text:
\section{Introduction}
\label{sec:introduction}

In this Letter we apply the two-tensor-pomeron model \cite{Britzger:2019lvc}
to deeply virtual Compton scattering (DVCS) on the proton
($\gamma^{*} p \to \gamma p$).
This Regge type model can be used
for large $\gamma^{*} p$ center-of-mass energy
$W \gg m_{p}, \sqrt{|t|}$, $|t| \lesssim 1$~GeV$^{2}$,
and small Bjorken-$x$, say
$x = Q^{2}/(W^{2} + Q^{2} - m_{p}^{2}) < 0.02$.
Here $m_{p}$ is the proton mass,
$Q^{2}$ is the photon virtuality, and
$t$ is the squared momentum transfer at the proton vertex.

The DVCS has been a subject of extensive experimental
and theoretical research; see e.g.
\cite{Ji:1996nm,Diehl:1997bu,Bartels:1981jh,
Diehl:2003ny,
Diehl:2005pc,Machado:2008tp,Kopeliovich:2008ct,
Braun:2014sta,Kriesten:2019jep,Dutrieux:2021nlz}
for theory papers. For a review and many references see \cite{Diehl:2003ny}.
Recently, the possibility of probing
the nonlinear (saturation) effects on the DVCS process
was considered
in \cite{Bendova:2022xhw,Xie:2022sjm,Goncalves:2022wzq}.
A comprehensive experimental overview of DVCS
is presented in \cite{dHose:2016mda};
see also Fig.~5 of \cite{Aschenauer:2013hhw}
which shows the kinematic coverage in the $x$-$Q^{2}$ plane
for existing DVCS measurements, as well as planned ones.
The H1 \cite{H1:2005gdw,H1:2009wnw} and ZEUS
\cite{ZEUS:2003pwh,ZEUS:2008hcd} Collaborations
measured the DVCS cross section over a broad range of
$W$ and $Q^{2}$ at low Bjorken-$x$.
Also the differential cross section $d\sigma/dt$ was extracted.
In \cite{H1:2009wnw}
the beam charge asymmetry due to the interference between
the DVCS and the purely electromagnetic Bethe-Heitler process
was measured.
The HERMES Collaboration \cite{HERMES:2011bou,HERMES:2012gbh,HERMES:2012idp}
performed measurements of single- and double-spin DVCS asymmetries.
The Jefferson Lab CLAS Collaboration measured
DVCS beam-spin asymmetries and cross sections
\cite{CLAS:2007clm,CLAS:2015uuo,CLAS:2018bgk,CLAS:2021gwi}
and longitudinally polarized target-spin asymmetries
\cite{CLAS:2006krx,CLAS:2014qtk,CLAS:2015bqi}.
The COMPASS Collaboration \cite{COMPASS:2018pup}
 measured the DVCS cross section by
studying exclusive single-photon production
in muon-proton scattering, $\mu p \to \mu p \gamma$.
The Jefferson Lab Hall A Collaboration measured
the photon electroproduction cross section
in the valence quark region
\cite{JeffersonLabHallA:2006prd,JeffersonLabHallA:2015dwe},
as well as the DVCS off the neutron \cite{JeffersonLabHallA:2007jdm},
and recently \cite{JeffersonLabHallA:2022pnx}
the DVCS off the proton at high values of Bjorken-$x$.
Experimental programs at Jefferson Lab,
at the Electron Ion Collider (EIC) under construction at BNL
\cite{Accardi:2012qut,Aschenauer:2013hhw,Aschenauer:2017jsk},
and, if realized, at a Large Hadron Electron Collider (LHeC)
at the LHC \cite{LHeCStudyGroup:2012zhm,LHeC:2020van},
are expected to improve our knowledge of DVCS
in a wide kinematic range.

DVCS is a prime playground for the application
of the generalized parton-distribution (GPD) concept
based on perturbative QCD (pQCD),
cf.~\cite{Diehl:2003ny} for a review.
However, for small values of Bjorken $x$ the application of pQCD
concepts faces the problem that the effective expansion
parameter is there $\alpha_{s} \ln(1/x)$
which is large for small $x$.
More phenomenological models are then frequently used,
based e.g. on the color-dipole approach
or the Regge approach.
In our present work we use the latter approach where
the scattering is described using exchange objects.
For high energies $W$ the pomeron will be the most important
exchange object. For general reviews of pomeron physics
see, e.g., \cite{Donnachie:2002en,Caneschi}.
Applications of Regge theory to DVCS have been given in
\cite{Capua:2006ij,Fazio:2013hza}
and we shall comment on these papers below.
In our work we shall use the model of \cite{Britzger:2019lvc}
comprising a soft and a hard pomeron which couple to particles
like symmetric spin 2 tensors.
It was shown in \cite{Britzger:2019lvc} that a vector nature
of these couplings is excluded.
In \cite{Ewerz:2016onn} it was shown
that scalar couplings of the pomeron can be excluded
through the experimental results of \cite{Adamczyk:2012kn}.
For some remarks on the history of tensor-pomeron concepts
and corresponding references see \cite{Ewerz:2016onn}.

In the following we shall use the tensor-pomeron model
as introduced for soft high-energy reactions
in \cite{Ewerz:2013kda}
and extended to hard reactions in \cite{Britzger:2019lvc}.
The soft and hard pomeron and the charge-conjugation
$C = +1$ reggeons ($\mathbb{R}_{+} = f_{2 \mathbb{R}}, a_{2 \mathbb{R}}$)
are described as
effective rank-2 symmetric tensor exchanges,
the odderon and the $C = -1$ reggeons
($\mathbb{R}_{-} = \rho_{\mathbb{R}}, \omega_{\mathbb{R}}$)
are described as effective vector exchanges.
Applications of this tensor-pomeron concept
were given for photoproduction of pion pairs
in \cite{Bolz:2014mya},
for a number of exclusive central-production reactions,
see for instance
\cite{Lebiedowicz:2013ika,Lebiedowicz:2014bea,
Lebiedowicz:2016ioh,
Lebiedowicz:2019boz,Lebiedowicz:2019jru,Lebiedowicz:2020yre},
and for soft-photon radiation (bremsstrahlung)
in hadronic collisions
\cite{Lebiedowicz:2021byo,Lebiedowicz:2022nnn}.

In \cite{Britzger:2019lvc} the two-tensor-pomeron approach
was applied to describe simultaneously
the experimental data on the inclusive cross sections for DIS
at low-$x$ \cite{H1:2015ubc} and for photoproduction.
The global fit performed there was
based on the hard and soft pomeron,
and the reggeon $f_{2 \mathbb{R}} + a_{2 \mathbb{R}}$.
The model describes the transition from the low-$Q^{2}$ regime,
including the $Q^{2} = 0$ photoproduction limit,
where the real or virtual photon behaves hadronlike
and the soft pomeron dominates,
to the large-$Q^{2}$ regime which is the domain of the hard pomeron.
For the photoproduction cross section
$\sigma_{\gamma p}(W)$
no significant contribution from the hard pomeron was found.
In this model $\sigma_{\gamma p}(W)$ is,
in the range 6~GeV~$< W < 209$~GeV,
dominated by soft-pomeron exchange with
a significant reggeon contribution for $W < 30$~GeV.
The transition region where both pomerons
contribute significantly was found to be roughly
$0 < Q^{2} < 20$~GeV$^{2}$.
But as $Q^{2}$ increases the hard pomeron component
becomes more and more important.

Our paper is organized as follows. In the next section
we introduce the tensor-pomeron approach
for real and deeply virtual Compton scattering.
Section~\ref{sec:results} presents
the comparison of our model results
with experimental data.
There we also discuss theoretical uncertainties
and limitations of the model.
Section~\ref{sec:conclusions} contains our conclusions.

\section{Theoretical formalism}
\label{sec:formalism}
We investigate the real and deeply virtual Compton scattering
on a proton
\begin{equation}
\gamma^{(*)}(q, \epsilon) + p (p,\lambda) \to
\gamma(q', \epsilon') + p (p',\lambda')\,.
\label{2.1}
\end{equation}
The momenta are indicated in brackets,
$\lambda, \lambda' \in \{1/2, -1/2 \}$
are the proton helicities,
and $\epsilon, \epsilon'$ are
the photon polarization vectors.

For an initial virtual photon $\gamma^{*}$ the reaction (\ref{2.1})
is extracted from $ep \to ep \gamma$ scattering
(see Fig.~\ref{fig:DVCS})
\begin{equation}
e(k) + p (p,\lambda) \to
e(k') + \gamma(q', \epsilon') + p (p',\lambda')\,.
\label{2.100}
\end{equation}
\begin{figure}[!h]
\includegraphics[width=7cm]{DVCS_ep.eps}
\caption{
DVCS contribution to $ep \to ep \gamma$ (\ref{2.100}).}
\label{fig:DVCS}
\end{figure}
Here the Bethe-Heitler process and DVCS contribute with
the latter corresponding to electroproduction of the $\gamma p$ state.
A detailed kinematic analysis of electroproduction reactions
can e.g. be found in \cite{Arens:1996xw} where, in particular,
many relations of variables in the rest system
of the original proton and in the system used in HERA
experiments are given.
The standard kinematic variables for
(\ref{2.1}) and (\ref{2.100}), setting the electron mass to zero,
are (see Table~1 of \cite{Arens:1996xw})
\begin{eqnarray}
&& q = k - k'\,, \quad q^{2} = -Q^{2}\,, \nonumber \\
&& s = (p + k)^{2}\,, \nonumber \\
&& t = (p - p')^{2}\,, \quad t \leqslant -|t|_{\rm min}\,,\nonumber \\
&& W^{2} = (p + q)^{2}\,, \nonumber \\
&& x = \frac{Q^{2}}{2 p \cdot q}
     = \frac{Q^{2}}{W^{2} + Q^{2} - m_{p}^{2}} \,, \nonumber \\
&& y = \frac{p \cdot q}{p \cdot k}
     = \frac{W^{2} + Q^{2} - m_{p}^{2}}{s - m_{p}^{2}}\,,\nonumber \\
&& \varepsilon = \frac{2 (1 - y) - 2 xy m_{p}^{2} (s - m_{p}^{2})^{-1}}
                    {1 + (1 - y)^{2} + 2 xy m_{p}^{2} (s - m_{p}^{2})^{-1}}\,.
\label{2.101}
\end{eqnarray}
Here $\varepsilon$ is the ratio of longitudinal and transverse
polarization strengths of the virtual photon $\gamma^{*}$
and $t = -|t|_{\rm min}$ corresponds to forward scattering.
For $Q^{2} = 0$ we have $|t|_{\rm min} = 0$
and for $W^{2} \gg Q^{2}, m_{p}^{2}$
we find (see also Appendix~A of \cite{Arens:1996xw})
\begin{equation}
|t|_{\rm min} = Q^{2} \left[ \frac{Q^{2} m_{p}^{2}}{W^{4}} +
{\cal O}\left(\frac{Q^{6}}{W^{6}}, \frac{Q^{4} m_{p}^{2}}{W^{6}}, \frac{Q^{2} m_{p}^{4}}{W^{6}},\frac{m_{p}^{6}}{W^{6}} \right) \right] \,.
\label{2.101a}
\end{equation}

We assume for (\ref{2.100}) unpolarized initial particles
and no observation of the polarization of final state particles.

