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\begin{document}


\title{Dual coherent particle emission as generalized 
two-component Cherenkov-like
effect}

\author{D. B. Ion} 
  \affiliation{
TH Division, CERN, CH-1211 Geneva 23,
Switzerland, and
\\
 NIPNE-HH,
 Bucharest, P.O. Box MG-6, Romania 
}

\author{E. K. Sarkisyan}
\altaffiliation[Now at ]{the University of Nijmegen on NWO/NATO
Fellowship B 64--29.}
  \affiliation{EP Division, CERN, CH-1211 Geneva 23, 
Switzerland}

\date{\today}%

\begin{abstract}
In this Letter we introduce a new kind of coherent
particle production mechanism  called \emph{dual
coherent particle emission (DCPE)} as generalized two-component 
Cherenkov-like 
effect, which takes place when the phase
velocity of emitted particle $v_{Mph}$ and the
particle source phase velocity $v_{B_1ph}$  satisfy a specific DCPE  
condition:
$v_{Mph}\leq v_{{B_1}ph}^{-1}$. The general signatures of the
DCPE in  dielectric, nuclear and hadronic media
are established and some experimental evidences are presented.
\end{abstract}
\pacs{25.40.-h, 25.70.-z, 25.75.-q, 13.85.-t}
\maketitle

Cherenkov radiation (CR) was first observed in the early 1900's by the 
experiments
developed by Mary and Pierre Curie when studying radioactivity emission. 
However, the
nature of such radiation was unknown until the experimental works (1934-1937)
of P.A. Cherenkov and the theoretical interpretation by I.E. Tamm and I.M.
Frank (1937) \ct{0}. 
Then, it was
discovered
that 
this phenomenon is produced 
by
charged
particles movings with superluminal speeds in  medium.
Now, the CR is one of the cornerstones of classical and quantum
electrodynamics and it is the subject of many studies related to the
extension to other
coherent particle emission via Cherenkov-like effects 
\ct{2,4,2a,14,3,14a}.
 The generalized Cherenkov-like effects based on four fundamental
interactions
 has been investigated and classified recently in \ct{3}.
In particular, this classification includes the nuclear (mesonic, $\gamma$,
 weak boson)-Cherenkov-like radiations as well as the high energy
component
 of the coherent particle emission via (baryonic, leptonic, fermionic)
Cherenkov-like effects.
 It is important to underline that the CR is extensively used in
experiments for counting and identifying relativistic particles.
 We note   that, by the experimental observation of the
subthreshold \ct{1} and  anomalous \ct{1a} CR, 
 it was clarified
that some of fundamental
 aspects of the CR are still open.
Therefore, more theoretical and experimental investigations on CR as well 
 on the generalized Cherenkov-like effects are needed.

In this Letter we introduce a new kind of {\it coherent} particle 
production 
mechanism
 in the medium called
\emph {dual coherent particle emission}
 (DCPE) which includes in a more general and exact form 
not only the
usual   
 CRs but also all kind of the generalized Cherenkov-like
 effects and unifies them as 
{\it two-component} Cherenkov-like effects.
The DCPE-effect is expected to take place when the
{\it
phase velocities of the emitted
particle, \( v_{Mph} \) (or \( v_{B_{2}ph}\)), 
and that of particle source
\( v_{B_1ph} \)
satisfy the dual coherence condition:
\( v_{B_{2}ph}v_{Mph}\leq 1 \)}  (\( v_{B_{1}ph}v_{B_{2}ph}\leq 1 \)).
The general signatures of DCPE effects in dielectric, nuclear and hadronic
 media are discussed.
 For illustration some recent theoretical predictions and experimental
results
 on the coherent meson production via mesonic Cherenkov-like radiation in
 nuclear \ct{5,6} and in hadronic \ct{9,16,11} media are included.


\emph{\noun{Dual coherent particle emission}}. Let us start
with a general B\( _{1}\rightarrow {\rm MB} \)\( _{2} \) decay, see 
Fig. \ref{fig:1}(a). Here
 a particle M [with energy \( \omega , \) momentum $k=$ Re\(
n_{M}(\omega ) \)\( \sqrt{\omega ^2-M^2_{\rm M}}\),
rest mass \(M_{\rm M} \) and refractive index \( n_{M}(\omega )] \) is
emitted
in a (nuclear, hadronic, dielectric, etc.) medium by  incident particle
B\( _{1} \) [with energy \(E_1 \), momentum \(p
_{1}={\rm Re} n(E_{1})\sqrt{E^{2}_{1}-M_{1}^{2}} \),
rest mass M\( _{1} \),  refractive index \( n_{1}(E_{1}) \){]} that
itself
goes over into a final particle B\( _{2} \) [with energy \( E_{2} \),
momentum
\(p _2={\rm Re}n(E_{2})\sqrt{E^{2}_{2}-M_{2}^{2}} \),  rest
mass 
\(M_{2} \),
 refractive index \(n_{2}(E_{2}) \){]}.


