%Title: ANGULAR DISTRIBUTIONS OF MASSIVE QUARKS AND LEPTONS
%       CLOSE TO THRESHOLD 
%Authors: S.J. Brodsky, A.H. Hoang, J.H. K\"uhn and T. Teubner,
%published:  <EM>Phys. Lett.</EM><B> B359 </B><EM>(1995) 355-361.</EM>
% Univ. of Karlsruhe preprint TTP95-26,
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\begin{document}
\noindent
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{\bf SLAC-PUB-95-6955} \\
{\bf TTP95-26}\footnote{The complete postscript file of this
preprint, including figures, is available via anonymous ftp at
ttpux2.physik.uni-karlsruhe.de (129.13.102.139) as
/ttp95-26/ttp95-26.ps or via www at
http://ttpux2.physik.uni-karlsruhe.de/cgi-bin/preprints/ 
Report-no: TTP95-26.}\\ 
{\bf July 1995}\\
hep-ph/9508274\\
\end{flushright}
\vspace{0.8cm}
\begin{center}
\begin{bf}
 \begin{large}
ANGULAR DISTRIBUTIONS OF MASSIVE QUARKS AND LEPTONS\\ 
CLOSE TO THRESHOLD\footnote{Work 
 supported by BMFT 056 KA 93P.}\\
 \end{large}
\end{bf}
  \vspace{0.5cm}
  \begin{large}
   S.J. Brodsky$\mbox{ }^b$, A.H. Hoang$\mbox{ }^{a, }
   $\footnotemark[3], 
   J.H. K\"uhn$\mbox{ }^{a, b}$ and 
   T. Teubner$\mbox{ }^{a, }$\footnote{e--mails: 
 hoang@ttpux2.physik.uni-karlsruhe.de and 
    tt@ttpux2.physik.uni-karlsruhe.de} \\
  \end{large}
  \vspace{0.8cm}
$\mbox{}^a$ Institut f\"ur Theoretische Teilchenphysik, 
            Universit\"at Karlsruhe,\\
            D--76128 Karlsruhe, Germany \\
\vspace{.3cm}
$\mbox{}^b$ Stanford Linear Accelerator Center\\
            Stanford University, Stanford, CA 94309\\
\vspace{2.0cm}
{\bf Abstract}\\
\vspace{0.8cm}
\noindent
\begin{minipage}{15.0cm}
\begin{small}
Predictions for the angular distribution of massive quarks and
leptons are presented, including QCD and QED corrections. Recent
results for the fermionic part of the two-loop corrections to the
electromagnetic form factors are combined with the BLM scale fixing
prescription. Two distinctly different scales arise as arguments of
$\alpha_s(\mu^2)$ near threshold: the relative momentum of the
quarks governing the soft gluon exchange responsible for the Coulomb
potential, and a large momentum scale approximately equal to twice
the quark mass for the corrections induced by transverse gluons.
Numerical predictions for charmed, bottom, and top quarks are given.
One obtains a direct determination of $\alpha_{\mbox{\it\scriptsize
V}}(Q^2)$, the coupling in the heavy quark potential, which can be
compared with lattice gauge theory predictions. The corresponding
QED results for $\tau$ pair production allow for a measurement of
the magnetic moment of the $\tau$ and could be tested at a future
$\tau$-charm factory.\\
\end{small}
\end{minipage}
\end{center}
\vfill
\centerline{(Submitted to Physics Letters B.)}
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\noindent 
{\bf 1.~Introduction}\\ 

\noindent
QCD predictions for the angular distributions of partons can be
tested in a variety of ways. For massless partons, \ie\ light quarks
or gluons, the canonical approach is based on angular distributions
of jets or related quantities, like the thrust axis. In fact, the
observation of a distribution $\propto(1+\cos^2\theta)$ for
two-jet-events has provided convincing evidence for the spin-1/2
nature of partons produced in $e^+e^-$ annihilation. Alternatively,
for charmed or bottom quarks one may directly identify the heavy
quark and heavy meson directions. This approach has been employed
frequently in the  analysis of the $b$ quark angular distributions
at high energies. In this kinematical region gluon radiation affects
the shape of the distribution, typically rendering the lowest order
predictions more isotropic. Real and virtual corrections must be
considered jointly, since only the sum leads to an infrared finite
prediction.
 
