\section{Projected Results}

To perform the simulation study and obtain the projected results, we developed
a DEMP generator, as discussed in Appendix-\ref{sec:generator}, and used it to
generate events within a kinematic phase space slightly larger than the
SoLID-SIDIS acceptance.  The Fermi motion of the neutron in $\mathrm{^{3}He}$,
radiation of the incident and scattered electrons, multiple scattering of the
final state particles, and energy loss due to the ionization have been taken in
account in this generator.  Then, for every detected particle in each event, we
added the acceptance profiles obtained from the GEANT4 simulation with the
SoLID-SIDIS configuration and smeared the momenta and angles of the final state
particles by the detector resolutions based on current knowledge from the
tracking reconstruction study.  We also compared the obtained distributions
with GEMC simulations taking into account the target and individual detector
geometries.  To better simulate the real experimental conditions, we generated
two sets of data with the target polarization up and down, respectively.

\subsection{Kinematic Coverage}

\begin{figure}[!ht]
 \begin{center}
      \includegraphics[width=0.7\textwidth]{./figures/Q2_vs_x_may19.pdf}
   \vspace{-5mm}
   \caption{\footnotesize{The triple coincidence $e$+$\pi^-$+$p$ kinematic
       coverage of DEMP events at 11~GeV within the SoLID acceptance.
       A $W>$2~GeV cut is applied.}}
  \label{fig:kin_cor}
  \end{center}
\end{figure}

The kinematic coverage in $Q^{2}$ vs. $x_{B}$ is shown in
Fig.~\ref{fig:kin_cor}, using the existing SoLID detectors to detect electrons,
pions and protons at $8^{\circ}\sim24^{\circ}$. These distributions were
weighted by the DEMP unpolarized cross sections and the SoLID acceptance
profiles for electrons, pions and protons.  A $W>2$~GeV cut was also applied to
exclude non-DIS events.

\begin{figure}[!ht]
 \begin{center}
   \includegraphics[type=pdf,
     ext=.pdf,read=.pdf,width=0.9\textwidth]{./figures/dvmp_acceptance_narrow}
   \caption[The acceptance of the momenta and scattering angles for electrons,
     $\pi^{-}$ and protons]{\footnotesize{The acceptance of the momenta and
       polar angles for electrons, $\pi^{-}$ and protons, respectively. Cuts of
       $Q^{2}>4~\mathrm{GeV^{2}}$ and $W>$2~GeV are applied.}}
  \label{fig:p_theta}
  \end{center}
\end{figure}

Fig.~\ref{fig:p_theta} shows the momentum and angular acceptance of $\pi^{-}$,
electrons and protons which form the DEMP events and can be detected with the
SoLID detectors.  The recoil protons shown
in Fig.~\ref{fig:p_theta} have low momenta ranging from 0.3~GeV/c up to
1.5~GeV/c, and are distributed in both the large- and forward-angle regions.
The panels have applied cuts of $Q^{2}>4~\mathrm{GeV^{2}}$ and $W>$2~GeV,
corresponding to the region of greatest physics interest.

\subsection{Estimated Rates
\label{sec:rates}}

\begin{table}[!ht]
\centering
\begin{tabular}{|c|c|}
 \hline
  $Q^2>$1~GeV$^2$ & $Q^2>$4~GeV$^2$\\
 \hline
\multicolumn{2}{|c|}{DEMP: $\vec{n}(e,e'\pi^{-}p)$ Triple-Coincidence (Hz)}\\
 \hline
 4.95   &  0.40 \\
 \hline
\multicolumn{2}{|c|}{SIDIS: $\vec{n}(e,e'\pi^{-})X$ Double-Coincidence (Hz)}\\
 \hline
   1424.62  & 35.77   \\
 \hline
\end{tabular}
\caption[Triple-Coincidence rates for
  neutron-DEMP]{\footnotesize{Triple-Coincidence rates for DEMP events compared
    with the SIDIS rates. A $W>$2~GeV cut was applied. The online production
    trigger will be the SIDIS double-coincidence trigger of which rates are
    also given.}}
\label{rate_table}
\end{table} 

