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\begin{document}
\thispagestyle{empty}
\begin{flushright}
MZ-TH/95-19 \\[-0.2cm]
FTUV/95-52 \\[-0.2cm]
IFIC/95-54 \\[-0.2cm]
January 1996 \\[-0.2cm]
\end{flushright}
\begin{center}

{\Large\bf Polar Angle Dependence of the}\\[.3cm]
{\Large\bf Alignment Polarization of Quarks}\\[.3cm]
{\Large\bf Produced in \boldmath{$e^+e^-$}-Annihilation}\\[.75cm]

{\large S.~Groote$^*$, J.G.~K\"orner\footnote{Supported in part
by the BMFT, FRG, under contract 06MZ566,\\\hbox{\qquad}
and by HUCAM, EU, under contract CHRX-CT94-0579}} \\[.4cm]
Institut f\"ur Physik, Johannes Gutenberg-Universit\"at \\
Staudingerweg 7, D-55099 Mainz, Germany. \\[.5cm]

{\large M.M.~Tung\footnote[7]{Feodor-Lynen Fellow}} \\[.4cm]
Departament de F\'\i sica Te\`orica, Universitat~de Val\`encia \\
and IFIC, Centre Mixte Universitat Val\`encia --- CSIC, \\
C/ Dr.~Moliner, 50, E-46100 Burjassot (Val\`encia), Spain.
\end{center}
\vspace{.75cm}
\centerline {\bf ABSTRACT}
\noindent
We calculate one-loop radiative QCD corrections to the three polarized and 
unpolarized structure functions that determine the beam-quark polar angle 
dependence of the alignment (or longitudinal) polarization of light and 
heavy quarks produced in $e^+e^-$-annihilations. We present analytical and 
numerical results for the alignment polarization and its polar angle 
dependence. We discuss in some detail the zero-mass limit of our results 
and the role of the anomalous spin-flip contributions to the polarization 
observables in the zero-mass limit. Our discussion includes transverse and 
longitudinal beam polarization effects.

\newpage

\section{Introduction}
This is the fifth and final paper in a series of papers devoted to the 
$O(\as)$ determination of the polarization of quarks produced in 
$e^+e^-$-annihilations. In the first paper of this series~\cite{forward1} 
we calculated the mean alignment polarization (sometimes also called 
longitudinal or helicity polarization) of the quark. i.e. the alignment 
polarization averaged w.r.t. the relative beam-event orientation. One of us 
derived convenient Schwinger-type representations for the structure 
functions appearing in the polarization expressions in~\cite{forward2}. In 
a third paper we determined the longitudinal component of the quark's 
alignment polarization, which vanishes at the Born term level~\cite{forward3}. 
The fourth paper~\cite{forward4} gave results on the two transverse 
components of the quark's polarization (perpendicular and normal, also 
sometimes referred to as in-the-plane and out-of-the-plane polarization). 
In the present final piece of work we present our results on the full polar 
angle dependence of the quark's alignment polarization w.r.t. the electron 
beam direction including beam polarization effects.

The full determination of the alignment polarization of the quark involves 
the calculation of three polarized and three unpolarized structure 
functions which are conveniently chosen as the helicity structure functions 
$H_{U,L,F}^\ell$ and $H_{U,L,F}$, respectively. The subscripts $U$, $L$ and 
$F$ label the relevant density matrix elements of the exchanged vector 
boson ($\gamma_V$,$Z$), where the polar angle dependence of the 
contributions of the three structure functions is given by $(1+\cos^2\theta)$ 
for $U$, $\sin^2\theta$ for $L$ and $\cos\theta$ for $F$. We mention that 
some of the unpolarized helicity structure functions have been calculated 
before, i.e. the vector/vector ($VV$) contribution to $H_U$ and 
$H_L$~\cite{forward5}\footnote{The $VV$-contribution to $H_{U+L}=H_U+H_L$ 
has been given a long time ago in the context of QED 
(see e.g.~\cite{forward6})} and the vector/axial-vector ($VA$) contribution 
to $H_F$~\cite{forward7}. We have verified the results of~\cite{forward5} 
and~\cite{forward7} and present new results on the
axial-vector/axial-vector ($AA$) contribution to $H_U$ and $H_L$ in this 
paper, where the $AA$-contribution to $H_{U+L}=H_U+H_L$ was already written 
down in~\cite{forward1,forward2}. As concerns the polarized structure 
functions our results on $H_{U+L}^\ell$ and $H_L^\ell$ where given 
in~\cite{forward1,forward2,forward3}. This paper includes new results on 
the polarized structure functions $H_U^\ell$ and $H_F^\ell$ that are 
necessary to determine the full $\ct$-dependence of the quark's alignment 
polarization.

The paper is structured such that we start in Sec.~2 by listing the four 
independent tree-graph components of the hadron tensor ($VV$, $AA$, $V\!A$ 
and $AV$) including its polarization dependence. The relevant helicity 
components of the structure functions $H_{U,L,F}$ and $H_{U,L,F}^\ell$ are 
obtained by covariant projections. We then proceed to integrate the 
projected tree-graph contributions over the full three-body phase-space and 
give analytical results for the once- and twice-integrated tree-graph cross 
sections. As is well familiar by now, the final expressions contain 
infrared (IR) divergences which we choose to regularize by introducing a 
small gluon mass. The tree-graph IR divergencies are cancelled by the 
corresponding IR divergencies of the one-loop contributions. Thus by adding 
in the loop contributions we finally arrive at finite results relevant for 
the total inclusive cross section integrated over the hard and the soft 
regions of the energy of the gluon as presented later on. In Sec.~2 we also 
detail the dependence of the polarization on the electroweak parameters 
including a discussion of beam polarization effects.

In Sec.~3 we first focus on the soft-gluon region and give results on the 
polar angle dependence of the alignment polarization of the quark with 
their typical logarithmic dependence on the gluon-energy cut. The hard-gluon
contribution is given by the complement of the soft-gluon contribution, 
i.e. as the difference of the full $O(\as)$ and the soft-gluon contribution. 
Since we provide numerically stable expressions for the latter two 
contributions, the hard-gluon contribution can be evaluated in a 
numerically stable way. In Sec.~4 we consider the zero quark mass case and 
calculate the three relevant polarized and unpolarized structure functions 
using helicity methods and dimensional reduction as regularization method. 
In this way one can keep out of the way of the notorious $\gamma_5$-problem 
when calculating e.g. the $V\!A$-contribution to the $F$-type unpolarized 
structure function. We compare the mass-zero results with the mass-zero 
limit of the corresponding structure function expressions in Sec.~3 and 
identify the global anomalous spin-flip contributions to the 
QCD$(m\rightarrow 0)$ polarized structure functions. Finally, Sec.~5 
contains our summary and our conclusions. In Appendix~A we catalogue some 
integrals that appear in the tree-graph integrations in Sec.~2, in 
Appendix~B we list some standard $O(\as)$ rate functions needed for the 
rate expressions in Sec.~3. In Appendix C, finally, we consider the case of 
polarized and unpolarized quark production from transversely and 
longitudinally polarized $e^+e^-$-beams.

\section{\boldmath{$O(\as)$} tree-graph contributions and total rates}
For the three body process
$(\gamma_V,Z)\rightarrow q(p_1)+\overline{q}(p_2)+g(p_3)$ (see Fig.~1) we 
define a polarized hadron tensor according to ($q=p_1+p_2+p_3$)
\be\label{eqn1}
H_{\mu\nu}(q,p_1,p_2,s)=\sum_{\overline{q},g\mbox{\scriptsize\ spins}}
  \langle q\,\overline{q}\,g|j_\mu|0\rangle\langle 0|j_\nu^\dagger|
  q\,\overline{q}\,g\rangle.
\ee
The hadron tensor depends on the vector ($V$) and axial-vector ($A$) 
composition of the currents in Eq.~(\ref{eqn1}). One has altogether four 
independent components $H_{\mu\nu}^i$ ($i=1,2,3,4$) which are defined 
according to ($V$: $\gamma_\mu$, $A$: $\gamma_\mu\gamma_5$)
\bea
H_{\mu\nu}^1&=&\frac12(H_{\mu\nu}^{VV}+H_{\mu\nu}^{AA})\qquad
H_{\mu\nu}^2\ =\ \frac12(H_{\mu\nu}^{VV}-H_{\mu\nu}^{AA})\label{eqn2}\\
H_{\mu\nu}^3&=&\frac i2(H_{\mu\nu}^{V\!A}-H_{\mu\nu}^{AV})\qquad
H_{\mu\nu}^4\ =\ \frac12(H_{\mu\nu}^{V\!A}+H_{\mu\nu}^{AV}).\nonumber
\eea
For the tree-graph contribution one calculates
\bea
H_{\mu\nu}^1&=&\frac{g_s^2\NC\CF}{2y^2z^2q^2}\Bigg[\ 
  q^2(-(2-\xi)(\xi(y+z)^2-4yz(1-y-z))+4yz(y^2+z^2))g_{\mu\nu}\nonumber\\&&
  +16y^2zp_{1\mu}p_{1\nu}+4(\xi y^2-4yz+2\xi yz+2y^2z+\xi z^2+2yz^2)
  (p_{1\mu}p_{2\nu}+p_{2\mu}p_{1\nu})\nonumber\\&&
  +16yz^2p_{2\mu}p_{2\nu}+4z(-2y+\xi y+2y^2+\xi z)
  (p_{1\mu}p_{3\nu}+p_{3\mu}p_{1\nu})\nonumber\\&&
  +4y(\xi y-2z+\xi z+2z^2)(p_{2\mu}p_{3\nu}+p_{3\mu}p_{2\nu})\ \Bigg]
  \nonumber\\&&
  +\frac{2img_s^2\NC\CF}{y^2z^2q^4}\Bigg[\ q^2(2-\xi)y^2\eps(\mu\nu p_1s)
  -q^2(4yz-2\xi yz+2y^2z-2yz^2)\eps(\mu\nu p_2s)\nonumber\\&&
  +q^2(2y^2-\xi y^2-2yz+2\xi yz+\xi z^2)\eps(\mu\nu p_3s)\nonumber\\&&
  -4y^2(p_{1\mu}+p_{3\mu})\eps(\nu p_1p_2s)
  +4y^2(p_{1\nu}+p_{3\nu})\eps(\mu p_1p_2s)\nonumber\\&&
  +4y(yp_{1\mu}+zp_{2\mu}+yp_{3\mu})\eps(\nu p_2p_3s)
  -4y(yp_{1\nu}+zp_{2\nu}+yp_{3\nu})\eps(\mu p_2p_3s)
  \ \Bigg]\\ \nonumber
  \\
H_{\mu\nu}^2&=&\frac{2m^2g_s^2\NC\CF}{y^2z^2q^4}\Bigg[ 
  -q^2(\xi(y+z)^2-4yz(1-y-z))g_{\mu\nu}+8yzp_{3\mu}p_{3\nu}\ \Bigg]
  \nonumber\\&& +\frac{2img_s^2\NC\CF}{y^2z^2q^4}\Bigg[\
  q^2(\xi y^2-2yz+\xi yz+\xi z^2+4yz^2)\eps(\mu\nu p_1s)\nonumber\\&&
  -q^2\xi z(y-z)\eps(\mu\nu p_2s)
  +q^2\xi y(y-z)\eps(\mu\nu p_3s)\nonumber\\&&
  -4yz(p_{1\mu}+p_{3\mu})\eps(\nu p_1p_2s)
  +4yz(p_{1\nu}+p_{3\nu})\eps(\mu p_1p_2s)\nonumber\\&&
  -4yz(p_{1\mu}-p_{2\mu}+p_{3\mu})\eps(\nu p_1p_3s)
  +4yz(p_{1\nu}-p_{2\nu}+p_{3\nu})\eps(\mu p_1p_3s)
  \ \Bigg]\\ \nonumber
  \\
H_{\mu\nu}^3&=&\frac{2img_s^2\NC\CF}{y^2z^2q^4}\Bigg[
  -4yz(p_3s)(p_{1\mu}p_{2\nu}-p_{2\mu}p_{1\nu})
  +4yz(p_2s+p_3s)(p_{1\mu}p_{3\nu}-p_{3\mu}p_{1\nu})\nonumber\\&&
  -q^2(\xi y^2-4yz+2\xi yz+2y^2z+\xi z^2+4yz^2)
  (p_{1\mu}s_\nu-s_\mu p_{1\nu})\nonumber\\&&
  -2q^2yz^2(p_{2\mu}s_\nu-s_\mu p_{2\nu})
  -q^2(\xi y^2-2yz+\xi yz+2y^2z)(p_{3\mu}s_\nu-s_\mu p_{3\nu})
  \ \Bigg]
\eea
and\goodbreak
\bea
H_{\mu\nu}^4&=&\frac{2ig_s^2\NC\CF}{y^2z^2q^4}\Bigg[ 
  -q^2(\xi y^2-4yz+2\xi yz-2y^2z+\xi z^2+2yz^2)\eps(\mu\nu p_1p_2)
  \nonumber\\&&
  -q^2(2y^2-\xi y^2-2yz+2\xi yz+\xi z^2)\eps(\mu\nu p_1p_3)
  +4q^2\xi y^2\eps(\mu\nu p_2p_3)\nonumber\\&&
  +8y(yp_{1\mu}+zp_{2\mu}+yp_{3\mu})\eps(\nu p_1p_2p_3)
  -8y(yp_{1\nu}+zp_{2\nu}+yp_{3\nu})\eps(\mu p_1p_2p_3)\ \Bigg]
  \nonumber\\&& +\frac{2mg_s^2\NC\CF}{y^2z^2q^4}\Bigg[
  -q^2((\xi y^2-4yz+2\xi yz+2y^2z+\xi z^2+2yz^2)(p_2s)
  \nonumber\\&&\qquad\qquad
  +(-2y^2+\xi y^2-2yz+2\xi yz+4y^2z+\xi z^2)(p_3s))g_{\mu\nu}\nonumber\\&&
  -4y^2(p_3s)(p_{1\mu}p_{2\nu}+p_{2\mu}p_{1\nu})
  +8yz(p_3s)p_{2\mu}p_{2\nu}\nonumber\\&&
  -4y(z(p_2s)+y(p_3s))(p_{2\mu}p_{3\nu}+p_{3\mu}p_{2\nu})
  +2q^2y^2z(p_{1\mu}s_\nu+s_\mu p_{1\nu})\nonumber\\&&
  +q^2(\xi y^2-4yz+2\xi yz+4y^2z+\xi z^2+2yz^2)(p_{2\mu}s_\nu+s_\mu p_{2\nu})
  \nonumber\\&&
  +q^2(-2yz+\xi yz+2y^2z+\xi z^2)(p_{3\mu}s_\nu+s_\mu p_{3\nu})
  \ \Bigg],
\eea
where we have accounted for the spin dependence of the hadron tensor by 
using the quark's spin projector
$u(p_1,s)\bar u(p_1,s)=(\slash p_1+m)\frac12(1+\gamma_5\slash s)$ when 
calculating the trace according to Eq.~(\ref{eqn1}). We have used the 
energy variables $y=1-2p_1\cdot q/q^2$ and $z=1-2p_2\cdot q/q^2$ and the 
abbreviation $\xi=4m^2/q^2$. Note that for the spin dependent 
contributions, $H_{\mu\nu}^{1,2,3}(s)$ are antisymmetric in $\mu$ and 
$\nu$, while $H_{\mu\nu}^4(s)$ is symmetric in $\mu$ and $\nu$. 
For the spin independent pieces, the nonvanishing contributions
$H_{\mu\nu}^{1,2}$ are symmetric, $H_{\mu\nu}^4$ is antisymmetric in~$\mu$ 
and~$\nu$, and there is no spin independent contribution to $H_{\mu\nu}^3$.