In this paper a different kinematical region will be considered:
heavy quark production close to threshold, where the cross section
is dominated by a few exclusive channels. In this case one expects
that the heavy mesons will essentially follow the heavy quark
direction---a simple consequence of Newton's law of inertia. To
employ the parton model in the case of hadron pair production may be
somewhat surprising, since the structure of form factors for
exclusive channels imposes stringent constraints on the angular
distributions of the hadrons. However, as a consequence of the near
degeneracy of the pseudoscalar and vector mesons $B$ and $B^*$, the
sum of the various channels ($B \bar B$, $B\bar B^*$ + c.c.,
$B^*\bar B^*$) may easily combine to saturate the prediction for the
angular distribution of the quarks. For definiteness, we will
subsequently discuss  the case of $b$ quarks. For certain
kinematical regions our calculation can be applied for charm quarks
and will be valid {\it a~forteriori} for top quarks.
 
Close to threshold, in the limit $\beta = |\vec q\,|/\sqrt{\vec
q\,^2+m_Q^2} = \sqrt{1-4m_Q^2/s} \to 0$, the cm angular distribution
for $e^+ e^- \to Q\,\bar Q$ is of course isotropic, a result of
$S$-wave dominance. The small admixture of $P$-waves slightly above
threshold provides a contribution $\propto \beta^2\cos^2\theta$
which will be studied in this paper. The tree graph prediction for
the angular distribution is trivial. The leading ${\cal
O}(\alpha_s)$ corrections have been evaluated in \cite{Jersak} for
arbitrary quark mass and energy. However, unavoidably, the
renormalization scale at which the strong coupling $\alpha_s(\mu^2)$
has to be evaluated in these formulae can only be fixed by a
two-loop calculation. In the high energy region one may convincingly
argue on physical grounds that the scale for the dominant correction
terms should be of order $\sqrt{s}$. In the threshold region,
however, both $\sqrt{s}$ and the quark momentum $|\vec q\,|$ are
viable options for the energy scale. In the BLM procedure \cite{BLM}
the renormalization scale is fixed by resumming all terms
proportional to the QCD beta-function into the running coupling. To
carry out the BLM procedure to leading order, it is sufficient to
identify the $n_f$ terms in the next to leading order coefficients.
Recent results for the fermionic part of the two-loop corrections to
the total rate~\cite{HKT1}, and in particular to the $\gamma^* \to
Q\,\bar Q$ form factors allow for a solution of the scale-setting
problem, at least in the threshold region. In fact, employing the
BLM scale-setting prescription, one may even fix not only the scale
but perhaps even accommodate the bulk of the two-loop corrections.

Real radiation is strongly damped in the threshold region,
decreasing $\propto \beta^2$ relative to the tree contributions.
Hence, dropping the terms of higher power in $\beta$, one may
dispose of real radiation (even if multiplied by an infrared cutoff)
and limit the discussion to virtual radiation; \ie\ to the
corrections to the form factors. Contributions from the
instantaneous Coulomb potential and from hard photon exchange can be
clearly distinguished, allowing for the resummation of the former.
The contribution from the Coulomb part of the heavy quark
rescattering yields a series in $(\pi\alpha_s/\beta)^n$. If one
utilizes the $\alpha_{\mbox{\it\scriptsize V}}$ scheme, defined for
the heavy quark potential $V(Q^2) =
-4\pi\,C_F\,\alpha_{\mbox{\it\scriptsize V}}(Q^2)/Q^2\,,$ then we
find that the scale of $\alpha_{\mbox{\it\scriptsize V}}$ is set to
a value proportional to $s\,\beta^2$, the square of the relative
momentum. Thus one can obtain a determination of
$\alpha_{\mbox{\it\scriptsize V}}(Q^2)$ from measurements of $e^+
e^- \to Q\,\bar Q$ near threshold. In particular we shall show that
the ``anisotropy'' $A(\beta^2)$ defined by
\begin{equation}
\frac{{\rm d}N}{{\rm d}\cos \theta} \propto 1 + A\,\cos^2\theta
\label{defa}
\end{equation}
is sensitive to $\alpha_{\mbox{\it\scriptsize V}}(s\,\beta^2)$.
Precise predictions for $\alpha_{\mbox{\it\scriptsize V}}(Q^2)$ have
been given in~\cite{Davies} using heavy quark lattice gauge theory
with constraints from the $\Upsilon$ spectrum.
 
