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\title{High--energy production of muoniums and anti--muoniums by
polarized electrons and positrons scattering by nuclei with $Z \sim
100$ as a laboratory for test of CPT invariance}

\author{E. A. Choban ~and~ V. A. Ivanova\,\thanks{E--mail:
 ivanov@kph.tuwien.ac.at, Department of Nuclear Physics of State
 Polytechnic University of St. Petersburg}}

\date{\today}

\maketitle

\begin{center}
{\it Department of Theoretical Physics, State Polytechnic University of
St. Petersburg,\\ Polytechnicheskaya 29, 195251 St. Petersburg,
Russian Federation}
\end{center}



\begin{center}
\begin{abstract}
The cross sections for the reactions of muonium(anti--muonium)
production in high energy electron(positron) scattering by nuclei
$e^-(e^+) + Z \to Z + M^0(\bar{M}^0) + \mu^-(\mu^+)$ are calculated in
dependence of an energy and polarization of an initial
electron(positron) and a polarization of a final
$\mu^-(\mu^+)$--meson. Due to coherent phenomenon the cross sections
are proportional to $Z^2$. For $Z \sim 100$ due to the factor $Z^2$
the cross sections are large enough to be measured at energies
available at HERA Collider in DESY. The results are discussed in
connection with a test of CPT invariance.
\end{abstract}
\end{center}


\newpage



\section{Introduction}
\setcounter{equation}{0}

\hspace{0.2in} The Standard Model [1] represents the Lagrangian
approach [2] to the description of strong, electromagnetic and weak
interaction of elementary particles, based on the assumptions of
locality and Lorentz invariance. Due to the L\"uders--Pauli theorem
(or the CPT theorem) [3] locality and Lorentz invariance of the
Lagrangian of a quantum system lead to the invariance of a quantum
system under CPT transformation which contains (i) a charge
conjugation (C), a replacement of all particles by their
anti--particles, (ii) a parity transformation (P), a reflection of
spatial coordinates $(t, \vec{x}\,) \to (t, -\vec{x}\,)$, and (iii) a
time reversal (T), a reflection of time $(t, \vec{x}\,) \to (- t,
\vec{x}\,)$. A simplest consequence of the CPT theorem is the equality
of masses and lifetimes of particles and their anti--particles. At
present these are the most well verified experimentally requirements
of the CPT theorem [4]. Nevertheless, theoretical and experimental
test for CPT invariance is still a well motivated problem of
Elementary particle and Nuclear Physics [5]. This is related to the
development of modern quantum field theories of strings and
superstrings [6], which are more fundamental than the Standard Model
and include it in the low--energy limit. Since string theories deal
with extended non--local objects, the L\"uders--Pauli theorem is not
valid for these theories. A direct consequence of this can be a
violation of CPT invariance for high energy reactions of elementary
particles and nuclei.

The problem of a test of CPT and Lorentz invariance has been recently
discussed by Kosteleck$\acute{\rm y}$ with co--workers [7]. They
suggested to check CPT and Lorentz invariance analysing a microwave
spectroscopy of muonium $M^0$ [8,9]. Muonium $M^0$ is a leptonic
hydrogenlike bound state of a positively charged muon $\mu^+$ and
electron $e^-$.  It was discovered in 1960 through the observation of
its characteristic Larmor precession in a magnetic field [9]. The mean
lifetime of muonium $\tau_{M^0}$ is approximately equal to the
lifetime of a positively charged muon $\tau_{M^0} \simeq \tau_{\mu^+}
= 2.197\times 10^{-6}\,{\rm s}$ [3]. Due to absence of strong
interactions muonium is an ideal system (i) for determining of the
properties of muons, (ii) for testing of quantum electrodynamics [10],
and (iii) for searching for effects of unknown interactions in the
electron--muon bound state [11]. Anti--muonium $\bar{M}^0$ is a
leptonic analog of anti--hydrogen. It is a bound state of a negatively
charged muon $\mu^-$ and positron $e^+$.

A hydrogenlike structure of muonium allows to use atomic notations for
the classification of its quantum states. For example, ${^{2S +
1}}L_J$ corresponds to the quantum state of muonium (or
anti--muonium) with a total angular momentum (or a total spin) $J$,
an angular momentum $L$ and a spin $S$.

The use of muoniums $M^0$ and anti--muoniums $\bar{M}^0$\,
\footnote{Anti--muonium $\bar{M}^0$ is a bound state of a negatively
charged muon $\mu^-$ and a positron $e^+$. It is a leptonic analogy of
the anti--hydrogen.} as a laboratory for a test of CPT invariance has
been recently suggested by Choban and Kazakov [12]. In their approach
muoniums and anti--muoniums are produced with a total angular momentum
$J = 0$ in the reactions $e^- + Z \to Z + M^0 + \mu^-$ and $e^+ + Z
\to Z + \bar{M}^0 + \mu^+$ of high energy scattering of electrons and
positrons by nuclei with a number of protons $Z$. According to atomic
classification muonium (or anti--muonium) with a total angular
momentum $J = 0$ can be in two bound states: (i) a ground $1s$ state
${^1}{\rm S}_0$ with $L = S = 0$ and (ii) an excited $2p$ state
${^3}{\rm P}_0$ with $L = S = 1$.

Due to principle of superposition muonium and anti--muonium should be
produced in the reactions $e^- + Z \to Z + M^0 + \mu^-$ and $e^+ + Z
\to Z + \bar{M}^0 + \mu^+$ in both states ${^1}{\rm S}_1$ and
${^3}{\rm P}_1$. The interference of these states should lead to
time--oscillations of a probability of muonium (anti--muonium)
detected at a moment $t$. A comparison of time--oscillations of
probabilities of the detected muonium and anti--muonium should testify
whether CPT invariance conserved or not. This is Choban--Kazakov's
idea of a test of CPT invariance in the high--energy reactions $e^- + Z
\to Z + M^0 + \mu^-$ and $e^+ + Z \to Z + \bar{M}^0 + \mu^+$. In terms
of formulas it can be represented as follows.

Let the wave function of muonium produced in the reaction $e^- + Z \to
Z + M^0 + \mu^-$ be defined by
%
\begin{eqnarray}\label{label1.1}
\Psi_{M^0}(t,\vec{x}\,) =\sqrt{\frac{m_{M^0}}{|\vec{k}\,|}}\,
e^{\textstyle i\,\vec{k}\cdot \vec{x} - i\,Et}\,\Psi_{M^0}(t),
\end{eqnarray}
%
where $E$ and $\vec{k}$
are an energy and 3--momentum of muonium, $m_{M^0}$ is a mass of
muonium. Note, that the energy $E$ does not contain the contributions
of the binding energies $E_{1s}$ and $E_{2p}$ of the bound $1s$ and
$2p$ states. The wave function $\Psi_{M^0}(t)$ can be described by
%
\begin{eqnarray}\label{label1.2}
\Psi_{M^0}(t) = C_{1s}\,\exp\Big( -i\,\frac{m_{M^0}}{E}\,E_{1s}t\Big)
+ C_{2p}\,\exp\Big( -i\,\frac{m_{M^0}}{E}\,E_{2p}t\Big).
\end{eqnarray}
%
The coefficients $C_{1s}$ and $C_{2p}$ describe the contributions of
the $1s$ and $2p$ states, respectively, and normalized by $|C_{1s} +
C_{2p}|^2 = 1$.