Adapting (3.20) and (3.21) of \cite{Arens:1996xw}
to the reaction (\ref{2.100})
and integrating over the azimuthal angle $\varphi$
defined in (2.1) of \cite{Arens:1996xw}
we get for the DVCS part of (\ref{2.100})
\begin{eqnarray}
\frac{d\sigma (ep \to ep \gamma)}{dydQ^{2}dt}
= \Gamma_{e p \gamma}
\left(
\frac{d\sigma_{\rm T}}{dt}(Q^{2},W^{2},t)
+ \varepsilon \frac{d\sigma_{\rm L}}{dt}(Q^{2},W^{2},t)
\right)\,.
\label{2.102}
\end{eqnarray}
Here, with Hand's convention \cite{Hand:1963bb},
the $\gamma^{*}$ flux factor reads
\begin{eqnarray}
\Gamma_{e p \gamma} =
\frac{\alpha}{\pi y Q^{2}}
\left( 1 - y + \frac{1}{2}y^{2}
+ xy \frac{m_{p}^{2}}{s - m_{p}^{2}}\right)
\frac{W^{2} - m_{p}^{2}}{W^{2} - m_{p}^{2} + Q^{2}
+ 4 x m_{p}^{2}}
\,.
\label{2.103}
\end{eqnarray}
Furthermore, $d\sigma_{\rm T}/dt$ and
$d\sigma_{\rm L}/dt$ are the differential cross sections
for $\gamma^{*} p \to \gamma p$ for transverse and longitudinal
polarization of the $\gamma^{*}$,
\begin{eqnarray}
&&\frac{d\sigma_{\rm T}}{dt}(Q^{2},W^{2},t) =
\frac{1}{2} \left( \frac{d\sigma_{++}}{dt}(Q^{2},W^{2},t)
                 + \frac{d\sigma_{--}}{dt}(Q^{2},W^{2},t) \right)
                 \,, \nonumber\\
&&\frac{d\sigma_{\rm L}}{dt}(Q^{2},W^{2},t) =
\frac{d\sigma_{00}}{dt}(Q^{2},W^{2},t)\,,
\label{2.104}
\end{eqnarray}
where
\begin{eqnarray}
\frac{d\sigma_{mm}}{dt}(Q^{2},W^{2},t)
&=& \frac{1}{16 \pi (W^{2} - m_{p}^{2})
\sqrt{(W^{2} - m_{p}^{2} + Q^{2})^{2} + 4 m_{p}^{2} Q^{2})}} \nonumber\\
&& \times
\frac{1}{2}\sum_{\lambda, \lambda', a}
\big\vert \Braket{\gamma(q', \epsilon'_{a}),p (p',\lambda'), {\rm out}
|e J_{\mu}(0) \epsilon_{m}^{\mu}| p (p,\lambda)} \big\vert^{2}
\,.
\label{2.105}
\end{eqnarray}
Here $e J_{\mu}$ is the electromagnetic current,
$\epsilon_{m} (m = \pm, 0)$ are the standard $\gamma^{*}$
polarization vectors for right and left circular
and longitudinal polarization
[see (3.11)--(3.14) of \cite{Arens:1996xw}]
and $\epsilon'_{a} (a = 1, 2)$
are the polarization vectors of the real photon in the final state.

So far, everything is completely general. Now we consider
the small $x$ region of DVCS:
\begin{eqnarray}
W \geqslant m_{p}, \sqrt{|t|}\,, \quad
|t| \lesssim 1 \; {\rm GeV}^{2}\,, \quad
x \leqslant 0.02\,.
\label{2.106}
\end{eqnarray}
There we describe the amplitude for (\ref{2.1})
in terms of the exchanges of soft ($\mathbb{P}_{1}$)
and hard ($\mathbb{P}_{0}$) pomeron and the reggeons
$f_{2 \mathbb{R}}$ and $a_{2 \mathbb{R}}$.
Only $C = +1$ exchanges can contribute in (\ref{2.1}).
The properties of $\mathbb{P}_{0}, \mathbb{P}_{1}, f_{2 \mathbb{R}}$,
and $a_{2 \mathbb{R}}$, and their couplings to protons and photons
will be taken as in \cite{Britzger:2019lvc}.
In this way we get
\begin{eqnarray}
&&\Braket{
\gamma(q', \epsilon'),p (p',\lambda'), {\rm out}
|e J_{\nu}(0) \epsilon^{\nu}| p (p,\lambda)}
\equiv
(\epsilon'^{\mu})^* {\cal M}_{\mu \nu, \lambda' \lambda}\, \epsilon^{\nu}
\nonumber \\
&&
=-(\epsilon'^{\mu})^*
\sum_{j = 0, 1}
\Gamma_{\mu \nu \kappa \rho}^{(\mathbb{P}_{j} \gamma^{*} \gamma^{*})}(q',q)\, \epsilon^{\nu}\,
\Delta^{(\mathbb{P}_{j})\,\kappa \rho, \alpha\beta}(W^{2},t)\,
\bar{u}_{\lambda'}(p')
\Gamma_{\alpha\beta}^{(\mathbb{P}_{j} pp)}(p',p)
u_{\lambda}(p)
\nonumber \\
&& \quad + (\mathbb{P}_{j} \to f_{2 \mathbb{R}}, a_{2 \mathbb{R}})\,.
\label{2.3}
\end{eqnarray}
Here $\Delta^{(\mathbb{P}_{j})}$ and $\Gamma^{(\mathbb{P}_{j} pp)}$
denote the effective propagator and proton vertex function,
respectively, for the tensor pomeron $\mathbb{P}_{j}$.
The corresponding expressions,
as given in
Appendix~A of \cite{Britzger:2019lvc}, are as follows
\begin{eqnarray}
&&i\Delta^{(\mathbb{P}_{j})}_{\mu \nu, \kappa \lambda}(W^{2},t) =
\frac{1}{4W^{2}} \left( g_{\mu \kappa} g_{\nu \lambda}
                  + g_{\mu \lambda} g_{\nu \kappa}
                  - \frac{1}{2} g_{\mu \nu} g_{\kappa \lambda} \right)
(-i W^{2} \tilde{\alpha}'_{j})^{\alpha_{j}(t)-1}\,,
\label{2.31}\\
&&i\Gamma_{\mu \nu}^{(\mathbb{P}_{j} pp)}(p',p)
=-i 3 \beta_{jpp} F_{1}^{(j)}(t)
\left\lbrace
\frac{1}{2}
\left[ \gamma_{\mu}(p'+p)_{\nu}
     + \gamma_{\nu}(p'+p)_{\mu} \right]
- \frac{1}{4} g_{\mu \nu} (\slash{p}' + \slash{p})
\right\rbrace. \nonumber \\
\label{2.32}
\end{eqnarray}
The coupling constants $\beta_{j pp}$ of the pomerons
to protons are
$\beta_{0 pp} = \beta_{1 pp} = 1.87$~GeV$^{-1}$.
The \textit{Ans{\"a}tze}
for effective propagators and vertices
for the $f_{2 \mathbb{R}}$ and $a_{2 \mathbb{R}}$ reggeons
are analogous to (\ref{2.31}) and (\ref{2.32}),
respectively.
We shall assume identical trajectories for $f_{2 \mathbb{R}}$
and $a_{2 \mathbb{R}}$ and combine their contribution in (\ref{2.3})
to a reggeon $\mathbb{R}_{+}$ term labeled $j = 2$
as explained in Appendix~A of \cite{Britzger:2019lvc};
see (A20)--(A31) of \cite{Britzger:2019lvc}.

The pomeron and reggeon trajectory functions
are assumed to be of linear form
\begin{eqnarray}
\alpha_{j}(t) = \alpha_{j}(0)+\alpha'_{j}\,t\,,
\quad
\alpha_{j}(0) = 1 + \epsilon_{j}\,,
\quad
(j = 0, 1, 2)\,.
\label{trajectories}
\end{eqnarray}
In (\ref{2.31}) the parameters multiplying the squared energy
$W^{2}$
in the effective propagators
are taken as
$\tilde{\alpha}'_{j} = \alpha'_{j}$.
The slope parameters $\alpha'_{j}$
are taken as the default values from \cite{Britzger:2019lvc}:
$\alpha'_{1} = \alpha'_{0} = 0.25\;{\rm GeV}^{-2}$,
$\alpha'_{2} = 0.9\;{\rm GeV}^{-2}$.
The intercept parameters of the trajectories (\ref{trajectories})
were determined from detailed comparison
of the two-tensor-pomeron model with the DIS HERA data
and photoproduction data in Ref.~\cite{Britzger:2019lvc}:
\begin{eqnarray}
{\rm soft \; pomeron} \;\mathbb{P}_{1}:\quad &&\epsilon_{1} = 0.0935(^{+76}_{-64})\,,
\label{FIT_parameters_P1}
\\
{\rm hard \; pomeron} \;\mathbb{P}_{0}:\quad &&\epsilon_{0} = 0.3008(^{+73}_{-84})\,,
\label{FIT_parameters_P0}\\
{\rm reggeon} \;\mathbb{R}_{+}:\quad &&\alpha_{2}(0) = 0.485(^{+88}_{-90})\,.
\label{FIT_parameters_R}
\end{eqnarray}

The \textit{Ans{\"a}tze} for the $\mathbb{P} \gamma^{*} \gamma^{*}$ and
$\mathbb{R}_{+}\gamma^{*} \gamma^{*}$
coupling functions for both real and virtual photons
are discussed in \cite{Britzger:2019lvc}.
The $\mathbb{P}_{j} \gamma^{*} \gamma^{*}$ vertex,
coupling the soft ($j = 0$) and hard ($j = 1$) pomeron
to two (virtual or real) photons, reads
[see (A18) of \cite{Britzger:2019lvc}]
\begin{eqnarray}
&&i\Gamma^{(\mathbb{P}_{j} \gamma^{*} \gamma^{*})}_{\mu \nu \kappa \rho}(q',q) =
i\left[
2a_{j \gamma^{*} \gamma^{*}}(q^{2},q'^{2},t)\,\Gamma^{(0)}_{\mu \nu \kappa \rho}(q',-q)\,
     - b_{j \gamma^{*} \gamma^{*}}(q^{2},q'^{2},t)\,\Gamma^{(2)}_{\mu \nu \kappa \rho}(q',-q)
     \right], \nonumber \\
&&{\rm where} \; t = (q - q')^{2}, \quad j = 0,1\,.
\label{2.5}
\end{eqnarray}
The coupling functions $a_{j \gamma^{*} \gamma^{*}}$
and $b_{j \gamma^{*} \gamma^{*}}$ have dimensions
GeV$^{-3}$ and GeV$^{-1}$, respectively.
The rank-4 tensor functions
$\Gamma^{(i)}_{\mu \nu \kappa \rho}$ ($i = 0, 2$)
are defined in (A13)--(A17) of \cite{Britzger:2019lvc}.
The $\mathbb{R}_{+} \gamma^{*} \gamma^{*}$ vertex
for real and virtual photons has the same structure
as (\ref{2.5}) and is labeled with $j = 2$;
see (A27)--(A31) of \cite{Britzger:2019lvc}.

