text:
\section{Introduction}
\label{sect1}

\vspace{1mm}
\noindent
Polarized deep-inelastic lepton-hadron scattering provides one of the
cleanest methods to investigate the spin-structure of the nucleons.
This field attracted much interest  after the finding of
the EMC experiment~\cite{EMC} in 1988 that the quarks
appear to carry only a small
fraction of the nucleon spin. During the last years precise
measurements of both the polarized structure function $g_1(x,Q^2)$
of the proton and neutron by the SMC, SLAC, and HERMES
experiments~\cite{REV} were performed. Very recently also
the structure function $g_2(x,Q^2)$ was measured  for
the first time~\cite{G2}.

The unfolding of the
structure functions from the deep-inelastic scattering cross section
requires the detailed knowledge of the electromagnetic radiative
corrections, which are very large in some parts of the kinematic
domain. A first calculation of the leptonic QED corrections for the
case of pure photon exchange was performed in ref.~\cite{Minsk} for
the range of small values of $Q^2$ and a simple ansatz for the
hadronic structure functions $g_1(x)$ and $g_2(x)$~\cite{MIN1}.

In the present paper the leptonic $O(\alpha)$ QED corrections for the
neutral current deep-inelastic scattering cross sections are calculated
in a model independent approach for the case of longitudinal and
transverse nucleon polarization. We  account for both photon and
$Z$ boson exchange. The resulting corrections are therefore applicable
irrespectively of the $Q^2$ range.
This is of particular importance for  polarized
deep-inelastic scattering experiments operating in the range of
larger values of $Q^2$, e.g. possible future experiments
in the kinematic regime being accessible at HERA, cf.~\cite{JB95A},
but also for the
CERN fixed target experiments for the range of large $Q^2$.

The $O(\alpha)$ hadronic QED corrections are understood to be absorbed
into the parton densities, as also the  QCD
corrections~\cite{HADR1}--\cite{HADR2},
leading to $Q^2$-dependent structure functions.
In this way the hadronic tensor can be formulated quite generally
in a model independent way. Particularly the structure functions need not
necessarily  to be represented using a partonic description.

Besides the complete $O(\alpha)$ results, we also derive
the corrections in the leading logarithmic approximation (LLA).
Due to their simpler analytical structure they are particularly
suited for fast numerical estimates.

The paper is organized as follows. In section~2 the neutral current
Born cross sections are derived. The leptonic $O(\alpha)$ QED corrections
are presented in section~3. In section~4 we derive
the different contributions to the $O(\alpha)$ QED corrections in the
leading logarithmic approximation for the polarized case.
Numerical results for the kinematic
range of the HERMES experiment and possible future polarized  $ep$
collider experiments in the kinematic   regime of HERA are presented
in section~5. Section~6 contains the conclusions. An appendix
summarizes  kinematic   relations used in the calculation.
\section{The Born Cross Section}
\label{sect2}
\noindent
The Feynman diagram describing  neutral current deep-inelastic
lepton--nucleon scattering
\begin{equation}
l(k_1) + p(p) \rightarrow l(k_2) + X(p')
\label{eq1}
\end{equation}
is shown in Figure~1. The particle 4--momenta are given in parentheses.

The matrix element for the Born cross section reads
\begin{eqnarray}
{\cal M}_{\rm Born}
&=& i {e^2}
         \langle p' | {\cal J}_{\mu} |p \rangle \frac{1}{Q^2}
{\bar{u}}(k_2) \left[ Q_l  \gamma^{\mu} + s_l \gamma^{\mu}
\left(v_l+a_l \gamma_5 \right) \chi(Q^2) \right] u(k_1)
\frac{1}{(2\pi)^3} \frac{1}{\sqrt{2 k^0_1  2 k^0_2}}. \nonumber \\
\label{matrixBorn}
\end{eqnarray}
Here $e$ denotes the electromagnetic coupling,
$q = k_1 - k_2$,
$m$ and $M$ are the lepton and nucleon masses,
${\cal J}_{\mu}$ is the hadronic current, $Q_{l}=s_l|Q_l|$
the charge of the electron,
with $s_l = 2 I^{(3)}_l$
twice the third component of the weak isospin of the electron,
$v_{l}$ and $a_{l}$ are the vector and axial-vector
couplings of the $Z$~boson to the electron
\begin{eqnarray}
v_{l}=1-4 |Q_{l}| \sin^{2}\theta_{W}, \hspace{3.cm} a_{l}=1,
\label{veae}
\end{eqnarray}
and
$Q^2 = -q^2$ denotes the four momentum transfer squared.

\vspace{3cm}
\begin{picture}(120,20)(100,100) \centering
\put(135,-250){\epsfig{file=dia1.ps,width=16cm}}
\end{picture}

\vspace{24mm}
\begin{center}
{\sf Figure~1: Born diagram for neutral current deep-inelastic
lepton--proton scattering.}
\end{center}

\vspace{2mm}
\noindent
The calculation is performed in the on--mass--shell (OMS) scheme,
in which
the weak mixing angle
$\theta_{W}$ is related to the weak boson masses by
$\sin^2 \theta_W = 1 - M_W^2/M_Z^2$.
Furthermore,
\begin{eqnarray}
 \chi (Q^2) =
{G_\mu \over\sqrt{2}}{M_{Z}^{2} \over{8\pi\alpha}}{Q^2 \over
{Q^2+M_{Z}^{2}}}
\label{chiq}
\end{eqnarray}
is the $\gamma$-$Z$ propagator ratio, with
$G_{\mu}= 1.16637\cdot 10^{-5}$ GeV$^{-2}$ the Fermi
constant, and $\alpha~=~1/137.06$ the fine structure constant.

The differential Born cross section is given by:
\begin{eqnarray}
d\sigma_{\rm Born} =
   \frac{1}{j}  (2\pi)^4  \sum_{\rm {spins}}
   \left| {\cal M}_{\rm Born} \right|^2
 {d{\vec k}_2}
\prod_i d {\vec{p}_i}\,' \delta^4 (k_1+p-k_2- \sum_i p'_i),
\label{sBorn1}
\end{eqnarray}
where
\begin{eqnarray}
 p'&\equiv& \sum_i p_i' = p+k_1-k_2 \qquad  \mbox{and} \qquad
 {p'}^2 = M^2_h.
\end{eqnarray}
$j$ denotes the flux factor
\begin{eqnarray}
 j= \frac{\sqrt{(k_1.p)^2-\Me\Mp}}{k^0_1 p^0 }  \frac{1}{(2\pi)^6}
  = \frac{\SLS             }{2k^0_1 p^0            (2\pi)^6},
\end{eqnarray}
with
\begin{eqnarray}
\lambda_S = S^2 - 4m^2M^2,~~~~~~~~~~~~S = (p + k_1)^2 - m^2 - M^2.
\label{lambdas}
\end{eqnarray}
For later use we rewrite
the differential Born cross section~(\ref{sBorn1})
in terms of the
leptonic and hadronic tensors,
$L^{\mu\nu}$ and $W_{\mu\nu}$, integrating over
the phase space up to two
variables:
\begin{eqnarray}
d\sigma_{\rm Born}
&=&  \frac {2\alpha^2}{\sqrt{\lambda_S}}\frac{1}{Q^4}
                    \Biggl[ L^{\mu\nu} W_{\mu\nu} \Biggr]
\frac{d{\vec k}_2}{k_2^0}
 =  \frac {2\pi \alpha^2}{\lambda_S}
 \frac{S^2 y}{Q^4}
 \Biggl[ L^{\mu\nu} W_{\mu\nu} \Biggr]
  dx dy.
\label{sBorn2}
\end{eqnarray}
$x$ and $y$ are the Bjorken variables
\begin{equation}
x = \frac{Q^2}{2 p.q},~~~~~~~~~~~y = \frac{p.q}{p.k_1}.
\end{equation}
The calculation is performed for incoming
longitudinally polarized  leptons.
We used the spin density matrix
\begin{eqnarray}
\rho(k_1)= \sum_s u^{\small s}(k_1) {\bar u}^{\small s}(k_1) = \frac{1}{2}
           \left( 1 - \gamma_5 \hat \xi_l \right) (\hat{k_1} + m).
\end{eqnarray}
The 4--vector of the lepton
polarization is given by
\begin{eqnarray}
\xi_l = \frac{\lambda_l}{m} \left (k_1 - \frac{2 m^2}{S} p \right)
\frac{S}{\sqrt{\lambda_S}},
\end{eqnarray}
with
\begin{eqnarray}
\xi^2_l = - \lambda^2_l,~~~~~~~~{\rm and}~~~~~~~~~\xi_l.k_1 =0.
\end{eqnarray}
${\lambda_l}$ denotes the degree of lepton
polarization.
The leptonic tensor for neutral current polarized lepton  (antilepton)
scattering reads
\begin{eqnarray}
 L^{\mu\nu} &=&   \Bigl [
     2\left(k^{\mu}_1 k^{\nu}_2+k^{\nu}_1
 k^{\mu}_2 \right )
- g^{\mu \nu} Q^2  \Bigr ]
   L_{S} (Q^2,\lambda_l)
 -2 i  k_{ 1 \alpha} k_{ 2 \beta} \varepsilon^{\alpha\beta\mu\nu}
   L_{A} (Q^2,\lambda_l) \nonumber\\
& &  + 4   \frac{m^2}{S} \Biggl \{
i~\varepsilon^{\alpha \beta \mu \nu}
\Biggl  [
p_{\alpha} q_{\beta}
L_v(Q^2,\lambda_l)
-p_{\alpha} \left (k_{1\beta} + k_{2\beta}\right)
 L_a(Q^2,\lambda_l)  \Biggr ]
\nonumber\\
& &     +   \Biggl  [ p^{\mu} \left ( k_{1}^{\nu} +  k_{2}^{\nu} \right )
          + p^{\nu} \left ( k_{1}^{\mu} +  k_{2}^{\mu} \right )
          - p^{\mu} q^{\nu}
          - p^{\nu} q^{\mu}  \nonumber\\
& &     - g^{\mu\nu} \Bigl [ p.(k_1 +
           k_2) - p.q \Bigr ] \Biggr ]
\Biggr \}
L_{\chi}(Q^2,\lambda_l)
+ 4 g^{\mu\nu} m^2
\Bigl [L_{\chi}(Q^2,\lambda_l) - L_a(Q^2,\lambda_l)\Bigr ]~.
\end{eqnarray}
The symmetric $(S)$ and  antisymmetric $(A)$ parts are
\begin{eqnarray}
   L_{S} (Q^2,\lambda_l)  =          Q^2_{l} + 2|Q_l|
               \left( v_l - \Pe \lambda_l a_l  \right) \chi(Q^2)
      + \left( v^2_l + a^2_l
- 2 \Pe  \lambda_l v_l a_l \right) \chi^2(Q^2),
\end{eqnarray}
\begin{eqnarray}
   L_{A} (Q^2,\lambda_l)=
       - \lambda_l
        Q^2_l + 2 |Q_l|
        \left( p_l a_l - \lambda_l v_l \right) \chi(Q^2)
 +\Biggl(2 p_l v_l a_l -  \lambda_l \left( v^2_l + a^2_l\right)
             \Biggr) \chi^2(Q^2)~.
\label{lnrten}
\end{eqnarray}
The formfactors contributing to the terms
$\propto m^2$ are
\begin{eqnarray}
L_v(Q^2,\lambda_l) &=& \lambda_l
\left[|Q_l|+v_l\chi(Q^2)\right]^2,
\nonumber \\
L_a(Q^2,\lambda_l) &=& \lambda_l
a^2_l\chi^2(Q^2),
\nonumber \\
L_{\chi}(Q^2,\lambda_l) &=& \lambda_l p_l
a_l \chi(Q^2)\left[|Q_l| + v_l \chi(Q^2)\right].
\end{eqnarray}
The particle label  $\Pe$ takes the values
$\Pe = 1$
for particles  and
$-1$ for antiparticles.

The hadronic tensor reads
\begin{eqnarray}
 W_{\mu\nu}     &=& p^0 (2\pi)^6 \sum   \int
          \langle p'| {\cal J}_{\mu} |p     \rangle
          \langle p    | {\cal J}_{\nu} |p' \rangle
          \delta^4 (p' - \sum_i p'_i)
                            \prod_i d{\vec{p_i}\,'}.
\end{eqnarray}
For its representation in terms of structure functions we follow the
convention of ref.~\cite{BK},
\begin{eqnarray}
W_{\mu\nu}  &=& \left(-g_{\mu\nu}+ \frac{q_\mu q_\nu }{q^2} \right)
                                              {\cal F }_1(x,Q^2)
 +\frac{\widehat{p_\mu}\widehat{p_\nu}}{p.q}
                                              {\cal F }_2(x,Q^2)
     - i \varepsilon_{\mu\nu\lambda\sigma} \frac{q^\lambda p^\sigma}
                                { 2 p.q}  {\cal F }_3(x,Q^2)       \nonumber \\
  && + i \varepsilon_{\mu\nu\lambda\sigma}
\frac{q^\lambda {s}^\sigma} {p.q}{\cal G }_1(x,Q^2)
     + i \varepsilon_{\mu\nu\lambda\sigma}
          \frac {q^{\lambda}(p.q {s}^\sigma
                      -{s}.q p^\sigma)}{(p.q)^2}
                                            {\cal G }_2(x,Q^2)      \nonumber \\
  && +      \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{1}{p.q}                {\cal G }_3(x,Q^2)\nonumber \\
  && +   {s}.q \frac{\widehat{p_\mu} \widehat{p_\nu}}{(p.q)^2}
                                                  {\cal G }_4(x,Q^2)
               +  \left( - g_{\mu\nu}+ \frac{q_\mu q_\nu }{q^2}\right)
                  \frac{ {s}.q  }{p.q}       {\cal G }_5(x,Q^2),
\label{hadten}
\end{eqnarray}
where
\begin{eqnarray}
 \widehat {p_\mu}   =   p_\mu - \frac{p.q }{q^2}  q_\mu,\qquad
 \widehat {{s}_\mu}  =  {s}_\mu - \frac{s.q}{q^2}  q_\mu.
\end{eqnarray}
$s$  denotes the polarization 4--vector of the nucleon.
In the nucleon rest frame it is given by
\begin{eqnarray}
    s= M(0,{\vec{n}}_{\lambda}).
\end{eqnarray}
${\vec n}_\lambda $
is an unit $3$-vector.

