\subsection{Strange quark distribution function and quark fragmentation functions} 
\label{subsec:physics.unpol.sect1}
Hadron multiplicities can be measured
at \compass\ for a large variety of produced hadrons, $\pi^+$, $\pi^-$, $\pi^0$,  
$K^+$, $K^-$, $K^0$, $\Lambda$ and $\bar{\Lambda}$ as a function of  $x$, $z$ and 
$Q^2$. 
Hadron multiplicities represent the number of hadrons of type $h$ produced per 
DIS event. At LO, they can be written as 
\begin{equation} 
{\frac{\di N^{h}(x,z,Q^2)}{\di N^{DIS}}}= \frac{\sum_q e_{q}^2 q(x,Q^2) D_q^h(z,Q^2)}
                                    {\sum_q e_{q}^2 q(x,Q^2)}, 
\label{unpol_mul}
\end{equation}
where $q(x,Q^2)$ is the unpolarised PDF for flavour $q$, $D_q^h$ is the FF
for a quark $q$ fragmenting into a hadron $h$, and $z$ is the fraction of 
virtual photon energy carried by the hadron. The multiplicities involve a 
product of PDFs and FFs. The possibility to disentangle them bases on 
factorisation,
%%%~\cite{} add some Collins Ref here later
\ie, PDFs depend only on $x$ and  FFs only on $z$.

%Hadron multiplicities can be measured at \compass\ for a large variety of 
%produced hadrons, $\pi^+, \pi^-, \pi^0, K^+, K^-, K^0, \Lambda$ and 
%$\bar{\Lambda}$, as a function of  $x, z$ and $Q^2$. 
An extensive and precise mapping of hadron multiplicities will be achieved 
for values of $x$ as low as 0.004, \ie, to a value ten times smaller than 
the lower $x$ limit of \hermes\ which performed the only other SIDIS 
measurement. The data will be taken at higher average values of $Q^2$ or 
$W$, which are a signature of the current fragmentation region. The 
particle identification by the RICH detector and by electromagnetic 
calorimeters will make it possible to identify various channels so that also
the strange quark sector will be covered, for example $K^+, K^-$ and $K^0$.

The data will provide input to global analyses of FFs such as the one by
DSS\cite{deFlorian:2007aj}. The advantage of SIDIS data is that they are 
sensitive to individual quark and antiquark flavours in the fragmentation 
process, which is not the case for $e^+e^-$ annihilation data.
In addition, the new data will be used on their own in a 
simple LO analysis to extract for instance $s(x)+\bar s(x)$ and some 
poorly known FFs, as described below.

\begin{figure}[tbp]
\centering
%\includegraphics[height=8cm,width=15cm,angle=90]{proj_s.pdf}
\includegraphics[width=0.8\hsize]{unpol_strange+frag_proj_s}
\caption{Projected statistical errors for $s(x) + \bar{s}(x)$ for 
one week of data taking with a 2.5~m long liquid hydrogen target. Systematic 
errors are estimated to be of the order of 25\% of the value of $s$. Also 
shown are the MSTW and CTEQ parameterisations at LO and the \hermes\ result.}
\label{unpol:s_compass}
\end{figure}

\subsubsection{Strange quark distribution function}
While light quark PDFs are well constrained, this is not the case for the 
strange quark and antiquark distribution functions $s(x)$ and $\bar s(x)$,
especially in the \compass\ $x$ range. A better knowledge of $s(x)$ is needed 
for the extraction of $\Delta s$ from the measured quantity $\Delta s(x)/s(x)$
and for physics issues relevant to LHC experiments.

Our present knowledge of PDFs comes from global analyses at 
NNLO (see Refs.~\cite{Martin:2009iq,Nadolsky:2008zw} and references therein)
of inclusive data on hard scattering from fixed-target experiments and the
\hera\ and Tevatron colliders. No semi-inclusive data are included in these 
analyses yet. The lack of knowledge on $s(x)$ was reflected in the common 
practise of past PDF analyses to adopt the simplifying ansatz of $s+\bar s$ 
being proportional to $\bar u+\bar d$. Only recently, PDF analyses 
started to treat $s$ and $\bar{s}$ separately
including recent dimuon cross section data from NuTeV, which are sensitive
to $s$.
%%% ~\cite{}  add a Ref here later

\hermes~\cite{Airapetian:2008qf} determined $s(x)+\bar s(x)$ using 
SIDIS kaon data on a deuteron target and a simplified formula for the 
multiplicities at LO. The results disagree substantially with existing 
parameterisations. At $x=0.03$, the \hermes\ value is twice higher than the 
CTEQ6L one and at $x=0.1$ it is close to zero, much below the CTEQ6L value 
(Fig.~\ref{unpol:s_compass}). 
\compass\ has taken similar data using the $^6$LiD target covering the 
range $0.004<x<0.6$ that extends considerably to lower values than explored
before. However, the proposed measurement on a pure hydrogen target will be 
substantially cleaner, as there will be no additional material in the target 
and secondary interactions will be less important. 