{\par
\begin{figure}[t]
\rotatebox{-90}{
\centering \resizebox*{3.8cm}{8.3cm}{\includegraphics{db_xfig.eps}  }
}
\caption{
The schematic description of DCPE phenomenon as the generalized
two-component
Cherenkov-like effect:
(a)  mesonic Cherenkov-like radiation (MCR) limit
\ct{3}:
\( v_{Mph}\leq v_{1}\);
(b)  two-body decay process B$_{1}$ $\rightarrow {\rm MB}_2$;
 (c)  baryonic Cherenkov-like
radiation (BCR)
limit \ct{3}: \( v_{B_{2}ph}\leq v_{1} \).
}
\label{fig:1}
\vspace*{-.45cm}
\end{figure}
}

Here we prove that in order to obtain a genuine spontaneous particle emission
in a given medium {\it the two general conditions} are necessary to be
fulfilled: 

(i) 
\hangindent=1.cm \hangafter=1 {$\!\!$ 
The incident particle-source must be coupled
to a specific
radiation
field (RF) (see Fig. \ref{fig:1}) and that the particles
propagation properties
in medium must be modified.
}
\smallskip

\noindent
(ii.1) 
\hangindent=.95cm \hangafter=1 {$\!$ The 
particle source must be moving in the medium with a dual
phase
velocity \( v^{-1}_{B_{1}ph} \) 
higher than the phase velocity \(v_{Mph}
\)
of the RF-quanta.}
\smallskip

\noindent
(ii.2) 
\hangindent=.95cm \hangafter=1 {$\!$
The particle source must be moving in medium with a dual
phase
velocity \( v^{-1}_{B_{1}ph} > v_{B_{2}ph} \).
}
\smallskip


\emph{Proof:} The propagation properties of particles in a medium are changed
in agreement with their elastic scattering with the constituents of that medium.
So, the phase velocity \(v_{Xph}(E_X) \) of any particle X 
(with the
total
energy $E_X$ and rest mass \(M_{X} \)) 
in medium is
modified:
\begin{equation}
\label{Eqo}
v_{Xph}(E_{X})=\frac{E_{X}}{p_{X}}=\frac{1}{{\rm 
Re}n_{X}(E_{X})}\cdot
\frac{E_{X}}{\sqrt{E_{X}^{2}-M^{2}_{X}}}.
\end{equation}
The \emph{refractive index} \( n_{X}(E_X) \) 
in a medium composed from the
constituents
``c'' can be calculated in standard way by using the Foldy-Lax formula 
\ct{12}
(we work in the units system \( {\hbar}=c=1 \)) \begin{equation}
\label{Eq0}
n^{2}_{X}(E_{X})=1+\frac{4\pi \rho
}{E_{X}^{2}-M^{2}_{X}}\cdot C(E_{X})
\overline{f}_{Xc\rightarrow Xc}(E_{X}),
\end{equation}
 where \( \rho  \) is the density of the constituents, $C(\omega)$
is
a \emph{coherence factor} [$C(\omega )=1$ when  the medium
constituents
are randomly distributed], \( \overline{f}_{Xc\rightarrow Xc}(E_X) \) is
the
averaged
forward Xc-scattering amplitude. 