We also will discuss the application of the anisotropy formalism to
the QED process $e^+ e^- \to \tau^+ \tau^-$. We shall show that the
anisotropy of the angular distribution provides a new way to measure
the Pauli form factors of the leptons, the analog of the anomalous
magnetic moment in the timelike region. In addition, the radiative
corrections we discuss here change the normalization of the cross
section near threshold.

\vspace{0.8cm}
\noindent 
{\bf 2.~Form factors, angular distributions, 
and infrared singularities}\\ 
  
\noindent
The amplitude for the creation of a massive fermion pair from a
virtual photon is characterized by the Dirac ($F_1$) and Pauli
($F_2$) form factors:
\begin{equation}
\bar u\,\Lambda_\mu\, v = 
  ieQ_f\,\bar u \,\big[\, \gamma_\mu\,F_1(q^2) + 
      \frac{i}{2\,m}\,\sigma_{\mu\nu}\,q^\nu\,F_2(q^2) \, \big]\,v
\end{equation}
where  $\sigma_{\mu\nu} = \frac{i}{2}[\gamma_\mu,\gamma_\nu]$. The
photon momentum flowing into the vertex is denoted by $q$, the
fermion mass by $m$. The resulting angular distribution is
conveniently expressed in terms of the electric and magnetic form
factors $G_e$ and $G_m$ \cite{Renard}:
\begin{equation}
\frac{\rm d \,\sigma(e^+ e^- \to f\bar f)}{\rm d\,\Omega} = 
 \frac{\alpha^2\,Q_f^2\,\beta}{4\,s}\bigg[
   \frac{4\,m^2}{s}\,|G_e|^2\,\sin^2\theta +
   |G_m|^2\,(1+\cos^2\theta)\, \bigg]
\end{equation}
with
\begin{equation}
G_e = F_1+\frac{s}{4\,m^2}F_2\,,\qquad
G_m = F_1+F_2\,.
\end{equation}
The anisotropy is thus given by 
\begin{eqnarray}
A & = & \frac{|G_m|^2 - ( 1- \beta^2 ) |G_e|^2}
             {|G_m|^2 + ( 1- \beta^2 ) |G_e|^2}
\nonumber\\
  & = & \frac{\widetilde A}{1-\widetilde A}\,, 
\label{ani}
\end{eqnarray}
where
\begin{equation}
\widetilde A  =  \frac{\beta^2}{2}\,
             \frac{|F_1|^2\,(1-\beta^2) - |F_2|^2}
                  {|F_1 + F_2|^2\,(1-\beta^2)}\,.
\label{atil}
\end{equation}
    
In Born approximation $A_{\rm Born} = \beta^2/(2-\beta^2)$. Note that
for $F_2 = 0$, the anisotropy is identical to the Born prediction,
independent of $F_1$. Thus the form
\begin{equation}
\frac{\widetilde A}{\widetilde A_{\rm Born}} - 1 = 
-\,2\,F_2(s)\,\bigg[\,1+{\cal O}(\frac{\alpha}{\pi})\,\bigg]\,,\qquad 
\widetilde A_{\rm Born} = \frac{\beta^2}{2}
\end{equation}
isolates $F_2(s)$. This provides a way to experimentally determine
the timelike Pauli form factor of the $\tau$ lepton. The QED
prediction is
\begin{equation}
F_2(4\,m^2) = -\,\frac{\alpha}{2\,\pi} + 
{\cal O}\left(\left(\frac{\alpha}{\pi}\right)^2\right)
\end{equation}
which is, up to the sign, equal to the familiar Schwinger result
$F_2(0) = \alpha/2\pi$. Away from threshold the one-loop QED
prediction is
\begin{equation}
F_2(\beta) = \frac{\alpha}{\pi}\,\bigg[\,\frac{1-\beta^2}{4\,\beta}\,
\ell n\frac{1-\beta}{1+\beta}\,\bigg]\,.
\end{equation}
We neglect this type of higher twist corrections in the following. 
   
In Born approximation $F_1=1$ and $F_2=0$. The impact of one- and
two-loop radiative corrections on the form factors and angular
distributions will be discussed in the context of QED in the
remainder of this section; the case of immediate interest, namely
QCD, will be discussed in section~3. To demonstrate the line of
reasoning, we first shall present the arguments for the leading
order calculation in some detail.
 