Table~\ref{rate_table} lists the triple-coincidence rate of the DEMP
events. The rates were calculated with the simulated events weighted by the
target luminosity, the SoLID acceptances and unpolarized cross sections.  The
``raw'' rates are not corrected by the beam and target polarization, target
dilution and so on.  Our conservatively estimated rate is around 4.95~Hz at
$Q^{2}>$1~GeV$^{2}$, or 0.40~Hz at $Q^{2}>$4~GeV$^{2}$. For comparison, the
table also gives the SIDIS rate which will be the online production trigger
rate and is the main background of DEMP events.

\subsection{Asymmetry Projections
\label{sec:asym}}

\begin{figure}[!ht]
 \begin{center}
      \includegraphics[width=0.75\textwidth]{./figures/Q2_vs_t_may19.pdf}
   \vspace{-5mm}
   \caption{\footnotesize{$Q^{2}$ vs. $-t$ coverage of triple coincidence
       $e$+$\pi^-$+$p$ DEMP events, where the black dashed lines specify the
       boundaries of 7 $-t$ bins.  $Q^2>$4~GeV$^2$ and $W>$2~GeV cuts are
       applied.}}
  \label{fig:Q2_t_bin}
  \end{center}
\end{figure}

The proposed experiment will run in parallel with E12-10-006, which has already
been approved to run 48 days at $E_{0}$=11~GeV.  As shown in
Fig.~\ref{fig:Q2_t_bin}, we defined 7 $-t$ bins of which the boundaries are
defined by the array:
 \begin{equation}
-t[8] = [0.05, 0.20, 0.30, 0.40, 0.50, 0.70, 1.0, 3.0]~~~~(\mathrm{in~GeV^{2}})
 \end{equation}
The number of events ($N^{\uparrow\downarrow}_{i}$) in the $i^{th}$ bin is
calculated from the total simulated events after applying cuts on important
kinematic variables, e.g. $Q^{2}>$4~GeV$^{2}$, $W>$2~GeV, 0.55$<\epsilon<$0.75
and $-t_{min}<-t<-t_{max}$. Two simulated data sets with target polarization up
and down follow exactly the same cuts and binning.  As shown in
Eqn.~\ref{eqn:ncount}, each event surviving the cuts is then weighted by the
polarized cross section, together with the acceptance of the electron, pion and
proton. $N^{\uparrow\downarrow}_{i}$ is further corrected by the phase-space
factor ($PSF$) defined in the event generator, the total number of randomly
generated events ($N_{gen}$), beam-time ($T$), the target luminosity
($L=10^{36}$~cm$^{-2}$s$^{-1}$), and the overall detector efficiency
($\epsilon_{eff}$):
 \begin{equation}
     N^{\uparrow\downarrow}_{i} = \bigl(\sum_{j\in i-bin}
     \sigma^{\uparrow\downarrow}_{j}\cdot A^{e}_{j} \cdot
     A^{\pi^{-}}_{j} \cdot A^{p}_{j}\bigr) \cdot (PSF/N_{gen}) \cdot T \cdot L
     \cdot \epsilon_{eff},
     \label{eqn:ncount}
 \end{equation}
where $j$ is the $j^{th}$ event in the $i^{th}$ bin,
$\sigma^{\uparrow\downarrow}_{j}$ is the cross section of the $j^{th}$ event with
the target polarization up or down. $A^{e(\pi^{-},p)}_{j}$ is the acceptance
weight of the electron (pion, proton) in this event. The detector efficiency,
$\epsilon_{eff}$, is approximately fixed at 85\% as was used in the SIDIS
proposals. $N^{\uparrow\downarrow}_{i}$ corresponds to the raw experimental
count of electrons scattering on neutrons in $\mathrm{^{3}He}$ after taking
into account the target polarization ($P\sim60\%$), the effective polarization
of neutrons ($\eta_{n}\sim0.865$), and the dilution effect from other reaction
channels when electrons scatter on $\mathrm{^{3}He}$ ($d \sim 0.9$).