Now, by using the \emph{energy-momentum conservation law} for the decay
\( {\rm B}_1 \rightarrow {\rm MB}_{2} \)
in medium,
$
%\begin{equation}
%\label{10}
E_{1}=E_{2}+\omega ,\; \;
\overrightarrow{p}_{1}=\overrightarrow{p}_{2}+\overrightarrow{k},
%\end{equation}
$, we obtain (see angles definition in Fig. \ref{fig:1}):
\begin{equation}
\label{Eq10a}
\cos\theta
_{1k}=v_{Mph}v_{B_{1}ph}+\frac{1}{2p_{1}k}[-D_{B_{1}}+D_{B_{2}}-D_{M}],
\end{equation}
 \vspace*{-.4cm}
 \begin{equation}
\label{Eq10c}
\cos\theta _{12}=v_{B_{1}ph}v_{B_{2}ph}+\frac{1}{2p_{1}p_2}[-D_{B_{1}}
-D_{B_{2}}+D_{M}],
\end{equation}
 \vspace*{-.4cm}
\begin{equation}
\label{Eq10b}
\cos\theta _{2k}=
v_{Mph}v_{B_{2}ph}+\frac{1}{2p_{2}k}[D_{B_{1}}-D_{B_{2}}-D_{M}].
\end{equation}
Here \(D_{X} \),\( \; X\equiv B_{1},B_{2},M, \)  are departures
off
 mass shell, 
$$
%\begin{equation}
%\label{Eq1c}
D_{X}\equiv
E^{2}_{X}-p^{2}_{X}=M^{2}_{X}+[1-({\rm 
Re}n_{X}(E_{X}))^{2}][E_{X}^{2}-M^{2}_{X}].
%\end{equation}
$$

We note that the second terms in the right side of Eqs.
(\ref{Eq10a})--(\ref{Eq10b}) can be
considered
as quantum corrections to the first ``classical'' terms \ct{3}. 

A rigorous proof of statement (ii) is obtained from the fact that
the respective emission angles must be the physical ones. The
\emph{coherence
quantum conditions,} \( \cos \theta _{ij} \) \( \leq 1, \) $i,j=1,2,k,$
Eqs. (\ref{Eq10a})--(\ref{Eq10b}),
at high incident particle energies transform into \emph{classical
coherence conditions},
e.g., \begin{equation}
\label{Eq1d}
\cos\theta _{1k}=v_{Mph}v_{B_{1}ph}\leq 1,
\end{equation}
 which is equivalent to
\begin{equation}
\label{Eq1e}
v_{Mph}\leq v^{-1}_{B_{1}ph}\; \: \: {\rm or}\; \: \: 
v_{B_{1}ph}\leq v^{-1}_{Mph}.
\end{equation}

It is worth to note that from the dual coherence conditions
(\ref{Eq1e}) we obtain the following {\it limits of the DCPE condition}: 
(iii) 
\smallskip
In case when \( v^{-1}_{B_{1}ph} =v _{1} \),
from
the condition (ii) the  two important generalized Cherenkov-like limits
follow:
(iii.1) the MCR limit \( v_{Mph}  
\leq v_{1}
\), and
(iii.2) the BCR limit \( v_{B_{2}ph}\leq v_{1}
\) (see Fig. 1)

The proof of the statement (iii) is obtained immediately if one observes
that when
the particle \emph{B\( _{1} \)} is \emph{on the mass shell in medium
{\rm (Re}n\(
_{1}=1) \)
then \( v^{-1}_{B_{1}ph} \)=v\( _{1} \),} and the \emph{dual coherence 
conditions}
(ii.1) and (ii.2) are the \emph{Cherenkov-like coherence conditions}
(iii.1) and (iii.2), respectively. 


Now, we can obtain a classification of these DCPE effects not only on the
basis of four fundamental (strong, electromagnetic, weak,
and gravitational)  interactions but also using the above
\emph{``M-B duality''} as well as the \emph{crossing
symmetry.}

The main signatures of  the DCPE as the generalized 
two-component Cherenkov-like effects
are
as follows:  
\smallskip

\noindent
$\bullet$ 
\hangindent=.4cm \hangafter=1 {$\!$
The  differential cross sections posses the bumps in the energy
bands where the
DCPE  conditions are fulfilled.
}
\smallskip
\vspace*{-.3cm}

\noindent
$\bullet$
\hangindent=.4cm \hangafter=1 {$\!$
 The DCPE-effects are threshold mechanisms.}
In the classical limit the DCPE-threshold velocity is related  
with Cherenkov-like threshold velocity via the relation
$
v^{thr}_{B_{1}}(DCPE)=v^{thr}_{B_{1}}(XCR)/{\rm Re}n_{B_{1}}$.
where $X\equiv M,B_{2}$.

