(\Phi_1=\bar p_{\max}=\mathrm{const}\); in the case of decay into two particles \(\Phi_1=\Phi_2\); in decay into three particles \(\Phi_2=0\).

In these cases the curve corresponding to (11a) is described by a second-degree equation. Investigation shows that this curve is always an ellipse. Thus, if in the coordinate system \((\bar p,\bar\vartheta)\) the curve bounding a certain region is a circle, then transformation to the coordinate system \((p,\vartheta)\) transforms it into an ellipse.

The major axis of this ellipse lies on the straight line that is the continuation of the velocity vector \(V\), and is equal to \(2\bar p\gamma\); the minor axis is equal to \(2\bar p\); the center is located at a distance \(EV\gamma^2\) from the origin of coordinates*).

Solving equation (11a) with respect to \(p\), we obtain:

\[ p= \frac{\bar E V\cos\vartheta \pm \sqrt{\mu^2\gamma^2 V^2\cos^2\vartheta-\mu^2\gamma^2+E^2}} {\gamma(1-V^2\cos^2\vartheta)}. \tag{12} \]

The sign before the radical in formula (12) is chosen as follows.

1. If the expression under the radical is always positive, then the sign “\(+\)” should be chosen. Indeed, in this case, as the momentum changes from \(0\) to \(\bar p_{\max}\), the sign before the radical cannot change, since the function must be continuous. Therefore throughout the whole interval there must be either the sign “\(+\)” or the sign “\(-\)”.

But since there are always positive values of \(p\) (for example, at \(\vartheta=0\)), in the present case the sign “\(+\)” should be chosen for the whole interval \((0-\bar p_{\max})\). Thus, the first crite-

* ) The ellipse has been investigated for the case of elastic collision in \(^4\); the general case was analyzed by Bletton \(^6\).

the criterion for choosing the sign is the positive definiteness of the expression

\[ D(\vartheta)=\mu^2\gamma^2 V^2\cos^2\vartheta-\mu^2\gamma^2+\bar E^2. \tag{13a} \]

It is easy to show that \(D(\vartheta)>0\), if \(\bar\beta>V\). Consequently, if the velocity of the particle \(\bar\beta>V\), then the sign “\(+\)” is always chosen.

1. If \(\bar\beta<V\), then for some value \(\vartheta=\vartheta_{\max}\), \(D(\vartheta)=0\), and it is necessary to take both signs into account. Since for \(\vartheta>\vartheta_{\max}\) we have \(D(\vartheta)<0\), \(\vartheta_{\max}\) corresponds to the greatest possible value of the angle \(\vartheta\)* ) ( \(2\vartheta_{\max}\) is the angle at which the ellipse described by equation (11a) is seen from the origin).

From the condition \(D(\vartheta_{\max})=0\) one can obtain

\[ \sin\vartheta_{\max}= \sqrt{\frac{1-V^2}{\frac{1}{\bar\beta^2}-V^2}} . \tag{14} \]

The existence of a maximum angle means that in this case the particles move only within the forward hemisphere.

Fig. 1a. Transformation \((\bar p,\bar\vartheta)\to(p,\vartheta)\); \(\bar\beta<V\), \(\bar E=3;\ V=0.9\). In Figs. 1a, 1b, 2a, 2b, and 3 the same scale is used.

Fig. 1b. Transformation \((\bar p,\bar\vartheta)\to(p,\vartheta)\); \(\bar\beta>V;\ \bar E=2;\ V=0.9\).

— In Fig. 1a a diagram is shown of the transformation of the circles \(\bar p=\mathrm{const}\) under a Lorentz transformation for the case \(\bar\beta<V\). Between the radius vectors of the circle and the ellipse there is a mutual—

* ) The existence of the maximum angle \(\bar\vartheta_{\max}\) under the condition \(\bar\beta<V\) was noted in a number of works \(^{5,6,7}\).

but a one-to-one correspondence. The elliptic curve was calculated by formula (12) for the following parameter values: \(\mu=1\); \(E=3\); \(V=0.9\). The case in which the maximum possible energy of the particles in the \(u\)-system is numerically equal to three times its mass can occur in the decay of one of the types of \(\chi\)-particles according to the scheme

\[ \chi \to \mu + \nu + \nu \tag{15} \]

(\(\mu\)—\(\mu\)-meson; \(\nu\)—neutrino).

For this it is necessary (since \(\mu_\nu=0\)) that the mass of the \(\chi\)-particle be \(M_\chi=5.7\mu_\mu\) (see formula (54)). A characteristic feature of the transformation described is the absence of forbidden angles in the \(\lambda\)-system.

In Fig. 16 a scheme of the transformation of a circle is presented for the case \(\bar{\beta}>V\). The ellipse is determined by the quantities: \(\mu=1\); \(E=2\); \(V=0.9\). The maximum possible energy of the \(\mu\)-meson in the \(u\)-system will be numerically equal to twice the value of its mass if the following decay scheme is realized:

\[ \chi \to \mu + \nu + \pi^0 \tag{16} \]

and the particle masses are: \(M_\chi=4.2\); \(\mu_{\pi^0}=1.3\). \(\vartheta_{\max}\) is the maximum permissible emission angle of the \(\mu\)-meson for this decay scheme and for the given value of the velocity.

It is expedient to use the described method of transformations for obtaining angular and energy dependences in the \(\lambda\)-system (see § 3).

Another form of analysis was used by Bradt, Kaplan, and Peters\(^5\), who employed a mixed coordinate system \((\bar p,\vartheta)\). In order to compute the Jacobian in this case, we shall use formula (2a), from which it follows that

\[ \cos \bar{\vartheta} = \frac{-\bar{E}V\gamma^2 \operatorname{tg}^2 \vartheta \pm \sqrt{\bar p^2 + [\bar p^2 - V^2 \bar{E}^2]\gamma^2 \operatorname{tg}^2 \vartheta}} {\bar p \,[\gamma^2 \operatorname{tg}^2 \vartheta + 1]} . \tag{17} \]

The sign in formula (17) is chosen from considerations analogous to those given earlier (see p. 409):

1. If the expression under the radical is always positive, then the sign “\(+\)” should be chosen throughout the entire interval of variation of the angle \(\vartheta\) \([0-\pi]\).

The criterion for choosing the sign is the positive definiteness of the expression

\[ f(\vartheta)=\bar p^2+\gamma^2 \operatorname{tg}^2 \vartheta \,[\bar p^2 - V^2 \bar{E}^2], \tag{13б} \]

which also holds under the condition \(\bar{\beta}>V\).

1. If \(\bar{\beta}<V\), then there is always a maximum permissible angle \(\vartheta_{\max}<\pi\). (It is clear that \(\vartheta\) may in this case also take any values from \(0\) to \(\pi\).) From considerations of continuity, it follows—

it follows that in the intervals \(0<\bar\vartheta<\bar\vartheta_{\max}\) and \(\bar\vartheta_{\max}<\bar\vartheta<\pi\) (\(\bar\vartheta_{\max}\) is the angle corresponding to \(\vartheta_{\max}\)) a constant sign is chosen in front of the radical. Since for the value \(\vartheta=0\), \(\cos 0=1>0\), throughout the interval \(0<\bar\vartheta<\bar\vartheta_{\max}\) the sign “\(+\)” should be chosen; analogously, in the interval \(\bar\vartheta_{\max}<\bar\vartheta<\pi\) the sign “\(-\)” is chosen.

Using the condition \(f[\vartheta(\bar\vartheta)]=0\), one may write:

\[ \cos \bar\vartheta_{\max}=-\frac{\bar\beta}{V}. \tag{18} \]

\(\bar\vartheta_{\max}\) is the angle in the \(u\)-system corresponding to the maximum possible angle \(\vartheta_{\max}\) in the \(l\)-system. In the \(l\)-system the angle \(\vartheta_{\max}\) is determined by formula (14).