We note another interesting possibility to determine $s(x)$ and $\bar{s}(x)$
separately by a measurement of the longitudinal spin transfer from polarised 
muons to $\Lambda$ and $\bar{\Lambda}$ hyperons, a measurement shown to be
feasible at \compass\ \cite{Alekseev:2009tv}.
%%% \cite{xxx} add a Ref here.

\subsubsection{Quark fragmentation functions} 
Our present knowledge on FFs is based on NLO analyses of single-inclusive 
hadron production which combine data from $e^+e^-$ annihilation, $pp$ 
collisions and deep inelastic lepton--proton scattering. The recent inclusion 
of SIDIS (\hermes) and $pp$ (\rhic) data with charge 
identification~\cite{deFlorian:2007aj} leads to significant progress, 
permitting to disentangle quark and antiquark fragmentation. However, there
exist large discrepancies between various FF parameterisations. In particular, 
the huge uncertainties of FFs for fragmentation of light quarks into kaons and 
$s$ quarks into any hadron are presently a major obstacle to the extraction 
of the strange-quark
polarisation from polarised SIDIS data. An important piece of the nucleon spin 
puzzle is the contribution of strange quarks $\Delta s$ to the 
total nucleon spin of 1/2. The first $x$-moment of $\Delta s(x)$ was estimated 
to be significantly negative, of the order of $-0.10$, from {inclusive} 
polarised DIS measurements, while present {semi-inclusive} data from \compass\ 
and \hermes\ indicate values of the strange quark polarisation 
$\Delta s(x)/s(x)$ that are very small or even compatible with zero in the 
measured $x$ ranges \cite{Alekseev:2009ci}. This dilemma already caused 
theoretical debates~\cite{Bass:2009ed}. However, \compass\ has shown that the 
SIDIS result on $\Delta s(x)$ and the associated error can change by a factor 
of three depending on the choice of the quark-to-kaon FFs..
%%%~\cite{} may add a Ref here later
In the next section, we show that with the proposed measurement on the
hydrogen target kaon FFs can be determined with high statistical precision 
and a systematic uncertainty of the order of 25\% of the signal value, already 
from a very short measurement. This will remove the above mentioned factor of 
three in the kaon FF uncertainty. 
Later, with the large data sets to be collected in the
two years of the GPD programme, a multidimensional mapping of FFs as a 
function of (at least) $x$ and $z$ will be possible so that the systematic 
uncertainty will be drastically reduced.

When included in combined NLO analyses, the future SIDIS data on the
proton for various identified hadrons will substantially increase the range in
$x$, $Q^2$, and $z$ as well as the sensitivity of the FF parameterisation to 
the partonic flavours. Several additional results will be obtained on FFs. For 
instance, a model-independent method~\cite{Christova:2008te} can be used to 
derive non-singlet FFs using cross-section differences between hadrons of 
opposite charges from a combination of proton and deuteron data and .
%In addition, the FF are also needed for the study of flavour separation in 
%the Sivers function.

Finally, it should be noted that the measurements of multiplicities are 
not limited by statistics but by systematic uncertainties. The dominant error 
comes from the determination of the acceptance of the apparatus. There will be
mutual benefits with the GPD programme, for which an excellent understanding of 
the acceptance is mandatory.

