\documentclass[11pt]{article}
%\documentstyle[11pt,twoside,ihepconf]{article}
%\usepackage{sw20lart}
\usepackage{amsmath}
\usepackage{graphicx}
%\usepackage{geometry}
\usepackage[english]{babel}
\RequirePackage {hyperref}
\usepackage{amsfonts}
\usepackage{amssymb}
%\usepackage{fancyheadings}
%\pagestyle{fancy}
\usepackage{graphicx}
\usepackage[dvips]{epsfig}
\usepackage{wrapfig}
\tolerance=7000
\begin{document}
W.~ Heisenberg, Zeit. Phys., {\bf Bd. 133, 3} 65 \hfill {\large \bf 1952}\\
\vspace{2cm}
\begin{center}
{\Large \bf Production of Mesons as a Shock Wave Problem}\\
\end{center}
\vspace{0.5cm}
\begin{center}
W. Heisenberg\\
With 6 figures in text\\
Received on 5 May 1952\\
\end{center}
\vspace{0.5cm}
\centerline{--- ---~~~$\diamond~\diamondsuit~\diamond$~~~--- ---}
\noindent
Translated by Herman Boos \\
\centerline{--- ---~~~$\diamond~\diamondsuit~\diamond$~~~--- ---}
\vspace{0.5cm}
\begin{abstract}
Multi-meson production in two-nucleon collision is described as a shock wave
process which is governed by a non-linear wave equation. Since one deals with
big quantum numbers, these quantum processes may be approximately described
by means of the correspondece principle. Analysing solutions to the
non-linear wave equation one can get the energy and angular distribution for
different meson sorts.
\end{abstract}
An analysis of experimental data on the $\pi$-meson production obtained in
last years tells us that a collision of two nucleons of high energy produces
a big number of mesons. The fact that the strong interaction of nucleons
with mesons and, in particular, of mesons with themselves results in such a
multiple production was established long time ago \cite{Heisenberg 1939}. In oder to
give a quantitative estimation of the energy dissipation
in meson field one can compare this process with
a turbulence flow \cite{Heisenberg 1949}. Some other way as Fermi
did \cite{Fermi 1950}
is to consider the temperature balance in a moment of collision. It allows to
calculate the energy distribution of mesons.
The present consideration of the problem is
based on the point of view that was suggested by author in
1939 in connection with the Yukawa theory \cite{Heisenberg 1939}. The meson production is
considered as a shock wave process which is described by a non-linear wave
equation. It will be shown that in this way it is possible to get
quantitative results for the spectral and spatial distributions of
different meson sorts.
\section{\it Visual description of a shock wave}
~~~~Below the meson production will be considered in the center of mass frame.
A transition to the laboratory frame can be easily
done \cite{Heisenberg 1949}. This question will not be discussed here.
a) In the center of mass frame two nucleons move to meet each other
in opposite directions (Fig.1) until they cross in some region (shaded area
in Fig.1). In this region they strongly interact. The nucleons can be
imagined as discs with a thickness which is less than their cross section
because of the Lorentz factor $\sqrt{1-\beta^2}$ ($\beta$ is a speed of
mass center). One can take this cross section to be of order
$1.4\cdot 10^{-13}cm$. In a moment of collision the speed of nucleons
change in such a way that in their intersection area the energy is transfered
to the meson field. So, in the very beginning
when a shock wave just appears the whole energy
of the meson field is concentrated in a thin flat layer which was filled by
both nucleons in the moment of collision.
\begin{figure}[h]
\centerline{\resizebox{12cm}{!}{\includegraphics{fig1.gif}}}
\caption{}
\end{figure}
b) If one could neglect the interaction between the mesons then they
would propagate in accordance with a wave equation
\begin{equation}
\square \varphi - \kappa^2 \varphi = 0
\end{equation}
(or more complicated linear wave equation that contains different
meson sorts). Then also the angular distribution of this wave would not
change during it's propagation. These distributions could be determined
already in the first moment by means of an expansion of the wave into the
Fourier series. Thus we can conclude that in the frequency region from
$k_0$ to $k_0+dk_0$ ($k_0$ is energy of a single meson) the energy of
the meson wave does not depend on $k_0$ up to the frequencies corresponding
to such wave lengthes which are comparable with the thickness of the layer
where the collision takes place. This thickness is of the order
${\sqrt{1-\beta^2}\over \kappa}$ ($\kappa$ is the meson mass). When
$k_0>k_{0 m} = {\kappa\over \sqrt{1-\beta^2}}$ the intensity abruptly
decreases as a function of $k_0$.
\begin{equation}
d \varepsilon = {\rm constant} \cdot d k_0 \quad {\rm for} \quad
k_0 \le k_{0m}.
\end{equation}
For a number of mesons with frequences lying in the region $dk_0$ one gets
\begin{equation}
d n = {\rm constant} ~ \frac{dk_0}{k_0} \quad {\rm for} \quad
k_0 \le k_{0m}.
\end{equation}
In Fig.2 is shown the behaviour of the function $\varphi$ on the
axis which is normal to the radiation plane (right after the act
of emission). Also shown are functions ${d\epsilon\over dk_0}$ and
$dn\over dk_0$ under condition that the equation (1) is fulfilled. The
spectrum (3) corresponds to the known spectrum of the R{\"o}ntgen
Bremsstrahlung of the electron. It is also valid in case when the most
part of energy of mesons is transfered to the meson field in such a way
that the number of outgoing mesons is not too big. It means that the
energy of a single meson is approximately $\ge{1\over 2}k_{0m}$.
\begin{figure}[h]
\centerline{\resizebox{13cm}{!}{\includegraphics{fig2.gif}}}
\caption{a - c}
\end{figure}
c) But in reality the interaction of mesons can not be neglected i.e. the
wave propagation is going on according to the same non-linear wave equation
which only approximately becomes linear in case of a small intensity.
As will be shown below the non-linearity results in a slight smoothing
the singularity
of a wave crest. Due to this, during the wave propagation the energy
is transfered from shorter to longer waves. Therefore in the
end of this propagation the spectral distribution goes down steeper in
comparison with the previous case when the equation (1) is valid. Thus
we come to the behaviour shown in Fig.3. The spatial distribution is
shown in Fig.4(a-d).
\begin{figure}[h]
\centerline{\resizebox{13cm}{!}{\includegraphics{fig3.gif}}}
\caption{a - c.}
\end{figure}
In the moment of collision the whole energy is concentrated in the
intersection area of two nucleons (a). After this two shock wave fronts
start to propagate to the right and to the left respectively.
The most part of the
energy is still concentrated in these two fronts. But a wave perturbation
also appears in a region between them where the rest of the energy is
concentrated (b). When the propagation of the shock wave fronts goes
further the perturbation on their wakes captures more and more space which
in it's turn becomes a source of a new wave propagation.
Then the front energy becomes
lower. It is transfered to other wave lengthes, in particular, to longer
waves (c). During the further propagation the perturbation in a central
area decreases. Thus actually the produced wave moves faster in a
direction of the shock wave fronts than in transverse direction.
Therefore short waves have a higher group speed. Now perturbation of a very
low intensity spreads out into all directions also with the speed of light.
The energy of shock wave fronts becomes so low that the non-linearity does
not play a big role any more here also. So the further propagation
continues according to the usual linear wave equation (d).
\begin{figure}[h]
\centerline{\resizebox{10cm}{!}{\includegraphics{fig4.gif}}}
\caption{a - d.}
\end{figure}
Up to the moment we ignored the quantum effects. This approximation is
acceptable because we deal with production of a big number of mesons i.e.
with a process with big quantum numbers. How the correspondence principle
is applied in quantum theory was described in detail in above mentioned
paper \cite{Heisenberg 1939}. Here it is enough for us to accept some qualitative
consideration based on Fig.4d. Namely, the most part of energy is emitted
in all directions in form of mesons whose wave lengthes are comparable
with the disc cross section i.e. ${1/ \kappa}$.
The transverse momentum can be greater
than $\kappa$ only in rare cases so that the
Fourier coefficients of that waves are very small. On the other hand,
the longitudinal momentum can be greater so that one can face
the shorter waves in the shock wave front. The mesons with the
energy $k_0$ will be emitted mostly only into the angle interval of order
$\kappa /{k_0}$ with regard to both dominating directions. Also heavier mesons
are mostly emitted only into the front of the shock wave.
\section{\it Solution of the shock wave equation}
~~~~The propagation of the shock wave depends on the form of the non-linear
wave equation on which the description of mesons is based. But it is
possible to show that there is a special case of ``strong'' interaction
for which one can accept that the spectral distribution of mesons does not
depend on the concrete form of the wave equation. The solution of the
non-linear equation can be performed by means of a simplification if to be
interested only in the spectral and not in the angular distribution.
Namely, if the plane where the emission happens tends to be infinite
then the layer tends to be infinitely thin. Then because of the Lorentz
invariance the function $\varphi$ can depend only on $s=t^2-x^2$. Thus
the partial differential equation becomes an ordinary differential
equation which is simpler to analyse.
Let us discuss two examples of non-linear wave theories:
1. The equation suggested by Shiff \cite{Schiff 1951} and
Thirring \cite{Thirring 1952} in connection
with description of nuclear forces:
\begin{equation}
\square \varphi - \kappa^2 \varphi - \eta \varphi^3 = 0.
\end{equation}
2. The wave equation which analogously to earlier works by
Born \cite{Born 1933} follows from the Lagrange function
\begin{equation}
L = l^{-4}~ \sqrt{1 + l^4 \left[ \sum~ \left( \frac{\partial
\varphi}{\partial \kappa_{\nu}} \right)^2 + \kappa^2
\varphi^2 \right]}
\end{equation}
Long time ago Born noted that the non-linear theories of that type have
solutions that are less singular in comparison with the solutions to
linear equations. That time this fact was used for the description of
the electron self-energy. But the same consideration is also acceptable
for the multiple production of mesons.
In the earlier works \cite{Heisenberg 1939} the investigation of the meson production
was already based on the Lagrange function \cite{Heisenberg 1949}.