In this paper we are only concerned with the alignment polarization. The 
covariant form of the polarization four-vector associated with the 
alignment polarization is given by
$s^{\ell\mu}=\xi^{-1/2}(\sqrt{(1-y)^2-\xi},0,0,1-y)$ 
($s^\ell_\mu s^{\ell\mu}=-1$) which reduces to $s^{\ell\mu}=(0,0,0,1)$ in 
the rest system of the quark when $y=1-\sqrt\xi$. We then define unpolarized 
and polarized structure functions $H_{\mu\nu}^i$ and $H_{\mu\nu}^{i\ell}$ 
($i=1,2,3,4$) according to
\bea
H_{\mu\nu}^i&=&H_{\mu\nu}^i(s^\ell)+H_{\mu\nu}^i(-s^\ell),\label{eqn3}\\
H_{\mu\nu}^{i\ell}&=&H_{\mu\nu}^i(s^\ell)-H_{\mu\nu}^i(-s^\ell).\label{eqn4}
\eea
In order to determine the polar angle dependence of the polarized and 
unpolarized cross section one turns to the helicity structure functions 
$H_U=H_{++}+H_{--}$, $H_L=H_{00}$ and $H_F=H_{++}-H_{--}$ which can be 
obtained from the covariant structure functions in Eqs.~(\ref{eqn3}) 
and~(\ref{eqn4}) by the appropiate helicity projections of the gauge boson,
\be
H_{\lambda_{Z,\gamma}\lambda_{Z,\gamma}}=\eps^\mu(\lambda_{Z,\gamma})
  H_{\mu\nu}\eps^{\nu*}(\lambda_{Z,\gamma})
\ee
and the same for $H_{\mu\nu}^\ell$.

A convenient way of obtaining the helicity structure functions $H_\alpha$ 
($\alpha=U,L,F$) is by covariant projection,
\be\label{eqn5}
H_U=\left(-\hat g^{\mu\nu}-\frac{\hat p_1^\mu\hat p_1^\nu}{p_{1z}^2}
  \right)H_{\mu\nu}\qquad
H_L=\frac{\hat p_1^\mu\hat p_1^\nu}{p_{1z}^2}H_{\mu\nu}\qquad
H_F=i\eps^{\mu\nu\alpha\beta}
  \frac{\hat p_{1\alpha}q_\beta}{p_{1z}\sqrt{q^2}}H_{\mu\nu},
\ee
where $\hat g_{\mu\nu}=g_{\mu\nu}-q_\mu q_\nu/q^2$ and
$\hat p_{1\mu}=p_{1\mu}-(p_1\cdot q)q_\mu/q^2$ are the four-transverse 
metric tensor and the four-transverse quark momentum, respectively. Note 
that the covariant projectors defined in Eq.~(\ref{eqn5}) are 
$(y,z)$-independent and can thus be freely commuted with the $y$- and 
$z$-integrations to be done later on.

After all these preliminaries let us write down the differential polarized 
and unpolarized three-body $e^+e^-$-cross sections, differential in $\ct$ 
and in the two energy variables $y$ and $z$. One has
\bea
\frac{d\sigma^\ell}{d\ct\,dy\,dz}
  &=&\frac38(1+\cos^2\theta)g_{14}\frac{d\sigma_U^{4\ell}}{dy\,dz}
  +\frac34\sin^2\theta\ g_{14}\frac{d\sigma_L^{4\ell}}{dy\,dz}\nonumber\\&&
  +\frac34\cos\theta\left(g_{41}\frac{d\sigma_F^{1\ell}}{dy\,dz}
  +g_{42}\frac{d\sigma_F^{2\ell}}{dy\,dz}\right)\qquad\mbox{and}
  \label{eqn6}\\
\frac{d\sigma}{d\ct\,dy\,dz}
  &=&\frac38(1+\cos^2\theta)\left(g_{11}\frac{d\sigma_U^1}{dy\,dz}
  +g_{12}\frac{d\sigma_U^2}{dy\,dz}\right)\nonumber\\&&
  +\frac34\sin^2\theta\left(g_{11}\frac{d\sigma_L^1}{dy\,dz}
  +g_{12}\frac{d\sigma_L^2}{dy\,dz}\right)
  +\frac34\ct\ g_{44}\frac{d\sigma_F^4}{dy\,dz}\label{eqn7}
\eea
where, in terms of the helicity structure functions defined above, one has
\be
\frac{d\sigma_\alpha^i}{dy\,dz}=\frac{\pi\alpha^2v}{3q^4}\left\{
  \frac{q^2}{16\pi^2v}H_\alpha^i\right\}.
\ee
($\alpha=U,L,F$) and the same for 
$\sigma_\alpha^i\rightarrow\sigma_\alpha^{i\ell}$ etc. The $g_{ij}$ coupling 
factors appearing in Eqs.~(\ref{eqn6}) and~(\ref{eqn7}) specify the 
electroweak structure of the lepton-hadron interaction and are listed in 
Appendix~C. The $y$, $z$ and $\ct$-dependent polarization $\Pl(\ct,y,z)$ is 
then given by the ratio of the polarized and unpolarized cross sections in 
Eqs.~(\ref{eqn6}) and~(\ref{eqn7}).

The generalization of the above cross section expressions to the case where 
one starts with ``longitudinally'' polarized beams is straightforward and 
amounts to the replacement (see Appendix C)\\
{\it polarized:}
\vspace{-1cm}
\bea
g_{14}&\rightarrow&[(1-h^-h^+)g_{14}+(h^--h^+)g_{44}]\\
g_{4i}&\rightarrow&[(h^--h^+)g_{1i}+(1-h^-h^+)g_{4i}]\quad(i=1,2)\nonumber
\eea
\noindent{\it unpolarized:}
\vspace{-1cm}
\bea
g_{1i}&\rightarrow&[(1-h^-h^+)g_{1i}+(h^--h^+)g_{4i}]\quad(i=1,2)\\
g_{44}&\rightarrow&[(h^--h^+)g_{14}+(1-h^-h^+)g_{44}]\nonumber
\eea
where $h^-$ and $h^+$ ($-1\le h^\pm\le+1$) denote the helicity polarization 
of the electron and the positron beam. Clearly there is no interaction 
between the beams when $h^+=h^-=\pm1$.