\subsubsection{Expected statistical precision}
The final goal is to combine the measurements of all hadron multiplicities
as a function of $x, z$ and $Q^2$ obtained during the two years of data taking 
for the GPD programme, together with the existing deuteron data. The FFs and the 
PDFs will be extracted from global analyses. Already from a first sample of 
data collected in one week and using some reasonable assumptions, first physics
results can be obtained. For instance, the unpolarised strange quark 
distribution function $s(x) + \bar{s}(x)$ can be extracted from 
Eq.~(\ref{unpol_mul}) using the charged kaon multiplicities as input and 
assuming that the non-strange PDFs $u$, $\bar{u}$, $d$ and $\bar{d}$ and all FFs are 
known.
%Note that by using two separate equations for $K^+$ and $K^-$ instead 
%of a single one for $K^{+}+K^{-}$, the $s$ and $\bar s$ functions could be 
%determined separately in principle; however this separation will likely be 
%limited by the size of various systematic errors.
The expected statistical errors on $s(x)+ \bar s(x)$ were calculated using a 
Monte Carlo simulation based on LEPTO assuming one week of data taking with 
the 2.5~m long liquid hydrogen target in parallel to the data taking for the
GPD programme. The values of the FFs $D_u^{K}$, $D_d^{K}$ and $D_s^{K}$ were 
taken from DSS \cite{deFlorian:2007aj} and the non-strange PDFs $u$, 
$\bar{u}$, $d$ and $\bar{d}$ from MRST~\cite{Martin:2002dr}, all in LO. 
The resulting projections are shown in Fig.~\ref{unpol:s_compass} 
together with the MSTW08 and CTEQ6L parameterisations and the \hermes\ 
results~\cite{Airapetian:2008qf} obtained from kaon multiplicities on a 
deuterium target. It can be seen that \compass\ can reach small statistical 
errors already in one week. However, at that moment the dominant uncertainties 
will be the systematic ones which are essentially coming from the uncertainties 
on the fragmentation functions. If FFs will be known with an uncertainty of,
\eg, 25\% as it will be discussed further below, this will affect the
systematic uncertainty on $s$ by the same amount. As a pure proton target will
be used, no systematic uncertainties from nuclear effects exist.

Using the same initial sample of data and integrating over the variable $z$, 
one can extract the FFs at LO assuming that all PDFs are known. As an example, 
we show below how to extract quark-to-kaon FFs from the measured kaon 
multiplicities on the proton. In the following, $D_q^K$ stands for 
$\int D_q^K(z) dz $ for simplicity. Using Eq.~(\ref{unpol_mul}) to describe 
separately the $K^+$ and the $K^-$ multiplicities, we obtain two equations 
involving twelve FFs. Assuming charge conjugation invariance in the 
fragmentation and assuming that all unfavoured FFs are equal, 
we are left with only three unknown FFs, namely

$$D_1= D_{\bar{s}}^{K^+} = D_s^{K^-}$$
$$D_2= D_u^{K^+}= D_{\bar{u}}^{K^-}$$
$$D_3= D_{\bar{u}}^{K^+} =  D_d^{K^+} = D_{\bar{d}}^{K^+} =  
D_u^{K^-} =  D_d^{K^-} = D_{\bar{d}}^{K^-} = D_s^{K^+}  = D_{\bar{s}}^{K^-}$$

\noindent
Since FFs do not depend on $x$ while the multiplicities and all PDFs do, 
the measured multiplicities of $K^+$ and $K^-$ in 12 $x$ bins provide 
24 equations with 3 unknowns, $D_1, D_2$ and $D_3$. Note that the known 
$Q^2$ dependence of the FFs can be taken into account in the calculation.

In order to estimate the statistical errors obtained on FFs from one week of 
data taking, the kaon multiplicities were calculated using a Monte Carlo 
simulation based on LEPTO with non-strange FFs taken from DSS and all PDFs 
from MRST. The simulation gives 
$D_1=0.39 \pm 0.03$, 
$D_2=0.11 \pm 0.002$ and
$D_3=0.03 \pm 0.012$.
As can be seen, very small statistical errors  will already be obtained in one 
week, especially for the favoured FFs $D_1$ and $D_2$. However, systematic 
errors on the acceptance determination are expected to dominate here. As an 
example, an uncertainty of the order of 25$\%$ is expected for $D_1$ if kaon 
multiplicities are determined with a precision of 5$\%$. Nevertheless, the
result of such a short run has already an impact because presently the various 
predictions for $D_1$ differ by a factor of three. The uncertainty on the 
unpolarised strange quark distribution function will propagate directly into 
that of $D_1$. We note that a similar extraction can be done for the 
$s$-to-pion FF. Once high-statistics data will be available after two years
of running, it will be possible to extract for the first time the FFs in a
multidimensional binning in $x$, $z$, and $Q^2$. Combining the new proton 
data with the existing deuteron data, more FFs will be determined in the 
analysis without assumptions.