\smallskip 

\noindent
$\bullet$
\hangindent=.4cm \hangafter=1 {$\!$ 
Coherent particles emitted via the DCPE must be coplanar
with
the incoming
and outgoing projectiles: strong \( (\theta _{1k},\omega ) \) and
\( (\theta _{1k},E_{p})\) correlations.
\smallskip 

\noindent
$\bullet$
\hangindent=.4cm \hangafter=1 {$\!$
The intensities as well as the absorption effects can be calculated  
as in the case
of generalized Cherenkov-like effects \ct{3} by using Feynman diagrams in
medium.}
\smallskip 

\noindent
$\bullet$
\hangindent=.4cm \hangafter=1 {$\!$ 
 Any two-body decay process B$_1$ $\rightarrow$ MB$_2$  in
medium
via DCPE posses two limiting modes: the nuclear MCR (NMCR) mode in which 
the particle M
is
emitted
with the coherence condition \( v_{Mph}\leq v_{1} \), and
the nuclear BCR (NBCR) mode
with  \( v_{B_{2}ph}\leq v_{1}\).}
\vspace*{.2cm}

\noun{The electromagnetic DCPE.}
To be more specific let us consider a charged particle (e.g.
\( e^{\pm },\mu ^{\pm }$,p, etc.) moving in a (dielectric, nuclear or 
hadronic) medium and to 
explore
 the
\( \gamma  \)
 coherent emission via DCPE in that media.
 In the case of dielectric medium the CR is well established phenomenon
 widely used in physics and technology.
 Also, experimentally,  the high energy $\gamma$-emission via
coherent bremsstrahlung (CB) is
 well known as a channeling effect.
 So, the high energy component of the DCPE phenomenon can be identified
 with the CB-radiation.
 Hence, more experimental and theoretical investigations are needed since
 CR and CB radiations can be described in an unified way via the DCPE as 
two-component generalized
 Cherenkov-like effects.
 Then, the same final interaction Hamiltonian \( H_{\rm fi} \)
just as in the quantum theory of CR with some modifications of the source
 fields in medium can also describe the coherent $\gamma$-emission in the
CB
 sector.
 Therefore, the subthreshold rings observed experimentally can also be
interpreted as DCPE signatures since in the CR sector (Fig.1(a)) the
number of
photons
 emitted per time unit in the interval
\( (\omega ,\omega +d\omega)  \)  in an nonabsorbent medium will be given by
\begin{equation}
\frac{dN_{\gamma }}{d\omega }(DCPE)\simeq \alpha
Z_{B_{1}}^{2}v_{B_{1}}\left( 
1-v^{2}_{\gamma ph}v^{2}_{B_{1}ph}\right)
\end{equation}
 where \( \alpha =1/137 \) is the fine structure constant and \(Z_{B_{1}} \)
is the electric charge of the
B\(_1 \)
 particle.


Now, it is easy to see that the DCPE-coherence condition (\ref{Eq1d})
includes
 in a general and exact form the subthreshold Cherenkov-like radiation 
\ct{1} 
since:
\( 
v^{thr}_{B_{1}}(DCPE)=v^{thr}_{B_{1}}(CR)/{\rm Re}n_{B_{1}} \).
So, 
if   
\( {\rm Re}n_{B_{1}}\geq 1,\: {\rm then}\: 
v^{thr}_{B_{1}}(DCPE)\leq v^{thr}_{B_{1}}(CR) \).

The second major result is  connected with the fact that the
physical domain
 of the DCPE-radiation includes a high energy component in which the 
charged
 particle source is
{\it ``coherently''}
 stopped in medium like in the channeling radiation process.
 Hence, in the CR-case, these secondary charged particles  produce
{\it secondary rings}
 in the same refractive medium.
 This DCPE-effect completely solves the mystery of the
{\it anomalous Cherenkov radiation}
\ct{1a}. In fact, we obtain a 
Cherenkov-Channeling (or coherent
 bremsstrahlung) radiation duality.

It is worth to note that  a two-component $\gamma$-CR can
explain $\gamma$-rays emission from cosmic sources \ct{2a}.

\vspace{.2cm}

{\sc The strong DCPE effects}. 
Now, let us consider a baryon (e.g. p, n, $\Lambda$, $\Sigma$, etc.)
moving in a (nuclear or hadronic) medium and to explore the coherent
meson emission (e.g. $\pi$, K, $\eta$, etc.) via the strong DCPE
radiation in that medium.

