\subsection{Simulations and projections}
\label{subsec:physics.gpd.simu_proj}

In order to calculate projected statistical accuracies for the proposed 
measurements, the following parameters and assumptions were used:
\bi
\item  a polarised muon beam with an energy of 160~GeV,
\item  a 48~s SPS period with a 9.6~s spill duration (flat top), 
\item  a $\mu^+$ beam intensity of $4.6 \times 10^8$ muons per spill,
\item  a three times lower intensity for the $\mu^-$ beam,
\item  a new liquid hydrogen target of 2.5~m length
       (Sect.~\ref{subsec:upgrades.LH2_RPD.LH2}), yielding a luminosity of 
       about $10^{32}~\mathrm{cm}^{-2}\mathrm{s}^{-1}$ for the $\mu^+$ beam,
\item  a new Recoil Proton Detector (RPD) surrounding the target 
       (Sect.~\ref{subsec:upgrades.LH2_RPD.RPD}), 
\item  an as continuous as possible polar and azimuthal photon detection 
       coverage provided by the two existing electromagnetic calorimeters 
       ECAL1 and ECAL2 (Sect.~\ref{sec:upgrades.ECALs}),
\item  a new large-angle calorimeter ECAL0 
       (Sect.~\ref{subsec:upgrades.ECALs.ECAL0}),
\item  an overall ``global'' efficiency of $\epsilon_{global}=0.1$
       including detection, tracking and reconstruction of events comprising 
       an incoming and scattered muon, a high-energy photon and a 
       recoiling proton within the RPD acceptance as well as beam and 
       spectrometer availabilities 
       (Sect.~\ref{subsec:physics.gpd.validation}),
\item  a running time of 280~days (70~days with $\mu^+$ and 210~days with
       $\mu^-$, in order to have as close as possible integrated luminosities 
       for both data sets).
\ei
%
The acceptance for exclusive single-photon production is determined using
the standard \compass\ program for reconstructing the simulated DVCS and BH 
events. 
%
Hard exclusive single-photon production, \ie\ the interfering Bethe--Heitler 
and DVCS processes, is simulated employing two separate codes using two 
generators for the DVCS amplitude: one is based on the VGG 
model~\cite{Vanderhaeghen:1999xj} and the other one on the 
Frankfurt--Freund--Strikman (FFS) model, which was modified for the proposed 
experiment~\cite{A-Sandacz}. 

The conditions listed above will allow for an accumulation of at least 300
DVCS events for those bins, in which the number of DVCS events reaches at least
10\% of the number of BH events and for which $Q^2 \le 8~\GeV^2$, as shown in
Table~\ref{tab:physics.gpd.simu_proj_nbevts}.
Such statistics are sufficient to perform a 2-dimensional analysis of the
$t$-slope measurement with 6~bins in $t$ and 5~bins in $\phi$ 
(Sect.~\ref{subsubsec:physics.gpd.simu_proj.t-slope}). It is 
worth noting that an increase in the number of muons per spill by a factor of
4 would extend the range in $Q^2$ up to about 16~GeV$^2$.


%%%%%%%%%[Tables for BH, DVCS and acceptance for VGG]%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[tbp]
\caption{Number of BH/DVCS events in $6 \times 4$~bins in $x_B$ and $Q^2$ for 
$0.05 \leq y =E_{\gamma^\ast}/E_{\mu} \leq 0.9$ projected for 280~days 
(70~days with $\mu^+$ and 210~days with $\mu^-$) with a global efficiency of 
0.1 and using ECAL0, ECAL1 and ECAL2. The numbers of DVCS events are obtained 
using the VGG model. The modified FFS model yields numbers of DVCS events 
smaller by a factor 0.7. The geometrical acceptance for single-photon detection
is indicated for each bin between parentheses. Bins containing at least 
300~DVCS events, which also have to represent not less than 10\% of the number 
of BH events, are marked by an asterisk.}
\label{tab:physics.gpd.simu_proj_nbevts}
\begin{center}
\scalebox{0.90}{
   \begin{tabular}{|c||c|c|c|c|c|c|} \hline
   \raisebox{-1mm}{$Q^2/$}
   &  $0.005 \leq x_B $\strut & $0.01 \leq x_B $ & $0.02 \leq x_B $ 
   & $0.03  \leq x_B $  & $0.07 \leq x_B $ & $0.13 \leq x_B $\\      
   $\GeV^2$
   & $ \leq 0.01$ & $ \leq 0.02$ & $ \leq 0.03$ 
   & $ \leq 0.07$ & $ \leq 0.13$ & $ \leq 0.27$\\      
   \hline
   \hline
   $8 \leq Q^2 $ & - & - & - & 3280 / 51  & 1032 / 84 & 190 / 120 \\
   $ \leq 16 $ &   &   &   &   (0.88)   &   (0.91) &   (0.93)  \\
   \hline
   $4 \leq Q^2 $ & - & 2512 / 18 & 4176 / 78 
         &  2371 / 309$^*$  &  269 / 347$^*$ &  66 / 501$^*$ \\
   $ \leq 8 $   &   &   (0.86)  &   (0.87)  &   (0.90)   &   (0.94)  &   (0.94)  \\
   \hline
   $2 \leq Q^2 $ & 6577 / 36 & 15258 / 368 &  1848 / 374$^*$    
        &   1050 / 1257$^*$ & 153 / 1466$^*$ &  29 / 933$^*$ \\
   $\leq 4 $  &  (0.86)   &   (0.86)  &   (0.90)  &   (0.93)   &   (0.94)  &   (0.97)  \\			  
   \hline
   $1 \leq Q^2 $ & 44749 / 627 &  7684 / 1338$^*$ &  1187 / 1270$^*$ 
         &  771 / 4420$^*$ &  68 / 2000$^*$ & - \\
   $\leq 2 $   &  (0.86)   &   (0.86)  &   (0.90)  &   (0.93)   &   (0.94)  &  -  \\			  
   \hline
   \end{tabular}
}
\end{center}
\end{table}