Inserting (\ref{2.31}), (\ref{2.32}) and (\ref{2.5})
in (\ref{2.3}) we get
\begin{eqnarray}
{\cal M}_{\mu \nu, \lambda' \lambda}
&=&-i \sum_{j = 0, 1, 2}
\left[ 2a_{j \gamma^{*} \gamma^{*}}(q^{2},0,t)\,\Gamma^{(0)}_{\mu \nu \kappa \rho}(q',-q)\,
- b_{j \gamma^{*} \gamma^{*}}(q^{2},0,t)\,\Gamma^{(2)}_{\mu \nu \kappa \rho}(q',-q) \right] \nonumber \\
&&\times \frac{1}{2W^{2}}
(-i W^{2} \alpha'_{j})^{\alpha_{j}(t)-1} \;
3 \beta_{jpp} F_{1}^{(j)}(t)\,
\bar{u}_{\lambda'}(p')
\gamma^{\kappa}(p' + p)^{\rho}
u_{\lambda}(p) \,.
\label{2.200}
\end{eqnarray}
From (\ref{2.200}) we can read off the parameters of our model.
Taking the pomeron and reggeon parameters as specified above
for granted ${\cal M}_{\mu \nu, \lambda' \lambda}$
is parametrized by the functions
\begin{eqnarray}
&&a_{j \gamma^{*} \gamma^{*}}(q^{2},0,t)\,
\beta_{jpp} F_{1}^{(j)}(t)\,,
\nonumber \\
&&b_{j \gamma^{*} \gamma^{*}}(q^{2},0,t)\,
\beta_{jpp} F_{1}^{(j)}(t)\,,
\nonumber \\
&&j = 0, 1, 2\,.
\label{2.201}
\end{eqnarray}
Here we consider the $\beta_{jpp}$ as normalization constants
with values
$\beta_{0 pp} = \beta_{1 pp} = 1.87$~GeV$^{-1}$,
$\beta_{2 pp} = 3.68$~GeV$^{-1}$,
as given in Appendix~A of \cite{Britzger:2019lvc}.

For forward virtual Compton scattering,
$q^{2} = q'^{2} = -Q^{2}$ and $t = 0$,
we have from (A19), (A30), and (A31) of \cite{Britzger:2019lvc}
\begin{eqnarray}
&&a_{j \gamma^{*} \gamma^{*}}(-Q^{2},-Q^{2},0) = e^{2} \hat{a}_{j}(Q^{2})\,,
\nonumber \\
&&b_{j \gamma^{*} \gamma^{*}}(-Q^{2},-Q^{2},0) = e^{2} \hat{b}_{j}(Q^{2}) \,,
\nonumber \\
&&j = 0, 1, 2\,.
\label{2.7_DIS}
\end{eqnarray}
The coupling functions
$\hat{a}_{j}(Q^{2})$ and $\hat{b}_{j}(Q^{2})$
were determined from the global fit to HERA inclusive DIS data
for $Q^{2} < 50$~GeV$^{2}$ and $x < 0.01$
and the ($Q^{2} = 0$) photoproduction data;
see Appendix~C and Table~IV of~\cite{Britzger:2019lvc}.

The $Q^{2}$ and $t$ dependencies of the coupling functions
in the $\mathbb{P}_{j} \gamma^{*}\gamma^{*}$
and $\mathbb{R}_{+} \gamma^{*}\gamma^{*}$ vertices
in (\ref{2.200})
must be determined from a comparison to experimental data.
As we shall show in Sec.~\ref{sec:results} we obtain quite
satisfactory descriptions of
the DVCS HERA and elastic $\gamma p$-scattering FNAL data
with the following assumptions:
\begin{eqnarray}
&&a_{j \gamma^{*} \gamma^{*}}(q^{2},0,t) =
e^{2} \hat{a}_{j}(Q^{2}) F^{(j)}(t)\,, \quad (j = 0, 1, 2)\,,
\nonumber \\
&&b_{j \gamma^{*} \gamma^{*}}(q^{2},0,t) =
e^{2} \hat{b}_{j}(Q^{2}) F^{(j)}(t)\,, \quad (j = 2)\,,
\label{2.7}
\end{eqnarray}
and for alternative fits 1 and 2
\begin{eqnarray}
{\rm FIT~1}: \;
&&b_{1 \gamma^{*} \gamma^{*}}(q^{2},0,t) = e^{2} \hat{b}_{1}(0)
(1 + Q^{2}/\Lambda_{1}^{2})^{-1.2}\, F^{(1)}(t)\,,
\quad \Lambda_{1} = 1.4\;{\rm GeV}\,, \nonumber\\
&&b_{0 \gamma^{*} \gamma^{*}}(q^{2},0,t) = e^{2} \hat{b}_{0}(Q^{2})
F^{(0)}(t)\,,
\label{FIT1} \\
{\rm FIT~2}:\;
&&b_{1 \gamma^{*} \gamma^{*}}(q^{2},0,t) = e^{2} \hat{b}_{1}(0)
(1 + Q^{2}/\Lambda_{2}^{2})^{-2.0} \, F^{(1)}(t)\,,
\quad \Lambda_{2} = 2.0\;{\rm GeV}\,, \nonumber \\
&&
b_{0 \gamma^{*} \gamma^{*}}(q^{2},0,t) =
\left\{ \begin{array}{lr}
e^{2} \hat{b}_{0}(Q^{2}) \, F^{(0)}(t)
& {\rm for}\ Q^{2} < 1.5~{\rm GeV}^{2} \\
e^{2} \Lambda_{0} (1 + Q^{2}/\Lambda_{3}^{2})^{-0.6} \, F^{(0)}(t)
& {\rm for}\ Q^{2} \geq 1.5~{\rm GeV}^{2}
\end{array}\right. \,, \nonumber \\
&& \Lambda_{0} = 9.46 \times 10^{-3} \; {\rm GeV}^{-1}\,, \quad
\Lambda_{3} = 2.3\;{\rm GeV}\,.
\label{FIT2}
\end{eqnarray}

The coupling functions as a function of $Q^{2}$
used in the calculations are shown in Fig.~\ref{fig:FIT}.
Shown are
only the middle values of the
pomeron- and reggeon-$\gamma^{*}\gamma^{*}$ coupling functions
up to $Q^{2} = 50$~GeV$^{2}$
determined from a global fit
in \cite{Britzger:2019lvc}.
According to \cite{Britzger:2019lvc},
for the $\mathbb{R}_{+}$-reggeon term,
the function $\hat{a}_{2}(Q^{2})$ is assumed to be zero
while the function $\hat{b}_{2}(Q^{2})$
(see the green dotted line)
vanishes rapidly with increasing $Q^{2}$.
We see that the soft pomeron function
$\hat{b}_{1}$ is larger than the corresponding hard one $\hat{b}_{0}$
up to $Q^{2} \simeq 20$~GeV$^{2}$.
The function $\hat{b}_{0}$
(see the red long-dash-dotted line)
is very small for $Q^{2} \to 0$
and reaches a maximum at $Q^{2} = 1.27$~GeV$^{2}$
with amplitude $\hat{b}_{0} = 0.0082$~GeV$^{-1}$.
This function is important in the large-$Q^{2}$ region.
Thus, it can be expected that the soft pomeron contribution
will be essential for understanding the HERA DVCS data.
We consider two different parametrizations:
(1) FIT~1 (\ref{FIT1}) with the modification of only one
pomeron-$\gamma^* \gamma^{*}$ coupling function
$\hat{b}_{1}(Q^{2})$,
corresponding to a ``soft term'';
(2) FIT~2 (\ref{FIT2}) with
the modification of both $\hat{b}_{1}(Q^{2})$
and $\hat{b}_{0}(Q^{2})$.
Here we increase the hard component.
All other coupling functions,
in both our fits,
are fixed in accord with those from the fit to DIS
of \cite{Britzger:2019lvc}.

We assume, furthermore, the following combined function
to describe the $t$ dependence
of the $\gamma^{(*)} p \to \gamma p$ cross sections
\begin{eqnarray}
F_{\rm eff}^{(j)}(t) =
F^{(j)}(t) \times  F_{1}^{(j)}(t) =
\exp(-b_{j}|t|/2 )\,.
\label{t_dependence_ff}
\end{eqnarray}
For our study here we assume $b_{1} = b_{2}$.
We take, for both FIT~1 and FIT~2 the same parameters,
$b_{1} = b_{2} = 5.0\; {\rm GeV}^{-2}$
and
$b_{0} = 1.0\; {\rm GeV}^{-2}$
found by comparison of the theoretical curves
to the experimental data for $d\sigma/dt$.

With this we have specified our model and given the parameter
values for our two fits which we shall compare to the data
in Sec.~\ref{sec:results}.

\begin{figure}[!ht]
\includegraphics[width=0.9\textwidth]{Q2_depend.eps}
\caption{\label{fig:FIT}
\small
The pomeron- and reggeon-$\gamma^{*}\gamma^{*}$ coupling functions
$\hat{a}_{j}(Q^{2})$ and $\hat{b}_{j}(Q^{2})$
from (\ref{2.7_DIS}),
for $\mathbb{R}_{+}$ reggeon ($j = 2$, green dotted line)
soft pomeron ($j = 1$, black solid and dashed lines),
and hard pomeron
($j = 0$, red long-dash-dotted and short-dashed lines),
as determined in \cite{Britzger:2019lvc}.
The experimental uncertainties of the fit are not shown here.
For this we refer to Figs.~12--15 of \cite{Britzger:2019lvc}.
The blue dash-dotted line
corresponds to the $b_{1}(-Q^{2},0,0)/e^{2}$ coupling function
used in the FIT~1 (\ref{FIT1}).
The azure dash-dot-dotted line
and the magenta long-dashed line are for
$b_{1}(-Q^{2},0,0)/e^{2}$ and $b_{0}(-Q^{2},0,0)/e^{2}$,
respectively,
in the FIT~2 (\ref{FIT2}).
The $a$ and $b$ parameters are plotted in units
GeV$^{-3}$ and GeV$^{-1}$, respectively.}
\end{figure}

\section{Results}
\label{sec:results}

In this section we compare
the tensor-pomeron model to the FNAL data
\cite{Breakstone:1981wm}
on real-photon-proton scattering ($\gamma p \to \gamma p$),
and to HERA data
\cite{ZEUS:2003pwh,H1:2005gdw,ZEUS:2008hcd,H1:2009wnw}
on DVCS ($\gamma^{*}(Q^{2}) p \to \gamma p$)
for different averaged $W$ and $Q^{2}$.
We shall restrict our discussion to experimental results
that satisfy the conditions $x \approx Q^{2}/W^{2} < 0.02$
and $|t| \lesssim 1$~GeV$^{2}$ where our model should be reliable.
We also have to make a comment on \textit{what} quantity
is measured in \cite{H1:2005gdw}.
For this we compare Eqs.~(6)--(8) of \cite{H1:2005gdw}
with Eqs.~(\ref{2.102})--(\ref{2.104}).
We cannot see that longitudinally polarized $\gamma^{*}$'s
do not contribute after integration over azimuthal angles,
as is claimed in footnote~7 of \cite{H1:2005gdw}.
Thus, to the best of our understanding, the cross section
measured in \cite{H1:2005gdw} must be
\begin{equation}
\frac{d\sigma}{dt}
= \frac{d\sigma_{\rm T}}{dt} + \varepsilon \frac{d\sigma_{\rm L}}{dt}\,,
\label{3.100}
\end{equation}
where $\varepsilon \approx 1$ for the HERA kinematic region.
In the following we suppose that this is the case.
We shall use the notation
\begin{equation}
\sigma(Q^{2},W^{2})
= \int dt \, \frac{d\sigma}{dt}(Q^{2},W^{2},t)
\label{3.101}
\end{equation}
and similarly for the $\rm T$ and $\rm L$ components.