In the QED calculation the order $\alpha$ correction to the Dirac
form factor $\delta F_1$ in the timelike region exhibits an infrared
singularity which can be regulated using a nonvanishing photon mass
$\lambda$:
\begin{equation}
\delta F_1 = \delta F_1^{\mbox{\it\scriptsize fin}} + 
\delta F_1^{\mbox{\it\scriptsize IR}}\,\ell n\frac{s}{\lambda^2}
\end{equation}
with
\begin{eqnarray}              
\delta F_1^{\mbox{\it\scriptsize fin}}  & = &
\frac{\alpha\,\pi}{4\,\beta} - 
\frac{3}{2}\,\frac{\alpha}{\pi} + 
\frac{\alpha\,\pi}{4}\,\beta  +{\cal O}(\beta^2)\,, \\
\delta F_1^{\mbox{\it\scriptsize IR}}  & = &
-\,\frac{2}{3}\,\frac{\alpha}{\pi}\, \beta^2 +{\cal O}(\beta^4)\,. 
\end{eqnarray}
The leading term of $\delta F_1^{\mbox{\it\scriptsize fin}}$ is
proportional $\pi\alpha/\beta$ and exhibits the familiar Coulomb
singularity. Also the constant term and the term linear in $\beta$
are infrared finite. The infrared singular part of $F_1$ is strongly
suppressed at threshold $\propto \beta^2$, giving rise to a
$\beta^3$ contribution to the rate. The correction to the Pauli form
factor $\delta F_2$ is infrared finite and approaches a constant
value at threshold:
\begin{equation}
\delta F_2 = 
-\,\frac{1}{2}\,\frac{\alpha}{\pi}  +{\cal O}(\beta^2)\,.
\end{equation}
Real radiation, in contrast, vanishes as $\beta^3$ in the threshold
region, where two powers of $\beta$ result from the square of the
dipole matrix element, and one power of $\beta$ comes from phase
space. It  exhibits the same logarithmic dependence on the infrared
cutoff as the $F_1$ form factor and the same leading $\beta$
dependence as the infrared singular part of the virtual correction.
As a  consequence of the strong suppression $\propto \beta^3$ it can
be neglected in the threshold region, together with the
corresponding infrared divergent part of the form factor. The
angular distribution and, similarly, the correction to the total
cross section in the threshold region are therefore determined by
the the infrared finite parts of the form factors. To order $\alpha$
one thus finds for the coefficient describing the angular dependent
piece
\begin{equation} 
A = A_{\rm Born}\, \left( \, 1 + \frac{2}{2 - \beta^2}\, 
\frac{\alpha}{\pi} \,\right) \,.
\end{equation} 
As we shall show there are interesting modifications of the
anisotropy due to the running of the QCD coupling, and the
dependence of the renormalization scale on $\sqrt{s}$ and $|\vec
q\,|$ will be crucial.
  
The ${\cal O}(\alpha^2)$-QED corrections to the form factors,
induced by light fermion loops, have been calculated analytically in
\cite{HKT1}. In the threshold region one obtains
\begin{eqnarray}
F_1 & = &
1 + \frac{\alpha\,\pi}{4\,\beta}\,
\left[\, 1 + \bigg(\frac{\alpha}{\pi}\bigg)\,
\sum_{i = 1}^{n_f}\frac{1}{3}\,\left( \ell n\frac{s\,{{\beta}^2}}{m_i^2
} - \frac{8}{3} \right) \,  \right] \,  \nonumber\,\\ 
 & & \mbox{}\,\,\, - 
  \frac{3}{2}\,\frac{\alpha}{\pi}\,
   \left[\, 1 + \bigg(\frac{\alpha}{\pi}\bigg)\,
       \sum_{i = 1}^{n_f}
         \frac{1}{3}\,\left( \ell n\frac{s}{4\,m_i^2} - 
            \frac{1}{2} \right)  \,  \right] \,,\\
F_2 & = &
\quad \ \ \frac{\alpha\,\pi}{4\,\beta}\,\left[ \,
   \bigg(\frac{\alpha}{\pi}\bigg)\,\frac{n_f}{3}\,\right]
   \nonumber\,\\ 
 & & \mbox{}\,\,\, - 
  \frac{1}{2}\,\frac{\alpha}{\pi}\,
   \left[\, 1 + \bigg(\frac{\alpha}{\pi}\bigg)\,
       \sum_{i = 1}^{n_f}
         \frac{1}{3}\,\left( \ell n\frac{s}{4\,m_i^2} - 
            \frac{13}{6} \right)  \, \right] \,.
\end{eqnarray}
The calculation has been performed in the limit where  the mass of
the light virtual fermion $m_f$ is far smaller than $m$, a situation
appropriate for the subsequent translation to QCD. The factor $n_f$
is introduced to allow for several light fermions and, in our case,
to single out the fermion-induced terms. These formulae provide the
first step on the way to a full two-loop calculation in order
$\alpha^2$. As we shall see, the results require two conceptionally
different scales in the argument of the running coupling, a scale of
order $s$ from the hard virtual correction from transverse photons
and a soft scale of order $s\,\beta^2$ from the Coulomb
rescattering. Supplemented by the BLM prescription they even
determine the dominant two-loop gluon-induced terms in QCD.
   