The acceptance for single-photon detection is seen to be always larger than 
86\%. It has to be noted that without ECAL0 the acceptance is smaller by a 
factor of two in the bin $0.07 \leq x_B \leq 0.13$ and by a factor of seven 
in the bin $0.13 \leq x_B \leq 0.27$.

The choice of 160~GeV for the muon beam energy bears the advantage that there 
will be a sufficiently large number of ($x_B,Q^2$) bins where the DVCS 
contribution dominates over the BH one. The price to pay is the relatively 
low intensity of the $\mu^-$ beam at this energy.

At a later date, measurements are envisaged using a transversely polarised 
NH$_3$ target (Sect.~\ref{subsubsec:physics.gpd.simu_proj.TTSA}) which are 
planned to be described in more detail in an addendum to this proposal.

%================== ===========================================================
\subsubsection{The $t$-slope of the DVCS cross section}
\label{subsubsec:physics.gpd.simu_proj.t-slope}
%
For the parametrisation of the $x_B$ dependence of the $t$-slope parameter
$B(x_B)$ introduced in Sect.~\ref{subsec:physics.gpd.intro}, the simple 
ansatz $B(x_B) = B_0 + 2 \, \alpha' \, \log(\frac{x_0}{x_B})$ is sufficient 
when disregarding the case $x_B \rightarrow 1$ that is of no interest here. 
In this parametrisation, the $x_B$-slope $\alpha'$ is a measure for the 
decrease in nucleon size with increasing $x_B$.
% Note that $x$ and $x_B$ coincide here.
%
%In dipole models,
%~\cite{xxx}, 
%$B(x_B)$ is related to the transverse 
%distribution of partons carrying the longitudinal momentum fraction 
%$x$, $B(x) \sim  1/2 \, \langle {\rm r} _\perp^2 \rangle (x)$. 
%This relation does 
%not apply to non-factorised models if the amplitude has a non-vanishing real 
%part.

The \compass\ $x_B$ region is uncharted territory also for the $t$-slope
parameter $B(x_B)$. Data on $B(x_B)$ exist only for smaller $x_B$ values, 
namely for the \hera\ collider range 
$10^{-4} < x_B <10^{-2}$~\cite{Aktas:2005ty,:2007cz,Chekanov:2008vy}. Above the
\compass\ $x_B$ range, in the valence region, no experimental determinations 
of $B(x_B)$ exist. The only information comes from fits adjusted to form factor
data which give $\alpha' \simeq 1~\GeV^{-2}$~\cite{Diehl:2004cx,Guidal:2004nd}.
In this situation, the values $B_0=5.83~\GeV^{-2}$, 
$\alpha' = 0.125~\GeV^{-2}$ and $x_0=0.0012$ are chosen for the simulation 
of a $t$-slope measurement inspired by the \hera\ data.

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=160mm,clip=true]{gpd_simul_t_slope_280days_3prct_global_bis}
\end{center}
\caption{Projections for measuring the $x_B$ dependence of the $t$-slope 
parameter $B(x_B)$ of the DVCS cross section, calculated for $1<Q^2<8~\GeV^2$.
For comparison some \hera\ results with similar $\langle Q^2 \rangle$ are 
shown~\cite{Aktas:2005ty,:2007cz,Chekanov:2008vy}, for which the
horizontal dashed lines indicate their $x_B$ range. The left vertical bar on 
each data point indicates the statistical error only while the right one 
includes also the quadratically added systematic uncertainty,
using only ECAL1 and ECAL2 (first row) and also ECAL0 (second row). 
Two different parametrisations are shown using $\alpha'=0.125$~GeV$^{-2}$ and 
0.26~GeV$^{-2}$.
}
\label{fig:physics.gpd.tslope}
\end{figure}
In Figure~\ref{fig:physics.gpd.tslope} are shown the projected statistical and
systematic uncertainties for a measurement of the $x_B$ dependence of the 
$t$-slope parameter $B(x_B)$ for the  range $1 \leq Q^2 \leq 8~\GeV^2$. 
As $B(x_B)$ is obtained from the $\phi$-integrated beam 
charge~\&~spin sum (Eq.~(\ref{eq:ssc})) after BH subtraction, it constitutes 
a direct measurement of the DVCS cross section. The statistical error
in a ($x_B,Q^2$) bin is given by $\sqrt{N_{BH} + N_{DVCS}} / N_{DVCS}$ where
$N_{BH}$ and $N_{DVCS}$ are the numbers of BH and DVCS events collected in this
bin (Table~\ref{tab:physics.gpd.simu_proj_nbevts}). For this purpose, only 
those bins indicated by an asterisk in 
Table~\ref{tab:physics.gpd.simu_proj_nbevts} 
are used for which $N_{DVCS} \geq 0.1 \times N_{BH}$ is satisfied. 