In Fig.~\ref{fig:1} we show
the $\gamma^{(*)}  p \to \gamma p$ total cross sections
as a function of the c.m. energy $W$.
In the top panels we compare our fits,
FIT~1 (left panel) and FIT~2 (right panel),
to the FNAL result from \cite{Breakstone:1981wm}\footnote{Note
that in \cite{Breakstone:1981wm} the value of
$\sigma = 88 \pm 4$~nb for the elastic $\gamma p$ cross section
was given.
This value was obtained by summing
the beam-energy-averaged differential cross sections
measured at c.m. energies $W = 9.73-15.645$~GeV
for $|t| \geqslant 0.07$~GeV$^{2}$
with those from the preferred fit~II of \cite{Breakstone:1981wm}
for $|t| < 0.07$~GeV$^{2}$.}
where $Q^{2}  = 0$ and to the HERA data from
\cite{ZEUS:2003pwh,H1:2005gdw,ZEUS:2008hcd,H1:2009wnw}
for different values of $Q^{2}$ (the average values).
The complete cross section is a coherent sum of
soft and hard components in the amplitude.
These components have different energy dependence.
For real Compton scattering ($Q^{2} = 0$)
the cross section is dominated by soft-pomeron exchange with
an additional contribution from reggeon exchange at lower energies.
The hard-pomeron contribution is negligibly small there.
The cross sections rise with energy
as $W^{2 \epsilon_{1}}$ at $Q^{2} = 0$
and change to $W^{2 \epsilon_{0}}$ for very high $Q^{2}$.
Here $\epsilon_{1} \approx 0.09$
and $\epsilon_{0} \approx 0.30$
are the intercept parameters
of the soft and hard pomerons, respectively,
see~(\ref{FIT_parameters_P1}), (\ref{FIT_parameters_P0}).
In our model the dominant contribution
comes from the $b$-type coupling functions
$b_{j \gamma^{*}\gamma^{*}}$ ($j =0,1$).
Their size, as shown in Fig.~\ref{fig:FIT},
differs in the two fits.
We see from the bottom panels of Fig.~\ref{fig:1} that
for higher $Q^{2}$ the soft component slowly decreases
relative to the hard one.
A significant constructive interference effect
between the soft and hard components is clearly visible.
Here and in the following, the interference term
is calculated as the difference of
coherent and incoherent cross sections
of the $\mathbb{P}_{1}$, $\mathbb{P}_{0}$, and $\mathbb{R}$ contributions.
The Fits~1 and 2 hardly differ for the $W$ region where
there are data. But for higher $W$ values FIT~2,
where the contribution from the hard pomeron is enhanced,
gives a steeper rise of the cross section with $W$
and especially so for larger $Q^{2}$.
Only future experiments will tell us what happens in nature.
\begin{figure}[!ht]
\includegraphics[width=0.49\textwidth]{DVCS_sig_vs_W_Fit1.eps}
\includegraphics[width=0.49\textwidth]{DVCS_sig_vs_W_Fit2.eps}
\includegraphics[width=0.49\textwidth]{DVCS_sig_vs_W_deco_fit1.eps}
\includegraphics[width=0.49\textwidth]{DVCS_sig_vs_W_deco_fit2.eps}
\caption{\label{fig:1}
\small
Top panels:
Total cross sections as a function of the c.m. energy $W$
for FIT~1 (left) and FIT~2 (right).
Comparison of theoretical results
to the FNAL data from \cite{Breakstone:1981wm}
for real Compton scattering ($Q^{2} = 0$)
and to the DVCS HERA data is shown.
The upper black solid line is for $Q^{2} = 0$,
the orange long-dashed-dotted line is for $Q^{2} = 1$~GeV$^{2}$.
The remaining lines correspond to the values
\mbox{$Q^{2} = 2.4, 3.2, 4,
 6.2, 8, 10, 15.5, 25$~GeV$^{2}$}
(from top to bottom) and should be compared with
the HERA data from
\cite{ZEUS:2003pwh,H1:2005gdw,ZEUS:2008hcd,H1:2009wnw}.
Bottom panels:
Our fit results for $Q^{2} = 8$~GeV$^{2}$
together with the H1 data \cite{H1:2009wnw}.
We show the contributions for soft and hard pomeron separately,
see the blue long-dashed line and the red dashed line,
and their coherent sum (total), see the black solid line.
The interference term is shown separately by the green dotted line.
}
\end{figure}

In Fig.~\ref{fig:2} we show how
the cross section depends on $Q^{2}$
for two c.m. energies $W = 82$ and 104~GeV.
Complete results for our two fits
and the soft and hard components separately
are shown together with the HERA data.
From the top panels (for FIT~1) we see that
the soft pomeron survives to relatively large $Q^2$
and at $Q^2 \simeq 50$~GeV$^{2}$ the interference term
plays an important role in the description of the data.
From the bottom panels (for FIT~2) one can observe that
at $Q^2 \simeq 50$~GeV$^{2}$ the hard pomeron alone
is able to describe the data.
Here the contribution from the interference term
is considerable for intermediate values of $Q^{2} \sim 10$~GeV$^{2}$.
It is interesting to note that
the interference effect is also large
when the hard component reaches a maximum.
For our fits this is at $Q^{2} \cong 1.3$~GeV$^{2}$.
\begin{figure}[!ht]
\includegraphics[width=0.49\textwidth]{DVCS_sig_vs_Q2_W82.eps}
\includegraphics[width=0.49\textwidth]{DVCS_sig_vs_Q2_W104.eps}
\includegraphics[width=0.49\textwidth]{DVCS_sig_vs_Q2_W82_fit2.eps}
\includegraphics[width=0.49\textwidth]{DVCS_sig_vs_Q2_W104_fit2.eps}
\caption{\label{fig:2}
\small
Comparison of the cross sections
for the $\gamma^{(*)} (Q^{2}) p \to \gamma p$ reaction
as a function of $Q^{2}$
to the experimental data for $\langle W \rangle = 82$ GeV from
\cite{H1:2005gdw,H1:2009wnw} (left panels)
and for $\langle W \rangle = 104$ GeV from \cite{ZEUS:2008hcd} (right panels).
Results on top panels correspond to FIT~1 and
those on the bottom panels to FIT~2.
The meaning of the lines is the same as in the bottom panels
of Fig.~\ref{fig:1}.}
\end{figure}

In Fig.~\ref{fig:3} we show
the differential cross sections $d\sigma/dt$
from (\ref{3.100}) with $\varepsilon = 1$
for different $\langle W \rangle$ and $\langle Q^{2} \rangle$.
In the top panels,
the upper line corresponds to $W = 12.7$~GeV
and $Q^{2} = 0$ and should be compared
to the averaged FNAL data (top data points)
for the $\gamma p \to \gamma p$ reaction
\cite{Breakstone:1981wm}.
In this kinematic range, for $Q^{2} = 0$
and at intermediate $W$,
the reggeon plus soft-$\mathbb{P}$ contributions
dominate and the hard-$\mathbb{P}$ exchange
gives negligible contribution.
As expected, there is a significant interference between
the reggeon and soft-pomeron components.
The slope parameters $b_{2}$ and $b_{1}$ in (\ref{t_dependence_ff})
are adjusted to the FNAL $d\sigma/dt$ data on
the real-photon-proton scattering.
At higher $W$ and $Q^{2}$ measured at HERA
the hard pomeron plays an increasingly important role.
The slope parameter $b_{0}$ for the hard-pomeron exchange
is adjusted to the DVCS HERA data.
As we noted above, $d\sigma(\gamma^{*} p \to \gamma p)/dt$
is the sum of two contributions
$d\sigma_{\rm T}/dt$ and $d\sigma_{\rm L}/dt$ with the latter term
becoming very small for $|t| \to 0$.
This is understandable since for $\gamma^{*} p \to \gamma p$
forward scattering only double-helicity-flip amplitudes
can contribute to $d\sigma_{\rm L}/dt$.
Furthermore, $d\sigma_{\rm T}/dt$ is dominated
by the $b$-type couplings
and $d\sigma_{\rm L}/dt$ is dominated by the $a$-type couplings;
see Eq.~(\ref{2.5}).
\begin{figure}[!ht]
\includegraphics[width=0.49\textwidth]{dsig_dt_gamp_gamp_DVCS_fit1.eps}
\includegraphics[width=0.49\textwidth]{dsig_dt_gamp_gamp_DVCS_fit2.eps}
\includegraphics[width=0.49\textwidth]{dsig_dt_gamp_gamp_DVCS_Q2_10_fit1.eps}
\includegraphics[width=0.49\textwidth]{dsig_dt_gamp_gamp_DVCS_Q2_10_fit2.eps}
\caption{\label{fig:3}
\small
Top panels: The differential cross sections $d\sigma/dt$
compared to experimental data
for different $\gamma^{*} p$ c.m. energy
and photon virtuality.
The upper line corresponds to $W = 12.7$~GeV
and $Q^{2} = 0$ and averaged FNAL data \cite{Breakstone:1981wm}.
The theoretical results for $\gamma^{*} p \to \gamma p$
at higher $W$ and $Q^{2}$
for transverse (long-dashed lines) and longitudinal (short-dashed lines)
polarization of the $\gamma^{*}$ and their sum (solid lines)
are shown together with experimental data.
Data for $\langle W \rangle = 104$~GeV are from \cite{ZEUS:2008hcd}
and for $\langle W \rangle = 82$~GeV from \cite{H1:2009wnw}.
Bottom panels:
Comparison of the fit results
to the H1 data \cite{H1:2009wnw}
for $\langle Q^{2} \rangle = 10$~GeV$^{2}$
and $\langle W \rangle = 40$, 70, and 100~GeV
(from bottom to top).
Here we show the results scaled by a factor $\rm k$
(specified in the figure legend)
for displaying purposes.}
\end{figure}

In Fig.~\ref{fig:4} we show the complete theoretical result
and individual components contributing to
the cross section $d\sigma/dt$, see (\ref{3.100}),
for $W = 82$~GeV and $Q^{2} = 8$~GeV$^{2}$
together with the H1 data \cite{H1:2009wnw}.
The constructive interference of the soft
and hard pomeron terms is again a salient feature.