Thus, if \(\bar\beta<V\), then for \(\bar\vartheta<\bar\vartheta_{\max}\) the sign “\(+\)” is chosen, and for \(\bar\vartheta>\bar\vartheta_{\max}\) the sign “\(-\)” is chosen.

In the present case the circle corresponding in the coordinate system \((\bar p,\bar\vartheta)\) to the equation \(\bar p=\mathrm{const}\) is transformed in the system \((\bar p,\vartheta)\) either into a circle (\(\bar\beta>V\)) or into a sector (\(\bar\beta<V\)).

From (16) one may calculate

\[ J_1=\left.\frac{\partial \cos \bar\vartheta}{\partial \cos \vartheta}\right|_{p=\mathrm{const}} \]

\[ J_1= \frac{\pm \gamma^2\left[V\bar E+\sqrt{\bar p^{\,2}+\gamma^2\tg^2\vartheta\left(\bar p^{\,2}-V^2\bar E^{\,2}\right)}\right]^2} {\bar p\cos^3\vartheta\left(\gamma^2\tg^2\vartheta+1\right)^2 \sqrt{\bar p^{\,2}+\gamma^2\tg^2\vartheta\left(\bar p^{\,2}-V^2\bar E^{\,2}\right)}} ; \tag{19} \]

the sign “\(+\)” is chosen if \(\bar\beta>V\), or if \(\bar\beta<V\) and \(0<\bar\vartheta<\bar\vartheta_{\max}\); the sign “\(-\)” is chosen if \(\bar\beta<V\) and \(\bar\vartheta_{\max}<\bar\vartheta<\pi\).

Fig. 2a. Transformation
\[ (\bar p,\bar\vartheta)\to(\bar p,\vartheta);\quad \bar\beta_{\max}>V; \]
\[ \bar E=3;\quad V=0.9. \]

In Fig. 2a the transformation is presented of the region bounded in the \(u\)-system by the equations \(\bar p_{\max}=\mathrm{const}\); \(\bar p_{\min}=0\) for the case \(\bar\beta_{\max}>V\). Since the momentum is not transformed, naturally,

the circle \(\bar p_{\max}=\mathrm{const}\) does not change. However, for small momenta, in accordance with formula (14), forbidden regions appear, situated inside two tangent circles. The shaded region corresponds to the allowed values of momenta and angles in the \(l\)-system. In Fig. 2b the transformation of an analogous region is shown for \(\bar\beta_{\max}<V\).

In the case when the momentum in the \(u\)-system has a strictly definite value (for example, in decay into two particles), then for \(\bar\beta>V\) the circle \(\bar p=\mathrm{const}\) passes into an arc. In Fig. 2 the same scale and the same parameter values have been chosen as in Fig. 1.

The indicated method was used by Bradt et al.\(^5\) for calculating angular distributions in the \(l\)-system under various particular assumptions about the character of the distributions in the \(u\)-system.

In individual cases, in order to obtain the momentum distribution it is convenient to solve the problem in the system \((p,\bar\vartheta)\). In order to obtain the Jacobian \(J_2\) corresponding to the transformation from the system \((\bar p,\bar\vartheta)\) to the system \((p,\bar\vartheta)\), it is necessary to compute the explicit expression \(\bar p(p,\bar\vartheta)\).

Fig. 2b. Transformation \((\bar p,\bar\vartheta)\to(p,\vartheta)\); \(\bar\beta_{\max}<V;\ \bar E=2;\ V=0.9\)

Using formula (12), we obtain:

\[ J_2=\left.\frac{\partial \bar p}{\partial p}\right|_{\bar\vartheta=\mathrm{const}}= \]

\[ =\frac{p}{\gamma(1-V^2\cos^2\bar\vartheta)} \left[ \frac{1}{\sqrt{p^2-\gamma^2V^2\mu^2\sin^2\bar\vartheta}} - \frac{V\cos\bar\vartheta}{\sqrt{p^2+\mu^2}} \right]. \tag{20} \]

The intervals of variation of the quantity \(p(\bar\vartheta)\) are easily determined from the equations

\[ p_{\min}=\gamma\sqrt{(\bar E_{\min}+\bar p_{\min}V\cos\bar\vartheta)^2-\frac{\mu^2}{\gamma^2}} \tag{21a} \]

and

\[ p_{\max}=\gamma\sqrt{(\bar E_{\max}+\bar p_{\max}V\cos\bar\vartheta)^2-\frac{\mu^2}{\gamma^2}}. \tag{21б} \]

The quantities \(p_{\min}\) and \(p_{\max}\) take an especially simple form in the case of interest to us, when \(\bar p_{\min}=0,\ \bar p_{\max}=\mathrm{const}\).

The transformation of the region under the transformation \((\bar p,\bar\vartheta)\to(p,\vartheta)\) is shown schematically in Fig. 3.

Fig. 3. Transformation
\((\bar p,\bar\vartheta)\to(p,\vartheta);\quad E=3,\ V=0.9.\)

It is necessary to emphasize that in the extreme relativistic case the principal formulas given above, (1a), (2a), (6), (7), (9), (16), (17), (19), and (20), are substantially simplified.

If

\[ |\cos\vartheta - V|\gg \frac{\mu^2}{p^2}, \]

then

\[ \tan \bar\vartheta \sim \frac{1}{\gamma}\, \frac{\sin\vartheta}{\cos\vartheta - V}, \tag{16} \]

and if

\[ |\cos\vartheta - V|\ll \frac{\mu^2}{p^2}, \]

then

\[ \tan \bar\vartheta \sim -\frac{1}{\gamma}\tan\frac{\vartheta}{2}. \tag{26} \]

For

\[ p\gg \mu,\qquad \gamma\gg 1 \]

\[ \bar p \sim \gamma(E-pV\cos\vartheta). \tag{6a} \]

If, in addition,

\[ 1-\cos\vartheta \gg \frac{\mu^2}{p^2}, \]

then

\[ \bar p \sim \gamma p(1-V\cos\vartheta)\sim \gamma E(1-V\cos\vartheta). \tag{6b} \]

Similarly, for

$$ \bar p \gg \mu, \quad \gamma \gg 1 $$

$$ p \sim \gamma(\bar E+\bar p V \cos \bar\vartheta). \tag{7a} $$

If, moreover,

$$ 1+\cos \bar\vartheta \gg \frac{\mu^2}{p^2}, $$

then

$$ p \sim \gamma \bar p(1+\cos \bar\vartheta)\sim \gamma \bar E(1+\cos \bar\vartheta). \tag{7b} $$

If

$$ p \gg \mu,\quad \gamma \gg 1, $$

then

$$ J \sim \frac{1}{\gamma(E-pV\cos\vartheta)}. \tag{9a} $$

Under the additional condition

$$ 1-\cos\vartheta \gg \frac{\mu^2}{p^2} $$

$$ J \sim \frac{1}{\gamma(1-V\cos\vartheta)}. \tag{9b} $$

If

$$ \gamma \gg 1 \ \text{and}\ \vartheta \gtrsim 1, $$

then

$$ \cos \bar\vartheta \sim \frac{-V\gamma^2 \operatorname{tg}^2\vartheta \pm 1}{\gamma^2 \operatorname{tg}^2\vartheta+1} \tag{16a} $$

and

$$ J_1 \sim \frac{\pm \gamma^2 (V \pm 1)^2}{\cos^3\vartheta\,(\gamma^2 \operatorname{tg}^2\vartheta+1)^2}. \tag{18a} $$

If additionally

$$ \vartheta \ll \frac{1}{\gamma}, $$

then

$$ \cos \bar\vartheta \sim 1-2\gamma^2\vartheta^2 \tag{16b} $$

and

$$ J_1 \sim 4\gamma^2(1-\gamma^2\vartheta^2). \tag{18b} $$

If

$$ p \gg \gamma\mu, $$

then

$$ J_2 \sim \frac{1}{\gamma(1+V\cos\vartheta)}. \tag{20a} $$

In conclusion of this section it should be noted that all the relations derived so far described transformations of im-

of momenta and angles for any two coordinate systems. However, in the case where the transition from the \(u\)-system to the \(l\)-system is considered, the formulas are substantially simplified, since in this case the transformation coefficient \(\gamma\) is expressed directly in terms of the most important characteristic of the processes of interest to us (decay, collision of identical particles)—the energy of the primary particle.