Introducing a parameter $\varepsilon = |C_{2p}|^2/|C_{1s} + C_{2p}|^2
= |C_{2p}|^2$, related to a fraction of the excited $2p$ state in the
wave function of muonium $\Psi_{M^0}(t,\vec{x}\,)$ [12], the
probability to find muonium at the moment $t$ can be given by
%
\begin{eqnarray}\label{label1.3}
P_{M^0}(t) = P_{M^0}(0)\,[1 - 4\sqrt{\varepsilon}\,(1 -
\sqrt{\varepsilon})\sin^2(\Omega t)],
\end{eqnarray}
%
where $\Omega = m_{\mu}(E_{2p} - E_{1s})/2E = 5.103\times
10^{-6}\,(m_{\mu}/E)\,{\rm MeV}$ [7]\,\footnote{The account for a
constant relative phase $2\varphi$ of coefficients $C_{1s}$ and
$C_{2p}$ changes the probability (\ref{label1.3}) as follows
$P_{M^0}(t) = P_{M^0}(0)\,[1 - 4\sqrt{\varepsilon}\,(\sqrt{1 -
\varepsilon\,\sin^2\varphi}-\sqrt{\varepsilon}\,\cos\varphi)\,
\sin(\Omega t + \varphi)\sin(\Omega t)]$}.

It is seen that the probability $P_{M^0}(t)$ is an
oscillating function.  A period of oscillations $T_{M^0}$ is
determined by
%
\begin{eqnarray}\label{label1.4}
T_{M^0} = \frac{2\pi}{\Omega} =\frac{4\pi}{E_{2p} -
E_{1s}}\,\Big(\frac{E}{m_{\mu}}\Big) = 1.232\times
10^6\,\Big(\frac{E}{m_{\mu}}\Big)\,{\rm MeV^{-1}}.
\end{eqnarray}
%
In order to get $T_{M^0}$ in seconds we have to multiply the r.h.s. of
(\ref{label1.4}) by $\hbar = 6.582\times 10^{-22}\,{\rm MeV\,s}$
[7]. This yields $T_{M^0}= 8.106\times 10^{-16}\,(E/m_{\mu})\,{\rm
s}$.  The period of oscillations $T_{M^0}$ should be compared with the
lifetime of muonium in the laboratory frame $t_{M^0}$ which is related
to the mean lifetime $\tau_{M^0}$ by the relativistic relation
%
\begin{eqnarray}\label{label1.5}
t_{M^0} = \Big(\frac{E}{m_{\mu}}\Big)\,\tau_{M^0}.
\end{eqnarray}
%
Taking into account that $\tau_{M^0} \simeq 2.197\times 10^{-6}\,{\rm
s} $ we are able to estimate the number of oscillations $\nu_{M^0}$:
%
\begin{eqnarray}\label{label1.6}
\nu_{M^0} = \frac{t_{M^0}}{T_{M^0}} \simeq 2.710\times 10^{\,9}.
\end{eqnarray}
%
The analogous expression can be written down for the probability
$P_{\bar{M}^0}(t)$ to detect anti--muonium at moment t with parameters
$\bar{\varepsilon}$ and $\bar{\Omega}$. The result reads
%
\begin{eqnarray}\label{label1.7}
P_{\bar{M}^0}(t) = P_{\bar{M}^0}(0)\,[1 -
4\sqrt{\bar{\varepsilon}}\,(1 -
\sqrt{\bar{\varepsilon}})\sin^2(\bar{\Omega} t)].
\end{eqnarray}
%
From a comparison of the probabilities $P_{M^0}(t)$ and
$P_{\bar{M}^0}(t)$\,\footnote{The account for a constant relative
phase $2\bar{\varphi}$ of coefficients $\bar{C}_{1s}$ and
$\bar{C}_{2p}$ changes the probability (\ref{label1.7}) as follows
$P_{\bar{M}^0}(t) = P_{\bar{M}^0}(0)\,[1 -
4\sqrt{\bar{\varepsilon}}\,(\sqrt{1 -
\bar{\varepsilon}\,\sin^2\varphi}-\sqrt{\bar{\varepsilon}}\,
\cos\bar{\varphi})\, \sin(\bar{\Omega} t +
\bar{\varphi})\sin(\bar{\Omega} t)]$.} one could make a certain
conclusion concerning a violation of CPT invariance for bound $M^0 =
(e^-,\mu^+)$ and $\bar{M}^0 =(e^+, \mu^-)$ states. The coincidence of
the probabilities $P_{M^0}(t)$ and $P_{\bar{M}^0}(t)$,
i.e. $P_{M^0}(t) = P_{\bar{M}^0}(t)$, should testify CPT invariance
for the production of bound states, muonium and anti--muonium, in the
high energy reactions $e^- + Z \to Z + M^0 + \mu^-$ and $e^+ + Z \to Z
+ \bar{M}^0 + \mu^+$. Any distinction of probabilities $P_{M^0}(t)
\neq P_{\bar{M}^0}(t)$ should imply a violation of CPT invariance. Of
course, this is only a qualitative test.

A practical realization of an experimental test of CPT invariance in
high--energy reactions $e^- + Z \to Z + M^0 + \mu^-$ and $e^+ + Z \to
Z + \bar{M}^0 + \mu^+$ is related to statistics of favourable outcomes
which can be detected during a certain time of observation, i.e. the
number of events $N$ determined by
%
\begin{eqnarray}\label{label1.8}
N = \sigma L T,
\end{eqnarray}
%
where $\sigma$ is the cross section of the reaction under
consideration, $L$ is a luminosity of the accelerator and $T$ is a
time of observation.

Nowadays the HERA Collider at DESY operates $27.5\,{\rm GeV}$ electron
and positron beams with luminosities $L_{e^-} = (15 - 17)\times
10^{30}\,{\rm cm^{-2}\,s^{-1}} = (15 - 17)\,{\rm pb^{-1}}$(H1 $-$
ZEUS) and $L_{e^+} = (65 - 68) \times 10^{30}\,{\rm cm^{-2}\,s^{-1}} =
(65 - 68)\,{\rm pb^{-1}}$ (H1 $-$ ZEUS), respectively [13]. For these
luminosities the number of events detected during one year for the
production of muonium and anti--muonium are equal to $N_{M^0} =
0.5\,\sigma_{M^0}$ and $N_{\bar{M}^0} = 2.1\,\sigma_{\bar{M}^0}$,
where cross sections $\sigma_{M^0}$ and $\sigma_{\bar{M}^0}$ are
measured in $1\,{\rm fb} = 10^{-39}\,{\rm cm^2}$.

Thus, the problem of an experimental realization of a test of CPT
invariance suggested by Choban and Kazakov [12] is related to (i) the
values of the cross sections for the reactions $e^- + Z \to Z + X^0 +
\mu^-$ and $e^+ + Z \to Z + \bar{X}^0 + \mu^+$, defining total number
of favourable events and (ii) a distinct signal that in the reactions
$e^- + Z \to Z + X^0 + \mu^-$ and $e^+ + Z \to Z + \bar{X}^0 + \mu^+$
the states $X^0$ and $\bar{X}^0$ should be identified with muonium
$M^0$ and $\bar{M}^0$ anti--muonium, i.e. $X^0 = M^0$ and $\bar{X}^0 =
\bar{M}^0$, respectively.