In the nuclear medium, two 
kinds of the NMCR have been intensively investigated 
in details in \ct{3}, namely 
the \emph{nuclear pionic Cherenkov-like radiation (NPICR)} and the 
nuclear kaonic Cherenkov-like radiation (NKCR).
The characteristic features of the NPICR-pions
predicted
in  \ct{3} are the  following. 
(i) The NPICR-coherence condition, \( v_{ph}(\omega )\leq v \),
 was
found to be {\it fulfilled in the three energy bands}:
(CB1) 190 MeV\(\leq \omega \leq \) 315 MeV, for all \( \pi ^{\pm ,0} \);
(CB2) 910 MeV \( \leq \omega \leq  \) 960 MeV, only for \(\pi \)\( ^{+}\);
(CB3) \( \omega  \geq  \) 80 GeV, for all \( \pi ^{\pm ,0} \).
(ii) The NPICR-pions must be complanar with the incoming and outgoing 
projectile possessing strong correlations, ($\theta_{1k},w$) and  
($\theta_{1k},T_p$).
(iii)
The \emph{NPICR differential cross
sections (DCS)} are peaked
at
the energy \( \omega _{\rm m}=244 \) MeV for CB1-emission band when
absorption
in medium is taken into account. The CB1-peak width in the DCS is
predicted
to be \( \Gamma _{\rm m}  \leq  \) 25 MeV.
(iv) The NPICR-peak position is predicted to behave with energy as 
$T_p^{-2}$, while the $A$-target dependence of the DCS is also predicted
\ct{3}.   


Therefore, the physical domain of the {\it low-energy component} of the 
DCPE 
phenomenon
can be identified with the NPICR-predicted pionic bands, CB1, CB2, and
CB3.
 Hence, more experimental and theoretical investigations are needed since
 NPICR and NBCR radiations can be described in an unified way via DCPE as
the generalized
 two-component Cherenkov-like effects with the same 
interaction Hamiltonian \ct{3}.


On the other hand, we must underline that the existence of the 
NPICR-CB1-emission band  has been 
experimentally confirmed
in the studies of dense
groups, or spikes, of negative pions in  
central  Mg-Mg collisions at 4.3$A$ GeV/$c$ from the
2m Streamer Chamber SKM-200  (JINR, Dubna) \ct{5,6}. 
In about 14420 events, 
the spikes have been extracted in each event from the ordered
pseudorapidities scanned with  the 
rapidity window of the fixed  
size $\delta {\stackrel{\sim}{\eta}}$
with the
definite
number $n$ of pions required to hit in the window. An example of the 
 c.m. energy distributions is shown 
in Fig.~\ref{fig:3}. So, from Fig. \ref{fig:3} and the $E^*$-spectra
at other $\delta
{\stackrel{\sim}{\eta}}$ and $n$ \ct{5,6}, one finds a 
significant peak in the energy spectra of
emitted
pions over the inclusive background.
The
experimental values of the peak position, 
$E^*_{\rm m}=238\pm 3({\rm stat})\pm 8({\rm syst})\: {\rm MeV}$, and its 
width, $\Gamma _{\rm m}=10\pm 3({\rm stat})\pm 5({\rm syst}) \: {\rm MeV}$,
 are found to be in a good agreement with the absolute NPICR 
predictions for $\omega_{\rm m}$ and $\Gamma_{\rm m}$
\ct{2,3}.
It is important to note that the  value of \( E^*_{\rm m} \) is
similar
to the position of the peak found in \ct{15} in the study
of \( \pi ^{+} \)-production in coincidence measurements of (p,n)
reactions
at 0\( ^{\circ} \) on C, the effect connected with NPICR 
\ct{3}.



{\par \centering \begin{figure}
\resizebox*{0.27\textwidth}{0.2\textheight} {\includegraphics{g111.eps} }
\caption
{The c.m.s. energy spectrum of dense groups (spikes) of \( \pi
^{-}\)s with required multiplicity $n=4$ within a rapidity window $\delta
{\stackrel{\sim}{\eta}}=0.05$ obtained in 
central Mg-Mg collisions at 4.3$A$ GeV/$c$ \ct{5,6}. The dashed and solid 
lines 
show
the 
inclusive 
background and  the fitted curve, respectively. 
} 
\label{fig:3} 
\vspace*{-.3cm}
\end{figure} }


\noindent

Finally, we can conclude that DCPE  
intensities are large enough in order
to be experimentally measured in exclusive experiments by rapid 
coincidence techniques. 
\vspace{.3cm}



