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\title{BOUNDS ON TRANSVERSE MOMENTUM DEPENDENT DISTRIBUTION FUNCTIONS}        %use uppercase letters
\authori{A. Henneman}      %all authors with the first common affiliation & address
\addressi{Division of Physics and Astronomy, 
		  Faculty of Science, 
		  Vrije Universiteit\\
		  De Boelelaan 1081,
		  1081 HV Amsterdam, 
		  The Netherlands}     %their affil. & address
\authorii{}     %all authors with the second affil. & address or {}
\addressii{}    %their affil. & address
\authoriii{}    %all authors with the third affil. & address or {}
\addressiii{}   %their affil. & address   + (if such a case) further authors
                % [{\normalsize\sc ...}\\ \medskip] and their affil. &
                % address [...]; or {}
\headtitle{BOUNDS ON TRANSVERSE MOMENTUM DEPENDENT DISTRIBUTION FUNCTIONS}            %page heading on the odd pages
\headauthor{A. Henneman}           %page heading on the even pages
\specialhead{A. Henneman:BOUNDS ON TRANSVERSE MOMENTUM DEPENDENT DISTRIBUTION FUNCTIONS  \ldots} %page heading on the last page if even
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\evidence{A}
\daterec{XXX}    %;\\ final version }
\cislo{0}  \year{1999}
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\maketitle

\begin{abstract}
%
When more than one hadron takes part in a hard process, an
extended set of quark distribution and fragmentation functions 
becomes relevant.
%
In this talk, the derivation of Soffer-like bounds 
for these functions, in the case of a spin-$\frac{1}{2}$ 
target \cite{boundsEigen}, is sketched and some of their 
aspects are discussed.
%
\end{abstract}

\section{Introduction}
%--------------------
In hard inclusive electro-weak processes the soft physics
is described by light-cone correlators of quark fields.
%
For a target hadron, for instance, all the relevant soft physics
resides in the correlator \cite{Soper77,Jaffe83,Manohar90}
%
\begin{equation}
\Phi_{ij}(x) = \left. \int \frac{d\xi^-}{2\pi}\ e^{ip\cdot \xi}
\,\langle P,S\vert \overline \psi_j(0) \psi_i(\xi)
\vert P,S\rangle \right|_{\xi^+ = \xi_\st = 0}.
\end{equation}
%
Here, $P$ and $S$ denote the parent hadron momentum and spin,
and the relevant component of the quark momentum is $x=p^+/P^+$, 
the light-cone momentum fraction. 
%
The components $a^\pm = a\cdot n_\mp$ stem from vectors $n_+$ and 
$n_-$, satisfying $n_+^2 = n_-^2$ = 0 and $n_+\cdot n_-$ = 1, which
are fixed by the momentum that introduces the large scale $Q$,
together with a (soft) hadron momentum.
%
When only the leading part in orders of $1/Q$ is considered, just
the $\Phi \gamma^+$ part of the correlator suffices.
%
This part is usually parametrized in terms of the following
quark distribution functions~\cite{remark}
%
\begin{equation}
\Phi(x) \gamma^+ = \Bigl\{
f_1(x) + S_L\,g_1(x)\,\gamma_5  + h_1(x)\,\gamma_5\,{S\!\!\!\slash}_\st
\Bigr\}{\cal P}_+ ,
\label{phiint}
\end{equation}
%
where ${\cal P}_+$ stands for the projector of good fields 
$\psi_+ \equiv {\cal P}_+\psi = \frac{1}{2}\gamma^-\gamma^+\psi$
\cite{KS70}.
%
For these functions some trivial bounds and the, less trivial,
so-called Soffer bound have been derived \cite{Soffer73}.
%

If, now, one regards processes involving more than one
hadron \cite{MT96,TM95}, quark transverse momentum
becomes relevant \cite{RS79}.
%
A correlator with transverse momentum leads to an extended set 
of distribution functions \cite{kTparam}.
%
The purpose of this talk is to sketch the derivation of bounds 
for the additional functions in this extended set.
%
\section{Light-front densities}
%-----------------------------
%
A first observation in the derivation of the bounds is that
the leading part of the correlator, now being non-diagonal in
target spin space in contrast to (\ref{phiint}),
%
\begin{equation}
(\Phi\gamma^+)_{ij,s^\prime s} = 
\left. \int \frac{d\xi^-}{2\pi\sqrt{2}}\ e^{ip\cdot \xi}
\,\langle P,s^\prime\vert \psi^\dagger_{+j}(0) \psi_{+i}(\xi)
\vert P,s\rangle \right|_{\xi^+ = \xi_T = 0},
\end{equation}
%
after inserting a complete set of intermediate states, can be written 
in the following way
%
\begin{equation}
(\Phi\gamma^+)_{ij,s^\prime s} =
\frac{1}{\sqrt{2}}\sum_n
\langle P_n\vert \psi_{+j}(0)\vert P,s^\prime\rangle^\ast
\langle P_n\vert \psi_{+i}(0)\vert P,s\rangle
\,\delta\left(P_n^+ - (1-x)P^+\right) ,
\label{dens}
\end{equation} 
%
which is a positive semi-definite quantity.
%
This property is not affected by inclusion of transverse momentum.