The linear combination appearing in the denominator of $\widetilde A$ in
Eq.~(\ref{atil}) is thus given by
\begin{eqnarray}
F_1+F_2 & = & 
1 + \frac{\alpha\,\pi}{4\,\beta}\,
   \left[\, 1 + \bigg(\frac{\alpha}{\pi}\bigg)\,
      \sum_{i = 1}^{n_f}
         \frac{1}{3}\,\left( \ell n\frac{s\,{{\beta}^2}}{m_i^2
              } - \frac{5}{3} \right)  \,  \right] \,
   \nonumber\,\\ 
 & & \mbox{}\,\,\, - 
  2\,\frac{\alpha}{\pi}\,
   \left[\, 1 + \bigg(\frac{\alpha}{\pi}\bigg)\,
      \sum_{i = 1}^{n_f}
         \frac{1}{3}\,\left( \ell n\frac{s}{4\,m_i^2} - 
            \frac{11}{12} \right) \,  \right] \,.
\end{eqnarray}
The $n_f$ terms arise from the vacuum polarization insertions and
thus can be resummed into the QED running coupling:
\begin{equation}              
\alpha(Q^2) = \alpha\,\left[\,
 1 + \bigg(\frac{\alpha}{\pi}\bigg) 
      \sum_{i = 1}^{n_f}
         \frac{1}{3}\,\left( \ell n\frac{Q^2}{m_i^2} - 
            \frac{5}{3} \right) \,\right]\,.
\end{equation}
The constant 5/3 is the usual term in the Serber-Uehling vacuum
polarization $\Pi(Q^2)$ at large $Q^2$. This corresponds to the
usual QED scheme where $V(Q^2)=-\,4\pi\,\alpha(Q^2)/Q^2$ is the QED
potential for the scattering of heavy test charges. One thus obtains
\begin{eqnarray}
F_1 + F_2 & = & 
1 + \frac{\alpha(s\,\beta^2)\,\pi}{4\,\beta} -
2\,\frac{\alpha(s\,e^{3/4}/4)}{\pi} \nonumber \\
& \cong & \mbox{} \left(1- 2\,\frac{\alpha(s\,e^{3/4}/4)}{\pi}\right)\,
\left( 1 + \frac{\alpha(s\,\beta^2)\,\pi}{4\,\beta} \right)\,.
\end{eqnarray}
Two distinctly different correction factors arise. The first
originates from hard transverse photon exchange, with the scale set
by the short distance process; the second from the instantaneous
Coulomb potential. It is remarkable and non-trivial that the
non-logarithmic terms in the $\pi\alpha/\beta$ corrections are
absorbed if the relative momentum is adopted as the scale for the
coupling. Up to two loops the running coupling governing the Coulomb
singularity is thus identical to the running coupling in the
potential. This will provide an important guide for the application
of these results to QCD.