For a short-hand notation we have introduced the
combined neutral current structure functions ${\cal F}_i$ and ${\cal G}_
i$
in eq.~(\ref{hadten}).
In terms of the structure functions  $F^{{J_1 J_2}}_i$ and
$g^{{J_1 J_2}}_i$,
which are associated with the respective currents, they read:

\begin{eqnarray}
{\cal  F }_{1,2}(x,Q^2)
 &=& Q^2_{l} F^{\gamma\gamma}_{1,2}(x,Q^2)
      + 2 |\Ql|
 \left( v_l -\pe \lambda_l a_l \right)\chi(Q^2) F^{\gamma Z}_{1,2}(x,Q^2
) \nonumber \\\nll
      &&+
 \left( v^2_l + a^2_l - 2\pe\lambda_l v_l a_l \right) \chi^2(Q^2)
                                                  F^{ZZ}_{1,2}(x,Q^2),
  \\  \nonumber \\
{\cal F }_{3}(x,Q^2)   &=&
 2|Q_l|\left( p_l a_l - \lambda_l v_l \right) \chi(Q^2)
                                                  F^{\gamma Z}_3(x,Q^2)
  \nonumber \\
      &&+ \left[2 p_l v_l a_l -  \lambda_l
          \left( v^2_l + a^2_l\right)\right] \chi^2(Q^2)
                                                  F^{ZZ}_3(x,Q^2) ,
  \\  \nonumber \\
{\cal G }_{1,2}(x,Q^2) &=&
 -Q^2_{l}\pe \lambda_l  g^{\gamma \gamma}_{1,2}(x,Q^2)
     + 2 |Q_l| \left( p_l a_l - \lambda_l v_l \right)\chi(Q^2)
     g^{\gamma Z}_{1,2}(x,Q^2)  \nonumber \\
  && +  \left[2 p_l v_l a_l - \lambda_l
         \left( v^2_l + a^2_l \right)\right] \chi^2(Q^2)
    g^{ZZ}_{1,2}(x,Q^2) , \\  \nonumber \\
{\cal G }_{3,4,5}(x,Q^2) &=&
              2 |Q_l|\left( v_l -\pe \lambda_l a_l \right) \chi(Q^2)
         g^{\gamma Z}_{3,4,5}(x,Q^2)   \nonumber \\\nll
  && +  \left( v^2_l + a^2_l - 2\pe\lambda_l v_l a_l \right)
                                         \chi^2(Q^2)
                                        g^{ZZ}_{3,4,5}(x,Q^2).
\end{eqnarray}
The twist-$2$ contributions to the structure functions can be expressed
in terms of parton densities.
In lowest order QCD one obtains~\cite{R1,R3,BK}~:
\begin{eqnarray}
 F^{{J_1 J_2}}_1(x,Q^2) =
 \sum_q \alpha^q_{{J_1 J_2}} \left[ q(x,Q^2)+\bar q(x,Q^2) \right],
\end{eqnarray}
\begin{eqnarray}
 F^{{J_1 J_2}}_2(x,Q^2) = {2x}{F^{{J_1 J_2}}_1(x,Q^2)},
\end{eqnarray}
\begin{eqnarray}
 F^{{J_1 J_2}}_3(x,Q^2) =
 \sum_q \beta^q_{{J_1 J_2}} \left[ q(x,Q^2)-\bar q(x,Q^2) \right],
\end{eqnarray}
\begin{eqnarray}
\label{eqg1}
 g^{{J_1 J_2}}_1(x,Q^2) &=&
\frac{1}{2}
 \sum_q \alpha^q_{{J_1 J_2}}
\left[\Delta q(x,Q^2) + \Delta \bar q(x,Q^2) \right],
\\ \nonumber \\
 g^{{J_1 J_2}}_2(x,Q^2) &=& -g^{{J_1 J_2}}_1(x,Q^2)
                       +\int_x^1\frac{dy}{y} g^{{J_1 J_2}}_1(y,Q^2),
\\ \nonumber \\
 g^{{J_1 J_2}}_3(x,Q^2) &=& 4x
                          \int_x^1\frac{dy}{y} g^{{J_1 J_2}}_5(y,Q^2),
\\ \nonumber \\
 g^{{J_1 J_2}}_4(x,Q^2) &=& 2 x g^{{J_1 J_2}}_5(x,Q^2),
\\ \nonumber \\
\label{eqg5}
 g^{{J_1 J_2}}_5(x,Q^2) &=&
 \sum_q \beta^q_{{J_1 J_2}}
\left[\Delta q(x,Q^2)-\Delta\bar q(x,Q^2) \right],
\end{eqnarray}
where
\begin{eqnarray}
\alpha^q_{{J_1 J_2}} &=&
\alpha^q_{{\gamma\gamma,\gamma Z ,ZZ}}
=\left[e^2_q,~~2e_q v_q,~~v_q^2+a_q^2\right], \\
 \beta^q_{{J_1 J_2}} &=&
\beta^q_{{\gamma\gamma,\gamma Z ,ZZ}}
=\left[0 ,~~2e_q a_q,~~2v_q a_q \right],
\end{eqnarray}
and the electroweak couplings are
\renewcommand{\arraystretch}{2}
\begin{equation}
\begin{array}{ll}
 e_u  = + {\displaystyle \frac{ 2}{ \displaystyle 3} },  &
 e_d  =
 - {\displaystyle
\frac{    1}{    3} },  \\
 v_u  =  {\displaystyle
  \frac{    1}{    2}}
- {\displaystyle
\frac{    4}{    3}} \sin^2\theta_W,   &
v_d  =  - {\displaystyle
\frac{    1}{    2}}
+ {\displaystyle
\frac{    2}{    3}}
\sin^2\theta_W,   \\
 a_u  = {\displaystyle
  \frac{    1}{    2} }, &
 a_d  =  - {\displaystyle
\frac{    1}{    2}}.
\end{array}
\end{equation}
For the numerical results, which are presented in section~5 below,
the structure functions~eq.~(\ref{eqg1}--\ref{eqg5}) are parametrized
using a partonic description for the structure functions $F_1(x,Q^2),
F_3(x,Q^2), g_1(x,Q^2)$
and $g_5(x,Q^2)$. The other structure functions are calculated using
the relations above.
$q(x,Q^2)$, $\overline{q}(x,Q^2)$, $\Delta q(x,Q^2)$, and
$\Delta \overline{q}(x,Q^2)$ denote the unpolarized and
polarized quark and antiquark densities, respectively.

It appears to be convenient to rewrite
the Born cross section in terms of the following two
contributions:
\begin{eqnarray}
\frac{ d^2\sigma_{{\rm Born}}}{ dx dy}  =
\frac{ d^2\sigma^{{\rm unpol}}_{{\rm Born}}}{ dx dy}
+\frac{ d^2\sigma^{{\rm  pol}}_{{\rm Born}}}{ dx dy},
\end{eqnarray}
with
\begin{eqnarray}
\frac{ d^2\sigma^{{\rm unpol}}_{{\rm Born}}}{ dx dy}
 =    \frac{2\pi \alpha^2 }{Q^4} S
\sum_{i=1}^{3} \;  S_i^U(x,y) {\cal F}_i(x,Q^2),
\label{equpol}
\end{eqnarray}
and
\begin{eqnarray}
\frac{ d^2\sigma^{{\rm   pol}}_{{\rm Born}}}{ dx dy}
 =    \frac{2\pi \alpha^2 }{Q^4}
     \lambda^p_N f^{p} S
     \sum_{i=1}^{5} \;  S^p_{gi}(x,y) {\cal G}_i(x,Q^2).
\label{eqpol}
\end{eqnarray}
$\lambda_N^p$ denotes the degree of nucleon polarization.
For unpolarized deep-inelastic scattering only the first term,
$d^2 \sigma^{\rm unpol}$, contributes.
Eq.~(\ref{eqpol}) applies both to the case of
longitudinal $(L)$ and
transversal $(T)$ nucleon polarization, where
\begin{eqnarray}
f^L &=& 1,                                            \\
f^T &=&               \COSPHI \;\frac{ d\varphi}{2\pi}
\sqrt{\frac{4\Mp x}{Sy}\left(1-y-\frac{\Mp x y}{S}\right)}
\equiv \COSPHI \;\frac{ d\varphi}{2\pi}
\frac{1-y}{y}\sin\theta_2.
\label{Borntran}
\end{eqnarray}
$\theta_2$ is the angle between the incoming and outgoing leptons.
$\varphi$ denotes
 the angle between the nucleon spin vector $s$ and
the plane of the incoming  and outgoing
lepton  in the nucleon rest frame (see  figure~4 in appendix~A).
The polarization 3--vectors for longitudinal and transverse polarization
are given by
\begin{eqnarray}
\vec{n}^{{L }} &=& \lambda_N^L\frac{\vec{k}_1}{|\vec{k}_1|},\\
\vec{n}^{{T }} &=&
\lambda_N^T
\vec{n}_{\bot }, \quad \mbox{with}  \quad
\vec{n}_{\bot }
\vec{k}_{1    }=0.
\end{eqnarray}
Finally,
the kinematic   coefficients in eqs.~(\ref{equpol}) and (\ref{eqpol})
are:
\begin{eqnarray}
  S_1^U(x,y) &=&  2 x y^2,                              \nonumber \\
  S_2^U(x,y) &=&   2 \left[ (1-y)-xy \frac{\Mp}{S}\right],    \nonumber \\
  S_3^U(x,y) &=&   x [1 - (1 - y)^2],
\end{eqnarray}
\begin{eqnarray}
   S^{L}_{1}(x,y) &=&
                          2xy
        \Biggl[ ( 2 - y )-2 \frac{\Mp}{S} xy \Biggr]  ,
                                                             \nonumber \\
   S^{L}_{2}(x,y) &=&
                             -8 x^2 y \frac{\Mp}{S}  ,
                                                             \nonumber \\
   S^{L}_{3}(x,y) &=&
                   4\frac{\Mp}{S}
        \Biggl[(1 - y) - xy \frac{\Mp}{S} \Biggr],
                                                             \nonumber \\
   S^{L}_{4}(x,y) &=&
                        -2
        \left(1+\frac{2x\Mp}{S}\right)
        \Biggl[  \left( 1 - y \right)-xy \frac{\Mp}{S} \Biggr],
                                                             \nonumber \\
   S^{L}_{5}(x,y)&=&  -2xy \left(  y + 2 \frac{\Mp}{S} x y \right),
\end{eqnarray}
\begin{eqnarray}
   S^{T}_{1}(y,Q^2) &=&    S^{T}_{5}(y,Q^2)  =
   S_1^U(y,Q^2)  ,                                 \nonumber \\
   S^{T}_{2}(y,Q^2) &=& 4 x y,                         \nonumber \\
   S^{T}_{3}(y,Q^2) &=&-\frac{S^{L}_{1}(y,Q^2)}{2xy}, \nonumber \\
   S^{T}_{4}(y,Q^2) &=& S_2^U(y,Q^2).
 \end{eqnarray}
\section{The $O(\alpha)$ Leptonic Correction}
\label{sect3}

\vspace{1mm}
\noindent
The $O(\alpha)$ leptonic radiative corrections to the scattering cross
sections are given by
\begin{equation}
\frac{d^2 \sigma^{\rm QED, 1}_{\rm rad}}{d x_l d  y_l} =
\frac{\alpha}{\pi} \delta_{\rm VR}
\frac{d^2 \sigma_{\rm Born}}{d x_l d y_l} +
\frac{d^2 \sigma_{\rm Brems}}{d x_l dy_l} =
\frac{d^2 \sigma_{\rm rad}^{\rm unpol}}{d x_l d y_l} +
\frac{d^2 \sigma_{\rm rad}^{\rm pol}}{d x_l dy_l}.
\label{COR1}
\end{equation}
The corrections are represented by the finite parts of the
virtual and soft (VR) and Bremsstrahlung terms, respectively.
Since we integrate over the phase space of the radiated photon, the
differential cross sections~(\ref{COR1}) depend on the way in which
the Bjorken variables $x$ and $y$ are determined kinematically.
Because in all the polarized deep-inelastic scattering experiments
performed so far the kinematic   variables were measured using the
scattered lepton\footnote{Other choices of kinematic   variables were
also
investigated for the case of unpolarized deep-inelastic
scattering~\cite{JB1,JB2,R1,R3}.} the present calculation refers
to this set of variables. The kinematic variables are defined by
\begin{equation}
x_l = \frac{Q^2_l}{S y_l},~~~~y_l =
\frac{p_1.(k_1 - k_2)}{p_1.k_1},~~~~{\rm and}~~Q_l^2 = (k_1 - k_2)^2.
\label{eqKIN1}
\end{equation}
The hadronic
structure functions depend  on the hadronic variables
\begin{equation}
x_h = \frac{Q^2_h}{S y_h},~~~~y_h =
\frac{p_1.(p_2 - p_1)}{p_1.k_1},~~~~{\rm and}~~Q_h^2 = (p_2 - p_1)^2,
\label{eqKIN2}
\end{equation}
over which is integrated.

