text:
\section{Introduction}

As is known, some experimental discoveries, such as the neutrino
oscillation, the existence of the dark matter (its nature and
interaction are unknown today) and some astrophysical data, lead to
the necessity of the consideration the physics beyond the SM (see
the reviews \cite{Rev12, Rev13, Rev15, Rev16}). One of the possible
new particle is the so-called dark photon, $A'$. It is massive vector
boson that can mix with the ordinary photon via "kinetic-mixing"
\cite{H86}. Its mass and interaction strength are not predicted
unambiguously by the theory since the mass of the dark photon can
arise via different mechanisms. Various theoretically motivated
regions of the dark photon mass are shown in Figs. 6-2, 6-3 in the
Ref. \cite{Rev12}. For the dark photon with mass larger than 1 MeV,
it is possible its production in electron (proton) fixed-target
experiments or at hadron or electron-positron colliders (see the
references in review \cite{Rev12}).

The experimental investigation of the dark photon effects are
planned or performed in various laboratories: APEX \cite{ESTW11,
ABet11}, HPS \cite{HPS}, DarkLight \cite{FOT10} (JLAB), MAMI
\cite{Met11} (fixed-target experiments) and VEPP-3 \cite{W09}
(electron-positron collider). The manifestation of the dark photon
was searched also in the decay of the known particles. The authors of
Ref. \cite{Fet03} have studied radiative pion decays $\pi^ +
\rightarrow e^+\nu\gamma$. The measurements were performed in the
$\pi$E1 channel at the Paul Scherrer Institute (PSI), Switzerland.
The dark photon was searched in the decay of $\pi^0$-meson
($\pi^0\rightarrow \gamma A'\rightarrow \gamma e^+e^-$)\cite{Xet13}
which produced in proton nuclei collisions at HIAF facility (China).
The decay of $\pi^0$-meson was also used to search for the dark photon
in WASA-at-COSY experiment (J\"{u}lich, Germany) \cite{M14} ($\pi^0$-mesons were produced in the reaction
$pp\rightarrow pp\pi^0$) and at CERN ($\pi^0$-mesons were produced
in the decays of K-mesons, $K^{\pm}\rightarrow \pi^{\pm}\pi^0$) \cite{G14}.
A search for a dark photon signal in inclusive dielectron spectra in
proton-induced reactions on either a liquid hydrogen target or
nuclei performed at the GSI in Darmstadt \cite{Aet13}. An upper
limit on the dark photon mixing parameter in the mass range m(A')=
0.02 - 0.6 GeV/c$^2$ has been established. The current status of the
limits on the dark photons parameters from electron beam dump experiments is
summarized in \cite{A12}. It has been demonstrated at JLab that
electron-beam fixed-target experiments would have powerful discovery
potential for a dark matter in the MeV-GeV mass range \cite{IKST14}.

Theoretically, the production of the dark photon in various
reactions has been investigated in a number of papers.
Bjorken et al. \cite{BEST09} discussed several possible
experimental schemes for the search for a $A'$ in the most likely mass
range of a few MeV/c$^2$ up to a few GeV/c$^2$. They stated that the
fixed-target experiments are ideally suited for discovering
few MeV-GeV mass dark photon.
The
production of the dark photon in the process of the electron
scattering on the proton or heavy nucleus has been investigated in
Ref. \cite{BMV13} (\cite{BV14}) for the experimental conditions of
the MAMI (JLab) experiment \cite{Met11} (\cite{ABet11}). The authors
of Ref. \cite{CT16} proposed to use rare leptonic decays of kaons
and pions, $K^+(\pi^+)\rightarrow \mu^+\nu_{\mu}e^+e^-$, to study the
light dark photon (with mass about 10 MeV). The constraints on the
dark photon in the 0.01 - 100 keV mass range are derived in Ref.
\cite{APPR14} (indirect constraints from $A'\rightarrow 3\gamma $
decay are also revisited). The proposal to search for light dark
photon using the Compton-like process, $\gamma e\rightarrow A'e$, in
a nuclear reactor was suggested in Ref. \cite{P17}. Using the
existed experimental data, the limits on the kinetic mixing
parameter were derived. Some results on the phenomenology of the
dark photon in the mass range of a few MeV to GeV have been
presented in \cite{P09}, where g-2 of muons and electrons together
with other precision QED data, as well as radiative decays of
strange particles, were analyzed.

At proposed project IRIDE (Frascati, Italy) \cite{IRIDE}, the
physical program consider a search for dark photon via the lepton
triplet production process in the electron-photon collision. The
main QED process of the lepton triplet production is determined by
Bethe-Heitler diagrams and the virtual Compton scattering diagrams.
Therefore, the virtual Compton scattering part of the QED process is
of the most importance, as it is intimately related to the dark
photon production, while the Bethe-Heitler contribution must be
reduced as much as possible. This can be done by specific angular
criteria for the event selection.

In this paper, we consider the lepton triplet production in the
photon scattering by electrons. The dark photon production is taken
into account only in the Compton-like diagrams where
electron-positron pair from the decay $A'\rightarrow e^+e^-$ can be
in resonance. The advantage of this consideration is that the
background, in this case, is a pure QED process $\gamma
e^-\rightarrow e^+e^-e^-$ which can be calculated with the
required precision. We calculate the distributions over the
invariant masses of the produced $e^+e^-$ pairs taking into account
the identity of the final electrons. All eight Feynman diagrams,
that determine the background reaction, was taken into account when
calculating distributions in $\gamma e^-\rightarrow e^+e^-e^-$.
We estimate what value of the parameter $\epsilon$,
as a function of the dark photon mass, can be obtained at given
number of the measured events.

In Sec. II the formalism of the calculation of the distribution
is given. Sec. II\,A is devoted to the description of the kinematical
variables. The calculation of the double differential distribution
over the masses of the produced $e^+e^-$ pairs, caused by the QED
mechanism, is given in Sec. II\,B. The dark photon contribution to the
distribution is calculated in Sec. II\,C. Sec. III is devoted to the
analysis to the dark photon effects in the considered reaction and
to the estimation of the parameter $\epsilon $ as a function of the
dark photon mass and the number of the measured events.
The detailed analysis of the kinematics for the restricted phase space is given in Appendix A.
The Appendix B contains the analytical
expressions for the contributions into distribution over invariant masses of $e^+e^--$pairs which are caused by the separate diagrams and their
interferences in the case of nonrestricted phase space.

\section{Formalism}

As we mentioned in the Introduction, the dark photon can manifest
itself as some resonance state decaying in the electron-positron
pair. In
this case, the process of the triplet production,
 \bge\label{eq:1}
\gamma(k)+e^-(p)\rightarrow e^+(p_3)+e^-(p_1)+e^-(p_2)\,, \ene
can
be used to search for the dark photon signal due to the creation of
two electron-positron systems, with the invariant masses squared
$$s_1=(p_3+p_1)^2 \,, \ \ s_2=(p_3+p_2)^2,$$
through the dark photon intermediate state. The double differential
distribution over the $s_1$ and $s_2$ variables, in the process
(\ref{eq:1}), is the most suitable for this goal since it takes into
account the identity of the final electrons.

The pure QED distribution is the background which exceeds
significantly the dark photon effect. Thus, it has to be calculated
as precise as possible and accounted for in the searches for the
dark photon signal. The QED amplitude, for the triplet production
process, is described by eight diagrams, four of them are presented
in Fig.\,1, and the another four can be derived from these diagrams
by permutation the electron 4-momenta $(p_1\leftrightarrows p_2).$

\begin{figure}
\includegraphics[width=0.44\textwidth]{diagran.eps}
\hspace{0.4cm}
\includegraphics[width=0.44\textwidth]{diagrbn.eps}
\caption{The Feynman diagrams for the QED amplitude of the process
(\ref{eq:1}). The diagrams ({\it a}) are the so-called Borsellino
diagrams and the diagrams ({\it b}) are the Compton-like ones. To
take into account the final electron identity, one has to add
corresponding diagrams with permutation of the 4-momenta $p_1$ and
$p_2$.}
\end{figure}

In present paper, we calculate the double differential distribution
over the $s_1$ and $s_2$  variables to calculate the QED background
and to find the regions of the $s_1$ and $s_2$ where the single
photon amplitudes (Compton-like) contribute  on the level of the
two photon amplitudes (Borsellino) and even more.

\subsection{Kinematics}

The process (1) is the $2\to 3$ one, and for its description we can
use a few sets of the kinematical variables \cite{BK1973}. To obtain
the distribution over the $s_1$ and $s_2$ variables, we use the
description in terms of five invariants as follows
$$s=(k+p)^2=(p_1+p_2+p_3)^2\,, \ s_1=(p_1+p_3)^2=(k+p-p_2)^2\,,
\ \ s_2=(p_2+p_3)^2=(k+p-p_1)^2\,,$$
\bge\label{eq:2}
t_1=(k-p_1)^2=(p_2+p_3-p)^2\,,\ \ t_2=(p-p_2)^2=(p+p_3-k)^2\,.
\ene
The scalar products of the 4-momenta in the reaction are expressed in
terms of these invariants
$$2(k\,p)=s-m^2\,, \ 2(p_1\,p_3)=s_1-2m^2\,, \ 2(p_2\,p_3)=s_2-2m^2\,,
\ 2(k\,p_1)=m^2-t_1\,, $$
\bge\label{eq:3}
2(p\,p_2)=2m^2-t_2\,, \ 2(k\,p_2)=s-s_1+t_2-m^2\,, \ 2(k\,p_3)=s_1+t_1-t_2-m^2\,,
\ene
$$2(p_1\,p_2)=s-s_1-s_2+m^2\,, \ 2(p\,p_1)=s-s_2+t_1\,, \ 2(p\,p_3)=s_2-t_1+t_2-m^2\,.$$

Bearing in mind the azimuthal symmetry relative to the photon beam
direction, the phase space of the final particles in (1) can be written
as \cite{BK1973}
\bge\label{eq:4}
d\,R_3=\frac{d^3p_1}{2\,E_1}\,\frac{d^3p_2}{2\,E_2}\,\frac{d^3p_3}{2\,E_3}\,\delta(k+p-p_1-p_2-p_3)=\frac{\pi}{16(s-m^2)}
\frac{dt_1\,dt_2\,ds_1\,ds_2}{\sqrt{-\Delta}}\,,
\ene
where $\Delta$ is the Gramian determinant. In terms of the used variables,
it can be written as
$$\Delta=\frac{1}{16}\left|\begin{array}{c c c c} ~0~~~~~~~s-m^2~~~~~~~~~m^2-t_1~~~~~~~~~~s-s_1+t_2-m^2~~~~\\
s-m^2~~~~~~~2m^2~~~~~~~~~~s-s_2+t_1~~~~~~~~~2m^2-t_2~~~~\\
m^2-t_1~~~~~s-s_2+t_1~~~~~~2m^2~~~~~~~s-s_1-s_2+m^2~~~~\\
s-s_1+t_2-m^2~~~2m^2-t_2~~~~s-s_1-s_2+m^2~~~2m^2\end{array}\right|\,.$$

To derive the studied distribution, we have to integrate the
differential cross section with respect to the variables $t_1$ and
$t_2.$ The limits of integration can be obtained from the condition
of the positivity of $(-\Delta).$ Solving equation $\Delta=0$ with
respect to the variable $t_1,$ we obtain \bge\label{eq:5}
t_{1-}<t_1<t_{1+}\,, \ \ \ t_{1\pm}=\frac{A\pm
2\sqrt{B}}{(s-s_1)^2-2(s+s_1)m^2+m^4}\,, \ene
$$A=s_1s_2(s_1-t_2)+s^2t_2-s[s_2t_2+s_1(s_2+t_2)]+
m^2[s^2+(s_1+s_2)(s_1+t_2)]-m^4(2s+4s_1+t_2)+m^6\,,$$
$$B=\big[st_2(s-s_1+t_2)+m^2(s_1^2-2st_2-s_1t_2)+
m^4t_2\big]\big[s_1s_2(s_1+s_2-s)+m^2(s^2-3s_1s_2)-2m^2s+m^6\big]\,.$$

The limits of the second integration over the variable $t_2$, at fixed
$s_1$ and $s_2$, are determined as the roots of the first factor in
the expression B, namely
\bge\label{eq:6}
t_{2-}\,<t_2<t_{2+}\,, \ \ \ t_{2\pm}=\frac{C\pm (s-m^2)\lambda_1}{2s}\,,
\ene
$$ C=s_1(s+m^2)-(s-m^2)^2\,, \ \lambda_i=\sqrt{(s-s_i)^2-2m^2(s+s_i)+m^4}\,, \ i=1\,, 2\,.$$

The roots of the second factor in the expression B define the region
of variation of $s_1$ and $s_2$ which is shown in the left panel of Fig.~2.

\begin{figure}
\includegraphics[width=0.4\textwidth]{reg3.eps}
\hspace{0.4cm}
\includegraphics[width=0.4\textwidth]{regint.eps}

\caption{The allowed invariant masses $s_1$ and $s_2$ for the
unrestricted (left panel, the total region) and the restricted by
the inequalities (\ref{eq:24}) (left panel, the shaded region) at
$s=10^{-4}$GeV$^2$. The six regions with independent integration
boundaries over the variables $t_2$ and $t_1$ are shown in right
panel.}

\end{figure}

\vspace{0.5cm}

This symmetrical region is restricted by the lines
\bge\label{eq:7}
s_{2-}<s_2<s_{2+}\,, \ \ 4m^2<s_1< (\sqrt{s}-m)^2\,, \ \ s_{2\pm}=\frac{1}{2}\bigg[s-s_1+3m^2\pm \lambda_1\sqrt{1-\frac{4m^2}{s_1}}\,\,\bigg]\,.
\ene

The boundaries of the integration with respect to the variables
$t_2$ and $t_1$, given by Eqs.\,(\ref{eq:5}, \ref{eq:6}), are valid
only for the unrestricted phase space in Fig.\,2 defined by
Eq.\,(\ref{eq:7}).

\subsection{Calculation of the QED cross-section}

For unpolarized particles we must average over the initial particles
polarizations and sum with respect to the final ones . The
differential cross section, in this case, can be written in the form
\bge\label{eq:8}
d\sigma=\frac{1}{4}\,\frac{e^6}{4(k\,p)\,(2\pi)^5}\,\frac{1}{2}\,\sum_{pol}|M|^2\,dR_3\,,
\ene where $M$ is the matrix element of the process (1) which takes
into account the contributions of eight diagrams.  To derive the
distribution over the variables $s_1$ and $s_2$, one has to detect
both final electrons, and additional factor 1/2 in Eq.~(\ref{eq:8})
before $\sum_{pol}|M|^2$ arises due to their identity.

The total matrix element squared of the process can be written as follows
\bge\label{eq:9}
\sum_{pol}|M|^2=|M_b|^2+|\overline{M}_b|^2+|M_c|^2+|\overline{M}_c|^2-2\mathfrak{Re}\big(M_b\overline{M}^*_b\Big)-
\ene
$$-2\mathfrak{Re}\big(M_c\overline{M}^*_c\big)-2\mathfrak{Re}\big(M_b\overline{M}^*_c\big)-2\mathfrak{Re}\big(\overline{M}_bM^*_c\big)+
2\mathfrak{Re}\big(M_bM^*_c\big)+2\mathfrak{Re}\big(\overline{M}_b\overline{M}^*_c\big),
$$ where index $b\,\,(c)$ corresponds to the diagrams of the
Borsellino (Compton-like) type and the bar means the permutation of
the final electrons in the corresponding diagrams.