Indeed, in the case of decay,

\[ \gamma=\frac{E_0}{M}. \tag{22} \]

In the collision of two identical particles the following relations hold:

\[ 2\bar E_0\gamma=E_0+M, \]
\[ 2\bar E_0\gamma V=p_0 \tag{23} \]

and, consequently,

\[ \gamma=\sqrt{\frac{E_0+M}{2M}}=\frac{\bar E_0}{M}. \tag{24} \]

2. DECAY INTO TWO PARTICLES

The most important feature of decay into two particles is the independence of the kinematic characteristics of the process from the type of interaction responsible for the decay. In decay into two particles they are determined only by the masses of the primary and secondary particles and by two parameters that do not depend on the type of interaction (for example, the momentum of the primary particle and the emission angle of the secondary particle in the \(u\)-system). It must be emphasized that this feature belongs to a considerably broader class of phenomena than decay into two particles. Indeed, as follows from what follows, it occurs in any transformations as a result of which there are only two particles in the final state (for example, in collisions of two particles).

In the experimental study of decay into two particles, the following problems usually arise:

establishing the criterion to which cases observed by means of a photographic emulsion or Wilson chamber must conform, if they are in fact caused by decay;

determining the mass of the primary particle, if the masses and momenta of both secondary particles are known;

determining the mass of one of the secondary particles, if the masses and momenta of the primary and the other secondary particle are known;

investigating angular and energy distributions.

a) Investigation of the criterion for the correctness of the interpretation of various cases as consequences of decay

At the present time there are no methods which, in the analysis of decay processes, would make it possible to show unambiguously that a given individual event should be interpreted as a decay. In considering individual cases one usually seeks to prove that they cannot be ascribed to other known processes (elastic or inelastic interactions of various types).

Owing to this circumstance, the investigation of the existence of decay processes is usually based on a statistical approach. First, with the aid of a small number of cases that are interpreted as decay, the principal dynamical characteristics of the decay are determined approximately (type of decay, masses of the primary and secondary particles). Then, for the analysis of a large body of decay cases according to the assumed scheme, the approximate values of the characteristics obtained earlier are used. The agreement of the observed features of the set of cases under study with the conclusions that follow from the assumed decay scheme confirms its correctness. Naturally, with such a statistical approach it is impossible to avoid the erroneous inclusion in the set under study of a certain number of cases that do not belong to it and do not affect the general picture.

Of course, the method described is used not only to verify the fact that decay exists, but also to check the correctness of the chosen decay scheme.

In this section we shall apply the statistical approach to the analysis of decay into two particles. It is thereby naturally assumed that the masses of the primary and secondary particles and the type of decay are given.*)

For what follows it is useful to keep in mind two types of decay into two particles: one of them, which we shall call \(H\)-decay, is the transformation of a neutral particle into two charged ones; the second type (\(З\)-decay) corresponds to the decay of a charged particle into one charged and one neutral particle (Fig. 4).

Fig. 4. Decay schemes of neutral (\(H\)-) and charged (\(З\)-) particles.

Turning to the investigation of one or another decay process, we shall first consider the simplest case—the decay of a particle at rest.

*) Some methods for determining the masses of particles participating in the decay are given below.

In this case one can write the following equations:

\[ \overline{E}_1+\overline{E}_2=M=\overline{E}_0, \tag{25a} \]

\[ \overline{p}_1=\overline{p}_2=\overline{p}_c. \tag{25b} \]

Using the relations

\[ \overline{E}_1=\sqrt{\overline{p}_1^{\,2}+\mu_1^2} \quad \text{and} \quad \overline{E}_2=\sqrt{\overline{p}_2^{\,2}+\mu_2^2}, \]

it is easy to obtain:

\[ \overline{E}_1=\frac{M^2+\mu_1^2-\mu_2^2}{2M}, \tag{26a} \]

\[ \overline{E}_2=\frac{M^2-\mu_1^2+\mu_2^2}{2M}. \tag{26b} \]

It follows from formulas (26) that, in the decay of a particle at rest into two secondary particles, the latter possess a constant energy and, consequently, also a range. Thus, the existence of breaks with a constant magnitude of the segment from the break point to the end of the track (\(\Xi\)-decay), or the presence of characteristic “forks” with constant magnitudes of both branches (\(H\)-decay), definitely testifies in favor of the decay of stopped particles. It was precisely this circumstance that served as the first basis for asserting the existence of \((\pi-\mu)\)-decay (see \(^{8}\)).

More complicated is the study of the decay of moving particles. The basis for one of the methods of verification is the invariance of the transverse components of the momenta \(\overline{p}_T\) with respect to the Lorentz transformation.

Indeed, since the secondary particles move isotropically in the \(\mu\)-system with one and the same momentum \(\overline{p}_c\), the probability that a particle is emitted in the interval \(\cos \overline{\vartheta},\ \cos \overline{\vartheta}+d\cos \overline{\vartheta}\) is

\[ \overline{N}(\overline{p}, \cos \overline{\vartheta})\,d\overline{p}\,d\cos \overline{\vartheta} = \frac{1}{2}\,\delta(\overline{p}-\overline{p}_c)\sin \overline{\vartheta}\,d\overline{\vartheta}\,d\overline{p} \tag{27} \]

(\(\delta\) is the Dirac delta function).

Since

\[ \sin \overline{\vartheta}=\frac{\overline{p}_T}{\overline{p}_c}, \]

then \(^{9}\)

\[ N(\overline{p}_T)\,d\overline{p}_T \sim \frac{\overline{p}_T}{p_c\sqrt{(p_c^2-\overline{p}_T^{\,2})}}\,d\overline{p}_T, \tag{28} \]

where, in accordance with (25a, b) and (26a, b),

\[ \bar p_c=\frac{\sqrt{M^4+\mu_1^4+\mu_2^4-2(M^2\mu_1^2+M^2\mu_2^2+\mu_1^2\mu_2^2)}}{2M}. \tag{29} \]

Thus, if the set of cases under study is due to decay into two particles, then the distribution of the transverse components of the momenta obeys relation (28).

Another method of analyzing decay into two particles is based on the existence of extremal values of the emission angles of the particles in the \(l\)-system. Generally speaking, the extremal values of the angles \(\vartheta\) are determined by two parameters (for example, by the momentum \(p_0\) of the primary particle in the \(l\)-system and the momentum of the secondary particle \(p_1\) in the \(u\)-system). However, in decay into two particles the momentum \(p_1\) is completely determined by the values of the masses (see (29)). Therefore one can calculate the functional dependence \(\vartheta_{\mathrm{extr}}(p_0)\) and, consequently, also determine the limits within which the emission angles of the secondary particles formed in the decay of a particle with momentum \(p_0\) must lie.