Next, target spin dependence is incorporated using 
a spin density matrix formalism.
%
\be
M(S) = \mbox{Tr}\left[ \rho(P,S)\,\tilde M(P) \right] 
\label{spindens}
\ee
%
All target polarization information is in $\rho(P,S)$,
while the spin dependence resides in the higher dimensionality
of $\tilde{M}(P)$.
%
For a spin-$1/2$ target, $S$ is just a
vector with properties $P\cdot S = 0$ and 
$-1 \le S^2 \le 0$ (being equal to $=-1$ for a pure state) and 
$\tilde{M}$ is just $2 \times 2$ in target spin space.
%
In the target rest frame $\rho(P,S)$ simplifies to 
$1+{\bf S}\cdot {\bm \sigma}$ and $\tilde{M}$ assumes
the form
%
\begin{equation}
\tilde M_{ss^\prime} =
\left\lgroup \begin{array}{cc}
M_O + M_L & M_\st^1 - i\,M_\st^2 \\
& \\
M_\st^1 + i\,M_\st^2 & M_O - M_L \\
\end{array}\right\rgroup
\label{spinexplicit}
\end{equation}
%
where the subscripts refer to target polarization $S=(0,{\bf S}_T,S_L)$,
where $S_L = M S^+/P^+$.
%
From the diagonal elements of this matrix one sees that it lives in 
the space spanned by states with $S_L = 1$ and $S_L = -1$.
%
In order to describe transverse target polarization one needs the 
off-diagonal elements.

Now, we turn our attention to quark spin.
%
The analogon of (\ref{phiint}) when quark transverse momentum is 
taken into account, is given by the sum of three parts
%
\bea
\Phi_O(x,\bm p_\st)\,\gamma^+ & = &
\Biggl\{
f_1(x,\bm p_\st^2)
+ i\,h_1^\perp(x,\bm p_\st^2)\,\frac{{p \!\!\!\slash}_\st}{M}
\Biggr\} {\cal P}_+
\\
\Phi_L(x,\bm p_\st)\,\gamma^+ & = &
\Biggl\{
S_L\,g_{1L}(x,\bm p_\st^2)\,\gamma_5
+ S_L\,h_{1L}^\perp(x,\bm p_\st^2)
\gamma_5\,\frac{{p\!\!\!\slash}_\st}{M}
\Biggr\} {\cal P}_+
\\
\Phi_T(x,\bm p_\st)\,\gamma^+  & = &
\Biggl\{
f_{1T}^\perp(x,\bm p_\st^2)\,\frac{\epsilon_{\st\,\rho \sigma}
p_\st^\rho S_\st^\sigma}{M}
+ g_{1T}(x,\bm p_\st^2)\,\frac{\bm p_\st\cdot\bm S_\st}{M}
\,\gamma_5
\nonumber \\ & &\mbox{}
+ h_{1T}(x,\bm p_\st^2)\,\gamma_5\,{S\!\!\!\slash}_\st
+ h_{1T}^\perp(x,\bm p_\st^2)\,\frac{\bm p_\st\cdot\bm S_\st}{M}
\,\frac{\gamma_5\,{p\!\!\!\slash}_\st}{M}
\Biggr\} {\cal P}_+.
\eea
%
Choosing for the above objects a convenient (Weyl) representation, 
one sees that they are effectively $2 \times 2$ in quark spin space.
%
The leading part of the correlator is spanned by just two types
of quarks; left and right-handed (good) quarks.
%