The double differential cross section with respect to the variables
$s_1$ and $s_2$ (or the $s_1,\,\,s_2-$distribution) reads
\bge\label{eq:10}
\frac{d\,\sigma}{d\,s_1\,d\,s_2}=\frac{\alpha^3}{64\,\pi(s-m^2)^2}\int\int\frac{d\,t_1\,d\,t_2}{\sqrt{-\Delta}}\frac{1}{2}\sum_{pol}|M|^2\,,
\ene where the limits of integration over the variables $t_1$ and
$t_2$ are determined by the choice of the event selection cuts. If
the whole kinematic region is allowed, these limits are defined by
the restrictions (\ref{eq:5}) and (\ref{eq:6}), and the
$s_1,\,\,s_2-$distribution is symmetrical under the permutation
$s_1\rightleftarrows s_2$, provided the identity of the final
electrons is taken into account.

Every matrix element is the contraction of the corresponding current
$j_\mu$ with the photon polarization 4-vector $A^\mu$
\bge\label{eq:11}
M_b=\frac{1}{t_2}A_{\mu}j_b^{\mu}\,, \ \ \overline{M}_b=\frac{1}{t}A_{\mu}\bar{j}_b^{\mu}\,,
\ene
$$M_c=\frac{1}{s_1}A_{\mu}j_c^{\mu}\,, \ \ \overline{M}_c=\frac{1}{s_2}A_{\mu}\bar{j}_c^{\mu}\,, \ t=(p-p_1)^2=2m^2-s-t_1+s_2\,.$$

The currents corresponding to these four types of the diagrams can be
written as
\bge\label{eq:12}
j_b^{\mu}=\bar u(p_2)\gamma_{\lambda}u(p)\bar u(p_1)\widehat{Q}^{\mu\lambda}v(p_3)\,, \ \
\widehat{Q}^{\mu\lambda}=\frac{1}{2d_1}\gamma^{\mu}\hat
k\gamma^{\lambda}-\frac{1}{2d_3}\gamma^{\lambda}\hat
k\gamma^{\mu}+e\,^{\mu}_{(31)}\gamma^{\lambda}\,,
\ene
$$\bar{j}_b^{\mu}=\bar u(p_1)\gamma_{\lambda}u(p)\bar u(p_2)
\widehat{R}^{\mu\lambda}v(p_3)\,, \ \
\widehat{R}^{\mu\lambda}=\frac{1}{2d_2}\gamma^{\mu}\hat
k\gamma^{\lambda}-\frac{1}{2d_3}\gamma^{\lambda}\hat
k\gamma^{\mu}+e\,^{\mu}_{(32)}\gamma^{\lambda}\,, $$
$$j^c{\mu}=\bar u(p_1)\gamma_{\lambda}v(p_3)\bar u(p_2)
\widehat{K}^{\mu\lambda}u(p), \ \
\widehat{K}^{\mu\lambda}=\frac{1}{2d_2}\gamma^{\mu}\hat
k\gamma^{\lambda}+\frac{1}{2d}\gamma^{\lambda}\hat
k\gamma^{\mu}+e\,^{\mu}_{(02)}\gamma^{\lambda}, $$
$$\bar{j}_c^{\mu}=\bar u(p_2)\gamma_{\lambda}v(p_3)\bar u(p_1)
\widehat{L}^{\mu\lambda}u(p), \ \
\widehat{L}^{\mu\lambda}=\frac{1}{2d_1}\gamma^{\mu}\hat
k\gamma^{\lambda}+\frac{1}{2d}\gamma^{\lambda}\hat
k\gamma^{\mu}+e\,^{\mu}_{(01)}\gamma_{\lambda}, $$
where $d=(k\, p)\,, \ d_i=(k\,p_i)$ and
$$e\,^{\mu}_{(0i)}
=\frac{p^{\mu}}{d}-\frac{p^\mu_i}{d_i}, \ \ e\,^{\mu}_{(ij)}
=\frac{p^\mu_i}{d_i}-\frac{p^\mu_j}{d_j}, \ \ i,j=1,2,3. $$

It is easy to verify that every current in the relations
(\ref{eq:12}) satisfies the condition $j^\mu\,k_\mu=0.$ If to define
\bge\label{eq:13}
J^\mu=\frac{j_b^\mu}{t_2}-\frac{\bar{j}_b^\mu}{t}+\frac{j_c^\mu}{s_1}-\frac{\bar{j}_c^\mu}{s_2}\,,
\ene then, for the case of unpolarized particles, we can write
\bge\label{eq:14} \sum_{pol}|M|^2=-g_{\mu\nu}\,J^\mu\,J^{\nu^*}\,.
\ene

The right hand side of Eq.~(\ref{eq:9}) includes three different
structures: four squares of the separate matrix elements, four their
interferences, which enter with negative sign, and two interferences
entering with positive sign. To find the total matrix element
squared, it is enough to calculate only one contribution for every
structure and the other ones can be obtained by means of definite
substitutions. We calculate directly $|M_b|^2\,, \
M_b\,\overline{M}_b^*$ and $M_b\,M_c^*.$ Result reads

 $$ |M_b|^2=\frac{8 }{t_2^2}\bigg\{-4\,m^2 \Big(-\frac{2 d^2}{d_3^2}+\frac{d_3}{d_1}+\frac{d_1}{d_3}\Big) +8 \,(p\,p_3)^2
   \Big[\Big(\frac{1}{d_3}+\frac{1}{d_1}\Big)^2 m^2-\frac{t_2}{d_1\, d_3}\Big]+$$
   $$ +4\,(p\,p_3) \Big[ \frac{t_2^2}{d_1\,d_3}+ t_2 \Big(\frac{2 d}{d_1 d_3}+\frac{1}{d_3}-\frac{1}{d_1}\Big)-
 \frac{4m^2\,d }{d_3}\Big(\frac{1}{d_3}+\frac{1}{d_1}\Big)-m^2\,t_2 \Big(\frac{1}{d_3}+\frac{1}{d_1}\Big)^2\Big]+$$
   $$+t_2^2 \Big[m^2\Big(\frac{1}{d_1}-\frac{1}{d_3}\Big)^2 -\frac{2(d+d_1)}{d_1\,d_3}\Big]+4m^2 t_2
   \Big(\frac{1}{d_3}+\frac{1}{d_1}\Big)\Big[m^2\Big(\frac{1}{d_3}+\frac{1}{d_1}\Big)+\frac{d}{d_3}-2\Big]-$$
\bge\label{eq:15}
   -\frac{t_2^3}{d_1\,d_3}-2\,t_2
   \Big(\frac{d_1}{d_3}+\frac{d_3}{d_1}+\frac{2 d \left(d-d_3\right)}{d_3 d_1}\Big)\bigg\}\,,
\ene

     $$M_b\,\overline{M}_b^*=\frac{8 }{t_2\,t}\bigg\{t_2\Big[\frac{5 d^2-10 d_1 d+d_3^2}{d_1\,d_2}-\frac{m^2(3 d_2^2+4 d_3 d_2+3 d_3^2+2 d d_1)+2m^4(d_2+d_3)}{d_1\,d_2\,d_3}+$$
  $$ +\frac{8 d}{d_2}+\frac{(d_1+d_2){}^2}{d_1\,d_3}\Big]+t\Big[-\frac{2 d^2 \left(d_1-d_3\right)}{d_1
   d_2 d_3}-\frac{2 (d_1+d_3) m^4}{d_1 d_2 d_3}+ t_2\Big(\frac{(d_1+d_2){}^2+6 d\,d_3}{d_1\,d_2\,d_3}-$$
   $$-2 m^2
   \Big(\frac{1}{d_3}+\frac{1}{d_1}\Big)\Big(\frac{1}{d_3}+\frac{1}{d_2}\Big)\Big)-\frac{\left(3 \left(d_1+d_3\right){}^2+2 d d_2-2 d_1
   d_3\right) m^2}{d_1 d_2 d_3}+$$
   $$+\frac{t_2^2}{2} \Big(\frac{2}{d_1 d_3}+\frac{3}{d_1 d_2}+\frac{1}{d_3 d_2}\Big)+\Big(\frac{3 d}{d_2
   d_3}+\frac{1}{d_1}\Big) \left(d-d_3\right)+\frac{4 d \left(d-d_2\right)}{d_1 d_2}\Big]+$$
      $$+4\,m^6 \Big(\frac{1}{d_3}+\frac{1}{d_1}\Big)
   \Big(\frac{1}{d_3}+\frac{1}{d_2}\Big) -\frac{4 \left(\left(d_1+d_2\right){}^2+3 d d_3\right) m^4}{d_1 d_2 d_3}+t_2^2
   \Big[\frac{3 d}{d_1 d_2}-\frac{1}{d_2}+\frac{d}{d_1
   d_3}-$$
      $$-m^2\Big(\frac{1}{d_3}+\frac{1}{d_1}\Big)\Big(\frac{1}{d_3}+\frac{1}{d_2}\Big)\Big] +t^2 \Big[-m^2\Big(\frac{1}{d_3}+\frac{1}{d_1}\Big) \Big(\frac{1}{d_3}+\frac{1}{d_2}\Big)+\frac{t_2(2d_1+d_2+3d_3)}{2\,d_1\,d_2\,d_3}- $$
   $$-\frac{1}{d_1}+\frac{3 d}{d_1 d_2}+\frac{d}{d_2 d_3}\Big]-2m^2 \Big(3
   -\frac{d}{d_2}+\frac{d}{d_3}-\frac{d}{d_1}\Big)+\frac{t_2^3(d_2+d_3)}{2d_1\,d_2\,d_3}+$$
\bge\label{eq:16}
   +\frac{t^3(d_1+d_3)}{2d_1\,d_2\,d_3}+
   \frac{2 d \left(\left(d_1+d_2\right){}^2+2 d d_3\right)}{d_1 d_2}\bigg\}\,,
\ene

    $$M_b\,M_c^*=\frac{1}{t_2\,s_1}\bigg\{\frac{4 \left(4 m^2+t_2\right) \left(d d_1-d_2 d_3\right) t_2^2}{d\,d_1\,d_2\,d_3}+\frac{8 [ d_1(2d^2+dd_1-d_2d_3
    -dd_2)-dd_2d_3 ]t_2^2}{d\,d_1\,d_2\,d_3}+$$
   $$+\frac{8t_2 \left[3 d_1 d^3+\left(2 d_1^2-d d_2\right) \left(d \left(d+d_1\right)+\left(d_1-d\right) d_2\right)+d_2^2 \left(4
   d_1^2+2 d_2 d_1-d_2 d_3\right)\right] }{d\,d_1\,d_2\,d_3}+$$
   $$+\frac{16 m^2t_2 \left(d-d_2\right) \left(-d_1^2+4 d d_1+d_2^2\right)}{d\,d_1\,d_2\,d_3}-\frac{64
   \left(p\,p_3\right){}^3 \left(d_1+d_3\right){}^2}{d\,d_1\,d_2\,d_3}+$$
   $$+\frac{16 \left(d_1+d_3\right) \left[\left(d_1+d_3\right){}^2+2 d d_2+2 d_1
   \left(d_1+d_2\right)\right]}{d_2\,d_3}+$$
   $$+\frac{32 m^2 \left[\left(d_1+d_2\right){}^3+d_1^2 d_3+d \left(d_2^2+\left(3 d-d_3\right) d_3\right)\right]}{d d_2
   d_3}+$$
   $$+\frac{32(p\,p_3)^2}{d\,d_1\,d_2\,d_3} \Big[t_2\big((d+d_1)(d_1+d_3)-2 d_2 d_3\big)+(d-d_2)(4dd_1-2d_1d_2+d^2-d_2^2)\Big]+$$
   $$+\frac{8(p\,p_3)}{d\,d_1\,d_2\,d_3}\Big[t_2^2(d_2d_1+3d_2d_3-dd_3-3dd_1)-4m^2(d-d_2)(d^2+d_2^2+2d_1d_3)-$$
   $$-4m^2t_2(d_1+d_3)^2 -2t_2(2dd_3^2+2d_2d_3^2+7d^2d_1+d^2d_2-3dd_2d_3-7dd_1d_2-d_2^3-d^2d_3)-$$
\bge\label{eq:17}
   -2(d-d_2)\big(4dd_1(d_1+d_2)+(d-d_2)(d+d_1)^2+(d-d_2)(d_1+d_2)^2\big)\Big]\bigg\}\,.
\ene

To obtain the expressions for $|\overline{M}_b|^2$ and
$\overline{M}_b\,\overline{M}_c^*$, we have to perform the
permutation
 $$\widehat{P}_{12}=( p_1\rightleftarrows p_2)$$
 in Eqs.~(\ref{eq:15}) and (\ref{eq:17}), respectively. Such permutation,
evidently, means the substitutions
\bge\label{eq:18}
d_1\rightleftarrows d_2\,, \ t_2\rightleftarrows t\,,  \  s_1\to
s_2\,,
\ene
in these equations, leaving the quantities $d\,, \ d_3$
and $(p\,p_3)$ unchanged.
 Note that the expression $M_b\,\overline{M}_b^*$ is invariant under these substitutions.

The rest of the contributions in Eq.~(\ref{eq:9}) are obtained similarly
by using another substitutions. If we define operators
\bge\label{eq:19}
\widehat{P}_{03}=(p\rightleftarrows-p_3)\,, \ \widehat{P}=(p\rightleftarrows p_1\,,  \ p_2\rightleftarrows-p_3\,,  \ k\to-k)\,,
\ene
then Eq.~(\ref{eq:9}) can be written as
\bge\label{eq:20}
\sum_{pol}|M|^2=\big(1+\widehat{P}_{12}+\widehat{P}_{03}+\widehat{P}_{12}\,\widehat{P}_{03}\big)|M_b|^2
+2\big(1+\widehat{P}_{12}\big)\overline{M}_b\,\overline{M}_c^*
\ene
$$-2\big(1+\widehat{P}_{12}\,\widehat{P}_{03} +\widehat{P}+\widehat{P}_{12}\,\widehat{P}\big)M_b\,\overline{M}_b^*\,.$$

\subsection{Dark photon contribution}

The effective
interaction Lagrangian of the dark photon with the S\,M
 electromagnetic current \cite{H86} can be written as
$$\mathfrak{L}=i\varepsilon\,e\overline{\psi}(x)\gamma^\mu\,\psi(x)\,A'_\mu(x)$$
where $A'_{\mu}$ is the 4$-$potential of the field $\gamma\,'$ and
the small parameter $\varepsilon$ characterizes the coupling
strength relative the electric charge $e.$ In this approach, the
dark photon has to manifest itself as an intermediate state in the
Compton-like Feynman diagrams with the ordinary Breit-Wigner
propagator for the spin-one particle
$$V^{\mu\nu}(q)=\Big(-g^{\mu\nu}+\frac{q^\mu\,q^\nu}{M^2}\Big)P^{BW}(q^2)\,, \ \ P^{BW}(q^2)=\frac{1}{q^2-M^2+iM\,\Gamma}\,,$$
where M ($\Gamma$) is the mass (total decay width) of the dark
photon.

The width of the dark photon decay to the SM lepton pair is
\begin{equation}\label{eq:21}
\Gamma (\gamma\,'\rightarrow
l^+l^-)=\epsilon^2\,\frac{\alpha}{3M^2}(M^2+2m_l^2)\sqrt{M^2-4m_l^2}= \epsilon^2\,\Gamma_0,
\end{equation}
where $m_l$ is the SM lepton mass. In our numerical calculations we restrict ourselves with analysis of the light dark photon signal and
suppose that its mass $M<200\,MeV$. At this condition the decay $A'\to\,\mu^+\,\mu^-$ is closed and, therefore, $m_l$ in Eq.\,(21) is the electron mass.

