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\begin{document}
\sloppy
\thispagestyle{empty}
\begin{flushleft}
DESY 96--040 \\
March 1996\\
\end{flushleft}

\setcounter{page}{0}

\mbox{}
\vspace*{\fill}
\begin{center}
{\LARGE\bf On the Twist-2 Contributions to
} \\
\vspace{2.5mm}
{\LARGE\bf Polarized Structure Functions
}\\
%\vspace{4em}
\vspace{2.5mm}
{\LARGE\bf and New Sum Rules
}\\
\vspace{4em}
\large
J. Bl\"umlein$^a$ and N. Kochelev$^{a,b}$

\vspace{4em}
\normalsize
{\it   $^a$DESY--Zeuthen}\\
{\it   Platanenallee 6, D--15735 Zeuthen, Germany}\\
\vspace{5mm}
{\it   $^b$Bogoliubov
Laboratory of Theoretical Physics, JINR,
}\\
{\it   RU--141980 Dubna, Moscow Region,
Russia}\\
\end{center}
\normalsize
\vspace*{\fill}
%
\begin{abstract}
\noindent
The twist-2 contributions to the polarized structure functions
in deep inelastic lepton--hadron scattering are calculated including
the exchange of weak bosons and using both
the operator product expansion and the covariant parton model.
A new relation between two structure functions leading to a
sequence of new sum rules is found.
The light quark mass corrections to the structure functions are derived
in lowest order QCD.
%
\end{abstract}
\vspace*{\fill}
\newpage
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sect1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The study of polarized deep inelastic scattering off polarized targets
has revealed a rich structure of phenomena during the last
years~\cite{REV}. So far mainly the case of deep inelastic photon
scattering has been studied experimentally. A future polarized proton
option at RHIC and HERA, however, would allow to probe the spin
structure of nucleons at much higher $Q^2$~(cf.~\cite{JB95A}) also.
In this
range $Z$--exchange contributions become relevant and one may investigate
charged current scattering as well.
For this general case
the scattering cross section is  determined by (up to) five polarized
structure
functions per current combination, if lepton mass effects are
disregarded.

In previous investigations different techniques have been used to derive
relations between these structure functions and discrepancies between
several derivations were reported~(cf. e.g.~\cite{a1}).
In refs.~\cite{a1,a0} the structure functions were calculated in
the parton model. Some of the investigations deal with the
case of longitudinal polarization only~\cite{a3}.
In other studies light-cone current algebra~\cite{DIC,A1C} and
the operator product expansion were
used~\cite{AR}--\cite{a77}. Furthermore  the structure
functions $g_1^{em}$
and $g_2^{em}$ were also
calculated in the covariant parton model~\cite{a1B,a17}.
Still a thorough agreement between  different approaches has not
been obtained.

It is the aim of
the present paper to derive
the relations for the complete
set of the polarized structure functions including weak interactions
which are not associated with terms in the
scattering cross section
vanishing as $m_{lepton} \rightarrow 0$.
The calculation is performed applying two different
techniques:~the operator product
expansion and  the covariant parton model~\cite{LP}.
The latter method is
furthermore used
to obtain also
the quark mass corrections in lowest
order QCD.

As it turns out the twist-2 contributions for
only
two out of the five polarized structure
functions, corresponding to the respective current combinations, are
linearly independent. Therefore three
linear operators have to exist which
determine the remaining three structure
functions over a basis of two in lowest order QCD.
Two of them are given by the
Wandzura--Wilczek\cite{WW} relation  and a
relation by Dicus\footnote{This relation corresponds
to the
Callan--Gross~\cite{CG}
relation for unpolarized structure functions since
the spin dependence enters the tensors of $g_4$ and $g_5$ in
$W_{\mu\nu}^{ij}$,~eq.~(\ref{eqz4}), in terms of a factor
$S.q$.}~\cite{DIC}.
A third {\it new}
relation is found.

New sum rules based on this relation are derived and  discussed
in the context of quark mass corrections. Extending a recent analysis
carried out for the case of photon scattering~\cite{a18}
to the complete set
of neutral and charged current interactions
we also investigate the validity
of known relations, as the Burkhardt--Cottingham~\cite{BC}
sum rule and other
relations, in the presence of quark mass effects.

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic Notation}
\label{sect2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The hadronic tensor for polarized deep inelastic scattering is given by
%----------------------------------------------------------------------
\be
W_{\mu\nu}^{ab}=\frac{1}{4\pi}\int d^4xe^{iqx}
\langle pS\mid[J_\mu^a(x),J_\nu^b(0)]\mid pS\rangle,
\label{eqHAD}
\ee
%-----------------------------------------------------------------------
where in framework of the quark model the currents are
%-----------------------------------------------------------------------
\be
J_\mu^a(x)=\sum_{f,f'}
U_{ff'} \overline{q}_{f'}(x)\gamma_\mu(g_V^a+g_A^a\gamma_5)q_f(x).
\ee
%-----------------------------------------------------------------------
In terms of structure functions the hadronic tensor reads:
%-----------------------------------------------------------------------
\ba
W_{\mu\nu}^{ab}
&=&(-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2})F_1^i(x,Q^2)+
\frac{\widehat{p}_\mu\widehat{p}_\nu}{p.q} F_2^i(x,Q^2)-
 i\epm\frac{q_\lambda p_\sigma}{2 p.q}  F_3^i(x,Q^2)\nn\\
&{+}& i\epm\frac{q^\lambda S^\sigma}{p.q} g_1^i(x,Q^2)+
i\epm\frac{q^{\lambda}(p.q S^\sigma - S.q p^\sigma)}
{(p.q)^2} g_2^i(x,Q^2)\nn\\
&{+}& \left[ \frac{\widehat{p_\mu} \widehat{S_\nu}
+ \widehat{S_\mu} \widehat{p_\nu}}{2}-
S.q \frac{\widehat{p_\mu} \widehat{p_\nu}}{(p.q)} \right]
\frac{g_3^i(x,Q^2)}{p.q}\nn\\
&+&
S.q \frac{\widehat{p_\mu}\widehat{p_\nu}}{(p.q)^2}
g_4^i(x,Q^2)+
(-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2})\frac{(S.q)}{p.q} g_5^i(x,Q^2),
%label{e6}
\label{eqz4}
\ea
%-----------------------------------------------------------------------
with  $ab \equiv i$
and
%-----------------------------------------------------------------------
\be
\widehat{p_\mu} = p_\mu-\frac{p.q}{q^2} q_{\mu},~~~~~~\widehat{S_\mu}
= S_\mu-\frac{S.q}{q^2} q_{\mu}.
\label{eqz5}
\ee
%-----------------------------------------------------------------------
Here $x = Q^2/2p.q \equiv Q^2/2M\nu$ and $Q^2 = -q^2$ is the transfered
four
momentum  squared.
$p$ and $S$ denote the four vectors of the
nucleon momentum and spin, respectively, with
$ S^2=-M^2$ and, $S.p = 0 $.
$g_{V_i}$ and $g_{A_i}$ are the vector and axialvector
couplings of the bosons exchanged in the respective subprocesses.
For charged current interactions
$U_{ff^\prime}$ denotes the
Cabibbo-Kobayashi-Maskawa matrix.
The hadronic tensor (\ref{eqz4})
was constructed using both Lorentz and time reversal
invariance and current conservation.

In previous analyses
partly different notations for the hadronic tensor
have been used. To allow for  direct comparisons
with earlier results
we relate the definition
of structure functions given in eq.~(\ref{eqz4}) to that of other
authors in table~1 for convenience~\footnote{A more comprehensive
comparison is given in~\cite{BK}.}.

\begin{center}
\begin{tabular}{||c||c|c|c|c||}\hline \hline
{\sf our notation}      & $\ct{a1}$
    & $\ct{a3}$ & $\ct{a7}$ & $\ct{a77}$\\ \hline \hline
$g_1$ & $g_1$ &  $g_1$      & $g_1$     & $g_1$\\
$g_2$ & $g_2$ &  $g_2$      & $g_2$  & $g_2$\\
$g_3$ & $-g_3$ &  $
(g_4-g_5)/2$      & $b_1+b_2$  & $(A_2-A_3)/2$ \\
$g_4 $
& $g_4-g_3$
& $g_4$
&$a_2+b_1+b_2$ &$A_2$
\\
$g_5$
& $-g_5$
&$ g_3$
&$a_1$ & $A_1$

\\  \hline
\end{tabular}
\end{center}
\small

\vspace{3mm}
\noindent
\small
{\sf Table~1:}~The definition of polarized deep inelastic scattering
structure functions in different conventions.\footnote{Note that
in part of the
above papers only the structure functions being related to longitudinal
nucleon polarization were delt with.}
\normalsize

\vspace{2mm}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Operator Product Expansion}
\label{sect3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The forward Compton amplitude, $T_{\mu\nu}^{ij}$,
is related to the hadronic tensor
by
%-----------------------------------------------------------------------
\be
W_{\mu\nu}^{ij} = \frac{1}{2\pi} Im T^{ij}_{\mu\nu}
\label{eqB1}
\ee
%-----------------------------------------------------------------------
where
%-----------------------------------------------------------------------
\be
T_{\mu\nu}^{ij}
=i\int d^4xe^{iqx}\langle pS\mid(T{J_\mu^i}^\dagger (x)J_\nu^j(0))\mid
pS\rangle.
\label{eqB2}
\ee
%-----------------------------------------------------------------------
It may be represented in terms of the amplitudes
$\left . T_k^i(q^2, \nu) \right|_{k=1}^3$ and
$\left . A_k^i(q^2, \nu) \right|_{k=1}^5$ analogously to (\ref{eqz4})
substituting
%-----------------------------------------------------------------------
\be
F_1 \rightarrow T_1,~~~~~~~F_{2,3} \rightarrow \frac{p.q}{M^2} T_{2,3}
\label{eqB3}
\ee
%-----------------------------------------------------------------------
and
%-----------------------------------------------------------------------
\be
%%g_{1,5} \rightarrow \alpha_{1,5} \equiv
g_{1,5} \rightarrow
\frac{p.q}{M^2} A_{1,5},~~~~~~~g_{2,3,4}  \rightarrow
%%%\rightarrow \alpha_{2,3,4} \equiv
\frac{(p.q)^2}{M^4} A_{2,3,4}.
\label{eqB3A}
\ee
%-----------------------------------------------------------------------
Near the light cone the forward Compton amplitude has the representation
%-----------------------------------------------------------------------
\ba
T^{ab}_{\mu\nu, ij}   &=&
   \frac{2i}{(2\pi)^2(x^2 - i0)^2}  \left [
\bar q(x)\gamma_\mu(g_{V_i}+g_{A_i}
\gamma_5)
\not \!{x}
\gamma_\nu(g_{V_j}+
g_{A_j}\gamma_5)\lambda^a\lambda^b q(0) \right.
\nn\\
&{-}&~~~~~~~~~~~~~~~~~~~~ \left .
\bar q(0)\gamma_\nu(g_{V_j}+g_{A_j}\gamma_5)
\not \!{x}
\gamma_\mu(g_{V_i}+
g_{A_i}\gamma_5)\lambda^b\lambda^a q(x) \right ],
\label{eqB4}
\ea
%-----------------------------------------------------------------------
where  $\lambda^a$ denote the $SU(N_f)$ matrices.
The spin dependent part of $T_{\mu\nu, ij}^{ab}$ is
%-----------------------------------------------------------------------
\be
\begin{array}{cl}
T_{\mu\nu, ij}^{spin, ab} &
=  \frac{\displaystyle
2 x^\alpha}{\displaystyle
(2\pi)^2(x^2-i0)^2}  \\
\times &
 \left \{
(g_{V_1}g_{V_2}+g_{A_1}g_{A_2}) \epa  \left [
                        (if^{abc}+\widetilde{d}^{abc})
\bq(x)\gb\g\lc q(0)
                       -(if^{abc}-\widetilde{d}^{abc})
\bq(0)\gb\g\lc q(x) \right ] \right.
 \\
 +& \left. (g_{V_1}g_{A_2}+g_{A_1}g_{V_2})
                       \Sa \left[
 (if^{abc} + \widetilde{d}^{abc})\bq(x)\gb\g\lc q(0)
              + (if^{abc} - \widetilde{d}^{abc})
\bq(0)\gb\g\lc q(x) \right ] \right \},\\
\end{array}
\label{eqB5}
\ee
%-----------------------------------------------------------------------
with
%-----------------------------------------------------------------------
\be
S_{\mu \alpha \nu \beta} = g_{\mu \alpha} g_{\nu \beta}
                         + g_{\mu \beta } g_{\nu \alpha}
                         - g_{\mu \nu   } g_{\alpha \beta}
\ee
%-----------------------------------------------------------------------
and
%-----------------------------------------------------------------------
\be
\widetilde{d}^{abc} \lambda_c = \frac{2}{N_f} \delta^{ab} +
d^{abd} \lambda_c.
\ee
%-----------------------------------------------------------------------
We further represent (\ref{eqB5}) in terms of a Taylor series around
$x=0$. The amplitudes $A_k(q^2, \nu)|_{k=1}^5$ can be related
to the expectation values of
a symmetric and an
antisymmetric operator
emerging in the Taylor expansion,
$\langle pS|\Theta_{S,A}^{\beta
\left\{\mu_1 ... \mu_n\right\}}|pS \rangle$,
and obey the following crossing relations:
%-----------------------------------------------------------------------
\ba
A_{1,3}(q^2, -\nu) &=& ~A_{1,3}(q^2, \nu) \label{eqXMP}
\\
A_{2,4,5}(q^2, -\nu) &=& -A_{2,4,5}(q^2, \nu)
\label{eqAMP}
\ea
for neutral current interactions.
%-----------------------------------------------------------------------
%%%======================================================================
One finally obtains the following expressions for the
moments of structure functions
using standard techniques.
%-----------------------------------------------------------------------
\ba
\int_0^1 dx x^n g_1^j(x,Q^2) &=& \frac{1}{4}
\sum_q \alpha_j^q a_n^q,~~~{\ } n=0,2...,
\label{eqYMP}
\\
%-------------------------------
\int_0^1 dx x^n g_2^j(x,Q^2) &=&  \frac{1}{4}
\sum_q \alpha_j^q
      \frac{n (d_n^q -a_n^q)}{n + 1} ,~~~{\ } n=2,4...,   \\
%-------------------------------
\int_0^1 dx x^n g_3^j(x,Q^2) &=&
 \sum_q  \beta_j^q
      \frac{a_{n+1}^q}{n + 2} ,~~~{\ } n=0,2...,   \\
%-------------------------------
\int_0^1 dx x^n g_4^j(x,Q^2) &=&
\frac{1}{2} 
  \sum_q 
\beta_j^q
      a_{n+1}^q ,~~~{\ } n=2,4...,   \\
%-------------------------------
\int_0^1 dx x^n g_5^j(x,Q^2) &=&
\frac{1}{4} 
  \sum_q 
\beta_j^q
      a_{n}^q ,~~~{\ } n=1,3...~~.
\label{eqg1M}
\ea
%-----------------------------------------------------------------------
Here we adopt
the notation of~\cite{RLJ} and
$a_n^q$ and $d_n^q$ are the matrix elements which are
related to the expectation
values of
$\langle pS|\Theta_{S}^{\beta\left\{
\mu_1 ... \mu_n\right\}}|pS \rangle$
and
$\langle pS|\Theta_{A}^{\beta\left\{
\mu_1] ... \mu_n\right\}}|pS \rangle$,
respectively. The factors $\alpha_j^q$ and $\beta_j^q$ are given by
%-----------------------------------------------------------------------
\ba
\left(
\alpha_{|\gamma|^2}^q, \alpha_{|\gamma Z|}^q, \alpha_{|Z|^2}^q \right)
&=& \left [
e_q^2, 2 e_q g_V^q, (g_V^q)^2 + (g_A^q)^2 \right ] \\
\left(
\beta_{|\gamma Z|}^q, \beta_{|Z|^2}^q \right)
&=& \left [
2 e_q g_V^q,
2 g_V^q g_A^q \right ]
\label{eqAlBe}
\ea
%-----------------------------------------------------------------------
Analogous relations to (\ref{eqXMP}--\ref{eqg1M})
 are derived for the charged current structure
functions~( cf.~\cite{BK}).