One version of this method, which is applicable to the analysis of both types of decay, is based on the use of formula (14). Substituting into (14) the values

\[ \bar p=\frac{\bar p_c}{\sqrt{\bar p_c^{\,2}+\mu_1^2}} \quad\text{and}\quad V=\frac{p_0}{\sqrt{p_0^2+M^2}}, \]

we obtain:

\[ \vartheta_{\max}=\operatorname{arctg} \frac{\bar p_c M}{\sqrt{p_0^2\mu_1^2-\bar p_c^{\,2}M^2}}, \tag{30} \]

if

\[ p_0^2\mu_1^2>\bar p_c^{\,2}M^2, \]

and

\[ \vartheta_{\max}=\frac{\pi}{2}, \]

if

\[ p_0^2\mu_1^2<\bar p_c^{\,2}M^2 \]

(\(\bar p_c\) is found from (29)).

Fig. 5. Dependence \(\vartheta_{\max}(p_0)\) for the case of decay of a meson of mass \(1000m_e\) into two \(\pi\)-mesons \((m_\pi=280m_e)\).

Consequently, in the decay of a particle of mass \(M\) and momentum \(p_0\) into two secondary particles of mass \(\mu_1\), it cannot deviate by an angle greater than \(\vartheta_{\max}\), determined by (30). Figure 5 shows the dependence \(\vartheta_{\max}(p_0)\) for the case of decay of a neutral particle of mass \(1000m_e\) into two \(\pi\)-mesons of mass \(280m_e\).

The shaded region corresponds to the allowed angles. In this figure, as in all subsequent ones, the unit of momentum is taken to be the quantity equal to \(2000m_e\).

Another method of using the extremal values of the angles will be applied only to \(H\)-decay and consists in finding the functional dependence \(\psi_{\mathrm{extr}}(p_0)\) (see Fig. 4).

The point is that, for certain numerical values of the parameters, the angle between the directions of the secondary particles lies within definite intervals. Therefore, if in a large number of cases this angle lies within the calculated limits, this is an important additional proof that the ensemble under investigation is due to decay of a definite type.

In accordance with (2a) one may write:

\[ \psi=\vartheta_1+\vartheta_2= \]

\[ =\operatorname{arctg}\frac{1}{\gamma}\, \frac{\overline{p}_1\sin\overline{\vartheta}_1} {\overline{p}_1\cos\overline{\vartheta}_1+\sqrt{E_1}} + \operatorname{arctg}\frac{1}{\gamma}\, \frac{\overline{p}_2\sin\overline{\vartheta}_2} {\overline{p}_2\cos\overline{\vartheta}_2+\sqrt{E_2}}. \tag{31} \]

Bearing in mind that \(\overline{\vartheta}_2=\pi-\overline{\vartheta}_1\), and restricting ourselves, for simplicity, to analysis of the case when \(\mu_1=\mu_2=\mu\), one may rewrite (31) in the following form:

\[ \psi=\operatorname{arctg}\frac{1}{\gamma}\, \frac{\overline{p}_c\sin\overline{\vartheta}} {\overline{p}_c\cos\overline{\vartheta}+\sqrt{E_c}} + \operatorname{arctg}\frac{1}{\gamma}\, \frac{\overline{p}_c\sin\overline{\vartheta}} {\sqrt{E_c}-\overline{p}_c\cos\overline{\vartheta}}, \tag{32} \]

where, according to (29),

\[ \overline{p}_c=\sqrt{\frac{M^2-4\mu^2}{4}}. \]

The function \(\psi(\cos\overline{\vartheta})\) has extremal values at \(\cos\overline{\vartheta}=0\) (a maximum), and also at

\[ \cos^2\overline{\vartheta} = \frac{p_0^2M^2-8p_0^2\mu^2+M^4-4\mu^2M^2} {p_0^2(M^2-4\mu^2)}. \tag{33} \]

Depending on the sign of the expression \(M^2-8\mu^2\), equation (33), which determines the minimum value \(\psi_{\min}\), has solutions in the real domain \((0<\cos^2\overline{\vartheta}<1)\) in different intervals of variation of \(p_0\):

a) \(M^2-8\mu^2>0\).

Equation (33) has a solution for

\[ p_0^2>\frac{M^2(M^2-4\mu^2)}{4\mu^2}. \]

b) \(M^2-8\mu^2<0\).

The equation (33) has a solution if

\[ \frac{M^2(M^2-4\mu^2)}{4\mu^2}<p_0^2< \frac{M^2(M^2-4\mu^2)}{8\mu^2-M^2}. \]

Figure 6 presents the curves \(\varphi_{\max}(p_0)\) for the case of \(H\)-decay of particles with mass \(1000m_e\) into two charged \(\pi\)-mesons (\(\mu_\pi\) was taken equal to \(280m_e\)). The shaded region corresponds to

Fig. 6. Dependence of \(\varphi_{\max}(p_0)\) for the case of decay of a meson with mass \(1000m_e\) into two \(\pi\)-mesons.

allowed values of the angles \(\varphi\). Analogous curves were published in Butler’s work\(^{10}\), where he used them to analyze the decay scheme of \(V^0\)-particles.

In conclusion it should be noted that both methods of analysis (the methods of transverse momentum components and of extreme angles) are, in essence, applicable not only to decay processes, but also to other processes as a result of which two particles are present in the final state (for example, elastic scattering).

However, the probability of erroneously interpreting other processes as decays is small, since the masses of particles arising as a result of decay, as a rule, differ from the masses of the scattering particles.

It is worth dwelling on one more criterion (the coplanarity criterion), which, however, has the drawback that it is applicable not only to decay processes, but also to elastic scattering; moreover, since the masses do not enter into the consideration, it cannot be recommended for proving the decay character of processes. But if it has already been established that the given set of cases is due to decay, then the coplanarity method can serve as a criterion for determining the number of secondary particles, i.e., it makes it possible to decide how many particles are formed in the decay.

The coplanarity criterion is based on the simple relation:

\[ \mathbf{p}_0=\mathbf{p}_1+\mathbf{p}_2 . \tag{34} \]

Multiplying this vector equation scalarly by \(\mathbf{n}\) (\(\mathbf{n}\) is a vector normal to the plane formed by two of the three vectors, for example \(\mathbf{p}_0\) and \(\mathbf{p}_1\)), we obtain \((\mathbf{n}\mathbf{p}_2)=0\), i.e., the third vector lies in the same plane.

In conclusion of this section*) it is necessary to note that in some particular cases, for example, when the mass of one of the secondary particles is close to the mass of the primary particle, the methods indicated above may lead to an erroneous interpretation of the cases under investigation as a decay into two particles, whereas in reality a larger number of particles will participate in the processes. These special cases are discussed in more detail in Sec. 3.

b) Determination of the mass of the primary particle**)

From the conservation equations

\[ \begin{aligned} \mathbf{p}_1+\mathbf{p}_2&=\mathbf{p}_0, \tag{34}\\ E_1+E_2&=E_0, \tag{34a} \end{aligned} \]

if one uses the relations

\[ E_1=\sqrt{p_1^2+\mu_1^2},\qquad E_2=\sqrt{p_2^2+\mu_2^2}, \tag{35} \]

we obtain:

\[ M^2=\mu_1^2+\mu_2^2+2\left\{\left[(p_1^2+\mu_1^2)(p_2^2+\mu_2^2)\right]^{1/2}-p_1p_2\cos\psi\right\}. \tag{36} \]

*) Methods for proving the decay character of processes, based on determining mass values, are considered in the following two sections.

**) In this and the following sections we consider kinematic methods for determining masses. Various methods for determining the masses of individual particles, based on the characteristics of their tracks, have been investigated in a number of works (see, for example, 8).