Due to the contribution of the dark photon, the matrix elements
$M_c$ and $\overline{M}_c$ are modified as
\begin{equation}\label{eq:22}
M_c \to M_c\,R(s_1)\,, \ \ \overline{M}_c \to \overline{M}_c\,R(s_2)\,, \ \ R(s)=1+s\,\varepsilon^2\,P^{BW}(s)\,,
\end{equation}
and this modification leads to the enhancement of the cross section
in two resonance regions: near $s_1\approx M^2$ and $s_2\approx
M^2$. Just in the resonance, the parameter $\varepsilon$ disappears
in the modification factor $R$ because the decay width $\Gamma
(\gamma\,'\rightarrow l^+l^-)$ is proportional to $\varepsilon^2$
$$R(s=M^2)=1-i\frac{M}{\Gamma_0}\,.$$

Taking into account the dark photon contribution, the modified matrix
element squared can be written as
$$\left|M\right|^2=\left|M_b-\overline{M}_b\right|^2+\left|M_c\right|^2\,\left|R(s_1)\right|^2+\left|\overline{M}_c\right|^2\,\left|R(s_2)\right|^2-
2\,\mathfrak{Re}\big[M_c\overline{M}_c\,^*\,R(s_1)\,R^*(s_2)\big]+$$
$$+2\mathfrak{Re}\big\{\big(M_b-\overline{M}_b\big)\big(M_c^*\,R^*(s_1)-\overline{M}_c^*\,R^*(s_2)\big)\big\}\,,$$
where
$$|R(s)|^2=1+\frac{s\epsilon^2}{D(s)}[2(s-M^2)+s\epsilon^2], \ \
D(s)=(s-M^2)^2+M^2\Gamma^2\,,$$
$$\mathfrak{Re}[R(s_1)R^*(s_2)]=1+\epsilon^2\Big\{\frac{s_2}{D(s_2)}(s_2-M^2)+
\frac{s_1}{D(s_1)}(s_1-M^2)+$$
$$+\frac{s_1s_2\epsilon^2}{D(s_1)D(s_2)} [(s_1-M^2)(s_2-M^2)+M^2\Gamma^2]\Big\}\,.$$

\section{Analysis of the dark photon signal}

At the photon energies more than 10\,MeV, the main contribution to
the cross section arises from the Borsellino diagrams due to the events
with small values of $t_2$ and $t$ \cite{BVMP1994}.
In Fig.\,3 we show the ratios $R^c_b$ (the upper row) and
$\overline{R}^c_b$ (the lower row) of the contributions, into
$s_1\,\,,s_2-$distribution caused by the Compton-like diagrams and the
Borsellino ones, defined as
$$
R^c_b=\frac{d\,\sigma_c}{d\,\sigma_b}\,, \ \ d\,\sigma_{c,\,b}=\frac{d\,\sigma}{d\,s_1\,d\,s_2}\Big(\frac{1}{2}\sum_{pol}|M|^2\,\to |M_{c,\,b}|^2\Big)\,,
$$
\begin{equation}\label{eq:23}
\overline{R}^c_b=\frac{d\,\overline{\sigma}_c}{d\,\overline{\sigma}_b}\,,
\ \ d\,\overline{\sigma}_{c,\,b}=
\frac{d\,\sigma}{d\,s_1\,d\,s_2}\Big(\sum_{pol}|M|^2\,\to
|M_{c,\,b}-\overline{M}_{c,\,b}|^2\Big)\,,
\ene
at $s=10^{-4}$\,GeV$^2$, 0.01\,GeV$^2$ and 1\,GeV$^2$ provided the
whole kinematical region is allowed. In this case, the double
differential cross section can be derived analytically by the
integration with respect to the $t_1$ and $t_2$ variables in the
limits defined by the restrictions (\ref{eq:5}) and (\ref{eq:6}).
The quantity $R^c_b$ does not take into account the identity effects
and it is not symmetrical under $s_1\leftrightarrows s_2$, whereas
the quantity $\overline{R}^c_b$ takes into account the identity
effects and it is symmetrical under this permutation.

\begin{figure}
\includegraphics[width=0.32\textwidth]{Rcb_1.eps}
\includegraphics[width=0.32\textwidth]{Rcb_01.eps}
\includegraphics[width=0.32\textwidth]{Rcb_0001.eps}

\vspace{1cm}

\includegraphics[width=0.32\textwidth]{Rtcb_1.eps}
\includegraphics[width=0.32\textwidth]{Rtcb_01.eps}
\includegraphics[width=0.32\textwidth]{Rtcb_0001.eps}

\caption{The ratio of the double differential distributions over the
invariant masses $s_1$ and $s_2$ for the unrestricted kinematical
region caused by the Compton-like diagrams and the Borsellino ones. The quantity
$R^c_b$  does not take into account the identity effects of the
final electrons and the quantity $\overline{R}^c_b$  takes into
account this effects.}

\end{figure}

In a wide, physically interesting, range of the variables $s_1$ and
$s_2$ the ratio $\overline{R}^c_b$ is rather small (does not exceed
2$\cdot$10$^{-2}$) and it is obvious that, in the case of the
nonrestricted phase space, the Borsellino contribution leads to a very
large QED background to search the not great dark photon signal,
which modifies the Compton-like contribution.

To reduce the Borsellino contribution, we suggest to remove the events,
with small values of the variables $t_2$ and $t$, by the kinematical
cuts
\begin{equation}\label{eq:24}
t_2\,<\eta\,s\,, \ \ t\,<\eta\,s\,, \ \eta<0\,,
\end{equation}
where $\eta$ is a parameter and in our
numerical calculations we use $\eta=-0.2$. It means that, in the rest
frame of the the initial electron, the energy of each final
electrons is more than $0.2\,\omega+1.1\,m\,,$ where $\omega$ is the
photon energy in this frame.

The corresponding symmetrical region of the invariants $s_1$ and
$s_2$ is shown in Fig.\,2 (the shaded region). In this reduced
region
$$4m^2<s_1\,, \, s_2\, <\frac{\eta s(s+m^2)+(s-m^2)\sqrt{\eta s(\eta\,s-4m^2)}}{2\,m^2}\,.$$

In this case, the shaded region in Fig.\,2 is divided into six
independent regions, and each region has its own integration
boundaries over the $t_2$ and $t_1$ variables. The detailed analysis
of the respective kinematics is given in Appendix A. One integration
can be performed analytically and the other one numerically, but the
$s_1\,\,,s_2-$distribution remains symmetrical due to the
symmetrical cuts on $t_2$ and $t$.

The event selection, in accordance with the constraints
(\ref{eq:24}) (with the restricted phase space), decreases
essentially the Borsellino contribution, whereas the Compton-like one is decreased
very little. Their ratio $\widetilde{R}^c_b$, for the restricted
phase space (that is analogue of the ratio $\overline{R}^c_b$), is
shown in Fig.\,4. Due to the final electrons identity,
 there is no possibility to distinguish between the
created electron and the recoil one. Thus, it is necessary to take
into account the effects of the identity.

\begin{figure}
\includegraphics[width=0.32\textwidth]{Rcutcb_1.eps}
\includegraphics[width=0.32\textwidth]{Rcutcb_01.eps}
\includegraphics[width=0.32\textwidth]{Rcutcb_0001.eps}

\caption{The ratio $\widetilde{R}^c_b$ for the restricted
kinematical region, defined by the inequalities (\ref{eq:24}), at
$\eta=-$0.2.}
\end{figure}

In Fig.\,5 we show the double differential cross section (that is
the QED background in search for the dark photon signal in the
process (1)) taking into account all the contributions in the matrix
element squared (\ref{eq:9}) and the constraints (\ref{eq:24}). We
use the dimensionless variables $x_1=s_1/s$ and $x_2=s_2/s$ to see
clearly how quickly this cross section decreases with the increase
of the invariant $s.$ As we noted, the cross section is symmetrical with respect to
the permutation $s_1\leftrightarrows s_2$ and this circumstance
removes the ambiguity of the interpretation of the variables $s_1$
and $s_2$ (due to the final electron identity) in the real
measurement: the event number does not depend what one takes as
$s_1$ or as $s_2.$ To demonstrate this more clearly, we show also
the 3-dimensional plot of the differential cross section at
$s$=10$^{-2}$\,GeV$^2$. In fact, the curves in the middle upper
panel are the intersections of the 3-dimensional plot with the
planes $x_1=0.1, \ x_1=0.3$ and $x_1=0.6.$

\begin{figure}
\includegraphics[width=0.32\textwidth]{x1.eps}
\includegraphics[width=0.32\textwidth]{x2.eps}
\includegraphics[width=0.32\textwidth]{x3.eps}
\includegraphics[width=0.5\textwidth]{BC1.eps}
\caption{The double differential cross section for the restricted
kinematical region, defined by the inequalities (\ref{eq:24}), at
$\eta=-$0.2 as a function of the variable $x_2$ at fixed values of the
variable $x_1.$
The 3-dimensional plot shows the $x_1,\,x_2-$symmetry.}
\end{figure}

Let us estimate the limits for the parameter $\epsilon$ following
Refs. \cite{BEST09,CT16}. We use the definition of the standard deviation
\begin{equation}\label{eq:25}
\sigma =\frac{S}{\sqrt{B}},
\end{equation}
where S(B) is the number of signal (background) events (
$\sigma =2$ corresponds to $\approx $ 95 $\%$ confidence
limit). The event number of any process $i$ is the product of the corresponding
cross section  and the integral luminosity  of the experimental device
$$N_i=d\,\sigma_i\cdot L\cdot T\,,$$
where $L$ is the luminosity, $T$ is the total event accumulation time and all differentials in $d\,\sigma_i$ are dimensionless.

We have also the following relation
\begin{equation}\label{eq:26}
\frac{S}{B}=\frac{d\sigma_{A'}(\varepsilon\,, M^2)}{d\sigma_Q}\,,
\end{equation}
where $d\sigma_{A'}$ is the calculated double differential distribution caused by
the dark photon mechanism
$$d\sigma_{A'}(\varepsilon\,, M^2)=\frac{\varepsilon^2\,s_1[2(s_1-M^2)^2+\varepsilon^2\,s_1]}{D(s_1)}d\,\sigma_c\,,$$
and $d\sigma_Q$ is pure QED contribution with accounting identity of the final electrons. Eqs.~(25) and (26) mean
\begin{equation}\label{eq:27}
\sigma\,d\,\sigma_Q=\sqrt{N}\,d\sigma_{A'}(\varepsilon\,, M^2)\,,
\end{equation}
where $N$ is the number of detected events at particular experimental conditions.
The experimental event selection is: the invariant mass $\sqrt{s_1}$ of the detected $e^+e^--$pair falls in the
energy region
$$M-\delta m/2 < \sqrt{s_1}  < M+\delta m/2\,,$$
where $\delta m$ is the experimental invariant mass resolution, i.e.,
the bin width containing almost all events of possible signal, whereas the invariant mass $s_2$ is fixed.
Then assuming $\Gamma\ll\delta m\ll M$ we can rewrite quantity $D(s_1)$ (see Eq.~(22) and text below) in the approximate form
$$\big[D(s_1)\big]^{-1}=\frac{\pi}{M\,\varepsilon^2\Gamma_0}\delta(s_1-M^2)\,.$$
After integration of the both parts of Eq.\,(27) over the variable $s_1$ within the bin interval $\delta m$ we obtain
\begin{equation}\label{eq:28}
\epsilon^2 =\frac{2\,\sigma}{\pi\sqrt{N}}\frac{\delta
m\Gamma_0}{M^2}\frac{d\sigma_Q (M^2, s_2)}{d\sigma_c (M^2,s_2)}\,.
\end{equation}

The last relation defines the constraints on the parameters $\varepsilon^2\,,\, M^2$ and the detected triplet event number $N$ at given
standard deviation $\sigma$.

In Fig.\,6 we illustrate these constraints at $\sigma=2$ for the energy bin value $1\,MeV$ and the event number $N=10^4.$ For every point in $(\varepsilon^2\,, \ M)$ region (at given values of $s$ and $s_2$) below curves, $\sigma<2$ and above curves,  $\sigma>2$. If the real $A'$ signal corresponds, at least, to three (or more) standard deviations, then the quantities $\varepsilon^2$ (at fixed $M$), when this signal can be recorded, are increased by 1.5 times
(or more) as compared with the corresponding points on the curves in Fig.\,6.

\begin{figure}
\includegraphics[width=0.32\textwidth]{dark1.eps}
\includegraphics[width=0.32\textwidth]{darkm2.eps}
\includegraphics[width=0.32\textwidth]{darkm4.eps}
\caption{The correlation between the $A'$ parameters $\varepsilon^2$ and $M$
at standard deviation $\sigma=2$ and fixed number of recorded triplet $N=10^4$ in the case
of event selection described in text.}
\end{figure}

It is easy to see from Eq.\,(28) that increasing of the energy bin value $\delta m$ decreases the sensitivity of the $A'$ signal detection in the process (1).
The reason is transparent because such experimental device increases the QED background which is, in fact, proportional to $\delta m$ and leaves changeless
the events number caused by the narrow $A'$ resonance.
The dependence of this sensitivity on the
dark photon mass $M$ is not smooth (especially at not great values of $s$) and is defined by interplay of the $M$-dependences of $\Gamma_0,\, d\sigma_Q$ and $d\sigma_c$ entering Eq.\,(28). The account for the kinematical restrictions (24) also increases the sensitivity due to the suppression of the QED background.
Our estimates show that at $s$=1\,GeV$^2$ this suppression is about two orders.

The events number $N$ in the denominator of the right hand side of Eq.\,(28), at described event selection, can be written with a good approximation as
\begin{equation}\label{eq:29}
N=\frac{2\,\delta m\,M}{s}\frac{d\sigma_Q}{d\,x_1\,d\,x_2}\Big(x_1=\frac{M^2}{s}\,, x_2=\frac{s_2}{s}\Big)\,L\cdot\,T \,.
\end{equation}
Using this formula, it is easy to estimate the necessary integral luminosity to accumulate 10$^4$  events. Taking for $d\,\sigma_Q$ values (10$^{-28}$$-$
10$^{-32})$\,cm$^{2}$ (as it follows from the curves in Fig.\,5) and values 10$^{-5}$$-$10$^{-1}$ for $2\,\delta\,m\,M/s$, we obtain the range
$$L\cdot\,T\approx (10\,^{32} - 10\,^{40})\, {\rm cm}^{-2}$$
at considered values of $s$ between 100\,MeV$^2$ and 1\,GeV$^2$.
The largest energies require the largest integral luminosity and vice versa.

The similar estimation were performed by the members of the IRIDE Collaboration (Frascati, Italy) for the
electron-photon collider with the photon range energies (1$-$100)\,MeV and the electron one (100$-$1000)\,MeV, taking into account only the Compton-like
contribution (without final electron identity) \cite{IRIDE}.
Assuming a conservative detector resolution 5\,MeV for the invariant mass of $e^+e^-$-pair
they analysed the sensitivity on $\epsilon^2$ as function of the dark photon mass at different $e^-\,\gamma$ collision energies and in Fig.\,9.8 have
shown their result at integral luminosity 10$^{37}\,{\rm cm}^{-2}$. It indicates an increase of sensitivity for the energy-beam configuration with
photons and electrons of lower energies and this is agreed, at least on the quality level, with our results.

Unfortunately, the search for the light dark photons in the energy region of a few to hundreds MeV in the triplet production,
are not possible at present electron-photon colliders because of the low photon intensities of the machines. But we hope that it will stand  feasible in
a near future.

\section{Conclusion}

In this paper, we have investigated the process of the triplet
photoproduction on a free electron, $\gamma e^-\to e^+e^-e^-,$ which
can be used in the search for a dark photon $A'$. It characterized by
unknown both the mass $M$ and parameter $\epsilon$, describing the coupling
strength relative the electrical charge e, and  can manifest itself in
this reaction being produced as a virtual state with subsequent
decay into $e^+e^-$- pair. The advantage of this
process is that the background to the $A'$ signal is a pure QED process
and it can be calculated with the required precision.

