%Title: The Perturbative QCD Potential and the tt-bar Threshold
%Author: M. Jezabek, J.H. Kuehn, M. Peter, Y. Sumino and T. Teubner
%Published:     Proc. <EM>29th ICHEP' </EM> Vancouver, CA, 23-29 July 1998, edited by A. Astbury, D. Axen, Jacob Robinson (World Scientific, Singapore, 1999) Vol.2, 1613-1617. 
%UNIX --- UPDATED ON 14/8/97 
%====================================================================%
%                                                                    %
%                  ltwol.tex     27-Feb-1995                         %
%                                                                    %
%   Modified to ltwol2e.tex by M. Comyn 12-June-1998 for ICHEP'98    %
%                                                                    %
% This latex file rewritten from various sources for use in the      %
% preparation of the (larger) two-column proceedings Volume, latest  %
% version by Susan Hezlet with acknowledgements to Lukas Nellen.     %
% Some changes are due to David Cassel.                              %
%                                                                    %
% Changes by M. Comyn for use by ICHEP'98:                           %
%                                                                    %
% Preamble changed to LaTeX2e format.                                %
% Added (but commented out) epsfig and graphics packages, as well as %
% existing psfig package, for use with embedded figures.             %
% Additional minor typing changes.                                   %
%                                                                    %
%====================================================================%

%\documentstyle[ltwol]{article}

\documentclass{article}
\usepackage{ltwol2e}

\usepackage{epsf}
%\usepackage{psfig}
%\usepackage{epsfig}
%\usepackage{graphics}

\font\eightrm=cmr8

%\input{psfig}

\arraycolsep1.5pt 

% A useful Journal macro
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}

% Some useful journal names
\def\NCA{\em Nuovo Cimento}
\def\NIM{\em Nucl. Instrum. Methods}
\def\NIMA{{\em Nucl. Instrum. Methods} A}
\def\NPB{{\em Nucl. Phys.} B}
\def\PLB{{\em Phys. Lett.}  B}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.} D}
\def\ZPC{{\em Z. Phys.} C}

% Some other macros used in the sample text
\def\st{\scriptstyle}
\def\sst{\scriptscriptstyle}
\def\mco{\multicolumn}
\def\epp{\epsilon^{\prime}}
\def\vep{\varepsilon}
\def\ra{\rightarrow}
\def\ppg{\pi^+\pi^-\gamma}
\def\vp{{\bf p}}
\def\ko{K^0}
\def\kb{\bar{K^0}}
\def\al{\alpha}
\def\ab{\bar{\alpha}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\CPbar{\hbox{{\rm CP}\hskip-1.80em{/}}}%temp replacemt due to no font

\newcommand{\simgt}{\hbox{ \raise3pt\hbox to 0pt{$>$}\raise-3pt\hbox{$\sim$} }}
\newcommand{\simlt}{\hbox{ \raise3pt\hbox to 0pt{$<$}\raise-3pt\hbox{$\sim$} }}
\def\rms#1{_{\rm #1}}
\def\qcut{\rm q_{cut}}

\bibliographystyle{unsrt}    %for BibTeX - sorted numerical labels

%\input psfig

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                      BEGINNING OF TEXT                           
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\title{
THE PERTURBATIVE QCD POTENTIAL AND THE $t\bar{t}$ THRESHOLD$^*$
}

\author{
M.~JE\.ZABEK$^{a)}$, J.~H.~K\"UHN$^{b)}$, M.~PETER$^{c)}$,
  Y.~SUMINO$^{b) \, **}$\\ and
  T.~TEUBNER$^{d)}$
}
\address{
$a)$ Institute of Nuclear Physics, Cracow, Poland, and\\
Department of Field Theory and Particle Physics, University of 
      Silesia, Katowice, Poland\\ 
  $b)$ Institut f\"ur Theoretische Teilchenphysik,
       Universit\"at Karlsruhe, Germany \\ 
  $c)$ Institut f\"ur Theoretische Physik,
       Universit\"at Heidelberg, Germany \\ 
  $d)$ Deutsches Elektronen-Synchrotron (DESY), Hamburg, Germany
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\twocolumn[\maketitle\abstracts{ 
We include the full second-order corrections to the
static QCD potential in the analysis of the $t\bar{t}$
threshold cross section.
There is an unexpectedly large difference 
between the QCD potential improved by the renormalization-group equation 
in momentum space and the potential
improved by the renormalization-group equation in coordinate space.
This difference remains even at a fairly short distance
$1/r \simeq 100$~GeV and its origin can be understood within perturbative QCD.
We scrutinize the theoretical uncertainties of the QCD potential
in relation to the $t\bar{t}$ threshold cross section.
In particular there exists a theoretical uncertainty which limits our present 
theoretical accuracy of the $t\bar{t}$ 
threshold cross section at the peak to be
$\delta \sigma_{\rm peak}/\sigma_{\rm peak} \simgt 6\%$
within perturbative QCD.
}]