As well--known, the structure function $g_2(x,Q^2)$ contains also
twist--3 contributions corresponding to the matrix elements $d_n^q$.
On the other hand, all the  remaining structure functions are {\it not}
related to $d_n^q$, and contain at
lowest twist
contributions  of twist--2 only.
We will disregard the terms $d_n^q$  in the
subseqent discussion.
The twist--2 contributions are
related by the equations:
%-----------------------------------------------------------------------
\ba
g_2^i(x) &=& -g_1^i(x) + \int_x^1\frac{dy}{y}g_1^i(x),
\label{qq7}  \\
g_4^j(x) &=& 2xg_5^j(x),
\label{qq8} \\
g_3^j(x)&=&4x\int_x^1\frac{dy}{y}g_5^j(y),
\label{qq9}
\ea
%-----------------------------------------------------------------------
where $i=\gamma,\gamma Z, Z, W $ and $j=\gamma Z, Z, W $.
Eqs.~(\ref{qq7}) and (\ref{qq8}) are the Wandzura--Wilczek~\cite{WW}
and Dicus~\cite{DIC} relations, and eq.~(\ref{qq9}) is a {\it new}
relation. 

Recently the first two moments of $g_3$
were calculated in~\cite{FRANKF}. They agree with our general relation
eq.~(\ref{qq9}). We do not confirm a corresponding relation for the
structure function $A_3$~(cf.~table~1)
given
in~\cite{a77} previously, which also
disagrees with the lowest moments
given in~\cite{FRANKF}.

Eqs.~(\ref{qq8},\ref{qq9}) yield the sum rules
%-----------------------------------------------------------------------
\be
\int_0^1 dx x^n \left [ g_3^k(x,Q^2) - \frac{2}{n+2} g_4^k(x,Q^2)
\right] = 0.
\label{qq10}
\ee
%-----------------------------------------------------------------------
For $n = 0$ one obtains
%-----------------------------------------------------------------------
\be
\int_0^1 dx   g_3^k(x,Q^2)  =
\int_0^1 dx   g_4^k(x,Q^2).
\label{qq11}
\ee
%-----------------------------------------------------------------------

Two of the five spin--dependent structure functions $\left. g_k^j\right
|_{k=1}^5$ are linearly independent. We will express the remaining ones
using $g_1^j$ and $g_5^j$ as a basis given by
%-----------------------------------------------------------------------
\ba
g_1^j(x,Q^2)  &=&
\frac{1}{2} \sum_q \alpha_j^q \left [ \Delta q(x,Q^2)
+ \Delta \overline{q}(x,Q^2) \right ],\\
g_5^j(x,Q^2) &=&
\frac{1}{2} \sum_q \beta_j^q
\left [\Delta q(x,Q^2)  - \Delta \overline{q}(x, Q^2) \right ]
\ea
%-----------------------------------------------------------------------
for the neutral current reactions. For charged current $lN$ scattering
one obtains:
%-----------------------------------------------------------------------
\ba
g_1^{W^{-(+)}}(x,Q^2) &=&
\sum_q \left [\Delta q_{u(d)}(x,Q^2)
+ \Delta \overline{q}_{d(u)}(x, Q^2)
\right ],
\\
g_5^{W^{-(+)}}(x,Q^2) &=& - \sum_q
\left [\Delta q_{u(d)}(x,Q^2)
- \Delta \overline{q}_{d(u)}(x, Q^2)
\right ].
\ea
%-----------------------------------------------------------------------

In figure~1 the behaviour of the twist--2 contributions to the
structure functions
$\left. g_k^j(x,Q^2) \right|_{k=1}^j$ are compared
for
$j = |\gamma|^2, |\gamma Z|$ and $|W^-|^2$ in leading order QCD
for the range $10^{-4} < x$  and $10 \GeV^2 \leq Q^2 \leq 10^4 \GeV^2$.
Here we refer to the
parametrization~\cite{GRVS} of the parton densities as one example.
Whereas  the absolute values of the
structure functions $g_{1,2,5}(x,Q^2)$ grow for $x \rightarrow 0$
$g_{3}^j$ and $g_4^j$ are predicted to vanish as
$x \rightarrow 0$. 
In the parametrization~\cite{GRVS} the structure functions $g_3^j$ to
$g_5^j$
are found to be positive for $j = \gamma Z$ and  negative for
$j = W^-$, while $g_1$ takes
negative values for $x \lsim 10^{-3}... 3 \cdot 10^{-4}$. For larger
values of $Q^2$ the twist--2 contribution to $g_2^k$ is predicted to be
positive, while for some current combinations it can take negative
values in the small $x$ region again.

Currently the experimental data on $g_1^n$ and
$g_1^p$ constrain the parton densities $\Delta q$ and
$\Delta \overline{q}$ in the kinematical range $10^{-2} \lsim x$
and the predictions for the small $x$ range result from extrapolations
only. Other parametrizations (see~\cite{LADIN} for a recent compilation)
agree in the range of the current data but differ in size in the
range of small~$x$. Clearly
more data, particularly in the low $x$ region, are
needed to yield better constraints on the flavour structure of
polarized structure functions.


%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Covariant Parton Model}
\label{sect4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
In the covariant parton model the hadronic
tensor for deep inelastic scattering is given by
%-----------------------------------------------------------------------
\be
W_{\mu\nu, ab}(q,p,S)=\sum_{\lambda, i} \int d^4k
f_{\lambda}^{q_i}(p,k,S)
w_{\mu\nu ,ab, \lambda}^{q_i}(k,q) \delta[(k+q)^2-m^2].
\label{eq1}
\ee
%-----------------------------------------------------------------------
Here $w_{\mu\nu, ab, \lambda}^{q_i}(k,q)$
denotes the hadronic tensor at the
quark level,
$f_{\lambda}^{q_i}(p,k,S)$ describes
the quark and antiquark
distributions  of the hadron,
$\lambda$ is the quark helicity,  $k$ the virtuality
of the initial state parton, and $m$ is the quark mass.

The spin-dependent part of the hadronic tensor at the
quark level takes the
following form:
%-----------------------------------------------------------------------
\ba
w_{\mu\nu, ab, \lambda}^{q_i, spin}(k,q)
%&=&\frac{1}{4}Tr[(1+\frac{\lambda\g \not \!
%  {n}}{m_q})
%(\not \! {k} +m_q)\gamma_\mu(
%g_{V_a}^{q_i}+g_{A_a}^{q_i}\g)(\not \! {k}+\not \!
%{q}+m_q)\gamma_\nu(g_{V_b}^{q_i}+g_{A_b}^{q_i}\g)]\nn\\
&=&\lambda \left\{
2i\epa [g_{A_a}^{q_i}g_{A_b}^{q_i}k_\alpha
n_\beta+(g_{A_a}^{q_i}g_{A_b}^{q_i}+
g_{V_a}^{q_i}g_{V_b}^{q_i})q_\alpha n_\beta] \right.
\nn\\
&+& \left.
g_{V_a}^{q_i}g_{A_b}^{q_i}[2k_\mu n_\nu-(n.q)g_{\mu\nu}]+
g_{A_a}^{q_i}g_{V_b}^{q_i}[2n_\mu k_\nu-(n.q)g_{\mu\nu}]
\right \},
%%%\label{q33}
\label{eq2}
\ea
%-----------------------------------------------------------------------
where $n$ is the spin vector of the
off-shell parton~\cite{a17}
%-----------------------------------------------------------------------
\be
n_\sigma=\frac{m   p.k}{\sqrt{(p.k)^2 k^2 - M^2 k^4}}(k_\sigma -
\frac{k^2}{p.k} p_\sigma).
%%%\label{q2}
\label{eq3}
\ee
%-----------------------------------------------------------------------
For massless quarks the spin dependent quark densities
$\Delta f^{q_i}  = f_+^{q_i} - f_-^{q_i}$ obey, due to covariance
(cf. \cite{a18}),
%-----------------------------------------------------------------------
\be
\Delta f^{q_i}
(p.k,S.k,k^2) =  - \frac{S.k}{M^2} \hat{f}^{q_i}(p.k,k^2).
\label{eq43}
\ee
%-----------------------------------------------------------------------
We further decompose  the spin dependent part of the hadronic
tensor $W_{\mu\nu}$ into a longitudinal and a transverse component
with respect to the nucleon spin $S^{\mu}_{\parallel} = p^{\mu}
+ {\cal O}(M^2/\nu)$ and $S^{\mu}_{\perp} = M(0,1,0,0)$ in the infinite
momentum frame $p = (\sqrt{M^2 + \pvec^2},0,0,\pvec)$:
%-----------------------------------------------------------------------
\ba
W_{\mu\nu}^{j,\|}
=i\epa \frac{q_\alpha p_\beta}{\nu}g_1^j(x)+\frac{p_\mu
  p_\nu}{\nu}g_4^j(x)-g_{\mu\nu}g_5^j(x) ,\nn\\
W_{\mu\nu}^{j, \bot}=
i\epa \frac{q_\alpha S_\beta^\bot}{\nu} \left [
 g_1^j(x)+g_2^j(x) \right ]
+\frac{p_\mu S_\nu^\bot+p_\nu S_\mu^\bot}{2\nu}g_3^j(x).
%%\label{q5}
\label{eq4}
\ea
%-----------------------------------------------------------------------
with $j \equiv ab$.
Using the Sudakov representation for
%-----------------------------------------------------------------------
\be
k = xp + \frac{k^2 + k_{\perp}^2 - x^2 M^2}{2 x \nu} (q + xp) + k_{\perp}
\label{eq5}
\ee
%-----------------------------------------------------------------------
the structure functions $g_k^j(x)$
are obtained from (\ref{eq1},\ref{eq2}) in the Bjorken limit
$Q^2, \nu \rightarrow \infty,$\newline
$x = const.$ by
%-----------------------------------------------------------------------
\ba
g_1^j(x)&=&\frac{\pi xM^2}{8}
\sum_q \alpha_q^j
\int_x^1dy(2x-y)
\widehat{h}_{q}(y) ,\nn\\
g_2^j(x)&=&
\frac{\pi xM^2}{8}
\sum_q \alpha_q^j
\int_x^1dy(2y-3x)
\widehat{h}_{q}(y) ,\nn\\
g_3^j(x)&=&
\frac{\pi x^2M^2}{2}
\sum_q \beta_q^j
\int_x^1dy(y-x)
\widehat{h}_{q}(y) ,\nn\\
g_4^j(x)
&=&\frac{\pi x^2M^2}{4}
\sum_q \beta_q^j
\int_x^1dy(2x-y)
\widehat{h}_{q}(y) ,\nn\\
g_5^j(x)
&=&\frac{\pi xM^2}{8}
\sum_q \beta_q^j
\int_x^1dy(2x-y)
\widehat{h}_{q}(y).
\label{eq6}
\ea
%-----------------------------------------------------------------------
for neutral current interactions,
where
$ y=x+k^2_\bot/(xM^2)$ and
$\widehat{h}_{q}(y)=\int dk^2 \hat{f}_{q}(y,k^2)$.
The corresponding relations for
charged current scattering are given in~\cite{BK}.
The expressions
for $g_1^{em}$ and $g_2^{em}$
have been obtained in \cite{a17,a18}
already.