We include the intermediate $A'$ state in two Compton-like diagrams when the virtual dark photon
is time-like, and near the resonance these diagrams can give the observable contribution.
Because near resonance the $A'$ amplitude is mainly imaginary, it practically does not interfere
with pure real QED amplitudes.
Thus, the $A'$ signal is proportional to Compton-like diagram contribution into cross section. As concerns
QED background, in our calculations we take into account all eight Feynmann diagrams.

After trivial azimuthal integration, we performed integration over two squared transferred
momenta $t_1$ and $t_2$, defined by the relations (2), and calculated double differential distribution
over the invariant masses $s_1$ and $s_2$ of two $e^+e^-$ pairs. The boundaries of the variables $t_1$ and $t_2$
for the total phase space of the final particles are obtained from analysis of the Gramian determinant entering
Eq.\,(4). In this case, both integrations are performed analytically (all necessary intermediate results are given in Appendix B) and
the main contribution into cross section, which does not decrease (and even increases logarithmically) when the collision energy grows,
is caused by Borsellino diagrams.

In such situation, the QED background is very large and
to decrease it we restrict the phase space
of the final particles to suppress mainly the Borsellino contribution (see the inequalities (24)). In fact, these inequalities
exclude two regions where the nondecreasing contribution into the cross section, with the growth of the collision energy, is accumulated.
This procedure requires for the detailed study of the kinematics based on the combined analysis of the Gramian determinant and
above inequalities.  The results of this combined analysis are given in Appendix A. In this case, we perform analytically only one integration
and the other one $-$ numerically. The corresponding effect (due to the restriction of the phase space)
can be seen comparing the Compton-like and Borsellino contribution ratios $\bar{B}^c_b$ and $\tilde{B}^c_b$
shown in Fig.\,3 and Fig.\,4, respectively.

The precise analytical results, taking into account all eight diagrams, for the triplet production, given in Appendixes A and B, are new.

As we noted, the considered double differential cross section is symmetrical with respect to the
permutation $s_1 \rightleftarrows s_2$ and this circumstance removes
the ambiguity of the interpretation of the variables $s_1$ and $s_2$
(due to the final electron identity) in the real measurement: the
event number does not depend what one takes as $s_1$ or as $s_2.$  This cross section is shown in Fig.\,5
as a function of the dimensionless variables $x_1=s_1/s$ and $x_2=s_2/s.$

We estimate what value of the strength coupling parameter $\epsilon$, as a function of
the dark photon mass $M$, can be obtained at given number of the measured events $N$ and the value of the standard deviation $\sigma$
assuming some ideal (it may be not realized experimentally but may be used in Monte Carlo simulation) event selection device.
The correlation between $A'$ parameters at $\sigma=2$ and $N=10^4$, at considered rule for the event selection, is shown in Fig.\,6. The curves in
Fig.\,6 indicate the values of  $\epsilon^2$ and $M$ at which the $A'$ signal can manifest itself on the level of two standard deviations at
different collision energies.
Two standard deviations is not enough to interpret the corresponding effect as a manifestation of a new physics, and ordinary one can speak about it on the level three and more.
It easy to recalculate the curves in Fig.\,6 for arbitrary values of $\sigma$ bearing in mind that, in accordance with Eq.\,(28), $\varepsilon^2\sim\sigma.$
As concerns the radiative corrections in the process of the triplet production, as far as we know, they where calculated in the Weizsacker-Williams approximation for the positron spectrum and the total cross section \cite{MO65,VKM74} and never has been considered in suggested experimental setup.  But we think that radiative corrections can not essentially shift the curves in Fig.\,6 because they are no more than a few percent for both $d\,\sigma_Q$ and $d\,\sigma_c.$

The approximate formula (29) allows to estimate (at chosen event selection) the necessary integral luminosity needed to accumulate $N$ triplet events. For $N=10^4$ we received for it $(10^{32}-10^{40}){\rm cm^{-2}}.$

On the quality level our results relative to the sensitivity on $\epsilon^2$ as function of the dark photon mass at different $e^-\,\gamma$ collision energies are agreed with the IRIDE Collaboration estimations, although only Compton-like diagrams has been considered there. Both calculations indicate an increase of sensitivity for the lower $\gamma\,e^-$ collision energies.

\begin{center}
{\Large{\bf{Acknowledgments}}}
\end{center}

This work was partially supported by the Ministry of Education and
Science of Ukraine (Projects No. 0115U000474 and No. 0117U004866).

\section*{Appendix A}
\setcounter{equation}{0}
\def\theequation{A\arabic{equation}}

In this Appendix, we present the integration region with respect to
the variables $t_1$ and $t_2$ taking into account the constraints
(\ref{eq:24}). Firstly, we introduce the dimensionless variables
$t_{1n}$ and $t_{2n}$ in such a way that
$$t_1=\frac{t_{1n}-z_2}{z_1}\,, \ \ t_2=\frac{t_{2n}-z_4}{z_3}\,, $$
$$z_1=-\frac{1}{t_{1-}-t_{1+}}\,, \ z_2=\frac{t_{1-}}{t_{1-}-t_{1+}}\,, \ z_3=-\frac{1}{t_{2-}-t_{2+}}\,, \ z_4=\frac{t_{2-}}{t_{2-}-t_{2+}}\,,$$
where $t_{1\pm}$ and $t_{2\pm}$ are defined in the relations
(\ref{eq:5}) and (\ref{eq:6}), respectively. If the phase space is
unrestricted, we have
$$\int\limits_{t_{2-}}^{t_{2+}}\int\limits_{t_{1-}}^{t_{1+}}d\,t_1\,d\,t_2=\int\limits_0^1\int\limits_0^1 \frac{d\,t_{1n}\,d\,t_{2n}}{z_1\,z_3}$$
for every point $s_1,\,s_2$ from the total region in Fig.\,1 (left panel).

In the case of the restricted phase space, we have six regions of
the variables $s_1,\,s_2$ with well defined boundaries (see Fig.\,1
right panel), and every point $(s_1,\,s_2)$, from each of these
regions, has its own region of integration over the variables $t_1$
and $t_2$. In Fig.\,7, we show all six different $s_1,\,s_2$ regions
and the corresponding variation regions of $t_{1n}$ and $t_{2n}$ at
$s=0.01\,$GeV$^2$ and $\eta=-0.2$.

The $(s_1,\,s_2)$ region {\bf{1}} is defined as
\begin{equation}\label{eq:A1}
B_1(\eta,\,s,\,s_1)<s_2<T_{1+}(\eta,\,s,\,s_1)\,, \ T_{1\pm}=\frac{A_1\,\pm 2\sqrt{C_1\,D_1}}{F_1}\,, \ B_1=\frac{\overline{A}_1}{2\,\overline{F}_1}\,,
\end{equation}
$$A_1=s^2\big[s(1+3\eta+2\eta^2)+s_1(\eta^2-2\eta-1)\big]-m^2\big[s(3+4\eta)+6\eta ss_1-2s_1^2\big]+m^4\big[s(3+\eta)+s_1\big]-m^6\,,$$
$$C_1=\eta s^2(s+\eta s-s_1)+m^2\big[s_1^2-\eta s(2s+s_1)\big]+\eta sm^4\,, \ F_1=(1+\eta)^2s^2-2m^2(s+\eta s+2s_1)+m^4\,,$$
$$D_1=s^2s_1\eta(1+2\eta)+m^2\big[(1+\eta)^2s^2-ss_1(2+7\eta)+s_1^2\big]-2m^4(s+\eta s-s_1)+m^6\,,$$
$${\overline{A}_1}=s^2(s-s_1)\big[\eta(s-s_1)-s_1\big]-sm^2\big[s_1(\eta+2s_1)+\eta(s^2+4ss_1-s_1^2)\big]+m^4\big[s_1(5s+s_1)-\eta s(s+2s_1)\big]+$$
$$+m^6(\eta s-3s_1)+\lambda_1(s-m^2)\big\{s\big[s_1+\eta(s+s_1)\big]-m^2\big[s(2+\eta)+s_1\big]+2m^4\big\}\,,$$
$${\overline{F}_1}=-s^2s_1+m^2(s^2+s_1^2)-m^4(2s+3s_1)+m^6\,.$$

If the point $s_1,\,s_2$ belongs to the region {\bf{1}}, the
corresponding $(t_{1n},\,t_{2n})$ region {\bf{1}} in Fig.\, 7 is
splitted up into two parts
\begin{equation}\label{eq:A2}
\big\{0\,<t_{1n}\,<1,\,\, \ 0\,<t_{2n}\,<X_{1-}\big\}; \ \big\{Z_1\,<\,t_{1n}\,<1,\, \ X_{1-}\,<t_{2n}\,<Y_1,\big\}\,,
\end{equation}
$$Z_1=\frac{s+\eta s-s_2+t_{1-}-2m^2}{t_{1-}-t_{1+}}\,, \  \ Y_1=\frac{-\eta s+t_{2-}}{t_{2-}-t_{2+}}$$
$$X_{1\pm}=\frac{1}{2}-\frac{G_1\,H_1 \pm 4s\,K_1}{2(s-m^2)\lambda_1\,\lambda_2^2}\,,$$
$$G_1=s^2-s_1s_2-s_{12}(s-m^2)-m^4\,, \ H_1=s(s+2\eta s-s_2)-m^2(2s+s_2)+m^4\,,$$
$$K_1=\sqrt{-C_1(s_1\to s_2)\,K}\,, \ K=s_1s_2(s-s_{12})-m^2(s^2-3s_1s_2)+2m^4s -m^6\,, \ s_{12}=s_1+s_2\,.$$

The constraints on the $(s_1,\,s_2)$ region {\bf{2}} in Fig.\,7
reads
\begin{equation}\label{eq:A3}
T_1(\eta,s,s_1)<s_2<s_{2+}\,, \ B_1(\eta,s,s_1)<s_2<s_{2+}\,, \ s_1<\frac{s-m^2}{2}\,,
\end{equation}
and the $(t_{1n},\,t_{2n})$ integration region {\bf{2}} is
\begin{equation}\label{eq:A4}
\big\{0\,<t_{1n}<\,1,\,\, 0\,<t_{2n}<\,X_{1-}\big\}; \ \ \big\{Z_1\,<t_{1n}<\,1,\,\, X_{1-}\,<t_{2n}<\,X_{1+} \big\}\,.
\end{equation}

In the $(s_1,\,s_2)$ region {\bf{ 3}}, the restrictions on the
variable $s_2$ are the same as in the region {\bf{2}} but they apply
at $s_1>(s-m^2)/2.$ The respective $(t_{1n},\,t_{2n})$ integration
region {\bf{3}} is very simple
\begin{equation}\label{eq:A5}
0\,<t_{1n}<\,1,\,\,0\,<t_{2n}\,<Y_1\,.
\end{equation}

In the $(s_1,\,s_2)$ region {\bf{4}} in Fig.\,7, we have
\begin{equation}\label{eq:A6}
T_{1-}<\,\,s_2\,<\,T_{1+}\,,
\end{equation}
provided an additional condition
$$ G_1\sqrt{-C_1(s_1\to s_2)}-H_1\sqrt{K}\,>\,0$$
is satisfied. The $(t_{1n},\,t_{2n})$ integration region {\bf{4}} is
$$Z_1\,<\,t_{1n}\,<\,1\,, \ \ X_{1-}\,<\,t_{2n}\,<\,Y_1\,.$$

The symmetrical $(s_1,\,s_2)$ region {\bf{5}} in Fig.\,7 can be
defined as
\begin{equation}\label{eq:A7}
s_{2-}\,<\,\,s_2\,<\,T_{1-}\,, \ \ s_1\,<\,T_{1-}(s_1\to s_2)\,.
\end{equation}
The $(t_{1n},\,t_{2n})$ integration region {\bf{5}} reads
\begin{equation}\label{eq:A8}
\big\{Z_1\,<\,t_{1n}\,<\,1\,, \ X_{1-}\,<\,t_{2n}\,<\, X_{1+}\big\}\,; \ \ \big\{0\,<\,t_{1n}\,<\,1\,, \ X_{1+}\,<\,t_{2n}\,<\, Y_1\big\}\,.
\end{equation}

At last, the $(s_1,\,s_2)$ region {\bf{6}} in Fig.\,7 is
\begin{equation}\label{eq:A9}
T_{1+}(s_1\to s_2)\,<\,s_1\,<B_2(\eta,s,s_2))\,, \ s_2\,>\,T_{1-}\,, \ B_2=\frac{A_2+C_2\sqrt{K_2}}{2\,F_2}\,,
\end{equation}
$$A_2=s^2\big[2\eta^2s^2+(s-s_2)(2\eta s-s_2)\big]+s^2m^2\big[2\eta^2s-s_2-2\eta(s+2s_2)\big]+m^4\big[s_2(5s+3s_2)-2\eta s(s+s_2)\big]+$$
$$+m^6(2\eta s-3s_2)\,, \ \
C_2=s(s+2\eta s-s_2)-m^2(2s+s_2)+m^4\,,$$
$$K_2=s^2s_2^2+2sm^2\big[2\eta s^2+2\eta s(s-s_2)-s_2(2s+s_2)+m^4\big[s_2(8s+5s_2)-4\eta s(2s+s_2)\big]+4m^6(\eta s-s_2)\big]\,,$$
$$F_2=(\eta s-s_2)\big[(1+\eta)s^2-m^2(2s+s_2)+m^4\big]\,.$$
The corresponding $(t_{1n},\,t_{2n})$ integration region {\bf 6}
\begin{equation}\label{eq:A10}
Z_1\,<\,t_{1n}\,<\,1\,, \ \ X_{1-}\,<\,t_{2n}\,<\,X_{1+}\,.
\end{equation}

\begin{figure}
\includegraphics[width=0.3\textwidth]{1s.eps}
\hspace{0.2cm}
\includegraphics[width=0.3\textwidth]{2s.eps}
\hspace{0.2cm}
\includegraphics[width=0.3\textwidth]{3s.eps}

\vspace{0.15cm}

\includegraphics[width=0.3\textwidth]{4s.eps}
\hspace{0.2cm}
\includegraphics[width=0.3\textwidth]{5s.eps}
\hspace{0.2cm}
\includegraphics[width=0.3\textwidth]{6s.eps}

\vspace{0.15cm}

\includegraphics[width=0.31\textwidth]{1t.eps}
\includegraphics[width=0.31\textwidth]{2t.eps}
\includegraphics[width=0.31\textwidth]{3t.eps}

\vspace{0.15cm}

\includegraphics[width=0.31\textwidth]{4t.eps}
\includegraphics[width=0.31\textwidth]{5t.eps}
\includegraphics[width=0.31\textwidth]{6t.eps}

\caption{The six kinematical regions, defined by the inequalities
(\ref{eq:24}), at $\eta=-$0.2 defined by the inequalities in Appendix A.}
\end{figure}

\section*{Appendix B}
\setcounter{equation}{0}
\def\theequation{B\arabic{equation}}