We have presented the results of calculating the full ($U+L$) and the 
longitudinal piece ($L$) of the polarized hadron tensor 
in~\cite{forward1,forward3}. Here we add the last building block, namely 
the polarized hadron tensor projected on its forward/backward 
component~($F$), where the requisite covariant projector has been given in 
Eq.~(\ref{eqn5}). A straightforward calculation of the $O(\as)$ tree-graph 
contributions leads to
\begin{eqnarray}
H_F^{1\ell}(y,z)&=&\frac{8\pi\as\NC\CF}{(1-y)^2-\xi}\Bigg[\
  8-4\xi-\xi^2-2(2-\xi)(2-3\xi)\frac1y\nonumber\\&&
  -\xi(1-\xi)(2-\xi)\left(\frac1{y^2}+\frac1{z^2}\right)
  -2\xi y-4yz-6\xi z-4(2-\xi)(3-2\xi)\frac1z\nonumber\\&&
  +2(1-\xi)(2-\xi)^2\frac1{yz}+(4-2\xi-\xi^2)\frac zy
  +(28-14\xi+\xi^2)\frac yz\nonumber\\&&
  +2\xi(2-\xi)\frac y{z^2}-\xi(4-\xi)\frac{y^2}{z^2}+4\frac{y^3}z
  +2\xi\frac{y^3}{z^2}-2(8-3\xi)\frac{y^2}z\Bigg]
  \label{eqn8}\\[12pt]
H_F^{2\ell}(y,z)&=&\frac{8\pi\as\NC\CF}{(1-y)^2-\xi}\ \xi\Bigg[\
  4+\xi-2(2-3\xi)\frac1y-\xi(1-\xi)\left(\frac1{y^2}+\frac1{z^2}\right)
  \nonumber\\&&
  +4z-4(3-2\xi)\frac1z+2(1-\xi)(2-\xi)\frac1{yz}+\xi\frac zy
  \nonumber\\&&
  +(12-\xi)\frac yz+2\xi\frac y{z^2}-\xi\frac{y^2}{z^2}-4\frac{y^2}z\Bigg]
  \\[12pt]
H_F^4(y,z)&=&\frac{16\pi\as\NC\CF}{\sqrt{(1-y)^2-\xi}}
  \Bigg[-(4-5\xi)\frac1y-\xi(1-\xi)\left(\frac1{y^2}+\frac1{z^2}\right)
  \nonumber\\&&
  -2(4-3\xi)\frac1z+\xi\frac y{z^2}+2z+2\frac zy+6\frac yz-2\frac{y^2}z
  +2(1-\xi)(2-\xi)\frac1{yz}\Bigg]
  \label{eqn9}
\end{eqnarray}
In all three above expressions the denominator vanishes when the quark's 
three-momentum is zero, i.e. when $\vec p_1=0$ or, in terms of the 
$y$-variable, when $y=1-\sqrt\xi$. However, a careful limiting procedure 
shows that the three structure functions in 
Eqs.~(\ref{eqn8})--(\ref{eqn9}) tend to finite limiting values (and not 
to zero as in the case of the longitudinal structure functions $H_L^{i\ell}$ 
treated in~\cite{forward3}). In contrast to this, the singularities at 
$y=z=0$ constitute true IR-singularities. They can be regularized by 
introducing a small gluon mass~$m_g=\sqrt{\Lambda q^2}$ which has the effect 
to deform the phase-space boundary to
\bea
y_-&=&\sqrt{\Lambda\xi}+\Lambda,\qquad y_+\ =\ 1-\sqrt\xi\\
z_\pm(y)&=&\frac{2y}{4y+\xi}\left\{1-y-\frac{1}{2}\xi+\Lambda+\frac{\Lambda}{y}
  \pm\frac{1}{y}\sqrt{(y-\Lambda)^2-\Lambda\xi}\sqrt{(1-y)^2-\xi}\right\}.
\eea
Note that the introduction of a small gluon mass is required only to deform 
the integration boundary. In the calculation of the matrix elements we need 
not pay regard to the gluon mass. With the gluon-mass regulator one can then 
perform the finite integration over the two phase-space variables $y$ and $z$. 
The integration over $z$ gives the cross section's dependence on the quark 
energy variable $y$. We obtain
\bea
H_F^{1\ell}(y)&=&\frac{8\pi\as\NC\CF}{(1-y)^2-\xi}\Bigg[\ 2\Bigg\{
  2\frac{(1-\xi)(2-\xi)^2}y-4(3-2\xi)(2-\xi)\nonumber\\&&\qquad\qquad
  +(28-14\xi+\xi^2)y-2(8-3\xi)y^2+4y^3\Bigg\}
  \ln\left(\frac{z_+(y)}{z_-(y)}\right)\nonumber\\&&
  +\frac1y\sqrt{(y-\Lambda)^2-\Lambda\xi}\sqrt{(1-y)^2-\xi}\Bigg\{
  -\frac{8\xi(1-\xi)(2-\xi)y}{\xi y^2+4\Lambda(1-y-\xi+\Lambda)}
  \nonumber\\&&\qquad\qquad
  -8\frac{(1-\xi)(2-\xi)}y+32(2-\xi)-2(18-5\xi)y+20y^2
  \nonumber\\&&\qquad\qquad
  -\frac{(4-\xi)(40+3\xi)y}{2(4y+\xi)}+\frac{(4-\xi)^3y}{(4y+\xi)^2}\Bigg\}
  \Bigg]\\[12pt]
H_F^{2\ell}(y)&=&\frac{8\pi\as\NC\CF}{(1-y)^2-\xi}\ \xi\Bigg[\ 2\Bigg\{
  2\frac{(1-\xi)(2-\xi)}y-4(3-2\xi)\nonumber\\&&\qquad\qquad
  +(12-\xi)y-4y^2\Bigg\}\ln\left(\frac{z_+(y)}{z_-(y)}\right)\nonumber\\&&
  +\frac1y\sqrt{(y-\Lambda)^2-\Lambda\xi}\sqrt{(1-y)^2-\xi}\Bigg\{
  -\frac{8\xi(1-\xi)y}{\xi y^2+4\Lambda(1-y-\xi+\Lambda)}
  \nonumber\\&&\qquad\qquad
  -8\frac{1-\xi}y+32-12y-4\frac{(4-\xi)y}{4y+\xi}\Bigg\}
  \Bigg]\\[12pt]
H_F^4(y)&=&\frac{16\pi\as\NC\CF}{\sqrt{(1-y)^2-\xi}}\Bigg[\ 2\Bigg\{
  \frac{(1-\xi)(2-\xi)}y-(4-3\xi)+3y-y^2\Bigg\}
  \ln\left(\frac{z_+(y)}{z_-(y)}\right)\nonumber\\&&
  +\frac1y\sqrt{(y-\Lambda)^2-\Lambda\xi}\sqrt{(1-y)^2-\xi}\Bigg\{
  -\frac{4\xi(1-\xi)y}{\xi y^2+4\Lambda(1-y-\xi+\Lambda)}
  \nonumber\\&&\qquad\qquad
  -4\frac{1-\xi}y+8-y-\frac{16y}{4y+\xi}+\frac{(4-\xi)^2y}{(4y+\xi)^2}\Bigg\}
  \Bigg].
\eea

Let us briefly pause to discuss the $y$-dependence of the forward/backward 
($F$)-polarization component $\Pl_F$ which we define as the $(2\ct)$-moment 
of the full $\theta$-dependent polarization $\Pl(\ct)$, i.e.
\be\label{eqn10}
\langle2\ct\,\Pl(\ct)\rangle=\frac{g_{41}H_F^{1\ell}+g_{42}H_F^{2\ell}}%
  {g_{11}H_{U+L}^1+g_{12}H_{U+L}^2}=:\Pl_F.
\ee
The IR-limit $y\rightarrow 0$ in Eq.~(\ref{eqn10}) is well defined since 
the potentially singular term proportional to $1/y$ in the numerator is 
cancelled by the $1/y$-terms of the denominator giving a finite polarization 
value for $y\rightarrow 0$. In fact the limiting value of $\Pl_F$ can be 
calculated to be
\be\label{eqn11}
\Pl_F(y\rightarrow 0)
  =\frac{2(2-\xi)g_{41}+2\xi g_{42}}{(4-\xi)g_{11}+3\xi g_{12}}.
\ee
Turning to the other corner of phase-space $y\rightarrow 1-\sqrt\xi$, the 
limiting value of the polarization expressions can be obtained by expanding 
numerator and denominator around $y=1-\sqrt\xi$. As mentioned before, the 
$(U+L)$- and $(L)$-components of the polarization vanish in this limit, 
whereas the $(F)$-component of the polarization tends to the finite limiting 
value
\be\label{eqn12}
\Pl_F(y\rightarrow 1-\sqrt\xi)=-\frac{2((2-2\sqrt\xi-\xi)g_{41}
  +\xi g_{42})}{3((4-4\sqrt\xi+3\xi)g_{11}-\xi g_{12})}.
\ee
In Fig.~2 we have plotted the $y$-dependence of $\Pl_F$ for the top quark 
case for the three values $\sqrt{q^2}=380$, $500$ and $1000$~GeV using a 
top quark mass of $m_t=180$ GeV~\cite{forward8}.\footnote{We have chosen 
the lowest energy $\sqrt{q^2}=380$ GeV to lie well above the nominal 
threshold value of $\sqrt{q^2}=360$ GeV for a perturbative calculation to 
make sense. As is well known, the production dynamics in the threshold 
region requires the consideration of non-perturbative effects. For a 
discussion of top quark polarization effects in the threshold region 
see~\cite{forward9}.} The polarization $\Pl_F$ can be seen to tend to the 
two limiting values given in Eqs.~(\ref{eqn11}) and~(\ref{eqn12}) as 
$y\rightarrow 0$ and $y\rightarrow 1-\sqrt\xi$, respectively.

Finally, with the integration over $y$ we include the relative 
three-body/two-body phase-space factor $q^2/16\pi^2v$ and express the 
result in terms of the rate functions
\be\label{eqn13}
\hat H_\alpha^{i(\ell)}(tree)=\frac{q^2}{16\pi^2v}\int dy\,dz\,
  H_\alpha^{i(\ell)}(y,z)
\ee
for $\alpha=F$, which gives
\bea
\hat H_F^{1\ell}({\it tree})&=&\frac{\as\NC\CF q^2}{2\pi v}\bigg[\
  (8-4\xi-\xi^2)J_1-2(2-\xi)(2-3\xi)J_2\nonumber\\&&
  -\xi(1-\xi)(2-\xi)(J_3+J_9)
  -2\xi J_4-4J_6-6\xi J_7-4(2-\xi)(3-2\xi)J_8\nonumber\\&&
  +2(1-\xi)(2-\xi)^2J_{10}+(4-2\xi-\xi^2)J_{11}
  +(28-14\xi+\xi^2)J_{12}\nonumber\\&&
  +2\xi(2-\xi)J_{13}-\xi(4-\xi)J_{14}+4J_{15}
  +2\xi J_{16}-2(8-3\xi)J_{17}\bigg]\\[12pt]
\hat H_F^{2\ell}({\it tree})&=&\frac{\as\NC\CF q^2}{2\pi v}\ \xi\bigg[\
  (4+\xi)J_1-2(2-3\xi)J_2-\xi(1-\xi)(J_3+J_9)\nonumber\\&&
  +4J_7-4(3-2\xi)J_8+2(1-\xi)(2-\xi)J_{10}+\xi J_{11}\nonumber\\&&
  +(12-\xi)J_{12}+2\xi J_{13}-\xi J_{14}-4J_{17}\bigg]\\[12pt]
\hat H_F^4({\it tree})&=&\frac{\as\NC\CF q^2}{\pi v}\bigg[
  -(4-5\xi)S_2-\xi(1-\xi)(S_3+S_5)-2(4-3\xi)S_4\nonumber\\&&
  +\xi S_6+2S_8+2S_9+6S_{10}-2S_{11}+2(1-\xi)(2-\xi)S_{12}\bigg].
\eea
The integrals $S_i$ have been computed and listed in~\cite{forward1}. 
Similar techniques allow one to calculate the integrals $J_i$ (for details 
see~\cite{forward1,forward3}). They are listed in Appendix~A. Remember that 
the IR-singularities in $S_3$, $S_5$, $S_{12}$, $J_3$, $J_9$ and $J_{10}$ 
are cancelled by the corresponding IR-singularities of the virtual 
contributions. This is a consequence of the Lee-Nauenberg theorem.

\section{Soft- and hard-gluon regions}
For some applications it is desirable to split the three-body phase-space into 
a soft and a hard gluon region. The two regions are defined with respect to 
a fixed cut-off value of the gluon energy $E_g^{\mbox{\scriptsize cut-off}}
=\lambda\sqrt{q^2}$ such that $E_g/\sqrt{q^2}\le\lambda$ 
and $E_g/\sqrt{q^2}>\lambda$ define the soft and the hard-gluon regions, 
respectively. Technically the integration over the soft-gluon region is 
quite simple. The hard-gluon contribution can then be obtained as the 
complement of the soft-gluon contribution. Since we provide numerically 
stable expressions for both the total $O(\as)$ contribution and the 
soft-gluon contribution the computation of the hard-gluon contribution as 
the difference of the two constitutes a numerically stable procedure 
without that differences of large numbers are encountered.

In as much as the soft-gluon contribution factorizes into a Born term part 
and a universal soft-gluon function we begin our discussion by listing the 
relevant nonvanishing Born term components of the hadron tensor. One has
\smallskip\\
{\it Born term:}
\vspace{-0.5truecm}
\bea
H_U^{4\ell}&=&4\NC q^2v\nonumber\\
H_L^{4\ell}&=&0\nonumber\\
H_F^{1\ell}&=&2\NC q^2(1+v^2)\qquad H_F^{2\ell}\ =\ 2\NC q^2(1-v^2)
  \nonumber\\
H_U^1&=&2\NC q^2(1+v^2)\qquad H_U^2=2\NC q^2(1-v^2)\\
H_L^1&=&\NC q^2(1-v^2)\ =\ H_L^2\nonumber\\
H_F^4&=&4\NC q^2\nonumber
\eea
where $v=\sqrt{1-4m^2/q^2}$ is the velocity of the quark in the 
c.m.~system. For completeness we note that the polarized and unpolarized Born 
term cross sections can be obtained from Eqs.~(\ref{eqn6}) and~(\ref{eqn7}) 
by the replacement
\be
\frac{d\sigma_\alpha^i}{dy\,dz}\ \rightarrow\
  \sigma_\alpha^i=\frac{\pi\alpha^2v}{3q^4}H_\alpha^i({\it Born})
\ee
and the same for the polarized cross sections.