It is well--known that a more detailed information about nuclear
reactions can be obtained investigating polarizations of coupled
particles. Therefore, in this paper we focus on the calculation of the
cross sections for the high--energy reactions $e^- + Z \to Z + M^0 +
\mu^-$ and $e^+ + Z \to Z + \bar{M}^0 + \mu^+$ in dependence on
polarizations of initial electron and positron and final muons $\mu^-$
and $\mu^+$. Following [14] we denote these reactions as
$\vec{e}^{\,\,-} + Z \to Z + M^0 + \vec{\mu}^{\,\,-} $ and
$\vec{e}^{\,\,+} + Z \to Z + \bar{M}^0 + \vec{\mu}^{\,\,+}$. We
suppose that a dependence of polarizations of final muons relative to
polarizations of initial electrons and positrons should provide a
necessary distinct signal confirming the production exactly muonium
and anti--muonium in the reactions $\vec{e}^{\,\,-} + Z \to Z + X^0 +
\vec{\mu}^{\,\,-} $ and $\vec{e}^{\,\,+} + Z \to Z + \bar{X}^0 +
\vec{\mu}^{\,\,+}$.

The paper is organized as follows. In section 2 we calculate the
energy spectrum of the final muon and the cross section for the
reaction $\vec{e}^{\,\,-} + Z \to Z + M^0 + \vec{\mu}^{\,\,-} $. Since
the cross sections are needed for the prediction of the number of
events $N$, they can be calculated assuming CPT invariance. Therefore,
the cross section for the reaction $\vec{e}^{\,\,-} + Z \to Z + M^0 +
\vec{\mu}^{\,\,-} $ amounts to the cross section for the reaction
$\vec{e}^{\,\,+} + Z \to Z + \bar{M}^0 + \vec{\mu}^{\,\,+}$,
i.e. $\sigma_{M^0}(\vec{e}^{\,\,-} Z) = \sigma_{\bar{M}^0}(
\vec{e}^{\,\,+} Z)$. In the Conclusion we discuss the obtained
results.

\section{Cross sections for reactions $\vec{e}^{\,\,-} + Z 
\to Z + M^0 + \vec{\mu}^{\,\,-}$ and $\vec{e}^{\,\,+} + Z 
\to Z + \bar{M}^0 + \vec{\mu}^{\,\,+}$} 
\setcounter{equation}{0}

\hspace{0.2in} Feynman diagrams describing the amplitude of the
reaction $\vec{e}^{\,\,-} + Z \to Z + M^0 + \vec{\mu}^{\,\,-} $ are
depicted in Fig.1. The amplitude of the reaction $\vec{e}^{\,\,-} + Z
\to Z + M^0 + \vec{\mu}^{\,\,-} $ has been calculated in Ref.[12] and
reads
\begin{figure}

\begin{center}
 \includegraphics[width=340pt,height=141pt]{figure1}

 Fig. 1.

 Feynman diagrams of the amplitude of the reaction $\vec{e}^{\,\,-} +
 Z \to Z + M^0 + \vec{\mu}^{\,\,-} $.
\end{center}

\end{figure}
%
\begin{eqnarray}\label{label2.1}
\hspace{-0.3in}M(\vec{e}^{\,\,-}(p_1)Z(p_2) \to Z(p\,'_2)
M^0(k)\vec{\mu}^{\,\,-}(p\,'_1)) = \frac{\alpha^2}{q^2}\,
\frac{16\pi^2}{m_e}\,\frac{\Psi_{1s}(0)}{m^{3/2}_{\mu}}\,
\frac{{\ell}^{\mu}\, L_{\mu}}{(q^2 - 2q\cdot k)},
\end{eqnarray}
%
where ${\ell}^{\mu}$ is the electromagnetic current of a nucleus and
$L_{\mu}$ denotes the leptonic current
%
\begin{eqnarray}\label{label2.2}
L_{\mu} = \bar{u}(p\,'_1,\sigma\,'_1)\,\gamma_5\,(\hat{q}\,p\,'_{1\mu}
- q\cdot p\,'_1\,\gamma_{\mu})u(p_1,\sigma_1),
\end{eqnarray}
%
where $u(p_1,\sigma_1)$ and $\bar{u}(p\,'_1,\sigma\,'_1)$ are
bispinorial wave functions of an initial electron and final muon
$\mu^-$, $\Psi_{1s}(0) = 1/\sqrt{\pi a^3_0}$ is the wave function of
the muonium in the ground state, $a_0 = 1/m_e\alpha$ is the Bohr
radius, $\alpha = e^2/4\pi = 1/137.036$ is the fine structure
constant. 

We would like to emphasize that the leptonic current $L_{\mu}$ is
calculated in the ultra--relativistic limit, when masses of leptons
are set zero. According to [12] this corresponds to the kinematical
region, where the squared invariant mass of the pair $M^0\mu^-$,
$\omega^2 = (p\,'_1 + k)^2$, is much greater than the squared mass of
the $\mu^-$--meson $m^2_{\mu}$, i.e. $\omega^2 \gg m^2_{\mu}$. In this
kinematical region muonium with a total spin $J = 0$ behaves like a
massless neutral scalar point--like particle.

The cross section for the reaction $\vec{e}^{\,\,-} + Z \to Z + M^0 +
\vec{\mu}^{\,\,-}$ is defined by
%
\begin{eqnarray}\label{label2.3}
\hspace{-0.3in}\sigma(E_1) = \frac{\alpha^7}{4\pi^2}\frac{m_e}{
m^3_{\mu}}\frac{1}{m_ZE_1}\int
\frac{T_{\mu\nu}R_{\mu\nu}}{q^4(p_1\cdot
p\,'_1)^2}\,\delta^{(4)}(p\,'_2 + p\,'_1 + k - p_2 -
p_1)\,\frac{d^3k}{E}\frac{d^3p\,'_1}{E\,'_1}\frac{d^3p\,'_2}{E\,'_2},
\end{eqnarray}
%
where $E_1$ is the energy of the initial electron in the laboratory
frame coinciding with the rest frame of a target nucleus $p_{2\mu} =
(m_Z, \vec{0}\,)$, then $E$, $E\,'_1$ and $E\,'_2$ are the energies of
muonium, a $\mu^-$--meson and a final nucleus, respectively. The
tensors $R_{\mu\nu}$ and $T_{\mu\nu}$ are determined by
%
\begin{eqnarray}\label{label2.4}
\hspace{-0.3in}&&R_{\mu\nu} = \frac{1}{4}\,{\rm Sp}\{(\hat{p}_2 +
m_Z){\ell}^{\dagger}_{\mu}(\hat{p}\,'_2 + m_Z){\ell}_{\nu}\} =
F^2_{1Z}(q^2)\, \Big[2\,p_{2\mu}p_{2\nu} - (p_{2\mu}q_{\nu} +
p_{2\nu}q_{\mu}) + \frac{1}{2}\,q^2\,g_{\mu\nu}\Big]\nonumber\\
\hspace{-0.3in}&&+F^2_{2Z}(q^2)\,\Big[2\,q^2\,m^2_Z\,g_{\mu\nu} +
q^2\,(p_{2\mu}q_{\nu} + p_{2\nu}q_{\mu}) -
q_{\mu}q_{\nu}\,\Big(\frac{1}{2}\,q^2 + 2\,m^2_Z\Big) -
2\,q^2\,p_{2\mu}p_{2\nu}\Big].
\end{eqnarray}
%
and 
%
\begin{eqnarray}\label{label2.5}
\hspace{-0.5in}T_{\mu\nu} &=& \frac{1}{4}\,{\rm Sp}\{(\hat{p}_1 -
\gamma_5\hat{w}_1)L^{\dagger}_{\mu}(\hat{p}\,'_1 -
\gamma_5\hat{w}\,'_1)L_{\nu}\} =\nonumber\\
\hspace{-0.5in}&=& \frac{1}{4}\,{\rm Sp}\{(\hat{p}_1 -
\gamma_5\hat{w}_1)\gamma_5(q\cdot p\,'_1\,\gamma_{\mu} -
\hat{q}\,p\,'_{1\mu})(\hat{p}\,'_1 -
\gamma_5\hat{w}\,'_1)\gamma_5(q\cdot p\,'_1\,\gamma_{\nu} -
\hat{q}\,p\,'_{1\nu})\},
\end{eqnarray}
%
where $F_{1Z}(q^2)$ and $F_{2Z}(q^2)$ are form factors of a nucleus
with a number of protons $Z$.