To the item 1. The first of two these equations gives for $\varphi=\varphi(s)$:
$$
4~ \frac{d}{ds}~ \left( s~ \frac{d \varphi}{ds} \right) + \kappa^2
\varphi + \eta \varphi^3 = 0. \eqno(4a)
$$
When $\eta=0$ one comes again to the linear equation (1) and then
it's solution looks as follows:
$$
\left.
\begin{array}{lll}
\varphi = a J_0 \left( \kappa \sqrt{s} \right)&~~~{\rm for}~~~&
s > 0\\
\varphi = 0& ~~~{\rm for}~~~&s < 0,
\end{array}
\right\} \eqno(6a)
$$
where $a$ is an integration constant; also compare it with Fig.2. For
$\eta\ne 0$ one can write down a power series at the point $s=0$.
$$
\left. \begin{array}{ll}
q = a [1 - (\kappa^2 + \eta a^2) s + \frac{1}{4} (\kappa^2 + 3 \eta
a^2)( \kappa^2 + \eta a^2)& s^2 - + \ldots ]=0\\
& {\rm for}~~~ s > 0\\
& {\rm for}~~~ s < 0.
\end{array} \right\} \eqno(6b)
$$
Let us note that the equation (4) corresponds to some ``weak''
interaction which does not change the shock wave front
if the wave function is continuous. It is connected with the fact
that the theory based on eq. (4) belongs to the group of
renormalizable theories because the coupling constant $\eta$ is dimensionless.
It was established in different ways that
renormalizable theories describe only ``weak'' interactions which
in general do not result in the multiple production of mesons.
To the item 2.
However the situation with the wave equation that follows from the formula (5)
is different. For $\varphi=\varphi(s)$ this wave equation looks as follows:
$$
4~ \frac{d}{ds}~ (s \varphi') + \kappa^2 \varphi = 8 l^4 s
\varphi'^2 ~ \frac{\varphi' + \kappa^2 \varphi}{1 + l^4 \kappa^2
\varphi^2}. \eqno(7)
$$
If $\kappa=0$ (zero meson mass) then the solution may be readily found:
$$
\left.
\begin{array}{llll}
\varphi&=\frac{\displaystyle 1}{\displaystyle a}~ {\rm lg} \left( 1
+ \frac{\displaystyle a^2}{\displaystyle 2 l^4} ~ s +
\frac{\displaystyle a}{\displaystyle 2l^4}~ \sqrt{4l^4s + a^2 s^2}
\right)&~~{\rm for}~~& s \ge 0\\
&= 0&~~{\rm for}~~&s \le 0.
\end{array}
\right\} \eqno(8)
$$
In general case ($\kappa\ne 0$) one can apply again the power series expansion.
Let us take
$$
\varphi = \frac{1}{\kappa l^2} ~ f(\zeta); \qquad \zeta = s \kappa^2
\eqno(9)
%\end{equation}
$$
and we write
$$
\left.
\begin{array}{llll}
f(\zeta) &= \sqrt{\zeta} \left( 1 + a \zeta + \frac{\displaystyle
27 a^2 + 2a - 1}{\displaystyle 10}~ \zeta^2 + - \ldots \right)&
~~{\rm for}~~& \zeta \ll 1\\
& \approx \gamma \zeta^{-1/4} \cos (\sqrt{\zeta} + \delta
)& ~~{\rm for}~~& \zeta \gg 1\\
& = 0&~~{\rm for}~~& \zeta \le 0.
\end{array}
\right\} \eqno(10)
$$
The constants $\gamma$ and $\delta$ are completely fixed via the
integration constant $a$. However it's value was not evaluated.
One can note that the non-linearity essentialy changes the character of the
solution. Namely, at $s=0$ the discontinuity of $\varphi$ disappears.
More exactly, $\varphi '$ has a discontinuity while
$\varphi$ behaves as $\sqrt{s}$ in a small vicinity of $s=0$.
Let us assume that to a given moment of evolution the function
\mbox{$\varphi(s)=\varphi(x,t)$}
is defined by the Fourier integral over the wave number $k$.
Then for the Fourier coefficient $\varphi(k,t)$ with $k\sim k_0 \gg\kappa$
one obtains up to some constant
$$
q(k,t) \sim k^{-3/2} t^{1/2} e^{\pm ik_0t}.
\eqno(11)
%\end{equation}
$$
It is easy to understand that the multiplier $t^{1/2}$ appears due to
the fact that during the propagation process the energy is transfered
from the head
part of the shock wave to other parts of the wave and so to lower
frequences also. Actually the energy potential in the head part of the
shock wave is infinite. This is just a consequence of our assumption
that the shock wave appears inside an infinitely thin layer. By means of this
assumption we have achieved that the solution $\varphi(x,t)$ depends only on
combination $t^2-x^2$ which is invariant under the Lorentz transformation
in the $x,t$-space. Some finite energy-momentum would define some
direction in this space. Therefore in this case it could not lead to
some invariant solution.
However in reality the shock wave appears in a layer with a
finite thickness
$\sim{\sqrt{1-\beta^2}\over k}$. Therefore the energy-momentum vector is
finite. The increasing of the Fourier amplitudes in (11) starts with some
delay. When the energy potential concentrated in the head part of
the shock wave is exhausted it comes to the state of rest.
Then for a big $t$ the Fourier coefficients as a function of $k$
for $k>k_{0m}={\kappa\over\sqrt{1-\beta^2}}$ fall down faster than $k^{-3/2}$.
Thus for the intensity distribution one gets
$$
\frac{d \varepsilon}{d k_0} = {\rm const}~ \frac{dk_0}{k_0} \quad
{\rm for} \quad \kappa \le k_0 \le k_{0m} = \frac{\kappa}{\sqrt{1 -
\beta^2}} \eqno(12)
$$
and
$$
%\begin{equation}
\frac{dn}{dk_0} = {\rm const}~ \frac{dk_0}{k^2_0}. \eqno(13)
$$
%\end{equation}
for the same region.
This is the form of the spectrum that was earlier suggested
in connection with the multiple
production of mesons \cite{Heisenberg 1939}, \cite{Heisenberg 1949} and it was
also shown in Fig.3.
The wave equation (7) borrowed from the Born theory \cite{Born 1933} is the typical
case of ``strong'' interaction and describes the multiple production
of mesons. Here the coupling constant has a dimention of length in the fourth power.
b) Now it is necessary to show that the spectrum (12) and (13) in general
corresponds to the strong interaction independent of a concrete form of
the Lagrange function and of the properties of participating particles.
We start with an arbitrary Lagrange function which depends on some scalar wave
function $\varphi$ and it's first derivatives ${\partial\over \partial x_{\nu}}$.
Because of the Lorentz
invariance $L$ can depend only on $\varphi$ and on
$\sum_{\nu}({\partial\over \partial x_{\nu}})^2$.
For very small
values of $\varphi$ and ${\partial\over \partial x_{\nu}}$
$L$ should reduce to the Lagrange function of
the usual wave equation (1). Now we would like to know the value of
${\partial\over \partial x_{\nu}}$ in a small vicinity of $s=0 (s>0)$.
For $s \rightarrow 0$ $\sum_{\nu}({\partial\over \partial x_{\nu}})^2$
can be either infinite or finite or even
tend to zero.
First one can exclude the last of these three possibilities because
in this case right
at the critical point $s=0$ the non-linearity would not play any role.
But this is impossible because for the usual wave equation (1)
$\sum_{\nu}({\partial\over \partial x_{\nu}})^2$ can never be equal zero
in the critical point but is infinite.
Among the both residual possibilities actually only the second one
gives a smooth behaviour of $\varphi$ at the singular point.
Therefore it may correspond to the strong interaction.
In the vicinity of $s=0$ it becomes
$$
\sum \left( \frac{\partial \varphi}{\partial x_{\nu}}\right)^2 = -
4s \left( \frac{\partial \varphi}{ds} \right)^2 = {\rm const} ~(\ne
0 \quad {\rm and} ~ \ne \infty). \eqno(14)
$$
from which one gets
$$
%\begin{equation}
\varphi(s) \sim {\rm const} ~\sqrt s, \eqno(15)
$$
and also expressions like (7) and (10)-(14).
c) But for (12) and (13) one can give more general reasons that are acceptable
also for arbitrary particles with higher spin. Already in item 11a
it was mentioned that in that special case
where the shock wave appears within an infinitely
thin layer it's total energy should be infinite. In this case the wave
function is invariant with regard to the rotations in the \mbox{$x,t$-space.}
The bigger is the energy dissipation caused by the interaction
the steeper falls down the energy spectrum of mesons.
Since the spectrum has a form that exactly corresponds to the
potential law (and it should be true for a majority of simple wave equations)
it can not fall down faster than in (12) and (13).
The reason for this is that the total energy
is divergent for (namely logarithmically)
for $k_{0m}\rightarrow\infty$.
Thus the spectrum (12) and (13) just corresponds
to the special case of the strong interaction. As was mentioned above
the Lagrange function (5) that comes from the Born theory
gives only one special example of a theory for the strong interaction.
But also for many much more complicated Lagrange functions
which in the case of a weaker interaction
contain as a solution different sorts of mesons the
spectrum (12) and (13) remains valid when one deals with a theory of strong
interaction.
\section{\it Application to the meson production.}
~~~~Now the multiple meson production should be quantitatively described
under the condition that we deal with the strong interaction.
a) One of the most important values for characterizing the meson showers
is the average energy of mesons in the center of mass frame.
In a very rough approximation one can take the spectrum (12), (13) as the
exact one between $k_0=\kappa_i$ (the rest meson mass for the
corresponding sort of a meson) and $k_0=k_{0m}$.