Again the structure functions given in eqs.~(\ref{eq6}) may be
expressed in terms
of two independent
structure functions in lowest order QCD:
%-----------------------------------------------------------------------
\ba
g_2^i(x)&=& -g_1^i(x) + \int_x^1\frac{dy}{y}g_1^i(y),
\label{eq7}   \\
g_4^j(x)&=& 2x g_5^j(x),
\label{eq8}  \\
g_3^j(x)&=&4x\int_x^1\frac{dy}{y}g_5^j(y).
\label{eq9}
\ea
%-----------------------------------------------------------------------
These relations agree with those found using the operator product
expansion in section~2, eqs.~(\ref{qq7}--\ref{qq9}).

As examples
one may derive from (\ref{eqYMP}--\ref{eqg1M},\ref{eq6}) the relations
%-----------------------------------------------------------------------
\be
  \left [ g_3^{\nu n}(x,Q^2) - g_3^{\nu p}(x,Q^2) \right ]
=
12x \left [ \left ( g_1(x,Q^2)
                  + g_2(x,Q^2) \right )^{ep}
          - \left ( g_1(x,Q^2)
                  + g_2(x,Q^2) \right )^{en}  \right ]^{|\gamma|^2}
\label{DIC1}
\ee
%-----------------------------------------------------------------------
%-----------------------------------------------------------------------
\ba
6x \left [         g_2^{en}(x,Q^2)
                  - g_2^{ep}(x,Q^2) \right ]^{|\gamma|^2}
=  \left [ \left ( g_4(x,Q^2)
                  - \frac{g_3(x,Q^2)}{2} \right )^{\nu n}
         - \left ( g_4(x,Q^2)
                  - \frac{g_3(x,Q^2)}{2} \right )^{\nu p}
 \right ]. \nonumber\\
\label{DIC2}
\ea
%-----------------------------------------------------------------------
Eqs.~(\ref{DIC1})
 and (\ref{DIC1}, \ref{DIC2}) were
given first in~\cite{a1B} and \cite{DIC}, respectively.
In a similar way various other relations follow
for other current combinations.

Let us now derive the light quark mass corrections to the structure
functions $g_j(x)\left|_{j=1}^5 \right.$.
We follow the treatment of ref.~\cite{a18} where it was shown that
as in the massless case the polarized structure functions can be
expressed in terms of  functions $\tilde{h}_q(y, \rho)$, with
$\rho = m^2/M^2$,
and the corresponding perturbative coefficient functions.
Due to the flavour dependence of the couplings, $g_{V_i}$ and $g_{A_i}$,
one has in general to introduce the functions
$\tilde{h}_{q}(x, \rho)$
even if the ratio of $m/M$ is treated as a single parameter.
In most of the cases given below a {\it single} function,
however, suffices for an {\it effective}
 parametrization.
The  mass dependent structure functions are given by:
%-----------------------------------------------------------------------
\ba
g_1^j(x,\rho) &=&
\frac{\pi M^2x}{8} \sum_q \alpha_q^j
\int_{x+\frac{\rho}{x}}^{1+\rho}dy
\left [ x(2x-y) + 2 \rho \right ]
\tilde{h}_{q}(y,\rho),
\label{eqAA}
\\
g_2^j(x,\rho) &=&
\frac{\pi M^2}{8} \sum_q \alpha_q^j
\int_{x+\frac{\rho}{x}}^{1+\rho}dy
\left [x (2y-3x) -  \rho \right ]
\tilde{h}_{q}(y,\rho)
-
\frac{\pi m^2}{4} \sum_q \gamma_q^j
\int_{x+\frac{\rho}{x}}^{1+\rho}dy
\tilde{h}_{q}(y,\rho),
\label{eqAC}
\label{eqVIO}
\\
g_3^j(x,\rho) &=& \frac{\pi M^2 x^2}{2}   \sum_q  \beta_q^j
\int_{x+\frac{\rho}{x}}^{1+\rho}dy
(y-x) \tilde{h}_{q}(y, \rho),
\\
g_4^j(x,\rho) &=& 2x g_5(x),
\label{eqAD}
\\
g_5^j(x,\rho) &=&
\frac{\pi M^2}{8} \sum_q \beta_q^j
\int_{x+\frac{\rho}{x}}^{1+\rho}dy
\left [x (2x-y) + 2 \rho \right ] \tilde{h}_q(y,\rho),
\label{eqAB}
\ea
%-----------------------------------------------------------------------
with $\gamma_q^j = g_{A_a}^q g_{A_b}^q, j~\equiv~ab$, and
%-----------------------------------------------------------------------
\be
\tilde{h}_{q}(y, \rho)
=  \int dk^2 \hat{f}_{q}(y,k^2, \rho).
\label{eq11}
\ee
%-----------------------------------------------------------------------
Corresponding relations are obtained for charged current scattering.
The last definition applies a slightly different convention than used in
ref.~\cite{a18}.
The functions $\tilde{h}_{q}(y, \rho)$ can be determined
from the different measured
structure functions
in phenomenological analyses.
In the case of the non--photonic structure functions
the direct determination of $\tilde{h}_q$
is complicated due to the fact that these structure functions are
difficult to unfold from the measured scattering cross sections.
However, one may still use the relations~(\ref{eqAA}--\ref{eqAB})
 in  global
analyses of polarization asymmetries at large $Q^2$ as corrections
in the determination of $g_1(x,Q^2)$ in this kinematical range.

The relations (\ref{eqAA}) and (\ref{eqAC}) agree with those derived
in ref.~\cite{a18} recently for photon exchange, where $\gamma_q^j
 = 0$.
Note that for the contributions due to $Z$ or $W^{\pm}$ exchange
a new contribution to $g_2 \propto m^2/M^2$ emerges.
In a different context similar terms were obtained in \cite{a1}
as the only contributions to $g_2^B$, $B=Z, W^{\pm}$.
The
Burkhardt--Cottingham sum rule
%------------------------------------------------------------------------
\begin{equation}
\int_0^1 dx g_2^k(x) = 0
\end{equation}
%------------------------------------------------------------------------
is valid for $\rho \neq 0$
iff $\gamma_{q}^j
= g_{A_a}^{q_i} g_{A_b}^{q_i} = 0$, i.e.
for pure $Z$ and $W^{\pm}$
it is violated due to the second term in
(\ref{eqVIO}).
For charged current interactions,
on the other hand,
%------------------------------------------------------------------------
\begin{equation}
\int_0^1 dx~x \left [  g_1^k(x)+ 2 g_2^k(x) \right ] = 0
\end{equation}
%------------------------------------------------------------------------
is valid  for all values of $\rho$.
The sum--rule eq.~({\ref{qq11})
\begin{equation}
\int_0^1 dx g_3^k(x, \rho) = \int_0^1 dx g_4^k(x, \rho)
\end{equation}
holds also
for massive quarks.
Finally
also  the  Dicus relation
between $g_4(x)$ and $g_5(x)$~(\ref{eqAD}) obtains no quark mass 
corrections.
%%
%%   Note that terms of the type $m^2/M^2$ were not considered
%%   in~\cite{FRANKF}.
%%
%%






An illustration of the relative
size of the mass terms for the different
structure functions is given in figure~2 for $m/M = 0.005$~and~0.010
in terms of  relative correction factors.
These mass values mark the typical range for the light current
quark masses $m_u = 5.1 \pm 0.9 \MeV$
and $m_d = 9.3 \pm 1.4 \MeV$~\cite{BERN}.

To obtain a first estimate we use the same parametrization for
all the functions $\tilde{h}(y,\rho)$~\footnote{We
would like to
thank R.G. Roberts  for providing us with the fit parameters of the
function $\tilde{h}(y,\rho)$ determined in ref.~\cite{a18}.}.
Due to the proportionality of $g_1^j(x, \rho)$, $g_4^j(x, \rho)$, and
$g_5^j(x, \rho)$ only the ratios $\left .
g_k^j(x, \rho)/g_k^j(x)\right|_{k=1}^3$ are different.
The present data constrain these ratios to a range around unity
in the region of $x \gsim 0.02$~\footnote{ The spike in
$g_2^j(x, \rho)/g_2^j(x)$ is due to
the zero in $g_2(x)$.}. The ratio for $g_3^j$ is somewhat closer
to unity than that for $g_1$ and $g_2$ at lower $x$ values.
At smaller values of $x$
and larger values of $\rho$
the correction factors differ for the various structure functions.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sect6}
%
We have derived the twist--2 contributions to the polarized structure
functions in lowest order QCD including weak currents. The results
obtained using the operator product expansion and the covariant
parton model agree. In lowest order two out of five structure functions
are independent
for the respective current combinations
and the
remaining structure functions are related by three linear operators.
A new relation between the structure functions $g_3^j$ and $g_5^j$ was
derived. As a consequence the first moment of $g_3^j$ and $g_4^j$ are
predicted to be equal.

The light quark mass corrections to the structure functions
$\left.
g^j_k \right|_{k=1}^5$ were calculated in the covariant parton model.
The first moments of the structure functions $g_3$ and $g_4$ are
equal also in the  presence of the
quark mass corrections. The Dicus relation remains to be valid.
The Burkhardt--Cottingham sum rule is broken by a term
$\propto g_{A_a} g_{A_b} m^2/M^2$, i.e. for pure $Z$~exchange and in
charged current interactions.

\vspace{3mm}
{\bf Acknowledgement} N.K.~would like to thank DESY for the hospitality
extended to him.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Scattering},~p.~25,~in:
Proc. of the {\sf
Workshop on the Prospects of Spin
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(DESY, Hamburg, 1995).
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\newpage
\vspace*{3cm}
\begin{center}

\vspace{-4cm}
\mbox{\epsfig{file=fig1a.eps,width=16cm}}
\small
\end{center}
\small
{\sf Figure~1:}~$x$ and $Q^2$ dependence of the structure functions
$\left . g^k_i(x,Q^2) \right|_{i=1}^5$ using the parton
parametrization~\cite{GRVS} (LO, STD). The lines correspond to
$Q^2 = 10 \GeV^2$~(a) and $Q^2 = 10^4  \GeV^2$~(b). The structure
functions for  photon exchange, $g_i^{em}$~(A),
$\gamma Z$ interference,
$g_i^{\gamma Z}$~(B),
and $W^-$ exchange in charged current $l N$ scattering,
$g_i^{W^-}$~(C),
are compared separately. Full lines: $g_1$, dashed lines:
$g_2$, dotted lines: $ 10 \times g_3$, dash-dotted line: $g_5$.
The structure function $g_4$ can be obtained by the
Dicus relation
$g_4 = 2x g_5$ directly.
\normalsize

\newpage
\vspace*{3cm}
\begin{center}

\mbox{\epsfig{file=fig1b.eps,width=16cm}}
\small
\end{center}
%%%%%{\sf Figure~1b:}

\newpage
\vspace*{3cm}
\begin{center}

\mbox{\epsfig{file=fig1c.eps,width=16cm}}
\small
\end{center}
%%%%%{\sf Figure~1c:}

\newpage
\vspace*{3cm}
\begin{center}

\mbox{\epsfig{file=fig2a.eps,width=16cm}}
\small
\end{center}
{\sf Figure~2:}~$x$ depedence for the relative mass corrections
$g_i(x, m/M)/g_i(x, m=0)$. (A)~$m/M = 0.005$, (B)~$m/M = 0.010$.
Full lines: $g_1$, dashed lines: $g_2$
($\gamma_q^j
 = 0$, cf. eq.~(\ref{eqAC})),  dotted lines: $g_3$.
The ratios for $g_4$ and $g_5$ are identical to the ratio for $g_1$.
\normalsize

\newpage
\vspace*{3cm}
\begin{center}

\mbox{\epsfig{file=fig2c.eps,width=16cm}}
\small
\end{center}
%%%{\sf Figure~2b:}
\end{document}
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%{\bf DESY  96-300\\
% March 1996}
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\begin{center}
{\bf Electroweak spin-dependent structure functions } \\[1cm]
\end{center}
\begin{center}
{\bf 1.Introduction}\\
\end{center}

\begin{center}
{\bf 2. Structure functions}\\
\end{center}



The cross section of the polarized lepton-nucleon scattering
is:
\be
\frac{d^3\sigma}{dxdyd\theta}=\frac{y\alpha^2}
{Q^4}\sum_{i}L^{\mu\nu}_iW_i^{\mu\nu}\eta_i,
\label{e1}
\ee
where  $ Q^2=-q^2$ is the squared momentum transfer, $x=Q^2/(2\nu)$,
$\nu=P\cdot q$, $P$ is the four-momentum of the nucleon,$\theta$ is the
azimuthal angle of the final lepton, $y=\nu/(ME)$,
$M$ is the nucleon mass, $E$ is the energy of the initial lepton in the
laboratory frame.
$L_{\mu\nu}^i$ is leptonic tensor and $W_{\mu\nu}^i$ is hadronic one and
$i=\gamma,\gamma Z, Z, W^\pm $. In \re{e1} factors $\eta^i$ are
\begin{eqnarray}
\eta^\gamma&=&1,\nonumber\\
\eta^{\gamma Z}&=&\frac{G_FM_Z^2}{2\sqrt{2}\pi\alpha}\frac{Q^2}{Q^2+M_Z^2},
\nonumber\\
\eta^Z&=&(\eta^{\gamma Z})^2,\nonumber\\
\eta^W&=&\frac{1}{2}\left( \frac{G_FM_W^2}{4\pi\alpha}
\frac{Q^2}{Q^2+M_W^2} \right) ^2.
\end{eqnarray}


The leptonic tensor has the following form:
\be
L_{\mu\nu}^i=\sum_{\lambda^\prime}[\bar
u(k^\prime,\lambda^\prime)\gamma_\mu(g_V^i+g_A^i
\gamma_5)u(k,\lambda)]^\ast\bar
u(k^\prime,\lambda^\prime)\gamma_\nu(g_V^j+g_A^j\gamma_5)u(k,\lambda),
\label{e2}
\ee
where $k (k^\prime)$ is initial (final) lepton momentum,
$\lambda (\lambda^\prime)$ is the helicity of the initial
(final) lepton, $i=Z$ and
$j=\gamma$ for
$\gamma Z$ interference and for the negative charge initial leptons
and $ \nu$
\begin{eqnarray}
g_V^\gamma &=&1, \hspace{2.8cm} g_A^\gamma=0; \nonumber\\
g_V^Z&=&-\frac{1}{2}+2 sin^2\theta_W, {\ }{\ } g_A^Z=\frac{1}{2};
\nonumber\\
g_V^{W^-}&=&1, \hspace{2.8cm}g_A^{W^-}=-1.
\label{ac}
\end{eqnarray}
For initial positive charge leptons and $\bar\nu$ one should replace in
\re{ac} sign of the $g_A$ to opposite.