\subsection{Virtual and Soft Corrections}
\label{sect31}

\vspace{3cm}
\begin{picture}(120,20)(100,100) \centering
\put(135,-250){\epsfig{file=dia2.ps,width=16cm}}
\end{picture}

\vspace{24mm}
\begin{center}
{\sf Figure~2: Diagram of the leptonic
virtual photon correction for deep-inelastic
scattering.}
\end{center}

\vspace{2mm}
\noindent
The virtual and soft correction, $\delta_{\rm VR}$, can be
written as
\begin{eqnarray}
\delta_{\rm VR}(y_l, Q^2_l)
 &=&  \delta_{\rm inf}(y_l, Q^2_l) - \frac{1}{2} \ln^2 \left [
\frac{1 - y_l(1 - x_l)}{1 - y_l x_l} \right ]
+ {\rm Li}_2 \left [ \frac{1 - y_l}{(1 - y_l x_l)(1 - y_l(1 -x_l))}
\right ] \nonumber\\ & &~~~~~~+ \frac{3}{2} \ln \left ( \frac{Q^2_l}{m^2} \right ) -
{\rm Li}_2(1)  - 2,
\end{eqnarray}
with
\begin{equation}
\delta_{\rm inf}(y_l, Q^2_l) =  \left [
\ln \left(
\frac{Q^2_l}{m^2}
\right ) - 1 \right ]
\ln \left [
\frac{y_l^2 (1 - x_l)^2}{(1 - y_l x_l)[1 - y_l(1 - x_l)]}
\right ]
\end{equation}
and
\begin{equation}
{\rm Li}_2(x) = - \int_0^1 dz \frac{\ln(1 - xz)}{z}.
\end{equation}
These expressions are the same in the unpolarized and polarized case.
They were derived in refs.~\cite{R1,R3}.
\subsection{Bremsstrahlung Corrections}
\label{sect32}

\vspace{1mm}
\noindent
The differential Bremsstrahlung
cross section for the scattering of polarized electrons
off polarized protons, originating from the  diagrams  in figure~3,
is
\begin{eqnarray}
\frac{\displaystyle d\sigma_{{\rm Brem}}}{\displaystyle d\xl d\yl}
=  { 2  \alpha^3 S\yl}
\int d\yh d\Qh \frac{1}{\QhS}
\Biggl[\frac{1}{2\pi}\frac{d\varphi_k}{{\SLQ}}
\frac{1}{4}\Biggl(L^{\mu\nu}_{{\rm rad}} W_{\mu\nu} \Biggr)\Biggr]
\label{cr_sect_rad}.
\end{eqnarray}

\vspace{3cm}
\begin{picture}(120,20)(100,100) \centering
\put(135,-250){\epsfig{file=dia3.ps,width=16cm}}
\end{picture}

\vspace{24mm}
\begin{center}
{\sf Figure~3: The leptonic Bremsstrahlung diagrams for
for deep-inelastic scattering.}
\end{center}

\vspace{2mm}
\noindent
$W_{\mu \nu}$ denotes the hadronic tensor~(\ref{hadten}). The invariant
$\lambda_q$ is defined in (\ref{eqLQ}).
The Bremsstrahlung correction to the leptonic tensor,
$L^{\mu\nu}_{\rm{rad}}$ has the following form~:
\begin{eqnarray}
L^{\mu\nu}_{\rm{rad}} &=&
        2 L_{_S}(Q^2_h)
\Biggl\{ 4 \Biggl( -\frac{2\Me}{z^2_2}
+\frac{\Qh}{z_1 z_2}
\Biggr) k_{1}^{\mu} k_{1}^{\nu}
       + 4 \Biggl( -\frac{2\Me}{z^2_1}
+\frac{\Qh}{z_1 z_2}
\Biggr) k_{2}^{\mu} k_{2}^{\nu}
\nonumber \\
&&
 +   \Biggl [ 2 \Me \Qh \Biggl( \frac{1}{z^2_1}
+ \frac{1}{z^2_2} \Biggl) - \frac{Q_l^4 + Q_h^4}{z_1 z_2}
- 2  \Biggr] g_{\mu\nu}
\Biggr\}
\nonumber \\
&&       +  4 i
\varepsilon^{\alpha\beta\mu\nu}  \Biggl \{ - L_A(Q^2_h,\lambda_l)
\Biggl [
  \Biggl( \frac{2\Me}{z^2_2}-\frac{\Ql}{z_1 z_2}
-\frac{1}{z_2} \Biggr)
k_{1\alpha} q_{ \beta}
 + \Biggl( \frac{2\Me}{z^2_1}-\frac{\Ql}{z_1 z_2}
+\frac{1}{z_1} \Biggl)
k_{2\alpha} q_{ \beta}
\Biggr ]
\nonumber \\
&& + 2 \left [ L_v(Q^2_h,\lambda_l) + L_a(Q^2_h,\lambda_l)\right]
\frac{\Me}{z^2_1}
\Biggl [
      \ylh
k_{1\alpha} k_{2\beta}
 +   \left(  1 +  \ylh \right)
k_{2\alpha} k_{ \beta}
\Biggr ]
\nonumber \\
&&
       -  2   \left[ L_v(Q^2_h,\lambda_l) - L_a(Q^2_h,\lambda_l)\right]
\frac{\Me}{z^2_1}
k_{1\alpha} k_{ \beta}  \Biggr \}
\nonumber \\
&&
       +   8 L_{\chi}(Q^2_h,\lambda_l) \frac{\Me}{z^2_1}
\Biggl [ 2 \left( k_{1}^{\mu} k_{2}^{\nu}+k_{1}^{\nu} k_{2}^{\mu} \right)
       - 4 \left( 1 + \ylh \right)  k_{2}^{\mu} k_{2}^{\nu}
       -   \left( \Qlh - \ylh \Qh \right) g^{\mu\nu}
\Biggr ]~.
\nonumber
\label{lrtaft}
\end{eqnarray}
Here we retained only those terms in $O(m^2)$, which yield non-vanishing
contributions after intgrating over $\varphi_k$ in the limit
$m \rightarrow 0$.
Similar to the leptonic tensor in the Born approximation,
the tensor $L^{\mu \nu}_{\rm    rad}$ can be decomposed into a symmetric
and an asymmetric term, and also
a  contribution $\propto m^2$.
Besides the invariants defined in
eqs.~(\ref{eqKIN1},\ref{eqKIN2}), it depends on
\begin{equation}
z_1 = 2 k.k_1,~~~~~~~~~~z_2 = 2 k.k_2.
\label{z12}
\end{equation}
The finite part of the Bremsstrahlung contributions are  expressed
in form of integrals over the radiator functions
$S_X^i(y_l, Q_l^2, y_h, Q^2_h)$
and
${\cal S}_Y^i(y_l, Q_l^2, y_h, Q^2_h)$.
Since the structure of the differential cross sections turns out to be
partly different
for  unpolarized and polarized deep-inelastic scattering,
we will describe the individual cases separately.
\subsubsection{Unpolarized Case}
\label{sect321}

\vspace{1mm}
\noindent
The finite contribution to the Bremsstrahlung cross section
reads
\begin{eqnarray}
\frac{ d\sigma_{{\rm Brem}}}{ d\xl d\yl}
&=&
  2  \alpha^3 S \int_{\bf R} d\yh d\Qh
\Biggl \{
\frac{1}{\QhS}
\sum_{i=1}^{3} \;
  {\cal S}_i^U(\yl,\Ql,\yh,\Qh) {\cal F}_i(\xh,\Qh)
\nonumber\\
& &~~~~~-\frac{1}{Q_l^4} \sum_{i=1}^3
  S_i^U(\yl,\Ql)
{\cal L}^{\rm IR}(\yl,\Ql,\yh,\Qh)
{\cal F}_i(\xl,\Ql) \Biggr \}~,
\label{eqBREM}
\end{eqnarray}
where
\begin{eqnarray}
{\cal L}^{\rm {IR}}(\yl,\Ql,\yh,\Qh)
&=&
\frac{\Ql+2m^2}{\Qlh} \Biggl(\ \frac{1}{\sqrt{C_1}}
   -\frac{1}{\sqrt{C_2}} \Biggr)\
   -m^{2}\Biggl(\ \frac{B_{1}}{C_{1}^{3/2}}
                +\frac{B_{2}}{C_{2}^{3/2}} \Biggr).
\label{eqn052a}
\end{eqnarray}
The integral over $\varphi_k$ in eq.~(\ref{cr_sect_rad})
can be performed analytically, see appendix~A. The remaining twofold
integral takes the form~\cite{R1}
\begin{eqnarray}
\int_{\bf R} d y_h d Q^2_h = \sum_{l=1}^2
\int_{\hat{y}_{h_l}}^{\hat{y}_{h_{l+1}}} d y_h
\int_{Q^2_{h_{1,d}}}^{Q^2_{h_{l,u}}} d Q^2_h,
\end{eqnarray}
with
\begin{eqnarray}
\hat{y}_{h_1} &=& \frac{y_l(y_l S - \sqrt{\lambda_q}) + 2 Q^2_l M^2/S}
               {    y_l S - \sqrt{\lambda_q}  +         M^2  } \nonumber \\
\hat{y}_{h_2} &=& \frac{y_l(y_l S + \sqrt{\lambda_q}) + 2 Q^2_l M^2/S}
               {    y_l S + \sqrt{\lambda_q}  +         M^2  } \nonumber \\
\hat{y}_{h_3} &=& y_l \nonumber \\
Q^2_{h_{1,d}} &=& Q^2_l + \frac{S}{2 M^2} y_{lh} (y_l S -
\sqrt{\lambda_q}) \nonumber \\
Q^2_{h_{2,u}} &=& Q^2_l + \frac{S}{2 M^2} y_{lh} (y_l S +
\sqrt{\lambda_q}) \nonumber \\
Q^2_{h_{1,u}} &=& y_h S,
\end{eqnarray}
and
\begin{eqnarray}
 \ylh  &=& \yl-\yh.
\end{eqnarray}
The functions   $S_i^U(\yl,\Ql,\yh,\Qh)$ are given by~:
\begin{eqnarray}
 S_1^U(\yl,\Ql,\yh,\Qh) &=& \yl \Biggl\{
       - 2\Me \Qh \left( \BCl+\BCll \right) \nonumber \\
       && + \frac{\QlS+\QhS}{\Qlh}
         \left(\lClS - \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}  \right)
       + \frac{2}{\SLQ} \Biggr\},
\\
 S_2^U(\yl,\Ql,\yh,\Qh)
 &=& \frac{\yl}{S\yh}
    \Biggl\{
- 2\Me\Biggl[
     \Biggl(S^2\yll (\yll+\yh)-\Mp \Qh\Biggr)   \BCl                  \nonumber \\
    && +\Biggl(S^2 \yhl-\Mp \Qh\Biggr)       \BCll \Biggr]            \nonumber \\
    && +\Biggl[S^2\Qh (\ylpl-\yl\yh)-\Mp (\QlS+\QhS)\Biggr]
        \frac{1}{\Qlh}  \left(\lClS - \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}  \right)              \nonumber \\
    && -S^2\yh\Biggl[ \lClS
                 + \frac{\yll}{\CllS} \Biggr]
       - \frac{2 \Mp }{\SLQ}
    \Biggr\},
\\
S_3^U(\yl,\Ql,\yh,\Qh) &=&
 \frac{\yl}{\yh} \Biggl\{
  - \Me  \Qh
      \Biggl[ (2 \yll+\yh)\BCl+(2-\yh)\BCll \Biggr]                   \nonumber \\
&&+ (2-\yl) \frac{\Ql\Qh}{\Qlh}\left(\lClS - \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}  \right)      \nonumber \\
&&- \Qh                   \left(\frac{\yll}{\ClS}  - \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right)
  - \frac{\yh(\Ql+\Qh)}{2}\left(\lClS              + \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}\right)
                              \Biggr\}~,
\end{eqnarray}
with
\begin{eqnarray}
 \yll  &=& 1-\yl,          \nonumber \\
 \yhl  &=& 1-\yh, \nonumber \\
 Q^2_{lh} &=& Q^2_l - Q^2_h.
\end{eqnarray}
The coefficients $B_{1,2}$ and $C_{1,2}$ are defined in
eqs.~(\ref{eqB1}--\ref{eqC2}).
These functions were given in refs.~\cite{R1,R3} before and are
presented here in a somewhat more compact form.

\subsubsection{Longitudinal Polarization}
\label{sect322}

\vspace{1mm}
\noindent
The Bremsstrahlung part of the differential cross section takes the form
\begin{eqnarray}
\frac{\displaystyle d\sigma^{{\rm {Long}}}_{{\rm {Brem}}}}{\displaystyle d\xl d\yl}
&=&
  2 \alpha^3 \lambda_N^L \int d\yh d\Qh \Biggl\{
\frac{1}{\QhS}
\sum_{i=1}^{5}\;S^{L}_{i}(\yl,\Ql,\yh,\Qh) {\cal G}_i(\xh,\Qh) \nonumber \\
& &~~~~-\frac{1}{Q^4_l}
\sum_{i=1}^{5}\;S^{L}_{i}(\yl,\Ql)
{\cal L}^{\rm IR}(y_l,Q^2_l,y_h,Q^2_h)
{\cal G}_i(\xh,\Qh) \nonumber \\
& &~~~~+ 2  \frac{\Me}{Q_h^4} \BCl \Biggl[
 \sum_{i=1}^{2}\Biggl(
   {\cal S}^{L}_{vi}(\yl,\Ql,\yh,\Qh){\cal G}^v_i(\xh,\Qh)
  +{\cal S}^{L}_{ai}(\yl,\Ql,\yh,\Qh){\cal G}^a_i(\xh,\Qh)\Biggr)
\nonumber \\
& &~~~~+\sum_{i=3}^{5}
   {\cal S}^{L}_{\chi i}(\yl,\Ql,\yh,\Qh){\cal G}^{\chi}_i(\xh,\Qh)
\Biggr]
\Biggr\}~.
\end{eqnarray}
As in the unpolarized case, to
each of the structure functions corresponds   a kinematic function
$S^{L}_{i}(\yl,\Ql,\yh, \Qh)$.
Furthermore, infrared-finite terms
$\propto m^2$  contribute, which are denoted by
${\cal S}^{L}_{vi,ai}(\yl,\Ql,\yh, \Qh)$, and
${\cal S}^{L}_{\chi i}(\yl,\Ql,\yh, \Qh)$, respectively.