The polarization matrices $(\hat{p}_1 - \gamma_5\hat{w}_1)$ and
$(\hat{p}\,'_1 - \gamma_5\hat{w}\,'_1)$ are obtained in the zero--mass
limit from the standard polarization matrices $(\hat{p}_1 + m_{\rm
e})(1 - \gamma_5\hat{a})$ and $(\hat{p}\,'_1 + m_{\mu})(1 -
\gamma_5\hat{b})$ [14], where $a_{\mu}$ and $b_{\mu}$, 4--vectors of
polarization of the initial electron and the final muon, are defined
by
%
\begin{eqnarray}\label{label2.6}
a_{\mu} &=& \Big(\frac{\vec{p}_1\cdot \vec{\xi}_1}{m_e}, \vec{\xi}_1
+\frac{\vec{p}_1(\vec{p}_1\cdot \vec{\xi}_1)}{m_e(E_1 +
m_e)}\Big),\nonumber\\ b_{\mu} &=& \Big(\frac{\vec{p}\,'_1\cdot
\vec{\xi}\,'_1}{m_{\mu}}, \vec{\xi}\,'_1
+\frac{\vec{p}\,'_1(\vec{p}\,'_1\cdot \vec{\xi}\,'_1)}{m_{\mu}(E\,'_1
+ m_{\mu})}\Big).
\end{eqnarray}
%
The 4--vectors of polarization $a_{\mu}$ and $b_{\mu}$ are normalized
by $a_{\mu}a_{\mu} = a^2_0 - \vec{a}^{\,2} = -1$ and $b_{\mu}b_{\mu} =
b^2_0 - \vec{b}^{\,2} = -1$.  In turn, the 3--vectors of polarization
$\vec{\xi}_1$ and $\vec{\xi}\,'_1$ are normalized by
$\vec{\xi}^{\,2}_1 = \vec{\xi}\,'^{\,2}_1 = 1$. Recall that $p_1\cdot
a = p\,'_1\cdot b = 0$.

According to definitions (\ref{label2.6}) the 4--vectors $w_{1\mu}$
and $w\,'_{1\mu}$ are equal to 
%
\begin{eqnarray}\label{label2.7}
w_{1\mu}&=& (\vec{p}_1\cdot \vec{\xi}_1, \vec{n}_1(\vec{p}_1\cdot
\vec{\xi}_1)) = (\vec{n}_1\cdot \vec{\xi}_1)\,p_{1\mu},\nonumber\\
w\,'_{1\mu} &=& (\vec{p}\,'_1\cdot \vec{\xi}\,'_1,
\vec{n}\,'_1(\vec{p}\,'_1\cdot \vec{\xi}\,'_1)) = (\vec{n}\,'_1\cdot
\vec{\xi}\,'_1)\,p\,'_{1\mu},
\end{eqnarray}
%
where $\vec{n}_1 = \vec{p}_1/E_1$ and $\vec{n}\,'_1 =
\vec{p}\,'_1/E\,'_1$ and $p_1\cdot w_1 = p\,'_1\cdot w\,'_1 = 0$ due
to $p^2_1 = p\,'\,^2_1 = 0$. The analytical expression of $T_{\mu\nu}$
is given by
%
\begin{eqnarray}\label{label2.8}
\hspace{-0.3in}T_{\mu\nu}&=&-\,2\,[1 + (\vec{n}_1\cdot
\vec{\xi}_1)(\vec{n}\,'_1\cdot \vec{\xi}\,'_1)]\nonumber\\
\hspace{-0.3in}&&\times\,(p_1\cdot p\,'_1) [(q\cdot
p\,'_1)^2g_{\mu\nu} - (q\cdot p\,'_1)(p\,'_{1\mu}q_{\nu} +
p\,'_{1\nu}q_{\mu}) + q^2p\,'_{1\mu}p\,'_{1\nu}].
\end{eqnarray}
%
Due to conservation of electric charge the tensors $T_{\mu\nu}$ and
$R_{\mu\nu}$ are gauge invariant
%
\begin{eqnarray}\label{label2.9}
q_{\mu}T_{\mu\nu} &=& T_{\mu\nu}q_{\nu} = 0,\nonumber\\
q_{\mu}R_{\mu\nu} &=& R_{\mu\nu}q_{\nu} = 0.
\end{eqnarray}
%
The cross section for the reaction under consideration is then defined
by
%
\begin{eqnarray}\label{label2.10}
\hspace{-0.3in}&&\sigma(E_1) =
\frac{\alpha^7}{\pi^2}\,\frac{m_e}{m^3_{\mu}}\frac{m_Z}{E_1}\int
\frac{(-1)}{q^4(p_1\cdot p\,'_1)}\,[1 + (\vec{n}_1\cdot
\vec{\xi}_1)(\vec{n}\,'_1\cdot \vec{\xi}\,'_1)]\Big\{(F^2_{1Z}(q^2)
-q^2F^2_{2Z}(q^2)) (q\cdot p\,'_1)^2\nonumber\\
\hspace{-0.3in}&& +
\frac{q^2}{m^2_Z}\Big[(F^2_{1Z}(q^2)-q^2F^2_{2Z}(q^2))\Big((p_2\cdot
p\,'_1)^2 - (q\cdot p\,'_1)(p_2\cdot p\,'_1)\Big) +
\frac{1}{2}(F^2_{1Z}(q^2) + 4m^2_ZF^2_{2Z}(q^2))\nonumber\\
\hspace{-0.3in}&& \times\,(q\cdot p\,'_1)^2\Big\}\delta^{(4)}(p\,'_2 +
p\,'_1 + k - p_2 -
p_1)\,\frac{d^3k}{E}\frac{d^3p\,'_1}{E\,'_1}\frac{d^3p\,'_2}{E\,'_2},
\end{eqnarray}
%
The integration over the phase volume of the final state $ZM^0\mu^-$
we suggest to carry out in the non--relativistic limit of motion of
a final nucleus [15]. In this approximation the 4--momentum of the
final nucleus is equal to $p\,'_{2\mu} = (m_Z +
\vec{q}^{\;2}/2m_Z,-\vec{q}\,) = (m_Z + T_2, -\vec{q}\,)$, then the
transferred 4--momentum $q_{\mu} = (- T_2, \vec{q}\,)$ and $q^2 =
-\vec{q}^{\;2}$.

In the non--relativistic limit of motion of a final nucleus the cross
section (\ref{label2.10}) reduces to the form
%
\begin{eqnarray}\label{label2.11}
\hspace{-0.5in}&&\sigma(E_1) =
Z^2\,\frac{\alpha^7}{\pi^2}\,\frac{m_e}{m^3_{\mu}}\frac{1}{E_1}\int
\frac{1}{E_1E\,'_1 - \vec{p}_1\cdot\vec{p}\,'_1}\,\Big[1 +
(\vec{n}_1\cdot \vec{\xi}_1)\Big(\frac{\vec{p}\,'_1\cdot
\vec{\xi}\,'_1}{E\,'_1}\Big)\Big]\nonumber\\
\hspace{-0.5in}&& \times\,\Big(E_1{'\,^2} - \frac{(\vec{q}\cdot
\vec{p}\,'_1 )^2}{\vec{q}^{\;2}}\Big)\delta(E\,'_1 + E + T_2 -
E_1)\,\delta^{(3)}(\vec{p}\,'_1 + \vec{k} - \vec{q} -
\vec{p}_1)\,\frac{d^3k}{E}\frac{d^3p\,'_1}{E\,'_1}
\frac{d^3q}{\vec{q}^{\;2}},
\end{eqnarray}
%
where we have taken into account that $F_{1Z}(0)=Z$ and that the main
contribution comes from transferred momenta $\vec{q}^{\;2}$
comeasurable with zero. The former corresponds to the
Weizs\"acker--Williams approximation [16--21].