The hadronic tensor are:
\ba
W_{\mu\nu}^{i,j}=\frac{1}{4\pi}\int d^4xe^{iqx}
\langle PS\mid[J_\mu^i(x),J_\nu^j(0)]\mid PS\rangle,
\nn
\ea
where in framework of the quark model
\ba
J_\mu^i(x)=\sum_{f,f^\prime}\bar q_f\prime
(x)\gamma_\mu(g_V^i+g_A^i\gamma_5)
q_f(x)U_{ff\prime},\nn
\ea
and $P$ is the momentum of the nucleon, $S$ is its spin
($ S^2=-M^2, S\cdot P=0$), $U_{ff^\prime}$ is Cabibbo-Kobayashi-Maskawa matrix.
One can connects this hadronic tensor with the imaginary  part of the
forward amplitude of the virtual
photon nucleon scattering
\ba
W_{\mu\nu}^{ij}=\frac{1}{2\pi}Im{\ }T_{\mu\nu}^{ij}\nn,
\nn
\ea
where
\ba
T_{\mu\nu}^{ij}=i\int d^4xe^{iqx}\langle PS\mid(T{J_\mu^i}^\dagger (x)J_\nu^j(0))\mid
PS\rangle,
\nn
\ea


Using the Lorentz and time-reversal invariance and the current
conservation
\footnote{ The discussion of structure functions which are
connected with the small current conservation violation effects have
been done in \ct{a7}.} the
general form for hadronic tensor is:
\begin{eqnarray}
W_{\mu\nu}&=&(-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2})F_1(x,Q^2)+
\frac{\hat{P}_\mu\hat{P}_\nu}{P\cdot q}F_2(x,Q^2)-
 i\epm\frac{q_\lambda P\sigma}{2P\cdot q}F_3(x,Q^2)\nn\\
&{+}& i\epm\frac{q^\lambda S^\sigma}{P\cdot q}g_1(x,Q^2)+
i\epm\frac{q^{\lambda}(P\cdot qS^\sigma-S\cdot qP^\sigma)}
{(P\cdot q)^2}g_2(x,Q^2)\nn\\
&{+}&[\frac{\hat{P_\mu}\hat{S_\nu}+\hat{S_\mu}\hat{P_\nu}}{2}-
S\cdot q\frac{\hat{P_\mu}\hat{P_\nu}}{(P\cdot q)}]
\frac{g_3(x,Q^2)}{P\cdot q}\nn\\
&+&
S\cdot q\frac{\hat{P_\mu}\hat{P_\nu}}{(P\cdot q)^2}
g_4(x,Q^2)+
(-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2})\frac{(S\cdot q)}{P\cdot q} g_5(x,Q^2),
\label{e6}
\end{eqnarray}
where
\begin{eqnarray}
\hat{P_\mu}=P_\mu-\frac{P\cdot qq_\mu}{q^2},{\ }
\hat{S_\mu}=S_\mu-\frac{S\cdot qq_\mu}{q^2},\nn
\end{eqnarray}
and  lepton masses have been neglected.

Connection between our structure functions and the structure
functions from the \ct{a1}- \ct{new} is given in the
Table 1.

\begin{center}
\begin{tabular}{|c|c|c|c|c|}\hline
$our{\ } notation$      & $\ct{a1},\ct{a2}$
    & $\ct{a3}-\ct{a6}$ & $\ct{a7}$ & $\ct{new}$\\ \hline
$g_1$ & $g_1$ &  $g_1$      & $g_1$     & $g_1$\\
$g_2$ & $g_2$ &  $g_2$      & $g_2$  & $g_2$\\
$g_3$ & $-g_3$ &  $\frac{g_4-g_5}{2}$      & $b_1+b_2$  & $\frac{A_2-A_3}{2}$ \\
$g_4 $
& $g_4-g_3$
& $g_4$
&$a_2+b_1+b_2$ &$A_2$
\\
$g_5$
& $-g_5$
&$ g_3$
&$a_1$ & $A_1$

\\  \hline
\end{tabular}
\end{center}


A similar decomposition for forward Compton scattering amplitude
has the form
\begin{eqnarray}
T_{\mu\nu}&=&(-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2})T_1(q^2,\nu)+
\frac{\hat{P}_\mu\hat{P}_\nu}{M^2}T_2(q^2,\nu)-
i\epm\frac{q_\lambda P\sigma}{2M^2}T_3(q^2,\nu)\nn\\
&{+}& i\epm\frac{q^\lambda S^\sigma}{M^2}A_1(q^2,\nu)+
i\epm\frac{q^{\lambda}(P\cdot qS^\sigma-S\cdot q
P^\sigma)}{M^4}A_2(q^2,\nu)\nn\\
&{+}&[\frac{\hat{P_\mu}\hat{S_\nu}+\hat{S_\mu}\hat{P_\nu}}{2}-
S\cdot q\frac{\hat{P_\mu}\hat{P_\nu}}{(P\cdot q)}]
\frac{A_3(q^2,\nu)}{M^2}\nn\\
&+&
S\cdot q\frac{\hat{P_\mu}\hat{P_\nu}}{M^4}A_4(q^2,\nu)+
(-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2})\frac{S\cdot q}{M^2}A_5(q^2,\nu).
\label{e7}
\end{eqnarray}

>From \re{e1},\re{e2} and \re{e6} one can obtain for cross section of
the scattering of the lepton with helicity $\lambda$ off  the nucleon
with the longitudinal
$S_L=(0,0,0,M)$ and transverse $S_T=M(0,cos\alpha,sin\alpha,0)$
the formules
\ba
\frac{d^2\sigma(\lambda,S_L)}{dxdy}&=&4\pi MEy\frac{\alpha^2}{Q^4}\sum_i
\eta_iC_i\{2xyF_1^i+\frac{2}{y}(1-y-\frac{xyM}{2E})F_2^i\nn\\
&-& 2\lambda x(1-\frac{y}{2})F_3^i-2\lambda x(2-y-\frac{xyM}{E})g_1^i+
4\lambda\frac{x^2M}{E}g_2^i\nn\\
&+&\frac{2xM}{yE}(1-y-\frac{xyM}{2E})g_3^i-\frac{2}{y}(1+\frac{xM}{E})
(1-y-\frac{xyM}{2E})g_4\nn\\
&-&2xy(1+\frac{xM}{E})g_5^i\},
\nn
\ea
\ba
\frac{d^3\sigma(\lambda,S_T)}{dxdyd\phi}&=&2MEy\frac{\alpha^2}{Q^4}\sum_i
\eta_iC_i\{2xyF_1^i+\frac{2}{y}(1-y-\frac{xyM}{2E})F_2^i\nn\\
&-& 2\lambda x(1-\frac{y}{2})F_3^i
+\frac{\sqrt{xyM[2(1-y)E-xyM]}}{E}\cos(\alpha-\phi)[
-2\lambda xg_1^i-
4\lambda\frac{x}{y}g_2^i\nn\\
&-&\frac{1}{y^2}(2-y-\frac{xyM}{E})g_3^i+\frac{2}{y^2}
(1-y-\frac{xyM}{2E})g_4
+2xyg_5^i\},
\nn
\ea
where $C^\gamma=1$, $C^{\gamma Z}=g_V+\lambda g_A $,
$C^Z=(g_V+\lambda g_A)^2$ and $C^{W^\pm}=(1\pm\lambda)$.
 The spin asymmetries are
\ba
\Delta^L\sigma^i(\lambda)&=&
\frac{d^2\sigma(\lambda,S_L)}{dxdy}-
\frac{d^2\sigma(\lambda,-S_L)}{dxdy}\nn\\
&=&16\pi MEy\frac{\alpha^2}{Q^4}\sum_i
\eta_iC_i\{-\lambda x(2-y-\frac{xyM}{E})g_1^i+
2\lambda\frac{x^2M}{E}g_2^i\nn\\
&+&\frac{xM}{yE}(1-y-\frac{xyM}{2E})g_3^i-\frac{1}{y}(1+\frac{xM}{E})
(1-y-\frac{xyM}{2E})g_4
-xy(1+\frac{xM}{E})g_5^i\},
\label{e79}
\ea
\ba
\Delta^T\sigma^i(\lambda)&=&
\frac{d^3\sigma(\lambda,S_T)}{dxdyd\phi}-
\frac{d^3\sigma(\lambda,-S_T)}{dxdyd\phi}\nn\\
&=&4My\frac{\alpha^2}{Q^4}\sum_i
\eta_iC_i\{ \sqrt{xyM[2(1-y)E-xyM]} \cos(\alpha-\phi)[
-2\lambda xg_1^i-
4\lambda\frac{x}{y}g_2^i\nn\\
&-&\frac{1}{y^2}(2-y-\frac{xyM}{E})g_3^i+\frac{2}{y^2}
(1-y-\frac{xyM}{2E})g_4
+2xyg_5^i\}.
\label{e80}
\ea
>From \re{e79} it is  followed that  the contribution of the $g_2$ and
 $g_3$ structure function
to the longitudinal asymmetries is small.

The expressions for the asymmetries include a lot of the  structure functions
and therefore it is very difficult  to extract the information
on just one of the spin-dependent structure function.
 As very well know that  quark model allows us to reduce the number of
 the independent structure functions for unpolarized case.
  In the paper
\ct{a1}, \ct{a2} it was shown that in the framework of the
so called "naive" parton model with mass of the quark equal
to $m_q=xM$ one can obtain the analog of the
Callan-Gross relation $g_3=2xg_5$  and the relation
$g_4=0$(in the \ct{a1}, \ct{a2} notation).
The similar relations $g_3=2xg_4$ (in the \ct{a3}-\ct{a6} notation)
and $a_1=2xa_2$ (in the \ct{a7} notation)
were obtained early in the \ct{a3}-\ct{a7} for the massless quarks.

However from the Table 1 it is followed that
we have some contradiction between two results. So, if the
Callan-Gross relation for spin-dependent
structure functions in all approaches is correct, then we should have
$b_1+b_2=0$ in  \ct{a3}-\ct{a6} and therefore $g_3=0$ in \ct{a1}-\ct{a2}
(see Table 1). The equation $g_3=0$ and the Callan-Gross relation
leads to the conclusion that $g_5=0$ in \ct{a1}-\ct{a2}.

It is very well know for electromagnetic case
that one should be very careful to take into
account  the mass of the quarks to obtain the correct results
for spin-dependent structure functions, specialy for $g_2$ structure
function.

This is more difficult task for the electroweak interaction
due to existance of the  additional $\gamma_\mu\gamma_5$ vertex.
As the result, the different ways to introduce the value of the
quark masses  lead to the different expressions for spin-
dependent structure functions. So, for example, we will show
 that the relations from
\ct{a1},\ct{a2}:
\be
g_2^Z=-\frac{1}{2}\sum_q(g_A^2)_q(\Delta q+\Delta\bar q),
\nn
\ee
\be
2g_2^{W^\pm}=-g_1^{W^\pm}
\nn
\ee
are not correct.
 It will be shown  that the many others results of the papers \ct{a1},\ct{a2}
for $g_2^i$, $g_3^i$ and $g_4^i$ structure functions
is not valid due to noncorrect
treatment of the value of the quark mass.





\begin{center}
{\bf 3. Operator product expansion}\\
\end{center}

Operator product expansion is the most general formalism for
discussion of the properties of the DIS structure functions (see
\ct{mu}
and references therein).
In this section we will perform the operator product expansion for
 product of the two weak currents. We   not include the $\alpha_s$
 corrections
to the coefficient functions here. They will be taken into account at the
end  of the paper.


Let us consider the T-product of the two electroweak currents:
\ba
T_{ab}=T({J_\mu^a}^\dagger (x)J_\nu^b(0)),
\nn
\ea
where
\ba
J_\mu^a(x)=\bar q(x)\gamma_\mu(g_V+g_A\gamma_5)\lambda^a q(x),
\nn
\ea
$\lambda^a$ is some $SU(N_f)$ matrix, $N_f$ is a number of the
flavours considered to be light.


Then near light cone we have the following expansion:
\ba
T_{ab}&=&\bar q(x)\gamma_\mu(g_{V_1}+g_{A_1}
\gamma_5)S(x)\gamma_\nu(g_{V_2}+
g_{A_2}\gamma_5)\lambda^a\lambda^b q(0)\nn\\
&{+}&\bar q(0)\gamma_\nu(g_{V_2}+g_{A_2}\gamma_5)S(-x)
\gamma_\mu(g_{V_1}+
g_{A_1}\gamma_5)\lambda^b\lambda^a q(x),
\nn
\ea
where
\ba
S(x)\approx\frac{2i\not \!{x}}{(2\pi)^2(x^2-i0)^2}
\nn
\ea
is the free quark propagator.