These coefficients are given by~:
\begin{eqnarray}
S^{L}_{1}(\yl,\Ql,\yh, \Qh) &=&
     \frac{\yl}{S\yh} \Biggl\{
  \left[ S^2(2\yll+\yh)-2 \Mp \Ql\right] \Qh
        \nonumber \\
                  && \times
  \left[-2 \Me \BCl
 + \frac{\Ql}{\Qlh}\left(\lClS -\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right)  \right]
        \nonumber \\
  && + \left[ S^2 (2-\yh)- 2 \Mp \Qh\right]\Qh
        \nonumber \\
  && \times  \left[- 2 \Me \BCll
   + \frac{\Ql}{\Qlh} \left(\lClS -\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right) \right]
        \nonumber \\
  && - \left[ S^2\left( \yh \Ql
     +(2\yll+\yh)\Qh \right) + 2 \Mp \QlS \right]
         \frac{1}{ \ClS }
        \nonumber \\
  && - \left[ S^2\left(\yh\Ql+(-2+\yh)\Qh\right)
 + 2\Mp \QhS\right]
         \frac{1}{ \CllS}
        \nonumber \\
  && - \frac{4 \Mp ( \Ql + \Qh )}{ \SLQ}
                          \Biggr\},
\end{eqnarray}
\begin{eqnarray}
 S^{L}_{2}(\yl,\Ql,\yh, \Qh) &=&
   4 \frac{\yl}{S\yhS}\Mp \Qh \Biggl\{
      (\yll+\yh) \Ql
    \Biggl[ 2\Me\BCl+\lClS
            \nonumber \\
    &&  -\frac{\Ql}{\Qlh}
    \left(\lClS -\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right) \Biggr]
            \nonumber \\
    && + \Qh \left[
          2 \Me \BCll - \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}
 - \frac{\Ql}{\Qlh}\left(\lClS -\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right) \right]
         \nonumber \\
    && - \Ql          \left(\lClS +\frac{\yll}{\CllS}\right)
       - \frac{\yl}{\SLQ}
                                           \Biggr\},
\end{eqnarray}
\begin{eqnarray}
 S^{L}_{3}(\yl,\Ql,\yh, \Qh) &=&
     \frac{\yl}{S^2\yhS} {\Mp} \Biggl\{
        \Biggl[S^2 \Biggl(\yh \Qh-(\yll+\yh) (2\yll+\yh)\Ql\
\Biggr)
                                + 2\Mp \Ql \Qh\Biggr]
             \nonumber \\
    &&\times  \Biggl[ 2\Me \BCl
             - \frac{\Qh}{\Qlh} \left(\lClS - \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right)
\Biggr]   \nonumber \\
    &&   - 2\Qh \left( S^2\yhl-\Mp \Qh \right)
             \nonumber \\
    &&\times  \Biggl[ 2\Me  \BCll
             - \frac{\Qh}{\Qlh} \left(\lClS - \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right)
\Biggr]   \nonumber \\
    &&   +     { 2 \left[S^2 \left(-\yh \Ql+ \yhl\Qh\right)
                            -\Mp (\QlS+\Ql\Qh+\QhS)\right]}
               {\lClS}
             \nonumber \\
    &&   +     {\left[S^2\left[\yh\yhl \Ql+\left(2\yllS
                +\yh\right)\Qh\right] +2\Mp\Ql\Qh\right]}
              {\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}}
             \nonumber \\
    &&   +\left[S^2 (-4+\yl)\yh
         -4\Mp \left( \Qh+\frac{B_2}{\SLQS} \right) \right]
 \frac{1}{\SLQ}
                                         \Biggr\},
\end{eqnarray}
\begin{eqnarray}
 S^{L}_{4}(\yl,\Ql,\yh, \Qh)
 &=& \frac{\yl}{S^2\yhS} \Biggl\{
       \left(S^2\yh+2\Mp \Ql\right)
       \left[S^2\yll (\yll+\yh)-\Mp\Qh\right]
             \nonumber \\
 && \times \Biggl[ 2\Me \BCl
  - \frac{\Qh}{\Qlh} \left(\lClS - \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right) \Biggr]
          \nonumber \\
 &&  + \left(S^2\yh+2\Mp\Qh\right)
         \left[ S^2 \yhl-\Mp \Qh \right]
             \nonumber \\
 && \times  \Biggl[ 2\Me \BCll
       - \frac{\Qh}{\Qlh} \left(\lClS -\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}\right)   \Biggr]
    \nonumber \\
 &&  + \Biggl[\left(S^2\yh+2\Mp \Ql\right) \left[S^2\yh
     + \Mp (\Ql+\Qh)\right]
             \nonumber \\
 &&     -2 \Mp \Qh \left[S^2 \yhl - \Mp \Qh \right]\Biggr] \lClS
       \nonumber \\
 &&  + \Biggl[\left( S^2\yh+2\Mp \Qh\right)
             \left[S^2\yh \yll-\Mp (\Ql+\Qh) \right]
             \nonumber \\
 &&  - 2\Mp \Qh \left[S^2\yll (\yll+\yh)-\Mp\Qh\right]\Biggr
]\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}      \nonumber \\
 &&  + 2\Mp\left[ S^2 (2+\yll)\yh
     + 2\Mp\left(\Qh+\frac{B_2}{\SLQS}\right) \right]\frac{1
}{\SLQ}
                                         \Biggr\},
\end{eqnarray}
\begin{eqnarray}
 S^{L}_{5}(\yl,\Ql,\yh, \Qh)
 &=& \frac{\yl}{S \yh} \Biggl\{
       ( S^2 \yh  + 2\Mp \Ql)
             \nonumber \\
 && \times \Biggl[ 2 \Me \Qh \BCl
 -  \frac{\Ql \Qh}{\Qlh}\left(\lClS -\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right)
 -  \frac{\Ql}{\ClS}  \Biggr]
             \nonumber \\
 && +\left( S^2 \yh + 2\Mp \Qh \right)
             \nonumber \\
 &&\times   \Biggl[ 2 \Me \Qh \BCll
 -  \frac{\Ql \Qh}{\Qlh}\left(\lClS -\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right)
 -  \frac{\Qh}{\CllS}
 -  \frac{2}{\SLQ}\Biggr]
             \nonumber \\
 && +S^2\yh \left[ \frac{\Qh}{\ClS }
                        +  \frac{\Ql}{\CllS}\right]
         - 4\Mp\frac{B_2}{\lambda^{3/2}_q}
                                   \Biggr\},
\end{eqnarray}
and
\begin{eqnarray}
{\cal S}^{L}_{v1}(\yl,\Ql,\yh, \Qh) &=&
-\frac{\yl}{S\yh} \Biggl\{\Biggl(S^2\yl+2\Mp\Ql\Biggr)\Biggl
(\Qlh-\ylh\Qh\Biggr)
\nonumber \\ &&
            -S^2\ylh\Biggl[\Ql-\left(1-\ylh\right)\Qh\Biggr]
\Biggr\},
\\ \nonumber \\
{\cal S}^{L}_{a1}(\yl,\Ql,\yh, \Qh) &=&
-\frac{\yl}{S\yh} \Biggl\{\Biggl(S^2\yl+2\Mp\Ql\Biggr)\Biggl
(\Qlh-\ylh\Qh\Biggr)
\nonumber \\ &&
            +S^2\Biggl[\ylh\Biggl(\Ql+\left(1-\ylh\right)\Qh
\Biggr)-2\Qlh\Biggr]
\Biggr\},
\\ \nonumber \\
{\cal S}^{L}_{v2}(\yl,\Ql,\yh, \Qh) &=&
- 4 \frac{\yl}{S\yhS}\Mp \Ql \ylhS \Qh,
\\ \nonumber \\
{\cal S}^{L}_{a2}(\yl,\Ql,\yh, \Qh) &=&
- 4 \frac{\yl}{S\yhS}\Mp
\Ql\Biggl(\Qlh-\ylh\Ql+\ylhS\Qh\Biggr),
\\ \nonumber \\
{\cal S}^{L}_{\chi 3}(\yl,\Ql,\yh, \Qh) &=&
\frac{\yl}{S^2\yhS} {\Mp} \Biggl\{
            S^2 \Ql \ylhS \Biggl[ \yll+3 \left(1-\ylh \right
) \Biggr]
\nonumber \\
&&           -\Biggl(\Qlh-\ylh\Qh\Biggr)
  \Biggl[S^2\yh\Biggl(2+\left(1-\ylh\right)\frac{\Ql}{\Qh}
\Biggr)
+4\Mp\Ql\Biggr]
                          \Biggr\},
\\ \nonumber \\
{\cal S}^{L}_{\chi 4}(\yl,\Ql,\yh, \Qh) &=&
\frac{\yl}{S^2\yhS} \Biggl(S^2 \yh + 2 \Mp \Ql \Biggr) \Biggl
\{
               -S^2\ylhS\left(2\yll+\yh\right)
\nonumber \\
&&             +\Biggl(\Qlh-\ylh\Qh\Biggr)
           \Biggl[S^2\yh\left(1-\ylh\right)\frac{1}{\Qh}+2\Mp
\Biggr]
                                                 \Biggr\},
\\ \nonumber \\
{\cal S}^{L}_{\chi 5}(\yl,\Ql,\yh, \Qh) &=&
-2\frac{\yl}{S \yh}\Biggl(S^2\yh+2\Mp\Ql\Biggr)
                     \Biggl(\Qlh-\ylh\Qh\Biggr).
\end{eqnarray}
The combined structure functions ${\cal G}^{v,a,\chi}_i(x,Q^2)$
read
\begin{eqnarray}
{\cal G}^v_{1,2}(x,Q^2) &=& \lambda_l \left[
Q^2_{e}g^{\gamma \gamma}_{1,2}(x,Q^2)
            + 2|\Qe|v_e\chi(Q^2) g^{\gamma Z}_{1,2}(x,Q^2)
              + v^2_e\chi^2(Q^2)g^{ZZ}_{1,2}(x,Q^2) \right],
            \nonumber \\
{\cal G}^a_{1,2}(x,Q^2) &=& \lambda_l
a^2_e\chi^2(Q^2)g^{ZZ}_{1,2}(x,Q^2),  \nonumber \\
{\cal G}^{\chi}_{3,4,5}(x,Q^2) &=& \lambda_l \left [
|\Qe|a_e\chi(Q^2) g^{\gamma Z}_{3,4,5}(x,Q^2)
              + v_e a_e \chi^2(Q^2) g^{ZZ}_{3,4,5}(x,Q^2) \right]~.
\label{addgenstf}
\end{eqnarray}
\subsubsection{Transverse Polarization}
\label{sect323}

