Semi-inclusive deep inelastic scattering (SIDIS) has been
used extensively in recent years as an important testing ground
for QCD. Studies so far have concentrated on better determination
of parton distribution functions, distinguishing between the quark 
and antiquark contributions, and understanding the fragmentation of 
quarks into hadrons. The use of polarization in leptoproduction
provides an essential new dimension for testing QCD.
%The spin structure of the nucleon has been of particular interest since the 
%EMC~\cite{Ashman:1987hv} measurements implied that the helicity of the 
%constituent quarks account for only a fraction of the nucleon spin.  
%Possible interpretations of the small fraction of the spin carried by quark helicities 
%include the contribution of the orbital momentum of quarks and gluons 
%and significant polarization of either the strange 
%sea (negatively polarized) or gluons (positively polarized).  


Azimuthal distributions of final state particles in semi-inclusive 
deep inelastic scattering are sensitive to the orbital motion of quarks and 
play an important role in the study of transverse momentum distributions of 
quarks in the nucleon.
Correlations of spin and transverse momentum of quarks are  
by now universally recognized as essential ingredients 
of the structure of hadrons. They are described
by a number of Transverse Momentum dependent Distribution 
functions (TMDs) \cite{Mulders:1995dh,Bacchetta:2006tn} which give rise to various 
observables in hard hadronic 
processes \cite{Barone:2001sp}. 

Large Single Spin Asymmetries (SSAs), 
%appearing as azimuthal moments of the 
%cross section, observed for decades in hadronic reactions,  
have been among 
the most difficult phenomena to understand from first principles in QCD and
can also be related to TMDs.
Two fundamental mechanisms have been identified leading to SSAs in hard processes;
the Sivers mechanism \cite{Sivers:1990fh,Anselmino:1998yz,Brodsky:2002cx,Collins:2002kn,Ji:2002aa}, which generates 
an asymmetry in the distribution of quarks due to orbital motion
of partons, and the Collins
mechanism \cite{Collins:2002kn,Mulders:1995dh}, which generates 
an asymmetry  during the hadronization of quarks.

Significant progress has been made recently in understanding the role of 
partonic initial and final state interactions~\cite{Brodsky:2002cx,
Collins:2002kn,Ji:2002aa}.  The interaction between the active parton in
the hadron and the spectators was included in gauge-invariant TMD 
distributions~\cite{Brodsky:2002cx,Collins:2002kn,Ji:2002aa,Belitsky:2002sm,Boer:2003cm}.  
Furthermore, QCD factorization for 
semi-inclusive deep inelastic scattering at low transverse momentum in the 
current-fragmentation region has been established in Refs.~\cite{Ji:2004wu,
Collins:2004nx}.  This new framework provides a rigorous basis to study the 
TMD parton distributions from SIDIS data using different spin-dependent and 
independent observables.  
%TMD distributions (see Table~\ref{tab1}) describe 
%transitions of a nucleon with one polarization in the initial state to a 
%quark with another polarization in the final state.

%The diagonal elements of the table are the momentum, longitudinal and 
%transverse spin distributions of partons, and represent well-known parton
%distribution functions related to the square of the leading-twist, light-cone 
%wave functions. Off-diagonal elements require non-zero orbital angular 
%momentum and are related to the  overlap of light-cone wave functions 
%with $\Delta L \neq 0$ ~\cite{Ji:2002xn}.  The parton distributions 
%$f_{1T}^\perp$ and $h_1^\perp$ represent the imaginary parts of the corresponding
%interference terms, while the functions $g_{1T}$ and $h_{1L}^\perp$ represent their
%real parts.  The TMDs $f_{1T}^\perp$ (chiral-even) and  $h_{1}^\perp$ (chiral-odd)
%are known as the Sivers and Boer-Mulders 
%functions, respectively~\cite{Sivers:1990fh,Anselmino:1998yz,Brodsky:2002rv,
%Collins:2002kn,Ji:2002aa,Belitsky:2002sm}. They describe unpolarized quarks in the 
%transversely polarized nucleon and transversely polarized quarks in the 
%unpolarized nucleon respectively.  They vanish at tree-level in a $T$-reversal 
%invariant model ($T$-odd) and can only be non-zero when initial or final state 
%interactions cause an interference between different helicity states. These 
%%functions parametrize the correlation between the transverse momentum of 
%quarks and the spin of a transversely polarized target or the transverse spin 
%of the quark, respectively.  They require both orbital angular momentum, as 
%well as non-trivial phases from the final state interaction that survive in 
%the Bjorken limit.  
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{table}
%\begin{center}
%\begin{tabular}{|c|c|c|c|} \hline\hline
%N $\backslash$ q & U & L & T \\ \hline
%% {U} & ${\color{blue} \bf f_1}$   & & ${\color{red} h_{1}^\perp}$ \\
%\hline
% {L} & &${\color{green} \bf g_1}$ &    ${\color{red} h_{1L}^\perp}$ \\
%%\hline
% {T} & ${\color{blue} f_{1T}^\perp} $ &  ${\color{green} g_{1T}}$ &  ${\color{red} \bf h_1}$ \, ${\color{red} h_{1T}^\perp }$ \\
%\hline\hline
%\end{tabular}
%\end{center}
%%\caption{\small{{Leading-twist transverse momentum-dependent distribution 
%functions.  $U$, $L$, and $T$ stand for transitions of unpolarized, 
%longitudinally polarized, and transversely polarized nucleons (rows) to 
%corresponding quarks (columns), respectively.}}
%\label{tab1}} 
%\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%The off-diagonal TMD distributions arise from interference between amplitudes 
%with left- and right-handed polarization states, and only exist because of 
%chiral symmetry breaking in the nucleon wave function in QCD.  Their study 
%therefore provides a new avenue for probing the chiral nature of the partonic 
%structure of hadrons. 
%
%
%  The most simple mechanism that 
%can lead to a Boer-Mulders function, $h_{1}^{\perp}$, is a 
%correlation between the spin of the quarks and their transverse momentum. 
%The T-even counterpart of the
%Boer-Mulders function, $h_{1L}^{\perp}$, first introduced by Ralston and Soper \cite{Ralston:1979ys},
% describes the correlations of the transverse spin of quarks in
%the longitudinally polarized nucleon and their transverse momentum.

TMD parton distribution and fragmentation functions in DIS, Drell-Yan, and
electron-positron annihilation have different gauge links, which raised
a question of the  universality of those functions \cite{Collins:2004nx,Boer:2003cm,Bomhof:2007xt}.
It has been found that all six T-even TMD parton distributions are the same in
SIDIS and Drell-Yan.
The violation of universality for T-odd distributions appeared to be just
a sign reversal from DIS to Drell-Yan, an exciting 
prediction that has to be confirmed by future experiments. Universality of
TMDs in processes with hadrons in both initial and final states is violated, at
least in its standard
form~\cite{Collins:2007nk,Bomhof:2007xt,Vogelsang:2007jk}, with far-reaching
consequences in hadron collider physics.