By using of the relations:
\ba
\gamma_\mu\not \!{x}\gamma_\nu=x^\alpha (\Sa\gb-i\epa\gb\g),\nn
\ea
where
\be
\Sa=g_{\mu\alpha}g_{\nu\beta}+g_{\mu\beta}g_{\nu\alpha}-
g_{\mu\nu}g_{\alpha\beta},
\label{e9}
\ee
and for $SU(3)_f$
\ba
\lambda^a\lambda^b=(if^{abc}+d^{abc})\lc
\nn
\ea
one  get for spin-dependent part of the $T_{i,j}$
formula:
\ba
T_{ab}^{spin}&=&\frac{2ix^\alpha}{(2\pi)^2(x^2-i0)^2}
[(g_{V_1}g_{V_2}+g_{A_1}g_{A_2})\nn\\
&{\times}&(-i\epa(if^{abc}+d^{abc})\bq(x)\gb\g\lc q(0)
-i\epa(-if^{abc}+d^{abc})\bq(0)\gb\g\lc q(x))\nn\\
&{+}&(g_{V_1}g_{A_2}+g_{A_1}g_{V_2})
(\Sa (if^{abc}+d^{abc})\bq(x)\gb\g\lc q(0)\nn\\
&{-}&\Sa (-if^{abc}+d^{abc})\bq(0)\gb\g\lc q(x))]\nn
\ea
By using of the Taylor expansion for the matrix elements at $x=0$
\ba
\bq(x)\gb\g\lc q(0)&=&\sum_n \frac{(-1)^n}{n!}x_{\mu_1}...x_{\mu_n}\bq
\gb\g  D^{\mu_1}... D^{\mu_n}\lc q,\nn\\
\bq(0)\gb\g\lc q(x)&=&\sum_n \frac{(+1)^n}{n!}x_{\mu_1}...x_{\mu_n}\bq
\gb\g  D^{\mu_1}... D^{\mu_n}\lc q,\nn
\ea
we can find the final expression:

\ba
T_{ab}^{spin}&=&\frac{4ix^\alpha}{(2\pi)^2(x^2-i0)^2}
[-i\epa (g_{V_1}g_{V_2}+g_{A_1}g_{A_2})\nn\\
&\times&  \{ if^{abc}\sum_{n{\ } odd}\frac{x_{\mu_1}...x_{\mu_n}}{n!}\bq
\gb\g  D^{\mu_1}... D^{\mu_n}\lc q\nn\\
&{+ }& d^{abc}\sum_{n{\ } even}\frac{x_{\mu_1}...x_{\mu_n}}{n!}\bq
\gb\g  D^{\mu_1}... D^{\mu_n}\lc q )\} \nn\\
&{-}&\Sa (g_{V_1}g_{A_2}+g_{A_1}g_{V_2})
\{if^{abc}\sum_{n{\ } even}\frac{x_{\mu_1}...x_{\mu_n}}{n!}\bq
\gb\g  D^{\mu_1}... D^{\mu_n}\lc q\nn\\
&{+}& d^{abc}\sum_{n{\ }odd}\frac{x_{\mu_1}...x_{\mu_n}}{n!}\bq
\gb\g  D^{\mu_1}... D^{\mu_n}\lc q \}].
\label{e10}
\ea

\begin{center}
{\bf 3. Sum rules for electroweak spin-dependent structure
functions }
\end{center}

Let us consider  the operator product expansion of the
electromagnetic currents in first
\footnote{ This case have  been considered  in \ct{a8} very carefully. We
   repeat some detailes of this calculation  to
  use them for calculation of the electroweak  spin-dependent
  structure functions.}.
>From \re{e10} one have got
\ba
T_{\gamma}^{spin}=\epa\frac{e_q^2x^\alpha}{\pi^2(x^2-i0)^2}
\sum_{n-even}\frac{ix_{\mu_1}...ix_{\mu_n}}{n!}
\Theta^{\beta\{\mu_1...\mu_n\}},
\nn
\ea
where
\ba
\Theta^{\beta\{\mu_1...\mu_n\}}
=\frac{i^n}{n!}
\sum_{perm \{\mu_i\}}\bq
\gb\g  D^{\mu_1}... D^{\mu_n}q.
\nn
\ea

It is very useful to introduce the following decomposition
of the operator $\Theta^{\beta\{\mu_1...\mu_n\}}$
\ba
\Theta^{\beta\{\mu_1...\mu_n\}}={\Theta_S}^{\beta\{\mu_1...\mu_n\}}+
{\Theta_A}^{\beta\{\mu_1]\mu_2...\mu_n\}}
\nn
\ea
where
\be
{\Theta_S}^{\beta\{\mu_1...\mu_n\}}=\frac{1}{n+1}[
\Theta^{\beta\{\mu_1...\mu_n\}}+\Theta^{\mu_1\{\beta\mu_2...\mu_n\}}+
\cdots],
\nn
\ee
\ba
{\Theta_A}^{\beta\{\mu_1]\mu_2...\mu_n\}}&=&\frac{1}{n+1}[
\Theta^{\beta\{\mu_1...\mu_n\}}-\Theta^{\mu_1\{\beta\mu_2...\mu_n\}}+\nn\\
&{\  }&\Theta^{\beta\{\mu_1...\mu_n\}}-\Theta^{\mu_2\{\mu_1\beta...\mu_n\}}
+\cdots],
\nn
\ea
Nucleon matrix elements of these operators are:
\ba
\langle PS\mid{\Theta_S}^{\beta\{\mu_1...\mu_n\}}\mid PS\rangle=\frac{a_n}{n+1}
[S^\beta P^{\mu_1}P^{\mu_2}...P^{\mu_n}+
S^{\mu_1} P^{\beta}P^{\mu_2}...P^{\mu_n}+\cdots-{\sf traces}],
\label{e11}
\ea
and

\ba
\langle PS\mid{\Theta_A}^{\beta\{\mu_1\mu_2...\mu_n\}}\mid PS\rangle
&=&\frac{d_n}{n+1}
[(S^\beta P^{\mu_1}-S^{\mu_1}P^\beta ) P^{\mu_2}...P^{\mu_n}+\nn\\
&{\ }&(S^\beta P^{\mu_2}-S^{\mu_2}P^\beta ) P^{\mu_1}...P^{\mu_n}+
\cdots-{\sf traces}];
\label{e12}
\ea
For Compton forward scattering amplitude we have the following
formula
\ba
T_{\mu\nu}=i\int d^4xe^{iqx}T(J_\mu^\gamma(x)J_\nu^\gamma(0))&=&
\epa \frac{ie_q^2}{\pi^2}\int d^4xe^{iqx}\frac{x^\alpha}{(x^2-i0)^2}
\sum_{n=0,2\cdots}\frac{ix_{\mu_1}...ix_{\mu_n}}{n!}
\Theta^{\beta\{\mu_1...\mu_n\}}\nn\\
&=&-ie_q^2\epa q^\alpha \sum_{n=0,2\cdots}q^{\mu_1}...q^{\mu^n}
(-\frac{2}{q^2})^{n+1}\Theta^{\beta\{\mu_1...\mu_n\}},\nn
\ea
By using the \re{e11}, \re{e12} one can obtain
\ba
T_{\mu\nu}&=&-ie_q^2\epa \frac{q^\alpha S^\beta}{\nu}\sum_{n=0,2\cdots}
\frac{a_n+d_nn}{x^{n+1}(n+1)}\nn\\
&-&ie_q^2\epa \frac{q^\alpha P^\beta(Sq)}{\nu^2}\sum_{n=0,2\cdots}
\frac{n(a_n-d_n)}{x^{n+1}(n+1)},
\label{e122}
\ea
where $x=-q^2/2\nu$.
>From \re{e7} and \re{e122}  we get

\ba
\alpha_1(x,Q^2)+\alpha_2(x,Q^2)&=&\sum_{n=0,2\cdots}
\frac{e_q^2(a_n+d_nn)}{x^{n+1}(n+1)},\nn\\
\alpha_2(x,Q^2)&=&\sum_{n=0,2\cdots}
\frac{e_q^2n(d_n-a_n)}{x^{n+1}(n+1)},
\label{e13}
\ea
where
\be
\alpha_1(x,Q^2)=\frac{\nu}{M^2}A_1(q^2,\nu),{\ }
\alpha_2(x,Q^2)=\frac{\nu^2}{M^4}A_2(q^2,\nu).\nn
\ee
>From the other hand we can use dispersion relations to
connect the amplitudes $A_i(q^2,\nu)$ with moments of the structure functions
$g_i(x,Q^2)$.
The crossing  relations for electroweak current are:
\ba
 T^{\gamma,Z}_{\mu\nu}(-q,P)&=&T^{\gamma,Z}_{\nu\mu}(q,P),\nn\\
 T^\pm_{\mu\nu}(-q,P)&=&\pm T^\pm_{\nu\mu}(q,P),
\label{cr}
\ea
where
\be
T^\pm=T^{W^-}\pm T^{W^+}.\nn
\ee

Therefore  for the electromagnetic current one has
\be
A_1(q^2,-\nu)=A_1(q^2,\nu), {\ } A_2(q^2,\nu)=-A_2(q^2,\nu).
\nn
\ee
 By using Cauchy's theorem, dispersion relations are:
\ba
A_1(q^2,\nu)&=&\frac{2}{\pi}\int_{-q^2/2}^\infty\frac{\nu^\prime
d\nu^\prime}{\nu^{\prime 2}-\nu^2}ImA_1(q^2,\nu^\prime),\nn\\
A_2(q^2,\nu)&=&\frac{2\nu}{\pi}\int_{-q^2/2}^\infty\frac{
d\nu^\prime}{\nu^{\prime 2}-\nu^2}ImA_2(q^2,\nu^\prime)\nn
\ea
These relations can be rewriting in the form
\ba
\alpha_1(\omega,Q^2)&=&4\omega\int_{-q^2/2}^\infty\frac{
d\omega^\prime}{\omega^{\prime 2}-\omega^2}g_1(\omega^{\prime},Q^2),\nn\\
\alpha_2(\omega,Q^2)&=&4\omega^3\int_{-q^2/2}^\infty\frac{
d\omega^\prime}{\omega^\prime(\omega^{\prime 2}-\omega^2)}
g_2(\omega^\prime,Q^2),
\nn
\ea
where $\omega=1/x$.
Performing  the Taylor expansion on $\omega=1/x$, one obtain
\ba
\alpha_1(x,Q^2)=\frac{4}{x}\sum_{n=0,2...}\frac{1}{x^n}\int_0^1
dyy^ng_1(y,Q^2),\nn\\
\alpha_2(x,Q^2)=\frac{4}{x^3}\sum_{n=2,4...}\frac{1}{x^n}\int_0^1
dyy^{n+2}g_2(y,Q^2).
\label{e14}
\ea
>From \re{e13}, \re{e14} it follows
\ba
\int_0^1dxx^ng_1(x,Q^2)=\sum_q\frac{e^2_qa_n^q}{4},{\ } n=0,2...\nn\\
\int_0^1dxx^ng_2(x,Q^2)=\sum_q\frac{e^2_qn(d_n^q-a_n^q)}
{4(n+1)},{\ }
n=2,4...
\label{e15}
\ea

For the first moment of the $g_1(x,Q^2)$ and $g_2(x,Q^2)$ we get
\ba
\int_0^1dxg_1(x,Q^2)=\sum_q\frac{e_q^2}{2}(\Delta q+\Delta \bq),
\nn
\ea
where
\ba
\langle PS\mid\bq\gamma_\beta\g q\mid PS\rangle=S_\beta a_0=2S_\beta\Delta q,
\nn
\ea
and
\be
\int_0^1dxg^\gamma_2(x,Q^2)=0.
\label{e155}
\ee
if \re{e15} is also correct for $n=0$ case.
The last equation is the Burkhardt-Cottingham sum rule \ct{a133}.



 If the twist-3 contribution is very small $\{d_n\}=0$ \re{e15},
 then Wandzura-Wilczek relation should be  hold \ct{ww}:
\be
g_2^\gamma(x,Q^2)=-g_1^\gamma(x,Q^2)+\int_x^1\frac{dy}{y}g_1^\gamma(y,Q^2).
\nn
\ee

Now we will apply the same method to the Z-boson structure functions.
The current for Z-bozon exchange has the following form
\ba
J_\mu^Z(x)=\bq\gamma_\mu(g_V^q+g_A^q\g)q,\nn
\ea
where
\ba
g_V^q=\frac{1}{2}-\frac{4}{3}sin^2\theta_W,{\ }g_A^q=-\frac{1}{2},{\ }
 q=u,c;\nn\\
g_V^q=-\frac{1}{2}+\frac{2}{3}sin^2\theta_W,{\ }g_A^q=\frac{1}{2},{\ }
 q=d,s,
\label{e111}
\ea
and for the antiquarks the sign of the $g_A$ in \re{e111} should be
opposite.

>From \re{e10} one has
\ba
T(J_\mu^Z(x)J_\nu^Z(0)^{spin}&=&\frac{ix^\alpha}{(\pi)^2(x^2-i0)^2}
[-i\epa ((g_V^q)^2+(g_A^q)^2)\nn\\
&{\times }& \sum_{n{\ } even}\frac{x_{\mu_1}...x_{\mu_n}}{n!}\bq
\gb\g \vec D^{\mu_1}...\vec D^{\mu_n}q) \nn\\
&{-}&2\Sa g_V^qg_A^q\sum_{n{\ } odd}\frac{x_{\mu_1}...x_{\mu_n}}{n!}\bq
\gb\g  D^{\mu_1}... D^{\mu_n} q\}].
\label{e16}
\ea
The term in \re{e16} which includes tensor $\epa$ has the same structure as for
photon exchange and therefore
\ba
\int_0^1dxx^ng_1^Z(x,Q^2)=\sum_q\frac{((g_V^q)^ 2+(g_A^q)^2)a_n^q}{4},
{\ } n=0,2...\nn\\
\int_0^1dxx^ng_2^Z(x,Q^2)=\sum_q\frac{((g_V^q)^2+(g_A^q)^2)n(d_n^q-a_n^q)}{4(n+1)},{\ }
n=2,4...
\nn
\ea
The first moment of the $g_1^Z(x,Q^2)$ is
\ba
\int_0^1dxg_1^Z(x,Q^2)=\sum_q\frac{(g_V^q)^2+(g_A^q)^2}{2}(\Delta
q+\Delta \bq),
\nn
\ea
There are  also  analogues of the
Wandzura-Wilczek and Burhhardt-Cottingham relations:

\ba
g_2^Z(x,Q^2)=-g_1^Z(x,Q^2)+\int_x^1\frac{dy}{y}g_1^Z(y,Q^2),
\label{e177}
\ea
\ba
\int_0^1dxg^Z_2(x,Q^2)=0.
\label{e188}
\ea



The new information on the parton spin distribution functions can be
obtained from  the structure functions $g_3, g_4, g_5$ which are
are connected with the  parity violated part of the weak current.