The dominant systematic uncertainty of the resulting DVCS contribution 
arises from the subtraction of the BH contribution. The BH contribution 
dominates the one of DVCS over a wide range in $x_B$ and $Q^2$. In this 
region, the BH yield is obtained from the calculated BH cross section, 
taking into account radiative corrections, luminosity and detection 
efficiencies. Up to now, we assume that this BH yield will be known within 
3$\%$. It turns out that in such a case the resulting systematic uncertainty, 
which is $ 0.03 \times N_{BH}  / N_{DVCS}$, is of relevance only for the first 
selected point in $x_B$ for each $Q^2$ domain
where the BH contribution is more than twice as large as the DVCS contribution.
When extrapolating the BH yield into the kinematic region where it has to be
used for subtraction, an additional uncertainty can originate from 
possible kinematic-dependent detector efficiencies.

As the DVCS cross section dominates the one of the BH process at larger values 
of $x_B$ (Fig.~\ref{fig:dvcs_bh_int_mc_2009}), the accuracy of the 
$t$-slope measurement profits considerably from the additional acceptance of a 
possible new large-angle calorimeter ECAL0. This becomes evident when comparing
the two sets of points in Fig.~\ref{fig:physics.gpd.tslope} which describe the 
projected total uncertainties without and with ECAL0.

Altogether, the measurement of the $\phi$-integrated beam charge~\&~spin sum 
in DVCS will lead to a model-independent determination of the 
$t$-slope parameter $\langle B(x_B) \rangle$ with a total accuracy of better
than 0.1~GeV$^{-2}$, when averaged over the \compass\ $x_B$ range. The 
measurement of its $x_B$ dependence allows a direct determination of the $x_B$ 
slope $\alpha'$ with a total accuracy better than 2.5 sigma in the total 
projected uncertainty, should $\alpha'$ possess a value above 0.26~GeV$^{-2}$
and only the existing calorimeters ECAL2 and ECAL1 be used. With a new ECAL0 
detector (Sect.~\ref{subsec:upgrades.ECALs.ECAL0}), a 2.5-sigma 
determination of $\alpha'$ is already possible when its value is above 
0.125~GeV$^{-2}$. A visualisation of 
these two scenarios is also given in  Fig.~\ref{fig:physics.gpd.tslope}. 
In any case, new and significant information will be obtained in this
uncharted $x_B$ region, further elucidating the issue of ``nucleon tomography'' 
as it was described in Sect.~\ref{subsec:physics.gpd.intro}.


\begin{figure}[tbp]
\begin{center}
\includegraphics[width=160mm,clip=true]
                             {gpd_simul_BCSA}	
\end{center}
\caption{Projected statistical accuracy for a measurement of the $\phi$
dependence of the beam charge~\&~spin asymmetry.
%in 280 days at a beam energy of 160~GeV. 
As an example, the 2-dimensional bin $0.03 \leq x_B \leq 0.07$ and 
$1 \leq Q^2 \leq 4~\GeV^2$ is shown. Predictions are calculated using the 
VGG model~\cite{Vanderhaeghen:1999xj}. The black (solid) and blue (dash-dotted)
curves correspond to 
two different variants of the VGG model (see text). The green curves show 
predictions based on the first fit of world data~\cite{Kumericki:2009uq}
including \jlab\ {\sc Hall A} (dashed line) or not (dotted line).
}
\label{fig:gpd_simul_bsca}
\end{figure}

%============================================================================
\subsubsection{Beam charge~\&~spin asymmetry and difference}
\label{subsubsec:physics.gpd.simu_proj.BCSA_D}
%
In Figures~\ref{fig:gpd_simul_bsca} and \ref{fig:gpd_simul_bscd}, the 
projected statistical accuracies are shown for a measurement of the azimuthal 
dependence of the beam charge~\&~spin asymmetry ${\cal{A}}_{CS,U}$ 
(Eq.~(\ref{eq:acs})) and difference ${\cal{D}}_{CS,U}$ (Eq.~(\ref{eq:dcs})) 
in a particular ($x_B, Q^2$) bin, calculated using the VGG GPD 
model \cite{Vanderhaeghen:1999xj} that is meant to be applicable mostly in the 
valence region. Two choices exist for the $(x,t)$ dependence of GPDs, either 
factorised or 'reggeised', the latter corresponding to $\alpha' \approx 
0.8~\GeV^{-2}$.
Recent evidence from phenomenology (see, \eg\ Ref.~\cite{Diehl:2004cx}) and 
experiment (see, \eg\ Ref.~\cite{:2008jga}) indicates that the factorised 
ansatz is disfavoured. This favours larger values of the beam charge~\&~spin 
asymmetry (and difference), as can be seen by comparing the two VGG curves in 
the figures.