Similar correlations arise in the hadronization process.  One particular case 
is the Collins $T$-odd fragmentation function $H_1^{\perp}$ 
\cite{Collins:1992kk}, describing fragmentation of transversely polarized 
quarks into unpolarized hadrons.  
%T-odd  fragmentation functions \cite{Collins:1992kk,Bacchetta:2002tk,Boer:2003cm} unlike the 
% T-odd parton distributions may be non-zero even if the rescattering 
%associated with their gauge link is neglected.
The Collins function is one of the most fundamental quantities accessible in hard 
fragmentation processes. It is of essential importance for spin physics because 
it works as an analyzer of the spin of the quark, but it is also interesting on 
its own because it allows the exploration of spin and orbital degrees of freedom 
of the QCD vacuum. It is universal \cite{Metz:2002iz,Collins:2004nx,Yuan:2008yv}, 
once measured in $e^+e^-$, it can be used in SIDIS and pp collisions and vice versa.
For kaons in particular, the $u$ to kaons Collins fragmentation function allows for 
exploring the structure of the strange vacuum, while the $s$ to kaons Collins function
 allows for study the spin structure of the strangeness in the nucleon.



In recent years, measurements of azimuthal moments of hadronic cross sections in 
hard processes have emerged  as a powerful tool to probe nucleon structure through 
transverse single spin asymmetries.
Many experiments worldwide are currently trying to pin down various 
TMD effects through semi-inclusive deep-inelastic scattering (in~experiments such as  
HERMES at DESY~\cite{Airapetian:1999tv,Airapetian:2001eg,Airapetian:2004tw,Airapetian:2006rx}, 
COMPASS 
at CERN~\cite{Alexakhin:2005iw}, CLAS and Hall-A at Jefferson 
Lab~\cite{Avakian:2003pk,Avakian:2005ps}), 
polarized proton-proton collisions (PHENIX, STAR and BRAHMS at RHIC)
~\cite{Adams:2003fx,Chiu:2007zy,Arsene:2008mi}, 
and electron-positron annihilation (BELLE at KEK)~\cite{Abe:2005zx, BELLE:2008HH}.
In contrast to inclusive deep inelastic lepton-nucleon scattering where 
transverse momentum is integrated out, these processes are sensitive to 
transverse momentum scales on the order of the intrinsic quark momentum
$P_T\sim k_\perp$.  

\subsection{Azimuthal Asymmetries in unpolarized SIDIS}

Within the one photon exchange
approximation, the cross section for unpolarized SIDIS
processes, $\ell \, p \to \ell \, h \, X$, can have a dependence on the
azimuthal angle $\phi$ of the final hadron already at leading order.
The cross section (see for ex.\cite{Ji:2004wu,Bacchetta:2006tn}) has contributions from
several structure functions:

\begin{align}
\lefteqn{\frac{d\sigma}{dx \, dy\, d\psi \,dz\, d\phi\, d P_{h\perp}^2}
=}
\nonumber \\ & \quad
\frac{\alpha^2}{x y\slim Q^2}\,
\frac{y^2}{2\,(1-\varepsilon)}\,  \biggl( 1+\frac{\gamma^2}{2x} \biggr)\,
\Biggl\{
F_{UU ,T}
+
\varepsilon\slim
F_{UU ,L}
+ \sqrt{2\,\varepsilon (1+\varepsilon)}\,\cos\phi\,
F_{UU}^{\cos\phi}
\nonumber \\  & \quad \qquad
+ \varepsilon \cos(2\phi)\,
F_{UU}^{\cos 2\phi}
+ \lambda_e\, \sqrt{2\,\varepsilon (1-\varepsilon)}\,
           \sin\phi\,
F_{LU}^{\sin\phi}
 \Biggr\},
\label{e:crossmaster}
\end{align}

\noindent
where $\alpha$ is the fine structure constant and $\varepsilon$ the
ratio of longitudinal and transverse photon flux.
The kinematic variables $x$ and $y$ are defined as: 
$x = Q^2/{2(P_1\cdot q)}$, and $y={(P_1\cdot q)/(P_1\cdot k_1)}$, respectively.
The variable $q=k_1-k_2$ is the momentum of the virtual photon, $Q^2=-q^2$, 
$\phi$ is the azimuthal angle between the scattering plane 
and the hadron production plane (see Fig. \ref{fig:anglestrento}). 
The subscripts of the structure functions %in $F_{UL}$, $F_{LL}$,etc.,  
specify the beam (first index) and target (second index) 
polarizations, for longitudinal ($L$), transverse ($T$),
and unpolarized ($U$) case.

In the limit of $P_{h\perp} << Q^2$, the
structure functions factorize into TMD parton distributions and
fragmentation functions.
%, and hard parts \cite{Ji:2004wu,Bacchetta:2006tn},
%i.e. ${F_{UU}} \propto  {f_{1}\left({x}, k_{\perp}\right) 
%D_1\left({z_h},p_{\perp}\right)H_{ UU}}$
%
%\begin{eqnarray}
%\sigma_{UU} \propto {F_{UU}} \propto & {f_{1}\left({x}, k_{\perp}\right) D_1\left({z_h},p_{\perp}\right)H_{UU}\left(Q^2\right)} \nonumber\\
%\sigma_{LL} \propto {F_{LL}} \propto & {g_{1L}\left({x}, k_{\perp}\right) D_1 \left({z_h},p_{\perp}\right)H_{LL}\left(Q^2\right)}  \nonumber\\
%\sigma_{UL} \propto {F_{UL}} \propto & {h_{1L}^\perp\left({x}, k_{\perp}\right) H_1^\perp\left({z_h},p_{\perp}\right)H_{UL}\left(Q^2\right)}, \ 
%\label{eq:JiMaYu2}
%\end{eqnarray}
%
%\noindent

%where $z={(P_1P_h)/(P_1q)}$, $k_{\perp}$ and $p_{\perp}$ are quark transverse 
%momenta before and after scattering
%and $P_1$  and $P_h$ are the four momenta of the initial nucleon and the 
%observed final-state hadron, respectively. 
%The different hard factors, which are calculable in pQCD, are similar at one-loop 
%order \cite{Ji:2004wu} in SIDIS and may cancel to a large extent in asymmetry
%observables.
%
%In the case of the polarized beam and unpolarized or longitudinally polarized target 
%the cross section in the leading order has only two contributions that depend on 
%the azimuthal angle of the final state hadron, appearing as $\cos2\phi$ and  $\sin2\phi$ modulations,
%involving four leading twist TMD distribution functions \cite{Mulders:1995dh,Boer:1997nt,Kotzinian:1994dv}. 


\begin{figure}[h]
\centerline{\epsfxsize=10.0cm\epsfbox{plots/anglestrento.eps}}   
\caption{\small 
SIDIS kinematics. For an unpolarized target, only the angle $\phi$ enters the cross-section.}
 \label{fig:anglestrento}
\end{figure}


%During the last few years, first results on transverse 
%SSAs have become available~\cite{Airapetian:2004tw,Alexakhin:2005iw}. 
%Pioneering measurements by the HERMES Collaboration 
%for the first time directly indicated significant azimuthal 
%moments generated both by Collins $(F_{UT}^{\sin(\phi+\phi_S)})$ and Sivers ($F_{UT}^{\sin(\phi-\phi_S)}$) effects.
%Significant differences were observed between charged pion and kaon, both for Sivers and
%Collins effects (see Fig.\ref{fig:hermeskaons}). Currently no satisfactory explanation exists
%for these observed differences.
%
%\begin{figure}[h]
%\begin{minipage}[b]{6.2cm}
%\hspace{15mm}
%\epsfig{file=plots/ssa-0204-sivers-comparion.eps,width=6.2cm,height=6.3cm}
%\end{minipage}
%     \ \hspace{15mm} \hspace{0mm} \
%\begin{minipage}[b]{6.2cm}
%\epsfig{file=plots/ssa-0204-collins-vpa-comparison.eps,width=6.2cm,height=6.3cm}
%\end{minipage}
%\caption{ \small First measurements of Collins (right)  and Sivers(left) asymmetries for pions and kaons.}
%\label{fig:hermeskaons}
%\end{figure} 


%\subsubsection{Azimuthal Asymmetries in unpolarized SIDIS}

%Spin-orbit correlations are accessible in SIDIS with
%unpolarized target in measurements of azimuthal distributions of hadrons as well as 
%cross section measurements as a function of their transverse momentum. 