It is easy to show that these structure functions are given
 by the term which includes the tensor structure $\Sa $ in \re{e16}:
\ba
T(J_\mu^Z(x)J_\nu^Z(0)_{\Sa}^{spin}&=&-2\Sa g_V^qg_A^q\frac{ix^\alpha}
{\pi^2(x^2-i0)^2}
\sum_{n{\ }odd}\frac{x_{\mu_1}...x_{\mu_n}}{n!}\bq
\gb\g  D^{\mu_1}... D^{\mu_n}q.
\nn
\ea
By using the explicit form of the $\Sa $ \re{e9}, one can find out
the structure at tensor $g_{\mu\nu}$ is related to the $g_5$
structure function in \re{e6} and to the amplitude $A_5$ in \re{e7}:
\ba
T_{g_{\mu\nu}}=-2g_{\mu\nu} g_V^qg_A^q\frac{ix^\beta}{\pi^2(x^2-i0)^2}
\sum_{n{\ }odd}\frac{ix_{\mu_1}...ix_{\mu_n}}{n!}\Theta^{\beta\{\mu_1...\mu_n\}}.
\nn
\ea
Therefore we have got the following
contribution to the forward Compton amplitude:
\ba
(T)^{spin}_{g_{\mu\nu}}=-2g_{\mu\nu}g_A^qg_V^q\frac{(Sq)}{\nu}
\sum_{n{\ }odd}\frac{a_n}{x^{n+1}}\nn
\ea
The amplitude $A_5$ is
\be
\frac{\nu}{M^2}A_5(q^2,\nu)=2g_A^qg_V^q
\sum_{n{\ }odd}\frac{a_n}{x^{n+1}}.
\label{e18}
\ee
The crossing relation gives $A_5(q^2,\nu)=-A_5(q^2,-\nu)$.
Using the dispersion relation
\ba
A_5(q^2,\nu)&=&\frac{2\nu}{\pi}\int_{-q^2/2}^\infty\frac{
d\nu^\prime}{\nu^{\prime 2}-\nu^2}ImA_5(q^2,\nu^\prime))\nn
\ea

leads to
equation:
\be
\alpha_5(q^2,\nu)=4\sum_{n=1,3...}\frac{1}{x^{n+1}}\int_0^1
dyy^ng_5(y,Q^2),
\label{e19}
\ee
where  $ \alpha_5(x,Q^2)=\nu
A_5(q^2,\nu)/M^2$ .
>From \re{e18} and \re{e19} it follows:
\be
\int_0^1dxx^ng_5^Z(x,Q^2)=\sum_{q}\frac{g_A^qg_V^q}{2}a_n^q,{\ }{\ }
 n=1,3...;
\label{e199}
\ee

The structure functions $g_3$ and $g_4$ are connected with the
part of the $T_{\mu,\nu} $ which is  the  at the rest
part of the tensor $\Sa $:
\ba
T(J_\mu^Z(x)J_\nu^Z(0)^{g_3,g_4}&=&-2
(g_{\mu\alpha}g_{\nu\beta}+g_{\mu\beta}g_{\nu\alpha})
g_V^qg_A^q\frac{ix^\alpha}{\pi^2(x^2-i0)^2}\nn\\
&{\times }&
\sum_{n{\ }odd}\frac{x_{\mu_1}...x_{\mu_n}}{n!}\bq
\gb\g  D^{\mu_1}... D^{\mu_n}q.
\nn
\ea
>From this equation it is followed:
\ba
T_{g_3,g_4}=2g_V^qg_A^q\frac{ix^\mu}{\pi^2(x^2-i0)^2}
\sum_{n{\ }odd}\frac{ix_{\mu_1}...ix_{\mu_n}}{n!}\Theta^{\nu\{\mu_1...\mu_n\}}
+\nu\leftrightarrow\mu.
\nn
\ea
The contribution to the forward Compton amlpitude is
\ba
T_{3,4}&=&-\frac{g_V^qg_A^q}{\pi^2}\int \frac{d^4xe^{iqx}}{(x^2-i0)}
\sum_{n{\ }even}\frac{x_{\mu_1}...x_{\mu_n}}{n!}\Theta^{\nu\{\mu\mu_1...\mu_n\}}
+\nu\leftrightarrow\mu.
\nn
\ea
where  the equation $2x^\mu/(x^2-i0)^2=-\partial^\mu(1/(x^2-i0))$ was used.
After calculation we have
\ba
T_{3,4}=2g_V^qg_A^q
\sum_{n{\ } odd}q_{\mu_1}...q_{\mu_n}(-\frac{2}{q^2})^{n+1}
\Theta^{\nu\{\mu\mu_1...\mu_n\}}
+\nu\leftrightarrow\mu.
\nn
\ea
By using of the explicit form for the tensor
$\Theta^{\nu\{\mu\mu_1...\mu_n\}}$ \re{e11}, \re{e12}, we have
\ba
T_{3,4}&=&2g_V^qg_A^q\sum_{n{\ }even}q_{\mu_1}...q_{\mu_n}(-\frac{2}{q^2})^{n+1}
\frac{a_{n+1}}{n+2}[S_\mu P_\nu P_{\mu_1}...P_{\mu_n}\nn\\
&{+ }&S_\nu P_\mu P_{\mu_1}...P_{\mu_n}+S_{\mu_1} P_\mu P_{\nu}...P_{\mu_n}+...
{\ }\nu\leftrightarrow\mu]\nn\\
&=&2g_V^qg_A^q\sum_{n{\ }even}(-\frac{2}{q^2})^n\frac{a_{n+1}}{(n+2)}
[(S_\mu P_\nu+S_\nu P_\mu)\nu^n+nP_\mu P_\nu(S\cdot q)\nu^{n-1}]\nn\\
&=&4g_V^qg_A^q[(S_\nu P_\mu+P_\nu
S_\mu)\sum_{n{\ }odd}
\frac{a_n}{x^n(n+1)}+P_\mu P_\nu\frac{(S\cdot q)}{\nu}\sum_{n{\ }odd}
\frac{(n-1)a_n}{x^n(n+1)}]\nn\\
&=&\frac{8g_V^qg_A^q}{\nu}[\frac{S_\nu P_\mu+P_\nu S_\mu}{2}-P_\mu
P_\nu\frac{(S\cdot q)}{\nu}]\sum_{n{\ }odd}
\frac{a_n}{x^n(n+1)}\nn\\
&+&4g_V^qg_A^q P_\mu P_\nu\frac{(S\cdot q)}{\nu^2}\sum_{n{\ }odd}
\frac{a_n}{x^n}.
\nn
\ea
Therefore the amplitudes are:
\ba
\frac{\nu A_3(q^2,\nu)}{M^2}=8g_A^qg_V^q
\sum_{n{\ }odd}\frac{a_n}{(n+1)x^{n}},
\nn
\ea
\ba
\frac{\nu^2 A_4(q^2,\nu)}{M^4}=4g_A^qg_V^q
\sum_{n{\ }odd}\frac{a_n}{x^{n}}.
\nn
\ea
>From dispersion relations for these amplitudes one can obtain:
\ba
\alpha_3(q^2,\nu)=4\sum_{n=1,3...}\frac{1}{x^n}\int_0^1
dyy^{n-1}g_3(y,Q^2),
\nn
\ea
\ba
\alpha_4(q^2,\nu)=4\sum_{n=1,3...}\frac{1}{x^n}\int_0^1
dyy^{n-1}g_4(y,Q^2),
\nn
\ea
where $\alpha_3(q^2,\nu)=\nu A_3/M^2$, $\alpha_4(q^2,\nu)=\nu^2
A_4/M^4$, and the crossing properties $A_3(q^2,-\nu)=A_3(q^2,\nu)$,
$A_4(q^2,-\nu)=-A_4(q^2,\nu)$, have been used.
The final result for the structure functions is:
\be
\int_0^1dxx^ng_3^Z(x,Q^2)=2\sum_{q}g_A^qg_V^q\frac{a_{n+1}^q}{n+2},{\
  }{\ }
n=0,2...,
\label{e28}
\ee
\be
\int_0^1dxx^ng_4^Z(x,Q^2)=\sum_{q}g_A^qg_V^qa_{n+1}^q,{\ }
{\  }{n}=0,2... .
\label{e29}
\ee



>From \re{e29} and  \re{e199} the analog of the Callan-Gross relation for
the electroweak
structure functions \ct{a1}-\ct{a7} is followed:
\be
g_4^Z(x,Q^2)=2xg_5^Z(x,Q^2),
\label{e30}
\ee
By using of the \re{e28}, \re{e29} and \re{e199} the new sum rules
has obtained:
\be
{g_3}^Z(x,Q^2)=2x\int_x^1\frac{dy}{y^2}{g_4}^Z(y,Q^2)
\label{e300}
\ee
or
\be
{g_3}^Z(x,Q^2)=4x\int_x^1\frac{dy}{y}{g_5}^Z(y,Q^2)
\label{e31}
\ee



It should be mentioned that \re{e300}, \re{e31}  does not include any twist-3
contribution and therefore this is very interesting analog of the
Wandzura-Wilczek sum rule without high twist contribution.

It is  easy to show that the same method leads to the
following formulas for the photon-Z interference spin-dependent
structure functions:
\ba
\int_0^1dxx^ng_1^{\gamma Z}(x,Q^2)&=&
\sum_q\frac{e_qg_V^qa_n^q}{2},{\ } n=0,2...;\nn\\
\int_0^1dxx^ng_2^{\gamma
  Z}(x,Q^2)&=&\sum_q\frac{e_qg_V^qn(d_n^q-a_n^q)}
{2(n+1)},{\ }
n=2,4...;\nn\\
\int_0^1dxx^ng_3^{\gamma Z}(x,Q^2)&=&
2\sum_{q}e_qg_A^q\frac{a_{n+1}^q}{n+2},{\ }{\ }n=0,2...;
\nn\\
\int_0^1dxx^ng_4^{\gamma Z}(x,Q^2)&=&\sum_{q}e_qg_A^qa_{n+1}^q,{\ }{\
  }n=0,2...;\nn\\
\int_0^1dxx^ng_5^{\gamma Z}(x,Q^2)&=&\sum_{q}\frac{e_qg_A^q}{2}a_n^q,
{\ }{\ } n=1,3...  ;
\nn
\ea

There are also the analogues of the Callan-Gross \re{e30},
Wandzura-Wilczek \re{e177},\re{e300}, \re{e31}
and Burkhardt-Cottingham \re{e188}
relations for $\gamma Z$
spin-dependent
structure functions.



The main difference of the operator product expansion of
the product of the  charge currents is the contribution
from the term in the \re{e10}
which is proportional to tensor $f^{abc}$. It leads to the
additional contribution to the structure functions
which comes from the  operators (see \re{e10}) with $ n $ even.
The weak charge current has the following form:
\ba
J_\pm=\sum_{i=1,2}\bar q\gamma_\mu(1-\g)\tau_\pm q_i,
\nn
\ea
where
\begin{equation}
q_1 =
\left( \begin{array}{c}
u\\
d_c\\
 \end{array} \right) \,{\ } {\ } q_2 =
\left( \begin{array}{c}
c\\
s_c\\
\end{array} \right) \,
\end{equation}
and $d_c=d cos\theta_c+s sin\theta_c$ and
$s_c=s cos\theta_c-d sin\theta_c $.

For charge current product we have the expansion:
\ba
T^{W^\pm}&=&\bar q(x)\gamma_\mu(1-
\gamma_5)S(x)\gamma_\nu(1-\gamma_5)\frac{1\mp\tau_3}{2} q(0)\nn\\
&{+}&\bar q(0)\gamma_\nu(1-\gamma_5)S(-x)
\gamma_\mu(1-
\gamma_5)\frac{1\pm\tau_3}{2} q(x),
\nn
\ea


It is useful to introduce the amplitudes (and related structure
functions) \ct{a13}
\be
A_i^{\pm}={A_i}^{W^-}\pm {A_i}^{W^+}.
\nn
\ee
These amplitudes have definite crossing properties (see \re{cr}):
\be
A^\pm_{1,3}(q^2,-\nu)=\pm A^\pm_{1,3},{\ }{\ }
A^\pm_{2,4,5}(q^2,-\nu)=\mp A^\pm_{2,4,5}(q^2,\nu).
\nn
\ee

By using the same procedure as above, the following extension
 for structure functions can be
obtain:
\ba
\int_0^1dxx^ng_1^{+}(x,Q^2)&=&
\sum_q\frac{a_{n}^q}{2},{\ }{\ } n=0,2,...,\nn\\
\int_0^1dxx^ng_2^{+}(x,Q^2)&=&\sum_q\frac{n(d_n^q-a_n^q)}{2(n+1)},{\ }
 n=2,4...; \nn\\
\int_0^1dxx^{n-1}g_3^{+}(x,Q^2)&=&
-2\sum_{q}\frac{(-1)^qa_n^q}{n+1},{\ }{\ } n=1,3...;\nn\\
\int_0^1dxx^{n-1}g_4^{+}(x,Q^2)&=&-\sum_{q}(-1)^qa_n^q,{\ }{\ }
n=1,3...;\nn\\
\int_0^1dxx^ng_5^{+}(x,Q^2)&=&-\sum_{q}\frac{(-1)^qa_n^q}{2},{\ }{\ }
 n=1,3....\nn\\
\ea
\ba
\int_0^1dxx^ng_1^{-}(x,Q^2)&=&
\sum_q\frac{(-1)^qa_{n}^{q-}}{2},{\ } n=1,3...,\nn\\
\int_0^1dxx^ng_2^{-}(x,Q^2)&=&\sum_q\frac{(-1)^qn(d_n^{q-}-a_n^{q-})}
{2(n+1)},{\ }
 n=1,3...,\nn\\
\int_0^1dxx^{n-1}g_3^{-}(x,Q^2)&=&-
2\sum_{q}\frac{a_n^{q-}}{n+1},{\ }{\ } n=0,2,...,\nn\\
\int_0^1dxx^{n-1}g_4^{-}(x,Q^2)&=&-\sum_{q}a_n^{q-},{\ }{\ } n=0,2,...,\nn\\
\int_0^1dxx^ng_5^{-}(x,Q^2)&=&-\sum_{q}\frac{a_n^{q-}}{2},{\ }{\ }
n=0,2,...,\nn\\
\ea
where $(-1)^q=1(-1)$ for quark (antiquark) and
$a_n^{q-}, d_n^{q-}$ correspond to the matrix elements \re{e11}, \re{e12}
from the operator
\ba
\Theta^{\beta\{\mu_1...\mu_n\}}
=\frac{i^n}{n!}
\sum_{perm \{\mu_i\}}\bq
\gb\g \tau_3 D^{\mu_1}... D^{\mu_n}q.
\nn
\ea

We also have the analogues of the Callan-Gross \re{e30},
Wandzura-Wilczek \re{e177},\re{e31} and Burkhardt-Cottingham \re{e188}
relations for $g_i^\pm$
spin-dependent
structure functions.


\begin{center}
{\bf 4. Electroweak spin-dependent structure functions in the
framework of the covariant quark model}
\end{center}


In the previous section the different sum rules for electroweak
spin-dependent structure functions has been obtained. It would be
interesting to find out the posibility to derive the same sum rules
in the framework of the quark parton model.