The proper resummation of the $1/\beta$ terms based on Sommerfeld's
rescattering formula then leads to
\begin{equation}
|F_1+F_2|^2 = 
\left(1- 4\,\frac{\alpha(m^2\,e^{3/4})}{\pi}\right)\,
\frac{x}{1-e^{-x}}
\end{equation}
with
\begin{equation}
x = \frac{\alpha(4\,m^2\,\beta^2)\,\pi}{\beta}\,.
\end{equation}
In a similar way one finds for the relevant combination in the
numerator of (\ref{atil})
\begin{equation}
|F_1|^2 \cong
\left(1- 3\,\frac{\alpha(m^2\,e^{7/6})}{\pi}\right)\,
\frac{x^\prime}{1-e^{-x^\prime}}
\end{equation}
with
\begin{equation}
x^\prime = \frac{\alpha(4\,m^2\,\beta^2/e)\,\pi}{\beta}\,.
\end{equation}
The $|F_2|^2$ term in the numerator can actually be ignored in the
present approximation. The scales of the effective coupling differ
between the numerator and denominator of (\ref{atil}): In particular
in the factor arising from Coulomb exchange the scale is
significantly smaller in the numerator than in the denominator. This
behavior is consistent with  qualitative considerations based on the
relative distances relevant for $S$- versus $P$-waves in the Coulomb
part. In the factor arising from hard photon exchange the scales are
quite comparable, with a slightly larger value in the numerator.
    
One thus arrives at the prediction in the context of QED
for the anisotropy which involves four scales: 
\begin{equation}
A = \frac{\widetilde A}{1-\widetilde A}\,,\qquad 
\widetilde A = \frac{\beta^2}{2}\,
 \frac{\left(1-3\,
\frac{\alpha(m^2 e^{7/6})}{\pi}\right)}
      {\left(1-4\,
      \frac{\alpha(m^2 e^{3/4})}{\pi}\right)}\,
 \frac{1-e^{-x}}{1-e^{-x^\prime}}\,
 \frac{\alpha(4\,m^2\,\beta^2/e)}{\alpha(4\,m^2\,\beta^2)}\,.
\end{equation}
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=12cm
\epsffile[120 300 450 530]{anitau.ps}
\vskip -3mm
\caption[]{{Ratio between the anisotropy $A$ and the Born 
prediction $A_{\rm Born}$ as function of $\beta$ for the process 
$e^+ e^- \to \tau^+ \tau^-$. Dashed curve: constant 
$\alpha$; solid curve: including the running of $\alpha$.}} 
\label{fig1} 
\end{center}
\end{figure}
\noindent 
To display the effects more clearly, the ratio of the anisotropy to
the Born prediction $A/A_{\rm Born}$ is shown in Fig.~\ref{fig1} for the
case of $\tau$ pair production. The dashed curve gives the
prediction for constant $\alpha_{\rm QED}$; the solid curve shows
the effect of the lepton vacuum polarization $\Pi(Q^2)$ in the QED
running coupling. The vacuum polarization affects the anisotropy for
small $\beta$ because two different scales appear in the $S$- and
$P$-wave Coulomb rescattering corrections. Away from threshold $A$
essentially measures the anomalous magnetic moment.