%In the collinear limit of QCD, only integrated over transverse momentum PDFs 
%integrating over the transverse momentum, integrated 
%(i.e. $f_1^a(x)=\int\di^2{\bf k}_Tf_1(x,{\bf k}_T)$) 
%and FFs (i.e. $D_1^a(z)=\int\di^2{\bf p}_TD_1(z,{\bf p}_T)$) remain,
%so the azimuthal dependences due to correlations of
%transverse momentum of partons and their spin vanish.

%\begin{figure}[t!]
%        \includegraphics[height=5cm]{plots/fig04c-intrinsic-pT.eps}
%\caption{\label{zz-Fig01:pT-in-unpol-SIDIS}
%Cahn effect from intrinsic transverse parton momenta \cite{Cahn:1978se}}
%\label{peter-cahn}
%\end{figure}
%

For an unpolarized target the only azimuthal asymmetry
arising in leading order is the $\cos2\phi$ moment~\cite{Boer:1997nt},

\begin{equation}
\sigma^{\cos2\phi}_{UU} \propto 2(1-y)\cos 2\phi\sum_{q,\bar{q}} e^2_q {x h^{\perp q}_{1}(x)}\otimes {H_1^{\perp q}(z)}.
\end{equation}
where $z={(P_1\cdot P_h)/(P_1\cdot q)}$, $k_{\perp}$ and $p_{\perp}$ are quark transverse 
momenta before and after scattering
and $P_1$  and $P_h$ are the four momenta of the initial nucleon and the 
observed final-state hadron, respectively. 
The Boer-Mulders distribution function, $h_{1}^\perp$, giving rise to the asymmetry,  is related to the imaginary 
part of the interference of wave functions for different orbital momentum states,
and describes transversely polarized quarks in the unpolarized 
nucleon. 
%The physics of $\sigma_{UU}$, which involves the Collins fragmentation 
%function $H_1^\perp$ and the Boer-Mulders distribution function $h_{1}^\perp$, 
%was first discussed by Boer and Mulders in 1997 \cite{Boer:1997nt}.  
The same distribution function is accessible in the unpolarized Drell-Yan, where 
it gives rise to the $\cos2\phi$ azimuthal moment in the cross section 
\cite{Tangerman:1994eh} and is a candidate to explain the violation of the Lam-Tung 
relation~\cite{Boer:2002ju}.
The study of this modulation is the target of many
proposed Drell-Yan experiments, i.e. the planned E906 experiment at Fermilab~\cite{Reimer:2011}. 
The behavior of the Boer-Mulders distribution function was recently studied
in large-$x$ \cite{Brodsky:2006hj}, large $N_c$ \cite{Pobylitsa:2003ty}, and large $P_T$ \cite{Bacchetta:2008xw} limits of QCD.
There are Boer-Mulders asymmetry predictions for various 
models \cite{Barone:2005kt,Gamberg:2007wm,Barone:2008tn,Zhang:2008ez},
for all DIS and DY experiments.

%Within the leading order (zero-th order in $\alpha_s$) parton model with 
%twist-two distribution and fragmentation functions (FFs), the simplest 
At subleading order, a $\cos\phi$ azimuthal asymmetry arises in unpolarized 
SIDIS of pure kinematic origin (the so-called Cahn effect~ \cite{Cahn:1978se}). 
It accounts for the partons intrinsic
transverse momenta in the target (${\bf k}_T$)  and the fact that produced hadrons 
may acquire transverse  momenta from the fragmentation process (${\bf p}_T$), so that
final hadron transverse momentum ${\bf P}_{h\perp}$ is defined as their sum 
(${\bf P}_{h\perp}=z{\bf k}_T+{\bf p}_T$).
Within the same approach, kinematical corrections proportional to 
$(k_\perp/Q)^2$ lead to additional contributions 
in the  $\cos 2\phi_h$ moment. 


Data on azimuthal distribution in SIDIS are 
available from 
EMC experiments at CERN \cite{Aubert:1983cz,Arneodo:1986cf},
the E665 Fermilab experiment \cite{Adams:1993hs},
and the ZEUS experiment at DESY \cite{Breitweg:2000qh,Chekanov:2006gt}.
The data from ZEUS with average $Q^2$ value, $\langle Q^2 \rangle \simeq 750$ GeV$^2$, 
is dominated by the perturbative contribution. 
In order to highlight the effect of the non perturbative contributions
to the $\cos2\phi$ moment (Boer--Mulders and higher twist), one has to probe the 
kinematical region corresponding to $P_T < 1$ GeV 
and $Q^2$ of order of few GeV$^2$,   
such as  HERMES, COMPASS, and JLab facilities, where 
the gluon emission is quite irrelevant. 
So far the azimuthal moments were measured only for pions and charged hadrons.
Data for positive pions is available from CLAS \cite{Osipenko:2008rv}
and in limited range of transverse momenta from Hall-C \cite{Mkrtchyan:2007sr}. Both 
COMPASS\cite{Kafer:2008ud} and HERMES \cite{Giordano:2009hi} presented
preliminary measurements of the $\cos2\phi$ moments for positive and negative hadrons for hydrogen and deuterium targets,
indicating that the Boer-Mulders effect may be more significant than anticipated.
% (see Fig.~\ref{fig:hermescos2}).  

%\begin{figure}[h]
%\vspace{-2.0cm}
%\centerline{\epsfxsize=12.0cm\epsfbox{plots/auuhermes-compass.eps}}   
%\caption{\small Preliminary measurements of the $\cos2\phi$ moment from HERMES \cite{Giordano:2008} and 
%COMPASS\cite{Kafer:2008ud} for positive and negative hadrons. Filled squares show the CLAS measurement for $\pi^+$ at $5.7$ GeV \cite{Osipenko:2008rv}. }
% \label{fig:hermescos2}
%\end{figure}

The existing data on pions, however, are affected by large uncertainties 
and do not allow for drawing definite conclusions about the 
magnitude and the shape of the asymmetry and in particular, to the
different contributing mechanisms. The CLAS12 approved experiment 
E12-06-112 is designed to adress this issues using a large-statistics 
pion sample.