For simplification of the calculation of the phase volume we neglect
the contribution of a kinetic energy of a final nucleus, which is
small compared with typical transferred energies of coupled
leptons. Integrating over $\vec{k}$, a 3--momentum of muonium, we get
%
\begin{eqnarray}\label{label2.12}
\hspace{-0.5in}&&\sigma(E_1) =
Z^2\,\frac{\alpha^7}{\pi^2}\,\frac{m_e}{m^3_{\mu}}\frac{1}{E_1}\int
\frac{E\,'_1}{E_1E\,'_1 - \vec{p}_1\cdot\vec{p}\,'_1 }\,\Big[1 +
(\vec{n}_1\cdot \vec{\xi}_1)\Big(\frac{\vec{p}\,'_1\cdot
\vec{\xi}\,'_1}{E\,'_1}\Big)\Big]\nonumber\\
\hspace{-0.5in}&& \times\,\Big(1 - \frac{(\vec{q}\cdot \vec{p}\,'_1
)^2}{\vec{q}^{\;2}E_1{'\,^2}}\Big)\,\delta(E_1 - E\,'_1 -
|\vec{p}\,'_1 - \vec{p}_1 -\vec{q}\,|
)\,\frac{d^3p\,'_1}{|\vec{p}\,'_1 - \vec{p}_1
-\vec{q}\,|}\,\frac{d^3q}{\vec{q}^{\;2}}=\nonumber\\ 
\hspace{-0.5in}&& =
Z^2\,\frac{\alpha^7}{\pi^2}\,\frac{m_e}{m^3_{\mu}}\frac{1}{E_1}\int
\Big[1 + (\vec{n}_1\cdot \vec{\xi}_1)\Big(\frac{\vec{p}\,'_1\cdot
\vec{\xi}\,'_1}{E\,'_1}\Big)\Big]\,\frac{E\,'_1\,
I(\vec{p}_1,\vec{p}\,'_1)}{E_1E\,'_1 - \vec{p}_1\cdot\vec{p}\,'_1
}\,d^3p\,'_1,
\end{eqnarray}
%
where we have denoted
%
\begin{eqnarray}\label{label2.13}
I(\vec{p}_1,\vec{p}\,'_1) = \int\Big(1 - \frac{(\vec{q}\cdot
\vec{p}\,'_1 )^2}{\vec{q}^{\;2}E_1{'\,^2}}\Big)\,\delta(E_1 - E\,'_1 -
|\vec{p}\,'_1 - \vec{p}_1 -\vec{q}\,| )\,\frac{1}{|\vec{p}\,'_1 -
\vec{p}_1 -\vec{q}\,|}\,\frac{d^3q}{\vec{q}^{\;2}}.
\end{eqnarray}
%
The integration over $\vec{q}$ we carry out assuming that
$|\vec{p}\,'_1 - \vec{p}_1| \gg |\vec{q}\,|$ that is valid for the
Weizs\"acker--Williams approximation. Using a vector $\vec{z} =
\vec{q}/|\vec{p}\,'_1 - \vec{p}_1|$ we obtain
%
\begin{eqnarray}\label{label2.14}
I(\vec{p}_1,\vec{p}\,'_1) = \int\Big(1 - \frac{(\vec{z}\cdot
\vec{p}\,'_1 )^2}{\vec{z}^{\;2}E_1{'\,^2}}\Big)\,\delta\Big(E_1 -
E\,'_1 - |\vec{p}\,'_1 - \vec{p}_1| +(\vec{p}\,'_1 - \vec{p}_1)\cdot
\vec{z}\, \Big)\,\frac{d^3z}{\vec{z}^{\;2}}.
\end{eqnarray}
%
Now it is convenient to introduce new variables $x = E\,'_1/E_1$,
$\vec{n}\,'_1 = \vec{p}\,'_1/E\,'_1$ and $\vec{n}_1 =
\vec{p}_1/E_1$. In these variables the function
$I(\vec{p}_1,\vec{p}\,'_1)$ reads
%
\begin{eqnarray}\label{label2.15}
I(\vec{p}_1,\vec{p}\,'_1) = \frac{1}{E_1}\int\Big(1 -
\frac{(\vec{z}\cdot \vec{n}\,'_1
)^2}{\vec{z}^{\;2}}\Big)\,\delta\Big(1 - x - |x\vec{n}\,'_1 -
\vec{n}_1| +(x\vec{n}\,'_1 - \vec{n}_1)\cdot \vec{z}\,
\Big)\,\frac{d^3z}{\vec{z}^{\;2}}.
\end{eqnarray}
%
The next step in the integration over $\vec{z}$ is to rewrite the
integral in the following form
%
\begin{eqnarray}\label{label2.16}
\hspace{-0.7in}&&I(\vec{p}_1,\vec{p}\,'_1) = \nonumber\\
\hspace{-0.7in}&&=\frac{1}{\pi E_1}{{\cal
R}e}\int\limits^{\infty}_0d\lambda\,e^{\textstyle i\lambda(1-x -
|x\vec{n}\,'_1 - \vec{n}_1|)}\int\Big(1 +
\frac{1}{\lambda^2\vec{z}^{\;2}}\frac{\partial^2}{\partial
x^2}\Big)\,e^{\textstyle i\lambda(x\vec{n}\,'_1 - \vec{n}_1)\cdot
\vec{z}}\,\frac{d^3z}{\vec{z}^{\;2}}.
\end{eqnarray}
%
Since the integrals over $\vec{z}$ are equal to
%
\begin{eqnarray}\label{label2.17}
\hspace{-0.3in}&&\int e^{\textstyle i\lambda(x\vec{n}\,'_1 -
\vec{n}_1)\cdot \vec{z}}\;\frac{d^3z}{\vec{z}^{\;2}} =
\frac{4\pi}{\lambda|x\vec{n}\,'_1 -
\vec{n}_1|}\int\limits^{\infty}_0\frac{\sin z}{z}\,dz
=\frac{4\pi}{\lambda|x\vec{n}\,'_1 - \vec{n}_1|}\lim_{\alpha \to
1}\int\limits^{\infty}_0\frac{\sin z}{z^{\alpha}}\,dz =\nonumber\\
\hspace{-0.3in}&&= \frac{4\pi}{\lambda|x\vec{n}\,'_1 -
\vec{n}_1|}\lim_{\alpha \to 1}{{\cal
I}m}\int\limits^{\infty}_0dz\,e^{\textstyle i z}z^{-\alpha} =
\frac{4\pi}{\lambda|x\vec{n}\,'_1 - \vec{n}_1|}\lim_{\alpha \to
1}{{\cal I}m}\,\frac{\Gamma(1-\alpha)}{(-i)^{1-\alpha}} =\nonumber\\
\hspace{-0.3in}&&= \frac{4\pi}{\lambda|x\vec{n}\,'_1 -
\vec{n}_1|}\,\lim_{\alpha \to
1}\Gamma(2-\alpha)\,\frac{\displaystyle\sin\Big(\frac{\pi}{2}
(1-\alpha)\Big)}{1-\alpha} = \frac{2\pi^2}{\lambda|x\vec{n}\,'_1 -
\vec{n}_1|}
\end{eqnarray}
%
and 
%
\begin{eqnarray}\label{label2.18}
\hspace{-0.3in}&&\int e^{\textstyle i \lambda(x\vec{n}\,'_1 -
\vec{n}_1)\cdot\vec{z} }\;\frac{d^3z}{\vec{z}^{\;4}} = \nonumber\\
\hspace{-0.3in}&&= 4\pi \lambda|x\vec{n}\,'_1 - \vec{n}_1|
\int\limits^{\infty}_0\frac{\sin z}{z^3}dz = 4\pi
\lambda|x\vec{n}\,'_1 - \vec{n}_1|\lim_{\alpha \to 3}{{\cal
I}m}\int\limits^{\infty}_0dz\,e^{\textstyle i z}z^{-\alpha}
=\nonumber\\ \hspace{-0.3in}&&= 4\pi \lambda|x\vec{n}\,'_1 -
\vec{n}_1|\lim_{\alpha \to 3}{{\cal
I}m}\,\frac{\Gamma(1-\alpha)}{(-i)^{1-\alpha}} = 4\pi
\lambda|x\vec{n}\,'_1 - \vec{n}_1|\lim_{\alpha \to
3}\Gamma(1-\alpha)\sin\Big(\frac{\pi}{2}(1-\alpha)\Big) =\nonumber\\
\hspace{-0.3in}&&= 4\pi \lambda|x\vec{n}\,'_1 - \vec{n}_1|\lim_{\alpha
\to 3}\Gamma(4-\alpha)\frac{\displaystyle
\sin\Big(\frac{\pi}{2}(1-\alpha)\Big)}{(1-\alpha)(2-\alpha)(3 -
\alpha)}=\nonumber\\
\hspace{-0.3in}&&= - 4\pi \lambda|x\vec{n}\,'_1 -
\vec{n}_1|\lim_{\alpha \to 3}\Gamma(4-\alpha)\frac{\displaystyle
\sin\Big(\frac{\pi}{2}(3-\alpha)\Big)}{(1-\alpha)(2-\alpha)(3 -
\alpha)} = - \pi^2 \lambda|x\vec{n}\,'_1 - \vec{n}_1|,
\end{eqnarray}
%
the function $I(\vec{p}_1,\vec{p}\,'_1)$ is defined by the integral
over $\lambda$
%
\begin{eqnarray}\label{label2.19}
I(\vec{p}_1,\vec{p}\,'_1) =
\frac{\pi}{E_1}\Big(\frac{1}{|x\vec{n}\,'_1 - \vec{n}_1|} + \frac{(x
-\vec{n}\,'_1\cdot \vec{n}_1)^2 }{|x\vec{n}\,'_1 -
\vec{n}_1|^3}\Big)\int\limits^{\infty}_0\frac{d\lambda}{\lambda}\,
\cos(\lambda(1-x - |x\vec{n}\,'_1 - \vec{n}_1|)).
\end{eqnarray}
%
The integral over $\lambda$ is divergent. However, it can be
regularized by following the theory of generalized functions [22]. The
result reads
%
\begin{eqnarray}\label{label2.20}
I(\vec{p}_1,\vec{p}\,'_1) =
\frac{\pi}{E_1}\Big(\frac{1}{|x\vec{n}\,'_1 - \vec{n}_1|} + \frac{(x
-\vec{n}\,'_1\cdot \vec{n}_1)^2 }{|x\vec{n}\,'_1 -
\vec{n}_1|^3}\Big){\ell n}\Big(\frac{1}{|x\vec{n}\,'_1 - \vec{n}_1| -
(1 -x)}\Big).
\end{eqnarray}
%
Substituting (\ref{label2.20}) in (\ref{label2.12}) and proceeding to
variables $x$ and $\vec{n}\,'_1$ we define the energy spectrum of a
final $\mu^-$--meson
%
\begin{eqnarray}\label{label2.21}
\hspace{-0.5in}\frac{1}{x^2}\frac{d\sigma(E_1)}{dx} &=&
Z^2\,\frac{\alpha^7}{\pi}\,\frac{m_e}{m^3_{\mu}}\int \frac{1 +
(\vec{n}_1\cdot \vec{\xi}_1)(\vec{n}\,'_1\cdot \vec{\xi}\,'_1)}{1 -
\vec{n}_1\cdot\vec{n}\,'_1 }\,\Big(\frac{1}{|x\vec{n}\,'_1 -
\vec{n}_1|} + \frac{(x -\vec{n}\,'_1\cdot \vec{n}_1)^2
}{|x\vec{n}\,'_1 - \vec{n}_1|^3}\Big)\nonumber\\ &&\times\,{\ell
n}\Big(\frac{1}{ |x\vec{n}\,'_1 - \vec{n}_1|-
(1-x)}\Big)\,d\Omega_{\,\vec{n}\,'_1}.
\end{eqnarray}
%
For the subsequent integration over a unit vector $\vec{n}\,'_1$ we
introduce angular variables as follows
%
\begin{eqnarray}\label{label2.22}
\vec{n}_1\cdot\vec{n}\,'_1 &=& \cos\vartheta_1\,',\nonumber\\
\vec{n}\,'_1\cdot \vec{\xi}\,'_1&=& \cos\vartheta_1\,'\cos\Theta_1\,'
+\sin\vartheta_1\,'\sin\Theta_1\,'\cos(\varphi_1\,' -
\Phi_1\,'\,),\nonumber\\ d\Omega_{\,\vec{n}\,'_1} &=&
\sin\vartheta_1\,'d\vartheta_1\,'d\varphi_1\,',
\end{eqnarray}
%
where $\Theta_1\,'$ and $\Phi_1\,'$ are polar and azimuthal angles of
the polarization vector $\vec{\xi}\,'_1$ relative to the momentum
$\vec{p}_1$. In (\ref{label2.22}) we have taken into account that
$|\vec{\xi}\,'_1| = 1$. Integrating over $\varphi_1\,'$ we get
%
\begin{eqnarray}\label{label2.23}
\hspace{-0.5in}&&\frac{1}{x^2}\frac{d\sigma(E_1)}{dx} =
2\,Z^2\,\alpha^7\,\frac{m_e}{m^3_{\mu}}\int\limits^{\pi}_0 \frac{1
+ (\vec{n}_1\cdot \vec{\xi}_1)\cos\vartheta_1\,'\cos\Theta_1\,'}{1 -
\cos\vartheta_1\,'}\,\Big(\frac{1}{\sqrt{1 - 2x\cos\vartheta_1\,'
+ x^2}}\nonumber\\
\hspace{-0.5in}&& + \frac{(x -\cos\vartheta_1\,')^2 }{(1 -
2x\cos\vartheta_1\,' + x^2)^{3/2}}\Big)\,{\ell n}\Big(\frac{1}{\sqrt{1
- 2x\cos\vartheta_1\,' + x^2} -(1-x)}\Big)\,
\sin\vartheta_1\,'d\vartheta_1\,'
\end{eqnarray}
%
Now it is convenient to introduce a new variable $t = \sqrt{1 -
2x\cos\vartheta_1\,' + x^2}$, which varies in the limits $1-x \le t \le
1+x$. In terms of $t$ the energy spectrum (\ref{label2.23}) reads
%
\begin{eqnarray}\label{label2.24}
\hspace{-0.5in}\frac{1}{x}\frac{d\sigma(E_1)}{dx} &=&
2\,Z^2\,\alpha^7\,\frac{m_e}{m^3_{\mu}} \int\limits^{1+x}_{1-x}
\frac{2x + (1+ x^2 - t^2)(\vec{n}_1\cdot
\vec{\xi}_1)\cos\Theta_1\,'}{t^2 - (1-x)^2}\nonumber\\
&&\times\,\Big(1+ \frac{(1 - x^2 - t^2)^2}{4x^2t}\Big)\,{\ell
n}\Big(\frac{1}{t - (1-x)}\Big)\,dt.
\end{eqnarray}
%
It is seen that the integral over $t$ is concentrated in the vicinity
of the lower limit. The singularity of the integrand in the vicinity
of the lower limit can be easily regularized by making a change of the
lower limit $1 - x \to 1 - x +\Lambda/E_1$, where $\Lambda$ is a
cut--off restricting energies of a final $\mu^-$--meson from
below. According to the kinematical region $\omega^2 \gg m^2_{\mu}$
[12] the cut--off $\Lambda$ can be chosen of order of $\Lambda \simeq
1\,{\rm GeV}$.