\vspace{1mm}
\noindent
Here the differential Bremsstrahlung contribution is given by
\begin{eqnarray}
\frac{\displaystyle d\sigma^{{\rm {Trans}}}_{{\rm {Brem}}}}{\displaystyle d\xl
d\yl}
&=&  2  \alpha^3
     \lambda_N^T \COSPHI \;  \frac {d\varphi}{2\pi}
\sqrt{\frac{4\Mp \xl}{S\yl}\left(1-\yl-\frac{\Mp \xl \yl }{S
}\right)}  \nonumber \\
&& \times \int d\yh d\Qh
\Biggr\{
\frac{1}{\QhS}
     \sum_{i=1}^{5} \; S^{T}_{i}(\yl,\Ql,\yh,\Qh) {\cal G}_i(\xh,\Qh)
\nonumber \\
& &~~~~-\frac{1}{Q^4_l}
\sum_{i=1}^{5}\;S^{T}_{i}(\yl,\Ql)
{\cal L}^{\rm IR}(y_l,Q^2_l,y_h,Q^2_h)
{\cal G}_i(\xh,\Qh) \nonumber \\
& &~~~~+ 2  \frac{\Me}{Q_h^4}
 \frac{1}{{C_1}^{3/2}} \Biggl[
 \sum_{i=1}^{2}\Biggl(
   {\cal S}^{T}_{vi}(\yl,\Ql,\yh,\Qh){\cal G}^v_i(\xh,\Qh)
  +{\cal S}^{T}_{ai}(\yl,\Ql,\yh,\Qh){\cal G}^a_i(\xh,\Qh)\Biggr)
\nonumber \\
& &~~~~+\sum_{i=3}^{5}
   {\cal S}^{T}_{\chi i}(\yl,\Ql,\yh,\Qh){\cal G}^{\chi}_i(\xh,\Qh)
\Biggr]
\Biggr\}~.
\end{eqnarray}
The corresponding coefficients
$S^{T}_{i}(\yl,\Ql,\yh, \Qh)$,
${\cal S}^{T}_{vi,ai}(\yl,\Ql,\yh, \Qh)$, and
${\cal S}^{T}_{\chi i}(\yl,\Ql,\yh, \Qh)$, are~:
 \begin{eqnarray}
 S^{T}_{1}(\yl,\Ql,\yh,\Qh) &=&
      S{\frac{\ylS}{\yh}} \Biggl\{
      \frac{S^2\yl\ylh+2\Mp\Qlh}{\SLQS}                 \\
 && \times \Biggl[-2\Me\Qh\left(\BCl-\BCll\right)
    -(\Ql+\Qh)\left(\lClS+\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}\right)   \Biggr]
            \nonumber \\
 && - 2\Me\Qh\Biggl(\BCl+\BCll\Biggr)
    +\Qh\frac{\Ql+\Qh}{\Qlh}\left(\lClS-\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}\right)
         \nonumber \\
 &&      + \Ql\left(\lClS+\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}\right)
            \nonumber \\
 && +2 \SQSK\frac{S^2\yl+2\Mp\Ql}{\SLQS}
            \nonumber \\
 && \times
   \Biggl[ 2\Me\Qh \left(\lCl-\lCll\right)
            \nonumber \\
 &&  + (\Ql+\Qh)\left(\FCZl +\FCZll\right) \Biggr]
 \Biggr\},
            \nonumber
\end{eqnarray}
\begin{eqnarray}
S^{T}_{2}(\yl,\Ql,\yh,\Qh)&=&
                  { 2S \frac{\ylS}{\yhS} }
   \Biggl\{
   \Qh \left[ 1-\ylh-\yll\frac{S^2\yl\ylh+2\Mp \Qlh}{\SLQS}\right] \\
   &&\times \left[-2\Me  \BCl
     + \frac{\Ql}{\Qlh}\left( \lClS - \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right)-\lClS\right
]    \nonumber \\
   && +\Qh \left[1-\frac{ S^2\yl\ylh+2\Mp \Qlh}{\SLQS} \right
]     \nonumber \\
   &&\times
 \left[- 2\Me  \BCll
        +   \frac{\Ql}{\Qlh} \left( \lClS - \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right)
        +\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}  \right]
           \nonumber \\
   &&   +   2 \SQSK \Qh
         \frac{S^2\yl+2\Mp \Ql}{\SLQS} \nonumber \\
   && \times
 \Biggl[- 2 \Me \left( \frac{\yll}{\ClSS}+\lCll \right)
           \nonumber \\   &&   + \FCZl  - \yllFCZll
                          \nonumber \\
   &&   + \frac{(2-\yl)\Qh}{\Qlh} \left( \FCZl-\FCZll \right)
 \Biggr]
   \Biggr\},
           \nonumber
\end{eqnarray}
\begin{eqnarray}
  S^{T}_{3}(\yl,\Ql,\yh,\Qh) &=&
     \frac{\ylS}{\yhS} \Biggl\{
          \frac{ S^2\yh \ylh+2\Mp \Qlh}{\SLQS}
      \\
     &&\times     \Biggl[
         -\Me\Biggl( \left[S^2 \yll (2\yll + \yh) -2\Mp \Qh\right
] \BCl \nonumber \\\
     &&  + \left[S^2      (2     - \yh) -2\Mp \Qh\right] \BCll
\Biggr)  \nonumber \\
     &&   +\left[S^2 \Qh \left( \ylpl - \frac{\yl\yh}{2} \right)
           - \Mp (\QlS+\QhS)\right]
           \frac{1}{\Qlh} \left(\lClS - \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}  \right)
         \nonumber \\
     &&    -\frac{S^2 \yh}{2}\left( \lClS+\frac{\yll}{\CllS}
\right)
          -\frac{2\Mp}{\SLQ} \Biggr]
           \nonumber \\
     &&    + \Me\Biggl( \left[S^2 (1-\ylh) (2\yll+\yh)
                     -2\Mp \Qh\right]     \BCl
          \nonumber \\
     &&   +\left[S^2       (2     -\yh)-2 \Mp \Qh\right] \BCll
\Biggr)   \nonumber \\
     &&   -\Biggl[S^2 \Qh \Biggl(\ylpl
           +\frac{(3\yll-\yhl) \yh}{2}\Biggr)
           -\frac{\Mp (\QlS+\QhS)}{\Qlh}
            \left(\lClS - \mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}  \right) \Biggr]
          \nonumber \\
     &&    +\frac{S^2 \yh}{2} \left(\lClS
           +\frac{1-\ylh}{\CllS}\right)+\frac{2\Mp}{\SLQ}
          \nonumber \\
     &&+ \SQSK  \frac{S^2\yl+2\Mp\Ql}{\SLQS }
          \nonumber \\     &&    \Biggl[
          2\Me \Biggl( \left[S^2\yll(2\yll+\yh)-2\Mp\Qh\right]
   \lCl  \nonumber \\
     &&  +  \left[S^2      (2     -\yh)-2\Mp\Qh\right] \lCll
\Biggr)    \nonumber \\
     &&    -\Bigl[S^2 \Qh \left(2 \ylpl-\yl\yh\right)
                        -2 \Mp (\QlS+\QhS)\Bigr]
\nonumber \\ && \times
                                                 \FCZ
           \nonumber \\
     &&    +S^2 \yh \Biggl(\FCZl
                  +\yllFCZll \Biggr) \Biggr]
                                               \Biggr\},
           \nonumber
\end{eqnarray}
\begin{eqnarray}
  S^{T}_{4}(\yl,\Ql,\yh,\Qh) &=&
            { \frac{\ylS}{\yhS} }  \Biggl\{
        \Biggl[1-\frac{S^2\yl\ylh+2\Mp \Qlh}{\SLQS} \Biggr]
 \Biggl[
          -2 \Me \Biggl( \left[S^2 \yll (1-\ylh )-\Mp \Qh\right]
                   \BCl
        \nonumber \\
 &&               +      \left[S^2  \yhl  - \Mp \Qh\right]
                   \BCll  \Biggr)
        \nonumber \\
 &&         + \left[S^2 \Qh(\ylpl-\yl\yh) - \Mp(\QlS+\QhS)\right
]
            \frac{1}{\Qlh}
              \left( \lClS -\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $}  \right)
     \nonumber \\
 &&          -S^2\yh\left(  \lClS  +\frac{\yll}{\CllS} \right)
  -\frac{2 \Mp}{\SLQ}
\Biggr]
     \nonumber \\
&& + 2\SQSK\frac{S^2\yl+2\Mp\Ql} {\SLQS}\nonumber \\
&& \times \Biggl[
       -2\Me \Biggl(
     \left[S^2 \yll (1-\ylh)   -\Mp \Qh\right]  \lCl
     \\
&&  +\left[S^2  \yhl           -\Mp \Qh\right] \lCll \Biggr)
     \nonumber \\
&&  +\left[S^2 \Qh(\ylpl -\yl\yh) -\Mp(\QlS+\QhS) \right]
     \nonumber \\
&& \times  \FCZ
     \nonumber \\
&&  -S^2\yh\left(\FCZl
    +\yllFCZll\right)  \Biggr]
                                 \Biggr\},
     \nonumber
\end{eqnarray}
\begin{eqnarray}
S^{T}_{5}(\yl,\Ql,\yh,\Qh) &=&
                   S{ \frac{\ylS}{\yh} } \Biggl\{
      \Biggl[1-\frac{S^2 \yl \ylh + 2\Mp \Qlh}{\SLQS }\Biggr]
    \Biggl[- 2\Me \Qh   \left( \BCl+\BCll \right) \\
   &&   + \frac{\QlS+\QhS}{\Qlh}\left( \lClS-\mbox{${\ds \frac{1}{\sqrt{C_2}}}  $} \right)
          +  \frac{2}{\SLQ} \Biggr]
     \nonumber \\
   &&   +  2 \SQSK\frac{S^2\yl+2\Mp \Ql}{\SLQS}
             \Biggl[-2\Me \Qh \left(\lCl+\lCll\right)
     \nonumber \\
   &&   +  \frac{\QlS+\QhS}{\Qlh}
 \left( \FCZl -\FCZll \right)  \Biggr]
 \Biggr\},
     \nonumber
\end{eqnarray}
and
\begin{eqnarray}
{\cal S}^{T}_{v1}(\yl,\Ql,\yh, \Qh) &=&
S\frac{\ylS}{\yh}
\Biggl[ \Biggl( \Qlh-\ylh\Qh \Biggr) \BZl
       -\Biggl(\Ql + \Qh + \ylh \Qh \Biggr){\cal B}_1 \Biggr],
\\ \nonumber \\
{\cal S}^{^T}_{a1}(\yl,\Ql,\yh, \Qh) &=&
S\frac{\ylS}{\yh}
\Biggl( \Qlh-\ylh\Qh \Biggr)
\Biggl( \BZl + {\cal B}_1 \Biggr),
\\ \nonumber \\
{\cal S}^{T}_{v2}(\yl,\Ql,\yh, \Qh) &=&
 2S\frac{\ylS}{\yhS} \Qh
\Biggl[ \ylhS\BZl - \left(\yl - \yll \ylh\right) {\cal B}_1 \Biggr],
\\ \nonumber \\
{\cal S}^{T}_{a2}(\yl,\Ql,\yh, \Qh) &=&
 2S\frac{\ylS}{\yhS}
\Biggl(\Qlh-\ylh\Qh\Biggr)
\Biggl[\left(1 - \ylh\right)\BZl - \yll {\cal B}_1 \Biggr],
\\ \nonumber \\
{\cal S}^{T}_{\chi 3}(\yl,\Ql,\yh, \Qh) &=&
 \frac{\ylS}{\yhS}\frac{1}{2}
\Biggl\{
\Biggl[-S^2\ylhS\left[\yll+3\left(1-\ylh\right)\right]
\nonumber \\
&&\hspace{1cm}
  +\Biggl(\Qlh-\ylh\Qh\Biggr)
   \Biggl(S^2\yh\frac{1-\ylh}{\Qh}+4\Mp\Biggr)
\Biggr] \BZl
\nonumber \\
&&\hspace{7mm}
  +\Biggl[S^2
   \Biggl(\yl\left(2-\yl\right)-\left(1+\yllS\right)\ylh+3\yll
\ylhS\Biggr)
\nonumber \\
&&\hspace{1cm}
  -\Biggl(\Qlh-\ylh\Qh\Biggr)
   \Biggl(S^2\yh\frac{\yll}{\Qh}+4\Mp\Biggr)
\Biggr] {\cal B}_1
\Biggr\},
\nonumber \\ \\
{\cal S}^{T}_{\chi 4}(\yl,\Ql,\yh, \Qh) &=&
\frac{\ylS}{\yhS}
\Biggl[S^2\ylhS\left(2\yll+\yh\right)
\nonumber \\
&&\hspace{7mm}
  -\Biggl(\Qlh-\ylh\Qh\Biggr)
   \Biggl(S^2\yh\frac{1-\ylh}{\Qh}+2\Mp\Biggr)
\Biggr]
  \Biggl(\BZl - {\cal B}_1
\Biggr),
\\ \nonumber \\
{\cal S}^{T}_{\chi 5}(\yl,\Ql,\yh, \Qh) &=&
 2S\frac{\ylS}{\yh}
\Biggl(  \Qlh - \ylh \Qh  \Biggr)
\Biggl( \BZl - {\cal B}_1
\Biggr)~.
\end{eqnarray}
The quantity ${\cal B}_1$, and similarly ${\cal B}_2$, which
contain  the
denominator $\lambda_q$, may be simplified in the following way for
the $O(m^2)$ terms,
\begin{eqnarray}
 {\cal B}_1 &=& \BZl \frac{S^2 \yl\ylh + 2 \Mp \Qlh}{\SLQS}
     - 2 \SQSK \frac{S^2 \yl     + 2 \Mp \Ql} {\SLQS}
 \nonumber\\
&\equiv& S^2 y_{lh} (Q^2_{lh}-y_{lh} Q^2_l),\\
 {\cal B}_2 &=& B_2  \frac{S^2 \yl\ylh + 2 \Mp \Qlh}{\SLQS}
     - 2 \SQSK \frac{S^2 \yl     + 2 \Mp \Ql} {\SLQS}
 \nonumber\\
&\equiv& S^2 y_{lh} Q^2_{lh} (1 + y_h) - S^2 y_{lh}^2 Q^2_{h}
+ 2 M^2 (\Qlh)^2.
\end{eqnarray}
The explicit substitution of these terms in the above relations does
not lead to a further simplification.
\section{The Leading Log Approximation}
\label{sect4}

\vspace{1mm}
\noindent
In the leading logarithmic approximation the leptonic
$O(\alpha)$  corrections  to the differential scattering cross section
can be written as
\begin{eqnarray}
\frac{d^2 \sigma_{\rm {rad}}^{\rm LLA}}{dxdy}
&=&
\frac{d^2 \sigma_{i}}{dxdy}
+
\frac{d^2 \sigma_{f}}{dxdy}
+
\frac{d^2 \sigma_{\rm{Comp}}}{dxdy}.
\label{lla3}
\end{eqnarray}
The individual contributions are due to the corresponding
mass-singularities of initial-state
({\it i}) and final-state ({\it f}) radiation, and the Compton
contribution, for which the mass singularity results from the
low $Q^2$
range of the virtual photon in the sub-system Born term, in those
situations which may be described partonically.

The initial and final-state radiation terms have the following
structure:
\begin{eqnarray}
\frac{d^2 \sigma^{k}_{i,f}}{dxdy}
=
\frac{\alpha}{2\pi}
\left( \ln\frac{Q^2}{m^2}-1 \right)
\int\limits_{0}^{1}dz
\frac{1+z^2}{1-z}
\Biggl\{\theta(z-z_0)~{\cal J}\left.
\frac{d^2
  \sigma^{k}_{{\rm{Born}}}}{dxdy}\right|^{i,f}_{x={\hat x},
 y={\hat y}, S={\hat S}}
-\frac{d^2 \sigma^{k}_{{\rm{Born}}}}{dxdy}\Biggr\},
\label{LLA1}
\end{eqnarray}
where the rescaled variables are
\begin{eqnarray}
\hat{S} = zS,~~~\hat{y} = \frac{y+z-1}{z},~~~\hat{Q^2} = zQ^2,~~~{\hat x}
 &=& \frac{{\hat Q}^2}{{\hat y}{\hat S}},
\end{eqnarray}
for initial-state radiation and
\begin{eqnarray}
\hat{S} = S,~~~\hat{y} = \frac{y+z-1}{z},~~~\hat{Q^2} =
\frac{Q^2}{z},~~~{\hat x}
 &=& \frac{{\hat Q}^2}{{\hat y}{\hat S}},
\end{eqnarray}
for final-state radiation.
The Jacobian ${\cal J}$
is given by
\begin{eqnarray}
{\cal J} \; \equiv \; {\cal J}(x,y,Q^2) =
\left|\frac{\partial({\hat x},{\hat y})}{\partial(x,y)}\right|,
\label{llaif}
\end{eqnarray}
cf.~refs.~\cite{HADR1,lla1}--\cite{lla2,llA1}.
The lower integration boundaries  $z_0$ derive  from the conditions
\begin{eqnarray}
{\hat x}(z_0) &\leq& 1, \qquad
{\hat y}(z_0) \; \leq\;  1,
\label{llascal}
\end{eqnarray}
and are given by
\begin{eqnarray}
z_0^i =  \frac{1-y}{1-y x},~~~~~~~~~~~~~~~~~~~~z_0^f  = 1 - y + x y~.
\end{eqnarray}
The structure of eq.~(\ref{LLA1}) is the same in the unpolarized and
polarized case in leading order QED, since, as is
well--known~\cite{SPLI}, the fermion--fermion splitting function
\begin{eqnarray}
P_{ff}(z) = \left ( \frac{1 + z^2}{1 - z} \right )_+
\end{eqnarray}
does not depend on the polarization of the incoming fermion in this
order. Therefore, the Bremsstrahlung contributions for the polarized
case are simply obtained in LLA by  inserting in
the
relations~refs.~\cite{HADR1,lla1}--\cite{lla2,llA1}
the differential Born cross section for the polarized case
\cite{lla3}.
Similar relations were also calculated in $O(\alpha^2 \ln^2(Q^2/m^2))$
in refs.~\cite{kripf,JB2}.