%In addition, we further divide each $-t$-bin into two $Q^{2}$ bins with similar
%statistics, as indicated in Fig.~\ref{Q2_t_bin}.  By doing that, we are able to
%examine the $Q^{2}$-dependence of the asymmetries, and also check the model
%dependence of the other corrections that are directly related to the values of
%$Q^{2}$.

With the numbers of simulated events in each bin for two anti-parallel target
polarizations, one is able to reconstruct the average target single-spin
asymmetry in that bin, which is identical to the experimental extracted
asymmetry:
\begin{equation}
   <A_{UT}> = \frac{1}{P\cdot\eta_{n}\cdot d}
   \frac{N^{\uparrow}-N^{\downarrow}}{N^{\uparrow}+N^{\downarrow}}.
   \label{eqn:asym_exp}
\end{equation}
The statistical error of the target single spin asymmetry ($A_{UT}$) in each
bin can be given as:
  \begin{equation}
    \delta A_{UT} = \frac{1}{P\cdot\eta_{n}\cdot d} \sqrt{\frac{1-(P\cdot
        <A_{UT}>)^{2}}{N^{\uparrow}_{i}+N^{\downarrow}_{i}}},
    \label{eqn:stat_err}
 \end{equation}
 
%Because of not performing a L/T separation in this experiment, the asymmetry
%should be corrected by another dilution factor, which is defined as:
%\begin{equation}
%  f_{L/T} =\frac{\epsilon\sigma_{L} }{\sigma_{T}+\epsilon\cdot\sigma_{L} },
%\end{equation} 
%where $\epsilon=(1+\frac{2\nu^{2}}{Q^{2}}\tan^{2}(\theta))^{-1}$. Additional
%dilution due to $\sigma_{TT}$ is assumed to be small.  A factor of $-1$ is also
%applied after comparing Eq.~\ref{eqn:asy} and Eq.~\ref{eqn:sigtarg}. Hence,
%$A_{UT} = -f_{L/T}\cdot A_{L}^{\perp,model}$.

As shown in Eqn.~\ref{eqn:six_asym} of Sec.~\ref{sec:sinphiS}, $A_{UT}$ can be
further decomposed into six asymmetries with different azimuthal angular
modulations:
   \begin{eqnarray}
        A_{UT}(\phi, \phi_{S}) &=& A_{UT}^{\sin(\phi-\phi_{S})}
        \sin(\phi-\phi_{S})+ A_{UT}^{\sin(\phi_{S})} \sin(\phi_{S}) \nonumber \\
       &+& A_{UT}^{\sin(2\phi-\phi_{S})} \sin(2\phi-\phi_{S})+
        A_{UT}^{\sin(3\phi-\phi_{S})} \sin(3\phi-\phi_{S}) \nonumber \\
      &+& A_{UT}^{\sin(\phi+\phi_{S})}
        \sin(\phi+\phi_{S})+A_{UT}^{\sin(2\phi+\phi_{S})} \sin(2\phi+\phi_{S}).
   \label{eqn:six_asym2}
   \end{eqnarray}
In our generator, the first five different azimuthal modulations of $A_{UT}$
are predicted with a phenomenological model as discussed in
Appendix-\ref{sec:generator}, and the last modulation is fixed to be zero as
the model predicts a negligible asymmetry.  As discussed in
Sec.~\ref{sec:motivation}, two modulations, $A_{UT}^{\sin(\phi-\phi_{S})} $ and
$A_{UT}^{\sin(\phi_{S})}$, are particularly interesting and are the main
quantities this proposal aims to measure.