Keeping only the dominant contributions to the integral over $t$ we
get
%
\begin{eqnarray}\label{label2.25}
\frac{d\sigma(E_1)}{dx} = Z^2\,\alpha^7\,\frac{m_e}{m^3_{\mu}}\,{\ell
n}^2\Big(\frac{E_1}{\Lambda}\Big)\,\frac{x^2(2-x)}{1-x}\,[1 +
(\vec{n}_1\cdot \vec{\xi}_1)\cos\Theta_1\,'\,].
\end{eqnarray}
%
Introducing the angle $\Theta_1$, defined by $\vec{n}_1\cdot
\vec{\xi}_1 = \cos\Theta_1$, where we have taken into account that
$|\vec{\xi}_1| =1$, we obtain the energy spectrum of $\mu^-$--mesons
for the reaction $\vec{e}^{\,\,-} + Z \to Z + M^0 + \vec{\mu}^{\,\,-}
$ in dependence of the polarizations of the initial electron and the
final $\mu^-$--muon described by the angles $\Theta_1$ and
$\Theta\,'_1$
%
\begin{eqnarray}\label{label2.26}
\frac{d\sigma(E_1)}{dx} =
Z^2\,\alpha^7\,\frac{m_e}{m^3_{\mu}}\,{\ell
n}^2\Big(\frac{E_1}{\Lambda}\Big)\,\frac{x^2(2-x)}{1-x}\,(1 +
\cos\Theta_1\cos\Theta_1\,'\,).
\end{eqnarray}
%
Integrating over $x$ we arrive at the total cross section for the
reaction $\vec{e}^{\,\,-} + Z \to Z + M^0 + \vec{\mu}^{\,\,-} $
%
\begin{eqnarray}\label{label2.27}
\sigma(E_1) = Z^2\,\alpha^7\,\frac{m_e}{m^3_{\mu}}\,{\ell
n}^3\Big(\frac{E_1}{\Lambda}\Big)\,(1 +
\cos\Theta_1\cos\Theta_1\,'\,).
\end{eqnarray}
%
Assuming that electrons are longitudinally polarized electrons,
$\cos\Theta_1 = 1$, one can see that for the fixed electron energy the
cross section acquires the maximal value only for longitudinally
polarized muons $\cos\Theta_1\,' = 1$. This agrees with the production
of muonium with a total spin $J = 0$. Thus, we argue that the
appearance of longitudinally polarized muons in the final state of the
reaction $\vec{e}^{\,\,-} + Z \to Z + X + \vec{\mu}^{\,\,-} $ should
testify the production of muonium $X \equiv M^0$.