For leptonic variables the Compton
contributions depend on the type of nucleon polarization. These
contributions are given by
\begin{eqnarray}
\frac{d^2 \sigma_{{\rm{Comp}}}^{U}}{d\xl d\yl}
&=&
\frac{\alpha^3}{Sx^2_l } \frac{Y_{+}}{y_{l_1}}
  \int \limits_{x_l}^{1} d z
  \int\limits^{(Q^2_h)^{\rm{\small{max}}}}_{(Q^2_h)^{\rm{\small{min}}}}
  \hspace{-2mm}
  \frac{d\Qh}{\Qh}
  \Biggl[\frac{Z_{+}}{z} F^{\gamma\gamma}_2(\xh,\Qh)
-      z F^{\gamma\gamma}_{L}(\xh,\Qh)\Biggr],~~~~{\rm
cf.~\cite{lla4,lla2,lla1,coU}},
\label{compt_u}
\nonumber \\
\frac{d^2 \sigma_{\rm Comp}^{L}}{d\xl d\yl}
&=&
  \left(- 2 \lambda_l \lambda^{L}_N \right)
  \frac{\alpha^3}{S x^2_l }\frac{Y_-}{\yll}
  \int \limits_{x_l}^{1} {d z }
  \int\limits^{(Q^2_h)^{\rm{\small{max}}}}_{(Q^2_h)^{\rm{\small{min}}}}
\hspace{-2mm}
  \frac{d\Qh}{\Qh} \frac{Z_{-}}{z} x_h g^{\gamma\gamma}_1(\xh,\Qh),
\nonumber \\
\frac{d^2 \sigma_{{\rm{Comp}}}^{T}}{d\xl d\yl}
&=&
  \left(- 2 \lambda_l \lambda^{T}_N \right)
  \frac{\alpha^3}{S x^2_l }
  \cos \varphi \frac{d\varphi}{2 \pi}
  \frac{y_l}{y^2_{l_1}}
  \sqrt{\frac{4 M^2 x_l}{S y_l} \left(\yll-\frac{M^2 x_l y_l}{S} \right)}
\nonumber \\
&\times&
\int \limits_{x_l}^{1} \frac{dz}{z}
\int\limits^{(Q^2_h)^{\rm{\small{max}}}}_{(Q^2_h)^{\rm{\small{min}}}}
\hspace{-2mm}
  \frac{d\Qh}{\Qh}
  \Biggl\{
  \left( Y_{-} - y_l z \right) z
 x_h g^{\gamma\gamma}_1(\xh,\Qh)
+ 2  \left[ Y_{+}\left(1-z \right)
 + y_{l_1} \right]
  x_h g^{\gamma\gamma}_2(\xh,\Qh)
  \Biggr\}~.
\nonumber\\
\label{compt}
\end{eqnarray}
Here we used the abbreviations
\begin{eqnarray}
\ylpl &=& 1+(1-\yl)^2,~~~~~~~~~~~~~~~~~~~~~Z_{\pm} = 1 \pm (1-z)^2,
\end{eqnarray}
\begin{eqnarray}
z\;=\;\frac{\xl}{\xh}.
\label{yz}
\end{eqnarray}
Part of the structure of eqs.~(\ref{compt}) can be understood, as in the
case of Bremsstrahlung, in
terms of partonic splitting functions. In the unpolarized case
\begin{eqnarray}
P_{\gamma f}^{U}(z) = \frac{Z_+}{z} = \frac{1 + (1-z)^2}{z}
\end{eqnarray}
emerges~\cite{lla2,llA1}.
The longitudinal structure function is convoluted by the coefficient
function
\begin{eqnarray}
c_L^q(z) = z,
\end{eqnarray}
cf.~\cite{TWZ}.
In the case of longitudinal nucleon polarization one has~\cite{SPLI}
\begin{eqnarray}
P_{\gamma f}^{L}(z) = \frac{Z_-}{z} = \frac{1 - (1-z)^2}{z},
\end{eqnarray}
and the Compton term can as well be understood in the collinear parton
model. Such an interpretation is, however, not possible in the case of
transverse nucleon polarization, see~ref.~\cite{BK}.

For the boundaries of the $\Qh$-integral in eqs.~(\ref{compt}) we  used
\begin{eqnarray}
(Q^2_h)^{\rm{\small{max}}} &=& \frac{\xl}{\xh}(1-\yl)\Ql,
\nonumber \\ \nonumber \\
(Q^2_h)^{\rm{\small{min}}} &=&
\max
\left\{
(Q^2_h)^{\rm{\small{min}}}_{\rm{kin}},\overline{Q}^2_h,{\hat Q}^2_h
\right\},
\label{q2lim}
\end{eqnarray}
where
\begin{eqnarray}
{\hat Q}^2_h &=& \left(\overline{M}^2_h-\Mp\right)\frac{\xh}{1-\xh},
\nonumber \\
(\Qh)_{\rm kin}^{\rm min} &=& \frac{\xh(\xh \yl S - \Ql)(\yl S
-  \sqrt{\lambda_q}) + 2 \Ql (\xh M)^2}{2\left [ \xh \yl S - \Ql
+ (\xh M)^2 \right ]}.
\end{eqnarray}
$\overline{Q}^2_h$ and $\overline{M}^2_h$ denote respective cut values.
The choosen value of $(Q^2_h)^{\rm max}$ leads to a particularly good
agreement of the LLA results and the complete $O(\alpha)$ corrections.

The LLA formulae are remarkably compact. The initial and final--state
radiation contributions rely only on the respective Born cross sections
in rescaled kinematic variables.
Also the Compton terms exhibit a similarly simple structure.
A numerical comparison of the complete leptonic
$O(\alpha)$ corrections and the corresponding LLA results is  given in
section~5.
\section{Numerical Results}
\label{sect5}

\vspace{1mm}
\noindent
The QED corrections, which are discussed subsequently, are presented
in terms of the correction factors
\begin{eqnarray}
\delta = \frac{d^2 \sigma^{\rm pol}_{\rm rad} }
              {d^2 \sigma^{\rm pol}_{\rm Born}} \times 100 \%
\end{eqnarray}
for the different nucleon polarizations and for different kinematic
ranges. The calculations were performed with the code
{\tt HECTOR}~\cite{HEC2}. For the parametrization of the polarized
parton densities  we use the `standard' set (LO) of ref.~\cite{GRSV}.
The non-perturbative low $Q^2$-behavior of the polarized parton densities
is modeled following an  approach~\cite{PROKH} which was used
in ref.~\cite{R3} in  the unpolarized case
before.

In Figure~5 the complete leptonic $O(\alpha)$  QED correction to the
polarized part of the differential deep-inelastic scattering cross
section for longitudinally
polarized protons is shown for the kinematic   range of the HERMES
experiment at HERA at an electron beam energy of $E_e = 27.5 \GeV$.
Similarly to the case of unpolarized deep-inelastic
scattering, see e.g.~\cite{R1,R3},
the corrections rise towards small $x$ and high
$y$ values. The correction was calculated for the case of pure
photon exchange. The structure functions were parametrized referring to
the twist 2 contributions,
using the parton densities of ref.~\cite{GRSV}.
The complete $O(\alpha)$ corrections are compared with the corresponding
LLA results. A very good agreement of both descriptions is obtained
for larger values of $x$ and lower and medium values of $y$. In the
high~$y$ range the LLA contributions yield  somewhat larger corrections,
which are further enhanced by the Compton contribution.

In Figure~6 the same corrections are considered applying a cut of
$Q^2_h > 1 \GeV^2$. The Compton part of the LLA contributions is
widely removed by this condition. For smaller $x$ values the
agreement between the complete corrections and the LLA contributions
becomes somewhat worse.

Figure~7 shows the different contributions to the leptonic corrections
in LLA for the kinematic regime of the HERMES experiment for longitudinal
proton polarization. While the
initial state corrections behave rising for growing $y$ and smaller
values of $x$, the final state corrections are rather flat in $y$.
The Compton contribution is only essential at small $x$ and large $y$.

In Figure~8 a comparison is given for the complete leptonic
$O(\alpha)$ corrections and the corresponding LLA contributions for
the polarized part of the
differential scattering cross section in the case of transverse
proton polarization for the kinematic range of the HERMES experiment.
The $x$-dependence of the corrections is implied by the behavior
of the structure functions $g_1(x,Q^2)$ and $g_2(x,Q^2)$.
As in the case of longitudinally polarized protons,
 the complete and
LLA corrections agree very well for small $y$ and larger values of $x$.
In the high $y$ range both descriptions differ. Again the Compton
term increases the difference slightly.

The effect of a cut of $Q^2_h > 1 \GeV^2$ on the corrections for
the case of transverse proton polarization is illustrated in Figure~9.
The Compton contribution is removed by this cut as in the case
of longitudinal proton polarization (Figure~6). For smaller values
of $x$ the LLA terms deviate stronger from the complete corrections
than in the case of longitudinal proton polarization.

In Figure~10 the different contributions to the LLA corrections
are depicted for the case of transverse nucleon polarization.
Similarly to the case of longitudinal nucleon polarization, the
initial state radiation terms do widely determine the structure of the
LLA correction. The final state radiation terms are flat in $y$ and
the Compton terms contribute at small $x$ and large $y$ only.

The polarized contribution to the
complete leptonic $O(\alpha)$ correction to the
differential deep-inelastic scattering cross section for longitudinally
polarized protons in the kinematic range of the HERA collider
at $\sqrt{S} = 314~\GeV$ are shown in Figure~11.
Their shape is rather similar to the corrections for lower values of
$\sqrt{S}$, cf.~Figure~5. Lower values of $x$, $x \sim 10^{-3}$ can
be probed. Here the corrections are rather large. The comparison with
the LLA contributions is also shown. While the full LLA corrections
agree rather well with the results of the complete calculation in the
whole kinematic range, the contributions due to only initial and final
state Bremsstrahlung differ in the high $y$ and low $x$ range from
the complete $O(\alpha)$ corrections. The good
agreement of the complete LLA relations is due to the choice of the
Compton-logarithms, eq.~(\ref{compt},\ref{q2lim}), which is particularly
suited for larger values of $\sqrt{S}$.

In Figure~12 a comparison is shown of the polarized part of the
complete leptonic $O(\alpha)$  QED corrections for longitudinally
polarized protons at $\sqrt{S} = 314~\GeV$ parametrizing the respective
contributions to the polarized part of the scattering cross section
by either only the structure function $g_1(x,Q^2)$ or the complete
set of structure functions $g_1(x,Q^2), g_4(x,Q^2)$, and $g_5(x,Q^2)$,
which are not  suppressed kinematically
by factors of $O(M^2/S)$, as the contributions due to the
structure functions
$g_2(x,Q^2)$ and $g_3(x,Q^2)$. Because of
the different $y$ behavior of
the kinematic factors weighting the structure functions,
the correction
changes significantly if only the structure function $g_1(x,Q^2)$ is
considered in the scattering cross section. This applies also for the
small $x$ and low $y$ range and is caused by the structure functions
$g_4(x,Q^2)$ and $g_5(x,Q^2)$,  which contribute to the $\gamma$-$Z$
interference and $Z$-exchange contributions.
\section{Conclusions}
\label{sect6}

\vspace{1mm}
\noindent
The $O(\alpha)$ leptonic QED corrections to the polarized
differential neutral
current  deep-inelastic scattering cross sections were calculated
both for the case of longitudinal and transverse nucleon polarization.
This approach allows for a general description of
structure functions, and is therefore particularly suited for
experimental analyses, which aim on the unfolding of the different
polarized
structure functions from the measured cross sections.
In the explicit calculations we considered a hadronic tensor which obeys
Lorentz and time--reversal invariance, as well as current conservation.
Its polarized part is described by the five structure functions
$\left. g_i^p(x,Q^2)\right|_{i=1}^5$.
In the $O(\alpha)$ corrections the general structure of the Born
cross section is resembled. The corrections can be expressed in terms
of general kinematic factors and appropriate combinations of the
neutral current structure functions, including both photon and
$Z$-boson exchange.
Additional combinations of
structure functions contribute as well for terms $\propto m^2$.
The latter terms yield finite contributions after integration in
the limit $m \rightarrow 0$.

In the leading logarithmic approximation the $O(\alpha)$ leptonic
QED corrections are described by the initial and final state
Bremsstrahlung contributions and the Compton terms for pure
photon exchange. While the structure of the Bremsstrahlung contributions
is the same for the case of unpolarized and polarized deep-inelastic
scattering, different expressions are obtained for the Compton terms.
This is due to the behavior of the leading order QED
splitting and coefficient functions in situations, in which the
collinear parton model applies. The respective expressions are rather
compact and their structure is partly induced by  mass
singularities, which allows an easy deduction of the corresponding
expressions.

Numerically the $O(\alpha)$ corrections are rather large in the
case of $e N$-scattering. Since the corrections behave
$\sim \ln(Q^2/m^2)$,
smaller corrections are obtained for the case of polarized $\mu N$
scattering.
A numerical comparison with the corrections obtained by the
code~\cite{MIN1} was carried out in refs.~\cite{HWS}--\cite{CRAC}
for the simplified assumptions in~\cite{MIN1}.