Here, we write the analytical results for the
contributions of different diagrams and their interferences into the
matrix element squared, as defined by the Eq.\,(\ref{eq:9}),
integrated over the variables $t_1$ and $t_2$ with the weight
$$\frac{\pi}{64\,(s-m^2)\,\sqrt{-\Delta}}\,.$$ We introduce the short notation
$$\frac{\pi}{64\,(s-m^2)}\int\limits_{t_{1-}}^{t_{1+}}d\,t_1 \int\limits_{t_{2-}}^{t_{2+}}d\,t_2\frac{W}{\sqrt{-\Delta}} \equiv \overbrace{W}\,.$$
Then we have
\begin{equation}\label{eq:B1}
\overbrace{|M_c|^2}=\frac{\pi^2}{2s^2s_1^2(s-m^2)^2\lambda_1^4}\Big[A_c+\frac{4s^2\,B_c\,L_1}{\lambda_1}\Big]\,, \ \ L_1=\ln\frac{m^2+s-s_1+\lambda_1}{2m\sqrt{s}}\,,
\end{equation}
$$A_c=s^2s_1 \left(s-s_1\right)\bigl[s^4+2\, s^3
   \left(s_1-s_2\right)+2\,s^2 \left(-7 s_1^2+s_2 s_1+s_2^2\right) +2\, s\, s_1^2 \left(9 s_1+7 s_2\right)-$$
 $$  -7 s_1^2 \left(s_1^2+2 s_2 s_1+2 s_2^2\right)\bigr]+$$
$$+s\,m^2\bigl[2 s^6+17 s^5s_1 -2 s^4s_1 \left(41 s_1+16 s_2\right) +2 s^3s_1 \left(32 s_1^2-42 s_2 s_1+13 s_2^2\right) +$$
   $$+2 s^2s_1^2 \left(19 s_1^2+142 s_2
   s_1+58 s_2^2\right) -s\,s_1^3 \left(41 s_1^2+172 s_2 s_1+126 s_2^2\right) +2 s_1^4 \left(s_1^2+2 s_2 s_1+2 s_2^2\right)\bigr]+$$
$$+m^4\bigl[22 s^6-87 s^5s_1 +s^4s_1 \left(319 s_1-18 s_2\right) -4 s^3s_1
   \left(74 s_1^2+140 s_2 s_1+15 s_2^2\right) +$$
  $$ +2 s^2s_1^2 \left(69 s_1^2+262 s_2 s_1+76 s_2^2\right) +s\,s_1^3 \left(-3 s_1^2-4 s_2 s_1+6 s_2^2\right) -s_1^4 \left(s_1^2+2 s_2 s_1+2 s_2^2\right)\bigr]+ $$
$$+m^6\bigl[-110 s^5+153 s^4s_1 +36 s^3s_1 \left(17 s_1+4 s_2\right) -
4 s^2s_1 \left(74 s_1^2+104 s_2 s_1-9 s_2^2\right) -$$ $$
   -2\,s\,s_1^2 \left(3 s_1^2+14 s_2 s_1+2 s_2^2\right)+ s_1^3 \left(5 s_1^2+12 s_2 s_1+6 s_2^2\right)\bigr]+$$
$$+m^8\bigl[182 s^4-117 s^3s_1 +
   s^2s_1 \left(191 s_1-102 s_2\right) +6 s\,s_1 \left(4 s_1^2+2 s_2 s_1-s_2^2\right) -2
   s_1^2 \left(5 s_1^2+12 s_2 s_1+3 s_2^2\right)\bigr]+$$
$$+m^{10}\bigl[-138 s^3+27 s^2s_1 -2s\,s_1 \left(9 s_1-8 s_2\right)
   +2 s_1 \left(4 s_1^2+10 s_2 s_1+s_2^2\right)\bigr] +$$
$$+m^{12}\left[s\left(50 s+11 s_1\right)+s_1 \left(s_1-6 s_2\right)\right]-5m^{14}(2s+s_1)+2m^{16}\,,$$

$$B_c=\left(s-s_1\right){}^2 s_1 \left(s^2-2 s_1 s+2 s_1^2\right)\big[s^2-2 s \left(s_1+s_2\right)+s_1^2+2 s_2^2+2 s_1 s_2\big]+$$
$$+2\,m^2\bigl[s^6-9 s_1 s^5+s^4s_1 \left(23 s_1+5 s_2\right) -2\,s^3 s_1 \left(11 s_1^2-3 s_2 s_1+4 s_2^2\right) +$$
$$+s^2s_1^2 \left(s_1^2-49 s_2 s_1-2 s_2^2\right)
   +s\,s_1^3 \left(11 s_1^2+58 s_2 s_1+26 s_2^2\right) -s_1^4 \left(5 s_1^2+20 s_2 s_1+14 s_2^2\right)\bigr]+$$
$$+m^4\bigl[-20 s^5+63 s^4s_1 -4  s^3s_1
   \left(27 s_1-7 s_2\right)+
   4 s^2s_1 \left(37 s_1^2+32 s_2 s_1+5 s_2^2\right) -$$
   $$-2\,s\, s_1^2 \left(54 s_1^2+157 s_2 s_1+48 s_2^2\right) +s_1^3 \left(37 s_1^2+146 s_2 s_1+62
   s_2^2\right)\bigr]+$$
$$+2\,m^6 \bigl[27 s^4-46
   s^3s_1 -30 s^2s_1 \left(3 s_1+s_2\right) +2\,s\, s_1^2 \left(71 s_1+77 s_2\right)
   -s_1^2 \left(47 s_1^2+103 s_2 s_1+10 s_2^2\right)\bigr]+$$
$$+m^8\bigl[-56 s^3+63 s^2s_1 +s\left(6 s_1 s_2-238 s_1^2\right)
   +6 s_1 \left(18 s_1^2+9 s_2 s_1-s_2^2\right)\bigr]+$$
$$+2\,m^{10} \left[7 s^2-9 s s_1 +s_1 \left(9 s_2-13 s_1\right)\right]+m^{12}(12\,s+s_1)-6\,m^{14}\,.$$

The quantity $\overbrace{|\overline{M}_c|^2}$ can be derived from
the quantity$\overbrace{|M_c|^2}$ by the simple permutation
$s_1\leftrightarrows s_2.$ For the interference of the $M_c$ and
$\overline{M}_c$ amplitudes, which is symmetrical under the
permutation $s_1\leftrightarrows s_2$, we have
\begin{equation}\label{eq:B2}
\overbrace{M_c\overline{M}^*_c}=-\frac{\pi^2}{(s-m^2)^2\,s_1\,s_2}\Big\{2\,D_{c\bar c}\,L_a
-\frac{2\,B_{c\bar c}}{\lambda_1^5}\,L_1+
\end{equation}
$$\frac{s-m^2}{4s^2}\Big[c_1-\frac{12\,s\,m^2(s-m^2)}{\lambda_1^4}c_2-\frac{2}{\lambda_1^2}c_3 \Big] +(s_1\leftrightarrows s_2)\Big\}\,,$$
$$L_a=\ln\Big(\frac{\sqrt{a}+\sqrt{4m^2+a}}{2m}\Big)\,, \ \ a=s-s_{12}-m^2\,, $$

$$D_{c\bar c}=\frac{(s-s_{12})^3-m^2(3s^2-3ss_{12}+s_{12}^2)+m^4(4s_{12}-9s)+3m^6}{\sqrt{a(4m^2+a)}}\,,$$

$$B_{c\bar c}=(s-s_1)^5(s-s_{12})^2-m^2(s-s_1)\big[8s^5-6s^4(5s_1+2s_2)+s^3(38s_1^2+31s_1s_2+5s_2^2)-$$
$$-s^2s_1(15s_1^2+14s_1s_2+4s_2^2)-s\,s_1^2(4s_1^2+13s_1s_2+9s_2^2)+s_1^3(3s_1^2+8s_1s_2+5s_2^2)\big]+$$
$$+m^4\bigl[24 s^5-2 s^4\left(23 s_1+15 s_2\right) +2 s^3s_2 \left(22 s_1+5
   s_2\right) +
   s^2s_1 \left(41 s_1^2-15 s_2 s_1-5 s_2^2\right) +$$
$$+s\,s_1^2 \left(-19 s_1^2+18 s_2 s_1+16 s_2^2\right) -s_1^3 s_2 \left(13 s_1+8 s_2\right)\bigr]+m^6\bigl[-35 s^4+8 s^3\left(2 s_1+5 s_2\right) -$$
$$-2 s^2\left(13 s_1^2-3 s_2 s_1+5 s_2^2\right) +s\,s_1 \left(43 s_1^2-s_2 s_1+3 s_2^2\right)
   +s_1^2 \left(-13 s_1^2+7 s_2 s_1+s_2^2\right)\bigr]+$$
$$+m^8\left[25 s^3-15
   s^2\left(s_1+2 s_2\right) +s\left(-37 s_1^2-32 s_2 s_1+5 s_2^2\right) +s_1 \left(26 s_1^2+s_2 s_1-2 s_2^2\right)\right]-$$
$$-m^{10}\big[6s^2-6\,s(3s_1+2s_2)+18\,s_1^2-13\,s_1s_2+s_2^2\big]-2m^{12}(s+2s_1+s_2)+m^{14}\,,$$

$$c_1=s(6 s^2-10 s s_{12}+3 s_{12}^2)+m^2\Big[38\,s^2-11\,s\,s_{12}+5\,s_{12}^2+m^2(10\,s+3\,s_{12})-6\,m^4+\frac{4\,m^2(a^2-4\,m^2)}{s-m^2}\Big]\,,$$
$$c_2=s\big[3\,s^3-s^2(9s_1+4s_2)+s(9\,s_1^2+10\,s_1s_2+s_2^2)-3\,s_1\,s_{12}^2\big]+$$
$$+m^2\big[3\,s^3+s^2(-22\,s_1+4s_2)+2\,s(9\,s_1^2+8\,s_1s_2-s_2^2)-s_1\,s_{12}^2\big]+$$
$$+m^4\big(-11\,s^2-25\,s\,s_1+4\,s\,s_2+5\,s_1^2+6\,s_1\,s_2+s_2^2\big)+m^6\big[s-4(2s_1+s_2)\big]+4\,m^8\,,$$

$$c_3=s^2 (3 s-4 s_1) (s-s_{12}){}^2+$$
$$+m^2s\big[-7 s^3+s^2(30 s_2-28 s_1) +s\,(39 s_1^2+20 s_2 s_1-7 s_2^2) +s_1 (-5 s_1^2-4 s_1 s_2+s_2^2)\big]-$$
$$-m^4\big[69 s^3+s^2(70 s_1-2 s_2) - s\,(25 s_1^2+16 s_2 s_1+3 s_2^2)+s_1 s_{12}^2\big]+$$
$$+m^6 \big[39\, s^2-22\, s\,(2 s_1+s_2) +5\, s_1^2+6 s_1\, s_2+s_2^2\big]+m^8 \big[30\, s-4 (2 s_1+s_2)\big]+4 m^{10}\,.$$

The Borsellino-contribution is more complicated
$$\overbrace{|M_b|^2}=\frac{4\pi^2}{s-m^2}\Big\{\frac{L_1^{(b)}}{\lambda_1}+\frac{L_2}{2\,\lambda_2}-\frac{L_{12}}{\lambda_{12}}
-\frac{2\,B_{s_1}\,L_{s_1}}{(s-m^2)\sqrt{s_1^5(s_1-4m^2)}}-\frac{B_{s_2}\,L_{s_2}}{s_1^4\sqrt{s_2^5(s_2-4m^2)^5}}+$$
$$+\frac{D_b\,L_a}{s_1^4\sqrt{a^5(4m^2+a)^5}}-\frac{s-m^2}{2\,s_1^4s_2^2}\Big[b_1-\frac{96\,m^8(s-2s_1-m^2)^2}{(s_2-4m^2)^2}-\frac{8\,m^4\,b_2}{s_2-4m^2}
-\frac{6\,m^4\,b_3}{a^2}-$$
\begin{equation}\label{eq:B3}
-\frac{6\,m^4(s-m^2)^2(s-s_1+3m^2)}{(4m^2+a)^2}-\frac{2\,s_1^2s_2^2\,b_4}{(s-m^2)^2}+\frac{2\,s_1^2\,b_5}{s-m^2}+\frac{b_6}{2(4m^2+a)}+\frac{b_7}{2\,a}\Big]\Big\}\,,
\end{equation}
$$L_1^{(b)}=\ln\Big(\frac{s(s-s_1)-m^2(2\,s+s_1)+m^4-(s-m^2)\lambda_1}{2\,m\,s_1\sqrt{s}}\Big)\,, \ L_{s_{1,2}}=\ln\Big(\frac{\sqrt{s_{1,2}}+\sqrt{s_{1,2}-4\,m^2}}{2\,m}\Big)\,,$$

$$L_{12}=\ln\Big(\frac{s_{12}-2m^2-\lambda_{12}}{2\,m\,\sqrt{s}}\Big)\,, \ \
\lambda_{12}=\sqrt{s_{12}^2-4m^2(s+s_{12})+4\,m^4}\,,$$
where $L_2=L_1(s_1\to s_2)$ and
$$B_{s_1}=s_1\big[s^2+s_2^2+s_{12}^2-s(s_1+2s_2)\big]+m^2(4\,s\,s_{12}-s_1^2-10\,s_1s_2-4\,s_2^2)-m^4\big[8\,s-3(s_1+4s_2)\big]-8\,m^6\,,$$
$$B_{s_2}=-s_1^2 s_2^4 \bigl[2 s^2-2 s\left(s_1+2 s_2\right) +s_1^2+4 s_2^2+2 s_1 s_2\bigr]+$$
$$+2 m^2\,s_1 s_2^3 \bigl[4 s_1^3+5 s_2 s_1^2+26 s_2^2 s_1+8 s_2^3+s^2 \left(8 s_1-4 s_2\right)-s
   \left(7 s_1^2+14 s_2 s_1+4 s_2^2\right)\bigr]+$$
$$+2\,m^4s_2^2\bigl[4 s^3 s_2 +s^2\left(-18 s_1^2+24 s_2 s_1+8 s_2^2\right) +4 s s_1 \left(3
   s_1^2+9 s_2 s_1+10 s_2^2\right) -$$
   $$-s_1 \left(8 s_1^3+s_2 s_1^2+129 s_2^2 s_1+92 s_2^3\right)\bigr]-$$
$$-8\,m^6s_2\bigl[6 s^3 s_2 + s^2\left(-2 s_1^2+20 s_2 s_1+19 s_2^2\right)
  +s s_2 \left(3 s_1^2+24 s_2 s_1+4 s_2^2\right) +s_1 s_2 \left(3 s_1^2-65 s_2 s_1-95 s_2^2\right)\bigr]+$$
$$ + 4\,m^8s_2\left[40 s^3+4 s^2\left(12
   s_1+29 s_2\right) +s\left(-24 s_1^2+16 s_2 s_1+70 s_2^2\right) +s_2 \left(-81 s_1^2-348 s_2 s_1+4 s_2^2\right)\right]-$$
$$-8\,m^{10}\left[24 s^3+92 s^2 s_2 +2 s s_2 \left(49 s_2-8 s_1\right) +s_2 \left(-10 s_1^2-140 s_2 s_1+17 s_2^2\right)\right]+$$
$$+16\,m^{12} \left[36 s^2+62 s s_2 +s_2 \left(23 s_2-20 s_1\right)\right]-32\,m^{14}(18\,s+13\,s_2)+192\,m^{16}\,,$$