If we now put everything together to obtain $\Phi(x,p_T)\gamma^+$
(or it's transpose in dirac space $(\Phi\gamma^+)^T$, to be more 
precise), from an expression like (\ref{spindens}), one concludes 
that the $\tilde{M}$ needed for the description including transverse 
momentum, is the following.
%
\begin{equation}
\left\lgroup \! \! \! \! 
\begin{array}{cccc} f_1 + g_{1L} & \left( \!\!  \begin{array}{c}
\frac{\vert p_\st\vert}{M}\,e^{i\phi}\times \\
\left(g_{1T}\!+\!i\, f_{1T}^\perp\right)
 \end{array} \! \!  \right)
& \left( \! \!  \begin{array}{c}
\frac{\vert p_\st\vert}{M}\,e^{-i\phi}\times \\
\left(h_{1L}^\perp\!+\!i\,h_1^\perp\right) 
\end{array} \! \!  \right)
& 2\,h_1\\ & & & \\ \left( \! \!  \begin{array}{c}
\frac{\vert p_\st\vert}{M}\,e^{-i\phi}\times \\
\left(g_{1T}\!-\!i\,f_{1T}^\perp\right)
\end{array} \! \!  \right) & f_1 - g_{1L} &
\frac{\vert p_\st\vert^2}{M^2}e^{-2i\phi}\,h_{1T}^\perp &
\left( \! \!  \begin{array}{c} -\frac{\vert p_\st\vert}{M}\,e^{-i\phi}\times \\
\left(h_{1L}^\perp\!-\!i\,h_1^\perp\right) 
\end{array}
\! \!  \right) \\ & & & \\ \left( \! \!  \begin{array}{c}
\frac{\vert p_\st\vert}{M}\,e^{i\phi}\times\\ 
\left(h_{1L}^\perp\!-\!i\,h_1^\perp\right)
\end{array} \! \!  \right) & \frac{\vert p_\st\vert^2}{M^2}e^{2i\phi}\,h_{1T}^\perp &
f_1 - g_{1L} & \left( \! \!  \begin{array}{c} -
\frac{\vert p_\st\vert}{M}\,e^{i\phi}\times \\
\left(g_{1T}\!-\!i\,f_{1T}^\perp\right) 
\end{array} \! \!  \right) \\ & & & \\
2\,h_1 & \left( \! \!  \begin{array}{c}
-\frac{\vert p_\st\vert}{M}\,e^{i\phi}\times \\
\left(h_{1L}^\perp\!+\!i\,h_1^\perp\right) 
\end{array} \! \!  \right) &
\left( \! \!  \begin{array}{c} -\frac{\vert p_\st\vert}{M}\,e^{-i\phi}\times \\
\left(g_{1T}\!+\!i\,f_{1T}^\perp\right) 
\end{array} \! \!  \right) &
f_1 + g_{1L} \end{array} \! \! \!\! \right\rgroup 
\label{bigmatrix}
\end{equation}
%
This matrix lives in the product space of target helicity and
quark handedness.
%
The upper-right as the lower-left $2 \times 2$ submatrices are solely
populated by so called chiral-odd functions \cite{Artru,Cortes92,JJ92}, that 
involve the flipping of quark handedness, whereas the diagonal submatrices
contain merely chiral-even functions.
%
Whithin each of these $2 \times 2$ matrices, the diagonal elements involve 
no and longitudinal polarization, as these states can be expressed in the 
helicity eigenstates of the target, whereas the non-diagonal ones involve 
transverse polarization as expected from (\ref{spindens}).
%
In (\ref{bigmatrix}) all distribution functions particular to transverse
quark momentum are accompanied by an azimuthal dependence.
%
This dependence averages to zero after integration over azimuthal angle,
showing that taking into account transverse momentum is necessary to
access the full helicity structure of a polarized nucleon \cite{BoM99}.
%
The sought for bounds follow from the fact that for any vector $a$ the quantity 
$a \tilde{M} a \ge 0$.
%
If an integration over azimuthal angle is perfomed first and after that
positive semi-definiteness is demanded, one finds the Soffer bound.
%
Note that the T-odd functions $f_{1T}^{\perp}$ and 
$h_1^\perp$  can be considered as imaginary parts of 
$g_{1T}$ and $h_{1L}^{\perp}$, respectively.