In our calculation the cross section for the reaction $e^- + Z \to Z +
M^0 + \mu^-$ has turned out to be dependent on a cut--off $\Lambda
\simeq 1\,{\rm GeV}$. In this connection we would like to remind that
the problem of the appearance of a cut--off in cross sections for some
reactions calculated within the Weizs\"acker--Williams approximation
has been pointed out by Bertulani and Baur [18].

Now let us discuss the energy dependence of the cross section
(\ref{label2.27}). It is well--known that for the $e^+e^-$ pair
production in heavy--ion collisions [18--20] and $p\bar{p}$ collisions
[21] the cross section for a capture of a final electron in an atomic
$K$--shell orbit is proportional to ${\ell n}(\gamma_{\rm coll})$,
where $\gamma_{\rm coll}$ is a Lorentz factor of colliding particles
in the center of mass frame. This factor is related to the
corresponding Lorentz factor $\gamma_p$ of the projectile (for a fixed
target machine) by $\gamma_p = 2\gamma^2_{\rm coll} - 1$ [18,20],
where $\gamma_p \sim E_1$. 

In turn, the cross section for the production of a point--like neutral
scalar particle in high--energy heavy--ion collisions in the
Weizs\"acker--Williams approximation is proportional to ${\ell
n}^3(\gamma_{\rm coll})$ [18,29].

For very high energies, when masses of coupled leptons can be
neglected, muonium with a total spin $J = 0$ can be treated as a
point--like massless scalar neutral particle. Such a property of
muonium is caused by an addition pole--singularity appearing at $(q -
k)^2 = q^2 - 2k\cdot q = 0$ for $k^2 = m^2_{\mu} = 0$ (see
Eq.(2.1)). This makes the part of the diagram in Fig.1, responsible
for creation of muonium, equivalent to an amplitude of a process
$\gamma^* + \gamma^* \to M^0$, where $\gamma^*$'s are virtual
photons. That is why the obtained cross section for the reaction $e^-
+ Z \to Z + M^0 + \mu^-$ has turned out to be proportional to ${\ell
n}^3(\gamma_{\rm coll})$.

\section{Conclusion}

\hspace{0.2in} We have calculated the cross sections for the reactions
$\vec{e}^{\,\,-} + Z \to Z + M^0 + \vec{\mu}^{\,\,-} $ and
$\vec{e}^{\,\,+} + Z \to Z + \bar{M}^0 + \vec{\mu}^{\,\,+}$ of the
production of muonium $M^0$ and anti--muonium $\bar{M}^0$ with
polarized $\mu^-$ and $\mu^+$ mesons by polarized electrons and
positrons coupled at high energies to the nucleus $Z$.

The cross sections are calculated in dependence on (i) an energy $E_1$
of initial electron and positron in the laboratory frame, coinciding
with the rest frame of a target nucleus $Z$, and (ii) polarizations of
initial electron and positron and final muons in the kinematical
region $\omega^2 = (p_1\,' + k)^2 \gg m^2_{\mu}$ making the massless
limit of coupled leptons as a well--defined approximation.

For the numerical estimate of the cross sections at the energies
available for the HERA Collider at DESY [13], i.e $E_1 = 27.5\,{\rm
GeV}$, we suggest to use Radon, ${^{222}_{~86}}{\rm Rn}$, as a target
nucleus, since Radon has a spin 1/2. The the cross sections for
longitudinally polarized electrons and positrons scattering by
${^{222}_{~86}}{\rm Rn}$ are equal to $\sigma(\vec{e}^{\,\,\pm}{\rm
Rn}) = 100\,{\rm fb}$. Using these theoretical values of the cross
sections for the reactions $\vec{e}^{\,\,-} + Z \to Z + M^0 +
\vec{\mu}^{\,\,-} $ and $\vec{e}^{\,\,+} + Z \to Z + \bar{M}^0 +
\vec{\mu}^{\,\,+}$ one can make the following predictions for the
number of favourable events: $N_{M^0} = 50$ and $N_{\bar{M}^0} =
200$. Hence, the increase of luminosities of electron and positron
beams should make the experiment for a test of CPT invariance,
suggested by Choban and Kazakov in Ref.[12], feasible at DESY.