The extraction of the Boer-Mulders 
transverse momentum dependent distributions is complicated by the presence of 
an essentially unknown Collins 
function.  
Recently a significant asymmetry was measured by BELLE~\cite{Abe:2005zx, BELLE:2008HH} 
in $e^{+}e^{-}$ annihilation
to pions, indicating that the Collins function is indeed large. 
Based on leading order, a procedure has been developed  
recently~\cite{Anselmino:2007fs, Anselmino:2008jk} to extract the transversity 
distribution and Collins fragmentation functions, combining 
$e^+e^-$ and semi-inclusive DIS data~\cite{Airapetian:2004tw}.  

\section{Scientific Case and Recent Developements in the Kaon Sector}

A comprehensive study of the nucleon structure should consider the
role of the quark flavor. The use of different targets in conjunction 
with the detection of various hadrons in the final state provide access 
to statistical information about the flavor of the struck quark. 
In particular, kaons provide enhanced sensitivity on strangeness 
in the matter (partonic sea of the nucleon) and in the vacuum 
(through fragmentation). Kaon detection is generally challenged 
by the about one order of magnitude larger flux of pions.
Thus very little is known about the spin-orbit correlations related to the strange quark.
Only recently dedicated measurements have become available and,
despite the limited statistical accuracy, in most of the cases 
they show surprising results. 

In high-energy hadron-hadron collisions, large single-spin asymmetries
have been measured since the eigthies at large rapidities (positive Feynman-x) 
which are opposite in sign for opposite charge pions, see Fig.~\ref{Brahms}. 
This is not the case for charged kaons, whose SSA are recently measured to be 
non zero and of the same sign up 
to 200 GeV in center-of-mass energy~\cite{Arsene:2008mi,Lee:2009az}. 
In SIDIS, precise hadron identification has been exploited only by the 
second generation experiments. The Sivers effect, proposed as a possible 
explanation of the SSA in hadron-hadron collisions, generates SSA in SIDIS 
reactions for positive kaons which are found larger than for positive 
pions, see Fig.~\ref{SIDIS_Sivers}. The difference concentrate at low-Q, a 
possible indication of the presence of higher-twist effects in the kaon sector~\cite{Airapetian:2009ti}.
The results for negative kaons are not conclusive due to the 
limited statistics~\cite{COMPASS:2011di}.
% but there is a hint their asymmetries are of the
%same sign of positive kaons, whereas the asimmetries for negative
%pions are compatible with zero~\cite{Airapetian:2009ti}.

\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[height=.21\textheight,width=0.63\textwidth]{plots/BRAHMS_pion_200GeV.eps}
&
\includegraphics[height=.22\textheight,width=0.33\textwidth]{plots/BRAHMS_kaon_200GeV.eps}
\end{tabular}
\caption{Single spin asymmetries for identified hadrons in proton-proton reactions at 200 GeV
center-of-mass energy~\cite{Lee:2009az}.} 
\label{Brahms}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\begin{tabular}{ccc}
\includegraphics[height=.22\textheight,width=0.28\textwidth]{plots/HERMES_Sivers_pion_splitQ.eps}
&
\includegraphics[height=.22\textheight,width=0.26\textwidth]{plots/HERMES_Sivers_kaon_splitQ.eps}
&
\includegraphics[height=.23\textheight,width=0.42\textwidth]{plots/COMPASS_Sivers.eps}
\end{tabular}
\caption{Sivers asymmetries measured for positive pions and kaons in two $Q^2$ ranges at 
HERMES~\cite{Airapetian:2009ti} (Left) and measured for identified charged hadrons at 
COMPASS~\cite{COMPASS:2011di} (Right).}
\label{SIDIS_Sivers}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[height=.24\textheight,width=0.44\textwidth]{plots/HERMES_Collins_kaons.eps}
&
\includegraphics[height=.24\textheight,width=0.52\textwidth]{plots/COMPASS_Collins.eps}
\end{tabular}
\caption{Collins asymmetries measured for identified charged hadrons at HERMES~\cite{Airapetian:2010ds} 
(Left) and COMPASS~\cite{COMPASS:2011di} (Right).}
\label{SIDIS_Collins}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[height=.25\textheight,width=0.48\textwidth]{plots/H_hp_pp_kp.eps}
&
\includegraphics[height=.25\textheight,width=0.48\textwidth]{plots/H_had-pion-kaon-.eps}
\end{tabular}
\caption{Azimuthal asymmetries in unpolarized SIDIS for identified hadrons~\cite{Giordano:2010gq}.}
\label{cos2kaon}
\end{center}
\end{figure}

The Collins asymmetry has been measured to be non-zero and opposite in sign for 
opposite charge pions, see Fig.~\ref{SIDIS_Collins}. This
is an indication that favored and un-favored Collins FF are opposite
but of similar magnitude, a result compatible with the fragmentation-function measurement
at $e^+e^-$ machines~\cite{Abe:2005zx, BELLE:2008HH}. 
The SSA for positive kaons is similar to that
of positive pions in sign and magnitude, a result compatibe with the
dominace of the $u$-flavor in lepto-scattering over a proton target. 
However at HERMES, the signal for $\pi^-$ and $K^-$ are found to follow a 
different behavior, the former being large and negative, the latter 
being basically compatible with zero with a hint to be positive~\cite{Airapetian:2010ds}. 
The result would be interesting since the $K^-$ has no valence quarks in common with 
the target proton and sea quark transversity is expected to be small,
thus $K^-$ brings specific sensitivity on rank-2 Collins FF. Note that
the knowledge of the Collins function has an impact on the extraction of 
all the chirally-odd TMD parton distributions.
The result for $K^-$ is still controversial, since COMPASS data~\cite{COMPASS:2011di} can not 
prove or disprove it, although seem to not support the same finding, see 
Fig.~\ref{SIDIS_Collins}. The issue can be solved only by novel 
high-precision experiments.

Azimuthal asymmetries in unpolarized SIDIS cross-section are generated 
by the Cahn and Boer-Mulders effects, but may get not-negligible contributions from
quark-gluon-quark correlators (pure sub-leading twist terms). 
%The Boer-Mulders 
%effect is a candidate to explain the violation of the Lam-Tung relation 
%measured in Drell-Yan experiments~\cite{Boer:2002ju}. In generates 
%azimuthal asymmetries in SIDIS reactions in conjunction with the Collins
%effect. 
Only recently preliminary SIDIS results 
become available with hadron identification or charge separation. 
Opposite-charge pion signals show unexpected large 
differences which can be related to the peculiar flavor dependence of 
the Collins FF entering the Boer-Mulders term~\cite{Giordano:2009hi}. 
These reflect in analogous differences for opposite-charge unidentified
hadrons, dominated by the pion subsample~\cite{Collaboration:2010fi}.
The kaon results are definetely surprising, showing $\rm cos2\phi$ 
asymmetries much larger than pions but of the same sign for both 
charges~\cite{Giordano:2010gq}, see Fig.~\ref{cos2kaon}. These asymmetries
arise at leading-twist due to a convolution of Boer-Mulders with
Collins functions, thus bring important piece of information on
these two mechanisms. The sub-leading twist terms may also play a 
non-negligible role. The $\rm cos \phi$ asymmetry gets contributions 
only at sub-leading twist and can be used to constrain the related
terms. It shows a $K^+$ signal larger than $\pi^+$ but a $K^-$ signal 
compatible with $\pi^-$.
The kaon signals are a challenge for the 
present understanding of the underlying physics processes. Any
explanations is not possible without a disentanglement of the
different contributions, possible only with high-precision mapping
of the kinematical dependences.