\footnotetext
{$^*$Presented by J.\ H.\ K\"uhn.}
\footnotetext
{$^{**}$On leave of absence from Tohoku Univ., Sendai, Japan.}

In this paper, based on Ref.~\cite{prd}, 
we report on our present theoretical understanding of
the $t\bar{t}$ total cross section near the threshold.
Up to now, all the ${\cal O}(\alpha_s)$ corrections (also 
leading logarithms) have been included in
the calculations of various cross sections near threshold.
In order to take into account the QCD binding effects properly
in the cross sections,
we have to systematically rearrange the perturbative expansion near threshold.
Namely, we first resum all the leading Coulomb singularities 
$\sim (\alpha_s/\beta)^n$, take the result as the leading order contribution, 
and then calculate higher order corrections, which are essentially 
resummations
of the terms
$\sim \alpha_s^{n+1}/\beta^n$, $\alpha_s^{n+2}/\beta^n$, $\ldots$
It is also important to resum large logarithms arising from the large
scale difference involved in the calculation~\cite{sp}.
This is achieved by (first) calculating the Green function of the 
non-relativistic Schr\"{o}dinger equation with the QCD potential~\cite{fk,sp}.
Conventionally 
both the coordinate-space approach developed in Refs.~\cite{sp,sfhmn}
and the momentum-space approach developed in Refs.~\cite{JKT92,JT93}
have been used in solving the equation
by different groups independently.
It has recently been found \cite{ps} that 
there are discrepancies in the results obtained
from the two approaches reflecting the difference in the construction of 
the potentials in both spaces.
It was argued that the differences are formally of ${\cal O}(\alpha_s^2)$ but
their size turns out to be non-negligible.

Quite recently there has been considerable progress in the 
theoretical calculations
of the second-order corrections to the cross section at threshold and
the Coulombic bound-state problem.
New contributions have been calculated analytically~\cite{hoang,hl} and 
numerically~\cite{adkins} for QED bound-states.
Very important steps have been accomplished in QCD as well.
The full second-order correction to the static QCD potential was computed
in~\cite{MP97}.
Also, the ${\cal O}(\alpha_s^2)$ total cross section is known now~\cite{cm} 
in the
region $\alpha_s \ll \beta \ll 1$ as a series expansion in $\beta$.
All these results have to be included in the calculation of
the full ${\cal O}(\alpha_s^2)$
corrections to the threshold cross section, 
which has just been completed
(as far as the production process of top quarks are 
concerned)~\cite{ht}.
The full second-order corrections turned out to be anomalously
large, which may suggest a poor convergence of the perturbative
QCD in the $t\bar{t}$ threshold region.

\begin{figure*}[htb]\begin{center}
 \epsfxsize 8cm \mbox{\epsfbox{fig1.eps}}
 \end{center}
 \caption[]{\label{fig1}Comparison of the total cross sections 
  (normalized to $R$) calculated
  from the different potentials:
  $V_{\rm JKPT}$ (solid), $V_{\rm SFHMN}$ (dashed), and $V_{\rm new}$ (dotted
  line).
  We set $\alpha\rms{\overline{MS}}(M_Z^2) = 0.118$,
  $m_t = 175$~GeV, and $\Gamma_t = 1.421$~GeV.
  }
\end{figure*}

In this paper, we incorporate the full ${\cal O}(\alpha_s^3)$ corrections
(the second-order corrections to the leading contribution)
to the static QCD potential into our analyses. 
In principle this is a step towards an improvement of the theoretical 
precision in our analysis of the $t\bar{t}$ threshold cross 
section.
Then we scrutinize the problem of the
difference between the momentum-space and the coordinate-space potentials.
Contrary to our expectation, the inclusion of the above corrections
does not reduce the difference of the cross sections significantly, and
there still remains a non-negligible deviation.
We find that there is a theoretical uncertainty within
perturbative QCD which limits our present-day
theoretical accuracy of the threshold cross section.