If one considers the decay of a particle at rest

\[ (\bar p_1 = p_2 = \bar p_c \quad \text{and} \quad \psi = \pi), \]

then formula (36) is greatly simplified:

\[ M^2=\mu_1^2+\mu_2^2+2\left[(\bar p_c^2+\mu_1^2)(\bar p_c^2+\mu_2^2)\right]^{1/2}. \tag{36a} \]

Equation (36) is also simplified if \(p_1 \ll \mu_1\) or \(p_2 \ll \mu_2\); then the term \(p_1p_2\cos\psi \ll \left[(p_1^2+\mu_1^2)(p_2^2+\mu_2^2)\right]^{1/2}\), and it may be neglected.

c) Determination of the mass of the secondary particle (3-decay)

From equation (34) it follows that:

\[ M_2^2=M^2+\mu_1^2-2\left\{\left[(M^2+p_0^2)(\mu_1^2+p_1^2)\right]^{1/2}-p_0p_1\cos\vartheta\right\}. \tag{37} \]

As before, if \(p_0 \ll M\) or \(p_1 \ll \mu_1\), the term \(p_0p_1\cos\vartheta\) may be neglected. In particular, if the decaying particles are at rest, then

\[ \mu_2^2=M^2+\mu_1^2-2M\bar E_1. \tag{37a} \]

As an illustration, one can determine the mass of the neutral particle formed in \((\pi-\mu)\)-decay.

Using the mass values \(M=\mu_\pi=280m_e\); \(\mu_1=\mu_\mu=210m_e\) and \(\bar E_1=218.3m_e\) (\(E_{1\text{ kin}}=8.3m_e^{8}\)), we obtain that \(\mu_2\sim 15m_e\). Since in this case

\[ \frac{\delta\mu_2}{\delta M}\sim\frac{M-E_1}{\mu_2}\gg 1, \]

the value of \(\mu_2\) depends very strongly on the values of \(M\) and \(\mu_1\). Thus, if one sets \(m_\pi=278m_e\), then \(\mu_2\sim 3m_e\). Therefore it is natural to assume that \(\mu_2=0\). Such a value agrees entirely, within the limits of experimental error, with the experimental values of \(\mu_\pi\) and \(\mu_\mu\).

Sometimes the problem arises of determining the energy of the primary particle from known values of the energy of one of the secondary particles \(E_1\) and the angle of its emission \(\vartheta\). This problem and its solution were considered in detail by D. V. Skobeltsyn\(^7\).

For its solution one can make use of relation (25a). In accordance with (6), this equation may be written in the following form:

\[ M=\gamma\left\{(E_1-p_1V\cos\vartheta)+ \right. \]

\[ \left. +\sqrt{(E_1-p_1V\cos\vartheta)^2-(\mu_1^2-\mu_2^2)(1-V^2)}\right\}. \tag{38} \]

In particular, if \(\mu_2=0\), then

\[ M=\bar E_1+|\bar p_1|. \tag{39} \]

In this case, if \(\cos\vartheta=\pm 1\), then

\[ M=\gamma\{E_1 \mp p_1 V+|p_1 \mp E_1V|\}. \tag{39a} \]

If \(\cos\vartheta=1\), then, depending on the sign of the expression \(p_1+E_1V\), formula (39a) can be rewritten in the following form:

\[ M=\sqrt{\frac{1-V}{1+V}}\,[E_1+p_1], \tag{40} \]

\[ \frac{p_1}{E_1}>V, \]

\[ M=\sqrt{\frac{1+V}{1-V}}\,[E_1-p_1]. \tag{40a} \]

If \(\cos\vartheta=-1\), then always

\[ M=\sqrt{\frac{1+V}{1-V}}\,[E_1+p_1]. \tag{40б} \]

Formulas (40) were obtained earlier\(^7\).

It is necessary to note that (36), (37), and (39) can be used to check hypotheses concerning the presence of a decay with the emission of particles, if the masses \(\mu_1\) and \(\mu_2\) are known (equations (36) and (39)) or the masses \(M\) and \(\mu_1\) are known (equation (37)). Indeed, in that case the right-hand side of equations (36), (37), and (39) must be constant for all cases of the ensemble under study.

г) Angular and energy distribution of particles formed in decay\(^*\)

In decay into two particles the momentum \(\bar p_1=\bar p_2=\bar p_c\) is determined from equation (29). Therefore, using the formulas obtained in the preceding section, one can obtain general expressions for the angular and energy distributions.

Angular distribution. In the \(\mu\)-system the angular distribution of the secondary particles is isotropic; the particles have constant momentum. Therefore the probability of emission of a particle at an angle whose cosine lies between \(\cos\vartheta\) and \(\cos\vartheta+d\cos\vartheta\) is determined by equation (27). Then the angular distribution \(N(\vartheta)d\cos\vartheta\) is determined by the integral

\[ N(\vartheta)d\cos\vartheta = \frac{1}{2}\,d\cos\vartheta \int_{p_{\min}}^{p_{\max}} \delta[\bar p(p)-\bar p_c]\,J\,dp . \tag{41} \]

\(^*\) We shall consider the transformation of an isotropic distribution at constant momentum in the general form, since this simple case plays an important role in the analysis of collisions (see, for example, the work of Bradt et al.\(^5\)).

Having used the formula

\[ \delta[\bar p(p)-\bar p_c] = \sum_{\kappa} \frac{\delta(p-p_\kappa)} {\left|\dfrac{d\bar p}{dp}\right|_{p=p_\kappa}} \tag{42} \]

(\(p_\kappa\) are the roots of the equation \(\bar p(p)=\bar p_c\) in the interval over which the integration is carried out) and taking into account equations (6), (9), and (12), equation (41) can be written in the following form:

\[ N(\vartheta)\, d\cos\vartheta = \frac{p_\kappa^2\, d\cos\vartheta} {2\gamma p_c\,[p_\kappa-E_\kappa V\cos\vartheta]}, \tag{43a} \]

if \(\bar\beta>V\), and

\[ N(\vartheta)\, d\cos\vartheta = \]

\[ = \frac{d\cos\vartheta}{2\gamma p_c} \left[ \frac{p_{\kappa_1}^2}{(p_{\kappa_1}-E_{\kappa_1}V\cos\vartheta)} + \frac{p_{\kappa_2}^2}{(E_{\kappa_2}V\cos\vartheta-p_{\kappa_2})} \right] \quad (p_{\kappa_1}>p_{\kappa_2}), \tag{43b} \]

if \(\bar\beta<V\).

Using the right-hand side of relation (12) as the solution of the equation \(\bar p(p)=\bar p_c\), we obtain:

\[ N(\vartheta)\, d\cos\vartheta = \]

\[ = \frac{ \left[ E_c V\cos\vartheta + \sqrt{\mu^2\gamma^2 V^2\cos^2\vartheta+\bar E_c^2-\mu^2\gamma^2} \right]^2 d\cos\vartheta } { 2p_c\gamma^2(1-V^2\cos^2\vartheta)^2 \sqrt{\mu^2\gamma^2 V^2\cos^2\vartheta+\bar E_c^2-\mu^2\gamma^2} }, \tag{44a} \]

if \(\bar\beta>V\), \(\vartheta\) varies in the interval \(0-2\pi\).