We note that our complete result for $d\sigma/dt$ does not have
a simple exponential $t$ dependence
as is frequently assumed in other models.
This is caused by the interference
of soft- and hard-pomeron terms
(each with different $t$ dependence)
and by the longitudinal contribution
which is important for large $|t|$.
\begin{figure}[!ht]
\includegraphics[width=0.49\textwidth]{dsig_dt_gamp_gamp_DVCS_deco_fit1.eps}
\includegraphics[width=0.49\textwidth]{dsig_dt_gamp_gamp_DVCS_deco_fit2.eps}
\caption{\label{fig:4}
\small
Comparison of the FIT~1 (left) and the FIT~2 (right)
to DVCS H1 data \cite{H1:2009wnw}
for $W = 82$~GeV and $Q^{2} = 8$~GeV$^{2}$.
The results for $\gamma^{*} p \to \gamma p$
for transverse (${\rm T}$) and longitudinal (${\rm L}$)
polarization of the $\gamma^{*}$ individually
and their sum ${\rm T + L}$ (see the upper solid lines) are shown.
The contributions of soft $\mathbb{P}$ (the blue short-dashed lines),
hard $\mathbb{P}$ (the red long-dashed lines),
the interference term (the green dotted lines),
and their sum total (the thin full lines)
for ${\rm T}$ and ${\rm L}$ components individually are also shown.}
\end{figure}

Figure \ref{fig:ratios} shows the ratios
of the $\gamma^{*} p \to \gamma p$ cross sections
for longitudinally and transversely polarized
virtual photons,
\begin{eqnarray}
&&R(Q^{2},W^{2}) = \frac{\sigma_{\rm L}(Q^{2},W^{2})}{\sigma_{\rm T}(Q^{2},W^{2})} \,,
\label{ratio_Q2}\\
&&\tilde{R}(Q^{2},W^{2},t) = \frac{\frac{d\sigma_{\rm L}}{dt}(Q^{2},W^{2},t)}
                                  {\frac{d\sigma_{\rm T}}{dt}(Q^{2},W^{2},t)} \,,
\label{ratio_t}
\end{eqnarray}
as functions of $Q^{2}$ and $|t|$, respectively.
The cross section $\sigma_{\rm L}$
vanishes proportionally to $Q^{2}$ for $Q^{2} \to 0$.
The ratio $\tilde{R}(Q^{2},W^{2},t)$ strongly grows with $|t|$.
We must emphasize that this behaviour of
$\tilde{R}(Q^{2},W^{2},t)$
depends crucially on our (reasonable) assumption
that $a$ and $b$ couplings in the $\mathbb{P} \gamma^{*} \gamma^{*}$
vertex functions have the same $t$ dependence
for a given $j$~$(j = 0, 1, 2)$;
see (\ref{2.200}), (\ref{2.201}), and (\ref{t_dependence_ff}).
But it is clear from the right panel of Fig.~\ref{fig:ratios}
that a very small ratio $\tilde{R}(Q^{2},W^{2},t)$
for all $|t| \lesssim 1.0$~GeV$^{2}$ could, in our model,
only be  achieved if the $a$ and $b$ couplings
would have drastically different $t$ dependences.
\begin{figure}[!ht]
\includegraphics[width=0.49\textwidth]{ratio_Q2.eps}
\includegraphics[width=0.49\textwidth]{ratio_t.eps}
\caption{\label{fig:ratios}
\small
The ratios of cross sections
$R(Q^{2},W^{2})$ (\ref{ratio_Q2})
and $\tilde{R}(Q^{2},W^{2},t)$ (\ref{ratio_t})
for the $\gamma^{*} p \to \gamma p$ reaction
for longitudinally and transversely polarized
$\gamma^{*}$.
Note that the meaning of the lines on these two panels is different.}
\end{figure}

\section{Conclusions and discussion}
\label{sec:conclusions}

In the present paper we have applied the tensor-pomeron
approach to deeply virtual Compton scattering
(DVCS) for high c.m. energies $W$
and small Bjorken-$x$, $x \lesssim 0.02$.

We have made a comparison of the two-tensor-pomeron model
to the DVCS data measured at HERA.
As a starting point we have used the fit parameters of the intercepts
of the two pomerons and of the reggeon $\mathbb{R}_{+}$,
and their coupling functions
to real and virtual photons determined in \cite{Britzger:2019lvc}
from deep inelastic scattering (DIS)
and real photoabsorption cross sections.
To our surprise, with a 'minimal', but reasonable,
modification of the $Q^2$ dependence
of only one $\gamma^*(Q^2) \gamma \mathbb{P}$ coupling function
[FIT~1 (\ref{FIT1})]
we have already got a good fit to the experimental data.
We could describe the $W$, $Q^2$ and $t$ dependences of
$d\sigma(\gamma^{*} p \to \gamma p)/dt$ measured at HERA
and of the elastic photon-proton cross section measured at FNAL.
The good description of the DVCS data
is achieved due to a sizeable interference
of soft and hard pomeron contributions.
We have considered also FIT~2 (\ref{FIT2})
in which the size of the hard component was increased.
We kept here, on purpose, the same parameters of the form factors (\ref{t_dependence_ff}) as in FIT~1.
The FIT~2 better describes the data
at larger $|t|$ for $Q^2 \gtrsim 8$~GeV$^{2}$
(see Fig.~\ref{fig:3}).
Note that in our two-tensor-pomeron model
the soft component and also the interference of soft and hard terms
are very important up to at least $Q^{2} \simeq 20$~GeV$^{2}$.

Now we comment on Refs.~\cite{Capua:2006ij,Fazio:2013hza,Jenkovszky:2018itd}
where also Regge theory is applied to DVCS.
These authors consider, as far as we can see,
only leading helicity amplitudes.
For DVCS this means that only transversely polarized initial
$\gamma^{*}$'s are considered.
In contrast, in our tensor-pomeron approach
we present a complete model,
including \textit{all} helicity amplitudes.
Therefore, we could make, e.g., predictions for
$\sigma_{\rm L}/\sigma_{\rm T}$ which can and should be checked
by experiments.
Also, our model for DVCS gives different relations between
soft and hard terms compared to the analysis
of \cite{Fazio:2013hza};
see Sec.~IV therein and the review \cite{Jenkovszky:2018itd}
where more results are presented.
The contribution from the interference term
found in \cite{Fazio:2013hza} is considerable
for intermediate values of $Q^{2}$,
but smaller than our findings (FIT~1 and FIT~2).

In our calculations we have included the contributions
of both transverse and longitudinal virtual photons.
The longitudinal cross section $d\sigma_{\rm L}(Q^{2}, W^{2}, t)/dt$
is predicted to be very small for $|t| \to 0$ but
to be sizeable for
$0.5 \; {\rm GeV}^{2} \lesssim |t| \lesssim 1.0\; {\rm GeV}^{2}$
(see Figs.~\ref{fig:3} and~\ref{fig:4}).
The corresponding ratio of ${\rm L/T}$ grows strongly with $|t|$
(see the right panel of Fig.~\ref{fig:ratios}).
We have also shown the $Q^2$ dependence of this ratio
for different c.m. energies of the $\gamma^{*}p$ system.

To summarize, we have presented predictions for low-$x$ DVCS
of the two-tensor-pomeron model which previously
was successfully applied to low $x$ DIS in \cite{Britzger:2019lvc}.
The~model provides amplitudes for all helicity configurations
and, thus, can be checked by experimentalists in many ways.
We are looking forward to further tests of
the non-perturbative QCD dynamics embodied in our
tensor-pomeron exchanges
in future electron-hadron collisions
in the low-$x$ regime at the EIC and LHeC colliders.

\acknowledgments
The authors are grateful to Markus Diehl for
correspondence and comments and to Carlo Ewerz
for useful discussions.
This work was partially supported by
the Polish National Science Centre Grant
No. 2018/31/B/ST2/03537.
P. L. was supported by
the Bekker Program of the Polish National Agency for Academic Exchange, Project No. BPN/BEK/2021/2/00009/U/00001.

redpajama_set_name: RedPajamaArXiv


text:
Home Politics I'll not return to APGA, Ojukwu Jnr tells Obiano
I'll not return to APGA, Ojukwu Jnr tells Obiano
bvichannel1
First son of late Dim Chukwuemeka Odumegwu Ojukwu, Chief Emeka Ojukwu Jnr, has turned down overtures of Governor Willie Obiano to return to the All Progressives Grand Alliance (APGA).
Ojukwu Jnr, who barely 48 hours to the Anambra State governorship election defected to the All Progressives Congress (APC), said he is not going back to APGA.
He said he is no longer interested in playing emotional politics, and added that, by wanting to play politics at the centre, he is following the steps of his father.
Speaking to newsmen in Awka, Ojukwu Jnr. described the governor's overtures as too infantile and smacks of ignorance about the workings of politics in Nigeria.
He said like his father, he ensures that his actions are informed by the best interest of Ndigbo, recalling how, shortly after his return from exile in 1982, his father joined the National Party of Nigeria (NPN) rather than the Nigeria Peoples Party (NPP) seen then as Igbo party.
"I can afford to stay back (in APGA) and wallow in sentimental politics that 'Nkea bu Nkeanyi' (this is our own), but in the long run, what becomes of our tomorrow and place in the great Nigeria project, for which our forefathers laboured, invested their time, talent and even their blood?
"Which is preferable; to be a big fish in a small pond or a big fish in the ocean? Real champions are never afraid of the big league. That could be why late Dr. Chuba Okadigbo asserted that if you cannot reason beyond petty sentiments, you are a liability to mankind. Igbo are not better off with emotional politics."
Ojukwu Jnr. reacted to a statement credited to Governor Obiano, that he considers Ikemba's son as of no consequence, but has extended an olive branch to stakeholders and co-contestants in the recent Anambra governorship poll. He said the governor was still playing kindergarten politics.
He wondered why Obiano and other party chieftains should condemn him for identifying with APC while they struggle to impress President Buhari that they support him for a second term in office.
"So, while they seek to crucify me for joining APC, they express support and solidarity for the APC leader. Which is worse, to walk the talk or pander to politics of deceit and duplicity? What is the essence of being in a party that would rather support the presidential candidate of another party than choose its own?"
According to him, having supported President Olusegun Obasanjo and Goodluck Jonathan in office, Ndigbo have decided to support President Buhari in the spirit of fairness and political goodwill, so that when a president of Igbo extraction emerges, other ethnic groups would reciprocate.
source ; Sun
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redpajama_set_name: RedPajamaCommonCrawl