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=160mm,clip=true]
                             {gpd_simul_BCSD}	
\end{center}
\caption{Projected statistical accuracy for a measurement of the $\phi$
dependence of the beam charge~\&~spin difference.
% in 280 days at a beam energy of 160~GeV.
For further explanations see caption of Fig.~\ref{fig:gpd_simul_bsca}.
%As an example, the 2-dimensional bin $0.03 \leq x_B \leq 0.07$ and 
%$1 \leq Q^2 \leq 4~\GeV^2$ is shown. Predictions are calculated using the VGG 
%model~\cite{Vanderhaeghen:1999xj}. The black (solid) and blue (dash-dotted)
%curves correspond to 
%two different variants of the VGG model (see text). The green curves show 
%predictions based on the first fit on world data~\cite{Kumericki:2009uq}
%including \jlab\ {\sc Hall A} (dashed line) or not (dotted line).
}
\label{fig:gpd_simul_bscd}
\end{figure}
 
% Original place of Fig.~\ref{fig:gpd_simul_bsca_c1Int} is here!

The statistics expected from the proposed two years of running with an
unpolarised LH target will permit us to study a 2-dimensional dependence with,
\eg\ 6~bins in $x_B$ combined with 6~bins in $t$ or with 4~bins in $Q^2$. This 
choice is driven by the requirement to have enough statistics in each of the 
about 10~$\phi$ bins for a stable fit of the azimuthal dependence. 
In Figure~\ref{fig:gpd_simul_bsca_c1Int} is shown the projected statistical 
accuracy for a measurement of the 2-dimensional $(x_B,t)$ dependence of the 
$\cos \phi$ azimuthal modulation of the beam charge~\&~spin asymmetry, leaning 
towards the analysis and interpretation techniques developed for azimuthal 
asymmetries at \hermes\ and described, \eg\ in Ref.~\cite{:2008jga}. The
asymmetry amplitude $A^{\cos \phi}_{CS,U}$ is related through the coefficient 
$c_1^I$ to the real part of the CFF $\cal H$ (Eq.~(\ref{eq:ac1I})). Such 
data will provide strong constraints for reliable future GPD models that must 
be able to simultaneously and consistently describe the full range in $x_B$.
Presently, no such model exists. 

A recent theoretical development exploits dispersion relations for Compton form
factors. In this context, a fitting procedure including next-to-next-to-leading
order (NNLO) corrections was developed and successfully applied to describe 
DVCS observables at very small values of $x_B$, which are typical for the 
\hera\ collider~\cite{Kumericki:2007sa}. This technique was very recently 
extended~\cite{Kumericki:2009uq} to include DVCS data from \hermes\ and \jlab\
(\clas\ and {\sc Hall A}), the fit yielding the $x_B$-dependence of 
$x H(x,x,t)$ which was then used to predict the $\phi$-dependence of 
${\cal{A}}_{CS,U}$ shown as additional curves in 
Figs.~\ref{fig:gpd_simul_bscd}, \ref{fig:gpd_simul_bsca} and
\ref{fig:gpd_simul_bsca_c1Int} (with and without the impact of the \jlab\ 
{\sc Hall A} data). The asymmetry amplitude $A^{\cos \phi}_{CS,U}$ is related 
via the coefficient $c_1^I$ to the real part of the CFF $\cal H$ 
(Eq.~(\ref{eq:ac1I})). This real part was found positive at \hera\ and 
negative at \hermes\ and \jlab. The kinematic domain of \compass, in particular
the region $0.005 < x_B < 0.03$ (see the 3 first panels of 
Fig.~\ref{fig:gpd_simul_bsca_c1Int}), is expected to allow the determination 
of the $x_B$ position of the node of this function, which is important input 
for the fitting procedure.  


The quantification of the systematic errors for the beam charge \& spin 
difference (Eq.~(\ref{eq:dcs})) is a major issue. This difference of 
cross sections is calculated as

\[{\cal{D}}_{CS,U} = \frac{N^+}{F^+\epsilon^+} - \frac{N^-}{F^-\epsilon^-},\]

\noindent
where $N^+$ and $N^-$ are the number of events recorded using $\mu^+$ and 
$\mu^-$ beams, $F^{+}$ and $F^{-}$ are the corresponding integrated 
luminosities, and $\epsilon^{+}$ and $\epsilon^-$ are the factors taking into 
account acceptance and efficiencies for the two beam charges. The $\mu^+$ and 
$\mu^-$ measurements will be done successively by changing as often as possible
from one setting to the other. The quantities $F^+$ and $F^-$ will have to be 
kept as close as possible to one another ($F^+\sim F^-$), which means that 
the duration of a $\mu^-$ measurement will be about three times the duration
of a $\mu^+$ measurement in order to compensate for the different maximum
beam intensities available.