\section{Interpreting the bounds}
%-------------------------------
Regarding 2-dimensional subspaces in (\ref{bigmatrix})
starts giving us non-trivial bounds on the distribution functions.
%
Omitting the $(x,\bm p_\st^2)$ dependences of these functions
one finds
%
\begin{eqnarray}
&& \vert h_1 \vert \le
\frac{1}{2}\left( f_1 + g_{1L}\right)
\le f_1,
\\
&&
 \frac{{\bm p}_T^2}{2 M^2}\,
\vert h_{1T}^{\perp}\vert \le
\frac{1}{2}\left( f_1 - g_{1L}\right)
\le f_1,
\\
&& 
\frac{{\bm p}_T^4}{4 M^4} \left(
\left( g_{1T}\right)^2
+ \left( f_{1T}^{\perp}\right)^2 \right)
\le \frac{\bm p_\st^2}{4M^2}
\left( f_1 + g_{1L}\right)
\left( f_1 - g_{1L}\right)
\le \frac{\bm p_\st^2}{4M^2}\,f_1^2,
\\
&& 
\frac{{\bm p}_T^4}{4 M^4} \left(
\left( h_{1L}^{\perp}\right)^2
+ \left( h_{1}^{\perp}\right)^2 \right)
\le \frac{\bm p_\st^2}{4M^2}
\left( f_1 + g_{1L}\right)
\left( f_1 - g_{1L}\right)
\le \frac{\bm p_\st^2}{4M^2}\,f_1^2.
\end{eqnarray}
%
In order to incorporate the more elaborate bounds that are found 
considering higher dimensional subspaces of the matrix (\ref{bigmatrix}),  
it is convenient to introduce two positive-definite functions 
$A(x,\bm p_\st^2)$ and $B(x,\bm p_\st^2)$ such that $f_1 = A + B$ and 
$g_1 = A - B$ and also
%
\bea
h_1 &=& \alpha\,A ,
\\ 
\frac{{\bm p}_T^2}{2 M^2}h_{1T}^{\perp} &=& \beta\,B ,
\\ 
\frac{{\bm p}_T^2}{2 M^2}
\left( g_{1T} + i\,f_{1T}^{\perp} \right) &=&
\gamma\,\frac{\vert {\bm p}_\st\vert}{M}\,\sqrt{AB} ,
\\ 
\frac{{\bm p}_T^2}{2 M^2} \left(
h_{1L}^{\perp} + i\,h_{1}^{\perp} \right) &=&
\delta\,\frac{\vert {\bm p}_\st\vert}{M}\,\sqrt{AB} .
\eea
%
Here $\alpha$, $\beta$,$\gamma$ and $\delta$ all depend on both $x$ and 
${\bm p}_T^2$ and have absolute values in the interval $[-1,1]$.
%
Note that $\alpha$ and $\beta$ are real-valued whereas $\gamma$ and $\delta$ 
are complex-valued.
%
Their imaginary parts determine the strength of the T-odd functions.
%

In terms of these functions, what is required to be positive semi-definite
are the following four expressions
%
\bea
e_{1,2} = (1-\alpha)A + (1+\beta)B
\pm \sqrt{4AB\vert\gamma+\delta\vert^2+((1-\alpha)A-(1+\beta)B)^2},
\\
e_{3,4} = (1+\alpha)A + (1-\beta)B
\pm \sqrt{4AB\vert\gamma-\delta\vert^2+((1+\alpha)A-(1-\beta)B)^2},
\eea
leading to
\bea
&&
A+B\ge 0,
\\ &&
\vert \alpha\,A-\beta\,B \vert \le A+B,
\quad \mbox{i.e.} \ \vert h_{1T}\vert \le f_1,
\\ &&
\vert \gamma + \delta \vert^2 \le (1-\alpha)(1+\beta) ,
\\ &&
\vert \gamma - \delta \vert^2 \le (1+\alpha)(1-\beta) .
\eea

In figure \ref{fig3} one can see the a graphical representation of
the allowed values for $\alpha$ and $\beta$.
%
It is remarkable to see that to see that an inclusively measured 
function as $h_1$ is involved in a bound including functions as 
$g_{1T}$ and $h^{\perp}_{1L}$ which cannot be measured 
inclusively and are responsible for asymmetries \cite{MT96,BM98}.
% 
\begin{figure} \centering
\mbox{\epsfysize=40mm\epsffile{hennemanfig1.eps}}
\caption{ \label{fig3}
Allowed region (shaded) for $\alpha$ and $\beta$ depending on
$\gamma$ and $\delta$.}
\end{figure}
%
\section{Concluding remarks}
%
It is important to note that though in this talk only distribution 
functions have been addressed, an almost identical analysis can be
performed on fragmentation functions \cite{CS82}.
%
The non-vanishing of T-odd functions, though disputed in the case
of distribution functions yet a possibility \cite{Sivers90}, is
accepted in the case of fragmentation functions, such  as 
$D^{\perp}_{1T}$ \cite{MT96,J96} and $H^{\perp}_1$ \cite{Collins93}, 
as time-reversal invariance cannot be imposed on the final state 
\cite{RKR71,HHK83,JJ93}.
%
This work can also straightforwardly be extended to spin-$1$ hadrons
\cite{alessandro} and gluons \cite{joaoGluons}.
%
Though it should be supplemented with a study of the
factorization, scheme dependence and stability of the bounds under
$Q^2$ evolution \cite{scheme}, these bounds provide an estimate of 
the magnitudes of functions measured in asymmetries at SMC \cite{SMC}, 
HERMES \cite{HERMES} and LEP \cite{LEP}.

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\end{thebibliography}
\end {document}