Then one has
$$
\left.
\begin{array}{l}
\varepsilon_i = A_i~ \int \limits^{k_{0m}}_{\kappa_i} ~
\frac{\displaystyle dk_0}{\displaystyle k_0} =
A_i {\rm lg}~ \frac{\displaystyle k_{0m}}{\displaystyle \kappa_i}\\
n_i = A_i~ \int \limits^{k_{0m}}_{\kappa_i} ~
\frac{\displaystyle dk_0}{\displaystyle k_0} =
\frac{\displaystyle A_i}{\displaystyle \kappa_i}
~\left( 1 - \frac{\displaystyle \kappa_i}{\displaystyle k_{0m}} \right)
\end{array} \right\} \eqno(16)
$$
and therefore
$$
\bar{k_{0i}} = \frac{\varepsilon_i}{n_i} = \kappa_i ~ \frac{{\rm lg}
~\frac{\displaystyle k_{0m}}{\displaystyle \kappa_i}}{1 -
\frac{\displaystyle \kappa_i}{\displaystyle k_{0m}}} \quad {\rm
for} \quad k_{0m} > \kappa_i. \eqno(17)
$$
For $k_{0m}\le \kappa_i$ the meson of the corresponding sort would not
produced at all.
In reality the spectrum should contain the factor $kdk_0$ because of
the phase space volume. Therefore for small $k$ it does not have
the form (12),(13) at all. Moreover for $k_0>k_{0m}$ the influence
of it will not completely disappear but only
cause a faster falling down than in (12) and (13).
One can try some probably a bit better approximation
$$
d \varepsilon_i = A_i~ \frac{kd k_0}{k^2_0 \left( 1 +
\frac{\displaystyle k^2_0}{\displaystyle k^2_{0m}} \right)}. \eqno(18)
$$
Then one gets
$$
\left.
\begin{array}{l}
\varepsilon_i = A_i \left( - 1 + \sqrt{1 + \alpha^2} {\rm lg} ~ \frac{
\displaystyle 1 + \sqrt{1 + \alpha^2}}{\displaystyle \alpha} \right)\\
n_i = \frac{\displaystyle A_i}{\displaystyle \kappa_i} ~
\frac{\displaystyle \pi}{\displaystyle 4} ~ \left( 1 + 2 \alpha^2
- 2 \alpha \sqrt{1 + \alpha^2} \right).
\end{array} \right\} \eqno(19)
$$
$$
\bar{k_{0i}} = \kappa_i~ \frac{4}{\pi} \frac{- 1 + \sqrt{1 +
\alpha^2}~{\rm lg}~ \frac{\displaystyle 1 + \sqrt{1 + \alpha^2}}
{\displaystyle \alpha}}{1 + 2 \alpha^2 - 2 \alpha \sqrt{1 + \alpha^
2}}, \eqno(20)
$$
where we have put $\kappa_i/k_{0m}=\alpha$.
Both approximations (17) and (20) are shown in Fig. 5 as functions
of $\log(1/\alpha)$. The difference between both of these curves
shows the uncertainty of the whole estimation.
From these calculations follows that in the special case of the strong
interaction the average meson energy increases only logarithmically
as a function of the initial energy. Thus the meson number grows
almost proportionally to the energy
which was transfered to the meson field in the center of mass frame.
b) Nevertheless for the higher energies
the above behaviour becomes more complicated
because of new meson sorts that come to the game.
One can assume that for a big enough value of $k_0$
($k_0>>\kappa_i$) the relative share $g_i$ of the meson sort
$\kappa_i$ becomes independent of $k_0$ and depends only on
the form of the shock wave equation. Hereby in this region
it would be allowed for different meson sorts to appear in general
with comparable frequencies. However $g_i$ are not necessarily
proportional to the statistical weight of the corresponding meson
sort. Let us normalize them as follows
$$
\sum g_i = 1 \eqno(21)
$$
%\end{equation}
and then take
%\begin{equation}
$$
A_i = g_i A. \eqno(22)
$$
\begin{figure}[h]
\centerline{\resizebox{7cm}{!}{\includegraphics{fig5.gif}}}
\caption{}
\end{figure}
\begin{figure}[h]
\centerline{\resizebox{10cm}{!}{\includegraphics{fig6.gif}}}
\caption{}
\end{figure}
Then the following rough approximation for (16) and (17) is valid
$$
\varepsilon = A \sum g_i {\rm lg} ~ \frac{k_{0m}}{\kappa_i},
\eqno(23)
$$
%\end{equation}
%\begin{equation}
$$
n = A \sum \frac{g_i}{\kappa_i}~ \left( 1 - \frac{\kappa_i}{k_{
0m}} \right), \eqno(24)
$$
Thus
$$
\left.
\begin{array}{llll}
n_i& = \varepsilon~ \frac{\frac{\displaystyle g_i}{\displaystyle \kappa_i}~
\left( 1 - \frac{\displaystyle \kappa_i}{\displaystyle k_{0m}}
\right)}{\displaystyle \sum_l~ g_l ~{\rm lg}~\frac{\displaystyle k_0m}
{\displaystyle \kappa_i}}
&~~{\rm for}~~& \kappa_i \le k_{0m}
\\
& = 0& ~~{\rm for}~~& \kappa_i \ge k_{0m}.
\end{array}
\right\} \eqno(25)
$$
We get that for the value $k_{0m}$ ($k_{0m} \gg \kappa_i$) the multiplicities
in different meson groups behave in the same way as $g_i/\kappa_i$. Therefore
when $k_{0m}$ decreases the number of heavier mesons falls down faster than
the corresponding number of lighter mesons.
Once $k_{0m}$ becomes less than the value $\kappa_i$ the corresponding
meson sort completely disappears.
So in the approximations of eqs. (18)-(20) the following relation
$$
n_i = \varepsilon~ \frac{\frac{\displaystyle g_i}{\displaystyle \kappa_i}
\frac{\displaystyle 4}{\displaystyle \pi} \left( 1 +
2 \alpha^2_i - 2 \alpha_i \sqrt{1 + \alpha^2_i} \right)}
{\sum g_i \left( - 1 + \sqrt{1 + \alpha^2_i} ~{\rm lg}~ \frac{1 +
\sqrt{1 + \alpha^2_i}}{\alpha_i} \right)}. \eqno(26)
$$
would be valid instead of (25).
The factor before $g_i/\kappa_i$ which charaterizes the dependence
of $n_i$ on $k_{0m}$ in eqs. (25) and (26) respectively is graphically
shown in Fig. 6. For the second approximation formula when $k_{0m}<\kappa_i$
the number of remaining mesons of the sort $\kappa_i$ would become even
smaller. One may expect this from the physical point of view.
c) In order to give an estimation of the total number of the emitted
mesons one should also know the total energy $\epsilon$ of the meson
field in (25) and (26) respectively.
As a first step one can take only the maximal value. Namely,
the energy can not be greater than the kinetic energy of both
nucleons in the center of mass frame before the collision.
Since, generally speaking only a part of this energy is transfered
to the meson field it would be reasonable to call this part $\gamma$
a ``degree of the inelasticity'' of the collision.
Then it is valid ($M =$ the nucleon mass)
$$
\varepsilon = \gamma \cdot 2 M \left( \frac{1}{\sqrt{1 - \beta^2}} - 1
\right). \eqno(27)
$$
where
$$
0\le\gamma\le 1.
$$
One can expect that in the central collision $\gamma$ should be close
to $1$ while for the peripherical collision only a small part of the
total energy will be transfered to the meson field.
Let us take $b$ for the distance between the nucleons in a moment
of the collision then one can consider the overlapping integral of
the $\pi$-meson field for both of the nucleons
as a measure of the interaction intensity.
As a very rough approximation for the inelasticity degree
$\gamma$ one can just take it proportional to this overlapping integral.
Then one gets
$$
\gamma = e^{- b \kappa}, \eqno(28)
$$
where $\kappa$ is the mass of the $\pi$-meson.
From this one can get the differential cross section
for the interval between $\gamma$ and $\gamma+d\gamma$
$$
d \sigma = 2 \pi b d b = \frac{2 \pi}{\kappa^2}~ \frac{d \gamma}
{\gamma} ~ {\rm lg} \left( \frac{1}{\gamma} \right).
\eqno(29)
$$
In order to evaluate the total cross section one should define
a minimal value of $\gamma$. For instance, in order to determine
the total cross section for the multiple meson production
one should take
for a minimal value of $\gamma$ the value for which at least two
mesons can be produced.
$$
\gamma_{\rm min} = \frac{\bar{k_0}}{M \left( \frac{1}{\sqrt{1 -
\beta^2}} - 1 \right)}. \eqno(30)
$$
($\bar k_0$ corresponds here to the lightest meson sort, namely,
to the $\pi$-mesons.)
From (30) follows:
$$
\sigma = \frac{\pi}{\kappa^2}~ {\rm lg}^2 ~\gamma_{\rm min}
\eqno(31)
$$
%\end{equation}
and
$$
\bar{\gamma} = \frac{2}{~{\rm lg}^2 ~\gamma_{\rm min}} \left( 1 -
\gamma_{\rm min} + \gamma_{\rm min} ~{\rm lg} ~\gamma_{\rm min} \right).
\eqno(32)
$$
Let us note that the estimation for the $\gamma$-distribution
which was used in eqs. (28)-(30)
does not depend on previous observations for the propagation
of the shock wave. Therefore it should be considered as less
reliable. The experimental data that we have for the moment
are not enough for the experimental determination of the
$\gamma$-distribution.