Turning to the soft-gluon region of the $O(\as)$ tree-graph contribution, 
it is well-known that the soft hadronic tensor factorizes into the Born 
term structure times a universal function of the soft-gluon energy. 
Integrating the soft-gluon spectrum and adding the $O(\as)$ soft-gluon 
contribution to the Born term contribution, one obtains
\bea
H_\alpha^i&=&H_\alpha^i({\it Born})\Bigg[\ 1-\frac{\as\CF}\pi
  \Bigg\{\ln\left(\frac{2\lambda}{\sqrt\Lambda}\right)
  \left(2+\frac{1+v^2}v\ln\left(\frac{1-v}{1+v}\right)\right)\nonumber\\&&
  +\frac1v\ln\left(\frac{1-v}{1+v}\right)
  +\frac{1+v^2}v\left(\Li\left(\frac{2v}{1+v}\right)
  +\frac14\ln^2\left(\frac{1-v}{1+v}\right)\right)\Bigg\}\Bigg]
  \label{eqn14}
\eea
where the scaled maximal gluon energy $\lambda$ denotes the cut-off energy 
which separates the soft-gluon region from the hard-gluon region. The IR 
singularity present in the soft-gluon region has been regularized as before 
by introducing a small gluon mass $m_g=\sqrt{\Lambda q^2}$ which of course 
can be chosen to be arbitrarily small compared to the energy cut-off 
parameter~$\lambda$.

The total $O(\as)$ soft-gluon contribution is then obtained by adding in 
the one-loop contribution. For the latter one has (see 
e.g.~\cite{forward1,forward3})
\medskip\\
{\it one-loop real part:}
\bea
H_U^{4\ell}&=&4\NC q^2v(\real A+\real C)\nonumber\\
H_L^{4\ell}&=&0\nonumber\\
H_F^{1\ell}&=&4\NC q^2(\real A+v^2\real C)\qquad
H_F^{2\ell}\ =\ 4\NC q^2(\real A-v^2\real C)\nonumber\\
H_U^1&=&4\NC q^2(\real A+v^2\real C)\qquad
H_U^2\ =\ 4\NC q^2(\real A-v^2\real C)\label{eqn15}\\
H_L^1&=&2\NC q^2((1-v^2)\real A+v^2\real B)\ =\ H_L^2
  \nonumber\\
H_F^4&=&4\NC q^2v(\real A+\real C)\nonumber
\eea
\medskip\noindent
{\it one-loop imaginary part:}
\vspace{-0.5truecm}
\bea
H_U^{3\ell}&=&-4\NC q^2v(\imag A-\imag C)\nonumber\\
H_L^{3\ell}&=&0\\
H_F^3&=&-4\NC q^2v(\imag A-\imag C)\nonumber
\eea
where the one-loop form factors $A$, $B$ and $C$ are given by
\bea
\real A&=&-\frac{\as\CF}{4\pi}\Bigg[\left(2+\frac{1+v^2}v
  \ln\left(\frac{1-v}{1+v}\right)\right)
  \ln\left(\frac{\Lambda q^2}{m^2}\right)
  +3v\ln\left(\frac{1-v}{1+v}\right)+4\nonumber\\&&
  +\frac{1+v^2}v\left(\Li\left(\frac{2v}{1+v}\right)
  +\frac14\ln^2\left(\frac{1-v}{1+v}\right)-\frac{\pi^2}2\right)\Bigg]
  \nonumber\\
\real B&=&\frac{\as\CF}{4\pi}\frac{1-v^2}v\ln\left(\frac{1-v}{1+v}\right)
\qquad\real C\ =\ \real A-2\real B\\
\imag B&=&\frac{\as\CF}{4\pi}\frac{1-v^2}v\pi\qquad
\imag C\ =\ \imag A-2\imag B.\nonumber
\eea
The $\gamma_5$-even one-loop results for $H_{U,L}^{1,2}$ and $H_{U,L}^{4\ell}$ 
are taken from~\cite{forward1,forward3}. The $\gamma_5$-odd one-loop 
contributions $H_F^{1\ell}$, $H_F^{2\ell}$ and $H_F^4$ have been calculated 
in dimensional regularization with zero gluon mass using a naive 
anticommuting $\gamma_5$. The result was then converted to the gluon mass 
regularization scheme with the help of the substitution\hfil\break 
$1/\eps-\gamma_E+\ln(4\pi\mu^2/m^2)\rightarrow\ln(\Lambda q^2/m^2)$.

Although the imaginary part of the one-loop contribution is only needed in 
the case of transverse normal polarization (see \cite{forward4}), we have 
included it for completeness. It is quite apparent that the sum of the real 
gluon emission contributions in Eq.~(\ref{eqn14}) and the real one-loop 
contributions in Eq.~(\ref{eqn15}) are IR finite, i.e. the dependence on 
$\Lambda$ drops out in the sum.

We have numerically evaluated the polarization in the soft-gluon region as 
a function of the gluon energy cut-off. We did not plot the results since 
the $O(\as)$ corrections to the polarization are quite small. In the scale 
of our figures the corrections only become visible for $\lambda<0.1\%$ 
which is too low a cut-off-value for a fixed order perturbation series to 
make sense. In fact, the unpolarized $O(\as)$ rate crosses zero and becomes 
negative at around $\lambda=0.05\%$ which again highlights the fact that 
such low cut-off values do not make any sense. One concludes that the 
$O(\as)$ corrections go in the same direction for both polarized and 
unpolarized rates rendering the ratio unsensitive to $O(\as)$ corrections.

Let us now present our results for the fully integrated three-body 
tree-graph polarized rate functions in terms of the rate functions given in 
Eq.~(\ref{eqn13}). As mentioned before, we integrate over the full 
phase-space keeping a small gluon mass as IR-cutoff. The complete $O(\as)$ 
contribution is then given by adding in the $O(\as)$ real one-loop 
contributions listed in Eq.~(\ref{eqn15}). One has
\be\label{eqn16}
\qquad\qquad\hat H_\alpha^{i\ell}(\as)
  =\hat H_\alpha^{i\ell}(tree)+H_\alpha^{i\ell}(loop).
\ee
Note that the dependence on the IR-cutoff parameter $\Lambda$ drops out in 
the sum of the two contributions. One has\\
{\it $O(\as)$ real part:}
\bea
\hat H_U^{4\ell}(\as)&=&\frac{\as\NC\CF q^2}{4\pi v}\Big[
  -2(1-\sqrt\xi)(2-6\sqrt\xi+29\xi)-16v^2t_{11}-16v^2t_{10}\nonumber\\&&
  +8(2-\xi)v(t_7-t_8)-2(8+2\xi+3\xi^2)t_6+4(4+9\xi)vt_3\nonumber\\&&
  -(16-30\xi+\frac{17}2\xi^2)(t_1-t_2)\ \Big]\label{eqn17}
  \\[12pt]
\hat H_L^{4\ell}(\as)&=&\frac{\as\NC\CF q^2}{4\pi v}\Big[\
  4(1-\sqrt\xi)(2-6\sqrt\xi+13\xi)+2\xi(10+3\xi)t_6\nonumber\\&&
  -52\xi vt_3-\xi(24-7\xi)(t_1-t_2)\ \Big]
  \\[12pt]
\hat H_F^{1\ell}(\as)&=&\frac{\as\NC\CF q^2}{4\pi v}\Big[
  -4(2+3\xi)v-16(2-\xi)vt_{12}-8(2-\xi)vt_{10}\nonumber\\&&
  -4(2-\xi)^2(t_8-t_9)+2\xi(10-\xi)t_5
  +2\sqrt\xi(1-\sqrt\xi)(2-\sqrt\xi)(4+\sqrt\xi)t_4\nonumber\\&&
  +2(24-12\xi+\xi^2)t_3\ \Big]\label{eqn18}
  \\[7pt]
\hat H_F^{2\ell}(\as)&=&\frac{\as\NC\CF q^2}{4\pi v}\xi\Big[\
  12v-16vt_{12}-8vt_{10}-4(2-\xi)(t_8-t_9)\nonumber\\&&
  +2\xi t_5+2\sqrt\xi(1-\sqrt\xi)t_4+2(6-\xi)t_3\ \Big]
  \\[7pt]
\hat H_U^1(\as)&=&\frac{\as\NC\CF q^2}{4\pi v}\Big[\
  2(2+7\xi)v-16(2-\xi)vt_{12}-8(2-\xi)vt_{10}-4(2-\xi)^2(t_8-t_9)\nonumber\\&&
  -2\xi(2+3\xi)t_5+2\sqrt\xi(1-\sqrt\xi)(2+4\sqrt\xi-3\xi)t_4
  +(48-48\xi+7\xi^2)t_3\ \Big]
  \\[7pt]
\hat H_U^2(\as)&=&\frac{\as\NC\CF q^2}{4\pi v}\xi\Big[\
  12v-16vt_{12}-8vt_{10}-4(2-\xi)(t_8-t_9)\nonumber\\&&
  +2\xi t_5+2\sqrt\xi(1-\sqrt\xi)t_4
  +2(6-\xi)t_3\ \Big]
  \\[7pt]
\hat H_L^1(\as)&=&\frac{\as\NC\CF q^2}{4\pi v}\Big[\ (8-23\xi+\frac32\xi^2)v
  -8\xi vt_{12}-4\xi vt_{10}-2\xi(2-\xi)(t_8-t_9)\nonumber\\&&
  +2\xi(2+3\xi)t_5-2\sqrt\xi(1-\sqrt\xi)(2+4\sqrt\xi-3\xi)t_4\nonumber\\&&
  +\xi(22-8\xi+\frac34\xi^2)t_3\ \Big]
  \\[7pt]
\hat H_L^2(\as)&=&\frac{\as\NC\CF q^2}{4\pi v}\xi\Big[\
  \frac32(10-\xi)v-8vt_{12}-4vt_{10}-2(2-\xi)(t_8-t_9)\nonumber\\&&
  -2\xi t_5-2\sqrt\xi(1-\sqrt\xi)t_4
  +(6-4\xi-\frac34\xi^2)t_3\ \Big]
  \\[7pt]
\hat H_F^4(\as)&=&\frac{\as\NC\CF q^2}{4\pi v}\Big[
  -16\sqrt\xi(1-\sqrt\xi)-16v^2t_{11}-16v^2t_{10}\nonumber\\&&
  +8(2-\xi)v(t_7-t_8)-4(4-5\xi)t_6+8(2-3\xi)vt_3-16(t_1-t_2)\ \Big]
\label{eqn19}
\eea
Closed form expressions for the $O(\as)$ rate functions $t_i$ 
($i=1,\ldots,12$) appearing in Eqs.~(\ref{eqn17})--(\ref{eqn19}) can be 
found in Appendix~B. They contain the set of basic $I$, $J$, $S$ and $T$ 
integrals calculated in \cite{forward1} ($I,S$), \cite{forward3} ($T$) and 
in Appendix~A ($J$). They are associated with the various hadron tensor 
components in the following way: $U(S,T)$, $L(T)$, $[U+L](S)$ and $F(J)$ 
(polarized case) and $U(I,J)$, $L(J)$, $[U+L](I)$ and $F(S)$ (unpolarized 
case). We mention that, concerning the unpolarized $(F)$-component in 
Eq.~(\ref{eqn19}), closed form expressions for this component are also 
given in \cite{forward10} and \cite{forward11}. We agree with the results 
of these authors.

For our numerical evaluation we take the current world mean value for $\as$, 
i.e. $\as(M_Z)=0.118$ for five active flavours and evolve $\as$ by use of 
the one-loop renormalization group equation. For the bottom and top mass we 
use pole mass values $m_b=4.83$~GeV~\cite{forward12} and 
$m_t=180$~GeV~\cite{forward8}. We have repeated the numerical evaluation 
using $\overline{\mbox{MS}}$ running masses $\bar m_b(\bar m_b)=4.44$~GeV and 
$\bar m_t(\bar m_t)=172.1$~GeV where the running of the masses follows the 
momentum dependence of the corresponding one-loop renormalization group 
equation~\cite{forward13}. However, since the polarization expressions are 
rather insensitive to which sets of masses is used, our numerical results 
in Figs.~2 and~3 are given only in terms of the above pole masses.