$$D_b=s_1^2(s-s_{12})^4(2s^2-4\,s\,s_{12}+3\,s_1^2+4\,s_2^2+6\,s_1 s_2)+$$
$$+2\,m^2s_1(s-s_{12})^3\bigl[10 s_1^3+34 s_2 s_1^2+34 s_2^2 s_1+8 s_2^3+2 s^2 \left(5 s_1+6 s_2\right)
   -s \left(19 s_1^2+44 s_2 s_1+20 s_2^2\right)\bigr]-$$
$$-2 m^4(s-s_{12})^2 \bigl[12 s^4-4 s^3\left(3 s_1+5 s_2\right) +s^2\left(-45 s_1^2-76 s_2 s_1+8 s_2^2\right) +$$
   $$+2 s s_1 \left(39 s_1^2+121 s_2 s_1+68 s_2^2\right) -s_1 \left(29 s_1^3+143 s_2 s_1^2+165 s_2^2 s_1+52 s_2^3\right)\bigr]-$$
$$-2\,m^6\bigl[16 s^5+4 s^4\left(s_1-5 s_2\right)-4 s^3\left(45 s_1^2+70 s_2 s_1+6 s_2^2\right) +s^2\left(370 s_1^3+960 s_2 s_1^2+604 s_2^2
   s_1+44 s_2^3\right) -$$
   $$-s\left(291 s_1^4+1021 s_2 s_1^3+1116 s_2^2 s_1^2+408 s_2^3 s_1+16 s_2^4\right) +s_1 \bigl(81 s_1^4+356 s_2 s_1^3+543 s_2^2 s_1^2+
  348 s_2^3 s_1+80 s_2^4\bigr)\bigr]+$$
$$+m^8\bigl[40 s^4-16 s^3\left(13 s_1+15 s_2\right) +s^2\left(438 s_1^2+592 s_2 s_1+256 s_2^2\right) -4 s\bigl(131 s_1^3+113 s_2 s_1^2+10 s_2^2 s_1
   +10 s_2^3\bigr) + $$
   $$+269 s_1^4-16 s_2^4-256 s_1 s_2^3-276 s_1^2 s_2^2+224 s_1^3 s_2\bigr]+$$
$$+2 m^{10}\left[96 s^3-24 s^2\left(s_1+3 s_2\right) -2 s\left(87 s_1^2+210 s_2 s_1+44 s_2^2\right) +75 s_1^3+28 s_2^3+388 s_1 s_2^2+462 s_1^2
   s_2\right]-$$
$$-2\,m^{12}\left[68 s^2-4 s\left(49 s_1+51
   s_2\right) +245 s_1^2+4 s_2^2+156 s_1 s_2\right]-8\,m^{14}\left(20 s+25 s_1+19 s_2\right)+120\,m^{16}\,,$$

$$b_1=-s_1^2\left(s^2-2 s_1 s+s_1^2+4 s_2^2\right)-2 m^2 s_1 \left(8 s^2-19 s s_1 +17 s_1^2+8 s_2^2\right)-$$
$$-m^4(16s^2-64s s_1+149 s_1^2)+16 m^6(2 s-7 s_1)-16 m^8\,, $$
$$b_2= s_1^2 \left(2 s_1-s\right) +m^2(8 s^2-44 s_1 s+61 s_1^2)+44 m^4 s_1-8m^6- \frac{16 m^2 s_1^2 \left(2 m^2+s_1\right) }{s-m^2}\,,$$
$$b_3=s^2+2 s_1^2-3 s s_1-m^2(2 s-3 s_1)+m^4\,, \ b_4=s_2^2+s_{12}^2-8 m^2 s_2+8 m^4\,,$$
$$b_5=s_2^2(s_1+2 s_2)+4 m^2(s_1^2+s_2^2)+24 m^4 s_1+32 m^6\,,$$
$$b_6=
   s s_1^2\left(s-s_1\right){}^2+m^2\left(2 s^4+8 s^3 s_1 -21 s^2 s_1^2 +16 s s_1^3 -5 s_1^4\right)+$$
   $$+m^4\left(16 s^3+96 s^2 s_1 -153 s s_1^2 +50 s_1^3\right)
-3 m^6\left(20 s^2-120 s_1 s+49 s_1^2\right)-48 m^8(4 s-s_1)+234 m^{10}\,,$$
$$b_7=
\left(s-s_1\right){}^2 s_1^2 \left(2 s_1-s\right)+m^2\left(2 s^4-24 s^3 s_1 +81 s^2 s_1^2 -116 s s_1^3 +73 s_1^4\right)-$$
$$-m^4(16 s^3-160 s^2 s_1+367s s_1^2-272 s_1^3)
+m^6(36 s^2-248 s s_1+287 s_1^2)-16 m^8(2 s-7 s_1)+10 m^{10}-$$
$$-\frac{16 m^2 s_1^4(s_1+2 m^2)}{s-m^2}\,.$$

To obtain $\overbrace{|\overline{M}_b|^2}$ we have to permutate $s_1$ and $s_2$ in $\overbrace{|M_b|^2}.$
The contribution of the interference $\overbrace{M_b}$ and $M_b$ is defined as
\begin{equation}\label{eq:B4}
\overbrace{M_b\,\overline{M}_b^*}=\frac{\pi^2}{s-m^2}\Big\{\frac{(s-s_1-5m^2)L_1+(s-s_1+3m^2)L_1^{(b)}}{\lambda_1(s-s_1-m^2)}
-\frac{2\,D_{b\bar{b}}\,L_a}{\sqrt{a^5(4m^2+a)}}-
\end{equation}
$$-\frac{1}{[a(s-m^2)+s_1s_2]}\Big[\frac{2\,\overline{B}_{s_1}L_{s_1}}{(s-m^2)s_2^2\sqrt{s_1^3(s_1-4m^2)}}-
\frac{\overline{B}_{12}\overline{L}_{12}}{\sqrt{a^5[a(s-m^2)^2+4m^2s_1\,s_2]}}\Big]+$$
$$+\frac{12m^4-8m^2(4m^2+a)+(4m^2+a)^2}{(s-m^2)s_1^2\,s_2^2\,a^2(4m^2+a)}C_{b\bar{b}}+(s_1\rightleftarrows s_2)\Big\}\,,$$
$$\overline{L}_{12}=\ln\Big[\frac{(s-m^2)\sqrt{a}}{2m\sqrt{s_1\,s_2}}\Big(1+\sqrt{1+\frac{4m^2s_1\,s_2}{a(s-m^2)^2}}\Big)\Big]\,,$$
$$C_{b\bar{b}}=2m^2(s-m^2)^2(s_1^2+s_2^2)-s_1\,s_2(4m^2+a)(s_1+s_2)^2\,,$$
$$D_{b\bar{b}}=-3 s^2+2 s s_{12}-2 (s_1^2-s_2 s_1+s_2^2)+23 m^4+4 m^2 s+\frac{1}{s_1 s_2(s-m^2)}\big[8 m^6(s_1-s_2)^2-$$
$$-4 m^4s_{12}(s_{12}^2-6 s_1 s_2)-2m^2\big((s_1-s_2)^4-12 s_1^2s_2^2\big)+s_{12}(s_1^4-6 s_1^3s_2-4s_1^2s_2^2-6 s_1 s_2^3+s_2^4)\big]-$$
$$-\frac{s^4-4 s^3m^2-6 m^4s^2+28m^6 s-19m^8}{a(s-m^2)+s_1 s_2}+$$
$$+\Big\{\frac{s^3-6 s_2^3+7 s s_2^2-4 s^2 s_2+m^2(-9 s^2+18 s_2 s-7 s_2^2)+3 m^4(5 s-2 s_2)+73 m^6}{2 s_1}+$$
$$+\frac{2 m^2\big[(s-s_2)^3-3 m^2(s-s_2)^2+m^4(s_2-9 s)+11 m^6\big]}{s_1^2}+\frac{1}{(s-m^2)(s-s_1-m^2)}\big[-38 m^8+$$
$$+m^6(56 s +62 s_2)-m^4(12 s^2+26 s s_2+5 s_2^2)-2m^2(4 s^3+3 s^2s_2+7 s s_2^2+6 s_2^3)+2(s+s_2)(s^3+2 s_2^3)+$$
$$+3 s^2s_2^2\big]+\frac{24 m^6\big[(s-s_1)(s-s_{12})+m^2(s_2-2s_1)-m^4\big]}{s_1^2(4m^2+a)}+(s_1\rightleftarrows s_2)\Big\}\,,$$

$$\overline{B}_{s_1}=-s_1 s_2 \bigl[s^3(s_1-s_2)+s^2(-2 s_1^2+s_2 s_1+2 s_2^2) +s(2 s_1^3-2 s_2 s_1^2-5 s_2^2 s_1-2 s_2^3)+2 s_1 s_2 s_{12}^2\bigr]+$$
$$+m^2\big[2 s^3 s_1(s_2-2 s_1)+s^2(4 s_1^3-s_2 s_1^2-5 s_2^2 s_1+4 s_2^3)-2 s s_2 (2 s_1^3-s_2 s_1^2+4 s_2^2 s_1+2 s_2^3)+$$
$$+s_1 s_2 (2 s_1^3+2 s_2 s_1^2+3 s_2^2 s_1+2 s_2^3)\big]+m^4\big[8 s^3 s_1+2 s^2(6 s_1^2-3 s_1 s_2-4 s_2^2)+$$
$$+s(-8 s_1^3+13 s_2 s_1^2+15 s_2^2 s_1-8 s_2^3) +s_2 (2 s_1^3+7 s_2 s_1^2+10 s_2^2 s_1+4 s_2^3)\big]+$$
$$+m^6\big[-40 s^2 s_1+s(-12 s_1^2+6 s_1s_2+32 s_2^2)+4s_1^3-11 s_1^2s_2-27 s_1 s_2^2 +4 s_2^3\big]+$$
$$+2 m^8(28 s s_1+2s_1^2-s_1 s_2-12s_2^2)-24 m^{10} s_1\,,$$

$$\overline{B}_{12}=s(s-s_{12})^2[s^2+(s-s_{12})^2]-m^2\big[10 s^4 -28 s^3 s_{12}+s^2(29 s_{12}^2+4 s_1 s_2)-$$
$$-2 s s_{12}(6 s_{12}^2+5 s_1 s_2)+s_{12}^2(s_{12}^2+6 s_1 s_2)\big]+$$
$$+m^4\big[28 s^3 -64 s^2 s_{12}+s(53 s_{12}^2+8 s_1 s_2)-2 s_{12}(8 s_{12}^2+5s_1 s_2)\big]-$$
$$-m^6(44s^2 -68 s s_{12}+31 s_{12}^2-12 s_1 s_2)+m^8(34 s-26 s_{12})-10 m^{10}\,.$$

The corresponding contribution of the interference between Borsellino and Compton-like diagrams reads
\begin{equation}\label{eq:B5}
\overbrace{M_c\,M_b^*}=\frac{2 \pi^2}{s_1(s-m^2)}\Big\{\frac{1}{(a+s_2)}\Big[\frac{B_{bc} L_1}{\lambda_1^3}+\frac{s_1 B^{(b)} L_1^{(b)}}{\lambda_1^3(s-m^2)}
+\frac{1}{s_1(s-m^2)}\Big(\frac{(s_2-2m^2)B_{s_2} L_{s_2}}{\sqrt{s_2(s_2-4 m^2)}}-
\end{equation}
$$-\frac{a D_{bc} L_a}{\sqrt{a(4 m^2 +a)}}\Big)\Big] +\frac{1}{2(s-m^2)}\Big[\frac{\widetilde{B}_{bc} L_2}{\lambda_2^3}+
\frac{\widetilde{B}_{12} L_{12}}{\lambda_{12}^3}\Big] -(4m^2+a-s_2)C_{bc} \Big\}\,,$$
$$C_{bc}=\frac{s_1}{2 \lambda_2^2}-\frac{s_1}{\lambda_1^2}+\frac{(s+s_1-m^2)[-s_1 s_{12}+2m^2(s+2 s_1)-2m^4]}{2 \lambda_2^2\,\lambda_{12}^2}\,,$$
$$B_{bc}=(4m^2+a-s_2)[(s-s_1)^3-m^2(3 s^2-2s s_1+s_1^2)+3m^4(s+3 s_1)-m^6]\,,$$
$$B^{(b)}=(4m^2+a-s_2)[(s-s_1)^3+m^2(s^2-4 s s_1+5 s_1^2)-m^4(5 s+ s_1)+3 m^6]\,,$$
$$B_{s_2}=s_1(3 s^2-4ss_1-6ss_2+3s_{12}^2+5s_2^2)+m^2(8s^2-6ss_1+4s_1^2-26s_1s_2)+m^4(35s_1-16s)+8m^6\,,$$
$$D_{bc}=s_1(5s^2-8ss_1-10ss_2+5s_{12}^2+3s_2^2)+m^2(8s^2-2ss_1-22s_1s_2)+m^4(29s_1-16s)+8m^6\,,$$
$$\widetilde{B}_{bc}=(s-s_1-2 s_2)(s-s_2){}^3+m^2\big[-8 s^3+s^2(3 s_1+13 s_2) +4 s s_1 s_2 -s_2^2 (3 s_1+5 s_2)\big]+$$
$$+m^4\big[14 s^2-3 s s_{12}+s_2(s_1+7s_2)\big]+m^6(-8 s+s_1-5 s_2)+m^8\,,$$
$$\widetilde{B}_{12}=s_{12}^3(-s+s_1+2 s_2)+m^2\big[2 s^2(3 s_1+s_2)-2 s s_{12}(2 s_{12}+s_2)-s_{12}^2(13 s_1+19 s_2)\big]+$$
$$+2 m^4\big[6 s^2+22 s s_1+26s s_2+5 s_{12}(6 s_{12}+s_2)\big]-2 m^6(44 s+57 s_1+59 s_2)+76 m^8\,.$$

To derive the quantity $\overbrace{\overline{M}_c\,\overline{M}_b^*}$ one has to make the permutation $(s_1\leftrightarrows s_2)$ in the right hand side of Eq.~(B5).

The contribution of the interference $\overline{M}_c$ and $M_b$ can be written as
\begin{equation}\label{eq:B6}
\overbrace{\overline{M}_c\,M_b^*}=\frac{\pi^2}{s_2(s-m^2)^2}\Big\{\frac{2 \widetilde{B}_{s_1} L_{s_1}}{\sqrt{s_1^3(s_1-4 m^2)}}
-\frac{2 \widetilde{B}_{s_2} L_{s_2}}{s_1^2\sqrt{s_2^3(s_2-4 m^2)^3}}-\frac{2 D_{b\bar{c}} L_a}{s_1^2\sqrt{a(4 m^2+a)}}-
\end{equation}
$$-\frac{B_{b\bar{c}} L_2}{\lambda_2^3}-\frac{\overline{B}^{(b)} L_1^{(b)}}{\lambda_1^3}+\frac{2 m^2 B_{12} L_{12}}{\lambda_{12}^3}+C_{b\bar{c}}\Big\}\,,$$