\begin{figure}[tbp]
\begin{center}
\includegraphics[width=160mm,clip=true]
                                    {gpd_simul_BCSA_C1int}
\end{center}
\caption{Projected statistical accuracy for the amplitude of the $\cos \phi$ 
modulation of the beam charge~\&~spin asymmetry
% from a measurement in 280 days at a beam energy of 160~GeV and 
using ECAL2+ECAL1+ECAL0. Projections calculated with the reggeised variant of 
the VGG model are shown for the $t$ 
dependence in each of the six $x_B$ bins. The blue triangles at large $x_B$ 
show recent \hermes\ results~\cite{:2009rj}. The curves show the latest 
predictions based on the first fit on world data~\cite{Kumericki:2009uq} 
including \jlab\ {\sc Hall A} (solid line) or not (dotted line).
}
\label{fig:gpd_simul_bsca_c1Int}
\end{figure}

When calculating the systematic errors on ${\cal{D}}_{CS,U}$ one can 
distinguish between the normalisation factors which are independent of the beam
charge and those which depend on it, the latter having more impact on the 
systematics than the former. For this purpose we consider the factor 
$a^{+(-)} = F^{+(-)}\epsilon^{+(-)}$ and decompose the systematic errors on 
$a^+$ and $a^-$ into i) an error common to $a^+$ and $a^-$, denoted as 
charge-independent error $\Delta a_{ci}$, and ii) an error which affects only 
the difference, denoted as charge-dependent error $\Delta a_{cd}$

\[ (\Delta {\cal{D}}_{syst})^2 = (\frac{ \Delta a_{ci}}{a} {\cal{D}})^2 
+ (\frac{ \Delta a_{cd}}{a} {\cal{S}})^2. \]
  
\noindent
For simplicity, in this section the notation $\cal{D}$ is used for 
${\cal{D}}_{CS,U}$ and $\cal{S}$ for ${\cal{S}}_{CS,U}$.
%As expected from Eq.~\ref{eq:scs} and Eq.~\ref{eq:dcs} the sum $\cal{S}$ 
%is dominated by the BH whereas $\cal{D}$ contains only the pure DVCS 
%(which is zero at leading twist) and the BH-DVCS interference. The 
%dominance of the pure BH process over these two quantities is illustrated in 
The relative contributions of $|{\rm BH}|^2$, $|{\rm DVCS}|^2$ and 
interference terms were
shown in Fig.~\ref{fig:dvcs_bh_int_mc_2009}. Figures \ref{fig:xssum} and 
\ref{fig:xsdiff} show for 12 bins in ($x_{B},Q^2$) the $\phi$ variation of the 
sum $\cal{S}$ and the difference $\cal{D}$ of the hard-exclusive single-photon
cross section. It becomes apparent that the ratio of $\cal{D}$ over $\cal{S}$ 
varies from below the percent level to values close to unity, so that for
small values the determination of $\cal{D}$ becomes more difficult.

The $\mu^+$ and the $\mu^-$ data are recorded during separate data taking 
periods, with different beam intensities and different beam line and 
spectrometer settings (all magnetic fields are reversed). Presently, there is
no information on the magnitude of the charge-dependent error $\Delta a_{cd}$.
We therefore chose to estimate what we consider a tolerable level and define 
the specifications required to achieve this goal. For the proposed running time
and for most of the bins in $(x_{B},Q^2, \phi)$, the statistical accuracy 
$\Delta {\cal{D}}_{stat} = 1/\sqrt{N_{BH} + N_{DVCS}} \times \cal{S}$ is always
above $3\% \times \cal{S}$. We therefore assume a value of 3\% for the relative
accuracy $\frac{\Delta a_{cd}}{a}$ of the charge-dependent term, which results
in the systematic error 
$\Delta {\cal{D}}_{syst} = \frac{\Delta a_{cd}}{a} {\cal{S}}$ shown as the grey
 bands in Fig.~\ref{fig:xsdiff}. Except for small $\phi$ angles and this only 
in two ($x_{B},Q^2$) bins, the charge-dependent systematic error does not 
exceed the statistical one.
   
\begin{figure}[tpb]
\begin{center}
\includegraphics[width=145mm]{gpd_simul_xs-sum}
\end{center}
%\vspace*{-0.3cm}
\caption{Simulated results for the sum $\cal{S}$ (solid line), based on the 
reggeised variant of the VGG model,
% from a measurement at \compass\ in 280 days 
shown for 12 bins in $(x_{B},Q^2)$. The error bars show the projected 
statistical errors.}
\label{fig:xssum}
\end{figure}

\begin{figure}[tpb]
\begin{center}
\includegraphics[width=145mm]{gpd_simul_xs-diff}
\end{center}
%\vspace*{-0.3cm}
\caption{Simulated results for the difference $\cal{D}$ (solid line), based on
the reggeised variant of the VGG model,
% from a measurement at \compass\ in 280 days 
shown for 12 bins in $(x_{B}, Q^2)$. The error bars show the projected 
statistical errors, the grey bands show the systematic uncertainties expected 
when assuming 3\% for the charge-dependent errors (see text).}

\label{fig:xsdiff}
\end{figure}

An illustration of the present understanding of systematic errors from existing
data on absolute SIDIS cross sections is shown in Table~\ref{tab:systerrors}. 
The table lists the different corrections with the corresponding errors which 
need to be applied to calculate the cross sections for semi-inclusive high 
$p_T$ hadron muo-production at \compass. Concerning acceptance corrections, a 
present estimate for the muon detection (Source~7) gives a correction of 5--7\%.
Assuming a similar number for photon detection we expect for both a correction 
of 7--10\%. The quoted error of 50\% is conservative.