\emph{\noun{The DCPE in hadronic media.}}
The mesonic Cerenkov-like radiation in hadronic media was considered by many
authors \ct{9,16,11}. A systematic investigation of the classical and
quantum
theory of this kind of effects in hadronic media can be found in Ref.
\ct{9}. The classical variant \ct{2,9} of the Cherenkov mechanism was
applied
to the study of single meson production in hadron-hadron interactions at high
energies. This variant is based on the usual assumption that hadrons are composed
from a central core in which most of the hadron mass is concentrated surrounded
by a large and more diffuse mesonic cloud (hadronic medium). Then, 
it was shown [9,11]
that a \emph{hadronic mesonic Cherenkov-like radiation (HMCR)} with an
mesonic
refractive index given by a pole approximation, can be able to explain
the integrated
cross sections of a single meson production in hadron-hadron interactions.
 Indeed, in the pole approximation the integrated cross section reads:
\vspace*{-0.31cm}
$$
\sigma _{HMCR}(v)=\sigma _{0}\, \frac{G^{2}}{4\pi }\, 
\, \frac{a}{Y_v}
\: 
\times
\vspace*{-0.43cm}
$$
\begin{equation}
\left\{ 1-
{1\over{Y_v}}
\left[
\frac{m}{a}+\frac{a^{2}+m^{2}}{a^{2}}
\arctan
\left( \frac{aY_v-m}{mY_v+a}\right)  \right]
\right\}
\label{sig13}
\end{equation}
for the (pseudo)scalar mesons, and
  similar formula for 
for the vector mesons \ct{9,11}:. Here, $v$ is the projectile velocity, 
$Y_v=v/\sqrt{1-v^{2}}$,
$m$ is the rest mass of the produced meson,
$G$ is the usual coupling constant,
$\sigma_0=0.389$ mb, and \( a =0.350 $ GeV.
To illustrate, in Fig. \ref{fig:5}(a) we present the 
measured integrated
cross
sections of the process
\( \rm pp\rightarrow pp\pi ^{0} \), compared with the HPICR-predictions.
This result was very encouraging for the extension of the HMCR analysis
to other single meson production in hadron-hadron
collisions at high energies. Collecting the \( \chi ^{2}/dof \)s  for
all 139 reactions fitted with
the HMCR approach \ct{11}, we get the
surprisingly good description as shown in  Fig.  \ref{fig:5}(b). We must
underline that only reactions with single  meson
production was fitted (a single parameter fit) with the HMCR predictions
on the integrated cross sections. 





{\par
\begin{figure}
\centering
\resizebox*{5.cm}{4.2cm}{\includegraphics{sigmap.eps} }
\resizebox*{5.cm}{4.6cm}{\includegraphics{histo.eps} }
\caption{
(a) Comparison between the experimental integrated cross sections of the 
reaction pp$\to$pp$\pi^0$ and those predicted by the pion emission via 
HPICR effect, Eq. (\ref{sig13}). 
(b) Experimental statistical test of the
\emph{HMCR}-\emph{mechanism dominance}
in hadron-hadron collisions:
the number of reactions fitted with
the HMCR
for integrated cross sections vs. \( \chi ^{2}/dof
\) \ct{9,11}.}
\label{fig:5}
\vspace*{-.3cm}
\end{figure}
\par}

\par{} \vspace{0.3cm}


{\sc Conclusions.} A new kind of coherent
particle production mechanism,  called \emph{dual  
coherent particle emission (DCPE)}, is introduced.
The DCPE phenomenon as generalized two-component Cherenkov-like effects 
can
 be viewed as two body decays
B$_{1}\rightarrow$ B$_{2}$M in medium.
 They
 are expected to take place when the
\emph {
phase velocities of the emitted
 particle
\( v_{Mph} \)
 and that of particle source 
\( v_{B_1ph} \)
 satisfy the dual coherence condition:
\( v_{B_{1}ph}v_{Mph}\leq 1. \)}
The secondary
B\(_{2} \)-particles produced by the high-energy component (BCR) of the 
DCPE can produce
 secondary MCR-effects.
 Hence, under certain circumstances the DCPE phenomenon applied to the CR
 in dielectrics (or crystals) can explain not only subthreshold CR \ct{1} 
but
 also the observed secondary rings (or anomalous CR) \ct{1a}.
For illustration,  experimental evidences for DCPE effects 
in various media are 
presented. 


Finally, it is important to remark that more investigations of  DCPE
 effects are necessary in connections with  Cherenkov particle 
detectors
 since they may help to explain discrepancies between some experimental
 results and theoretical predictions in high-energy physics.

 

\begin{acknowledgments}
We would like to thank G. Altarelli for fruitful discussions. One of the
authors (D.B.I.) would like to thank TH Division for hospitality during
his stay at CERN. 
\end{acknowledgments}

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