$$C_{b\bar{c}}=\frac{s-m^2}{2 s}\Big\{-s+s_1-m^2+\frac{(s-s_1-m^2)(s^2-s s_{12}-s_1 s_2+m^2 s_{12}-m^4)}{\lambda_1^2}+$$
$$+\big[s(s_2-s_1)-s_2 s_{12}+m^2(2 s +s_1+3 s_2)-2 m^4\big]\Big(\frac{2 m^2}{\lambda_{12}^2}+\frac{s+s_2-m^2}{\lambda_2^2}\Big)\Big\}+$$
$$+\frac{2}{s_2}\Big\{(s-s_{12})^2-2 m^2(s-s_{12})-3 m^4+\frac{1}{(s_{12}-4 m^2)^2}\Big[-\frac{24 m^6(s-m^2)(s-2 s_{12}+7m^2)}{(s_2-4 m^2)}+$$
$$+\frac{1}{s_1}\big[(s-s_{12})s_{12}^4+m^2s_{12}^2(2 s^2 -4 s s_{12}+7 s_{12}^2)-8 m^4s_{12}(2 s^2+s s_{12}+s_{12}^2)+m^6[8 s^2+48 s s_{12}-26 s_{12}^2]-$$
$$-16m^8(s-2s_{12})+8m^{10}\big]\Big]-$$
$$-\frac{2 m^2(s-m^2)^2(s_{12}^2-4 m^4)}{s_1^2(s_{12}-4 m^2)}-\frac{s-m^2}{\lambda_2^2}\big[s(s-s_2)(s-s_{12})+m^2(s^2+2ss_1-6ss_2+3s_2 s_{12})-$$
$$-m^4(5 s +s_1+8 s_2)+3 m^6\big]\Big\}\,,$$
$$\widetilde{B}_{s_1}=s_1(s^2+2 s s_{12}+s_{12}^2)+2 m^2s_{12}(2 s-s_1-2s_2)+m^4(8 s-13 s_1-4 s_2)+8 m^6\,,$$
$$\widetilde{B}_{s_2}=s_1s_2^3[(s-s_{12})^2+s_2^2]-2m^2s_2^2[3s^2s_1-s(5s_1^2+3 s_1 s_2+2 s_2^2)+s_1(2 s_1^2+6 s_1 s_2+5 s_2^2)]+$$
$$+m^4s_2[8 s^2s_1-4 s(2 s_1^2-3 s_1s_2+4 s_2^2)+s_2(6 s_1^2+s_1 s_2-4 s_2^2)]+2m^6s_2[8 s^2-8 s(4 s_1+s_2)+20 s_1^2+$$
$$+21 s_1s_2+8 s_2^2]+8 m^8[4 s^2+4s s_2+s_2(2 s_2-s_1)]-16 m^{10}(4 s +3 s_2)+32 m^{12}\,,$$
$$B_{b\bar{c}}=(s-s_2)[(s^2-s_2^2)(s_1+2 s_2)-2 s s_1 s_2]-m^2[4 s^3+s^2(3 s_1-2 s_2)+2 s s_2(s_2-5 s_1)+s_2^2(9 s_1+8 s_2)]+$$
$$+m^4[12 s^2+3 s(s_1-6 s_2)+s_2(9 s_1+20 s_2)]-m^6(12 s+s_1+18 s_2)+4 m^8\,,$$
$$\overline{B}^{(b)}=(s-s_1)^3(s_1+2s_2)+m^2[4 s^3-s^2(11 s_1+2 s_2)+4 s s_1(s_1-s_2)+s_1^2(3 s_1+10 s_2)]-$$
$$-m^4(4 s^2+13 s s_1+2 s s_2+s_1^2+6 s_1 s_2)+m^6(-4 s-9 s_1+2 s_2)+4 m^8\,,$$
$$B_{12}=s_{12}(2s s_1+3 s s_2-2 s_1^2-3 s_1 s_2-s_2^2)-m^2(10 s^2+8 s s_2-14 s_1^2-23 s_1 s_2-9 s_2^2)-$$
$$-4 m^4(3 s+8 s_1+6 s_2)+22 m^6\,,$$
$$D_{b\bar{c}}=s_1(s^3-3 s^2 s_{12}+4 s s_{12}^2-2 s_{12}^3)+m^2[s^2(4 s_2-5 s_1)+2 s(4 s_1^2+ s_1 s_2-2 s_2^2)-2 s_1^2(2 s_1+s_2)]+$$
$$+m^4(8 s^2-17 s s_1-8 s s_2+19 s_1^2+9 s_1 s_2+ 4 s_2^2)+m^6(5 s_1+4 s_2)-8 m^8\,.$$

In accordance with Eq.~(10) and definition of quantity $\overbrace{W}$ given in this Appendix, we can write
$$\frac{d \sigma}{d\,s_1\,d\,s_2}=\frac{\alpha^3}{2\,\pi^2(s-m^2)}\overbrace{\sum_{pol}|M|^2}\,,$$
where $\sum_{pol}|M|^2$ is defined by Eq.~(9).

\input{dph12.bbl}

\end{document}

\

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Working within our professional negligence team, I have particular expertise in financial services negligence involving the mis-selling or mismanagement of a financial product or service leading to financial loss.
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text:
Imminent HRI Acute Care Ward Closures Stopped
Hands off HRI Campaign were recently tipped off about planned major ward closures at Huddersfield Royal Infirmary. The A&E Dept would no longer have been properly supported by vital Cardiology and Respiratory acute wards, as the Trust were, on the quiet, intending to relocate them to Calderdale Royal Hospital by the end of September.
These 'stealth changes' would have meant, the Campaign understands, an end to a fully-functioning A&E and Acute Services at HRI. It could have pre-determined the outcome of the legal process and the ongoing review of the closure proposals by the Independent Reconfiguration Panel. Once emergency and acute services were moved to Calderdale it would have been a classic case of a 'fait accompli' with little chance of resurrecting Huddersfield's A&E.
Hands off HRI took immediate action, instructing their solicitors, who warned the CCGs and the NHS Trust to "cease all actions which could render the continuation of current services unviable at any stage in the future". And such actions could make a mockery of the consultation process "undermining any consultation and decision making and rendering the commitment to consult and engage with stakeholders futile". They could also be "at odds with the reality of any final decision in the future, and… not take account of the serious reservations expressed by the [Kirklees and Calderdale] Joint Health Scrutiny Committee".
While there was no admission of knowledge of the planned closures by the CCGs or the Trust, Hands off HRI have full confidence that the information is well founded. In fact, they've just received further information: the closures will not, after all, be going ahead. This is great news and a small victory for the campaigners and the people of Huddersfield who depend on their local hospital.
However, Hands off HRI ask the people of Huddersfield to stay alert for any further stealth changes and would ask that any information be forwarded to the info.handsoffhri@gmail.com email address where any information will be handle in strict confidence.
A spokesman for the Legal and Finance Committee of Hands Off HRI commented that it is rumoured that the closures may now be rescheduled for later this year.
The campaign continues to fundraise for the impending Judicial Review and Hands Off HRI Upper Colne Valley will be present at the Marsden Jazz Festival over this coming weekend. Should you be unable to attend this event you can also catch up with the latest campaign news at www.Officialhandsoffhri.org, our new website which will officially launch soon. The web page syncs alongside the popular Facebook Group and photographs and posts of the groups recent visit to Manchester, last Sunday, at the Conservative Party Conference can be seen. Hands Off HRI joined with Health Campaigners from across the country; Barnsley, Leeds, Chorley, Lincolnshire, Bristol to name a few, to protest about Government cuts and privatisation of the NHS and the resulting closures in their local communities. The march encompassed some 50,000 protestors taking a route past the Conference Venue.
Hands Off HRI would like to thank the community of Kirklees for their continuing support of the campaign to save Huddersfield Royal Infirmary.

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text:
Stożne () ist ein Dorf in der polnischen Woiwodschaft Ermland-Masuren und gehört zur Landgemeinde Kowale Oleckie (Kowahlen, 1938 bis 1945 Reimannswalde) im Powiat Olecki (Kreis Oletzko, 1933 bis 1945 Kreis Treuburg).

Geographische Lage
Stożne liegt im Nordosten der Woiwodschaft Ermland-Masuren am Jezioro Stożne, neun Kilometer nordwestlich der Kreisstadt Marggrabowa (1928 bis 1945: Treuburg, ).

Geschichte
1560 war das Gründungsjahr des nach 1785 Stosznen und bis 1938 Stoosznen genannten kleinen Dorfes, das 1874 in den neu errichteten Amtsbezirk Schareyken () eingegliedert wurde. Dieser – 1938 umbenannt in "Amtsbezirk Schareiken" – gehörte bis 1945 zum Kreis Oletzko – 1933 bis 1945 "Landkreis Treuburg" genannt – im Regierungsbezirk Gumbinnen der preußischen Provinz Ostpreußen.

Im Jahre 1910 verzeichnete Stoosznen 344 Einwohner. Ihre Zahl stieg bis 1933 auf 364 und belief sich 1939 auf 412.

Aufgrund der Bestimmungen des Versailler Vertrags stimmte die Bevölkerung im Abstimmungsgebiet Allenstein, zu dem Stoosznen gehörte, am 11. Juli 1920 über die weitere staatliche Zugehörigkeit zu Ostpreußen (und damit zu Deutschland) oder den Anschluss an Polen ab. In Stoosznen stimmten 268 Einwohner für den Verbleib bei Ostpreußen, auf Polen entfiel keine Stimme.

Aus politisch-ideologischen Gründen zwecks Vermeidungs fremdländisch klingender Ortsnamen wurde Stoosznen 1938 in "Stosnau" umbenannt. In Kriegsfolge kam das Dorf 1945 mit dem südlichen Ostpreußen zu Polen und trägt seither die polnische Namensform "Stożne". Heute ist das Dorf eine Ortschaft im Verbund der Landgemeinde Kowale Oleckie im Powiat Olecki der Woiwodschaft Ermland-Masuren.

Kirche
Stoosznen resp. Stosnau war vor 1945 in das evangelische Kirchspiel Schareyken (1938 bis 1945: Schareiken, ) im Kirchenkreis Oletzko/Treuburg in der Kirchenprovinz Ostpreußen der Kirche der Altpreußischen Union eingepfarrt. Seit 1945 gehören die evangelischen Kirchenglieder Stożnes zur Kirchengemeinde in Gołdap, einer Filialgemeinde von Suwałki in der Diözese Masuren der Evangelisch-Augsburgischen Kirche in Polen.

Waren die katholischen Kirchenglieder vor 1945 nach Marggrabowa bzw. Treuburg () im Bistum Ermland orientiert, so sind sie heute in die neu errichtete Pfarrei Szarejki eingepfarrt. Dieser zugeordnet ist eine Kapelle in Stożne, die dem Hl. Josef geweiht ist. Die Pfarrei gehört zu einem der beiden Dekanate in Olecko im Bistum Ełk (Lyck) der Katholischen Kirche in Polen.

Verkehr
Stożne liegt unweit der bedeutenden polnischen Landesstraße DK 65 (einstige deutsche Reichsstraße 132) an einer Nebenstraße, die südlich von Kowale Oleckie (Kowahlen, 1938 bis 1945 Reimannswalde) in westlicher Richtung abzweigt, und nach Sokółki (Sokolken, 1938 bis 1945 Halldorf) führt. Stożne ist Bahnstation an der Bahnstrecke Ełk–Tschernjachowsk (Lyck–Insterburg), die seit 1993 für den Personenverkehr nicht mehr in Betrieb ist.

Persönlichkeiten
 Werner Kließ (1939–2016), Film- und Fernsehproduzent, Drehbuchautor und Bildhauer

Einzelnachweise

Ort der Woiwodschaft Ermland-Masuren
Gmina Kowale Oleckie

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A brief post to wish my father a Happy 60th Birthday. HAPPY BIRTHDAY DAD!! We are out for a family dinner tonight, then a wee celebration tomorrow afternoon. Posting largely depends on inclination and quantities of Southern Comfort (which my sister said she'd buy! Hope you remembered Jen!).


by Rob Manning Follow OPB March 20, 2019 5 a.m. | Portland, Ore.
Everyone is frustrated at the difficulty of shifting away from physical interventions and toward individualized support for children with special needs. But changing the rules is hard.
In late February, a congressional subcommittee heard testimony from a mother from Rhode Island and a teacher from Kansas.
The mom described how her 8-year-old son struggled when he had to change activities and had such severe conflicts early on, he was expelled from preschool.
The teacher described feeling overwhelmed when, shortly after starting her career, she was left largely on her own to teach 13 middle schoolers on the autism spectrum.
Both women spoke of the physical and psychological toll they suffered when classroom struggles led to physical interventions, like restraint and isolation.
House members thanked them for testifying and committed to work on making things better. But Congress has looked at changing the law around restraint and isolation before, and fixes have invariably stalled.
Closer to home, Oregon and Washington have passed laws in recent years to limit or better document the practices, but parents say the changes haven't gone far enough. They've asked for tighter timelines, clearer training guidelines and mandatory changes for schools with particularly high rates of restraints or isolations.
Parents, teachers and lawmakers are all frustrated at the complexity, sensitivity and high expense of moving schools in the direction that all sides agree is best for students and staff: less use of physical interventions and more individualized support for children with special needs.
Washington state Rep. Monica Stonier, D-Vancouver, understands parents' frustration.
That leaves oversight largely in the hands of schools and state agencies. In complaint-driven systems, such as those in Oregon and Washington, that puts the onus on parents like Joe and Nikki Putman.
Their daughter Jennifer has a big presence. She's tall for a 12 year-old, with broad shoulders, a wide grin and a booming voice. Jennifer is on the autism spectrum and, while she can be quite friendly, she also can be a handful at home and in class.
Jennifer, 12, swings from a tree in the backyard of her family's home in Hillsboro, Ore., Friday, March 1, 2019.
Nikki Putman and her daughter, Jennifer, 12, share a moment to calm down in the hallway of their home in Hillsboro, Ore., Friday, March 1, 2019.
Gelser says she made headway in Salem because teachers also see a benefit in clarifying laws around restraint and isolation.
Joel Nixon is a social worker at Oregon's Clackamas Education Service District and a point person on children with disabilities for the union. He says there's quite a bit of common ground between parents and teachers.
As legislators look to make potentially big investments in public schools this session, maintaining safe classrooms and supporting students' social-emotional needs, have become high priorities. There's growing consensus that adding counselors, aides and teachers could improve relationships with challenging students.
"It's just a time-intensive process to develop relationships, and some of our teachers have so many students, that just keeping their names straight is a challenge," Nixon said.
In Washington, parents are optimistic that lawmakers can find more money for special education, after they met a court mandate last year to spend more on education more broadly. Legislative staff have said that funding could help train more educators, both around which restraint and isolation tactics are appropriate and how to comply with relevant state and federal laws. Rep. Stonier says they're both important.
"It may be that educators, that school security officers, that principals don't recognize that they're breaking the law when they're not in compliance," Stonier said.
But Stonier said it will take more than money to get school administrators to follow the law.
Katherine Sheppard's daughter attended Hough Elementary School in downtown Vancouver, Wash., until she pulled her out over concerns with special education services.