The surprising and controversial pattern of TMDs results for kaons 
is an indication of a non trivial role of the sea quarks in the nucleon, 
or of a peculiar behaviour of the fragmentation mechanism in the presence 
of strange quark. Moreover
a hint exists that kaons provide enhanced sensitivity on higher-twist
effects~\cite{Airapetian:2009ti}.
The interpretation would become possible only in the presence of
high-luminosity large-acceptance experiments able to explore the
relevant kinematic dependences in conjunction with an efficient 
hadron identification. Examples of wanted informations are a
precise Q-dependence to isolate higher-twist effect and an extended
$P_{h\perp}$-dependence to map the transient between perturbative and 
non-perturbative regime. 

Nothing is known about polarization dependent FF (i.e. Collins) into 
kaons but an effort is being pursued to extract them from the
large sample of data collected at B-factories~\cite{BELLE:2008HH, Babar:2010HH}. Very precise 
information on the fragmentation process is anticipated in the next
future thanks to the approval of Super B-factories. The detailed 
study can be completed only by SIDIS measurements which constrain 
FF at much lower center-of-mass energy with specific flavor sensitivity
(not accessible in $e^+e^-$ reactions). The detailed knowledge of 
the fragmentation process would reflect on the precise determination 
of parton distributions only at experiments with enough statistical 
precision and flavor sensitivity, like CLAS12 with a RICH detector
dedicated to hadron identification.

The goal of our proposed experiment is to gather a data
set on kaon  SIDIS in the region $0.1 \le x \le 0.8$, $0\le P_T \le 1.2$, and
$0.2 \le z \le 0.8$. Global analysis of the data combined with the 
analogous measurement approved with pions and polarized target data
\cite{PACall} will provide access to the flavor-decomposition of TMDs.


\subsection{The flavor-dependence of Boer-Mulders and Collins functions}

So far no experimental information  is available about the
Collins fragmentation function for kaons. 
Recent direct calculations of kaon Collins function  \cite{Bacchetta:2007wc}
 indicated that it may be comparable with pion Collins function. %(see Fig.~\ref{colluK}). 
HERMES measurements of the Collins asymmetry for pions and kaons, 
though, with large uncertainties, indicate that the differences may be significant~\cite{Airapetian:2010ds}.


%\begin{figure}
%\begin{tabular}{cc}
%\includegraphics[height=.25\textheight,width=0.45\textwidth]{plots/ratio_uK+.eps}
%&
%\includegraphics[height=.25\textheight,width=0.45\textwidth]{plots/ratio_upi+.eps}
%\end{tabular}
%\caption{$H_1^{\perp (1/2)}/D_1$ for 
%$u \to K^+$ (left) and $\pi^+$ (right) in the model of Bacchetta et al.\cite{Bacchetta:2007wc}, 
%at the model scale (solid line) and at two other scales (dashed and dot-dashed
%lines)  under the assumption that the Collins function does not evolve with the scale.}
%\label{colluK}
%\end{figure}


Pions and kaons are both Goldstone bosons of chiral symmetry breaking. 
In the chiral limit one has
\be\label{App-Eq:H1perp-K-chiral-limit}
	\lim\limits_{m_K  \to 0}\frac{H_1^{\perp(1/2)a/  K}}{D_1^{a/  K}}=
	\lim\limits_{m_\pi\to 0}\frac{H_1^{\perp(1/2)a/\pi}}{D_1^{a/\pi}}\;,
\ee
where $H_{1}^{\perp (1/2)}$ means integration over the transverse momentum weighted with $k_T$.
%\noindent Predictions for kaon Collins asymmetries  assume the following relations to hold approximately
%\ba
%	\frac{H_1^{\perp(1/2)\bar s/K^+}}{D_1^{\bar s/K^+}}\approx
%	\frac{H_1^{\perp(1/2)     u/K^+}}{D_1^{     u/K^+}}\approx 
%	\frac{H_1^{\perp(1/2)   u/\pi^+}}{D_1^{   u/\pi^+}}\,,\nonumber\\
%	\phantom{a}\nonumber\\
%	\frac{H_1^{\perp(1/2){\rm unf}/  K^+}}{D_1^{{\rm unf}/  K^+}}\approx 
%	\frac{H_1^{\perp(1/2){\rm unf}/\pi^+}}{D_1^{{\rm unf}/\pi^+}}
%	\;,\label{App-Eq:H1perp-K}
%\ea
%where it is understood that the fragmentation of $d$- and $\bar u$-flavour 
%into $K^+$ is unfavoured. 

Simultaneous measurements of the Boer-Mulders asymmetry for pions and kaons on proton
and deuteron targets will provide an independent measurement of ratios of Collins 
functions of pions and kaons, providing complementary measurements
to $e^+e^-$ annihilation. Thanks to the different center-of-mass energy of fixed-target
with respect collider experiments,
the measurements can be used to test the evolution properties of the Collins function.
That measurement will also provide a check of chiral limit prediction, where
that ratio is expected to be at unity.
With the knowledge of the Collins function, one can study all involved TMD distributions.

%Measurements of transverse momenta of final state hadrons in SIDIS with 
%unpolarized targets will thus provide 
%information on the polarized Collins fragmentation of kaons complementary to transverse 
%target and future measurements in $e^+e^-$ by BELLE.

Measurement of the fraction coming from the Collins fragmentation of transversely
polarized quarks in the unpolarized nucleon will require separation of other
contributions, and in particular those due to the Cahn effect.
 Although, preliminary results from HERMES indicate
that the Cahn contribution is not dominant in $\cos2\phi$ as was expected before 
\cite{Anselmino:2005nn}.  In order to extract the contribution 
related to the Collins fragmentation one needs a reliable calculation of the 
kinematical corrections. Perturbative QCD contributions (at order $\alpha_s$ 
and possibly $\alpha_s^2$) to the kinematical $\cos\phi_h$ and $\cos 2\phi_h$ 
asymmetries also have to be evaluated. Such a study shows that the parton model
with TMD DFs and FFs dominates at $P_{T}$ values below 1 
(GeV/$c$)~\cite{Anselmino:2005nn}.