In Fig.~3 we show the polar angle dependence of the alignment polarization 
of top quarks in the continuum, and for bottom quarks on the $Z$. We want 
to emphasize that we define the $O(\as)$ polarization such that we keep the 
full $O(\as)$ dependence of the unpolarized rate functions in the 
denominator. Thus we do not expand the inverse of the rate functions in 
powers of $\as$ as is sometimes done in the literature.

The $\ct$-dependence of the alignment polarization of the top quark shown 
in Fig.~3a is quite strong for all three c.m.~energies $380$~GeV, $500$~GeV 
and $1000$~GeV and shows a strong forward-backward asymmetry signalling a 
strong $(F)$-component in the polarization. The energy dependence of the 
alignment polarization is not very pronounced where the asymmetry becomes 
somewhat larger as the energy increases. Shown are the full $O(\as)$ 
results for the polarization. We did not plot the corresponding Born term 
results since the respective $O(\as)$ and Born term results differ by less 
than 2\% in absolute value. Again the $O(\as)$ corrections in the numerator 
and in the denominator of the polarization expression tend to go in the 
same direction leaving the polarization practically unchanged under 
$O(\as)$ corrections. For the bottom quark case shown in Fig.~3b we plot 
the Born term results and the full $O(\as)$ results separately. Note, 
however, that the $O(\as)$ corrections amount to only $\cong2\%$ over the 
whole $\ct$ range. In order to be able to exhibit the size of the $O(\as)$ 
corrections we chose to use a ``suppressed zero'' plot in Fig.~3b. The 
$\ct$ dependence of the alignment polarization is very weak with a small 
asymmetry component. We have checked that using a running mass 
$\bar m_b(\MZ)=3.30$~GeV in Fig.~3b changes the results by less than 0.2\%.

\section{Massless QCD and the zero-mass limit of QCD}
It is well known by now that, as concerns spin-flip contributions induced 
by gluon radiation, the zero-mass limit of QCD and massless QCD do not 
coincide, at least in the perturbative 
sector~\cite{forward1,forward2,forward3,forward14,forward15}. Since 
familiarity with this statement is not so widespread we want to use this 
section to highlight the difference between the $m=0$ and 
$m\rightarrow 0$ $O(\as)$ results for the alignment polarization. 
Needless to say that this has important practical implications for the 
calculation of polarization observables for the process 
$Z\rightarrow\mbox{\it light quark-pairs}$ where one might be tempted to set 
the quark mass to zero from the very beginning because of the presence of 
the large $Z$-mass scale.

Let us begin our discussion by listing the zero-mass Born term 
contributions. These are
\bea
H_U^{4\ell}&=&H_U^1\ =\ H_F^{1\ell}\ =\ H_F^4\ =\ 4\NC q^2\\
H_L^{4\ell}&=&H_L^1\ =\ 0.
\eea

Next we write down the $O(\as)$ zero-mass tree-graph contributions, which 
read
\bea
H_{U+L}^{4\ell}(y,z)&=&H_{U+L}^1(y,z)
  \ =\ 32\pi\as\NC\CF\ \frac{(1-y)^2+(1-z)^2}{yz}\label{eqn20}\\
H_L^{4\ell}(y,z)&=&H_L^1(y,z)
  \ =\ 32\pi\as\NC\CF\ \frac{2(1-y-z)}{(1-y)^2}\\
H_F^{1\ell}(y,z)&=&H_F^4(y,z)
  \ =\ 32\pi\as\NC\CF\left(\frac{(1-y)^2+(1-z)^2}{yz}-2\,\frac{1-z}{1-y}
  \right).\label{eqn21}
\eea
Note that for massless QCD one has $H_\alpha^2=H_\alpha^{2\ell}=0$ and
$H_\alpha^3=H_\alpha^{3\ell}=0$ to any order in~$\as$. This is implicit in 
the following discussion.

The hadron tensors $H_\alpha$ ($\alpha=U+L,F$) become singular for 
$y\rightarrow 1$ and $z\rightarrow 1$. These singularities need to be 
regularized when integrating over $(y,z)$ phase space. A particularly 
convenient regularization scheme is regularization by dimensional 
reduction~\cite{forward16} where the spin degrees of freedom are kept in 
four dimensions and only scalar integrals are regularized dimensionally. 
In this scheme one can retain the matrix element expressions 
Eq.~(\ref{eqn20}--\ref{eqn21}) when doing the $n$-dimensional phase space 
integration. After adding in the loop contribution (also regularized by 
dimensional reduction) the finite result can be transposed to the 
dimensional regularization scheme by adding in a finite global 
counter-term~\cite{forward17} which will be specified later on. The 
dimensional reduction calculation of the rates proceeds along the lines 
discussed in \cite{forward18}.

The tree-graph expressions have to be integrated with respect to the 
$n$-dimensional integration measure ($n=4-2\eps$)
\be
\frac{q^2}{16\pi^2}\left(\frac{4\pi\mu^2}{q^2}\right)^\eps
  \frac1{\Gamma(1-\eps)}\int_0^1dy\int_0^{1-y}dz\,y^{-\eps}z^{-\eps}
  (y+z-1)^{-\eps}
\ee
where we have again included the relative two-body/three-body phase-space 
factor $q^2/16\pi^2$ as before ($v=1$ for massless quarks).

The integrations can be done in standard fashion, and one obtains the 
tree-graph contributions
\bea
\hat H_{U+L}^{4\ell}({\it tree})&=&\hat H_{U+L}^1({\it tree})
  \ =\ 4\NC q^2\frac{\as}\pi\CF C\left(\frac1{\eps^2}
  +\frac3{2\eps}+\frac{17}4\right)\\
\hat H_L^{4\ell}({\it tree})&=&\hat H_L^1({\it tree})
  \ =\ 4\NC q^2\frac{\as}\pi\CF C\left(\frac12\right)\\
\hat H_F^{1\ell}({\it tree})&=&\hat H_F^4({\it tree})
  \ =\ 4\NC q^2\frac{\as}\pi\CF C\left(\frac1{\eps^2}
  +\frac3{2\eps}+\frac{17}4-\frac34\right)\label{eqn22}
\eea
where
\be
C=\left(\frac{4\pi\mu^2}{q^2}\right)^\eps
  \frac{\Gamma^2(1-\eps)}{\Gamma(1-3\eps)}.
\ee
The right-most finite contributions in the round bracket of 
Eq.~(\ref{eqn22}) results from the integration of the right-most term in 
the round bracket of Eq.~(\ref{eqn21}).

The loop contributions in dimensional reduction are given by
($H_L^{4\ell}=H_L^1=0$)~\cite{forward18}
\be\label{eqn23}
H_U^{4\ell}=H_U^1=H_F^{1\ell}=H_F^4=4\NC q^2\frac{\as}\pi\CF C'\left(
  -\frac1{\eps^2}-\frac3{2\eps}-3\right)
\ee
where one has to remember to do all $\gamma$-matrix manipulations in four 
dimensions. $C'$ is given by
 \be
C'=\left(\frac{4\pi\mu^2}{-q^2}\right)^\eps
  \frac{\Gamma(1+\eps)\Gamma^2(1-\eps)}{\Gamma(1-2\eps)}.
\ee
For the real parts that are of interest here we have $C'=C+O(\eps^3)$.

In order to convert our results to the more familiar dimensional 
regularization result we have to add the global counter-term, which reads
($H_L^{4\ell}=H_L^1=0$) \cite{forward17}
\be
H_U^{4\ell}=H_U^1=H_F^{1\ell}=H_F^4=4\NC q^2\frac{\as}\pi\CF
  \left(-\frac12\right).
\ee

By adding up all the contributions including the counter-term we arrive at
the full $O(\as)$ result
\bea
\hat H_U^{4\ell}(\as)&=&\hat H_U^1(\as)
  \ =\ 4\NC q^2\frac{\as}\pi\CF\left(\frac14\right)\label{eqn24}\\
\hat H_L^{4\ell}(\as)&=&\hat H_L^1(\as)
  \ =\ 4\NC q^2\frac{\as}\pi\CF\left(\frac12\right)\\
\hat H_F^{1\ell}(\as)&=&\hat H_F^4(\as)\ =\ 0.\label{eqn25}
\eea
For easy reference one may add the Born term contribution to obtain the 
familiar result ($\CF=4/3$) $H_{U+L}^1=4\NC q^2(1+\as/\pi)$. The vanishing 
of $H_F^4$ and thereby $H_F^{1\ell}$ in massless QCD has been noted 
before~\cite{forward7,forward19}.\footnote{For a careful discussion of the 
$\gamma_5$-subtleties in dimensional regularization when deriving the zero 
result $\hat H_F^4(\as)=0$ we refer the reader to~\cite{forward19}. In 
dimensional reduction the troublesome $\gamma_5$-odd traces can be 
completely avoided by deriving the cross section expression~Eq.(\ref{eqn21}) 
and the one-loop contribution~Eq.(\ref{eqn23}) in a well-defined manner by 
use of e.g. helicity amplitudes in which the massless axial vector current 
helicity amplitudes are related to the massless vector current helicity 
amplitudes in canonical fashion. This is implicit in Eq.~(\ref{eqn23}) 
where we have used the relation $H_F^{1\ell}=H_F^4=H_U^1=H_U^{4\ell}$ to 
relate the troublesome $\gamma_5$-odd objects $H_F^{1\ell}=H_F^4$ to the 
unproblematic $\gamma_5$-even objects $H_U^{4\ell}$, $H_U^1$ using parity 
and $\gamma_5$-invariance.}

As it turns out one does not reproduce the polarized result in 
Eq.~(\ref{eqn24}) and in Eq.~(\ref{eqn25}) when taking the $m\rightarrow 0$ 
limit in the corresponding expressions Eqs.~(\ref{eqn17}) and~(\ref{eqn18}) 
in Sec.~3, respectively. One obtains instead
\bea
\hat H_U^{4\ell}(\as)&=&4\NC q^2\frac{\as}\pi\CF
  \left(\frac14-\left[\frac12\right]\right)\label{eqn26}\\
\hat H_F^{4\ell}(\as)&=&4\NC q^2\frac{\as}\pi\CF\left(\frac12\right)\\
\hat H_F^{1\ell}(\as)&=&4\NC q^2\frac{\as}\pi\CF
  \left(0-\left[\frac12\right]\right)\label{eqn27}
\eea
where, by hindsight, we have split the contributions in the round brackets 
into a normal spin non-flip piece and an anomalous spin-flip piece. Note 
the drastic changes in the $O(\as)$ results for $H_U^{4\ell}$ and 
$H_F^{1\ell}$ when the anomalous pieces are included.

The anomalous spin-flip contributions have their origin in the collinear 
limit where the spin-flip contribution proportional to $m$ survives since it 
is multiplied by the $1/m$ collinear mass singularity. Because the 
anomalous spin-flip terms are associated with the collinear singularity, the 
flip contributions are universal and factorize into the Born term 
contribution and an universal spin-flip bremsstrahlung 
function~\cite{forward15}. This explains why there is no anomalous 
contribution to $H_L^{4\ell}$ and why the anomalous flip contributions to 
$H_U^{4\ell}$ and $H_F^{1\ell}$ are equal. In fact the strength of the 
anomalous spin-flip contribution can directly be calculated from the 
universal helicity-flip bremsstrahlung function listed in~\cite{forward15}. 
One has
\be
\frac{dH_{hf}}{dy}=H({\it Born})\frac{\as}{2\pi}\CF y
\ee
where $y$ is the energy fraction of the quark after the radiative flip 
transition relative to that of the radiating quark ($0\le y\le 1$). 
Integrating over $y$ one finds
\be
\int_0^1dy\ \frac{dH_{hf}}{dy}=H({\it Born})\frac{\as}{2\pi}\CF\,\frac12.
\ee
Then considering the fact that the flip contribution contributes negatively 
to the alignment polarization (see Eq.~(\ref{eqn4})), and that there can be
a flip on the antiquark side also, one reproduces exactly the anomalous 
flip contributions in Eqs.~(\ref{eqn26}) and~(\ref{eqn27}).