\begin{figure}[hbt!]
\begin{center}
\includegraphics[width=0.9\textwidth]{./figures/asym_demp.pdf}
\end{center}
\caption{\label{fig:asym_comp}
\footnotesize{As an illustration of the validity of the asymmetry fitting
  method, the $A_{UT}$ asymmetries of Eqn.~\ref{eqn:asym_exp} binned as a
  2-dimensional scatter plot for $12\times 12$ $(\phi,\phi_S)$ bins, where
  dark (bright) color indicates negative (positive) single-spin asymmetry for
  that bin.
   {\it Left:} Average input model values per bin (Eqn.~\ref{eqn:six_asym2}), 
   {\it Right:} Average fit values from the UML analysis.
Note, the UML analysis uses the full power of the data, without binning, as
explained in the text.}}
\end{figure}

\begin{figure}[h!]
\begin{center}
\includegraphics[height=18cm]{./figures/t_mult_fit5.png}
\end{center}
\caption{\label{fig:asym1_t}
\footnotesize{Projected uncertainties of the five single-spin asymmetry
  azimuthal modulations $A_{UT}^{\sin(\mu\phi+\lambda\phi_S)}$, as determined
  in the UML analysis, as a function of $-t$.  The black squares are the
  extracted values for each bin, while the blue curves are the input values, as
  obtained from the model of Fig.~\ref{fig:gk16} but averaged over the $Q^2$
  acceptance of each bin.  The simulated data include all radiative, Fermi,
  multiple scattering and energy loss effects.  The difference between the
  input and output values is largely due to Fermi momentum, as discussed
  further in Sec.~\ref{sec:resp_fermi}.}}
\end{figure}

To demonstrate that the proposed measurement has the capability of extracting
these asymmetries, we adopted the procedure presented in
Ref.~\cite{hermes-thesis} to extract all five asymmetries by using a unbinned
maximum likelihood (UML) method. Compared with the regular extraction methods,
where the data in each $-t$ bin are further binned into two dimensional
($\phi$, $\phi_S$) bins, the UML method can perform much better fitting when
the statistics are limited.

The polarized cross sections with two target polarization directions are given
approximately by:
\begin{eqnarray}
     \sigma_{\uparrow} &=& \sigma_{UT}(\phi, \phi_{S}) = [1 +
       \frac{|P_{T}|}{\sqrt{1-\sin^2(\theta_{q})\sin^2(\phi_{S})}} A_{UT}(\phi,
       \phi_{S})] \cdot \sigma_{UU}(\phi) \\
     \sigma_{\downarrow} &=& \sigma_{UT}(\phi, \phi_{S}+\pi) = [1
       -\frac{|P_{T}|}{\sqrt{1-\sin^2(\theta_{q})\sin^2(\phi_{S})}} A_{UT}(\phi,
       \phi_{S})]\cdot \sigma_{UU}(\phi).
\end{eqnarray}
where $|P_{T}| = P\cdot \eta_{n} \cdot d $. Hence, the probability density
function can be constructed as:
\begin{equation}
  f_{\uparrow\downarrow}(\phi, \phi_{S}; \theta_{k}) =
  \frac{1}{C_{\uparrow\downarrow}} [1 \pm
    \frac{|P_{T}|}{\sqrt{1-\sin^2(\theta_{q})\sin^2(\phi_{S})}} \sum_{k=1}^{5}
    \theta_{k} \sin(\mu\phi+\lambda\phi_{S})],
\end{equation}
where $\theta_{k}$, $k=1-5$, are the values of asymmetries that can maximize
the likelihood function.  $C_{\uparrow\downarrow}$ is a normalization constant
and is set to one as it is not important in the UML fitting. Here, we have
dropped out the sixth asymmetry, which is zero. The UML function can be defined
as:
\begin{equation}
	L(\theta_{k}) = L_{\uparrow}(\theta_{k})\cdot
        L_{\downarrow}(\theta_{k})=\prod_{l=1}^{N_{MC}^{\uparrow}}[f_{\uparrow}(\phi_{l},
          \phi_{S,l};\theta_{k})]^{w^{\uparrow}_{l}}\cdot
        \prod_{m=1}^{N_{MC}^{\downarrow}}[f_{\downarrow}(\phi_{m},
          \phi_{S,m};\theta_{k})]^{w^{\downarrow}_{m}}.
\end{equation}
where $w^{\uparrow}_{l} = \sigma^{\uparrow}_{l}\cdot A^{e}_{l} \cdot
A^{\pi^{-}}_{l} \cdot A^{p}_{l} \cdot PSF/N_{gen} \cdot T \cdot L \cdot
\epsilon_{eff}$, is the weight of the $l^{th}$ simulated event. It takes into
account the fact that the Monte-Carlo events are generated uniformly, and it
also includes the experimental conditions, such as the acceptances of three
particles and the detector efficencies. For the real experimental data, the
weight will only takes into account the acceptance correction, detector
efficiencies correction and other experimental related corrections. From
Eq.~\ref{eqn:ncount}, one has $ N^{\uparrow}_{i} = \bigl(\sum_{l\in
  i-bin}w^{\uparrow}_{l})$. Note that $N_{MC}^{\uparrow}= \bigl(\sum_{l\in
  i-bin})$ is simply the total number of simulated events in the $i^{th}$ $-t$
bin without any weighting. In practice, we use the TMinuite package to minimize
the following negative log-likelihood function:
\begin{equation}
  -lnL(\theta_{k}) =-ln
  L_{\uparrow}(\theta_{k})-lnL_{\downarrow}(\theta_{k})=-\sum_{l=1}^{N_{MC}^{\uparrow}}
  w^{\uparrow}_{l}\cdot lnf_{\uparrow}(\phi_{l},
  \phi_{S,l};\theta_{k})-\sum_{m=1}^{N_{MC}^{\downarrow}}
  w^{\downarrow}_{m}\cdot lnf_{\downarrow}(\phi_{m}, \phi_{S,m};\theta_{k}).
\end{equation}