In the following Table 1 there are shown the data
on the total cross-section, expectation values
of $\gamma$, $n_{\pi}$ and $n_{\kappa}$ (the number of $\pi$-
and $\kappa$-mesons respectively),
\newpage
\begin{center}
Table 1\\
\end{center}
\begin{center}
{\footnotesize
\begin{tabular}{cc|c|c|c|c|c}
\hline
&E~~~~~~~~&10&$10^2$&$10^3$&$10^4$&BeV\\
\hline
&&&&&&\\
&$\sigma$&0.18&0.49&0.85&1.3&$10^{-24}$ cm$^2$\\
&$\bar{\gamma}$&0.34&0.19&0.13&0.09&\\
&$\bar n_{\pi}$&$3.6 \pm 0.7$&$4.2 \pm 0.8$&$5.2 \pm 0.8$&$8.0 \pm 1$&\\
&$\bar n_{\kappa}$&--&$0.9 \pm 0.2$&$2.0 \pm 0.4$&$3.4 \pm 0.6$&\\
&$\bar k_{0 \pi}$&$0.25 \pm 0.04$&$0.36 \pm 0.04$&$0.50 \pm 0.05$&$ 0.67
\pm 0.06$& BeV\\
&$\bar k_{0 \kappa}$&--&$1.0 \pm 0.2$&$1.4 \pm 0.15$&$2.0 \pm 0.18$&BeV\\
$\begin{array}{c}
\gamma = 1
\end{array}$&
$\biggl \{
\begin{array}{c}
\bar n_{\pi}\\
\bar n_{\kappa}
\end{array} \biggr.$&
$\begin{array}{c}
10.7 \pm 2\\
--
\end{array}$&
$\begin{array}{c}
22.1 \pm 4\\
4.7 \pm 1
\end{array}$&
$\begin{array}{c}
40.3 \pm 6\\
15 \pm 6
\end{array}$&
$\begin{array}{c}
89 \pm 12\\
38 \pm 6
\end{array}$&
\end{tabular}}
\end{center}
their average energy and also the number of mesons
in a special case $\gamma=1$
as the function of the initial energy $E$ (in the
laboratory frame). Other sorts of mesons, exept $\pi$- and $\kappa$-mesons,
were not taken into consideration. Then it was assumed that
$g_{\kappa}=2g_{\pi}$
i.e. $g_{\pi}={1\over 3}, g_{\kappa}={2\over 3}$ in order to take
into account a relatively more frequent appearance of $\kappa$-mesons
in accordance with the recent measurements in Bristol.
These values are still to be checked on the basis of more
precise measurements. For the mass of the $\kappa$-meson it was
taken $0.61$ BeV. In order to take into account an uncertainty
of the theoretical estimation each time (except the two first
columns) it was taken the mean-value of the expressions
(16), (17) or (18) until (20). For the error the
half-difference has been taken.
d) The angular distribution of the emitted mesons follows from
a visual consideration of the Section 1.
However the details of the angular distribution will also
depend on the shock wave equation.
Nevertheless generally speaking the transverse to the initial direction
component of the momentum of mesons can drastically exceed the value $\kappa$
only in rare cases.
The mesons with the energy $k_0$ are
emitted mostly into the interval of angles of order $k/k_0$ to the axes.
Therefore the distribution of $\kappa$-mesons is always anisotropic
while the distribution of slower $\pi$-mesons
in the center of mass frame can be to some extent isotropic.
\section{\it The comparison with the experiment.}
~~~~So far only some of meson showers without gray or
black tracks have been observed.
Thus only for these showers it is possible to assume
that one deals with the collision of only
two nucleons without participation of a bigger nucleus.
If the showers with a small
number (until 3) of thick tracks will be also involved
into the experimental consideration
then generally speaking the change of
shower due to nuclei should be small.
\begin{center}
Table 2
\end{center}
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\hline
$E$&30&40&40&90&130&1000&2000&30000 BeV\\
\hline
$n_{\pi} - n_{\kappa}$&9&18&25&10&18&9&12&21\\
$1/{\rm emp}$& 0.51&0.8&1.0&0.38&0.61&0.16&0.17&0.1
\end{tabular}
\end{center}
\vspace{0.4cm}
However, since the secondary scattering of the emitted mesons
on the atomic nuclei may happen
the determination of the initial energy from the angle distribution
and evaluation of the angle distribution itself become rather uncertain.
The observations of showers that are suitable for comparison with
the theory were done in
works of Teucher \cite{Teucher 1950}, group of Bristol \cite{Camerini 1950},
von Shein and coauthors \cite{Schein 1950},
Pickup and Voyvodic \cite{Pickup 1951} and Hopper, Biswas and
Derby \cite{Hopper 1951}.
If according to the published data one tries to
estimate initial energies from the angle distribution (this is
in some cases very uncertain) then one can gets the numbers of
mesons for the eight
observed showers which were showed in the second line of the Table 2
(these numbers may be slightly different because of neutral mesons).
Hereby it was assumed that
the relation of neutral mesons to the charged ones is $1:2$.
Assuming that two last lines in
the Table 1 are correct one gets some empirical
value of $\gamma$ for each of these showers.
These values are shown in the third line of the Table 2.
First one can see that these numbers of mesons are not well-defined
functions of the initial energy.
The values of $\gamma$ fluctuate very strongly, as
was expected. However they are on average a bit
bigger than it could be expected
from the Table 1.
This can be based on the fact that a small shower is easier
to miss than a big one.
But this may also mean that the estimation in the equation
(28) is still too rough\footnote{{\it Noted during proof-reading}
In the conference in Kopenhagen, June 1952,
Le Couteur showed that average value $\gamma$ in a heavier
matter (as, for
example, in a photo emulsion) should be essentialy bigger than
in the Hydrogen
(in the Table 1 are shown the data for the Hydrogen) because peripherical
collisions happen only for those nucleons which are placed on
the surface of an atomic
nucleus. Then Powell has reported about new experiments which indicated
that the particles called here $\kappa$-mesons decay into the
two groups of mesons
with masses 0.74 and 0.54 BeV respectively.
These mesons have absolutely different properties.
}.
On the other hand the empirical $\gamma$-values themselves in the
Table 2 are still rather uncertain because, for instance,
the part of $\kappa$-mesons is not exactly
known. Also Perkins \cite{Perkins} communicated about relatively
higher $\gamma$-values.
But it is still to wait for more experimental data.
Two showers (Teucher \cite{Teucher 1950} and Hopper, Biswas and
Derby \cite{Hopper 1951}) could be measured
so precisely that the average energy of mesons in
the center of mass frame could be found.
In the first case (40 BeV, approximately 25 mesons)
the average energy of mesons is 0.29 BeV which is comparable with
0.31 BeV from the Table 1. In the second case (1000 BeV, approximately
9 mesons) there is some uncertainty due to a possibility
that some of the observed particles can be $\kappa$-mesons.
This possibility was not taken into account by the authors
(from the Table 1 it could be expected that
among 9 mesons are to expect about 3 $\kappa$-mesons).
If one does not take this into consideration then
the measured average energy of $\pi$-mesons in the center mass frame
is 0.44 BeV which is comparable with 0.50 BeV from the Table 1 also.
Thus both of this measurements
confirm relatively low meson energies from the Table 1.
On the other hand Perkins \cite{Perkins}
takes the value 1.5 BeV as the average
energy of mesons from a group of showers
with initial energy from $10^2$ untill $10^3$ BeV.
It is essentialy higher.
However one needs to take into consideration some uncertainty
in the measurement of the initial energy. Any error of the initial
energy in general increases the average energy of mesons because
it has a minimum value just in the center of mass frame.
Concerning the frequency of $\kappa$-mesons
there exists only one result obtained by the Bristol group
which tells us
that at high energies it is comparable with the frequency
of $\pi$-mesons \cite{Perkins}. However it is still
impossible to determine this ratio from the theory (in the Table 1 it
was taken $g_{\kappa}/g_{\pi}=2$ just ad-hoc).
Concerning the angular distribution it is observed
that in the center of mass frame the angular distribution is almost
isotropic for showers of slow energy. Meanwhile
for showers of a higher energy
one can clearly observe the accumulation of events in the forward and
backward directions.
It exactly corresponds to the picture from the item Ic.
Actually the mesons at high energy seem to be distributed
always anisotropically,
in particular, the $\kappa$-mesons (Perkins \cite{Perkins}).
And also the degree of the anisotropy corresponds to the theoretical estimation.
In general, the impression is that for the ``strong'' interaction
the formulae from the Section
III satisfactory describe the experimental data. Thus
the interaction of elementary particles at high energy really
belongs to the group of ``strong'' interactions first studied by Born.\\\\
{\it G{\"o}ttingen}, Max-Planck-Institute for Physics
\newpage
\begin{thebibliography}{99}
\bibitem{Born 1933}
Born, M.: Proc. Roy. Soc. Long., Ser. A {\bf 143}, 410 (1933);
Born, M. and L.~Infeld: Proc. Roy. Soc. Long., Ser. A {\bf 144}, 425 (1934);
{\bf 147}, 522 (1934); {\bf 150}, 141 (1935). \\
\bibitem{Camerini 1950}
Camerini, U., P.H. Fowler, W.O. Lock and H. Muirhead; Phil. Mag.,\\
(7) {\bf 41}, 413 (1950)
\bibitem{Fermi 1950}
Fermi, E.: Progr. theor. Phys. {\bf 5}, 570 (1950).--Phys. Rev. {\bf 81},
683 (1951).\\
\bibitem{Heisenberg 1939}
Heisenberg, W.; Z. Physik {\bf 113}, 61 (1939)
\bibitem{Heisenberg 1949}
Heisenberg, W.; Z. Physik {\bf 126}, 519 (1949)
\bibitem{Hopper 1951}
Hopper, V.D., S. Biswas and J.F. Derby; Phys. Rev. {\bf 84}, 457 (1951)
\bibitem{Perkins}
Mr. Perkins has kindly informed us about the results of his recent
work before it's publication.
\bibitem{Pickup 1951}
Pickup, E., and L. Voyvodic; Phys. Rev. {\bf 82}, 293 (1951);
{\bf 84}, 1190 (1951).\\
\bibitem{Schein 1950}
Schein, M., J.J. Lord and J. Fainberg; Phys. Rev. {\bf 80}, 970 (1950);
{\bf 81}, 313 (1951)\\
\bibitem{Schiff 1951}
Schiff, L.J.; Phys. Rev., {\bf 84}, 1 (1951)
\bibitem{Teucher 1950}
Teucher, M.; Naturwiss, {\bf 37}, 260 (1950);
{\bf 39}, 68 (1952).\\
\bibitem{Thirring 1952}
Thirring, W.; L. Naturforsch. {\bf 7a}, 63 (1952)
\end{thebibliography}
\end{document}
%ENCODE December 2002 by Boos G. and Sukhikh N.