It is important to realize that the anomalous spin flip contribution does 
not come into play when calculating the transverse polarization of a single 
quark, because of the extra $m/\sqrt{q^2}$ factor in the transverse 
polarization expressions. However, when calculating quark-antiquark 
polarization correlation effects, the $O(\as)$ anomalous contribution does 
come in even when considering transverse polarization effects.

\section{Summary and Conclusion}
We have presented the results of an $O(\as)$ calculation of the full polar 
angle dependence of the quark's alignment polarization with respect to the 
electron beam direction including beam polarization effects. This completes 
a series of papers devoted to the full calculation of quark polarization 
effects on and off the $Z$-peak in $e^+e^-$-annihilations into a pair of 
massive quarks. When averaged over phase-space, the $O(\as)$ corrections to 
the polarization observables are generally small (typically of the order 
$2\%$) indicating that the $O(\as)$ corrections to the polarized and 
unpolarized rates tend to go in the same direction. This of course must not 
be true everywhere in phase-space and for every polarization component. We 
mention that our results are also of relevance for the final-state 
radiative QED corrections to lepton polarization in 
$e^+e^-\rightarrow\tau^+\tau^-$~\cite{forward20}, where, to our knowledge, 
a full $O(\alpha)$ calculation including quark mass effects has not yet 
been published.

We have not discussed how to measure the polarization of the quarks either 
when it decays as a free quark as in top quarks, or when it decays as a 
bound quark in e.g. polarized $\Lambda_b$-baryon decays. There exist a long 
list of papers devoted to this subject, among which are the papers listed 
in~\cite{forward21,forward22}. Of particular interest is the transverse 
normal polarization studied in~\cite{forward4,forward22,forward23,forward24}. 
It provides the SM background to possible $CP$-violating contributions to 
transverse polarization considered in~\cite{forward23,forward24}. In fact, 
by comparing the transverse normal polarization of the quark and the 
antiquark, the truly $CP$-violating contributions can be 
separated~\cite{forward24}.

Recently there has been a thorough numerical study of gluonic corrections 
to top quark pair production {\em and\/} decay (interference effects not 
included)~\cite{forward25}. By using helicity amplitudes for both the 
production and decay process, the full spin momentum correlation structure 
of the production and decay process has been retained in the calculation 
of~\cite{forward25}. The production-side information in~\cite{forward25} 
is identical to the information contained in our paper. However, we believe 
that our results are physically more transparent since we have chosen a 
two-step approach in which we first represent the production-side dynamics 
in terms of the polarization components of the produced top quarks 
including beam polarization effects as in \cite{forward25}. The 
polarization of the produced top quarks can then be probed by its 
subsequent decay. Also, by using analytical methods, we were able to effect 
large rate cancellations in sensitive areas of phase-space. These could 
lead to numerically unstable results in the purely numerical approach 
of~\cite{forward25} when not properly dealt with. At any rate, by comparing 
the production-side results of~\cite{forward25} and our results one has a 
powerful check on the technically rather involved calculation of gluonic 
corrections to the polarization of quarks in $e^+e^-$-annihilation.

\vspace{1cm}

\noindent{\bf Acknowledgement:} We would like to thank D.H.~Schiller for 
allowing us to freely quote from the unpublished material 
in~\cite{forward26} when writing Appendix~C.
\newpage

\section*{Appendix A}
\setcounter{equation}{0}
\def\theequation{A\arabic{equation}}
The list of $J$-integrals appearing in Section~2 had been calculated 
completely by an uniform substitution. The substitution 
$1-y=\sqrt\xi(1+\eta^2)/(1-\eta^2)$ leads to the replacement of the lower
boundary $y=0$ by $\eta=w:=\sqrt{(1-\sqrt\xi)/(1+\sqrt\xi)}$, which occurs 
in most of the expressions. Remember also that we have $v=\sqrt{1-\xi}$.
\bea
J_1&=&\jint\nonumber\\ 
  &=&2\,{1-\xi\over 4-\xi}\ln\left({1+v\over 1-v}\right)\\[12pt]
J_2&=&\jint{1\over y}\nonumber\\ 
  &=&{6\over 4-\xi}\ln\left({1+v\over 1-v}\right)\\[12pt]
J_3&=&\jint{1\over y^2}\nonumber\\
  &=&{4\over\xi\,v}\Biggl[\,-\ln\Lambda^{1\over 2}
     -\ln\xi+2\ln v+2\ln2-1\,\Biggr]
     -{24\over\xi(4-\xi)}\ln\left({1+v\over 1-v}\right)\\[12pt]
J_4&=&\jint y\nonumber\\
  &=&-\left(1+\Frac{1}{2}\xi-{6\over 4-\xi}\right)
     \ln\left({1+v\over 1-v}\right)-v\\[12pt]
J_5&=&\jint y^2\nonumber\\ 
  &=&\left(2+\Frac{1}{2}\xi+\Frac{1}{8}\xi^2-{6\over 4-\xi}\right)
     \ln\left({1+v\over 1-v}\right)-\Frac{1}{4}(6-\xi)v\\[12pt]
J_6&=&\jint yz\nonumber\\ 
  &=&\Frac{1}{2}\xi\left(-2+\Frac{1}{8}\xi+{9\over 4-\xi}
     -{12\over(4-\xi)^2}\right)\ln\left({1+v\over 1-v}\right)
     +\left(\Frac{3}{4}+\Frac{1}{8}\xi-{2\over 4-\xi}\right)v\\[12pt]
J_7&=&\jint z\nonumber\\ 
  &=&-{3\xi\over 4-\xi}\left(1-{2\over 4-\xi}\right)
     \ln\left({1+v\over 1-v}\right)+{2v\over 4-\xi}\\[12pt]
J_8&=&\jint{\:1\over z}\nonumber\\
  &=&\frac1{\sxi}\Bigg[\,\Li(w)-\Li(-w)
     +\Li\left({2+\sxi\over 2-\sxi}\;w\right)
     -\Li\left(-{2+\sxi\over 2-\sxi}\;w\right)\,\Biggr]\\[12pt]
J_9&=&\jint{1\over z^2}\nonumber\\ 
  &=&{4\over\xi\,v}\Biggl[\,-\ln\Lambda^{1\over 2}-\ln\xi+2\ln v+2\ln2
     -\frac{2-\xi}{2v}\ln\left(\frac{1+v}{1-v}\right)\,\Biggr]\\[12pt]
J_{10}&=&\jint{1\over yz}\nonumber\\
  &=&{1\over 2(1-\xi)}\ln\left(\frac{1+v}{1-v}\right)
     \Biggl[-4\ln\Lambda^{\frac12}-\Frac{5}{2}\ln\xi+5\ln(1+\sxi)
     +4\ln(1-\sxi)\nonumber\\&&\hphantom{-\ln(2+\sxi)+6\ln2\Biggr]}
     -\ln(2+\sxi)+6\ln2\Biggr]\nonumber\\&&
     +{2\over 1-\xi}\Biggl[\,\Li\left(\frac{1+v}2\right)
     -\Li\left(\frac{1-v}2\right)\,\Biggr]
     +{3\over 1-\xi}\Biggl[\,\Li\left(-{2v\over 1-v}\right)
     -\Li\left(\frac{2v}{1+v}\right)\,\Biggr]\nonumber\\&&
     +{1\over\sxi(1-\sxi)}\Bigg[\,\Li(w)-\Li(-w)
     +\Li\left({2+\sxi\over 2-\sxi}\;w\right)
     -\Li\left(-{2+\sxi\over 2-\sxi}\;w\right)\,\Biggr]\nonumber\\&&
     -{1\over 1-\xi}\Bigg[\Li\left({1+w\over 2}\right)
     -\Li\left({1-w\over 2}\right)
     +\Li\left((2+\sxi){1+w\over 4}\right)\nonumber\\&&
     -\Li\left((2+\sxi){1-w\over 4}\right)
     +\Li\left({2\sxi\over(2+\sxi)(1+w)}\right)
     -\Li\left({2\sxi\over(2+\sxi)(1-w)}\right)\Bigg]\nonumber\\ \\
J_{11}&=&\jint{z\over y}\nonumber\\
  &=&\left(-1+{12\over 4-\xi}-{24\over(4-\xi)^2}\right)
     \ln\left({1+v\over 1-v}\right)-\frac{2v}{4-\xi}\\[12pt]
J_{12}&=&\jint{y\over z}\nonumber\\
  &=&\Frac{1}{2}\ln\left(\frac{1+v}{1-v}\right)\Biggl[\,
     \Frac{1}{2}\ln\xi+\ln(2+\sxi)-\ln(1+\sxi)-2\ln2\,\Biggr]\nonumber\\&&
     +{1-\sxi\over\sxi}\Bigg[\,\Li(w)-\Li(-w)
     +\Li\left({2+\sxi\over 2-\sxi}\;w\right)
     -\Li\left(-{2+\sxi\over 2-\sxi}\;w\right)\,\Biggr]\nonumber\\&&
     +\Li\left({1+w\over 2}\right)-\Li\left({1-w\over 2}\right) 
     +\Li\left((2+\sxi){1+w\over 4}\right)
     -\Li\left((2+\sxi){1-w\over 4}\right)\nonumber\\&&
     +\Li\left({2\sxi\over(2+\sxi)(1+w)}\right)
     -\Li\left({2\sxi\over(2+\sxi)(1-w)}\right)\\[12pt]
J_{13}&=&\jint{y\over z^2}\nonumber\\
  &=&\frac2\xi\ln\left({1+v\over 1-v}\right)\\[12pt]
J_{14}&=&\jint{y^2\over z^2}\nonumber\\
  &=&\frac2\xi\left(\,\ln\left({1+v\over 1-v}\right)-2v\,\right)\\[12pt]
J_{15}&=&\jint{y^3\over z}\nonumber\\
  &=&\Frac{1}{2}(3+\xi)\ln\left({1+v\over 1-v}\right)\Biggl[\,
     \Frac{1}{2}\ln\xi+\ln(2+\sxi)-\ln(1+\sxi)-2\ln2\,\Biggr]\nonumber\\&&
     +{(1-\sxi)^3\over\sxi}\Bigg[\,\Li(w)-\Li(-w)
     +\Li\left(\frac{2+\sxi}{2-\sxi}\;w\right)
     -\Li\left(-\frac{2+\sxi}{2-\sxi}\;w\right)\,\Biggr]\nonumber\\&&
     +(3+\xi)\Bigg[\,\Li\left({1+w\over 2}\right)
     -\Li\left({1-w\over 2}\right) 
     +\Li\left((2+\sxi){1+w\over 4}\right)\nonumber\\&&
     -\Li\left((2+\sxi){1-w\over 4}\right)
     +\Li\left(\frac{2\sxi}{(2+\sxi)(1+w)}\right)
     -\Li\left(\frac{2\sxi}{(2+\sxi)(1-w)}\right)\,\Bigg]\nonumber\\&&
     +{40-16\xi+\xi^2\over16}\ln\left(\frac{1+v}{1-v}\right)
     -{26-\xi\over 8}\,v\\[12pt]
J_{16}&=&\jint{y^3\over z^2}\nonumber\\
  &=&\frac{2+\xi}\xi\ln\left({1+v\over 1-v}\right)-\frac{6}{\xi}v\\[12pt]
J_{17}&=&\jint{y^2\over z}\nonumber\\
  &=&\Frac{1}{2}\ln\left({1+v\over 1-v}\right)
     \Biggl[\,\Frac{1}{2}\ln\xi+\ln(2+\sxi)
     -\ln(1+\sxi)-2\ln2\,\Biggr]\nonumber\\&&
     +{(1-\sxi)^2\over\sxi}\Bigg[\,\Li(w)-\Li(-w)
     +\Li\left(\frac{2+\sxi}{2-\sxi}\;w\right)
     -\Li\left(-\frac{2+\sxi}{2-\sxi}\;w\right)\,\Biggr]\nonumber\\&&
     +\Li\left({1+w\over 2}\right)-\Li\left({1-w\over 2}\right) 
     +\Li\left((2+\sxi){1+w\over 4}\right)\nonumber\\&&
     -\Li\left((2+\sxi){1-w\over 4}\right)
     +\Li\left(\frac{2\sxi}{(2+\sxi)(1+w)}\right)
     -\Li\left(\frac{2\sxi}{(2+\sxi)(1-w)}\right)\nonumber\\&&
     +{2-\xi\over2}\ln\left(\frac{1+v}{1-v}\right)-v
\eea