Fig.~\ref{fig:asym_comp} compares the distribution of $A_{UT}$
vs. $(\phi,\phi_S)$ from the UML fitting (right) to the direct
statistical averaged model calculations (left).  The variations in intensity
from left to right and top to bottom of each panel indicate the presence of the
different $\sin(\mu\phi+\lambda\phi_S)$ azimuthal modulations in the
single-spin asymmetry.  The $\sin(\phi_S)$ asymmetry is the only one not
required to vanish at $-t_{min}$, giving a simple top-bottom modulation in the
top panels (lowest $-t$ bin).  In contrast, the fourth $t$-bin displays a
additional modulation structure reflecting primarily the contributions of the
$\sin(\phi-\phi_S)$ and $\sin(\phi+\phi_S)$ Fourier components.  The similarity
between the left and right panels for both $t$-bins is readily apparent,
confirming the validity of the fitting procedure.  The equivalent HERMES
figure, with lower statistics and coarser binning, can be found in
Ref.~\cite{hermes-thesis}.

Fig.~\ref{fig:asym1_t} presents our expected precision for the five
$A_{UT}(\phi,\phi_S)$ Fourier components extracted from the UML analysis
vs. $-t$.  Compared with the existing HERMES results (Fig.~\ref{fig:hermes6}),
the new measurement can provide more precision data to be directly compared
with theoretical predictions.  The detailed information for the two asymmetries
of main interest are listed in Table~\ref{asym_bin_table}.
\begin{table}[!ht]
\centering
 \small
\begin{tabular}{|c|ccc|cc|cc|}
\hline
$t$-bin   &  $<-t>$ & $<Q^{2}>$ & $<x_{B}>$ & 
$A_{UT}^{\sin(\phi-\phi_{S})}$ & $\delta A_{UT}^{\sin(\phi-\phi_{S})}$ & 
$A_{UT}^{\sin(\phi_{S})}$ & $\delta A_{UT}^{\sin(\phi_{S})}$ \\
  \hline
1 & 0.1625 & 4.378 & 0.3368 & -0.0390 & 0.0060 & 0.3817 & 0.0059 \\
2 & 0.2504 & 4.843 & 0.3886 & -0.0214 & 0.0038 & 0.3399 & 0.0038 \\
3 & 0.3478 & 5.301 & 0.4332 & -0.0289 & 0.0031 & 0.3403 & 0.0031 \\
4 & 0.4463 & 5.770 & 0.4711 & -0.0347 & 0.0028 & 0.3484 & 0.0028 \\
5 & 0.5854 & 6.324 & 0.5140 & -0.0419 & 0.0026 & 0.3543 & 0.0025 \\
6 & 0.8091 & 6.856 & 0.5595 & -0.0449 & 0.0025 & 0.3570 & 0.0024 \\
7 & 1.0783 & 7.002 & 0.5844 & -0.0466 & 0.0025 & 0.3585 & 0.0024 \\
\hline
\end{tabular}
\caption[Detailed information of projected bins]{\footnotesize{Detailed
    information of projected bins from the new DEMP measurements with SoLID.
    $<Q^{2}>$ and $<-t>$ are in units of GeV$^{2}$. The values are as obtained
    in the UML analysis, projected uncertainties are statistical only.}}
\label{asym_bin_table}
\end{table} 