In the parton approach to the deep-inelastic scattering the hadronic
tenzor is given by the convolution:
\ba
W_{\mu\nu}(q,P,S)=\sum_\lambda\int d^4kf_\lambda(q,k)w^q_{\mu\nu
  ,\lambda}\delta[(k+q)^2-m_q^2],
\nn
\ea
where $w^q_{\mu\nu,\lambda}$ is quark tenzor, $f_\lambda(P,k,S)$ is
some function which carried the information on the quark distribution
inside hadron, $\lambda$ is the quark helicity.

For electroweak interaction the spin-dependent part of the
quark tenzor has the following form:
\ba
{w^q_{\mu\nu,\lambda}}^{spin}&=&\frac{1}{4}Tr[(1+\frac{\lambda\g \not \!
  {n}}{m_q})
(\not \! {k} +m_q)\gamma_\mu(g_{V_1}+g_{A_1}\g)(\not \! {k}+\not \!
{q}+m_q)\gamma_\nu(g_{V_2}+g_{A_2}\g)]\nn\\
&=&i\lambda\epb [2g_{A_1}g_{A_2}k_\alpha n_\beta+(g_{A_1}g_{A_2}+
g_{V_1}g_{V_2})q_\alpha n_\beta]\nn\\
&+&\lambda g_{V_1}g_{A_2}[2k_\mu n_\nu-(n\cdot q)g_{\mu\nu}]+
\lambda g_{A_1}g_{V_2}[2n_\mu k_\nu-(n\cdot q)g_{\mu\nu}],
\nn
\ea
where $n$ is the off-shell parton spin vector \ct{a14}, \ct{a15}:
\ba
n_\sigma=\frac{m_qp\cdot k}{\sqrt{(p\cdot k)k^2-M^2k^4}}(k_\sigma-
\frac{k^2}{p\cdot k}p_\sigma).
\nn
\ea

For massless quarks the hadronic tenzor is:
\ba
W_{\mu\nu}^{spin}&=&\int d^4k\Delta f(p\cdot k,k\cdot S,k^2)[
i\epb q_\alpha k_\beta(g_{A_1}g_{A_2}+g_{V_1}g_{V_2})\nn\\
&+&2(g_{V_1}g_{A_2}+g_{A_1}g_{V_2})k_\mu k_\nu-
(g_{V_1}g_{A_2}+g_{A_1}g_{V_2})(k\cdot q)g_{\mu\nu}]\delta[(k+q)^2],
\label{q3}
\ea
where $\Delta f=f_+-f_- $.
For a spin $\frac{1}{2}$ particle $\Delta f $ must be has the form
\ct{a14}
\ba
\Delta f(p\cdot k,k\cdot S,k^2)=-\frac{(k\cdot S)}{M^2}\tilde{f}
(p\cdot k,k^2).\nn
\ea
By using of the Sudakov variables, parton momentum is
\be
k=xP+\frac{k^2+k_\bot^2-x^2M^2}{2x\nu}(q+xP)+k_\bot.
\label{q4}
\ee

To obtain the spin-dependent structure functions, we should project
the hadron tenzor to the states with the longitudinal $S_\|^\mu=P^\mu+
O(\frac{M^2}{\nu})$,
and transverse $S_\bot^\mu=M(0,cos\alpha,sin\alpha ,0)$
nucleon polarization in the infinite momentum frame $
P=(\sqrt{P^2+M^2},0,0,P)$.
The result is
\ba
W_{\mu\nu}^\|=i\epb \frac{q_\alpha P_\beta}{\nu}g_1(x)+\frac{P_\mu
  P_\nu}{\nu}g_4(x)-g_{\mu\nu}g_5(x) ;\nn\\
W_{\mu\nu}^\bot=i\epb \frac{q_\alpha S_\beta^\bot}{\nu}(g_1(x)+
g_2(x))+\frac{P_\mu S_\nu^\bot+P_\nu S_\mu^\bot}{2\nu}g_3(x).
\label{q5}
\ea
>From \re{q3}, \re{q4} and \re{q5} the spin-dependent
structure functions are:
\ba
g_1(x)&=&\frac{a\pi xM^2}{8}\int_x^1dy(2x-y)\tilde{h}(y) ;\nn\\
g_2(x)&=&\frac{a\pi xM^2}{8}\int_x^1dy(2y-3x)\tilde{h}(y) ;\nn\\
g_1(x)+g_2(x)&=&\frac{a\pi xM^2}{8}\int_x^1dy(y-x)\tilde{h}(y) ;\nn\\
g_3(x)&=&\frac{b\pi x^2M^2}{2}\int_x^1dy(y-x)\tilde{h}(y) ;\nn\\
g_4(x)&=&\frac{b\pi x^2M^2}{4}\int_x^1dy(2x-y)\tilde{h}(y) ;\nn\\
g_5(x)&=&\frac{b\pi xM^2}{8}\int_x^1dy(2x-y)\tilde{h}(y) ,
\label{q6}
\ea
where $a=g_{A_1}g_{A_2}+g_{V_1}g_{V_2}$,
$b=g_{V_1}g_{A_2}+g_{V_2}g_{A_1}$,
  $ y=x+k^2_\bot/(xM^2)$ and $\tilde{h}(y)=\int dk^2\tilde{f}(y,k^2)$.
The result \re{q6} for electromagnetic spin-dependent
structure functions $g_1(x), g_2(x)$ is in the agreement with
the result of \ct{a14}, \ct{a15}.
>From \re{q6} one can obtain the
sum rules for electroweak spin-dependent structure functions:
\be
g_1^i(x)+g_2^i(x)=\int_x^1\frac{dy}{y}g_1^i(x);
\label{q7}
\ee
\be
\int_0^1dxg_2^i(x)=0;
\label{q77}
\ee
\be
2xg_5^j(x)=g_4^j(x);
\label{q8}
\ee
\be
g_3^j(x)=2x\int_x^1\frac{dy}{y^2}g_4^j(y),
\label{q99}
\ee
\be
g_3^j(x)=4x\int_x^1\frac{dy}{y}g_5^j(y),
\label{q9}
\ee
where $i=\gamma,\gamma Z, Z, W $ and $j=\gamma Z, Z, W $.


The \re{q9} can be expressed in terms of moments
\be
\int_0^1 dxx^n(g_3^j(x)-\frac{4}{n+2}xg_5^j(x))=0.
\label{q10}
\ee
By using of the Callan-Gross relation \re{q8} one
can obtain the following sum rule
\be
\int_0^1 dxx^n(g_3^j(x)-\frac{2}{n+2}g_4^j(x))=0.
\label{q11}
\ee


Thus in the framework of the covariant quark model the
 same relations between spin-dependent structure functions   for
 massless quarks which
have been obtained above in the framework of the operator product
expansion are origined
(see \re{e30}, \re{e31},\re{e300}).

Taking into account  of the masses of quarks leads to some violation
of the sum rules \ct{a15}.
We can estimate this violation in the framework of the model which was
proposed in \ct{a15}.
The final result is:
\ba
g_1(x)&=&\frac{a\pi M^2}{8}\int_{x+\frac{\alpha}{x}}^{1+\alpha}dy
(x(2x-y)+2\alpha)
\tilde{h}(y);
\label{q12}\\
g_2(x)&=&\frac{a\pi M^2}{8}\int_{x+\frac{\alpha}{x}}^{1+\alpha}dy
(x(2y-3x)-\alpha)
\tilde{h}(y)-\frac{c\pi
  m^2}{4}\int_{x+\frac{\alpha}{x}}^{1+\alpha}dy
\tilde{h}(y);
\label{q133}\\
g_1(x)+g_2(x)&=&\frac{a\pi M^2}{8}\int_{x+\frac{\alpha}{x}}^{1+\alpha}dy
(x(y-x)+\alpha(
1-\frac{2c}{a}))\tilde{h}(y);
\label{q13}\\
g_3(x)&=&\frac{b\pi M^2x^2}{2}\int_{x+\frac{\alpha}{x}}^{1+\alpha}dy
(y-x)\tilde{h}(y);
\label{q14}\\
g_5(x)&=&\frac{b\pi M^2}{8}\int_{x+\frac{\alpha}{x}}^{1+\alpha}dy
(x(2x-y)+2\alpha)\tilde{h}(y);
\label{q15}\\
g_4(x)&=&2xg_5(x),
\label{q16}
\ea
where $\alpha=m^2/M^2$, $c=g_{A_1}g_{A_2} $.

>From \re{q16} it follows that
Callan-Gross relation  is justified for massive quarks.
Direct calculation leads to the conclusion that sum rule \re{q11}
for first moments of the $g_3(x)$ and $g_4(x)$
\be
\int_0^1 dxg_3^j(x)=\int_0^1dxg_4^j(x).
\label{q19}
\ee
is also correct  for the massive case.
Burkhard-Cottingham sum rule \re{q77} is  valid for massive
quarks for $\gamma$ and $\gamma Z $ exchanges. However for $Z$ and
$W^\pm$ exchange the last term in \re{q133} leads to the violation
of this sum rule.

It is interesting to mention that the structure function $g_5(x)$ is
given, in practicaly, by the same formule as $g_1(x)$. For the
case $c=a/2$, which is correct for the charge current, we have the
direct connection between shapes of the structure functions
$g_1(x)+g_2(x)$ and $g_3(x)$ also.




\begin{center}
{\bf 5. Radiative corrections}
\end{center}
Thanks to the recent next-to-leading order calculation of the polarized
splitting functions \ct{a9} the completed NLO analysis of the
spin-dependent structure function $g_1(x,Q^2)$ has been done
\ct{a10},\ct{a11}. As result the NLO sets of the parton polarized
distribution functions was suggested.
We will use these results to perform the LO and NLO analysis of the
electroweak spin-dependent structure functions.

These structure functions are determined by the convolution
formula:
\ba
g^i(x,Q^2)=\sum_{i,j} C_{i,j}(x,Q^2)\otimes \Delta q_j(x,Q^2)+
C_{i,g}(x,Q^2)\otimes \Delta g(x,Q^2),
\nn
\ea
where $C_{i,j}$ are coefficient functions and $\Delta q_j(x,Q^2)$,
$\Delta g(x,Q^2) $ are parton polarized distributions in the NLO
approximation and convolution means
\ba
C(x,Q^2)\otimes \Delta
q(x,Q^2)=\int_x^1\frac{dy}{y}C(\frac{x}{y})q(y,Q^2).
\nn
\ea
The sets of the NLO polarized distributions from the
paper \ct{a10} will used to calculate $g_i$ structure functions. Due
to the same tensorial structure in the front of the  spin-dependent
structure
functions $g_4$ and $g_5$ as in front of  the nonpolarized structure functions $F_2$ and
$F_1$ in the \re{e6}, we should have the same $\alpha_s$ corrections for
coefficient functions \ct{a6} in the both cases.
It is allow us to perform NLO calculation of the $g_4$, $g_5$ by using
the well knowed results NLO corrections to the $F_1$ and $F_2$ structure
functions \ct{a12}.
Furthermore, we can use the \re{e300} to calculate the $g_3$ structure
function also in LO approximation. To estimate $g_2$
at the same order  $\alpha_s $, Wandzura-Wilczek relation \re{q7}
will used.