\vskip .4cm  
\noindent 
{\bf 3.~From QED to QCD}\\ 
\par
\noindent
The QED coupling $\alpha(Q^2)$ translates into the the QCD coupling 
$\alpha_{\mbox{\it\scriptsize V}}(Q^2)$, defined as the effective 
charge in the potential 
\begin{equation}
V(Q^2) = -\,\frac{4}{3}\,\frac{4\,\pi\,
\alpha_{\mbox{\it\scriptsize V}}(Q^2)}{Q^2}
\end{equation}
for the scattering of two heavy quarks in a color-singlet state. In
the BLM procedure all terms arising from the non-zero beta-function
are resummed into $\alpha_{\mbox{\it\scriptsize V}}(Q^2)$. For
example, all $n_f$-dependent coefficients vanish in the
$\pi\alpha/\beta$ terms if the scale of the relative momentum is
adopted. This is, in fact, a result expected on general grounds:
threshold physics is governed by the nonrelativistic instantaneous
potential. Below threshold, the potential leads to bound states,
above threshold it affects the cross section through final state
interactions. It is, therefore, natural to take for the QCD case the
coupling governing the QCD potential at the momentum scale involved
in the rescattering. To relate $\alpha_{\mbox{\it\scriptsize V}}$ to
$\alpha_{\overline{MS}}$, we use
\begin{equation}
\alpha_{\overline{MS}}(Q^2) = 
\alpha_{\mbox{\it\scriptsize V}}(e^{+5/3}\,Q^2)\,\left[ 
\,1 + 2\,\frac{\alpha_{\mbox{\it\scriptsize V}}}{\pi} + 
{\cal O}(\alpha_{\mbox{\it\scriptsize V}}^2)\,\right]\,.
\label{alphav}
\end{equation}
In a similar way, BLM scale-fixing is adopted for the correction
from hard gluon exchange. In the radiative correction, there still
remain ${\cal O}(\alpha_s^2)$ terms, identical to the radiative
corrections for the theory with a fixed coupling constant. With the
same scheme convention for the coupling as above, one arrives at
\begin{equation}
\widetilde A = \frac{\beta^2}{2}\,
 \frac{\left(1-4\,
\frac{\alpha_{\mbox{\it\tiny V}}(m^2 e^{7/6})}{\pi}\right)}
      {\left(1-\frac{16}{3}\,
      \frac{\alpha_{\mbox{\it\tiny V}}(m^2 e^{3/4})}{\pi}\right)}\,
 \frac{1-e^{-x_s}}{1-e^{-x_s^\prime}}\,
 \frac{\alpha_{\mbox{\it\tiny V}}(4\,m^2\,\beta^2/e)}
      {\alpha_{\mbox{\it\tiny V}}(4\,m^2\,\beta^2)}
\end{equation}
where
\begin{equation}
x_s = \frac{4\,\pi}{3}\,
 \frac{\alpha_{\mbox{\it\scriptsize V}}(4\,m^2\beta^2)}{\beta}\,,
 \qquad 
x_s^\prime = \frac{4\,\pi}{3}\,\frac{\alpha_{\mbox{\it\scriptsize V}}
 (4\,m^2\beta^2/e)}{\beta}\,.
\end{equation}
%
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=12cm
\epsffile[120 300 450 530]{anicbt.ps}
\vskip -3mm
\caption[]{{Anisotropy for charmed, bottom and top quark 
production as a function of $\beta$. Also shown is the Born
prediction. We have assumed the effective quark masses $m_c = 1.7$
GeV, $m_b = 5$ GeV and $m_t = 175$ GeV.}}
\label{fig2} 
\end{center}
\end{figure}
The anisotropy $A$ is plotted in Fig.~\ref{fig2} versus the velocity
$\beta$ in the range  $0.2 <\beta < 0.5$ for charmed, bottom, and
top quarks. For comparison, the tree level prediction is also shown.
For charmed quarks, only $\beta$ values above $0.4$ are admitted in
order to allow for the simultaneous production of $D \bar D$ and
$D^*\bar D$. The charm prediction is particularly sensitive to the
QCD parameters, since very low scales are accessible. Measurements
of the anisotropy for $e^+ e^- \to c \bar c$ thus have the potential
of determining $\alpha_{\mbox{\it\scriptsize V}}$ in the regime
where perturbation theory begins to fail.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=12cm
\epsffile[120 300 450 530]{anibalph.ps}\\
\leavevmode
\epsfxsize=12cm
\epsffile[120 300 450 530]{anicalph.ps}
\vskip -3mm
\caption[]{{Sensitivity of the anisotropy $A$ for (a) 
$e^+ e^- \to b\,\bar b$ and (b) $e^+ e^- \to c\,\bar c$ to changes in 
$\alpha_{\overline{MS}}(M^2_Z)$.}} 
\label{fig3} 
\end{center}
\end{figure}
    
The curves are based on an input value
$\alpha_{\overline{MS}}^{(n_f=5)}(M_Z^2) = 0.115$. We use the
two-loop beta-function to evolve $\alpha_{\overline{MS}}$ to lower
momenta and then used Eq.~(\ref{alphav}) to calculate
$\alpha_{\mbox{\it\scriptsize V}}(Q^2)$. To investigate the
sensitivity of the predictions for bottom quarks, the input value
for $\alpha_{\overline{MS}}(M_Z^2)$ has been varied by $\pm 0.008$
from the central value of 0.115. As demonstrated in Fig.~\ref{fig3}a
the variation of the anisotropy parameter amounts to about 10\%, and
could therefore be accessible experimentally. The charm predictions
(see Fig.~\ref{fig3}b) are even more sensitive.
   