\subsection{Missing Mass and Background}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{./figures/missing_mom_cut}
\caption[Missing Momentum]{\footnotesize{Missing momentum spectra of recoil
    protons in DEMP (blue) and SIDIS (red) processes with a polarized $^{3}$He
    target. The dashed magenta curve is the DEMP missing mass only considering
    the Fermi motion, multiple scattering and the energy loss, while the blue
    and red curves have further taken into account the detector
    resolutions. The light-blue dashed line indicates a cut at
    $P_{miss}<1.2$~GeV/c to remove most of the SIDIS events.  Note that the
    SIDIS background is overestimated since we assume all SIDIS events contain
    protons in the final state. }}
  \label{fig:missing_mom}
  \end{center}
\end{figure}

\begin{figure}[!ht]
 \begin{center}
 \subfloat[Before cutting on missing momentum] {
      \includegraphics[width=0.5\textwidth]{./figures/Missing_Mass.pdf}
 }
  \subfloat[Before cutting on missing momentum] {
     \includegraphics[width=0.5\textwidth]{./figures/Missing_Mass_cut.pdf}
 }
   \caption[Missing Mass]{\footnotesize{Missing mass spectra of recoil protons
       in DEMP (blue) and SIDIS (red) processes with a polarized $^{3}$He
       target. The left (right) plot shows the background contamination from
       SIDIS events before (after) the missing momentum cut (shown in
       Fig.~\ref{fig:missing_mom}).  The dashed magenta curve is the DEMP
       missing mass only considering the Fermi motion, multiple scattering and
       the energy loss, while the blue and red curves have further taken into
       account the detector resolutions. Note that the SIDIS background is
       overestimated, since we assume all SIDIS events contain protons in the
       final state.}}
  \label{fig:missing_mass}
  \end{center}
\end{figure}

In the DEMP reaction on a neutron, all three charged particles in the final
state, $e^{-}$, $\pi^{-}$ and $p$, can be cleanly measured by the SoLID
detector system.  Hence, contamination from other reactions, including DEMP
with other two protons in $^{3}$He, can be greatly eliminated.  The dominant
background of the DEMP measurement comes from the SIDIS reactions of electrons
scattering on the neutron and two protons in $\mathrm{^{3}He}$.  In a SIDIS
event, besides the scattered electron and the hadron ($\pi^{\pm}$, $K^{\pm}$
etc.), there could be at least a proton in the target fragments, and in that
case, the SIDIS event will be possibly misidentified as a DEMP event.  In
addition to identifying the recoil protons, which should largely suppress most
of background, we will also rely on reconstructing the missing masses and
missing momentum spectra of recoil protons to ensure the exclusivity of the
DEMP events.