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Join the First Look Book Club!
We're thrilled to announce our new partnership with DearReader, the company that has been bringing you and your patrons daily excerpts from books for years. Beginning February 3, 2014, First Look Book Club will be delivering content from just-published Random House titles that we are most excited about here in library marketing.
If your library already works with DearReader, you'll already have a link to our club on your sign up page. If not, don't fret, you can sign up for free through this link. Once the club launches, you'll receive daily excerpts from new books. We'll bring you the best in Mystery, Suspense, Romance, Nonfiction, Literary Fiction, and more. It's like a matchmaking service for reading. Get a first look before you commit to a book! We've got some great titles lined up for you—A Star for Mrs. Blake, Under the Wide and Starry Sky and Red Rising to name a few. This club is perfect to share with your patrons, but it is an ideal Readers' Advisory tool too! It's a great way to dip in to some of the best new books coming out this year. Happy Reading!
I Want the First Look!
Tagged with:book club, book excerpts, DearReader, First Look Book Club, Readers Advisory
Favorite Science Fiction: Signed Copy Giveaway
December 11, 2014 Written by rherica
In celebration of our favorite and most lauded Science Fiction titles of 2014, we are giving away a limited number of signed copies of THE MARTIAN! Enter to Win One Free Signed Copy! We've also compiled a preview collection of our Spring 2015 Science Fiction Titles! What will you read next?
November is National Military Family Appreciation Month!
October 28, 2014 Written by rherica
The unforgettable story of a military family that lost two sons—one to suicide and one in combat—and channeled their grief into fighting the armed forces' suicide epidemic. "In his superb new book, The Invisible Front, Yochi Dreazen paints a deeply disturbing portrait of the overstretched United States Army, downsizing in Afghanistan while deploying against the […]
#GiveaBook
November 2, 2015 December 3, 2015 Written by cnewbould
Books make better gifts! GiveaBook is an online campaign to promote books as gifts and give back to children in need. For each use of the hashtag #GiveaBook on Facebook and Twitter between November 16th and December 24th, Penguin Random House will donate one book to the literacy nonprofit First Book, up to 35,000 books. […]
Nominate a Teacher for the Penguin Random House Teacher Awards for Literacy
April 7, 2015 Written by rhelizabeth
Do you know an educator who fosters in students a passion and love of reading? Nominate him or her for the 2015 Penguin Random House Teacher Awards for Literacy! The award recognizes the nation's most dynamic and resourceful teachers who use their creativity to inspire and successfully instill a love of reading in students. The […]
Terry Welch Jan 21, 2014
How Fab!!!
PRH Story Time Temporary Open License Extended to 6/30/21
#FridayReads: The War Widow by Tara Moss
#FridayReads: The Twelve Dates of Christmas by Jenny Bayliss


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Federal Judge Rejects Dismissal of Lawsuits Against Donald Trump Alleging A Violation Of The Reconstruction-Era Ku Klux Klan Act
Home » Federal Judge Rejects Dismissal of Lawsuits Against Donald Trump Alleging A Violation Of The Reconstruction-Era Ku Klux Klan Act
By AZ BlueMeanie|2022-02-21T15:32:38-07:00February 21st, 2022|Courts & Law, National Elections, National Politics|
Completing the trifecta of a bad week in court in last week, a federal judge once again rejected Donald Trump's authoritarian claim of absolute immunity, and will allow three lawsuits to proceed against him for inciting a violent seditious insurrection on January 6, 2021, a violation of the Reconstruction-era Ku Klux Klan Act, which bars violent interference in Congress's constitutional duties.
The Washington Post reports, U.S. judge rejects Trump claim of 'absolute immunity' from Jan. 6 lawsuits:
A federal judge on Friday sweepingly rejected former president Donald Trump's claim of "absolute immunity" from lawsuits accusing him of inciting the violent Jan. 6, 2021, Capitol riot, finding that there was evidence to plausibly suggest that he engaged in a conspiracy with organized groups to intimidate Congress into overturning the results of the 2020 election.
In a searing, 112-page opinion that quoted repeatedly and at length from the former president's own public statements, U.S. District Judge Amit P. Mehta refused to dismiss three lawsuits against Trump by Democratic House members and police officers seeking damages for physical and emotional injuries they incurred in the assault. The judge did or said he would drop as defendants Donald Trump Jr., attorney Rudolph W. Giuliani and Rep. Mo Brooks (R-Ala.), who joined Trump in addressing rallygoers at the Ellipse before they marched to the Capitol that day.
The cheese stands alone.
Mehta said Trump's own words and conduct in falsely alleging a "stolen" election were not immune on separation-of-powers grounds because they served only his personal aim of retaining office, falling beyond the "outer perimeter" of a president's official responsibilities.
"The President's actions here do not relate to his duties of faithfully executing the laws, conducting foreign affairs, commanding the armed forces, or managing the Executive Branch. They entirely concern his efforts to remain in office for a second term. These are unofficial acts," Mehta wrote.
At the same time, the judge said, Trump allegedly went beyond simply trying to pressure state and local officials, Congress, or Vice President Mike Pence to overturn lawful election results. Rather, the judge said, he raised the threat of violence, culminating in the attack that delayed Congress's certification of President Biden's election victory, led to assaults on nearly 140 officers and contributed to five deaths.
Trump's statements in the weeks, hours and minutes before the attack on the Capitol were plausibly "an implicit call for imminent violence or lawlessness," the judge ruled.
"He called for thousands 'to fight like hell' immediately before directing an unpermitted march to the Capitol, where the targets of their ire were at work, knowing that militia groups and others among the crowd were prone to violence," the judge said.
Trump's role "was to encourage the use of force, intimidation or threat to thwart the Certification from proceeding, and organized groups such as the Proud Boys and the Oath Keepers would carry out the required acts," the judge said. Such a "tacit agreement" is made all the more plausible by Trump's response to the Capitol breach, when he attacked Pence on Twitter for not having the courage to block the certification after rioters broke into the building.
"It is reasonable to infer that the President would have understood the impact of his tweet, since he had told rallygoers earlier that, in effect, the Vice President was the last line of defense against a stolen election outcome," Mehta wrote.
The decision by Mehta, an appointee of President Barack Obama who is also presiding over the seditious conspiracy prosecution of Oath Keepers founder Stewart Rhodes, was a defeat for the former president, who moved to dismiss three lawsuits: one by Rep. Karen Bass (D-Calif.) and nine other lawmakers; one from Rep. Eric Swalwell (D-Calif.), a former prosecutor and Trump impeachment manager; and another from Capitol Police officers James Blassingame and Sidney Hemby. The initial lawsuit was brought by Rep. Bennie G. Thompson (D-Miss.), who left his case in July after becoming co-chair of the Jan. 6 House select committee.
The opinion allows for further fact-finding — including depositions and document demands of the former president and other co-defendants, which include members of the Oath Keepers and the Proud Boys. Discovery might prove such connections "to be an important one," Mehta wrote.
Trump attorney Jesse Binnall in arguments last month cast the lawsuits as "chock full of propaganda that are meant to achieve political rather than legal objectives." Binnall urged the court to deny House Democrats the opportunity "to score points against a political rival at the expense" of the Constitution's bedrock principles of separation of powers and freedom of speech.
[Mehta] dropped Giuliani and Donald Trump Jr. as defendants from Swalwell's suit, and said he would do so similarly for Brooks, saying their alleged actions were limited to less incendiary statements at a fiery rally that morning at the White House Ellipse.
[H]ouse lawmakers' legal team, led by Joseph M. Sellers and joined by attorneys for the NAACP, argued in alignment with lead Swalwell attorney Phil Andonian, that Trump and others instigated and facilitated that day's attack in violation of the Reconstruction-era Ku Klux Klan Act, which bars violent interference in Congress's constitutional duties. Police officers asserted that Trump and others should also be legally liable for injuries sustained by lawmakers and police during the deadly, hours-long breach of the Capitol by Trump supporters angered by his unfounded claims of election fraud.
Mehta called [Trump's attorney's] argument "grossly misleading," noting that the federal government's role is to ensure election integrity through litigation, and that a president has absolutely no constitutional or statutory role in the counting or certification of the electoral college results.
A president's statements are not immune from a civil suit just because they are spoken by a president, Mehta added.
Let the discovery begin!
About the Author: AZ BlueMeanie
The Blue Meanie is an Arizona citizen who wishes, for professional reasons, to remain anonymous when blogging about politics. Armed with a deep knowledge of the law, politics and public policy, as well as pen filled with all the colors stolen from Pepperland, the Blue Meanie's mission is to pursue and prosecute the hypocrites, liars, and fools of politics and the media – which, in practical terms, is nearly all of them. Don't even try to unmask him or he'll seal you in a music-proof bubble and rendition you to Pepperland for a good face-stomping.
Buzz Davis February 21, 2022 at 6:10 pm
I hope that the higher courts will uphold the law upon appeal and support Judge Mehta's decisions. Thanks for the report Bluemeanie! Peace Buzz Davis Vets for Peace Tucson
Wileybud on The Prima Donna Davos Diva Kyrsten Sinema is Courted By Republicans
A Horse of a Different Color Sharpie on The Prima Donna Davos Diva Kyrsten Sinema is Courted By Republicans


Carol's Paperie: "You're Awesome & Gorgeous"
Hi, everyone. Today at Imaginarium Designs, we are creating Feminine projects. So I made a layout for my girl.
Made a simple and sweet layout. Tore 2 pieces of patterned papers and pop up with foam tape. Stamped these cute Love Cynthia stamp images and colored with color pencils. And added them to the page.
Painted the butterfly chipboards with light pink paint and glitter glue.
Painted the title chipboards with black shimmer acrylic paint. Added dew drops and half pearls randomly across page.
Hope you enjoy my sharing. Thanks for dropping by.


For drying, perming, colouring, hair setting and ozone purposes.
Light weight and powerful yet low noise hair dryer.
Powerful, Light weight and low noise. Reduced dimensions, easy handling. Anti-sliding heat dispersing grooves.
ECO Hair dryer using 1250W & output of 2500W.
Newly designed ionic generator emits over 2,000,000 negative ions per cm3.


Not only colleagues passed criticism on Fekter, but also international media did. The Austrian Minister of Finance presented the results of the EU summit untimely. Euro zone finance ministers have agreed to boost the currency bloc's debt crisis firewall to roughly € 800bn ($1.06tn), Austrian Finance Minister Maria Fekter said to journalists on Friday.


With the Peak District occupying a large portion of the county and the Pennines edging into the north, Derbyshire is one of England's greenest and most expansive counties. In fact, it is so sparsely populated that a reported 75% of the Derbyshire population lives in only 25% of the area. Much of this population is centred around its main city: Derby.
Some of the most beautiful parts of the Peak District, such as Mam Tow and Kinder Scout, can be found in Derbyshire, while Chatsworth House, Bolsocer Castle and Castleton are all within the county's borders. Other than the tourism brought in by the landscape, the primary industries of Derbyshire centre on the engineering industry, and so in Derby the two biggest employers are Rolls-Royce and Toyota.
Because Derbyshire is so sparse, there are few major roads which run through it. There are however, some major roads passing through, such as the M1 and the A38, which connects Derby to Birmingham.
AA Cars have 3,126 used cars listed in Derbyshire, 1,403 used cars are listed with approved franchised dealerships. 34% of second hand cars in Derbyshire are priced over the national average. This indicates there might be room for negotiation on any deal.


I first came across this tiny creeping herb in my garden where it had been planted as an ornamental. The bright green leaves, mostly less than 3 mm across form an attractive carpet. Until now I had never been able to spot its flowers despite having checked several times over the last few years. My guess is that this species has a relatively narrow flowering time in spring and the flowers are so tiny that they are only visible with a hand-lens. For several years the genus has intrigued me, not so much because of its small size and creeping habit but because of its distribution and evolutionary relationships.
Firstly because Soleirolia consists of a single species that in the wild is only known from the Mediterranean island of Corsica. This is the only genus of nettle I know that is restricted to a single island or to the Mediterranean and I am very keen to try and found out why this could be (the history of the Mediterranean basin is quite a turbulent one). Secondly Soleirolia has traditionally been grouped with the widespread Parietaria and intriguingly with Gesnouinia, which also includes a single species but is restricted to the Canary Islands in the Atlantic Ocean. Whilst they look very different as plants their female flowers share many similarities of form. It might be, therefore, that Soleirolia and Gesnouinia should be viewed as Parietaria species that have diverged morphologically as a consequence of being isolated on islands, a common phenomenon in evolutionary biology. I am currently testing this possibility using DNA sequence data and could have a better idea in a couple of months.


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Draper, Richard D.
Richard D. Draper is a professor emeritus at Brigham Young University who taught Old Testament, New Testament, Pearl of Great Price, Book of Mormon, and the Book of Revelation. Brother Draper grew up in the Utah Valley and graduated from Pleasant Grove High School. Right after graduation he joined the armed forces for a year. Upon his return he served a mission to the Central Atlantic States. He attended BYU, and as soon as he graduated he was hired by the Church Education System, serving as an institute instructor for the next twenty years. When he completed his PhD in ancient history from BYU, he was hired by the Religious Education Department. He was a professor of ancient scripture specializing in prophecy, apocalyptic, and New Testament background when this was written. Brother Draper can read Greek, Hebrew, French, and German.
Articles/Chapters by: Richard D. Draper
The Prophets of the Exile: Saviors of a People from Sperry Symposium Classics: The Old Testament
7. "What Therefore God Hath Joined Together, Let No Man Put Asunder" from The Sermon on the Mount in Latter-day Scripture
Teaching the Book of Revelation: Five Considerations from RE 14, no. 1 (2013)
18. Light, Truth, and Grace: Three Themes of Salvation (D&C 93) from Sperry Symposium Classics: The Doctrine and Covenants
The First Coming of the Lord to the Jews: A Book of Mormon Perspective from A Book of Mormon Treasury
He Has Risen: The Resurrection Narratives as a Witnessof Corporeal Regeneration from Sperry Symposium Classics
The Earliest "New Testament" from How the New Testament Came to Be
The Apocalyptic Witness of the Messiah from Sperry Symposium Classics
13. Understanding Images and Symbols in the Book of Revelation from Shedding Light on the New Testament
14. The Exalted Lord from Shedding Light on the New Testament
A Scriptural Basis for the Doctrine of Sexual Purity from An Eye of Faith


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Reengineering the Future: Why Bitcoin is Reshaping More than Money
, Finance
In the last decade we've seen innovations and inventions come and go. The Internet of Things and home hubs, "Smart" XYZ and the Tesla are just some of the developments that have reshaped the way we live. However, as useful and impressive as they are, none have or are likely to have as big of an influence as Bitcoin.
Originally developed in 2009 by an unknown programmer known only as Satoshi Nakamoto, Bitcoin is basically a monetary system that's manufactured, transferred and stored online. Through a series of complex equations and a network of servers and users, the peer-to-peer system is able to survive and function without the need for outside help.
Bitcoin Important for Businesses and Consumers
"Bitcoin Magazine at Barnes & Nobles" (CC BY-SA 2.0) by zcopley
Why is that important? Well, from an ideological perspective, it means the Bitcoin isn't controlled by a government like regular currencies and, therefore, users can trade it freely across borders. From a technical perspective, it means that the system powering Bitcoin has to be smart and that's what's really got tech fans interested in recent years.
When Bitcoin was initially launched, many fans thought it would change the way we pay and completely replace standard currencies. In some way, that has happened. For instance, online gamers can now enjoy their favourite activities without using their native currency. At VegasCasino, a new user can sign up, make a deposit and hit the play now button on something like Satoshi's Secret slot or European Roulette and only use Bitcoin.
By linking with Bitcoin exchanges such as Cubits.com, VegasCasino has made it possible for players to use their debit card as a means of depositing and playing in Bitcoin. Similarly, Xbox Live players can now purchase in-game credits, add-ons and upgrades through the Microsoft store. By logging into their account, a player can go through the purchase process in the usual way, but instead of choosing pounds, dollars or euros, they can pay in Bitcoin.
Thanks to this gradual adoption of Bitcoin by major businesses, the value of a single coin has risen over the years. However, the thing that's really made the cryptocurrency's value breach the $975 mark in the first quarter of 2017 is the blockchain technology powering the system. In short, a single Bitcoin is manufactured through a process known as mining which is basically a series of servers trying to solve a complex equation.
Bitcoin Gives Blockchain to the World
"Please Keep Door Locked and Students Sup" (CC BY 2.0) by stevendepolo
Once a server has processed and "solved" the equation, it gets rewarded with a Bitcoin. This coin is then logged by something known as the blockchain. Then, at every point the coin is transferred, sold or broken down, the information is recorded by the blockchain. In essence, the blockchain is a virtual ledger that records each transaction in a coded block that can be viewed but not modified. This process happens in real time and is, therefore, used to verify a Bitcoin transaction.
Since discovering the effectiveness of the blockchain, tech forms around the world have been looking at its potential applications. Today, every industry from healthcare to music is looking to harness the power of the blockchain. Why? Well, in a world where we live so much of our lives online, the ability to instantly verify and record each process and then store it in a coded block that can't be altered is invaluable.
This dynamic would dramatically reduce the risk of security leaks and fraud. For example, if a healthcare company was to use blockchain to store patient records, it would mean that they could be viewed by the necessary people but not tampered with. Moreover, any attempt to steal someone's records would be flagged up immediately, which would allow security experts to shut it down. Essentially, blockchain is the key to a more efficient, secure online ecosystem and that, in a nutshell, is why Bitcoin is the most significant innovation of the last decade. Although it's still early days, the next decade will see blockchain become an integral part of existing and new technology across a range of industries.