\subsection{Transverse momentum dependence of partonic distributions}

The width for different partonic distributions can be different. For example 
values for different T-even partonic distributions, computed~\cite{Boffi:2009sh} in the constituent
quark model~\cite{Pasquini:2008ax}, are listed in Table~\ref{Table:pT-model}.
Values normalized to the width of the unpolarized distribution function are listed
in the same Table.

\begin{table}
\begin{tabular}{clll|c}
\hline
  \ \hspace{3cm} \ 
& \ \hspace{2cm} \ 
& \ \hspace{2cm} \ 
& \ \hspace{2cm} \  
& \ \hspace{3cm} \cr
TMD & $\begin{array}{l}\la k_T  \ra \cr {\rm in \; GeV}   \end{array}$
    & $\begin{array}{l}\la k_T^2\ra \cr {\rm in \; GeV}^2 \end{array}$
    & $\displaystyle\frac{4\la k_T\ra^2}{\pi\la k_T^2\ra}$ 
    & $\displaystyle\frac{\la k_T^2\ra}{\la k_{Tf_1}^2\ra}$ \cr
& & & &\cr
\hline
$f_1$          & 0.239 & 0.080 & 0.909 & 1.00 \cr
$g_1$          & 0.206 & 0.059 & 0.916 & 0.74 \cr
$h_1$          & 0.210 & 0.063 & 0.891 & 0.79 \cr
$g_{1T}^\perp$ & 0.373 & 0.176 & 1.007 & 2.20 \cr
$h_{1L}^\perp$ & 0.373 & 0.176 & 1.007 & 2.20 \cr
$h_{1T}^\perp$ & 0.190 & 0.050 & 0.919 & 0.63 \cr
& & & & \cr
\end{tabular}
\caption{\label{Table:pT-model} Predictions for the transverse momentum
dependence of the T-even TMDs from the constituent quark model~\cite{Pasquini:2008ax}.
(Left) The mean transverse momenta and the mean square transverse 
momenta: if the transverse momenta in the TMDs were Gaussian, then the
result for the ratio in the last row would be unity. (Right)
Mean square transverse momenta of T-even TMDs 
in units of the mean square transverse momenta of
$f_1$, denoted as $\la k_{Tf_1}^2\ra$. These ratios are considered
to be a more robust model prediction.}
\end{table}


A common assumption is the Gaussian ansatz for the transverse momentum dependence of 
distribution and fragmentation functions 
%
%\begin{eqnarray} 
%D_1(z,{\bf p}_T) &=& D_1^a(z)\,\exp(-{\bf p}_T^2/\la p_{\!D_1}^2\ra)/\pi\la p_{\!D_1}^2\ra \\
%f^a_1(x,{\bf k}_T)&=& f^a_1(x)\,\exp(-{\bf k}_T^2/\la {k}_T^2\ra)/\pi\la { k}_T^2\ra
%\end{eqnarray}
with the average $P_{h\perp}$ of hadrons produced in SIDIS given by
\ba\label{Eq:Phperp-av}
        \la P_{h\perp}(z)\ra &=& \frac{\sqrt{\pi}}{2}\,
        \sqrt{z^2\la k_T^2\ra+\la p^2_T\ra}\;.
\ea
In the approximation  of flavor and $x$ or $z$-independent widths,
a satisfactory description of HERMES deuteron data on average 
$P_{h\perp}$~\cite{Airapetian:2002mf} was obtained~\cite{Collins:2005ie} with
\be\label{Eq:fit-pT2-KT2}
        \la k_T^2\ra = 0.33\,{\rm GeV}^2 \;,\;\;\;
        \la p^2_T \ra = 0.16\,{\rm GeV}^2 \;.\ee
Numerically very similar results were obtained in Ref.~\cite{Anselmino:2005nn} 
from a study of EMC data~\cite{Arneodo:1986cf} on the Cahn effect. Although the
ansatz seems to describe the present available data sets~\cite{Schweitzer:2010tt},
it has to be consider an approximation which is not necessary valid and
most of the models do not base on it.

The unpolarized azimuthal moment in kaon lepto-production will be studied  
as a function of $P_{h\perp}$, in different bins in $x,z$, and $Q^2$, and
complemented with the analogous measurements for pions. 
These measurements will provide access to widths in transverse momentum of 
different underlying partonic distributions, like the number density $f_1$ and the
Boer-Mulders $h_1$, and to their flavor dependence. %of transverse momentum distributions.

%Detailed studies of the $P_{h\perp}$-dependences of the unpolarized SIDIS 
%cross-section, joining the information of azimuthal dependencies and 
This measurements, in conjunction with the study of hadron 
multiplicities in a large range of kinematic variables ($x_B$, $Q^2$, $z$, $P_{h\perp}$), 
will provide important input for the flavor decomposition of the transverse
momentum dependence of all the others TMDs from polarized measurements.

\subsection{Vector meson contribution}

One of the main sources of uncertainties for pions is the fraction of pions coming from vector meson decays.
Since the Collins asymmetry has a significant dependence on the type of a produced hadron, pions
produced from rho decays will have very different moments compared to direct pions. This makes
measurements with kaons, which have a much smaller contribution from vector meson production,
very import in understanding the underlying dynamics of the part
of measured asymmetries due to
Collins fragmentation mechanism

\section{Technical Progress Towards Realizing the Experiment}

\section{The Beam request and Expected Results}

We propose a measurement of azimuthal moments of the single kaon cross section
in SIDIS using the  CLAS12 detector in Hall B at Jefferson Lab, a 6.6-11.0 GeV 
longitudinally polarized 
electron beam and  unpolarized hydrogen and deuterium targets. 
The focus of our proposal is to study the
$ \cos 2 \phi$ asymmetry, related to the correlation of intrinsic
transverse momentum  of quarks and their transverse spin. 
Competing mechanisms are also related to the transverse motion of quarks
and are also relevant
in the CLAS12 kinematic regime ($\langle Q^2 \rangle
\sim 2$ GeV$^2$).

The overall statistics of kaons is almost an order of magnitude less than for pions with
most of the relevant sources of systematic errors being the same, so the main focus will
be the new information we can access with kaon SIDIS measurements and in particular
studies of the Collins fragmentation of kaons.

The JLab 12-GeV upgrade will provide the unique combination of wide kinematic 
coverage, high beam intensity (luminosity), high energy, high polarization,
and advanced detection capabilities necessary to study the transverse momentum 
and spin correlations in  semi-inclusive processes both in the 
target and current fragmentation regions for a variety of hadron species.
In this proposal 
we focus on observables related to kaon production in DIS, accessible with
unpolarized  targets and new information on the structure 
of nucleon they can provide.
Measurements of spin-orbital structure of hadrons with unpolarized hydrogen and deuterium 
targets will use the beam time of an already approved electroproduction experiment to study the 
Neutron Magnetic Form-Factor at High $Q^2$ \cite{CLAS12Gilf}.
 
\subsection{Statistical and systematic errors}
%\input{kaonsoc-errors.tex}

The proposed spin asymmetry measurement is rather insensitive 
to  uncertainties in acceptances and charge normalization.
The overall statistics of kaons is, though, an order of magnitude less than for 
pions, with most of the relevant sources of systematic errors being the same.
One of the main systematic errors affecting the extraction of the Collins moment 
is due to possible contamination of the  single kaon sample 
with kaons from decays of exclusive $K^*$ mesons. The fraction of indirect
kaons, however, according to LUND studies, is significantly less than 
for pions (see Fig.\ref{fig:indir}).
The main difference from the pion
production case for the background is the  contribution from  
$K^*$ production (e.g., $K^* \rightarrow K\pi$) and the radiative tail
on exclusive Kaon production.  The contributions to the systematic error from these
backgrounds requires a detailed analysis once the requisite data are in hand,
but experience with pion data from CLAS at 6 GeV show that one can avoid 
most of them by judicious choice of kinematic cuts.


\begin{figure} [!htb]
\begin{center}
\includegraphics[scale=0.7]{plots/kappipsource11gev1.eps}
\caption{Fractions of direct (with a parent defined as "string") kaons and pions from PEPSI MC for $ehX$ events.}
\label{fig:indir}
\end{center}
\end{figure}