text:
Annette J. LaRocca, 86
Annette J. LaRocca, 86, of Selbyville, passed away at Beebe Medical Center in Lewes on Monday, Feb. 12, 2007.
She had been an administrator with the Department of Defense and was a member of St. Luke Catholic Church in Ocean City, Md.
Annette is survived by sisters Marie B. Somers of Selbyville, Bernice J. Duncan of Sanger, Calif; and several nieces and nephews.
She was predeceased by her parents, Louis and Christine Martin LaRocca.
A prayer service was to be held Friday, Feb. 16, 2007, at the Hastings Funeral Home in Selbyville, with interment on Monday, Feb. 19, 2007, at the National Memorial Park in Falls Church, Va., at 1 p.m.
In lieu of flowers, donations may be made to the Worcester County Humane Society, PO Box 48, Berlin, MD 21811.
Clarence G. 'Gib' Johnson, 90
Clarence G. "Gib" Johnson, 90, of Millsboro, passed away on Saturday, Feb. 10, 2007, at his home.
Johnson was a produce farmer, and a longtime member of Harmony United Methodist Church in Millsboro and the Nanticoke Indian Association. He enjoyed farming, cutting grass and just plain work.
He is survived by his son, Troy Johnson of Millsboro; grandsons Robert Duff Jr. of Smyrna, Keenan Harmon of Philadelphia, Pa., Kelvin Harmon of New Castle; granddaughters Karen Overton and Katrina Harmon of Philadelphia, Pa.; sisters Olive Walls of Wilmington, Louise Cangelose of Va., Maxine Hitchens of Millsboro; six great-grandchildren and three great-great-grandchildren.
He was predeceased by his parents, Dorsey E. and Anna E. Clark Johnson; wife, Fannie E. Johnson; sons Clarence H. and Alvin L. Harmon; one granddaughter; four brothers, Chester, Paris, Everett and Clyde Johnson; and sisters Hilda, Juanita and Sylvia.
Funeral services will be held Friday, Feb. 16, 2007, at Harmony United Methodist Church, Route 24, Millsboro, at 11 a.m. Interment will be at the Indian Mission United Methodist Church Cemetery in Millsboro.
In lieu of flowers, the family suggests contributions be made to Delaware Hospice, Southern Division, 20167 Office Circle, Georgetown, DE 19947.