If \(\bar\beta<V\), then

\[ N(\vartheta)\, d\cos\vartheta = \]

\[ = \frac{ \left[ \bar E_c^2 V^2\cos^2\vartheta + \mu^2\gamma^2 V^2\cos^2\vartheta + \bar E_c^2-\mu^2\gamma^2 \right] d\cos\vartheta } { p_c\gamma^2(1-V^2\cos^2\vartheta)^2 \sqrt{\mu^2\gamma^2 V^2\cos^2\vartheta+\bar E_c^2-\mu^2\gamma^2} }. \tag{44b} \]

In the latter case \(\vartheta\) varies in the interval \(0-\vartheta_{\max}\) (\(\vartheta_{\max}\) is determined by relation (14)). Formulas (44) are simplified if \(\bar E_c\gg \mu\gamma\); in this case[^11]

\[ N(\vartheta)\, d\cos\vartheta = \frac{d\cos\vartheta}{2\gamma^2(1-V\cos\vartheta)^2}. \tag{45} \]

The formulas presented in this section may be applied, in particular, to the analysis of decay into two particles. In this case, to determine the distribution parameter \(\bar p_c\), one should use the form-

mula (29). The angular distribution for a particle with mass \(\mu_1\) can be written in the form

\[ \begin{aligned} N(\vartheta)\,d\cos\vartheta &= -\frac{\left[(M^2+\mu_1^2-\mu_2^2)V\cos\vartheta+\right.} {2\gamma^2(1-V^2\cos^2\vartheta)^2 \sqrt{M^4+\mu_1^4+\mu_2^4-2(M^2\mu_1^2+M^2\mu_2^2+\mu_1^2\mu_2^2)}} \to \\[4pt] &\quad \left. \frac{\sqrt{4M^2\mu_1^2V^2\gamma^2\cos^2\vartheta+ (M^2+\mu_1^2-\mu_2^2)^2-4M^2\mu_1^2\gamma^2}} {2\gamma^2(1-V^2\cos^2\vartheta)^2 \sqrt{M^4+\mu_1^4+\mu_2^4-2(M^2\mu_1^2+M^2\mu_2^2+\mu_1^2\mu_2^2)}} \right] \\[4pt] &\quad \times \frac{d\cos\vartheta} {\sqrt{4M^2\mu_1^2V^2\gamma^2\cos^2\vartheta+ (M^2+\mu_1^2-\mu_2^2)^2-4M^2\mu_1^2\gamma^2}}, \end{aligned} \tag{46} \]

if \(\bar{\beta}>V\), and analogously for the case \(\bar{\beta}<V\). The formula assumes an especially simple form in the case of decay into two relativistic particles. For example, for the decay of a particle into two photons (such a case occurs in the decay of the neutral \(\pi\)-meson), formula (45) should be used.

Fig. 7. Dependences \(N(\vartheta)\). a) For the case of decay of a particle with mass \(1000\,m_e\) into two \(\pi\)-mesons; \(V=0.6\). b) For the case of decay \(V_1^0 \to p+\pi^{-}\). \(M_{V_1^0}=2230m_e\); \(\mu_p=1820m_e\); \(\mu_\pi=280m_e\); \(V=0.6\). c) For the case of decay \(\pi^0\to 2\gamma\); \(V=0.9\).

In Fig. 7 the angular distributions \(N(\vartheta)\) of secondary particles are presented for various cases of decay into two particles.

Energy distribution. The energy distribution \(N(p)\,dp\) is determined by the integral

\[ N(p)\,dp=\frac{dp}{2}\int_{0}^{\vartheta_{\max}} \delta\,[p(\vartheta)-p_c]\,J\sin\vartheta\,d\vartheta . \tag{47} \]

Using formulas (6), (9), and (42), it is easy to obtain:

\[ N(p)\,dp=\frac{p\,dp}{2\bar E\,p_c\gamma V}. \tag{48} \]

The interval of variation of \(p\) is determined by the bounds

\[ \gamma\left|\bar p_c-\bar E_c V\right|\leq p\leq \gamma\left(\bar p_c+\bar E_c V\right). \]

At relativistic velocities \((p\gg\mu)\)

\[ N(p)\,dp=\frac{dp}{2p_c\gamma V}. \tag{49} \]

In the case of decay into two particles, formulas (48) and (49) may be written in the form

\[ N(p)\,dp= \frac{pM\,dp}{ E\gamma V\sqrt{ M^4+\mu_1^4+\mu_2^4 -2\left(M^2\mu_1^2+M^2\mu_2^2+\mu_1^2\mu_2^2\right) }} \quad *) \tag{50} \]

and, in particular, if \(p\gg\mu_1\) and \(p\gg\mu_2\),

\[ N(p)\,dp=\frac{dp}{M\gamma V}. \tag{51} \]

An example of an energy distribution is given in Fig. 8. As before, the unit of momentum is taken to be \(2000m_e\).

Thus, in decay into two ultrarelativistic particles, these particles can have, with equal probability, any value of the energy in the interval from

\[ \frac{M}{2}\sqrt{\frac{1-V}{1+V}} \]

to

\[ \frac{M}{2}\sqrt{\frac{1+V}{1-V}}. \]

Fig. 8. Distribution over momenta of particles formed in the decay
\(V_1^0\to\rho+\pi^{-};\quad V=0.6.\)

The secondary particles cannot possess energy lying outside this interval. This feature of the energy distribution makes it possible to draw two conclusions about the character of the energy distribution

\[ \text{*) The energy distribution is identical for both particles.} \]

secondary particles arising in the decay of a nonmonoenergetic beam of mesons. It is essential that the conclusions drawn below practically do not depend on the form of the spectrum of the primary particles.

From the expression determining the limits of the interval it follows that, whatever energy the primary meson may have, the secondary particles can always have an energy equal to \(\dfrac{M}{2}\). If the spectrum of the primary particles extends from 0 to \(\infty\), then the value \(\dfrac{M}{2}\) is the only one possessing this property. For any definite energy of the primary particle, the secondary particles are distributed uniformly within the corresponding interval. Therefore, at the value \(\dfrac{M}{2}\), independently of the form of the spectrum of the primary particles, there will be a maximum of the energy spectrum of the secondary particles. Let us next consider two energy values \(E_1\) and \(E_2\) having the property that \(N(E_1)=N(E_2)\). Since the spectrum of the secondary particles has a maximum for any distribution of the primary particles, there is always an infinite number of pairs of energies (located on both sides of the maximum) for which this relation is satisfied.

It is clear that \(E_1\) and \(E_2\) must be limiting values of the interval corresponding to some value of the energy \(\dfrac{M}{\sqrt{1-V^2}}\) of the primary particle.

Therefore one may write:

\[ \frac{M}{2}\sqrt{\frac{1-V}{1+V}}=E_1, \]

\[ \frac{M}{2}\sqrt{\frac{1+V}{1-V}}=E_2, \]

and, consequently,

\[ M=2\sqrt{E_1E_2}. \tag{52} \]

The features of the energy spectrum of secondary particles considered above were first established by Carlson et al. \({}^{11}\).

3. DECAY INTO THREE PARTICLES

A characteristic feature of decay into three particles (as also of any transformation as a result of which three or more particles are formed), in comparison with decay into two particles, is the dependence of the energy and angular distributions of the secondary particles on the type of interaction responsible for the decay.

Indeed, in the \(u\)-system one may write the following equations:

\[ \bar E_1+\bar E_2+\bar E_3=M, \tag{53a} \]

\[ \mathbf p_1+\mathbf p_2+\mathbf p_3=0, \tag{53b} \]

and consequently, even for given values of the masses \(\mu_1,\ \mu_2,\ \mu_3\), a continuous spectrum of momenta of each of the particles is possible, from \(0\) to some \(p_{\max}\).

Therefore, since in the present work we disregard the investigation of concrete types of interaction, in describing decay into three particles only the solutions of certain special problems will be given.

a) Investigation of the extremal values of momenta and angles

First of all let us determine the maximum value, attainable in decay into three particles, of the momentum \(\bar p_1\) of the particle with mass \(\mu_1\).