\newpage

\section*{Appendix B}
\setcounter{equation}{0}
\def\theequation{B\arabic{equation}}
It is convenient to define the mass dependent variables $a:=2+\sqrt\xi$, 
$b:=2-\sqrt\xi$ and $w:=\sqrt{(1-\sqrt\xi)/(1+\sqrt\xi)}$. The functions 
$t_1,\ldots,t_{12}$ appearing in Eqs.~(\ref{eqn17})--(\ref{eqn19}) are then 
given by
\bea
t_1&:=&\ln\left(\frac{2\xi\sqrt\xi}{b^2(1+\sqrt\xi)}\right),\quad
t_2\ :=\ \ln\left(\frac{2\sqrt\xi}{1+\sqrt\xi}\right)\quad
  \Rightarrow\quad t_1-t_2\ =\ \ln\left(\frac\xi{b^2}\right)\\
t_3&:=&\ln\left(\frac{1+v}{1-v}\right)\\
t_4&:=&\Li(w)-\Li(-w)+\Li(\frac abw)-\Li(-\frac abw)\\
t_5&:=&\frac12\ln\left(\frac{a\sqrt\xi}{4(1+\sqrt\xi)}\right)
  \ln\left(\frac{1+v}{1-v}\right)
  +\Li\left(\frac{2\sqrt\xi}{a(1+w)}\right)
  -\Li\left(\frac{2\sqrt\xi}{a(1-w)}\right)\,+\nonumber\\&&
  +\Li\left(\frac{1+w}2\right)-\Li\left(\frac{1-w}2\right)
  +\Li\left(\frac{a(1+w)}4\right)-\Li\left(\frac{a(1-w)}4\right)\\
t_6&:=&\ln^2(1+w)+\ln^2(1-w)+\ln\left(\frac a8\right)\ln(1-w^2)
  \,+\nonumber\\&&
  +\Li\left(\frac{2\sqrt\xi}{a(1+w)}\right)
  +\Li\left(\frac{2\sqrt\xi}{a(1-w)}\right)
  -2\Li\left(\frac{2\sqrt\xi}a\right)\,+\nonumber\\&&
  +\Li\left(\frac{1+w}2\right)+\Li\left(\frac{1-w}2\right)
  -2\Li\left(\frac12\right)\,+\nonumber\\&&
  +\Li\left(\frac{a(1+w)}4\right)+\Li\left(\frac{a(1-w)}4\right)
  -2\Li\left(\frac a4\right)\\
t_7&:=&2\ln\left(\frac{1-\xi}{2\xi}\right)\ln\left(\frac{1+v}{1-v}\right)
  -\Li\left(\frac{2v}{(1+v)^2}\right)
  +\Li\left(-\frac{2v}{(1-v)^2}\right)\,+\nonumber\\&&
  -\frac12\Li\left(-\left(\frac{1+v}{1-v}\right)^2\right)
  +\frac12\Li\left(-\left(\frac{1-v}{1+v}\right)^2\right)\,+\\&&
  +\Li\left(\frac{2w}{1+w}\right)-\Li\left(-\frac{2w}{1-w}\right)
  -2\Li\left(\frac{w}{1+w}\right)+2\Li\left(-\frac{w}{1-w}\right)
  \,+\nonumber\\&&
  +\Li\left(\frac{2aw}{b+aw}\right)-\Li\left(-\frac{2aw}{b-aw}\right)
  -2\Li\left(\frac{aw}{b+aw}\right)+2\Li\left(-\frac{aw}{b-aw}\right)
  \nonumber\\
t_8&:=&\ln\left(\frac\xi 4\right)\ln\left(\frac{1+v}{1-v}\right)
  +\Li\left(\frac{2v}{1+v}\right)-\Li\left(-\frac{2v}{1-v}\right)-\pi^2\\
t_9&:=&2\ln\left(\frac{2(1-\xi)}{\sqrt\xi}\right)
  \ln\left(\frac{1+v}{1-v}\right)
  +2\left(\Li\left(\frac{1+v}2\right)-\Li\left(\frac{1-v}2\right)\right)
  \,+\nonumber\\&&
  +3\left(\Li\left(-\frac{2v}{1-v}\right)
  -\Li\left(\frac{2v}{1+v}\right)\right)\\
t_{10}&:=&\ln\left(\frac4\xi\right),\quad
t_{11}\ :=\ \ln\left(\frac{4(1-\sqrt\xi)^2}\xi\right),\quad
t_{12}\ :=\ \ln\left(\frac{4(1-\xi)}\xi\right)
\eea

\section*{Appendix C}
\setcounter{equation}{0}
\def\theequation{C\arabic{equation}}
In this Appendix we generalize the cross-section expressions in the main 
text to include transverse and longitudinal beam polarization 
effects.\footnote{Here we return to common usage and call the ``alignment'' 
polarization of the electron and positron beam their ``longitudinal'' 
polarization.} The cross section can be written in a matrix factorized form 
as~\cite{forward26}
\be\label{eqnc1}
d\sigma=\left(\frac{2\pi\alpha}{q^2}\right)^2\sum_{r',r=1}^4g_{r'r}(q^2)
  L^{r'}\cdot H^r dPS,
\ee
where $\alpha=1/137$ is the structure constant, $q^2=4E_{\rm beam}^2$ the 
total quared c.m.~beam energy, $dPS$ the Lorentz-invariant phase space 
element and $L\cdot H$ stands for $L_{\mu\nu}H^{\mu\nu}$. The indices $r$ 
and~$r'$ run over the four independent components of the hadron tensor 
$H^r_{\mu\nu}$ (defined in Eq.~(\ref{eqn2})) and the lepton tensor 
$L^{r'}_{\mu\nu}$, resp.\ (defined accordingly). $H^r_{\mu\nu}$ stands for 
the final hadron tensor regardless of whether one is considering polarized 
or unpolarized final states. In the case treated here (polarized quark), 
the hadron tensor would carry an additional index labelling the three 
independent polarization components of the quark. The coefficients of the 
energy dependent electro-weak coupling matrix $g_{r'r}(q^2)$ are given 
below. The factorized form of Eq.~(\ref{eqnc1}) has the advantage that the 
various parts determining the cross section are separated into three 
modular pieces: $L^{r'}_{\mu\nu}$ contains the angular dependences and the 
beam polarization parameters, $H^r_{\mu\nu}$ contains the hadron dynamics 
of the final state and $g_{r'r}(q^2)$ comprises the model dependent 
parameters of the underlying electro-weak theory.

The electro-weak coupling matrix $g_{r'r}(q^2)$ entering in Eq.~(\ref{eqnc1}) 
is given by
\bea
g_{11}&=&Q_f^2-2Q_fv_ev_f\real\cz+(v_e^2+a_e^2)(v_f^2+a_f^2)|\cz|^2,\nonumber\\
g_{12}&=&Q_f^2-2Q_fv_ev_f\real\cz+(v_e^2+a_e^2)(v_f^2-a_f^2)|\cz|^2,\nonumber\\
g_{13}&=&-2Q_fv_ea_f\imag\cz,\\
g_{14}&=&2Q_fv_ea_f\real\cz-2(v_e^2+a_e^2)v_fa_f|\cz|^2,\nonumber
\eea

\newpage

\bea
g_{21}&=&q_f^2-2Q_fv_ev_f\real\cz+(v_e^2-a_e^2)(v_f^2+a_f^2)|\cz|^2,\nonumber\\
g_{22}&=&q_f^2-2Q_fv_ev_f\real\cz+(v_e^2-a_e^2)(v_f^2-a_f^2)|\cz|^2,\nonumber\\
g_{23}&=&-2Q_fv_ea_f\imag\cz,\nonumber\\
g_{24}&=&2Q_fv_ea_f\real\cz-2(v_e^2-a_e^2)v_fa_f|\cz|^2,\nonumber\\
\nonumber\\
g_{31}&=&-2Q_fa_ev_f\imag\cz,\nonumber\\
g_{32}&=&-2Q_fa_ev_f\imag\cz,\nonumber\\
g_{33}&=&2Q_fa_ea_f\real\cz,\addtocounter{equation}{-1}\\
g_{34}&=&2Q_fa_ea_f\imag\cz,\nonumber\\
\nonumber\\
g_{41}&=&2Q_fa_ev_f\real\cz-2v_ea_e(v_f^2+a_f^2)|\cz|^2,\nonumber\\
g_{42}&=&2Q_fa_ev_f\real\cz-2v_ea_e(v_f^2-a_f^2)|\cz|^2,\nonumber\\
g_{43}&=&2Q_fa_ea_f\imag\cz,\nonumber\\
g_{44}&=&-2Q_fa_ea_f\real\cz+4v_ea_ev_fa_f|\cz|^2\nonumber
\eea
where $\cz(q^2)=gM_Z^2q^2/(q^2-M_Z^2+iM_Z\Gamma_Z)^{-1}$, with $M_Z$ and 
$\Gamma_Z$ the mass and width of the $Z^0$ and
$g=G_F(8\sqrt 2\pi\alpha)^{-1}\approx 4.49\cdot 10^{-5}\mbox{\rm GeV}^{-2}$.
$Q_f$ are the charges of the final state quarks to which the electro-weak 
currents directly couple; $v_e$ and $a_e$, $v_f$ and $a_f$ are the 
electro-weak vector and axial vector coupling constants. For example, in 
the Weinberg-Salam model, one has $v_e=-1+4\sin^2\theta_W$, $a_e=-1$ for 
leptons, $v_f=1-\frac83\sin^2\theta_W$, $a_f=1$ for up-type quarks 
($Q_f=\frac23$), and $v_f=-1+\frac43\sin^2\theta_W$, $a_f=-1$ for down-type 
quarks ($Q_f=-\frac13$). The left- and right-handed coupling constants are 
then given by $g_L=v+a$ and $g_R=v-a$, respectively. In the purely 
electromagnetic case one has $g_{11}=g_{12}=g_{21}=g_{22}=Q_f^2$ and all 
other $g_{r'r}=0$. The terms linear in $\real\cz$ and $\imag\cz$ come from 
$\gamma-Z^0$ interference, whereas the terms proportional to $|\cz|^2$ 
originate from $Z$-exchange.

\subsection*{C.1\quad Lepton tensor}
We first specify the lepton tensor in the laboratory frame denoted by a 
prime. The nonvanishing elements of the lepton tensor $L'^r_{\mu\nu}$ 
(with $r=1,2,3,4$ defined in analogy to Eq.~(\ref{eqn2})) are given by
\bea
L'^1_{11}\ =\ L'^1_{22}\ =\ (1-h^-h^+),&&\mbox{\qquad\qquad s}\nonumber\\
L'^1_{12}\ =\ -L'^1_{21}\ =\ -i(h^--h^+),&&\mbox{\qquad\qquad a}\nonumber\\
L'^2_{11}\ =\ L'^2_{22}\ =\ -(\xi^-_{x'}\xi^+_{x'}-\xi^-_{y'}\xi^+_{y'}),
  &&\mbox{\qquad\qquad s}\nonumber\\
L'^2_{12}\ =\ L'^2_{21}\ =\ -(\xi^-_{x'}\xi^+_{y'}+\xi^-_{y'}\xi^+_{x'}),
  &&\mbox{\qquad\qquad s}\label{eqnc2}\\
L'^3_{11}\ =\ -L'^3_{22}\ =\ (\xi^-_{x'}\xi^+_{y'}+\xi^-_{y'}\xi^+_{x'}),
  &&\mbox{\qquad\qquad s}\nonumber\\
L'^3_{12}\ =\ L'^3_{21}\ =\ -(\xi^-_{x'}\xi^+_{x'}-\xi^-_{y'}\xi^+_{y'}),
  &&\mbox{\qquad\qquad s}\nonumber\\
L'^4_{11}\ =\ L'^4_{22}\ =\ (h^--h^+),&&\mbox{\qquad\qquad s}\nonumber\\
L'^4_{12}\ =\ -L'^4_{21}\ =\ -i(1-h^-h^+).&&\mbox{\qquad\qquad a}\nonumber
\end{eqnarray}
Here we have indicated the $\mu\leftrightarrow\nu$ symmetry properties 
(s=symmetric, a=antisymmetric) of the various components of $L'_{\mu\nu}$ 
which will be of later convenience when considering the contraction with 
the hadronic tensor. The $\xi^\pm_{x'}$ and $\xi^\pm_{y'}$ denote the 
transverse beam polarization in and out of the accelerator plane, and 
$\xi^\pm_{z'}$($=\mp h^\pm$) the longitudinal polarization (helicity) of 
$e^\pm$. For natural transverse beam polarization of degree $P$ in the case 
of a circular accelerator one has $\vec\xi^\pm=(0,\pm P,0)$.