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text:
The First World War day-by-day
WWI as it happened 100 years ago
Italian Front
Mid East / Caucasus
Methods & Intentions
First day on the Somme | 1 July 1916
Posted on 1st July 2016 by makersley — 5 Comments ↓
On the signal, company will advance!
First day on the Somme
In the British imagination, 1st July 1916 looms over the rest of the First World War. Ask someone about the First World War, and if they know anything at all about the war, it is extremely likely that the first day on the Somme is the first thing they think of. It is also reasonably likely that it is the only thing they think of. Or rather, it is a particular sub-section of the first day of the Somme that they think of. It is, to say the least, a very arresting image, and a very easily distilled image, and a very easily understandable image.
However, it is not the entire story. This battle defies easy characterisation when looked at in any detail. So let's get down to the detail. Let us have, once again, the map, to remind ourselves of what is going on.
As ever, this bears only the slightest resemblance to what the front actually looks like
From Serre to the end of the French attack front is very approximately 23 miles. General Haig expects that by the end of the day, his men will have advanced to the pink line. This will set up an attack in a few days to break the German Second Line at Pozieres, send the BEF's cavalry through the gap to exploit it, and capture a critical railway junction at Bapaume.
Meanwhile, the diversions will go on to the last possible moments. All along the BEF front, zero hour at 7:30am will see "demonstrations"; artillery barrages, shouting and whistle-blowing by officers, and other such excitement to simulate an attack and keep the enemy confused as long as possible. And, about five miles north of Serre, there will be a full-scale diversionary attack at Gommecourt. And that's where we'll start the first day on the Somme.
Gommecourt
There is a small tactical advantage to be had by eliminating the Gommecourt salient, but that's not what the attack is about. This is about confusing German intelligence for as long as possible, disguising the main thrust of the Battle of the Somme. Even a delay of a few hours could prove important. The men here have been quite deliberately making no attempt to disguise their preparations for attack. This is, bluntly, going to suck for the people who have drawn this shortest of straws and who now are being ordered to sacrifice themselves for the greater good.
One of them is Captain Arthur Agius. We've been following him on and off since he arrived on the Western Front in January 1915. He was in the Battle of Neuve Chapelle, and he's already drawn the shortest straw once already; for the Battle of Loos, he survived being in a diversionary attack at Neuve Chapelle. This, however, is on a completely different scale.
The trouble was that Gommecourt stuck out in the middle of the line and we weren't to attack it directly. We were supposed to encircle it and link up behind. But what we didn't know is that the Germans had manoeuvred and organised their line that this part which we weren't to attack was really their strong-point. They simply had a clear field of fire on either side and nothing to bother about in front. The shellfire was appalling. We just couldn't get across. We didn't even get as far as [the jumping-off trench]. There was no trench left.
We got orders to turn and try to make our way back to the village. One of my subalterns was newly out. [He] jumped out of the trench to try to organise the men, and he was promptly killed. Just disappeared in an explosion. So many gone, and we'd never left our own front-line trench. And then we found we couldn't get back. We were simply treading on the dead. Eventually my Sergeant and I got out. I heard a shell coming. It burst just above my head. The Sergeant…was killed. I don't know how I got back. It was murder.
A few brave/lucky souls have managed to make it into the German trenches, but they've been quickly cut off. Nobody can cross No Man's Land to reinforce them. German artillery has cut all the BEF telephone lines, and once again the generals who are supposed to coordinate the attack have absolutely no idea what's going on. The end of the day sees the last stragglers dragging themselves back across No Man's Land to the BEF's lines. Arthur Agius has somehow found his way to Battalion Headquarters, where he is now sitting quietly in a corner, sobbing uncontrollably. In good time he will be invalided out as a shell shock casualty.
Attacking battalions at Gommecourt are lucky if they've only lost 50% of their men killed or wounded. So this is all horrible and disgusting and bloody. But it's the diversionary attack. Just in planning the attack they convinced the Germans to send a division away from the main battle to reinforce Gommecourt. If we can show that it came to something positive…
So this is where we are as now we turn to the main battle. I will now take it from south to north. And that means starting with the French contribution. The map's just up there a bit; and so far it is completely unchanged. Won't be for long, though!
The French Army on Day 1
This is because the French Army has achieved complete and total success. Taking a considerably more pessmistic view than the BEF, Generals Foch and Fayolle have planned a more cautious, step-by-step advance. The French contribution to the Somme is extremely shabbily treated in the English-language history, so about all I'm able to say is that south of the River Somme, the French are doing extremely well.
The infantry advanced hard on the heels of a well-planned rolling barrage, they walked almost right through the German First Line without much resistance, and by the end of the day they're digging in for the next bit. The Second Line will not be attacked until everybody is ready, and it will come after another extensive prepatory bombardment, in accordance with the new French doctrine. On top of that, because of their total air superiority, enforced by plentiful Nieuport Bebe fighter planes, their pre-battle fire has paid far more attention than at any time before to counter-battery fire, the fine art of using your artillery to shoot the enemy's artillery.
The men have consequently been able to advance almost unmolested. With the big guns destroyed by French artillery, and the machine guns suppressed by the rolling barrage, the French have done exactly what the BEF was hoping to do. They've just walked across No Man's Land and taken possession, and been happy with that and not tried to accomplish too much more. North of the River Somme the story is mostly the same. Although here the counter-battery work has been less successful, the French attackers have benefited from an extensive early morning river mist. They've captured the First Line, and come the end of the day they're digging in and another lot are looking forward to the hard work of marching on the Second Line.
And there's even a nice human moment to transition into the first BEF sector. Over the last few weeks, Commandant Le Petit of the 153rd Regiment has become rather friendly with his left-hand neighbour, Lt-Col Fairfax of the 17th King's Liverpools. Perhaps it was not an accident that brought them together in the last moments before zero hour. Arm in arm, the two men lead their battalions over the top and across No Man's Land in perfect order at a steady walk. The men have rifles slung over their shoulders, many of them are smoking, and they cross No Man's Land mostly without inconvenience. Johnny Crapaud's done his bit; over to you, Tommy Atkins.
This covers the southernmost part of the BEF's attack, centred on Montauban, which sits between the First and Second Lines. This is the sector where Lt-Col Fraser-Tytler has been stationed. The British artillery, with considerable French assistance, has been firing a relatively advanced version of the creeping barrage to support the advance. They're assisted by two large mines under a pair of German strong-points, and the relatively incoherent state of the German defences in many places owing to intense mining and counter-mining earlier in the year. Fraser-Tytler:
The line advanced steadily, scarcely meeting any opposition in the [German First Line]. By 8:20 the formidable Glatz Redoubt was captured. The infantry reported that the maze of strong-points, machine-gun emplacements, etc. had all been swept away, and that the trenches were crammed with dead. …[They] then continued the attack and captured the whole village of Montauban by 10am. … In the afternoon Hun shelling increased slightly, but he appeared to be completely demoralised and, for the moment, shaken out of his wits.
After dinner I went round to "swap lies" with the Brigade Commander of the 75mm batteries behind us. He said he had been twice at Verdun, but even there he had never seen such an intense bombardment. … We were sitting, cups in hand, when we saw three French gunners trolleying down the little railway line which runs through all the positions. A stray shell landed exactly on the trolley, which simply dissolved away. Some more men of their battery then collected any scraps [of human remains] they could find, butting them into different sandbags, amidst heated arguments as to which was the proper one, etc.
This reference to confusion among the German artillery is critical. The British counter-battery fire was not nearly so accurate as the French, but the German guns are mostly without orders or observation posts. German telephone wires are not immune to being cut, after all. This is where the men threw footballs over the top and found it very useful in calming their nerves as they walked across the open ground. There is one other personal account to bring in at this point. This is from Albert Andrews of the 19th Manchesters, shortly after entering the German trenches…
[We] turned round to go along the trench, when three fine Germans came running towards us with their hands up. They would be about 20 yards away. We both fired and two fell, my mate saying as we let go, 'That's for my brother in the Dardanelles!', and as he fired again and the third German fell, 'That's for my winter in the trenches!' We walked up to them and one moved. My mate kicked him and pushed his bayonet into him. That finished him.
This kind of thing was going on all along the line, no Germans being spared. Wounded were killed by us all. We hadn't exactly been told, 'No prisoners!' but we were given to understand that that was what was wanted.
This is, of course, a war crime. Sad to say, stories like this are pretty much ten-a-penny from both sides when there's a major offensive on. So too are stories of other men going out of their way to take prisoners. Andrews had himself just a few minutes before spared a shell-shocked German who he could easily have shot down. The perfect Balanced Sample? After securing the trench he and some mates attempt to settle down to some light looting and are then intercepted by their captain with more orders. "You all look very nice, but get some fucking digging done."
Wise words. Here then is the south of the Battle of the Somme at nightfall. It's all going quite well. There's a couple of attempted German counter-attacks, but they're beaten off by artillery fire and a little rifle fire and there is plenty of time to get some fucking digging done. It's not quite as good as it might have been, but it's still a good day, and one that mostly runs completely counter to the common picture of the Somme. They've achieved just about everything that they were supposed to and are in a good position to push on tomorrow.
Mametz and Fricourt
Moving west. Fricourt is where Bernard Adams was stationed until he was wounded a month ago. This is a nasty little part of the line. Fricourt and Mametz themselves are now gigantic fortified super-pillboxes, with a series of inter-connected dugouts running beneath them. The German fire trenches have been carefully laid along the winding hillside to create a number of miniature salients, with barbed wire entanglements designed to direct attackers into the teeth of machine-gun nests.
Against this, we have the BEF's jumping-off trenches. They've dug some of the best prepatory trenches in the battle, and almost all of them are the recommended 100 yards away from the enemy. XV Corps has also taken a lot of care over its artillery preparation, conducting several useful experiments (of which more later) with different counter-battery techniques. While not approaching French levels of success, they've still put a serious dent in the Germans' artillery capabilities. They've also enthusiastically adopted the creeping barrage along strict French lines, and battalion commanders have been emphasising the need to stay close to it, however silly it sounds.
Things have got rather difficult here. The early going is decent enough; No Man's Land is short, and the creeping barrage somewhat effective in suppressing German machine-guns. The trouble begins once the first wave has crossed without trouble. They've done a poor job of "mopping up" the trenches, the act of completing the equation of attack by killing Germans in their dugouts. Now they're spewing grenades as they come up to make a fight of it. Men are slipping through the Tommies in their trenches and manning the machine guns to shoot down the second and third waves who are supposed to be pushing on while the first wave consolidates. Private Burke of the 20th Manchesters:
Then the hand-to-hand fighting started. It was Hell. Bombing was the star turn; many of the Devils were taken unawares and were asleep in their dugouts. We threw bombs of every description down, smoke bombs especially and as the hounds came up, crawling half dead, we stuck the blighters and put them out of time. In one dugout there were about twenty-five in there and we set the place on fire and we spared them no mercy, they don't deserve it.
They continued sniping as we were advancing until we reached them and then they throw up their hands, 'Merci, Kamerad!' We gave them mercy, I don't think! We took far too many prisoners, they numbered about 1,000 and they didn't deserve being spared.
And all this trouble is throwing the artillery timetable out of sync. The creeping barrage is creeping off into the distance, and now the men are too busy fighting to send messages back to the rear. The Germans bring up reserves, and are sometimes able to feed them directly into the trenches. By the end of the day the attackers have captured most of the First Line, and Mametz itself. However, they're well behind schedule; a few parties have attempted to push towards the Second Line, but have run into intermediate trenches and taken heavy casualties. Importantly, Fricourt itself and Fricourt Wood just behind are still in German hands. We'll finish with an observation from 2nd Lt Probert, an artillery officer who spent most of the day watching the battle from an observation post.
The 2nd Gordons were deepening the communication trench … but we had to stop here some time as the sniping was continuing. One captain was sitting in the front line eating his lunch with one hand and shooting the snipers with the other as they came out to surrender. I thought that rather rough as some had their hands up, but he said that he had had several wounded Jocks shot on their stretchers.
War crimes for war crimes. Yay. This is still not the popular picture of the Somme, though. This is still success. It's limited, hard-fought, bloody success; and they're not at the Day 1 objective, but they are going forwards. Let's stop and see the map again.
Pink is the Day 1 objectives
Ovillers and La Boisselle
Ovillers and La Boisselle guard the Albert to Bapaume road. They've been fortified and incorporated into the First Line. They're supplemented by two redoubts, the Schwaben Height (not to be confused with the Schwaben Redoubt a little further north) and Sausage Redoubt (why not just call it "Huns-R-Us", guys?) The one-size-fits-all solution to these problems appears to have been "a bloody big bang", with vast amounts of underground explosive being brought in. This is possibly the most critical sector of all.
General Haig's immediate objective is a breakthrough at Pozieres, a few miles up the road, and then cavalry exploitation and the capture of Bapaume. This has to be done before the Germans realise what's going off and dispatch reinforcements to strengthen and occupy their Third Line. The attackers are on a serious timetable here; the men need to at the very least replicate the success at Montauban, if not exceed it. And now, here is where the stories start to sound depressingly familiar. Corporal James Tansley of the 9th York and Lancasters:
We had been told, 'There's no need for this short rushes and getting down on your stomach, go straight over as if you were on parade. That's the orders, there's no fear of enemy attack, that's been silenced by the British guns'. Up we went through the lanes cut in the wire, spread out and tried to follow this instruction. Myself, I was a bit sceptical about it. I and my section made for this slight ridge marked by an old farm implement. Looked around for where the line was, they seemed to disappear. Lying about on the ground. There was a severe machine-gun fire coming from the region of Pozieres, half-left.
The machine guns cut down Tansley and a friend. The friend dies; Tansley lies in No Man's Land for about seven hours, with his finger stuffed deep into his wounded groin to stop the bleeding. Then he crawls back into the BEF trenches and almost falls on top of a stretcher bearer. Meanwhile, Private Harry Baumber, 10th Lincolnshires, advancing on La Boisselle:
Line behind line of steadfast men walking grimly forward and wondering what was in store. We soon found out. I noticed men falling thick and fast about me and all the time the tremulous chatter of machine guns. It was akin to striding into a hailstorm and the further you went the less and less became your comrades. Jerry had not been obliterated, his wire had not been destroyed and we had been called upon to walk 800 yards across No Man's Land into Hell. A far cry from the walkover we had been promised.
Baumber is exaggerating about the distance he had to walk, but he arrives at the German wire to find it almost completely uncut; or possibly cut and then repaired. At any rate, he and his mates soon realise that they will have to spend 12 hours or more lying in No Man's Land until darkness, when they might be able to sneak back. Many more are spotted and shot or grenaded as they try to keep a low profile. 2nd Lt John Turnbull is trying to exert some command, but there's a problem. Either he's got lost, or the exact location of the Lochnagar Mine was told wrong to him.
Very puzzled with the rotten crater, which was in the wrong place. Used it to screen us from La Boisselle, got as far as the ridge, and goodness knows how many machine guns opened up on us. We all dropped, and I started to crawl to the crater to see who was there, when I got hit in the back. Corporal Turton helped me in. Unfortunately I couldn't move about much, and felt very dazed.