It is clear that the momentum \(\bar p_1\) will have its maximum value if the momenta of all three particles are collinear, and in such a way that the directions of the momenta of the other two particles \(\bar{\mathbf p}_2\) and \(\bar{\mathbf p}_3\) are antiparallel to the direction of the momentum \(\bar{\mathbf p}_1\):

\[ -\bar{\mathbf p}_1=\bar{\mathbf p}_2+\bar{\mathbf p}_3. \]

Substituting in (53a), we obtain:

\[ \sqrt{\bar p_1^{\,2}+\mu_1^2} +\sqrt{\bar p_2^{\,2}+\mu_2^2} +\sqrt{(\bar{\mathbf p}_1-\bar{\mathbf p}_2)^2+\mu_3^2} =M. \tag{53c} \]

Considering in (53c) the quantity \(\bar p_1\) as a function of \(\bar p_2\), one can find the maximum value of the function \(\bar p_1(\bar p_2)\), which is determined by the expression

\[ \sqrt{\bar p_{1\max}^{\,2}+\mu_1^2} = \frac{M^2+\mu_1^2-(\mu_2+\mu_3)^2}{2M}. \tag{54} \]

This relation was obtained by Michel \(^{12}\).

The maximum total energy of two particles is easily determined from (53a):

\[ (\bar E_1+\bar E_2)_{\max}=M-\mu_3. \tag{55} \]

Let us proceed to the determination of the extremal values of the angles. From formula (14) it follows that the maximum angle through which a particle with mass \(\mu_1\) is deflected increases with its momentum. Therefore the maximum admissible angle \(\vartheta_{\max}\) is determined by formula (30), where instead of \(\bar p_c\) one must substitute the quantity \(\bar p_{\max}\) according to expression (54).

If the mass of one of the particles is negligibly small in comparison with the masses of the other secondary particles, then \(\overline{p}_{\max}\) becomes \(\overline{p}_c\). Therefore the described method of analysis does not make it possible to draw an unambiguous conclusion as to whether the aggregate under investigation is due to decay into two particles or to decay into three particles, of which at least one has a mass negligibly small compared with the masses of the other particles.

The known caution must also be exercised in analyzing the admissible angles \(\psi\) (see § 2).

Let us consider special cases:

1) the mass of one of the particles \(\mu_1 \sim M\)*);

2) \(\mu_1+\mu_2 \sim M\), under the additional conditions \(\mu_1 \gg \overline{p}_{1\max}\) and \(\mu_2 \gg \overline{p}_{2\max}\).

In these special cases one may write:

\[ \psi=\operatorname{arc\,tg}\frac{1}{\gamma}\frac{\overline{p}_{\max}}{\mu_1 V} +\operatorname{arc\,tg}\frac{1}{\gamma}\frac{\overline{p}_{\max}}{\mu_2 V}, \tag{56} \]

where \(\overline{p}_{\max}\) is determined from (54).

Since in these cases the mass of the third particle \(\mu_3\) is negligibly small in comparison with the mass of at least one of the particles, \(\overline{p}_{\max}\sim \overline{p}_c\), and consequently it is again impossible to draw an unambiguous conclusion about the number of secondary particles. It is precisely for this reason that Butler’s\(^{10}\) analysis of the admissible angles of separation of secondary charged particles in the decay of a heavy \(V_1^0\)-particle cannot be considered convincing. Although the experimentally observed angles \(\psi\) agree with the calculated ones, this by no means excludes the possibility that a third light neutral particle appears in the decay.

A similar ambiguity may also arise in analyzing the coplanarity of the tracks of the primary and two charged secondary particles (see § 3).

Indeed, one can prove the following proposition: if the mass of one of the two secondary charged particles (for example, \(\mu_2\)) is close to the mass of the primary particle \(M\), and the velocity of the latter is close to 1, then the paths of both charged particles and of the primary particle are almost coplanar.

In the \(\Lambda\)-system the relation holds

\[ \mathbf{p}_0=\mathbf{p}_1+\mathbf{p}_2+\mathbf{p}_3. \tag{57} \]

Multiplying this equality scalarly by the unit vector \(\mathbf{n}\), nor-

*) The determination of the angle \(\psi\) for the general case of decay into three particles reduces to the solution of a system of five transcendental equations and, apparently, cannot be carried out in general form.

mal to the vectors \(\mathbf p_0\) and \(\mathbf p_1\), we obtain:

\[ (\mathbf n\mathbf p_2)=-(\mathbf n\mathbf p_3). \tag{58} \]

Since in the present case \(p_2\sim \sqrt{E_2}\gamma \gg (p_3+\sqrt{E_3})\gamma \sim p_3\), it follows that

\[ |\cos\theta_2|=\frac{p_3}{p_2}|\cos\theta_3|\ll 1 \tag{59} \]

(\(\theta_2\) and \(\theta_3\) are, respectively, the angles between the vectors \(\mathbf p_2,\mathbf n\) and \(\mathbf p_3,\mathbf n\)); consequently, the vector \(\mathbf n\) is close to the normal to the momentum \(\mathbf p_2\). Thus, if the primary particle decays into two charged particles (with \(\mu_2\sim M\)) and one neutral particle, then the vectors \(\mathbf p_0,\mathbf p_1,\mathbf p_2\) are almost coplanar.

b) Energy spectrum of secondary particles

As was already mentioned above, for a rigorous calculation of the energy spectrum of secondary particles produced in the decay of mesons into three particles, it is necessary to know the form of the interaction of the particles with the field. However, up to now it has not been possible in any case to establish unambiguously the character of such an interaction. Therefore, in obtaining estimates of the energy spectrum it is advisable, for the sake of greater generality, to sacrifice the rigor of the calculation. The simplest general method for calculating the characteristics of decay is based on the assumption that they are determined only by the statistical weight of the final states and do not depend on the form of the interactions. One may expect that such an approach is justified in the case of decay with not very strong interactions between the secondary particles. Thus, the forms of the energy spectra of electrons produced in the decay of \(\mu\)-mesons and in \(\beta\)-decay, calculated under this assumption, do not contradict the experimental data \(^{13}\). An analogous method has been successfully applied to the study of multiple processes at high energies of interacting particles \(^{14,15}\). In this case, however, the application of such a method is obviously justified by the fact that at considerable energies a large number of particles participate in the processes. Fermi \(^{13,14}\) also tried to extend this approach to the case of the interaction of nucleons with energy close to the threshold for \(\pi\)-meson production. It turned out that, whereas the magnitude of the total cross section agrees satisfactorily with the experimental data, in order to obtain the correct form of the energy spectrum of \(\pi\)-mesons it is necessary to take into account the specific character of the nucleon interaction, i.e. to allow for the dependence of the matrix element on the energy.

Calculations of the energy spectra under the assumption of the decisive influence of statistical factors were carried out in works \(^{12,16,17,18}\). It turned out that in the \(\mu\)-system the probability \(d\omega\) that a particle with mass \(\mu_1\) will have a momentum lying be-

between \(\bar p_1\), \(\bar p_1+d\bar p_1\), is determined by the expression

\[ d\omega \sim \frac{B^{1/2}\bar p_1^2}{(A^2-\bar p_1^2)^2} \left[\left(1-\frac{4A^2}{A^2-\bar p_1^2}\right)B+AC\right]d\bar p_1, \tag{60} \]

where

\[ \begin{aligned} A&=M+\sqrt{\bar p_1^2+\mu_1^2},\\ B&=(\bar p_1^2+\mu_2^2+\mu_3^2-A^2)^2-4\mu_2^2\mu_3^2,\\ C&=6A\,[A^2-(\bar p_1^2+\mu_2^2+\mu_3^2)]. \end{aligned} \]

\(\bar p_1\) varies within the limits \(0,\ \bar p_{1\max}\); \(p_{1\max}\) is determined by (54). Naturally, the distribution (60) is determined only by the values of the masses.