\vspace{0.8cm}
\noindent 
{\bf 4.~Conclusions}\\ 
\par
\noindent
An important consequence of heavy quark kinematics is that the
production angle of a heavy hadron follows the direction of the
parent heavy quark. This applies not only at Born approximation, but
also after QCD corrections have been applied. In this paper we have
shown that the anisotropy $A(\beta^2)$ in the cm angular
distribution ${\rm d}\sigma(e^+ e^- \to Q\bar Q)/{\rm d}\Omega
\propto 1+A\,\cos^2\theta$ of heavy quarks produced near threshold
is sensitive to the QCD coupling $\alpha_{\mbox{\it\scriptsize
V}}(Q^2)$ at specific scales determined by the quark relative
momentum $p_{\rm cm} = \sqrt{s}\,\beta$. The coupling
$\alpha_{\mbox{\it\scriptsize V}}(Q^2)$ is the physical effective
charge defined through heavy quark scattering. The predictions
provide a connection between observables and thus are independent of
theoretical conventions.
    
An important feature of our analysis is the use of BLM scale-fixing,
in which all higher-order corrections associated with the
beta-function are resummed into the scale of the coupling. The
resulting scale for $\alpha_{\mbox{\it\scriptsize V}}(Q^2)$
corresponds to the mean gluon virtuality. In the case of the soft
rescattering corrections to the $S$-wave, the BLM scale is
$s\,\beta^2 = p^2_{\rm cm}$. One thus has sensitivity to the running
coupling over a range of momentum transfers within the same
experiment. The anisotropy measurement thus can provide a check on
other determinations of $\alpha_{\mbox{\it\scriptsize V}}(Q^2)$,
e.g. from heavy quark lattice gauge theory, or from the conversion
of $\alpha_{\overline{MS}}$ determinations to
$\alpha_{\mbox{\it\scriptsize V}}$ as given in Eq.~(\ref{alphav}).
    
Our analysis also shows that the running coupling appears within the
cross section with several different scales. This is particularly
apparent at low $\beta$ where the physical origin of the ${\cal
O}(\alpha_s)$ corrections can be traced to gluons with different
polarization and virtuality.
    
In principle, the anisotropy of $\tau$ pairs produced in $e^+e^-
\rightarrow \tau^+\tau^-$ could be used to measure the Pauli form
factor $F_2(s)$ near threshold $s\
{\scriptscriptstyle\stackrel{\scriptscriptstyle >}{ \sim}}\ 4m^2_\tau$.
A highly precise measurement of the anisotropy thus could provide a
measurement of a fundamental parameter of the $\tau$ lepton, its
timelike anomalous magnetic moment.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{sloppy}
\begin{raggedright}
\def\app#1#2#3{{\it Act. Phys. Pol. }{\bf B #1} (#2) #3}
\def\apa#1#2#3{{\it Act. Phys. Austr.}{\bf #1} (#2) #3}
\def\lhc{Proc. LHC Workshop, CERN 90-10}
\def\npb#1#2#3{{\it Nucl. Phys. }{\bf B #1} (#2) #3}
\def\plb#1#2#3{{\it Phys. Lett. }{\bf B #1} (#2) #3}
\def\prd#1#2#3{{\it Phys. Rev. }{\bf D #1} (#2) #3}
\def\pR#1#2#3{{\it Phys. Rev. }{\bf #1} (#2) #3}
\def\prl#1#2#3{{\it Phys. Rev. Lett. }{\bf #1} (#2) #3}
\def\prc#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3}
\def\cpc#1#2#3{{\it Comp. Phys. Commun. }{\bf #1} (#2) #3}
\def\nim#1#2#3{{\it Nucl. Inst. Meth. }{\bf #1} (#2) #3}
\def\pr#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3}
\def\sovnp#1#2#3{{\it Sov. J. Nucl. Phys. }{\bf #1} (#2) #3}
\def\jl#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3}
\def\jet#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3}
\def\zpc#1#2#3{{\it Z. Phys. }{\bf C #1} (#2) #3}
\def\ptp#1#2#3{{\it Prog.~Theor.~Phys.~}{\bf #1} (#2) #3}
\def\nca#1#2#3{{\it Nouvo~Cim.~}{\bf #1A} (#2) #3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\elevenbf\noindent  Acknowledgments \hfil}
\vglue 0.4cm
One of the Authors (JHK) would like to thank the SLAC Theory Group
for hospitality and the Volkswagen-Stiftung Grant I/70452 for
generous support.
\bigskip

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\end{raggedright}
\end{sloppy}
\end{document}
 
 