Both in the kinematic domain
probed by the HERMES experiment and the
kinematic range which would be
accessible in
possible future polarized
deep inelastic scattering experiments in the HERA collider mode,
the corrections grow towards small $x$ and high $y$.
Except the range of large values of $y$ and small $x$ already
the LLA results provide a very good description of the corrections.
The structure of the corrections as a function of $x$ and $y$ is  widely
determined by the initial-state Bremsstrahlung terms.
If cuts, e.g.
on $Q^2_h$, are applied this agreement, however,
becomes worse.
While for longitudinal nucleon polarization the corrections are
structurally similar to those in the
unpolarized case, the corrections
in the case of transverse nucleon polarization show a different
$x$-behavior.
\newpage
\setcounter{equation}{0}

redpajama_set_name: RedPajamaArXiv


Ian Astbury's long-delayed solo debut shifts the focus from metal to the kind of ragged industrial trip-hop heard on the Cult's 1992 single "The Witch" (a new version of which is also included here). Spirit/Light/Speed is probably indistinguishable from what a Cult album would've sounded like circa 2000: rough and earthy, full of the mystic lyrics and extroverted performances that Astbury's been doing since early in the Cult's career. Chris Goss is an able stand-in for the Cult's Billy Duffy on the production and guitar-playing side, though it's clear that Duffy's distinctive songwriting skills could've come in handy. Though this is a solo album only a fan could really love, Astbury's transcendent voice -- one of the best of any British band during the '80s -- makes highlights out of a few songs here, including the ballad "Tonight (Illuminated)."
Spirit\Light\Speed 5 out of 5 based on 0 ratings. 1 reviews.
Being a 'Cult' fan I purchased Ian's album not knowing what to expect. I'm more than happy to tell you this is by far the most amazing work ever done by either Ian or The Cult. Ian's voice is still hypnotic and eerie but is now set against an even more electronic background than before. Songs like 'Sweet Soul Sister' (The Cult) always intrigued me because of the use of electronica with guitar. Ian's album is definately a giant leap forward artistically. If you're into the Hacker/Sci-Fi style electronic music you'll love it. Think The Cult meets Moby! This is the most original sounding music I've heard in a long time and if anything this guy influenced the electronic rock scene not the other way around. After all Ian's been doing this since the late seventies. What more can I say - Buy It! (If you don't you're jippin yourself).


text:
HomeAutoBuzz.TVVideo: Mercedes-Benz C200 & C300 facelift, Things You Need To Know
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/ Nov 2, 2018 11:50 am
/Pan Eu Jin
/Mercedes-Benz C-ClassMercedes-Benz MalaysiaThings You Need to Know
Video: Mercedes-Benz C200 & C300 facelift, Things You Need To Know
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Mercedes-Benz Malaysia has just introduced the facelifted C-Class in the form of three different models – the C43 AMG 4MATIC, the C300 AMG Line and the C200 Avantgarde Line and here's a quick walk-through on what's new with the latter two.
The new C200 Avantgarde is fitted with a new 1.5-litre turbocharged engine and unlike before, the new C300 AMG Line now gets the 9G-TRONIC 9-speed automatic transmission and the AIRMATIC suspension along with a bump in horsepower.
Find out what are the other new additions to one of the most successful models for Mercedes-Benz Malaysia and check out how the new Mercedes-AMG C43 4MATIC differs from the pre-facelift model here.
Tags:Mercedes-Benz C-ClassMercedes-Benz MalaysiaThings You Need to Know
Mercedes-Benz Malaysia launches the facelifted C43 AMG 4MATIC
Mercedes-Benz Malaysia launches the updated C-Class Coupe
Pan Eu Jin
Regularly spend countless hours online looking at cars and parts I can't afford to buy. How a car makes you feel behind the wheel should be more important than the brand it represents - unless resale value is your thing.

redpajama_set_name: RedPajamaCommonCrawl


text:
Dr. Tim Franson Rejoins FaegreBD in Indianapolis
Faegre Baker Daniels is pleased to announce that Dr. Tim Franson has rejoined the firm's Indianapolis office as a principal in the Health & FDA group. Dr. Franson also will work significantly from the Washington, D.C. office.
Dr. Franson has extensive experience in all pre- and post-approval phases of pharmaceutical development (small and large molecule) relating to interactions with FDA for policy and product issues, as well as interactions with global regulators. His practice provides clinical and regulatory professional services for drug and biologics development (strategy, planning, training and issues management) post-approval advice (REM's, labeling, compliance) and due diligence for venture capital firms.
"We are delighted that Tim is bringing his unparalleled regulatory expertise back to the firm," noted Health & FDA group leader Dave Zook. "Our clients in industry, research organizations, and patient advocacy communities will be able to apply his deep experience to their efforts to advance new therapies."
Dr. Franson previously served as Vice President of Global Regulatory Affairs at Eli Lilly & Co. after which he was a principal with Faegre Baker Daniels Consulting (2010-14). He also held the position of President of USP, the standards-setting organization for the biopharmaceutical sector and is currently the board chair of the Critical Path Institute. From 2014-19, Dr. Franson was Chief Medical Officer of YourEncore.
Timothy R. Franson, M.D.
Principal - Faegre Drinker Consulting
tim.franson@faegredrinker.com
Lindsay Gotwald
lindsay.gotwald@faegredrinker.com
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Dr. Tim Franson Rejoins FaegreBD in Indianapolis | News | About | Faegre Drinker Biddle & Reath LLP


Weekend Escape or year round home. Exquisite multi-dwelling family compound. Three homes, and a renovated Greenhouse provide over 5,400 square feet of living space. A tennis court, in-ground swimming pool & cabana, and gorgeous gardens adorn this manicured seven acre estate; artfully dotted with massive trees and old stone walls. Wonderful for entertaining! The large front lawn has hosted family weddings and large social gatherings. While the back yard, decks, gardens, pool, and tennis court provide a more intimate setting. Elegant yet understated home maintains an authentic mid-century design yet providing all the expected amenities. Sun-decks overlook a zen-like setting flowing gracefully to the heated pool, cabana, and tennis court below. A renovated 1890 Lord and Burnham Greenhouse, the estate's original Gatehouse and a delightful cottage complete the 4 building ensemble. Miles of gorgeous wooded hiking trails are easily accessed; including the Appalachian Trail.
This data has been updated as of 4/25/2019 2:21:58 PM.


Reports confirm that 30-year-old Natasha Harris, a New Zealand mother-of-eight who died of an alleged heart attack three years ago, ultimately passed away due to her addiction to Coca-Cola. Although Coca Cola Ltd. insists that there's no proof Harris' aggressive consumption of the soda beverage contributed to her unexpected death, Coroner David Crerar released a report on Tuesday (Feb. 12), claiming that Harris' dependency on caffeine and poor nutrition indeed led to her fatal cardiac arrest in 2010.
Harris' de facto partner Christopher Hodgkinson claims that the young mother would consume up to 10 litres of Coke a day - equivalent to more than twice the 400mg recommended safe limit of caffeine. "The first thing she would do in the morning was have a drink of Coke, and the last thing she would do in the day was have a drink of Coke by her bed... She had no energy and was feeling sick all the time ... She would get up and vomit in the morning...she would get moody and get headaches if she didn't have any Coke and also feel low in energy." Hodgkinson said in the death inquest. Harris also ate very little, and would smoke up to 30 cigarettes a day.
Hodgkinson found Harris slumped over on the toilet, gasping for air on February 25, 2010. She later died of cardiac arrest.
According to the death inquest, Harris' teeth had been pulled out because they became so rotten from decay. At least one of her eight children was born having no tooth enamel.
"The Coroner acknowledged that he could not be certain what caused Ms Harris' heart attack. Therefore we are disappointed that the Coroner has chosen to focus on the combination of Ms Harris' excessive consumption of Coca-Cola, together with other health and lifestyle factors, as the probable cause of her death. This is contrary to the evidence that showed the experts could not agree on the most likely cause."
Crerar agrees that Coca-Cola should not be held responsible "for the health of consumers who drink unhealthy quantities of the product," but he does urge that the popular soda company consider including the quantity of caffeine content on its labels. He also recommends including a warning label about excessive consumption.
Crerar also suggests that the New Zealand Ministry of Health review and revise nutrition labels on soft drinks, so consumers are more aware and protected.


text: Slurs related to low intelligence

redpajama_set_name: RedPajamaWikipedia


The Academy has confirmed that four of its awards will be announced during the show's advert breaks - a move against tradition, but in a bid to reduce the show's running time to three hours.
Those categories will be cinematography, film editing, live-action short and makeup and hairstyling.
As there is no host this year following the furore around Kevin Hart, the ceremony will feature a variety of Hollywood stars - including Jennifer Lopez, Daniel Craig, and Brie Larson - handing out awards.
In an email from Academy president John Bailey, obtained by The Hollywood Reporter, Bailey stressed that the organisation is "still honouring the achievements of all 24 awards on the Oscars", and that the winners' speeches will air later in the broadcast.
"I want to reiterate however, that all 24 Academy Award-winning presentations will be included in the broadcast," Bailey explained. "We believe we have come up with a great way to do this, and keep the show to three hours."
In addition, he added: "With the help of our partners at ABC, we also will stream these four award presentations online for our global fans to enjoy, live, along with our audience. Fans will be able to watch on Oscar.com and on the Academy's social channels. The livestream is a first for our show, and will help further awareness and promotion of these award categories.
"So, buckle up! We are committed to presenting a show which we all will be proud of."
Comedian Hart stepped down as host this year after a string of homophobic tweets from 2010 resurfaced on social media. The star immediately issued an apology to the LGBTQ community.
This year's Oscar nominees include Olivia Colman for The Favourite, Mahershala Ali for Green Book, Rami Malek for Bohemian Rhapsody, Regina King for If Beale Street Could Talk and Glenn Close for The Wife.
The ceremony will take place on Sunday, February 24.
Which tube line gets hottest during the summer?


text:
Midweek Herald > News
Axminster gymnasts bounce back from Covid closure
Inside Axminster Gymnastics Club new site on Woodmead Road - Credit: Emma Rowe
After a tricky 18 months, Axminster Gymnastics Club is back on track and winning medals once again.
The club was forced to close due to the Covid pandemic, but members and staff worked hard to run gymnastics sessions virtually, while also organising a move to a new site in Woodmead Road. Face-to-face sessions restarted in April this year, and the move to the bigger and better new venue took place in September.
October and November brought the first round of competitions the gymnasts had been able to take part in since February 2020. At the Falcons Hallowe'en Competition the club bought home an array of medals including golds, and the following gymnasts were placed first overall in their categories: Jasmine - 11yr Girls, Cally - 13yr Girls and Sam - O13 Boys.
Falcons Halloween Competition 2021 - Credit: Emma Rowe
At the Women's Artistic Devon Levels Championships Evie was placed 3rd overall at Level 5 U11s, Cally 2nd overall at Level 6 U13s and Fern 3rd overall at Level 6 O13s.
Head coach Emma Rowe said: "These were fantastic results as we are one of the smallest and newest artistic clubs in Devon."
Artistic Devon Levels Championship winners - Credit: Emma Rowe
Artistic Devon Levels Championship 2021 - Credit: Emma Rowe
Emma founded Axminster Gymnastics Club in 2017. She had moved to Axminster in 2014 after leaving a career with the Royal Marines, and found there was little local provision for artistic gymnastics and tumbling.
Axminster Gymnastics began from the back of a van with Emma running classes in small local halls. By summer 2017 the club had moved to the local leisure centre, enabling the storage of larger equipment and allowing expansion of the club to run five days a week.
9 Sophie sacrifices waist-length hair for charity
10 Four Devon locations make 'worst places to live' list for 2022
On January 1 2018 the club became a limited company, registered as not-for-profit, and its numbers continued to grow. It is affiliated to British Gymnastics, South West Gymnastics and Devon Gymnastics.
It now offers a variety of groups including free play for 0-3yrs, gymnastics lessons for age 2yrs upward to adults, children's parties, and competition options for girls and boys.
Emma said: 'We are excited to be back and performing and having the chance to offer gymnastics lessons to so many local children."
If you'd like to try out a class please get in touch via Facebook or 07717310198 or email info@axminstergymnastics.co.uk.
Devon woman shocks TV hosts after naming son LUCIFER


It's a positive early part of the week, though the bears have stopped the momentum, with Tron having seen red in 5 of the last 6 days and with Tron back in the red through the early hours of this morning.
It's been a mixed start to the week for Tron ("TRX"), with a bullish start to the week seeing Tron rally to an early intraday week high $0.027 on Tuesday, Tron falling short of the week's first major resistance level at $0.0312, before hitting reverse.
A current week gain of 15%, Monday through Tuesday, came on Monday, with Tron rallying by 21.36% to reverse last week's 8.63% slide, before falling by 5.25% on Tuesday to pullback to $0.24 levels.
Moves through the early part of the week have seen Tron manage to steer clear of the week's first major support level at $0.0165, while also coming up well short of last week's high $0.03585.
In spite of Monday's rally, it's been a bearish of late, with Tron seeing red on 5 of the last 6 days to cut the 2019 gain to 87%, still a whopping number when compared with the broader market.
In spite the bullish start to the year, the extended bearish trend, formed at an end of April swing hi $0.1004, remained intact following a late November swing lo $0.0111, with Tron continuing to fall short of the 23.6% FIB Retracement Level of $0.0322.
At the time of writing, Tron was down 0.25% to $0.024687, with moves through the early hours of Wednesday morning seeing Tron fall from a morning high $0.024803 to a low $0.024320 before steadying.
For the remainder of the week, a move back through to $0.026 levels would support a break through the current week's high $0.027 to bring $0.03 levels and the first major resistance level at $0.0312 into play before any pullback.
Following last week's high $0.035846, Tron will likely fall short of $0.04 levels and the second major resistance level at $0.0408, barring a major crypto rally. A move through to $0.03 levels and the 23.6% FIB Retracement Level of $0.0322 needed ahead of the weekend to raise the possibility of $0.04 levels by the end of the week.
Failure to move back through to $0.026 levels ahead of the weekend could see Tron hit reverse going into the weekend, a pullback through the current week low $0.0213 likely to bring sub-$0.02 levels into play before any recovery.
While the extended bearish trend remains intact, we would expect Tron to steer clear of the week's first major support level at $0.0165 through the remainder of the week, barring another crypto meltdown.
For the crypto bulls, last week's reversal brought to an end any near-term chance of a bearish trend reversal, with Tron needing a breakout from the 23.6% FIB Retracement Level of $0.0322 to $0.040 levels to begin forming a near-term bullish trend.