In order to pass to the energy representation, in (60) one must replace \(\bar p_1\) by \(\sqrt{\bar E_1^2-\mu_1^2}\) and \(d\bar p\) by \(\dfrac{\bar E}{\bar p}\,d\bar E\).

Let us consider separate special cases of relation (60):

1) \(\mu_i \ll \bar p_i,\quad i=1,2,3,\)

\[ d\omega \sim (3M^2-6M\bar p_1+2\bar p_1^2)\bar p_1^2\,d\bar p_1; \tag{61} \]

2) \(\mu_i \gg p_i,\)

\[ d\omega \sim \left[2(\mu_2+\mu_3)\bar T-\frac{\bar p_2^2M}{\mu_1}\right]^{1/2} \bar p_1^2\,d\bar p; \tag{62} \]

\(\bar T=M-(\mu_1+\mu_2+\mu_3)\) is the total kinetic energy of the secondary particles;

3) \(\mu_1 \ll \bar p_1,\ \mu_2 \gg \bar p_2,\ \mu_3 \gg \bar p_3,\)

\[ d\omega \sim (\bar T-\bar p_1)^{1/2}\bar p_1^2\,d\bar p_1; \tag{63} \]

4) \(\mu_1 \gg \bar p_1,\ \mu_2 \ll \bar p_2,\ \mu_3 \gg \bar p_3,\)

\[ d\omega \sim (\bar T^2+\bar p_1^2)\,p_1^2\,d\bar p_1; \tag{64} \]

5) \(\mu_1 \ll \bar p_1,\ \mu_2 \ll \bar p_2,\ \mu_3 \gg \bar p_3,\)

\[ d\omega \sim (\bar T^2-\bar p_1^2)\bar p_1^2\,d\bar p_1. \tag{65} \]

c) Energy and angular distributions of secondary particles in decay in flight

Let us calculate the energy and angular distributions in the \(l\)-system, if the distribution in the \(q\)-system corresponds to the distribution function (61). In this case, in the \(q\)-system

\[ \bar N(\bar p_1)\,d\bar p_1\,d\cos\vartheta \sim (3M^2-6M\bar p_1+2\bar p_1^2)\bar p_1^2\,d\bar p_1\,d\cos\vartheta. \tag{61a} \]

For the functions \(J(\vartheta)\) and \(\bar p(p)\) we shall use the approximate expressions (96) and (66). Then the momentum distribution is

\[ \begin{aligned} N(p_1)\,dp_1 \sim dp_1 \int_{\cos\vartheta_{\max}(p)}^{1} \frac{[3M^2-6M\bar p_1+2\bar p_1^{\,2}]\,\bar p_1^{\,2}} {1-V\cos\vartheta}\,\sin\vartheta\,d\vartheta &= \\ = dp_1 \int_{\cos\vartheta_{\max}(p)}^{1} [3M^2-6Mp_1(1-V\cos\vartheta) &+ \\ +2p_1^2(1-V\cos\vartheta)^2]\,p_1^2(1-V\cos\vartheta)\sin\vartheta\,d\vartheta. \end{aligned} \tag{66} \]

From (66) it follows that

\[ \cos\vartheta_{\max}= \begin{cases} \dfrac{1}{V}\left(1-\dfrac{\bar p_{1\max}}{\gamma p_1}\right) & \text{if } \dfrac{1}{V}\left(1-\dfrac{\bar p_{1\max}}{\gamma p_1}\right)>-1,\\[1.2em] -1 & \text{if } \dfrac{1}{V}\left(1-\dfrac{\bar p_{1\max}}{\gamma p}\right)<-1. \end{cases} \]

After integration we obtain:

\[ N(p_1)\,dp_1 \sim p_1^2\{3M^2[H_1^2-(1-V)^2]-4M\gamma p_1[H^3-(1-V)^3]+ \]

\[ +\gamma^2p_1^2[H^4-(1-V)^4]\}. \tag{67} \]

\[ H= \begin{cases} \dfrac{M}{2p_1\gamma} & \text{if } p_1>\dfrac{M}{2(1+V)\gamma},\\[1.2em] 1+V & \text{if } p_2<\dfrac{M}{2(1+V)\gamma}. \end{cases} \]

The values of the momentum \(p\) lie in the interval

\[ 0;\quad \frac{M}{2}(1+V)\gamma. \]

In the case of interest to us (ultrarelativistic particles), one can establish general relations for the mean momenta. In accordance with (51), the mean momentum \(p_1\) in the \(\lambda\)-system corresponding to the fixed momentum \(\bar p_1\) is equal to \(\gamma\bar p_1\); therefore, in the case of an arbitrary distribution in the \(u\)-system,

\[ p_\lambda = \gamma\int_{0}^{M/2} \bar N(\bar p_1)\,\bar p_1\,d\bar p_1 = \gamma\bar p_1. \]

Let us proceed to the calculation of the angular distribution

\[ N(\vartheta)\,d\cos\vartheta \sim d\cos\vartheta \int_{0}^{p_{\max}(\vartheta)} \left[3M^2 - 6Mp_1(1 - V\cos\vartheta) + 2p_1^2(1 - V\cos\vartheta)^2\right] p_1^2(1 - V\cos\vartheta)\,dp_1, \tag{68} \]

\[ p_{\max} = \frac{M}{2\gamma(1 - V\cos\vartheta)}. \]

After integration we obtain:

\[ N(\vartheta)\,d\cos\vartheta \sim \frac{d\cos\vartheta}{(1 - V\cos\vartheta)^2}. \tag{69} \]

This expression, if normalized, coincides with the angular distribution of ultrarelativistic particles produced in decay into two particles (see (45)). Such a coincidence is by no means accidental. A characteristic feature of expressions (45) and (69) is their independence of the energy of the secondary particles (provided only that they have sufficiently high velocities). Therefore, for any momentum distribution of ultrarelativistic particles in the \(ц\)-system, their angular distribution in the \(л\)-system will be determined by relation (45).

REFERENCES

1. C. D. Rochester, C. C. Butler, Rep. Prog. Phys. 41, 364 (1953).
2. A. I. Alikhanov, UFN 50, 481 (1953).
3. L. Michel, Progress in Cosmic Ray Physics, Amsterdam, 1952.
4. L. Landau and E. Lifshitz, Field Theory, Gostekhizdat, 1941.
5. H. Bradt, M. Kaplon, B. Peters, Helv. Phys. Acta 23, 24 (1950).
6. J. Blaton, Det Kgl. Vidensk. Selskab. Mat.-fys. Medd 24, No. 20 (1950).
7. D. V. Skobeltsyn, in In Memory of S. I. Vavilov, p. 292, 1952.
8. C. Powell, UFN 45, 15 (1951).
9. R. Armenteros, K. H. Barker, C. C. Butler, A. Cachon, Phil. Mag. 42, 1113 (1951).
10. C. C. Butler, UFN 48, 389 (1952).
11. A. G. Carlson, J. E. Hooper, D. T. King, Phil. Mag. 41, 701 (1950).
12. L. Michel, Proc. Phys. Soc. 63A, 514 (1950).
13. E. Fermi, Elementary Particles, IL, 1952.
14. E. Fermi, Prog. Theor. Phys. 5, 570 (1950).
15. L. D. Landau, Izv. AN SSSR, ser. fiz., 17, 51 (1953).
16. O. Kofoed-Hansen, Phil. Mag. 42, 1411 (1951).
17. K. A. Brueckner, R. W. Watson, Phys. Rev. 87, 390 (1952).
18. M. I. Podgoretskii, I. L. Rozental, ZhETF 26, No. 8 (1954).
19. I. S. Shapiro, UFN 53, 7 (1954).