\documentclass[11pt]{article}
%\documentstyle[11pt,twoside,ihepconf]{article}
%\usepackage{sw20lart}
\usepackage{amsmath}
\usepackage{graphicx}
%\usepackage{geometry}
\usepackage[english]{babel}
\RequirePackage {hyperref}
\usepackage{amsfonts}
\usepackage{amssymb}
%\usepackage{fancyheadings}
%\pagestyle{fancy}
\usepackage{graphicx}
\usepackage[dvips]{epsfig}
\usepackage{wrapfig}
\tolerance=7000
\begin{document}
W.~ Heisenberg, Zeit. Phys., {\bf Bd. 133, 3} 65 \hfill {\large \bf 1952}\\
\vspace{2cm}
\begin{center}
{\Large \bf Production of Mesons as a Shock Wave Problem}\\
\end{center}
\vspace{0.5cm}
\begin{center}
W. Heisenberg\\
With 6 figures in text\\
Received on 5 May 1952\\
\end{center}
\vspace{0.5cm}
\centerline{--- ---~~~$\diamond~\diamondsuit~\diamond$~~~--- ---}
\noindent
Translated by Herman Boos \\
\centerline{--- ---~~~$\diamond~\diamondsuit~\diamond$~~~--- ---}
\vspace{0.5cm}
\begin{abstract}
Multi-meson production in two-nucleon collision is described as a shock wave
process which is governed by a non-linear wave equation. Since one deals with
big quantum numbers, these quantum processes may be approximately described
by means of the correspondece principle. Analysing solutions to the
non-linear wave equation one can get the energy and angular distribution for
different meson sorts.
\end{abstract}
An analysis of experimental data on the $\pi$-meson production obtained in
last years tells us that a collision of two nucleons of high energy produces
a big number of mesons. The fact that the strong interaction of nucleons
with mesons and, in particular, of mesons with themselves results in such a
multiple production was established long time ago \cite{Heisenberg 1939}. In oder to
give a quantitative estimation of the energy dissipation
in meson field one can compare this process with
a turbulence flow \cite{Heisenberg 1949}. Some other way as Fermi
did \cite{Fermi 1950}
is to consider the temperature balance in a moment of collision. It allows to
calculate the energy distribution of mesons.
The present consideration of the problem is
based on the point of view that was suggested by author in
1939 in connection with the Yukawa theory \cite{Heisenberg 1939}. The meson production is
considered as a shock wave process which is described by a non-linear wave
equation. It will be shown that in this way it is possible to get
quantitative results for the spectral and spatial distributions of
different meson sorts.
\section{\it Visual description of a shock wave}
~~~~Below the meson production will be considered in the center of mass frame.
A transition to the laboratory frame can be easily
done \cite{Heisenberg 1949}. This question will not be discussed here.
a) In the center of mass frame two nucleons move to meet each other
in opposite directions (Fig.1) until they cross in some region (shaded area
in Fig.1). In this region they strongly interact. The nucleons can be
imagined as discs with a thickness which is less than their cross section
because of the Lorentz factor $\sqrt{1-\beta^2}$ ($\beta$ is a speed of
mass center). One can take this cross section to be of order
$1.4\cdot 10^{-13}cm$. In a moment of collision the speed of nucleons
change in such a way that in their intersection area the energy is transfered
to the meson field. So, in the very beginning
when a shock wave just appears the whole energy
of the meson field is concentrated in a thin flat layer which was filled by
both nucleons in the moment of collision.
\begin{figure}[h]
\centerline{\resizebox{12cm}{!}{\includegraphics{fig1.gif}}}
\caption{}
\end{figure}
b) If one could neglect the interaction between the mesons then they
would propagate in accordance with a wave equation
\begin{equation}
\square \varphi - \kappa^2 \varphi = 0
\end{equation}
(or more complicated linear wave equation that contains different
meson sorts). Then also the angular distribution of this wave would not
change during it's propagation. These distributions could be determined
already in the first moment by means of an expansion of the wave into the
Fourier series. Thus we can conclude that in the frequency region from
$k_0$ to $k_0+dk_0$ ($k_0$ is energy of a single meson) the energy of
the meson wave does not depend on $k_0$ up to the frequencies corresponding
to such wave lengthes which are comparable with the thickness of the layer
where the collision takes place. This thickness is of the order
${\sqrt{1-\beta^2}\over \kappa}$ ($\kappa$ is the meson mass). When
$k_0>k_{0 m} = {\kappa\over \sqrt{1-\beta^2}}$ the intensity abruptly
decreases as a function of $k_0$.
\begin{equation}
d \varepsilon = {\rm constant} \cdot d k_0 \quad {\rm for} \quad
k_0 \le k_{0m}.
\end{equation}
For a number of mesons with frequences lying in the region $dk_0$ one gets
\begin{equation}
d n = {\rm constant} ~ \frac{dk_0}{k_0} \quad {\rm for} \quad
k_0 \le k_{0m}.
\end{equation}
In Fig.2 is shown the behaviour of the function $\varphi$ on the
axis which is normal to the radiation plane (right after the act
of emission). Also shown are functions ${d\epsilon\over dk_0}$ and
$dn\over dk_0$ under condition that the equation (1) is fulfilled. The
spectrum (3) corresponds to the known spectrum of the R{\"o}ntgen
Bremsstrahlung of the electron. It is also valid in case when the most
part of energy of mesons is transfered to the meson field in such a way
that the number of outgoing mesons is not too big. It means that the
energy of a single meson is approximately $\ge{1\over 2}k_{0m}$.
\begin{figure}[h]
\centerline{\resizebox{13cm}{!}{\includegraphics{fig2.gif}}}
\caption{a - c}
\end{figure}
c) But in reality the interaction of mesons can not be neglected i.e. the
wave propagation is going on according to the same non-linear wave equation
which only approximately becomes linear in case of a small intensity.
As will be shown below the non-linearity results in a slight smoothing
the singularity
of a wave crest. Due to this, during the wave propagation the energy
is transfered from shorter to longer waves. Therefore in the
end of this propagation the spectral distribution goes down steeper in
comparison with the previous case when the equation (1) is valid. Thus
we come to the behaviour shown in Fig.3. The spatial distribution is
shown in Fig.4(a-d).
\begin{figure}[h]
\centerline{\resizebox{13cm}{!}{\includegraphics{fig3.gif}}}
\caption{a - c.}
\end{figure}
In the moment of collision the whole energy is concentrated in the
intersection area of two nucleons (a). After this two shock wave fronts
start to propagate to the right and to the left respectively.
The most part of the
energy is still concentrated in these two fronts. But a wave perturbation
also appears in a region between them where the rest of the energy is
concentrated (b). When the propagation of the shock wave fronts goes
further the perturbation on their wakes captures more and more space which
in it's turn becomes a source of a new wave propagation.
Then the front energy becomes
lower. It is transfered to other wave lengthes, in particular, to longer
waves (c). During the further propagation the perturbation in a central
area decreases. Thus actually the produced wave moves faster in a
direction of the shock wave fronts than in transverse direction.
Therefore short waves have a higher group speed. Now perturbation of a very
low intensity spreads out into all directions also with the speed of light.
The energy of shock wave fronts becomes so low that the non-linearity does
not play a big role any more here also. So the further propagation
continues according to the usual linear wave equation (d).
\begin{figure}[h]
\centerline{\resizebox{10cm}{!}{\includegraphics{fig4.gif}}}
\caption{a - d.}
\end{figure}
Up to the moment we ignored the quantum effects. This approximation is
acceptable because we deal with production of a big number of mesons i.e.
with a process with big quantum numbers. How the correspondence principle
is applied in quantum theory was described in detail in above mentioned
paper \cite{Heisenberg 1939}. Here it is enough for us to accept some qualitative
consideration based on Fig.4d. Namely, the most part of energy is emitted
in all directions in form of mesons whose wave lengthes are comparable
with the disc cross section i.e. ${1/ \kappa}$.
The transverse momentum can be greater
than $\kappa$ only in rare cases so that the
Fourier coefficients of that waves are very small. On the other hand,
the longitudinal momentum can be greater so that one can face
the shorter waves in the shock wave front. The mesons with the
energy $k_0$ will be emitted mostly only into the angle interval of order
$\kappa /{k_0}$ with regard to both dominating directions. Also heavier mesons
are mostly emitted only into the front of the shock wave.
\section{\it Solution of the shock wave equation}
~~~~The propagation of the shock wave depends on the form of the non-linear
wave equation on which the description of mesons is based. But it is
possible to show that there is a special case of ``strong'' interaction
for which one can accept that the spectral distribution of mesons does not
depend on the concrete form of the wave equation. The solution of the
non-linear equation can be performed by means of a simplification if to be
interested only in the spectral and not in the angular distribution.
Namely, if the plane where the emission happens tends to be infinite
then the layer tends to be infinitely thin. Then because of the Lorentz
invariance the function $\varphi$ can depend only on $s=t^2-x^2$. Thus
the partial differential equation becomes an ordinary differential
equation which is simpler to analyse.
Let us discuss two examples of non-linear wave theories:
1. The equation suggested by Shiff \cite{Schiff 1951} and
Thirring \cite{Thirring 1952} in connection
with description of nuclear forces:
\begin{equation}
\square \varphi - \kappa^2 \varphi - \eta \varphi^3 = 0.
\end{equation}
2. The wave equation which analogously to earlier works by
Born \cite{Born 1933} follows from the Lagrange function
\begin{equation}
L = l^{-4}~ \sqrt{1 + l^4 \left[ \sum~ \left( \frac{\partial
\varphi}{\partial \kappa_{\nu}} \right)^2 + \kappa^2
\varphi^2 \right]}
\end{equation}
Long time ago Born noted that the non-linear theories of that type have
solutions that are less singular in comparison with the solutions to
linear equations. That time this fact was used for the description of
the electron self-energy. But the same consideration is also acceptable
for the multiple production of mesons.
In the earlier works \cite{Heisenberg 1939} the investigation of the meson production
was already based on the Lagrange function \cite{Heisenberg 1949}.