Amid tight security arrangements, and guarded by nervous armed Egyptian officers, funeral services were held for the victims of the two terrorist attacks that targeted churches in Tanta and Alexandria.
Attendance at the funeral ceremonies was limited to the relatives of the victims and church officials.
A Sunday morning terrorist bomb attacks on two Coptic churches in Egypt killed over fifty people and left over one-hundred people wounded. Both churches were packed with worshipers attending Palm Sunday services.
The latest attack targeted the Mar Girgis church in Tanta in Lower Egypt, and the entrance of the St. Mark church in Alexandria on the Mediterranean Sea coast, where Pope Tawdros II led the service.
A policeman was also killed in the church blast in Alexandria.
The Islamic State (ISIS) has claimed responsibility for the two attacks that came about three hours apart.
Following the deadly attack, Egyptian President Abdulfatah Al-Sissi announced a three-day national mourning period and a three-month state of emergency which will take effect after fulfilling some required constitutional requirements.
Last week, Al-Sissi was on a state visit to the USA and met president Donald Trump.
On December 2016, a suicide bomber has killed 29 people and damaged the grand Coptic church in Cairo.
In 2010, as the Arab Spring demonstrations were getting stronger, another such church attack by extremists left scores of people dead and wounded. At the time, some observers suspected Mubarek's interior minister, General Habib Al-Adly, of complacency in the New Year's Day attacks on Saints church in Alexandria.
The alleged plot aimed at pressuring the late Pope Shenuda III and others to distance themselves from the Arab Spring demonstrations that aimed to topple the president Hosni Mobarek regime. The minister was accused "of deliberately releasing prisoners in order to show [there was] chaos in Egypt." The 2010 attack killed 24 people and wounded 90.
According to Egyptian reporter that when he came to power, Al-Sissi ordered the release of Habib Al-Adly from jail.
My sympathy and condolences to the bereaved families ,the senseless attacks targeted churchgoers attending Palm Sunday services, which mark the Sunday before Easter and the start of Holy Week for Christians.
In 628 C.E. Prophet Muhammad (s) granted a Charter of Privileges to the monks of St. Catherine Monastery in Mt. Sinai. It consisted of several clauses covering all aspects of human rights including such topics as the protection of Christians, freedom of worship and movement, freedom to appoint their own judges and to own and maintain their property, exemption from military service, and the right to protection in war. The protection that Christian communities were meant to receive under Islam was enshrined in a letter of protection from Prophet Muhammad to the Christian monks at St. Catherine's Monastery in Sinai in the early 7th century. This letter promised the monks that, under Islamic rule, the Christian community, as a "people of the book", shall have the freedom to practice their religion and be protected from any unlawful interference or molestation, whether in their communities or while traveling. Distinct from a war with Christianity, Prophet Muhammad further stated, "No one shall bear arms against [Christians], but, on the contrary, the [Muslims] shall wage war for them." Their houses of worship should be defended from attack and their right to worship according to their choice respected. The right to self-defense was granted in Islam in order to protect freedom of religion. No one is to destroy a house of their religion, to damage it, Their churches are to be respected. They are neither to be prevented from repairing them nor the sacredness of their covenants. it clearly protects the right to property, freedom of religion, freedom of work, and security of the person.
Indeed, the word is still in a much better situation, despite all the problems.
I find it difficult to believe that human beings are born evil and this is engraved in their dna. I do not think that our dna controls more than our biological functions and our phenotypes. The hunter and gatherer human being lived a shared life. When human beings acquired the concept of property and power, that was when they became evil. Remove these two concepts from their environment, you can make them innocent creatures. The environment plays the most important role. If a child is born into a criminal family, the probability is that it will becomes a criminal, because that is what he learns from the family, his micro-environment. The same holds true for most human behaviours in human society.
Who has the power has also the means, and he is the one who decides and takes the lion's share from everything, if he wants. Although fate has made the white man to have the upper hand and the future of the world is in his hands, nevertheless, he has acted out of greed up to now, and not in a responsible way. If it is going to continue with the concept of winner takes all, their might not be human society in the future as we know it today. We cannot take away his responsibilities by saying that if the roles were reversed, others would have acted the same way. What we know is that today he is the main actor, until somebody else comes to replace him.
What does it take to destroy the world? A thermonuclear war that could start by mistake by some dumb who has access to the codes, climate change and and global warming that might not make this world habitable anymore due to the fact that some fool would like to continue burning fossil fuel and coal, and a virulent viral epidemic for which the world is not prepared, are not science fictions anymore. The world is a better place for the time being, but not a safe place, unless responsible people take its fate into their hands.
The world is a small place. A fire in a one house is going to affect the other however far away it might be. The middle east crisis has shown us exactly that. That is why human beings should learn to exist together in peace, destroy all wmd, protect the environment and the climate for the good of all. If the ice cap in the arctic melts seaside cities around the world are going to be flooded without any exception, an epidemic in one country will be in western or eastern capitals the next day. Therefore, it should be all for all, for no one is safe alone.
Human society has lived through the two phases of progress and regression. Up to now it has managed to make pace in the technological field, but not in the social. That is why human development has always been precarious, marked by wars and destructions in between periods of progress. This is an interdependent world, and that is why it should be an inclusive world too. Human beings will either laugh together or cry together. In the very last, either all will be winners or all losers. Can the exploiter exist for ever by exploiting others, can the producer exist without the consumer, has technology and artificial intelligence any meaning if it does not serve all human beings? Or if AI is going to be superior to human intelligence and functions autonomously, let us say in 500 years or so, what will be the fate of the greedy human being, who never wanted to share?
May Egypt heal from this and live in peace. It has been home to many Eritreans both Muslims and Christians since the 50s of the last century. My condolence to the families of the victims.
Greetings, The challenge of Egypt seems to be multi-faceted. The Middle East is characterized by high concentration of political, spiritual, and monetary resource in very limited circles of people.
In the past few hundred years, Egypt was the uncontested leader of the middle east in military, economy, and intellectual capacity.
In the past couple of decades the region (including Turkey and Iran) changed beyond recognition. The GCC countries became the economic powerhouses of the middle east; Turkey and Iran also transomed their capabilities substantially.
In the meantime Egypt showed a net economic growth of 2-3% over the past six decades (the population growth evened out the other 3% of the growth).
The fracture within the Egyptian Muslim intellectuals (and others too) is also visible. Egypt is the origin of Muslim Brothers; the Alazar school of theology is there; The Wahhabi interpretation (with its origins in the KSA) is also finding more and more traction. Further, the Coptic and other minor religions (with atheist and Nasserist currents included) were also part and parcel of the Egyptian society.
The current ideological and military center of gravity looks also to be on the motion away from Egypt, and more towards the GCC countries, albite slowly and incoherently.
It looks to be challenge of the century for Egyptian to synchronize the various, at times antagonistic, political currents within the country; and find its new contemporary place in the Greater Middle East region.
We live in the age of religious fanaticism, ethnic and racial superiority complexes, and a new world order in the making, the nature of which nobody seems to know or is able to foresee. A world in turmoil, underlings at the helm of world power, and hope for a peaceful world seems as far away as it has ever been, with the possibility of a looming worldwide confrontations.
Christians are murdered in egypt simply because they happen to be christians, muslims are murdered in myanmar, again simply because they are muslims, and civilians are murdered simply because, unfortunately, they were at the wrong place in the wrong time, and more importantly, criminals have taken care that as many civilians as possible are gathered together in the same place, so that their crimes will have a broader impact on societies. Nothing is holy, and nobody is safe, anymore.
Is the egyptian government able to stop this chain of bombings that have cost hundreds of innocent lives, or is it out of its control? Myanmar's leader aung san suu kyi, seems to be not doing enough to control the murders and rapes of muslims. Human beings have made religion and race a worldwide problem that could one day destroy humanity.
An indian doctor and his wife were beaten and thrown out of a delta flight in the usa, because they refused to abandon their seat for a white american. This used to happen in the usa more than 50 yrs ago. It is again happening in this 21st century, in the age of worldwide information, when nothing under the sun is a secret anymore.
The economic crisis is not going to change in the immediate future, because the greed of the white bear is out of this world, with no prospect for any improvement. Human beings have turned against human beings, and love is dying and hate is mushrooming everywhere. It takes a messiah to save the world? A melancholic world, indeed.
I think the doctor is a Chinese descent on United Airline if we're speaking of the same passenger.
Thanks for the info. There appears to be some confusion with the identity of the person. Yesterday's initial reports mentioned that it was an Indian doctor. The mix up may have been due to what the person who posted the video thought the couple were.
Any way, Indian, Chinese or Vietnamese, who expects to end up being dragged along the aisle and with a bloody nose, when one has a valid airline ticket in one's hand. Why this fellow was chosen among so many was also a mystery, to say the least. Other passengers looked appalled when they saw the person being dragged. It has become a common seen on usa airlines.
Hey Semere; I think the Dr has contributed much of this incidence. I mean, the airlines were offering 800 USD just to skip that particular flight and four people took the offer and left. the reason I am saying is that this particular flight was full and in the last minute they were told they must take a crew so they needed few sits for the crew. My point is if the crew are delayed; there are many more passengers will be delayed but one Dr could have understood and exited with dignity. I know it is a matter principal but sometimes also about understanding and commonsense.
This is my thing. Delay (for hours and days) at the airport is a fact of life. I happens all the time, and people complain all the time.
It is not so much what they (the security personnel) did (made him wait) that infuriated the public, it is how they did it.
Just imagine – under any pretext (for their own "crazy" reasons), without offering him a dime for his delay, if they would've stopped him (the doctor) at the gate, and made him miss his flight how things would've turned out differently.
Good to see you around Kokhobay. I'm glad you're feeling better.
Like everyone in this forum, I miss your presence, I miss your poems, and I miss your contribution to this great Awate university. I hope, soon, you recover fully and join this war of ideas. Your presence is badly needed to knock some teeth.
Until then take it easy and take care of yourself.
It's gut wrenching to witness attack after attack on those helpless Coptic Christians in Egypt and there's nothing government can do to prevent it. It's a question of when for the next imminent attack to happen. Islamic extremists can't coexist, there's loads of evidences to justify it. I feel sad for the peaceful billions of Muslims out there. These filths are severely tarnishing the great faith.


text:
Celebrate Captain Marvel's Milestone 150th Issue with the BossLogic Variant Cover
'Captain Marvel' #16 by Kelly Thompson and Lee Garbett marks Carol Danvers' 150th Issue!
This March, Carol Danvers AKA Captain Marvel reaches a mighty milestone when CAPTAIN MARVEL #16 hits stands -- not only will the issue present the epic conclusion to the hit "The Last Avenger" arc written by Kelly Thompson with art by Lee Garbett, but it will also be the 150th issue for the character! To honor the occasion, take a look at the brand-new variant cover by renowned artist BossLogic and pick up the landmark issue on Wednesday, March 18!
Thompson began her acclaimed take on the character last year, just in time for Marvel Studios' record-breaking blockbuster film, and like so many Captain Marvel writers before her from Chris Claremont to Kelly Sue DeConnick, she's proven why Carol Danvers is one of Marvel's most inspiring heroes!
"It's so great that the last issue of our "The Last Avenger" arc is the Legacy 150th issue of CAPTAIN MARVEL because it's an issue that really highlights how and why Carol is such an incredible hero," promises Thompson. "That she will go to any lengths to save the day... that she NEVER stays down no matter how deep and dark the odds... and that's been in her since the beginning."
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Q: ValueError using recursive feature elimination for SVM with rbf kernel in scikit-learn I'm trying to use the recursive feature elimination (RFE) function in scikit-learn but keep getting the error ValueError: coef_ is only available when using a linear kernel. I am trying to perform feature selection for a support vector classifier (SVC) using a rbf kernel. This example from the website executes fine:
print(__doc__)

from sklearn.svm import SVC
from sklearn.cross_validation import StratifiedKFold
from sklearn.feature_selection import RFECV
from sklearn.datasets import make_classification
from sklearn.metrics import zero_one_loss

# Build a classification task using 3 informative features
X, y = make_classification(n_samples=1000, n_features=25, n_informative=3,
                       n_redundant=2, n_repeated=0, n_classes=8,
                       n_clusters_per_class=1, random_state=0)

# Create the RFE object and compute a cross-validated score.
svc = SVC(kernel="linear")
rfecv = RFECV(estimator=svc, step=1, cv=StratifiedKFold(y, 2),
          scoring='accuracy')
rfecv.fit(X, y)

print("Optimal number of features : %d" % rfecv.n_features_)

# Plot number of features VS. cross-validation scores
import pylab as pl
pl.figure()
pl.xlabel("Number of features selected")
pl.ylabel("Cross validation score (nb of misclassifications)")
pl.plot(range(1, len(rfecv.grid_scores_) + 1), rfecv.grid_scores_)
pl.show()

However, simply changing the kernel type from linear to rbf, as follows, produces the error:
print(__doc__)

from sklearn.svm import SVC
from sklearn.cross_validation import StratifiedKFold
from sklearn.feature_selection import RFECV
from sklearn.datasets import make_classification
from sklearn.metrics import zero_one_loss

# Build a classification task using 3 informative features
X, y = make_classification(n_samples=1000, n_features=25, n_informative=3,
                       n_redundant=2, n_repeated=0, n_classes=8,
                       n_clusters_per_class=1, random_state=0)

# Create the RFE object and compute a cross-validated score.
svc = SVC(kernel="rbf")
rfecv = RFECV(estimator=svc, step=1, cv=StratifiedKFold(y, 2),
          scoring='accuracy')
rfecv.fit(X, y)

print("Optimal number of features : %d" % rfecv.n_features_)

# Plot number of features VS. cross-validation scores
import pylab as pl
pl.figure()
pl.xlabel("Number of features selected")
pl.ylabel("Cross validation score (nb of misclassifications)")
pl.plot(range(1, len(rfecv.grid_scores_) + 1), rfecv.grid_scores_)
pl.show()

This seems like it could be a bug, but if anyone could spot something I'm doing wrong that would be great. Also, I'm running python 2.7.6 with scikit-learn version 0.14.1.
Thanks for the help!

A: This seems to the expected outcome. RFECV requires the estimator to have an coef_ which signifies the feature importances:

estimator : object
A supervised learning estimator with a fit method that updates a coef_ attribute that holds the fitted parameters. Important features must correspond to high absolute values in the coef_ array.

By changing the kernel to RBF, the SVC is no longer linear and the coef_ attribute becomes unavailable, according to the documentation:

coef_
array, shape = [n_class-1, n_features]
Weights asigned to the features (coefficients in the primal problem). This is only available in the case of linear kernel.

The error is raised by SVC (source) when RFECV is trying to access coef_ when the kernel is not linear.

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