Other sources of systematic errors include the longitudinal to 
transverse photo absorption cross section ratio, $R(x,Q^2)$ and the beam polarization (for $\sin\phi$). 
The main sources of systematic errors in measurements of 
azimuthal asymmetries are listed in the   Table~\ref{tab:errors}.
These errors are all scale errors, so they are proportional to the size
of the measured asymmetry. 

\begin{table}
\begin{center}
\caption{\small  Uncertainties for asymmetry measurements.}
\label{tab:errors}
\begin{tabular}{c|c|c|c} \hline\hline
Item & $A_{UU}^{\cos2\phi}$ & $A_{UU}^{\cos\phi}$ & $A_{LU}^{\sin\phi}$ \\ \hline
beam polarization & - & - & 3\% \\ \hline
acceptance corrections & 4\% & 4\% & 2\% \\ \hline
radiative corrections & 3\% & 3\% & 3\% \\ \hline
fitting procedure & 4\%   & 4\% & 3\% \\
\hline
\end{tabular}
\end{center}
\end{table}

Studies of other sources of systematics, related to physics background, 
including target fragmentation, semi-exclusive processes, exclusive vector 
meson contributions, and higher twist
require the data of this measurement.


We based our predicted statistical errors in the following sections on the assumption of
running 54 days on a hydrogen and deuterium \cite{CLAS12Gilf}. 
For our estimate of the total systematic error,
we have added the systematic errors from the various
contributions discussed in the previous Section in quadrature. 
They are listed in Table~\ref{tab:errors}.
 
%\subsubsection{Exclusive production of K$^*$ and GPDs}
%While understanding of K$^*$--production is important for extracting 
%the kaon moments studied in the present proposal, it is also interesting 
%in its own right, as the exclusive $K^*\Lambda$ channel provides 
%access to the Generalized Parton Distributions (GPDs) associated with the
%$p \rightarrow \Lambda$ transition. A QCD factorization theorem 
%\cite{Collins:1997hv} states that in the limit $Q^2 \rightarrow \infty$
%the meson is predominantly produced in configurations of small 
%transverse size $\sim 1/Q$, and the longitudinal cross section
%can be expressed in terms of the GPDs and the distribution amplitude 
%of the produced meson. In the $Q^2$ range covered by JLab one expects 
%significant finite--size (``higher--twist'') corrections to the 
%asymptotic production mechanism, making it difficult to interpret
%absolute cross sections within the GPD formalism \cite{Goloskokov:2007nt}. 
%The focus here is on the analysis of cross section ratios, which are less
%sensitive to finite--size effects and can be compared with GPD--based
%expressions already at $Q^2 \sim \textrm{few GeV}^2$ \cite{Strikman:2008pi}. 

%Figure~\ref{fig:ratios} shows a simple prediction for the ratio of 
%exclusive $K^\ast$ and 
%$\rho^+$ production, based on $SU(3)$ flavor symmetry and approximation
%of the GPDs by the ``usual'' parton densities; estimates based on 
%more sophisticated GPD models were presented in Ref.~\cite{Diehl:2005gn}. 
%Analysis of the $Q^2$--dependence of the measured ratio will serve
%as a test of the expected early scaling. The observed limiting 
%($Q^2$--independent) value can then be compared with the theoretical 
%predictions and  provide an interesting test of $SU(3)$ flavor 
%symmetry and present GPD models. The $K^\ast / \rho^+$ ratio addressed 
%here is particularly useful for this purpose because (a) both channels 
%involve non-singlet quark GPDs, eliminating the uncertainty associated 
%with modeling gluon and singlet quark exchange in $\rho^0$ and $\phi$ 
%production; (b) the final--state masses of the produced meson/baryon
%systems are similar, reducing $SU(3)$ breaking effects resulting
%from kinematic corrections in the cross section ratio; (c) the
%longitudinal cross section (for which the QCD factorization theorem
%applies) can be extracted by analyzing the decay of the vector meson
%and relying on $s$--channel helicity conservation, eliminating the
%need for Rosenbluth separation as is necessary in pseudoscalar
%($\pi, K$) meson production.

%It is important to note that detection of exclusive $K^*$s, unlike the case with
%exclusive kaons, where the missing mass technique could be used,
%will require kaon identification over the full kinematic range, which could be
%provided only by the RICH detector.
%
%\begin{figure}[htb]
%\begin{center}
%\mbox{ \psfig{file=plots/kstarrat.eps,height=8cm,width=10cm} }
%\end{center}
%\caption{ Theoretical estimate of the longitudinal cross section
%ratio for exclusive $K^\ast \Lambda$ and $\rho^+ n$ production,
%$\sigma (\gamma^\ast_L p\rightarrow K^*\Lambda)/\sigma (\gamma^\ast_L 
%p\rightarrow \rho^+n)$, based on $SU(3)$ flavor symmetry 
%for the GPDs and vector meson distribution amplitudes, and 
%approximation of the GPDs by the ``usual'' quark densities.
%The results for the ratio in this approximation are close to 
%those obtained with more sophisticated GPD models \cite{Diehl:2005gn}.}
%\label{fig:ratios}
%\end{figure}



\subsection{Results}
%\input{kaonsoc-results.tex}



The proposed experiment will simultaneously
collect data on 
${p},{d}(\vec{e},e' h)$, including kaons and pions. 
The charged kaons will be detected
in the forward spectrometer and the central tracker of CLAS12 in
coincidence with the scattered electrons and identified by the CLAS12-RICH detector. 
The following predicted results were obtained with a full simulation of the
hadronization process~\cite{Mankiewicz:1991dp} and the acceptance of CLAS12 for all particles.

The  expected counts of kaons in different kinematic bins 
for the hydrogen and deuteron targets for each of
the kaon charges were calculated using the PEPSI-MC kaon to pion ratios (Fig.\ref{fig:kapiratios}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\vspace{-2.0cm}
\begin{tabular}{cc}
\includegraphics[height=.4\textheight,width=0.45\textwidth]{plots/kaonpipratclas12.eps}
&
\includegraphics[height=.4\textheight,width=0.45\textwidth]{plots/kaonpipratclas12pt.eps}
\\
(a) & (b)
\end{tabular}
\caption{\small{ PEPSI-Lepto predictions for  $x$-dependence (left) and $P_T$-dependence 
of kaon pion ratios.}}
\label{fig:kapiratios}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


 
\subsubsection{Projected results for unpolarized moments}

The precision measurement of the Boer-Mulders asymmetry requires subtraction
of all  contributions to the $\cos2\phi$ moment discussed above. All contributions,
including the Cahn and Berger terms as well as the perturbative and radiative
contributions to first order, are expected to be ``flavor blind'', i.e. are the same 
for negative and positive kaons. The check performed with the $\cos\phi$ moment
can serve as a test for that. Contributions to $\cos\phi$ moment are also related
to contributions to $\cos2\phi$ and their extraction will provide an additional
check for the background contributions to $\cos2\phi$ being under control.
Extraction of $\cos\phi$ and $\cos2\phi$ moments for kaons will also provide
additional to pion measurements \cite{PACauu}, 
information on widths of $k_T$ distributions, and their flavor dependence.
The difference of the $\cos2\phi$ moments for $K^+$ and  $K^-$ from  $\pi^+$ and  $\pi^-$ 
will provide important information about sea orbital structure, described by 
corresponding Collins functions.
Simultaneous measurement of the $\cos2\phi$ and $\sin2\phi$ moment of the cross section 
will provide independent measurements of  the Collins effect, and provide 
complementary information to measurements of the Collins effect \cite{Collins:1992kk} with transversely polarized 
target  \cite{LOI34} as well as direct measurements, which may be
performed at BELLE \cite{Abe:2005zx}.
 