To the item 1. The first of two these equations gives for $\varphi=\varphi(s)$:
$$
4~ \frac{d}{ds}~ \left( s~ \frac{d \varphi}{ds} \right) + \kappa^2
\varphi + \eta \varphi^3 = 0. \eqno(4a)
$$
When $\eta=0$ one comes again to the linear equation (1) and then
it's solution looks as follows:
$$
\left.
\begin{array}{lll}
\varphi = a J_0 \left( \kappa \sqrt{s} \right)&~~~{\rm for}~~~&
s > 0\\
\varphi = 0& ~~~{\rm for}~~~&s < 0,
\end{array}
\right\} \eqno(6a)
$$
where $a$ is an integration constant; also compare it with Fig.2. For
$\eta\ne 0$ one can write down a power series at the point $s=0$.
$$
\left. \begin{array}{ll}
q = a [1 - (\kappa^2 + \eta a^2) s + \frac{1}{4} (\kappa^2 + 3 \eta
a^2)( \kappa^2 + \eta a^2)& s^2 - + \ldots ]=0\\
& {\rm for}~~~ s > 0\\
& {\rm for}~~~ s < 0.
\end{array} \right\} \eqno(6b)
$$
Let us note that the equation (4) corresponds to some ``weak''
interaction which does not change the shock wave front
if the wave function is continuous. It is connected with the fact
that the theory based on eq. (4) belongs to the group of
renormalizable theories because the coupling constant $\eta$ is dimensionless.
It was established in different ways that
renormalizable theories describe only ``weak'' interactions which
in general do not result in the multiple production of mesons.
To the item 2.
However the situation with the wave equation that follows from the formula (5)
is different. For $\varphi=\varphi(s)$ this wave equation looks as follows:
$$
4~ \frac{d}{ds}~ (s \varphi') + \kappa^2 \varphi = 8 l^4 s
\varphi'^2 ~ \frac{\varphi' + \kappa^2 \varphi}{1 + l^4 \kappa^2
\varphi^2}. \eqno(7)
$$
If $\kappa=0$ (zero meson mass) then the solution may be readily found:
$$
\left.
\begin{array}{llll}
\varphi&=\frac{\displaystyle 1}{\displaystyle a}~ {\rm lg} \left( 1
+ \frac{\displaystyle a^2}{\displaystyle 2 l^4} ~ s +
\frac{\displaystyle a}{\displaystyle 2l^4}~ \sqrt{4l^4s + a^2 s^2}
\right)&~~{\rm for}~~& s \ge 0\\
&= 0&~~{\rm for}~~&s \le 0.
\end{array}
\right\} \eqno(8)
$$
In general case ($\kappa\ne 0$) one can apply again the power series expansion.
Let us take
$$
\varphi = \frac{1}{\kappa l^2} ~ f(\zeta); \qquad \zeta = s \kappa^2
\eqno(9)
%\end{equation}
$$
and we write
$$
\left.
\begin{array}{llll}
f(\zeta) &= \sqrt{\zeta} \left( 1 + a \zeta + \frac{\displaystyle
27 a^2 + 2a - 1}{\displaystyle 10}~ \zeta^2 + - \ldots \right)&
~~{\rm for}~~& \zeta \ll 1\\
& \approx \gamma \zeta^{-1/4} \cos (\sqrt{\zeta} + \delta
)& ~~{\rm for}~~& \zeta \gg 1\\
& = 0&~~{\rm for}~~& \zeta \le 0.
\end{array}
\right\} \eqno(10)
$$
The constants $\gamma$ and $\delta$ are completely fixed via the
integration constant $a$. However it's value was not evaluated.
One can note that the non-linearity essentialy changes the character of the
solution. Namely, at $s=0$ the discontinuity of $\varphi$ disappears.
More exactly, $\varphi '$ has a discontinuity while
$\varphi$ behaves as $\sqrt{s}$ in a small vicinity of $s=0$.
Let us assume that to a given moment of evolution the function
\mbox{$\varphi(s)=\varphi(x,t)$}
is defined by the Fourier integral over the wave number $k$.
Then for the Fourier coefficient $\varphi(k,t)$ with $k\sim k_0 \gg\kappa$
one obtains up to some constant
$$
q(k,t) \sim k^{-3/2} t^{1/2} e^{\pm ik_0t}.
\eqno(11)
%\end{equation}
$$
It is easy to understand that the multiplier $t^{1/2}$ appears due to
the fact that during the propagation process the energy is transfered
from the head
part of the shock wave to other parts of the wave and so to lower
frequences also. Actually the energy potential in the head part of the
shock wave is infinite. This is just a consequence of our assumption
that the shock wave appears inside an infinitely thin layer. By means of this
assumption we have achieved that the solution $\varphi(x,t)$ depends only on
combination $t^2-x^2$ which is invariant under the Lorentz transformation
in the $x,t$-space. Some finite energy-momentum would define some
direction in this space. Therefore in this case it could not lead to
some invariant solution.
However in reality the shock wave appears in a layer with a
finite thickness
$\sim{\sqrt{1-\beta^2}\over k}$. Therefore the energy-momentum vector is
finite. The increasing of the Fourier amplitudes in (11) starts with some
delay. When the energy potential concentrated in the head part of
the shock wave is exhausted it comes to the state of rest.
Then for a big $t$ the Fourier coefficients as a function of $k$
for $k>k_{0m}={\kappa\over\sqrt{1-\beta^2}}$ fall down faster than $k^{-3/2}$.
Thus for the intensity distribution one gets
$$
\frac{d \varepsilon}{d k_0} = {\rm const}~ \frac{dk_0}{k_0} \quad
{\rm for} \quad \kappa \le k_0 \le k_{0m} = \frac{\kappa}{\sqrt{1 -
\beta^2}} \eqno(12)
$$
and
$$
%\begin{equation}
\frac{dn}{dk_0} = {\rm const}~ \frac{dk_0}{k^2_0}. \eqno(13)
$$
%\end{equation}
for the same region.
This is the form of the spectrum that was earlier suggested
in connection with the multiple
production of mesons \cite{Heisenberg 1939}, \cite{Heisenberg 1949} and it was
also shown in Fig.3.
The wave equation (7) borrowed from the Born theory \cite{Born 1933} is the typical
case of ``strong'' interaction and describes the multiple production
of mesons. Here the coupling constant has a dimention of length in the fourth power.
b) Now it is necessary to show that the spectrum (12) and (13) in general
corresponds to the strong interaction independent of a concrete form of
the Lagrange function and of the properties of participating particles.
We start with an arbitrary Lagrange function which depends on some scalar wave
function $\varphi$ and it's first derivatives ${\partial\over \partial x_{\nu}}$.
Because of the Lorentz
invariance $L$ can depend only on $\varphi$ and on
$\sum_{\nu}({\partial\over \partial x_{\nu}})^2$.
For very small
values of $\varphi$ and ${\partial\over \partial x_{\nu}}$
$L$ should reduce to the Lagrange function of
the usual wave equation (1). Now we would like to know the value of
${\partial\over \partial x_{\nu}}$ in a small vicinity of $s=0 (s>0)$.
For $s \rightarrow 0$ $\sum_{\nu}({\partial\over \partial x_{\nu}})^2$
can be either infinite or finite or even
tend to zero.
First one can exclude the last of these three possibilities because
in this case right
at the critical point $s=0$ the non-linearity would not play any role.
But this is impossible because for the usual wave equation (1)
$\sum_{\nu}({\partial\over \partial x_{\nu}})^2$ can never be equal zero
in the critical point but is infinite.
Among the both residual possibilities actually only the second one
gives a smooth behaviour of $\varphi$ at the singular point.
Therefore it may correspond to the strong interaction.
In the vicinity of $s=0$ it becomes
$$
\sum \left( \frac{\partial \varphi}{\partial x_{\nu}}\right)^2 = -
4s \left( \frac{\partial \varphi}{ds} \right)^2 = {\rm const} ~(\ne
0 \quad {\rm and} ~ \ne \infty). \eqno(14)
$$
from which one gets
$$
%\begin{equation}
\varphi(s) \sim {\rm const} ~\sqrt s, \eqno(15)
$$
and also expressions like (7) and (10)-(14).
c) But for (12) and (13) one can give more general reasons that are acceptable
also for arbitrary particles with higher spin. Already in item 11a
it was mentioned that in that special case
where the shock wave appears within an infinitely
thin layer it's total energy should be infinite. In this case the wave
function is invariant with regard to the rotations in the \mbox{$x,t$-space.}
The bigger is the energy dissipation caused by the interaction
the steeper falls down the energy spectrum of mesons.
Since the spectrum has a form that exactly corresponds to the
potential law (and it should be true for a majority of simple wave equations)
it can not fall down faster than in (12) and (13).
The reason for this is that the total energy
is divergent for (namely logarithmically)
for $k_{0m}\rightarrow\infty$.
Thus the spectrum (12) and (13) just corresponds
to the special case of the strong interaction. As was mentioned above
the Lagrange function (5) that comes from the Born theory
gives only one special example of a theory for the strong interaction.
But also for many much more complicated Lagrange functions
which in the case of a weaker interaction
contain as a solution different sorts of mesons the
spectrum (12) and (13) remains valid when one deals with a theory of strong
interaction.
\section{\it Application to the meson production.}
~~~~Now the multiple meson production should be quantitatively described
under the condition that we deal with the strong interaction.
a) One of the most important values for characterizing the meson showers
is the average energy of mesons in the center of mass frame.
In a very rough approximation one can take the spectrum (12), (13) as the
exact one between $k_0=\kappa_i$ (the rest meson mass for the
corresponding sort of a meson) and $k_0=k_{0m}$.