JRR Tolkien has a friend in all this mayhem. Lieutenant Rob Gilson is attacking the Sausage Redoubt, and he dies within minutes of going over the top, hit by a shell almost immediately after taking emergency command of his company. Of the 16 officers in Gilson's battalion, only one returns unhurt. Two thirds of the men are dead or wounded. G.B. Smith is luckier, over on the far left, and he escapes unhurt for now, with his battalion only having taken one-third casualties.
At nightfall, we have a picture of almost complete failure. Artillery fire was, apparently, inadequate. Wire was not cut. Machine-guns were not suppressed. The enemy had plenty of warning of the attack. There are a few men clinging to a few shreds of trench, and the new Lochnagar Crater is British. For whatever that's worth. Which isn't much at all. The line of the Albert to Bapaume Road is where the Somme that everyone knows about starts to happen.
And on we go further north.
This sector is dominated by Thiepval on top of its ridge, the Leipzig Ridge just to the south, and Schwaben Redoubt just to the north. Here we have a clear case of Staff Officer Optimism. The objectives are simple, in theory; capture the First Line in the morning and attack the Second Line in the afternoon. By the timetable it's just about possible, as long as no German in the sector offers more than token resistance, but it's going to be an enormous ask. It starts well enough; the German fire trenches are protected by up to sixteen rows of barbed wire, but a little luck and a little skill with the guns has cleared plenty of it away.
Then something odd happens. Attacking the Schwaben Redoubt is an Irish division. Benefiting from an extremely successful smoke deployment, they haul themselves and their 60 pounds of kit across No Man's Land in double-quick time, and through the trenches before the Germans know what's hit them. Messages for defensive artillery support go astray, and well ahead of schedule, the Irishmen are pushing the Germans out of the redoubt. An inspirational battalion commander pushes the attack home, and then, mostly on his own initiative, orders another advance up towards the top of Pozieres Ridge and the German Second Line.
Quite how they didn't all die in the attempt, I'm not quite sure. It soon becomes obvious to the few men who made it all the way that they are utterly isolated, and need to fall back into the Redoubt. Into the afternoon and evening, there are lots of awkward looks from the Redoubt across at Thiepval. The two positions are deeply interlinked. One position can be reinforced from the other, or attacked from the other. Thiepval also holds a dominating view of the route BEF men must take to come up and reinforce the newly-won trenches.
Thiepval is now one gigantic machine-gun nest. Even with the barbed wire mostly cleared, the battalions thrown directly at Thiepval at zero hour have all but ceased to exist. To their right, the Scottish battalions attacking towards Leipzig Redoubt managed to get across No Man's Land (including the 17th Highland Light Infantry, who also crossed while kicking footballs). Then their troubles begin. Some battalions ran into trouble, as we've seen, because they broke into a trench, but didn't spend long enough "mopping up" the Germans still alive down in the dugouts, and then started getting shot in the back.
These men are now having exactly the opposite problem. They've done a good job of work mopping up, but it took so long that the Germans have regained local command and control. Machine-guns from Thiepval, Leipzig Redoubt, and Wundtwerk Redoubt to the rear are now firing on them from three sides as they attempt to push towards the Leipzig Redoubt, and Thiepval and Leipzig are also raking No Man's Land at will. Private Bentley Meadows, 17th HLI:
The machine gun swept us down outside the Leipzig Redoubt. It became evident that we, who were working up between two communications trenches, after two or three rushes, that further advancing was impossible without support. We waited for our own reserve waves and the Lonsdales who should have come on behind. But no reserves reached us and we saw our only hope lay in the fact that they had rushed one of the communication trenches and might manage to bomb out the machine gun. But the bombers were checked out of the range of the gun.
I at length found myself the only living occupant of that corner. About twelve o'clock I managed to leap the parapet without being hit. I found my platoon officer, Lieutenant MacBrayne, lying shot through the head. Of the others of my platoon I could get no news, except those I saw lying dead or wounded
Another officer appears and leads a last, futile charge that ends in the order "Every man for himself". They cling on to a few German trenches until a counter-attack at 5pm encourages them to leave. One of the men in this bit of fighting is Private Eversmann, who gave us a German perspective during the barrage. His body was never found; his diary was taken from the trenches by a Scotsman and sent back to Blighty.
The position at the end of the day; the Irish have been evicted from Schwaben Redoubt but are holding a few German front-line trenches; the Scots are still clinging on by their fingernails to half a trench on the edge of Leipzig Spur. Nobody's anywhere near their objectives. I would describe it as complete and total failure, but there is still one more northward jump for us to make. Let's have the small map again, though.
All Day 1 objectives that have not been reached will, going forward, be marked in pink.
Serre and Beaumont Hamel
And so we arrive at Serre and Beaumont Hamel. This is where, almost unbelievably, it has been decided that the supporting mines will not be blown at 7:30am, precisely at zero hour, but at 7:20am, ten minutes before the infantry are due to go over the top. And, on top of that, some idiot has told the artillery that the meen will also be going over at 7:20. There will be ten minutes of unmolested silence for the Germans to leave their dugouts and prepare to be attacked, terrible rolling barrage notwithstanding. Private Pearson, 15th West Yorkshires, who is attacking Serre:
Every man climbed out of the trenches at the whistle of the officers and not a man hesitated. But I was lucky. I was in a part of the trench where the parapet had been battered down as Jerry sought for a trench mortar. When I ran up the rise out of the trench I was under the hail of bullets which were whizzing over my head. Most of our fellows were killed kneeling on the parapet. There was nobody coming forward, only one man, the reserves had been shelled in our lines and blown to smithereens.
I noticed higher up the trench one of our chaps laid there with a baulk of timber across his leg, one leg had been cut off—severed. This baulk of timber had cut across his leg and acted as a tourniquet and stopped the bleeding.
Private Glenn of the 12th York & Lancasters was supposed to be in one of the supporting waves.
The first line all lay down and I thought they'd had different orders because we'd all been told to walk. It appears they lay down because they'd been shot and either killed or wounded. They were just mown down like corn. Our line simply went forward and the same thing happened. You were just trying to find your way in amongst the shell holes. You can imagine walking through shell-pitted ground with holes all over the place, trying to walk like that. You couldn't even see where you were walking!
When you got to the line you saw that a lot of the first line were stuck on the wire, trying to get through. We didn't get to the German wire, I didn't get as far as our wire. Nobody did, except just a few odd ones who got through and got as far as the German wire. The machine-gun fire was all trained on our wire. Only a few crept along. I lay down. We weren't getting any orders at all; there was nobody to give any orders, because the officers were shot down.
It is possible that despite this utter ridiculousness, about 200 men may just have broken all the way through, taken a few lucky turns, and then blundered into Serre village itself. It would explain the messages that General Rees, he of the lack of optimism from a few days earlier, was receiving.
An aeroplane reported that my men were in Serre. The corps and the division urged me to support the attack with all the force at my disposal. I was quite sure that we had not got anyone in Serre except a few prisoners, but the 93rd Brigade on my right reported that their left had got on, whilst the 4th Division beyond them again claimed the first four lines of German trenches and were said to be bombing down our way. It was obviously necessary to attempt to get a footing in the German first trenches to assist these two attacks.
The hostile barrage had eased off by now and was no longer formidable, so I ordered two companies of the 13th York and Lancs to make the attempt. I did not know that the German barrage was an observed barrage, but as soon as this fresh attack was launched down came the barrage again.
There are no accounts of what might have happened in Serre. If anyone ever did got in, they just disappeared into the war. At Beaumont Hamel, German observation has prevented anyone digging trenches closer than about 500 yards. There is a general shortage of stories of Beaumont Hamel and the attack on Hawthorn Redoubt, for obvious reasons. The ones we do have are by this point familiar; confusion, barbed wire, shells, machine-guns, hiding in No Man's Land for hours until darkness. Men going over the top long after it should have been obvious that the attacks should be called off. And death, death, death, death everywhere.
And this is where our correspondent 2nd Lt Malcolm White is, right between the two strong-points, leading the 1st Rifle Brigade over the top like a good junior officer should. The good news is that we have an account, of sorts, of what happened next; although, as you'd expect, it is rather confused and lacking in crunchy detail.
The Battalion went into action in front of Mailly-Maillet. White was hit [presumably by bullet or shrapnel] in advance of his men. His servant, who had followed him in the attack, reached him, and asked if he was badly wounded. He said, "I'm all right; go on." At that moment a shell burst near them. His servant remembers nothing more till the time when he was in hospital.
The bad news is that it is not from Malcolm White. When the 1st RB drags itself back to the rear and takes a roll call, he will be missing, and reported to his family as such. It can take a month or more for prisoners' identities to be sent to Britain via the Red Cross, but. He will remain officially missing for about ten months. Then, after the war has finally moved away from Beaumont Hamel and Serre, there will be a pioneer battalion clearing out some disused trenches. The shell took White's life; however, it did not take his identity disc.
So he has a grave of his own, and he does not appear among the 72,246 names on the British Memorial to the Missing, whose bodies have never been found or identified. The Memorial to the Missing, by the way, is at Thiepval. I hope now it is clear why they put it there.
The General's view
Generally speaking, ahaha, the easier it was to advance, the easier it was to get information back to the rear (for a given value of "easy"; reports are still taking hours to reach anyone who can pass them back via field-telephone). The moment we get north of Thiepval, we find British command and control once more breaking down almost entirely. General Reed gave us a taste of the picture being painted to the rear. Sometimes there are spurious reports of men in places they never reached. More often, observers are unable to tell the difference (at several thousand feet's remove) between men who have easily captured a trench, and men who are having a bloody difficult time of it.
And then there are the runners. Signaller Dudley Meneaud-Lissenburg has one more story for us, as he sits in his OP in the old front line in front of the Hawthorn Crater (nee Redoubt). A knot of men has reached a sunken road unhurt. They send a man back to report that it's completely impossible.
I watched a lone figure, a runner no doubt, coming back towards our lines, dropping every now and then into shell holes for cover. On reaching our barbed wire he was about to jump into the trench when a shell burst at his feet and blew him sky high.
The picture filtering back to General Haig is extremely murky and full of half-truths and exaggerations and the odd piece of completely unrealistic bullshit. He's got some decisions to make, though; there's been success in the south and he does know about it. But I've talked far too much about this one day; time to see if it can be encouraged towards a conclusion. We'll go into the BEF's strategic thinking tomorrow.
There is chaos. Let us try to bring order to it. Here now is the full map at nightfall.
Even abstracting things this much makes me just go "ugh".
Never mind the north. The north we know about. Look at the south! The south has moved! At its furthest point, this is the biggest advance for the BEF since the Battle of Loos. You'd need a tape measure to tell whether they got further than the French did in some parts of Second Champagne, or in their attack today. And this is why it is wrong and inaccurate and incomplete to speak of the Somme simply in terms of Thiepval, Ovilliers, and Serre. This is even why it is wrong just to say that the BEF has lost near as dammit 20,000 dead and 37,500 more wounded as though it speaks for itself. (The French lost about 1,500 killed and wounded, attacking with much more caution.)
Well, okay. It stands if you just want to step back and think about the July Crisis, and the insane series of diplomatic pratfalls that made this stupid war happen in the first place, and go "it's all bonkers". It stands if you want to think of a particularly bloody single day of war. It stands as a profoundly traumatic moment for Britain; the casualty figures exceed those sustained in the Crimean, Boer, and Korean Wars put together. More than half the men who attacked and three-quarters of all the officers are casualties.
And, of course, this was the day that the BEF used a large number of its Pals battalions all at once. Men recruited from the same village, same street, same workplace. They've gone over the top together and died together. In a few hours of fighting, the heart has been ripped out of entire communities at once, and they don't even know it yet. Working-class communities sent to their deaths by, and in the company of, the upper classes. It's huge and heartbreaking and overwhelming and yet there is more to the story than just that. However callous it may seem, there is still a war tomorrow. We can't just write it off as senseless butchery.
The BEF has done something positive, for given values of "something" and "positive". And it won't be until tomorrow, as the battle begins to take shape, that we can put our fingers on why. Time to put this day in the books and move the hell on. There's a chance here. A small chance, and one that'll be very hard to take advantage of. But there is a chance.
I think that's all for today.
There is one more perspective that we now must take advantage of, a question far too often overlooked. What about the Germans? What do they think has just happened to them, in front of them, around them? Due to the way German casualties were reported it's impossible to know how many there were on the first day. Prisoners are also not included in that total, and there's been plenty of German prisoners taken, despite the bloodthirsty stories above. Reasonable estimates are in the region of 10,000 to 15,000.
I think the best description of German higher command in the evening is "equally confused, but in a different way to their opponents". It's hard to say, but from the reaction it's very probable that General von Falkenhayn saw this as 1915 all over again; a French attack with a British supporting component, and it's been launched far to the south of where it should have been. Hold that thought a few days for me. Why are the French attacking when they're taking a kicking at Verdun? It makes no sense. Perhaps this is just a limited offensive and the BEF is going to do the real attack somewhere else tomorrow, or the day after.
It must have been an extremely odd picture to make sense of. In the north of what appears to be the battle front which appears to be the wrong place, you've shot thousands and thousands of enemy troops down in No Man's Land. In the south, General von Below has been worried enough to commit his three reserve divisions down there, but the picture of where exactly the enemy attack is and what exactly it's achieved is highly unclear. It's like trying to repair a burst water main while wearing a blindfold.
Once you get to division level or lower, second and third-hand reports of what was going on seem all very panicky. No orders from on high, and they've been blown right out of the First Line. Of course, the Germans aren't paralysed; officers are encouraged to use their initiative, but that's not always a good thing. Some have used their initiative to order immediate counter-attacks. Some have used theirs to order a fall-back to the Second Line. This is not going to last for long. However, for the time being, part of von Below's army is pulling on the same rope from opposite directions.
And that is all.
Henri Desagneaux
Henri Desagneaux is very relieved, thank you very much, to be out of the Battle of Verdun.
After being relieved, we are quartered in the same camp where we stopped on the way here. We arrive at 2pm, exhausted. We fall into bed and sleep like brutes.
There is always, always, always something else going on. Fleury is changing hands at Verdun once more. The Sharif of Mecca has just met up with a British artillery battery which promises to turn the tide of the Battle of Mecca. The war continues.
Actions in Progress
Battle of Verdun
Brusilov Offensive
Battle of Mecca
Siege of Medina
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This entry was posted in Western Front and tagged Arthur Agius, Battle of the Somme, Beaumont Hamel, Fricourt, Gommecourt, Haig, Henri Desagneaux, Malcolm White, Mametz, Rees, Schwaben Redoubt, Serre, Thiepval, von Below, von Falkenhayn by makersley. Bookmark the permalink.
5 thoughts on "First day on the Somme | 1 July 1916"
james on 3rd July 2016 at 12:12 am said:
Goodbye Malcolm White. I will have "a very wonderful sort of breakfast" to remember him after staying up all night.
Matt on 6th July 2016 at 3:35 pm said:
Love the blog – is it possible to put links to the first of the year 1915 and 1916 for those of us who have bought and read the books and now need to jump back into the blog?
makersley on 6th July 2016 at 11:45 pm said:
It seems like it should be. I'll twiddle some knobs backstage and see what happens.
Peter Agius on 6th July 2016 at 11:37 pm said:
Capt Arthur Agius mentioned at the Centenary Memorial Service at Thiepval 1st July 2016!
Peter Agius on 11th July 2016 at 12:02 am said:
Find out more about Capt Arthur Agius (mentioned in the bit about Gommecourt above and elsewhere) in a blog of his daily letters home. http://agiusww1.com/
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