To calculate the missing mass and missing momentum of the recoil proton, we
reconstructed the four-momentum Lorentz vectors of the incoming electron, the
scattered electron and the pion.  We assumed the neutron target is at rest even
though its Fermi motion was simulated in the generator.  We then used the
momentum and energy conservation laws to calculate the missing mass and missing
momentum of the recoil proton.  Note that the energy loss, multiple scattering
effects, and the detector resolutions have been considered in our study.  Based
on the current tracking study, the SoLID-SIDIS system can provide a momentum
resolution of $2\%/\sqrt{E}$, a polar angle resolution of 0.6~mrad, an
azimuthal angle resolution of 5~mrad and a vertex target position of 0.5~cm.
In this measurement, we have conservatively proposed to only identify the
recoil proton. However, with improved time resolution and certain tracking
information, the momenta of the recoil protons will still be determined with
certain accuracies, which gives us room to further suppress any background.

The SIDIS events, $p(e,e'\pi^{-})X$ and $n(e,e'\pi^{-})X$, were simulated with
the same generator used for the SoLID-SIDIS proposals.  The same acceptance
profiles of scattered electrons and pions in DEMP were applied to the SIDIS
events, along with the same kinematic cuts, such as $Q^2>4$~GeV$^2$.  It is
difficult to estimate what percentage of the target fragments in SIDIS contain
at least a proton, so we have to assumed all target fragments ($``X''$) contain
proton. Such an assumption likely results in the SIDIS background being
significantly overestimated.

We followed exactly the same methods as used in DEMP to calculate the missing
masses and missing momenta of the recoil protons in SIDIS.
Fig.~\ref{fig:missing_mom} shows the reconstructed missing momenta of both
processes.  One immediately sees that the main peak of the DEMP missing
momentum spectrum is well separated from the SIDIS background, which can be
largely rejected with a loose cut of $P_{miss}<1.2$~GeV/c.

Fig.~\ref{fig:missing_mass} shows the reconstructed missing mass spectra of the
DEMP and SIDIS events w/ and w/o the missing momentum cuts. Before applying the
missing momentum cut, the tail of the SIDIS background significantly leaks into
the DEMP peak. Of course, keep in mind that the SIDIS rate is likely
overestimated.  After applying the missing momentum cut, the SIDIS background
is largely suppressed.  The total integrated SIDIS becomes 0.04~Hz, compared
with the DEMP rate of 0.5~Hz.  Further considering the fact that only a
fraction $``X''$ in SIDIS contains a proton, we conclude that the SIDIS
background is negligible after the cut.

The background sources, such as the random coincident background events, will
show up in the missing mass spectrum with more uniform distributions. We should
be able to suppress most of them with tight missing momentum and missing mass
cuts.  The residual background events will be largely suppressed or corrected
during the real asymmetry extraction. In general, we expect to have a clean
measurement of the DEMP process because all of the final particles being
detected.

\subsection{Systematic Uncertainties}

\begin{table}[!htp]
\centering
\begin{tabular}{|c|c|}
\hline
{\bf Sources}            & {\bf Relative Value} \\\hline
Beam Polarization        & $2\%$ \\\hline 
Target Polarization      & $3\%$ \\\hline 
Dilution Factor          & $1\%$ \\\hline 
Nuclear Effect           & $<4\%$ \\\hline 
Acceptance               & $3\%$ \\\hline
Radiation Correction     & $2\%$ \\\hline
Background Contamination & $<5\%$ \\\hline
\end{tabular}
\caption{\footnotesize{Expected systematic errors.}}\label{table:det_sys_err}
\end{table}

The systematic errors are expected to be close to those given in the
E12-10-006 proposal \cite{solid:e12-10-006}, as well as in other SIDIS
experiments with SoLID \cite{solid:e12-11-007,solid:e12-11-108,
  solid:e12-10-006A,solid:e12-11-008A}. The procedure of extracting DEMP
asymmetries is also expected to be similar to the SIDIS asymmetry extraction.
The contamination of background should be well controlled by the proton
detection and cuts on missing momenta and mass.  However, to be conservative,
we quote the overall systematic errors of background contamination to be $5\%$
level.  Here we list several major sources of systematic uncertainties as shown
in Table~\ref{table:det_sys_err}.