\subsection*{C.2\quad Hadron tensor}
In the overall c.m.~system that we are considering, only the three space 
components of the hadron tensors $H^r_{mn}$ ($m,n=1,2,3$) contribute to 
the annihilation cross section in Eq.~(\ref{eqnc1}). Also, since the hadron 
tensors $H^r_{mn}$ are hermitean, there are in general nine real 
independent components $H^r_{mn}$ for each $r=1,2,3,4$. We introduce six 
symmetric and three antisymmetric real combinations under $m\leftrightarrow n$ 
(dropping the superscript $r$ for compactness) in Table~1, with 
$H_{\sigma\sigma'}=\eps^\mu(\sigma)H_{\mu\nu}\eps^{*\nu}(\sigma')$, where
\be
\eps^\mu(\pm)=\frac1{\sqrt2}(0;\mp 1,-i,0),\qquad \eps^\mu(0)=(0;0,0,1).
\ee
Table~1 contains the sperical as well as the Cartesian components of the 
nine independent hadron tensor components in the c.m.\ system. We also list 
the $\mu\leftrightarrow\nu$ symmetry properties of the components as well 
as the parity nature of the product of currents in the helicity system 
($z$-axis in event plane!).

\subsection*{C.3\quad Angular correlations}
We are now in the position to consider the beam-event correlations that 
arise from the contraction of the lepton and hadron tensors. First we have 
to rotate the lepton tensors $L'_{mn}$ given in Eq.~(\ref{eqnc2}) to the 
event plane. This is achieved by the Euler rotation illustrated in Fig.~4,
\be
L_{kl}(\varphi,\theta,\chi)=R_{km}(\varphi,\theta,\chi)
  R_{ln}(\varphi,\theta,\chi)L'_{mn},
\ee
where $\varphi$, $\theta$ and $\chi$ are the Euler angles describing the 
rotation from the beam-system to the event-system, and the Euler rotation 
matrix is given by
\medskip
\bea
\lefteqn{R(\varphi,\theta,\chi)}\\[12pt]
&=&\pmatrix{\cos\varphi\cos\theta\cos\chi-\sin\varphi\sin\chi&
  \sin\varphi\cos\theta\cos\chi+\cos\varphi\sin\chi&-\sin\theta\cos\chi\cr
  -\cos\varphi\cos\theta\sin\chi-\sin\varphi\cos\chi&
  -\sin\varphi\cos\theta\sin\chi+\cos\varphi\cos\chi&\sin\theta\sin\chi\cr
  \cos\varphi\sin\theta&\sin\varphi\sin\theta&\cos\theta\cr}.\nonumber
\eea
\medskip\\
Let us note, though, that the contraction $L_{mn}(\varphi,\theta,\chi)H_{mn}$ 
is simplified by separately considering the symmetric and antisymmetric 
parts of $L_{mn}$ and $H_{mn}$ according to the classification given in 
Table~1. Such symmetry considerations are also very helpful when one wants 
to assess quickly the influence of beam polarization on the measurement of 
the various hadron tensor components. One obtains (omitting again the 
superscript $r$):
\bea
\frac34L^1_{mn}H_{mn}&=&(1-h^-h^+)H_A+(h^--h^+)H_D,\nonumber\\
\frac34L^2_{mn}H_{mn}&=&X(\varphi)H_B+Y(\varphi)H_C,\nonumber\\
\frac34L^3_{mn}H_{mn}&=&X(\varphi)H_C-Y(\varphi)H_B,\label{eqnc3}\\
\frac34L^4_{mn}H_{mn}&=&(1-h^-h^+)H_D+(h^--H^+)H_A,\nonumber
\eea
where
\bea
H_A&=&\frac38(1+\cos^2\theta)H_U+\frac34\sin^2\theta H_L
  +\frac34\sin^2\theta\cos 2\chi H_T\nonumber\\&&
  -\frac34\sin^2\theta\sin 2\chi H_4
  -\frac3{2\sqrt2}\sin 2\theta\cos\chi H_I
  +\frac3{2\sqrt2}\sin 2\theta\sin\chi H_5,\nonumber\\
H_B&=&\frac38\sin^2\theta H_U-\frac34\sin^2\theta H_L
  +\frac34(1+\cos^2\theta)\cos 2\chi H_T\nonumber\\&&
  -\frac34(1+\cos^2\theta)\sin 2\chi H_4
  +\frac3{2\sqrt2}\sin 2\theta\cos\chi H_I
  -\frac3{2\sqrt2}\sin 2\theta\sin\chi H_5,\nonumber\\
H_C&=&\frac32\cos\theta\sin 2\chi H_T+\frac32\cos\theta\cos 2\chi H_4
  \label{eqnc4}\\&&
  +\frac3{\sqrt2}\sin\theta\sin\chi H_I
  +\frac3{\sqrt2}\sin\theta\cos\chi H_5,\nonumber\\
H_D&=&\frac34\cos\theta H_F-\frac3{\sqrt2}\sin\theta\cos\chi H_A
  +\frac3{\sqrt2}\sin\theta\sin\chi H_9,\nonumber
\eea
with all hadron tensors referring to the event system and
\bea
X(\varphi)&=&(\xi^-_{x'}\xi^+_{x'}-\xi^-_{y'}\xi^+_{y'})\cos 2\varphi
  +(\xi^-_{x'}\xi^+_{y'}+\xi^-_{y'}\xi^+_{x'})\sin 2\varphi,\nonumber\\
Y(\varphi)&=&-(\xi^-_{x'}\xi^+_{x'}-\xi^-_{y'}\xi^+_{y'})\sin 2\varphi
  +(\xi^-_{x'}\xi^+_{y'}+\xi^-_{y'}\xi^+_{x'})\cos 2\varphi.
\eea
Note that $H_B$ and $H_C$ contain no more hadronic structure than $H_A$. 
This means that the presence of transverse beam polarization does not 
increase the number of possible hadronic measurements. On the other hand, 
transverse beam polarization is helpful in disentangling the hadronic 
structure.

Eq.~(\ref{eqnc3}) gives the most general structure of possible beam-event 
correlations in the presence of beam polarization and final state 
polarization effects. The whole angular dependence on the relative 
beam-event angles $\varphi$, $\theta$ and $\chi$ is explicitly exhibited in 
Eqs.~(\ref{eqnc3}) and~(\ref{eqnc4}), since the hadronic helicity structure 
functions $H_\alpha$ are evaluated in the event frame and thus are 
independent of these angles. We repeat that, when we have been referring 
to the hadron tensor in this Appendix, this means either the unpolarized 
hadron tensor or the polarized hadron tensor carrying any number of 
additional polarization indices (in our case the quark's polarization 
index $m$ which can take the three values $m=\ \perp$ (transversal 
perpendicular), $N$ (transverse normal) and $\ell$ (alignment)). Also, in 
this paper we have only considered the $(U)$-, $(L)$- and $(F)$-components 
of the hadron tensor in Eq.~(\ref{eqnc4}) which remain after azimuthal 
$\varphi$- and $\chi$-averaging.

\newpage

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\begin{table}[h]
\begin{center}
\begin{tabular}{|l||l|l|c|r|}
\hline
&spherical components&Cartesian components&&\\\hline\hline
$H_U$&$H_{++}+H_{--}$&$H_{11}+H_{22}$&s&$pc$($pv$)\\\hline
$H_L$&$H_{00}$&$H_{33}$&s&$pc$($pv$)\\\hline
$H_T$&$\frac12(H_{+-}+H_{-+})$&$\frac12(-H_{11}+H_{22})$
  &s&$pc$($pv$)\\\hline
$H_I$&$\frac14(H_{+0}+H_{0+}-H_{-0}-H_{0-})$
  &$\frac{-1}{2\sqrt 2}(H_{31}+H_{13})$&s&$pc$($pv$)\\\hline
$H_9$&$-\frac i4(H_{+0}-H_{0+}-H_{-0}+H_{0-})$
  &$\frac{-i}{2\sqrt 2}(H_{31}-H_{13})$&a&$pc$($pv$)\\\hline
$H_F$&$H_{++}-H_{--}$&$-i(H_{12}-H_{21})$&a&$pv$($pc$)\\\hline
$H_A$&$\frac14(H_{+0}+H_{0+}+H_{-0}+H_{0-})$
  &$\frac{-i}{2\sqrt 2}(H_{23}-H_{32})$&a&$pv$($pc$)\\\hline
$H_4$&$-\frac i2(H_{+-}-H_{-+})$&$-\frac12(H_{12}+H_{21})$
  &s&$pv$($pc$)\\\hline
$H_5$&$-\frac i4(H_{+0}-H_{0+}+H_{-0}-H_{0-})$
  &$\frac{-1}{2\sqrt 2}(H_{23}+H_{32})$&s&$pv$($pc$)\\\hline
\end{tabular}
\end{center}
\bigskip
\centerline{\Large\bf Table 1}
\end{table}

\newpage

\centerline{\bf\Large Figure Captions}
\vspace{0.5truecm}
\newcounter{fig}
\begin{list}{\bf\rm Fig.\ \arabic{fig}: }{\usecounter{fig}
\labelwidth1.6cm \leftmargin2.5cm \labelsep0.4cm \itemsep0ex plus0.2ex }

\item Born term and $O(\as)$ contributions to 
$e^+e^-\rightarrow q\bar q(g)$. One-loop wave function renormalization 
graphs are omitted

\item $y$-dependence of the forward-backward component $\Pl_F(y)$ of the 
alignment polarization of top quarks produced in the continuum for 
c.m.~energies of $380$~GeV (solid), $500$~GeV (dashed), $1000$~GeV 
(dashed-dotted) and $1000$~TeV (dotted). Also shown is the limiting curve 
of $\Pl_F(y)$ for maximal values of $y$ (dotted curve in the upper half of 
the figure)

\item Polar angle dependence of the alignment polarization of\\
a) top quarks in the continuum for c.m.~energies of $380$~GeV (solid),
$500$~GeV (dashed) and $1000$~GeV (dashed-dotted)\\
b) bottom quarks on the $Z$-peak. Shown are Born term results (dotted) and 
full $O(\as)$ results (solid)

\item Definition of the Euler angles $\varphi$, $\theta$ and $\chi$ relating 
the beam frame ${\cal O}_{x'y'z'}$ and the event frame ${\cal O}_{xyz}$. 
The three successive Euler rotations from the beam frame to the event frame 
are $R_{z'}(\varphi)$ followed by $R_{y_1}(\theta)$ and then by $R_z(\chi)$
\end{list}

\vspace{1truecm}
\centerline{\bf\Large Caption Table 1}
\vspace{0.5truecm}\noindent
Independent helicity components of the hadron tensor $H_{\mu\nu}$ in the 
spherical basis (second column) and in the Cartesian basis (third column). 
Column 4 gives the $\mu\leftrightarrow\nu$ symmetry of the nine components 
and column 5 lists the parity of the products of contributing currents 
($pc$: $VV,AA$ and $pv$: $V\!A,AV$) for the unpolarized case and in round 
brackets for the polarized case.

\end{document}