As a distinct signal for the production of muonium $M^0$ and
anti--muonium $\bar{M}^0$ with a total spin $J = 0$ in the reactions
$\vec{e}^{\,\,-} + Z \to Z + X + \vec{\mu}^{\,\,-}$ and
$\vec{e}^{\,\,+} + Z \to Z + \bar{X} + \vec{\mu}^{\,\,+}$ with
longitudinally polarized electrons and positrons we argue the
appearance of longitudinally polarized muons in the final state. This
should testify that $X \equiv M^0$ and $ \bar{X} \equiv \bar{M}^0$
with a total spin $J = 0$.

\newpage

\begin{thebibliography}{9}
\bibitem{[1]} 
D. E. Groom {\it et al.}, 
Eur. Phys. J. {\bf C15}, 85 (2000).
\bibitem{[2]} 
S. Weinberg, in {\it THE QUANTUM THEORY OF FIELDS}, {\it
Foundations} Vol. I, Cambridge University Press, 1995; {\it Modern
Applications} Vol. II, Cambridge University Press, 1996;
{\it Supersymmetry} III, Cambridge University Press, 2000.
\bibitem{[3]} 
R. F. Streater and A. S. Wightman,
in {\it PCT, SPIN AND STATISTICS, AND ALL THAT},
Third Edition, Princeton University Press, Princeton and Oxford, 
1980.
\bibitem{[4]}
(see [1] p.313)
\bibitem{[5]}
L. M. Sehgal,
Phys. Rev. {\bf 181}, 2151 (1969);
J. P. Hsu, 
Phys. Rev. D {\bf 5}, 981 (1972); Phys. Rev. D {\bf 9}, 304 (1974);
R. Morse, U. Nauenberg, E. Bierman, D. Sager, and A. P. Colleraine,
Phys. Rev. Lett. {\bf 28}, 388 (1972);
S. Barshay,
Phys. Lett. B {\bf 101}, 155 (1981);
W. Bernreuther, U. Law, J. P. Ma, and O. Nachtmann,
Z. Phys. C {\bf 41}, 143 (1988);
G. Gabrielse,
Nucl. Phys. Proc. Suppl. {\bf 8}, 448 (1989);
{\it Parity and Time Reversal Violation in Compound Nuclear 
States and Related Topics}, edited by A. Auerbach and J. D. 
Bowman, World Scientific, Singapore, 1996;
P. Colangelo and  G. Corcela,
Eur. Phys. J. C {\bf 1}, 515 (1998);
S. R. Coleman and  S. L. Glashow Phys. Rev. 
D {\bf 59}, 116008 (1999).
\bibitem{[6]}
{\it UNIFIED STRING THEORIES},
edited by M. Green and D. Gross,
World Scientific, Singapore, 1986;
{\it SUPERSTINGS, A Theory of Everything}?,
edited by P. C. W. Davies and J. Brown, Cambridge University Press, 
1988;
B. F. Hatfield,
in {\it QUANTUM FIELD THEORY OF POINT PARTICLES AND STRINGS},
 Frontiers in Physics, Addison--Wesley Publishing Co., Singapore,
1989;
B. M. Barbashov and V. V. Nesterenko,
in {\it INTRODUCTION TO THE RELATIVISTIC STRING THEORY},
World Scientific, Singapore, 1990;
L. Castellani, R. D'Auria, and P. Fr$\acute{\rm e}$,
in {\it SUPERGRAVITY AND SUPERSTRINGS, A Geometric Perspective}, 
{\it Superstrings}, Vol. 3, World Scientific, Singapore, 1991;
J. Polchinski,
in {\it STRING THEORY, Superstring Theory and Beyond}, Vol. II,
Cambridge University Press, 1998.
\bibitem{[7]}
V. A. Kosteleck$\acute{\rm y}$ and R. Potting,
Nucl Phys. B {\bf 359}, 545 (1991); Phys. Lett. B {\bf 381},
(1996); D. Colladay and V. A. Kosteleck$\acute{\rm y}$, 
Phys. Rev. D {\bf 55}, 6760 (1997), Phys. Rev. D {\bf 58}, 
116002 (1998); {\it CPT AND LORENTZ SYMMETRY}, edited by 
V. A. Kosteleck$\acute{\rm y}$, World Scientific, Singapore, 1999;
V. A. Kosteleck$\acute{\rm y}$ and C. D. Lane,
Phys. Rev. D {\bf 60}, 116010 (1999).
\bibitem{[8]}
G. P. Lepage,
Phys. Rev. A {\bf 16}, 863 (1977).
\bibitem{[9]}
V. W. Hughes,
Z. Phys. {\bf 56}, S35 (1992).
\bibitem{[10]}
D. Kawall, V. W. Hughes, M. G. Perdekamp, W. Liu, K. P. Jungmann,
and G. Zu Putlitz,
{\it Test of CPT and Lorentz Invariance from Muonium Spectroscopy},
.
\bibitem{[11]}
L. Willmann and K. P. Jungmann,
{\it The Muonium Atom as a Probe of Physics beyond the 
Standard Model}, Physics {\bf 499}, 43 (1997), ;
K. P. Jungmann,
{\it Searching New Physics in Muonium Atoms}, .
\bibitem{[12]}
G. A. Kazakov and E. A. Choban,
JETP Letters {\bf 74}, 216 (2001).
\bibitem{[13]}
Roberto Sacchi,
{\it Search for Physics Beyond the Standard Model at HERA}, 
DESY, 2002.
\bibitem{[14]}
 M. P. Rekalo, J. Arvieux, and E. Tomasi--Gustafsson,
Phys. Rev. C {\bf 56}, 2238 (1997);
 A. Ya. Berdnikov, Ya. A. Berdnikov, A. N. Ivanov, V. A. Ivanova,
V. F. Kosmach, M. D. Scadron, and N. I. Troitskaya, Eur. Phys. J. A
{\bf 12} (2000) 341, .
\bibitem{[15]}
A. I. Akhiezer, A. G. Sitenko, and V. K. Tartakovskii,
in {\it NUCLEAR ELECTRODYNAMICS}, Springer--Verlag, Berlin, 1993.
\bibitem{[16]}
C. F. von Weizs\"acker, Z. Phys. {\bf 88}, 612 (1934);
E. J. Williams, Phys. Rev. {\bf 45}, 729 (1934).
\bibitem{[17]} 
V. N. Gribov, V. A. Kolkunov,  L. B. Okun, and V. M. Shekhter, 
JETP {\bf 41}, 1839 (1961).
\bibitem{[18]} 
C. A. Bertulani and G. Baur,
Phys. Rep. {\bf 163}, 299 (1988).
\bibitem{[19]}
R. Anholt and U. Becker,
Phys. Rev. A {\bf 36}, 4628 (1987).
\bibitem{[20]}
J. Eichler,
Phys. Rep. {\bf 193}, 165 (1990).
\bibitem{[21]}
C. T. Munger, S. J. Brodsky, and I. Schmidt,
Phys. Rev. D {\bf 49}, 3228 (1994).
\bibitem{[22]} 
I. M. Gel'fand and G. E. Shilov,
in {\it GENERALIZED FUNCTIONS, Properties and Operations}, Vol. I, 
Academic Press, New York and London, 1964.
\end{thebibliography}

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