Then one has
$$
\left.
\begin{array}{l}
\varepsilon_i = A_i~ \int \limits^{k_{0m}}_{\kappa_i} ~
\frac{\displaystyle dk_0}{\displaystyle k_0} =
A_i {\rm lg}~ \frac{\displaystyle k_{0m}}{\displaystyle \kappa_i}\\
n_i = A_i~ \int \limits^{k_{0m}}_{\kappa_i} ~
\frac{\displaystyle dk_0}{\displaystyle k_0} =
\frac{\displaystyle A_i}{\displaystyle \kappa_i}
~\left( 1 - \frac{\displaystyle \kappa_i}{\displaystyle k_{0m}} \right)
\end{array} \right\} \eqno(16)
$$
and therefore
$$
\bar{k_{0i}} = \frac{\varepsilon_i}{n_i} = \kappa_i ~ \frac{{\rm lg}
~\frac{\displaystyle k_{0m}}{\displaystyle \kappa_i}}{1 -
\frac{\displaystyle \kappa_i}{\displaystyle k_{0m}}} \quad {\rm
for} \quad k_{0m} > \kappa_i. \eqno(17)
$$
For $k_{0m}\le \kappa_i$ the meson of the corresponding sort would not
produced at all.
In reality the spectrum should contain the factor $kdk_0$ because of
the phase space volume. Therefore for small $k$ it does not have
the form (12),(13) at all. Moreover for $k_0>k_{0m}$ the influence
of it will not completely disappear but only
cause a faster falling down than in (12) and (13).
One can try some probably a bit better approximation
$$
d \varepsilon_i = A_i~ \frac{kd k_0}{k^2_0 \left( 1 +
\frac{\displaystyle k^2_0}{\displaystyle k^2_{0m}} \right)}. \eqno(18)
$$
Then one gets
$$
\left.
\begin{array}{l}
\varepsilon_i = A_i \left( - 1 + \sqrt{1 + \alpha^2} {\rm lg} ~ \frac{
\displaystyle 1 + \sqrt{1 + \alpha^2}}{\displaystyle \alpha} \right)\\
n_i = \frac{\displaystyle A_i}{\displaystyle \kappa_i} ~
\frac{\displaystyle \pi}{\displaystyle 4} ~ \left( 1 + 2 \alpha^2
- 2 \alpha \sqrt{1 + \alpha^2} \right).
\end{array} \right\} \eqno(19)
$$
$$
\bar{k_{0i}} = \kappa_i~ \frac{4}{\pi} \frac{- 1 + \sqrt{1 +
\alpha^2}~{\rm lg}~ \frac{\displaystyle 1 + \sqrt{1 + \alpha^2}}
{\displaystyle \alpha}}{1 + 2 \alpha^2 - 2 \alpha \sqrt{1 + \alpha^
2}}, \eqno(20)
$$
where we have put $\kappa_i/k_{0m}=\alpha$.
Both approximations (17) and (20) are shown in Fig. 5 as functions
of $\log(1/\alpha)$. The difference between both of these curves
shows the uncertainty of the whole estimation.
From these calculations follows that in the special case of the strong
interaction the average meson energy increases only logarithmically
as a function of the initial energy. Thus the meson number grows
almost proportionally to the energy
which was transfered to the meson field in the center of mass frame.
b) Nevertheless for the higher energies
the above behaviour becomes more complicated
because of new meson sorts that come to the game.
One can assume that for a big enough value of $k_0$
($k_0>>\kappa_i$) the relative share $g_i$ of the meson sort
$\kappa_i$ becomes independent of $k_0$ and depends only on
the form of the shock wave equation. Hereby in this region
it would be allowed for different meson sorts to appear in general
with comparable frequencies. However $g_i$ are not necessarily
proportional to the statistical weight of the corresponding meson
sort. Let us normalize them as follows
$$
\sum g_i = 1 \eqno(21)
$$
%\end{equation}
and then take
%\begin{equation}
$$
A_i = g_i A. \eqno(22)
$$
\begin{figure}[h]
\centerline{\resizebox{7cm}{!}{\includegraphics{fig5.gif}}}
\caption{}
\end{figure}
\begin{figure}[h]
\centerline{\resizebox{10cm}{!}{\includegraphics{fig6.gif}}}
\caption{}
\end{figure}
Then the following rough approximation for (16) and (17) is valid
$$
\varepsilon = A \sum g_i {\rm lg} ~ \frac{k_{0m}}{\kappa_i},
\eqno(23)
$$
%\end{equation}
%\begin{equation}
$$
n = A \sum \frac{g_i}{\kappa_i}~ \left( 1 - \frac{\kappa_i}{k_{
0m}} \right), \eqno(24)
$$
Thus
$$
\left.
\begin{array}{llll}
n_i& = \varepsilon~ \frac{\frac{\displaystyle g_i}{\displaystyle \kappa_i}~
\left( 1 - \frac{\displaystyle \kappa_i}{\displaystyle k_{0m}}
\right)}{\displaystyle \sum_l~ g_l ~{\rm lg}~\frac{\displaystyle k_0m}
{\displaystyle \kappa_i}}
&~~{\rm for}~~& \kappa_i \le k_{0m}
\\
& = 0& ~~{\rm for}~~& \kappa_i \ge k_{0m}.
\end{array}
\right\} \eqno(25)
$$
We get that for the value $k_{0m}$ ($k_{0m} \gg \kappa_i$) the multiplicities
in different meson groups behave in the same way as $g_i/\kappa_i$. Therefore
when $k_{0m}$ decreases the number of heavier mesons falls down faster than
the corresponding number of lighter mesons.
Once $k_{0m}$ becomes less than the value $\kappa_i$ the corresponding
meson sort completely disappears.
So in the approximations of eqs. (18)-(20) the following relation
$$
n_i = \varepsilon~ \frac{\frac{\displaystyle g_i}{\displaystyle \kappa_i}
\frac{\displaystyle 4}{\displaystyle \pi} \left( 1 +
2 \alpha^2_i - 2 \alpha_i \sqrt{1 + \alpha^2_i} \right)}
{\sum g_i \left( - 1 + \sqrt{1 + \alpha^2_i} ~{\rm lg}~ \frac{1 +
\sqrt{1 + \alpha^2_i}}{\alpha_i} \right)}. \eqno(26)
$$
would be valid instead of (25).
The factor before $g_i/\kappa_i$ which charaterizes the dependence
of $n_i$ on $k_{0m}$ in eqs. (25) and (26) respectively is graphically
shown in Fig. 6. For the second approximation formula when $k_{0m}<\kappa_i$
the number of remaining mesons of the sort $\kappa_i$ would become even
smaller. One may expect this from the physical point of view.
c) In order to give an estimation of the total number of the emitted
mesons one should also know the total energy $\epsilon$ of the meson
field in (25) and (26) respectively.
As a first step one can take only the maximal value. Namely,
the energy can not be greater than the kinetic energy of both
nucleons in the center of mass frame before the collision.
Since, generally speaking only a part of this energy is transfered
to the meson field it would be reasonable to call this part $\gamma$
a ``degree of the inelasticity'' of the collision.
Then it is valid ($M =$ the nucleon mass)
$$
\varepsilon = \gamma \cdot 2 M \left( \frac{1}{\sqrt{1 - \beta^2}} - 1
\right). \eqno(27)
$$
where
$$
0\le\gamma\le 1.
$$
One can expect that in the central collision $\gamma$ should be close
to $1$ while for the peripherical collision only a small part of the
total energy will be transfered to the meson field.
Let us take $b$ for the distance between the nucleons in a moment
of the collision then one can consider the overlapping integral of
the $\pi$-meson field for both of the nucleons
as a measure of the interaction intensity.
As a very rough approximation for the inelasticity degree
$\gamma$ one can just take it proportional to this overlapping integral.
Then one gets
$$
\gamma = e^{- b \kappa}, \eqno(28)
$$
where $\kappa$ is the mass of the $\pi$-meson.
From this one can get the differential cross section
for the interval between $\gamma$ and $\gamma+d\gamma$
$$
d \sigma = 2 \pi b d b = \frac{2 \pi}{\kappa^2}~ \frac{d \gamma}
{\gamma} ~ {\rm lg} \left( \frac{1}{\gamma} \right).
\eqno(29)
$$
In order to evaluate the total cross section one should define
a minimal value of $\gamma$. For instance, in order to determine
the total cross section for the multiple meson production
one should take
for a minimal value of $\gamma$ the value for which at least two
mesons can be produced.
$$
\gamma_{\rm min} = \frac{\bar{k_0}}{M \left( \frac{1}{\sqrt{1 -
\beta^2}} - 1 \right)}. \eqno(30)
$$
($\bar k_0$ corresponds here to the lightest meson sort, namely,
to the $\pi$-mesons.)
From (30) follows:
$$
\sigma = \frac{\pi}{\kappa^2}~ {\rm lg}^2 ~\gamma_{\rm min}
\eqno(31)
$$
%\end{equation}
and
$$
\bar{\gamma} = \frac{2}{~{\rm lg}^2 ~\gamma_{\rm min}} \left( 1 -
\gamma_{\rm min} + \gamma_{\rm min} ~{\rm lg} ~\gamma_{\rm min} \right).
\eqno(32)
$$
Let us note that the estimation for the $\gamma$-distribution
which was used in eqs. (28)-(30)
does not depend on previous observations for the propagation
of the shock wave. Therefore it should be considered as less
reliable. The experimental data that we have for the moment
are not enough for the experimental determination of the
$\gamma$-distribution.