Putting aside the acceptance correction, which is still preliminary and should 
not depend on beam charge, the resulting systematics would reach a maximum of 
$\sim 7\%$, assuming it is similar for data taking in the GPD programme. Note 
that the 
dominant contributions are Source~2 and 4 (Table~\ref{tab:systerrors}). It is 
presently not possible to evaluate which fraction of this systematics is 
beam-charge dependent. Therefore, an obvious goal is to have this fraction 
equal (or less) to the 3\% assumed for the charge dependent systematic error 
$\frac{\Delta a_{cd}}{a}$.



\vspace*{0cm}
\begin{table}[tbp]
\caption{The corrections and corresponding errors affecting absolute SIDIS 
cross sections.}
\label{tab:systerrors}
\begin{center}
\begin{tabular}{|llcc|}
\hline
\multicolumn{2}{|l}{Source}				& Correction (relative error)	& Resulting systematics	\\
\hline
1 & Missing events		&			&				\\
 &(errors in data processing)	& $\le$10\% (3\%)		& $\le$0.3\%		\\
\hline
2 & Detector instabilities	&			&				\\
  &(not full diagnostics)	&			& 5\%				\\
\hline
3 & DAQ dead time		& 10\% (4\%)		& 0.4\%				\\
\hline
4 & Beam Flux determination	& 20\% (25\%)		& 5\%				\\
\hline
5 & Veto dead time		&			&				\\
  &needs precise spill structure& 20\% (5--10\%)		& 1--2\%		\\
\hline
6 & Radiative corrections	& 10--20\% (10\%)	& 1--2\%			\\
\hline
7 & Acceptance/Efficiency	 & 7--10\% ($\le$50\%)	& $\le$3--5\%			\\
\hline
\end{tabular}
\end{center}
\end{table}


In addition to a precise monitoring of the above mentioned corrections, the 
following quantities will need to be precisely controlled
\bi
\item the beam position when using $\mu^+$ or $\mu^-$ beams,
\item the symmetry of the two \compass\ spectrometer magnets with the two 
polarities,
\item the Moller electron contamination which affect different parts of the 
detectors for the two beam charges.
\ei

%===========================================================================
%%%%%%%%%[Tables for BH, DVCS and acceptance for VGG]%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[tbp]
\caption{Number of DVMP events for $\rho$ production, shown for $6 \times 4$
bins in $x_B$ and $Q^2$ for $0.05 \leq y =E_{\gamma^\ast}/E_{\mu} \leq 0.9$,
%projected  for 280~days (70~days with $\mu^+$ and 210~days with $\mu^-$) with 
using a global efficiency of 0.1 and the \compass\ set-up comprising the new 
large-angle calorimeter ECAL0 with small central hole, located directly behind 
the RPD.
%which reduces the acceptance for $\rho$ detection at larger values of $x_B$. 
%The numbers of DVMP events for $\rho$ production are obtained using a model 
%developed for \compass\ (see text).
The acceptance for $\rho$ detection is indicated for each bin between 
parentheses.
}
\label{tab:physics.gpd.simu_proj_rho_nbevts}
\begin{center}
%\scalebox{0.96}{
   \begin{tabular}{|c||c|c|c|c|c|c|} \hline
   \raisebox{-1mm}{$Q^2/$}
   &  $0.005 \leq x_B $ & $0.01 \leq x_B $ & $0.02 \leq x_B $ 
   & $0.03  \leq x_B $  & $0.07 \leq x_B $ & $0.13 \leq x_B $\\      
   $\GeV^2$
   & $ \leq 0.01$ & $ \leq 0.02$ & $ \leq 0.03$ 
   & $ \leq 0.07$ & $ \leq 0.13$ & $ \leq 0.27$\\      
   \hline
   \hline
   $8 \leq Q^2 $ & - & - & - & 538  & 569 & 103 \\
   $ \leq 20 $ &   &   &   &   (0.87)   &   (0.69) &   (0.10)  \\
   \hline
   $4 \leq Q^2 $ & - & 303 & 1140 
         &  3224  &  716 &  11 \\
   $ \leq 8 $   &   &   (0.83)  &   (0.86)  &   (0.80)   &   (0.21)  &   (0.003)  \\
   \hline
   $2 \leq Q^2 $ & 1138 & 8427 &  6491    
        &   8148 & 308 &  0 \\
   $\leq 4 $  &  (0.82)   &   (0.86)  &   (0.86)  &   (0.45)   &   (0.02)  &   (0.00)  \\			  
   \hline
   $1 \leq Q^2 $ & 31186 &  42496 &  18897 
         &  8763 &  38 & - \\
   $\leq 2 $   &  (0.84)   &   (0.84)  &   (0.57)  &   (0.11)   &   (0.002)  &  -  \\			  
   \hline
   \end{tabular}
%}
\end{center}
\end{table}