\begin{figure}
\begin{tabular}{cc}
\includegraphics[height=.28\textheight,width=7.5cm]{plots/plot12gevauuxkaon.eps}
&
\includegraphics[height=.28\textheight,width=7.5cm]{plots/plot12gevauuptkaon.eps}
\\
(a) & (b)
\end{tabular}
\caption{\small  The $\cos2\phi$ moment  for kaons as a function of $x$ (left)  and  
$P_T$ (right) for proton target
at 11 GeV from 80 days of CLAS12 running. 
The band corresponds to set-I of Ref.\cite{Zhang:2008ez} with Boer-Mulders function 
extracted \protect\cite{Zhang:2008nu} from the Drell-Yan data \protect\cite{Zhu:2007mja}. 
The Collins fragmentation function was approximated using the chiral limit. 
The black curve is from Ref. \protect\cite{Barone:2008tn}, with Collins 
function from parametrization
\cite{Anselmino:2007fs} based on the $e^+e^-$ and SIDIS data.}
\label{fig:aucos2prot}
\end{figure}

\begin{figure}
\begin{tabular}{cc}
\includegraphics[height=.28\textheight,width=7.5cm]{plots/plot12gevauuxkaondeut.eps}
&
\includegraphics[height=.28\textheight,width=7.5cm]{plots/plot12gevauuptkaondeut.eps}
\\
(a) & (b)
\end{tabular}
\caption{\small 
        The $\cos2\phi$ moment  for kaons as a function of $x$ (left)  and  
$P_T$ (right) for deuteron target
at 11 GeV from 80 days of CLAS12 running. 
Curves are calculated using assumptions for Collins fragmentation function discussed above.}
\label{fig:aucos2deut}
\end{figure}

Projections for some of the most important observables 
%for 2000 hours of running with the proposed configuration 
are shown on Figs. \ref{fig:aucos2prot},\ref{fig:aucos2deut} 
for proton and deuteron targets, respectively.
Measured $\cos2\phi$ moment for charged kaons for two targets, combined
with the measurements on pions \cite{PACauu}, will allow the extraction of the
Collins analyzing power ratios for pions and kaons, providing information on the 
polarized fragmentation function of kaons.
 


Proposed measurements of azimuthal asymmetries in SIDIS will measure the ratio of favored to 
unfavored polarized  fragmentation functions for kaons provide independent
to pion measurements constrains on  the corresponding TMD 
distributions.  The new data will also allow a more precise test of 
the factorization ansatz and the investigation of the $Q^2$ dependence of  
$\cos 2\phi$, $\cos \phi$, and $\sin \phi$ asymmetries. This will enable us 
to study the leading-twist and higher-twist nature of the corresponding 
observables~\cite{Cahn:1978se,Metz:2004je,Bacchetta:2006tn}.

Measurement of the $P_T$ dependence of the Boer-Mulders-asymmetry 
will also allow for checking of the high $P_T$ 
predictions  ~\cite{Ji:2004wu,Ji:2006vf,Bacchetta:2008xw} to 
study the transition from a non-perturbative to a perturbative 
description.  Both $\cos 2 \phi$ and   $\sin 2 \phi$ asymmetries  
for semi-inclusive deep inelastic scattering in the kinematic regions 
of {\tt CLAS12} are predicted to be 
significant (a few percent on average) and tending to be larger in 
the large-$x$ and large-$z$ region. 

The combined analysis of the future {\tt CLAS12} data on 
$\langle \cos 2 \phi \rangle$ and of the previous HERMES measurements in the 
high-$Q^2$ domain (where higher-twist effects are less significant) will provide 
information on the Boer-Mulders function, shedding light on the correlations 
between transverse spin and transverse momenta of quarks.  

  At lower $x$, this very large data set allows us to further subdivide the data
into bins in $P_T$ and $z$.
Once in hand, these data will be
combined with CLAS12 data on pions and  existing SIDIS data from HERMES, COMPASS and RHIC
for a full NLO analysis.

Measured azimuthal asymmetries for kaons
in a  large range  of kinematic variables ($x_B$, $Q^2$, $z$, $P_{\perp}$ 
and $\phi$) combined with measurements with polarized target measurements and data for pions \cite{PACall}, 
will provide detailed information on flavor and polarization dependence
of transverse momentum distributions of quarks in valence region 
and in particular on the $x_B$ and $k_{T}$ dependence of leading TMD parton distribution
functions.
Measurements of spin and azimuthal asymmetries 
across a wide range of $x,z,Q^2$, and $P_T$ would allow to perform
detailed tests of QCD dynamics in valence region.



\subsection{Summary and Request}
%\input{kaonsoc-summary.tex}


  Understanding of 
spin-orbit correlations, together with independent measurements related to the 
spin and orbital angular momentum of the quarks, will help to construct a more 
complete picture of the nucleon in terms of elementary quarks and gluons going 
beyond the simple collinear partonic representation.
The proposed set of measurements on unpolarized  proton 
and deuteron targets will
yield a comprehensive set of azimuthal moments in spin-dependent and independent
SIDIS providing access to corresponding
distribution and fragmentation  functions
in a wide range of $x$, $Q^2$, $z$, and $P_T$.
Our data, combined with
the data from HERMES, COMPASS, and BELLE, will provide independent
(complementary to $e^+/e^-$) measurement of polarized kaon Collins fragmentation function 
and will allow a study complementary to pion SIDIS
study of leading twist TMD parton distribution  
$h_{1}^{\perp}(x,k_T)$.

In addition, the proposed experiment will yield data on single beam 
spin asymmetries in SIDIS,
which can provide constraints on the higher-twist nucleon structure
functions. As a by-product,
the experiment will also provide interesting data on the
exclusive cross section ratio $K* Lambda / rho^+ n$,
which can be interpreted within the GPD formalism.

To achieve this goal, we request a total of 56 days of beam time with an 11~GeV electron beam in Hall B. We assume the same
target configuration as an already approved experiment on measurement of the neutron magnetic 
form-factors \cite{CLAS12Gilf}. 
 The breakdown of this beam time is shown in Table~\ref{tab:request}.
The number of days requested was chosen to allow a statistically significant measurement
of the T-odd distribution and fragmentation functions.

\begin{table}
\begin{center}\begin{tabular}{p{0.5\columnwidth}p{0.5\columnwidth}}
\hline 
\hline
Time & Activity\tabularnewline
\hline
2 days & Commissioning: Empty target, interchange of targets \tabularnewline
54 days & Production data taking on proton and deuterium (50\% with reversed field)\tabularnewline
\hline
\hline
\end{tabular}\end{center}
\caption{\label{tab:request} Requested beam time broken down by activity.}
\end{table}

We want to conclude by noting that while this proposed 
experiments requires a substantial commitment of
beam time (56 days total), we will simultaneously take 
data with an already approved experiment to study the neutron magnetic form-factor \cite{CLAS12Gilf}.