In the following Table 1 there are shown the data
on the total cross-section, expectation values
of $\gamma$, $n_{\pi}$ and $n_{\kappa}$ (the number of $\pi$-
and $\kappa$-mesons respectively),
\newpage
\begin{center}
Table 1\\
\end{center}
\begin{center}
{\footnotesize
\begin{tabular}{cc|c|c|c|c|c}
\hline
&E~~~~~~~~&10&$10^2$&$10^3$&$10^4$&BeV\\
\hline
&&&&&&\\
&$\sigma$&0.18&0.49&0.85&1.3&$10^{-24}$ cm$^2$\\
&$\bar{\gamma}$&0.34&0.19&0.13&0.09&\\
&$\bar n_{\pi}$&$3.6 \pm 0.7$&$4.2 \pm 0.8$&$5.2 \pm 0.8$&$8.0 \pm 1$&\\
&$\bar n_{\kappa}$&--&$0.9 \pm 0.2$&$2.0 \pm 0.4$&$3.4 \pm 0.6$&\\
&$\bar k_{0 \pi}$&$0.25 \pm 0.04$&$0.36 \pm 0.04$&$0.50 \pm 0.05$&$ 0.67
\pm 0.06$& BeV\\
&$\bar k_{0 \kappa}$&--&$1.0 \pm 0.2$&$1.4 \pm 0.15$&$2.0 \pm 0.18$&BeV\\
$\begin{array}{c}
\gamma = 1
\end{array}$&
$\biggl \{
\begin{array}{c}
\bar n_{\pi}\\
\bar n_{\kappa}
\end{array} \biggr.$&
$\begin{array}{c}
10.7 \pm 2\\
--
\end{array}$&
$\begin{array}{c}
22.1 \pm 4\\
4.7 \pm 1
\end{array}$&
$\begin{array}{c}
40.3 \pm 6\\
15 \pm 6
\end{array}$&
$\begin{array}{c}
89 \pm 12\\
38 \pm 6
\end{array}$&
\end{tabular}}
\end{center}
their average energy and also the number of mesons
in a special case $\gamma=1$
as the function of the initial energy $E$ (in the
laboratory frame). Other sorts of mesons, exept $\pi$- and $\kappa$-mesons,
were not taken into consideration. Then it was assumed that
$g_{\kappa}=2g_{\pi}$
i.e. $g_{\pi}={1\over 3}, g_{\kappa}={2\over 3}$ in order to take
into account a relatively more frequent appearance of $\kappa$-mesons
in accordance with the recent measurements in Bristol.
These values are still to be checked on the basis of more
precise measurements. For the mass of the $\kappa$-meson it was
taken $0.61$ BeV. In order to take into account an uncertainty
of the theoretical estimation each time (except the two first
columns) it was taken the mean-value of the expressions
(16), (17) or (18) until (20). For the error the
half-difference has been taken.
d) The angular distribution of the emitted mesons follows from
a visual consideration of the Section 1.
However the details of the angular distribution will also
depend on the shock wave equation.
Nevertheless generally speaking the transverse to the initial direction
component of the momentum of mesons can drastically exceed the value $\kappa$
only in rare cases.
The mesons with the energy $k_0$ are
emitted mostly into the interval of angles of order $k/k_0$ to the axes.
Therefore the distribution of $\kappa$-mesons is always anisotropic
while the distribution of slower $\pi$-mesons
in the center of mass frame can be to some extent isotropic.
\section{\it The comparison with the experiment.}
~~~~So far only some of meson showers without gray or
black tracks have been observed.
Thus only for these showers it is possible to assume
that one deals with the collision of only
two nucleons without participation of a bigger nucleus.
If the showers with a small
number (until 3) of thick tracks will be also involved
into the experimental consideration
then generally speaking the change of
shower due to nuclei should be small.
\begin{center}
Table 2
\end{center}
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\hline
$E$&30&40&40&90&130&1000&2000&30000 BeV\\
\hline
$n_{\pi} - n_{\kappa}$&9&18&25&10&18&9&12&21\\
$1/{\rm emp}$& 0.51&0.8&1.0&0.38&0.61&0.16&0.17&0.1
\end{tabular}
\end{center}
\vspace{0.4cm}
However, since the secondary scattering of the emitted mesons
on the atomic nuclei may happen
the determination of the initial energy from the angle distribution
and evaluation of the angle distribution itself become rather uncertain.
The observations of showers that are suitable for comparison with
the theory were done in
works of Teucher \cite{Teucher 1950}, group of Bristol \cite{Camerini 1950},
von Shein and coauthors \cite{Schein 1950},
Pickup and Voyvodic \cite{Pickup 1951} and Hopper, Biswas and
Derby \cite{Hopper 1951}.
If according to the published data one tries to
estimate initial energies from the angle distribution (this is
in some cases very uncertain) then one can gets the numbers of
mesons for the eight
observed showers which were showed in the second line of the Table 2
(these numbers may be slightly different because of neutral mesons).
Hereby it was assumed that
the relation of neutral mesons to the charged ones is $1:2$.
Assuming that two last lines in
the Table 1 are correct one gets some empirical
value of $\gamma$ for each of these showers.
These values are shown in the third line of the Table 2.
First one can see that these numbers of mesons are not well-defined
functions of the initial energy.
The values of $\gamma$ fluctuate very strongly, as
was expected. However they are on average a bit
bigger than it could be expected
from the Table 1.
This can be based on the fact that a small shower is easier
to miss than a big one.
But this may also mean that the estimation in the equation
(28) is still too rough\footnote{{\it Noted during proof-reading}
In the conference in Kopenhagen, June 1952,
Le Couteur showed that average value $\gamma$ in a heavier
matter (as, for
example, in a photo emulsion) should be essentialy bigger than
in the Hydrogen
(in the Table 1 are shown the data for the Hydrogen) because peripherical
collisions happen only for those nucleons which are placed on
the surface of an atomic
nucleus. Then Powell has reported about new experiments which indicated
that the particles called here $\kappa$-mesons decay into the
two groups of mesons
with masses 0.74 and 0.54 BeV respectively.
These mesons have absolutely different properties.
}.
On the other hand the empirical $\gamma$-values themselves in the
Table 2 are still rather uncertain because, for instance,
the part of $\kappa$-mesons is not exactly
known. Also Perkins \cite{Perkins} communicated about relatively
higher $\gamma$-values.
But it is still to wait for more experimental data.
Two showers (Teucher \cite{Teucher 1950} and Hopper, Biswas and
Derby \cite{Hopper 1951}) could be measured
so precisely that the average energy of mesons in
the center of mass frame could be found.
In the first case (40 BeV, approximately 25 mesons)
the average energy of mesons is 0.29 BeV which is comparable with
0.31 BeV from the Table 1. In the second case (1000 BeV, approximately
9 mesons) there is some uncertainty due to a possibility
that some of the observed particles can be $\kappa$-mesons.
This possibility was not taken into account by the authors
(from the Table 1 it could be expected that
among 9 mesons are to expect about 3 $\kappa$-mesons).
If one does not take this into consideration then
the measured average energy of $\pi$-mesons in the center mass frame
is 0.44 BeV which is comparable with 0.50 BeV from the Table 1 also.
Thus both of this measurements
confirm relatively low meson energies from the Table 1.
On the other hand Perkins \cite{Perkins}
takes the value 1.5 BeV as the average
energy of mesons from a group of showers
with initial energy from $10^2$ untill $10^3$ BeV.
It is essentialy higher.
However one needs to take into consideration some uncertainty
in the measurement of the initial energy. Any error of the initial
energy in general increases the average energy of mesons because
it has a minimum value just in the center of mass frame.
Concerning the frequency of $\kappa$-mesons
there exists only one result obtained by the Bristol group
which tells us
that at high energies it is comparable with the frequency
of $\pi$-mesons \cite{Perkins}. However it is still
impossible to determine this ratio from the theory (in the Table 1 it
was taken $g_{\kappa}/g_{\pi}=2$ just ad-hoc).
Concerning the angular distribution it is observed
that in the center of mass frame the angular distribution is almost
isotropic for showers of slow energy. Meanwhile
for showers of a higher energy
one can clearly observe the accumulation of events in the forward and
backward directions.
It exactly corresponds to the picture from the item Ic.
Actually the mesons at high energy seem to be distributed
always anisotropically,
in particular, the $\kappa$-mesons (Perkins \cite{Perkins}).
And also the degree of the anisotropy corresponds to the theoretical estimation.
In general, the impression is that for the ``strong'' interaction
the formulae from the Section
III satisfactory describe the experimental data. Thus
the interaction of elementary particles at high energy really
belongs to the group of ``strong'' interactions first studied by Born.\\\\
{\it G{\"o}ttingen}, Max-Planck-Institute for Physics
\newpage
\begin{thebibliography}{99}
\bibitem{Born 1933}
Born, M.: Proc. Roy. Soc. Long., Ser. A {\bf 143}, 410 (1933);
Born, M. and L.~Infeld: Proc. Roy. Soc. Long., Ser. A {\bf 144}, 425 (1934);
{\bf 147}, 522 (1934); {\bf 150}, 141 (1935). \\
\bibitem{Camerini 1950}
Camerini, U., P.H. Fowler, W.O. Lock and H. Muirhead; Phil. Mag.,\\
(7) {\bf 41}, 413 (1950)
\bibitem{Fermi 1950}
Fermi, E.: Progr. theor. Phys. {\bf 5}, 570 (1950).--Phys. Rev. {\bf 81},
683 (1951).\\
\bibitem{Heisenberg 1939}
Heisenberg, W.; Z. Physik {\bf 113}, 61 (1939)
\bibitem{Heisenberg 1949}
Heisenberg, W.; Z. Physik {\bf 126}, 519 (1949)
\bibitem{Hopper 1951}
Hopper, V.D., S. Biswas and J.F. Derby; Phys. Rev. {\bf 84}, 457 (1951)
\bibitem{Perkins}
Mr. Perkins has kindly informed us about the results of his recent
work before it's publication.
\bibitem{Pickup 1951}
Pickup, E., and L. Voyvodic; Phys. Rev. {\bf 82}, 293 (1951);
{\bf 84}, 1190 (1951).\\
\bibitem{Schein 1950}
Schein, M., J.J. Lord and J. Fainberg; Phys. Rev. {\bf 80}, 970 (1950);
{\bf 81}, 313 (1951)\\
\bibitem{Schiff 1951}
Schiff, L.J.; Phys. Rev., {\bf 84}, 1 (1951)
\bibitem{Teucher 1950}
Teucher, M.; Naturwiss, {\bf 37}, 260 (1950);
{\bf 39}, 68 (1952).\\
\bibitem{Thirring 1952}
Thirring, W.; L. Naturforsch. {\bf 7a}, 63 (1952)
\end{thebibliography}
\end{document}
%ENCODE December 2002 by Herman Boos and Natalia Sukhikh;
%ENCODED DECEMBER 2002 BY HEB;