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=140mm,clip=true]{gpd_simul_t_slope_rho}
\end{center}
\caption{Projections for measuring the $x_B$ dependence of the $t$-slope 
parameter $B(x_B)$ in $\rho^0$ vector meson production, calculated for 
$1 \leq Q^2 \leq 20~\GeV^2$ and compared to \zeus\ results with 
similar $\langle Q^2 \rangle$~\cite{Chekanov:2007zr}.
Only statistical uncertainties are shown.
}
\label{fig:gpd_simul_rho_t_slope}
\end{figure}
\subsubsection{The $t$-slope of the $\rho^0$ cross section}
\label{subsubsec:physics.gpd.simu_proj.rho}
%
	
Data on the exclusive production of vector mesons ($\rho, \phi$, \ldots), which
will be recorded simultaneously to the DVCS measurements, will be used to 
determine the corresponding cross sections and $t$-slope parameters $B(x_B)$. 
As an example, we show here projections for the $\rho^0$ vector meson. The 
simulations are based on a model developed for \compass~\cite{A-Sandacz_Rho}
where the $Q^2$ and $\nu$ dependencies are taken from the parametrisation of 
NMC data~\cite{Arneodo:1994id} and the absolute normalisation is done according
to predictions of Ref.~\cite{Goloskokov:2007nt}. The statistical accuracy 
expected for 280~days (70~days with $\mu^+$ and 210~days with $\mu^-$)at 
160~GeV muon beam energy is shown in 
Fig.~\ref{fig:gpd_simul_rho_t_slope} for the $x_B$ and  $Q^2$ bins 
given in Table~\ref{tab:physics.gpd.simu_proj_rho_nbevts}.
The data from \zeus~\cite{Chekanov:2007zr}, which cover a lower $x_B$ range, 
are shown for comparison. The only existing data in the \compass\ range are 
those from NMC, although with significantly lower precision. The $t$-slope 
parameter $B(x_B)$ measured at \hera\ was found to decrease with increasing 
$Q^2$ from $B(x_B)\sim 8~\GeV^{-2}$ at $Q^2 \sim 1~\GeV^2$ to 
$B(x_B) \sim 5.5~\GeV^{-2}$ at $Q^2$ larger than 10~GeV$^2$. The present 
simulation was performed assuming $B(x_B) = 8 ~\GeV^{-2}$; a smaller value 
would decrease the error bars. It can be seen that the future DVMP data from 
\compass\ will cover a $x_B$ range which is slightly shifted to lower values 
as compared to the DVCS data shown in Fig.~\ref{fig:physics.gpd.tslope}.
At $x_B$ above $0.1$, the statistics are reduced by the acceptance of the 
first spectrometer magnet SM1 and the central hole of the ECAL0 
calorimeter. Data on exclusive production of heavier mesons, like $\phi$, 
are expected to provide additional measurements of $B(x_B)$ in a similar 
kinematic range. The virtue of other mesons is to enhance the capability of 
disentangling not only different flavours, but also valence, sea and gluons. 
In particular, for $\phi$ only sea and gluons contribute, whereas for $\rho$
valence quarks have also to be taken into account, as explained in 
Sect.~\ref{subsubsec:physics.gpd.observables.DVMP}.



%===========================================================================
\subsubsection{Transverse target spin asymmetries}
\label{subsubsec:physics.gpd.simu_proj.TTSA}
%
Measurements of transverse target spin asymmetries for the proton are foreseen 
at a later stage. In this section, we present the corresponding projected 
statistical accuracies for the DVCS process in order to illustrate the 
potential of such measurements. The target will consist of transversely 
polarised ammonia, 
similar to the one used presently at \compass, with the density of the 
material in the target cells equal to 0.5~g/cm$^3$, the total length of 
the target cells equal to 120~cm and a diameter of up to 4~cm~\cite{Jaakko}.
The target cells will be surrounded by a recoil proton detector (RPD) of a 
length that matches the length of the target. The present polarised target 
set-up based on the combination of two superconducting magnets, i) a 2.5~T 
large-diameter highly uniform solenoid and ii) a 0.6~T transverse dipole,
leaves no space for a RPD of similar structure as the one foreseen for the 
measurements with a 2.5~m length liquid hydrogen target, as described in 
Sect.~\ref{subsec:upgrades.LH2_RPD.RPD}. Two hypothetical configurations are 
therefore considered here for polarised target and RPD. In configuration~1, 
a simplified and more compact version of an RPD could be installed inside
the solenoid of the existing \compass\ polarised target. In configuration~2, 
an RPD similar to the new one but of shorter length would be used which 
necessitates a new technology for the polarised target magnet(s). Possible 
solutions to integrate a transversely polarised target into a complete RPD 
are under discussion~\cite{W-Meyer}. They involve the construction of thin 
superconducting coils needed to perform the dynamic nuclear polarisation and 
to hold the transverse polarisation. Their structure would have to be 
compatible with that of the cryogenic and microwave equipments. We note that 
configuration~1 will most likely not provide the necessary resolution in 
recoil proton momentum $P$ and consequently in momentum transfer $t$ required 
for such measurements. Therefore it is very important to soon initiate R\&D 
activities for configuration~2 of polarised target and RPD.