In the framework of the $\overline{MS}$ scheme the expressions for the
electroweak spin-dependent structure functions are:
\ba
g_1^{\gamma}(x,Q^2)&=&\frac{1}{2}\sum_qe_q^2(\Delta q(x,Q^2)+\Delta
\bar{q}(x,Q^2)\nn\\
&+&\frac{\alpha_s(Q^2)}{2\pi}(C_q^{(1)}\otimes(\Delta q(x,Q^2)+\Delta
\bar{q}(x,Q^2))+\frac{1}{N_f}C_g\otimes \Delta g(x,Q^2)),
\label{e50}
\ea
 where the appropriate Wilson coefficients are \ct{a9}:
\ba
C_q^{(1)}&=&C_F[(1+x^2)(\frac{ln(1-x)}{1-x})_+
-\frac{3}{2}\frac{1}{(1-x)_+}\nn\\
&-&\frac{1+x^2}{1-x}ln
x+2+x-(\frac{9}{2}+\frac{\pi^2}{3})\delta(1-x)];
\label{e51}
\ea
\be
C_g=2T_F[(2x-1)(ln\frac{1-x}{x}-1)+2(1-x)],
\label{e52}
\ee
with $C_F=4/3$ and $T_F=N_f/2$.
The NLO running coupling is:
\ba
\frac{\alpha_s(Q^2)}{4\pi}\approx \frac{1}{\beta_0ln
  Q^2/\Lambda^2_{\overline{MS}}}-
\frac{\beta_1ln ln Q^2/\Lambda^2_{\overline{MS}}}{\beta_0^3(ln
  Q^2/\Lambda^2_{\overline{MS}})^2},
\nn
\ea
where $\beta_0=11-2N_f/3$, $\beta_1=102-38N_f/3$ and
$\Lambda^{N_f}_{\overline{MS}}$ is given by \ct{a50}
\ba
\Lambda^{3,4}_{\overline{MS}}=248,200 MeV.
\nn
\ea
To calculate $g_1^{\gamma Z}$ one should change in \re{e50}
$e_q^2/2\rightarrow e_q{g_V}^q$ and  for $g_1^Z(x,Q^2)$  $e_q^2/2
\rightarrow ((g_V^q)^2+(g_A^q)^2)/2$.

The structure functions for the charge current are given by
\ba
g_1^{W^-}(x,Q^2)&=&\Delta u(x,Q^2)+\Delta c(x,Q^2)+
\Delta
\bar{d}(x,Q^2)+\Delta
\bar{s}(x,Q^2)\nn\\
&+&\frac{\alpha_s(Q^2)}{2\pi}C_q^{(1)}\otimes (\Delta u(x,Q^2)+\Delta c(x,Q^2)+
\Delta
\bar{d}(x,Q^2)+\Delta
\bar{s}(x,Q^2)),
\nn\\
&+&\frac{\alpha_s(Q^2)}{4\pi}C_g\otimes \Delta g(x,Q^2),\nn\\
g_1^{W^+}(x,Q^2)&=&\Delta d(x,Q^2)+\Delta s(x,Q^2)+
\Delta
\bar{u}(x,Q^2)+\Delta
\bar{c}(x,Q^2)\nn\\
&+&\frac{\alpha_s(Q^2)}{2\pi}C_q^{(1)}\otimes (\Delta d(x,Q^2)+\Delta s(x,Q^2)+
\Delta
\bar{u}(x,Q^2)+\Delta\bar{c}(x,Q^2))
\nn\\
&+&\frac{\alpha_s(Q^2)}{4\pi}C_g\otimes \Delta g(x,Q^2).
\label{e544}
\ea

For structure function $g_5(x,Q^2)$ we have the formulas
\ba
g_5^\gamma(x,Q^2)&=&0;
\nn\\
g_5^{\gamma Z}(x,Q^2)&=&\sum_qe_q g_A^q(\Delta q(x,Q^2)-\Delta
\bar{q}(x,Q^2)
+\frac{\alpha_s(Q^2)}{2\pi}(C_q^{(5)}\otimes(\Delta q(x,Q^2)-\Delta
\bar{q}(x,Q^2));
\nn\\
g_5^{Z}(x,Q^2)&=&\sum_qg_V^q g_A^q(\Delta q(x,Q^2)-\Delta
\bar{q}(x,Q^2)
+\frac{\alpha_s(Q^2)}{2\pi}(C_q^{(5)}\otimes(\Delta q(x,Q^2)-\Delta
\bar{q}(x,Q^2)),
\nn\\
g_5^{W^-}(x,Q^2)&=&-[\Delta u(x,Q^2)+\Delta c(x,Q^2)-
\Delta
\bar{d}(x,Q^2)-\Delta
\bar{s}(x,Q^2)\nn\\
&+&\frac{\alpha_s(Q^2)}{2\pi}C_q^{(5)}\otimes (\Delta u(x,Q^2)+\Delta c(x,Q^2)-
\Delta
\bar{d}(x,Q^2)\Delta
-\bar{s}(x,Q^2))],
\nn\\
g_5^{W^+}(x,Q^2)&=&-[\Delta d(x,Q^2)+\Delta s(x,Q^2)-
\Delta
\bar{u}(x,Q^2)-\Delta
\bar{c}(x,Q^2)\nn\\
&+&\frac{\alpha_s(Q^2)}{2\pi}C_q^{(5)}\otimes (\Delta d(x,Q^2)+\Delta s(x,Q^2)-
\Delta
\bar{u}(x,Q^2)-\Delta
\bar{c}(x,Q^2))],
\label{e59}
\ea
where \ct{a6}
\be
C_q^{(5)}(z)=C_q^{(1)}+\frac{4}{3}(1-z).
\label{e57}
\ee

For $g_4(x,Q^2)$ structure function we have
\ba
g_4^\gamma&=&0;
\nn\\
g_4^{\gamma Z}(x,Q^2)&=&2x\sum_qe_q g_A^q[(\Delta q(x,Q^2)-\Delta
\bar{q}(x,Q^2))
+\frac{\alpha_s(Q^2)}{2\pi}(C_q^{(4)}\otimes(\Delta q(x,Q^2)-\Delta
\bar{q}(x,Q^2))];
\nn\\
g_4^{Z}(x,Q^2)&=&2x\sum_qg_V^q g_A^q[(\Delta q(x,Q^2)-\Delta
\bar{q}(x,Q^2))
+\frac{\alpha_s(Q^2)}{2\pi}(C_q^{(4)}\otimes(\Delta q(x,Q^2)-\Delta
\bar{q}(x,Q^2))],
\nn\\
g_4^{W^-}(x,Q^2)&=&-2x[\Delta u(x,Q^2)+\Delta c(x,Q^2)-
\Delta
\bar{d}(x,Q^2)-\Delta
\bar{s}(x,Q^2)\nn\\
&+&\frac{\alpha_s(Q^2)}{2\pi}C_q^{(4)}\otimes (\Delta u(x,Q^2)+\Delta c(x,Q^2)-
\Delta
\bar{d}(x,Q^2)-\Delta
\bar{s}(x,Q^2))],
\nn\\
g_4^{W^+}(x,Q^2)&=&-2x[\Delta d(x,Q^2)+\Delta s(x,Q^2)-
\Delta
\bar{u}(x,Q^2)-\Delta
\bar{c}(x,Q^2)\nn\\
&+&\frac{\alpha_s(Q^2)}{2\pi}C_q^{(4)}\otimes (\Delta d(x,Q^2)+\Delta s(x,Q^2)-
\Delta
\bar{u}(x,Q^2)-\Delta
\bar{c}(x,Q^2))].
\label{e599}
\ea
where \ct{a6}
\be
C_q^{(4)}(z)=C_q^{(1)}+\frac{4}{3}(1+z).
\label{e62}
\ee

\newpage
\begin{thebibliography}{99}

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Z.Phys.{\bf C64} (1994) 267.
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Phys.Rept.{\bf 261} (1995) 1.
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\bi{a4} B. Lampe, Phys.Lett. {\bf B227} (1992) 469.
\bi{a5}P.Mathews and V.Ravindran, Phys.Lett.{\bf B278}
(1992) 175; Int. J. Mod. Phys. {\bf A7} (1992) 6371.
\bi{a6}D. de Florian and R.Sassot, Phys. Rev. {\bf D51} (1995) 6052.
\bi{a7} X.Ji, Nucl. Phys. {\bf B402}(1993) 217.
\bi{new} V.Ravishankar, Nucl. Phys. {\bf B374}(1992) 309.
\bi{mu} M.Muta, Foundation of quantum chromodynamics (World
Scientific, Singapure, 1987).
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\bi{a133} H.Burkhardt and W.N.Cottingham, Ann. Phys. (NY) 56, (1970)
453.
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.
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Preprint DO-TH 95/13;RAL-TR-95-042, .
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CERN-TH/95-266,;\\
T.Gehrmann and W.J.Stirling, Preprint DTP/95/82, ;\\
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\end{thebibliography}

\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The  cross section for polarized deep inelastic scattering reads
%-----------------------------------------------------------------------
\be
\frac{d \sigma^3}{dx dy d\phi} = \frac{\alpha^2 y}{Q^4} \sum_{ab}
\kappa_{ab}(Q^2) L_{ab}^{\mu\nu} W_{ab,\mu\nu}
\label{eqz1}
\ee
%-----------------------------------------------------------------------
Here $x = Q^2/2p.q \equiv Q^2/2 M \nu$
and $y=p.l/p.q$, with $q^2 = -Q^2$ the four
momentum transfer squared, $p$ and $l$ the four momenta of the
incoming nucleon and lepton, respectively, $M$ denotes the nucleon mass,
and $\phi$ is the  {\tt ???}.
$L_{ab}^{\mu\nu}$ and $W_{ab,\mu\nu}$ denote the leptonic and hadronic
tensors corresponding to different current combinations, i.e.
$ab \equiv
i = |\gamma|^2  , |\gamma Z|$ and
$|Z|^2$ for neutral current interactions
and $ab = |W^{\pm}|^2$
for charged current scattering. Finally the factors
$\kappa_{ab}(Q^2)$ describe the propagators of the respective subprocess
normalized to the photon propagator and are given by
%-----------------------------------------------------------------------
\begin{equation}
\kappa_{|\gamma|^2}(Q^2) = 1,~~~\kappa_{|\gamma Z|}(Q^2)
= \frac{G_F M_Z^2}{2 \sqrt{2} \pi \alpha}
\frac{Q^2}{Q^2 + M_Z^2},~~~~\kappa_{|Z|^2}(Q^2) = \left [
\kappa_{|\gamma Z|}(Q^2) \right]^2,
\label{eqz21}
\end{equation}
%-----------------------------------------------------------------------
and
%-----------------------------------------------------------------------
\be
\kappa_{|W^{\pm}|^2}(Q^2) = \left (\frac{G_F M_W^2}{4 \pi \alpha}
\frac{Q^2}{Q^2 + M_Z^2} \right )^2.
\label{eqz2}
\ee
%-----------------------------------------------------------------------
The leptonic and hadronic
tensor are given by
%-----------------------------------------------------------------------
\be
L_{ab}^{\mu\nu} = \sum_{\lambda^\prime}[\bar
u(k^\prime,\lambda^\prime)\gamma_\mu(g_{V_a}+g_{A_a}
\gamma_5)u(k,\lambda)]^\ast\bar
u(k^\prime,\lambda^\prime)\gamma_\nu(g_{V_b}+g_{A_b}
\gamma_5)u(k,\lambda),
\label{e2}
\ee
%-----------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-----------------------------------------------------------------------
\ba
\bq(x)\gb\g\lc q(0)&=&\sum_n \frac{(-1)^n}{n!}x_{\mu_1}...x_{\mu_n}\bq
\gb\g  D^{\mu_1}... D^{\mu_n}\lc q,    \\
\bq(0)\gb\g\lc q(x)&=&\sum_n \frac{(+1)^n}{n!}x_{\mu_1}...x_{\mu_n}\bq
\gb\g  D^{\mu_1}... D^{\mu_n}\lc q.
\label{eqB6}
\ea
%-----------------------------------------------------------------------
Eq.~(\ref{eqB5}) can be expressed in terms of the operator
$\Theta^{\beta\{\mu_1...\mu_n\}}$ containing a symmetric and an
antisymmetric part.
%-----------------------------------------------------------------------
\be
\Theta^{\beta\{\mu_1...\mu_n\}}
=\frac{i^n}{n!}
\sum_{perm \{\mu_i\}}\bq
\gb\g  D^{\mu_1}... D^{\mu_n}q \equiv
{\Theta_S}^{\beta\{\mu_1...\mu_n\}}+
{\Theta_A}^{\beta\{\mu_1]\mu_2...\mu_n\}},
\ee
%-----------------------------------------------------------------------
with
%-----------------------------------------------------------------------
\be
{\Theta_S}^{\beta\{\mu_1...\mu_n\}}=\frac{1}{n+1}[
\Theta^{\beta\{\mu_1...\mu_n\}}+\Theta^{\mu_1\{\beta\mu_2...\mu_n\}}+
\cdots],
\ee
%-----------------------------------------------------------------------
%-----------------------------------------------------------------------
\ba
{\Theta_A}^{\beta\{\mu_1]\mu_2...\mu_n\}}&=&\frac{1}{n+1}[
\Theta^{\beta\{\mu_1...\mu_n\}}
-\Theta^{\mu_1\{\beta\mu_2...\mu_n\}}+\nn\\
&{\  }&\Theta^{\beta\{\mu_1...\mu_n\}}
-\Theta^{\mu_2\{\mu_1\beta...\mu_n\}}
+\cdots].
\ea
%-----------------------------------------------------------------------
The nucleon matrix elements of these operators are:
%-----------------------------------------------------------------------
\ba
\langle PS\mid{\Theta_S}^{\beta\{\mu_1...\mu_n\}}
\mid PS\rangle=\frac{a_n}{n+1}
[S^\beta P^{\mu_1}P^{\mu_2}...P^{\mu_n}+
S^{\mu_1} P^{\beta}P^{\mu_2}...P^{\mu_n}+\cdots-{\sf traces}],
\label{e11}
\ea
%-----------------------------------------------------------------------
and
%-----------------------------------------------------------------------
\ba
\langle PS\mid{\Theta_A}^{\beta\{\mu_1\mu_2...\mu_n\}}\mid PS\rangle
&=&\frac{d_n}{n+1}
[(S^\beta P^{\mu_1}-S^{\mu_1}P^\beta ) P^{\mu_2}...P^{\mu_n}+\nn\\
&{\ }&(S^\beta P^{\mu_2}-S^{\mu_2}P^\beta ) P^{\mu_1}...P^{\mu_n}+
\cdots-{\sf traces}].
\label{e12}
\ea
%-----------------------------------------------------------------------
\\ .......................




%-----------------------------------------------------------------------