In a later analysis of data on hard-exclusive single-photon production on a
transversely-polarised target, the transverse spin-dependent part of the DVCS 
cross section will be obtained by calculating the difference of cross sections 
with opposite values of $\phi_s$, the azimuthal angle between lepton scattering
plane and target spin vector, \ie, with two opposite orientations of the 
transverse target spin vector. In order to minimise systematic effects the 
different target cells will have to be polarised in opposite directions and 
their polarisation will be reversed periodically using the method of dynamic 
nuclear polarisation, as it was done at \compass\ in the past. 

In order to disentangle the contributions of the $\rm |DVCS|^2$ and the 
interference terms, which have the same azimuthal dependence, data with 
both $\mu ^+$ and $\mu ^-$ beams will have to be taken and the difference and 
sum of $\mu ^+$ and $\mu ^-$ cross sections will be calculated which permits 
to separate these contributions. The asymmetries for the difference and
the sum of $\mu ^+$ and $\mu ^-$ transverse spin-dependent cross sections,
$A^D_{CS,T}$ and $A^S_{CS,T}$ (Eqs.~(\ref{eq:DCST}) to 
(\ref{eq:ACST})), will be analysed separately.

Simulations were performed assuming the polarised ammonia target as 
described in Ref.~\cite{Jaakko}, in order to estimate the expected statistical 
accuracy of transverse target spin asymmetries. The spectrometer set-up,
parameters and fluxes of $\mu ^+$ and $\mu ^-$ beams, as well as the 
global efficiency were assumed to be the same as described earlier in 
this section. The fraction of all protons in the target, both polarised 
and unpolarised, was estimated to be equal to 0.57 and the dilution factor
(fraction of polarised protons to all protons) equal to 0.26. The value 
assumed for the target polarisation was 0.9. Simulations were performed for 
the two above assumed configurations of the target region. Due to the 
recoil-proton energy loss in the target, in the microwave cavity and in 
addition for configuration~2 also in the thin holding coil, the minimal
value of $|t|$ measured by the RPD is 0.10 (0.14)~GeV$^2$ for configuration~1 
(2). In the exclusive-photon event generator, the DVCS amplitude was calculated
according to the Frankfurt--Freund--Strikman (FFS) model which was modified
for the \compass\ experiment~\cite{A-Sandacz}.

As an example of the results from the simulations, the expected statistical 
accuracy of the asymmetry $A_{CS,T}^{D,\sin(\phi-\phi_s)\cos\phi  }$ is 
shown in Fig.~\ref{fig:gpd_simul_azimasy} as a function of $-t$, $x_B$ and 
$Q^2$ for the two hypothetical target/RPD configurations. This asymmetry is 
analogous to the asymmetry $A_{UT}^{\sin(\phi-\phi_s)\cos\phi }$ measured with
unpolarised electrons on a transversely polarised proton target, which is also
shown in the figure. It is related to the leading (\mbox{twist-2}) contribution 
$c_{1T-}^I$ (Eq.~(\ref{eq:abcsdt2})).

\begin{figure}[tbp]
\centering
\includegraphics[width=160mm,clip=true]{gpd_simul_AS_new_asywithhermes_2}
%\includegraphics[width=160mm,clip=true]{gpd_simul_AS_new_asywithhermes_2.png}
\caption{Expected statistical accuracy of 
$A_{CS,T}^{D,\sin(\phi-\phi_s)\cos\phi}$ as a function of $-t$, $x_B$ 
and $Q^2$ from a measurement in 280 days, using a 160~GeV muon beam
and ECAL1+ECAL2. Solid and open circles correspond to the simulations for the 
two hypothetical configurations of the target region (see text). Also shown is
the asymmetry $A_{U,T}^{\sin(\phi-\phi_s)\cos\phi  }$ measured at 
\hermes~\cite{:2008jga} with its statistical errors.}
\label{fig:gpd_simul_azimasy}
\end{figure}

Typical values for the statistical errors of
$A_{CS,T}^{D,\sin(\phi-\phi_s)\cos\phi  }$, as well as of the seven remaining
asymmetries related to the \mbox{twist-2} terms in the cross section, are expected to
be about 0.03. This projected statistical accuracy for possible future 
\compass\ measurements of transverse target spin asymmetries represents a 
significant improvement compared to existing results. One should also mention 
that using $\mu ^+$ and $\mu ^-$ beams \compass\ can measure the whole set of 
eight transverse-spin-dependent \mbox{twist-2} terms, whereas only four of them were
previously determined by \hermes.