Let us now briefly explain the construction of our potentials in momentum space
and in coordinate space, respectively.
More detailed descriptions including formulas can be found in \cite{prd}.
The large-momentum part of the momentum-space potential $V_{\rm JKPT}(q)$
is determined as follows.
First the potential has been calculated up to ${\cal O}(\alpha_s^3)$
in a fixed-order calculation.
The result is then improved using the
three-loop renormalization group equation in momentum space.
At low momentum, the potential is continued smoothly to a
Richardson-like potential.
On the other hand, the short-distance part of the
coordinate-space potential $V_{\rm SFHMN}(r)$ 
is calculated by taking the Fourier transform of the fixed-order 
perturbative potential
in momentum space, and
then is improved using the three-loop renormalization group
equation in coordinate space.
At long distance, the potential is continued smoothly
to a phenomenological ansatz.
Thus, it is important to note that the two potentials are {\it not}
the Fourier transforms of each other even in the large-momentum or
short-distance region.
They agree
only up to the next-to-next-to-leading logarithmic terms of 
the series expansion in a fixed $\overline{\rm MS}$ coupling.
The difference begins
with the non-logarithmic term in the three-loop fixed-order correction.

In Fig.~\ref{fig1} we show a comparison of the total cross sections 
(normalized to $R$) 
calculated from $V_{\rm JKPT}$ (solid) and from $V_{\rm SFHMN}$ (dashed line),
without any weak or hard-gluon corrections:
\bea
R = \frac{4}{\pi m_t^2}  \int_0^\infty dp \, p^2 |G(E,p)|^2  \, \Gamma_t .
\eea
For the physical parameters we used
$\alpha\rms{\overline{MS}}(M_Z^2) = 0.118$,
$m_t = 175$~GeV, and $\Gamma_t = 1.421$~GeV.
We find that the two cross sections differ by 7.8\% at the 
peaks and by 2.2\% 
at $E = 5$~GeV.\footnote{
In this paper we are not concerned with those differences of the
cross sections which can be absorbed into an additive constant to
the potential $V(r)$, or equivalently, into a redefinition of the top
quark mass.
The theoretical uncertainty in the pole mass for our problem
is discussed in~\cite{jps}.
}
Since the difference of the cross sections calculated from the
next-to-leading order potentials is 8.6\% at the peak and 2.4\%
at $E = 5$~GeV for the same value of $\alpha_s(M_Z^2)$,
%(see Figs.~\ref{scomp} and \ref{compsfhmn}),
the cross sections have come closer only slightly after the inclusion
of the second-order correction to the potential.
The remaining difference is much larger than what one would expect from
an ${\cal O}(\alpha_s^3)$ correction relative to the leading order,
which is not fully included in our analyses, even if we
take into account the high sensitivity to the coupling~\cite{sp}, 
$\sigma_{\rm peak} \propto \alpha_s^2$.
The purpose of this paper is to understand the 
origin of this unexpectedly large difference. 

In order to compare the asymptotic behavior of the potentials 
more clearly,
we plot in Fig.~\ref{eff}(a) the coordinate-space effective couplings 
defined by
\bea
\bar{\alpha}_{\rm V} ( 1/r ) 
= \left( - C_F / r \right)^{-1} V(r) 
\label{cooreff}
\eea
for $V_{\rm JKPT}(r)$ and $V_{\rm SFHMN}(r)$ as
solid and dashed lines, respectively.
\begin{figure*}
\begin{center}
  \leavevmode
  \epsfxsize=8cm 
  \mbox{\epsffile{fig3a.eps}} 
  \epsfxsize=8cm 
  \mbox{\epsffile{fig3b.eps}}\\
  (a)
\hspace{8cm}
  (b)
\end{center}
  \caption[]{
        \label{eff}
(a) Comparison of the coordinate-space effective charges defined
    from $V_{\rm JKPT}$ (solid) and from $V_{\rm SFHMN}$ (dashed line).
(b) Comparison of the coordinate-space effective charges defined
    from the various terms of Eq.~(\ref{rel}).
    See the text for the description of each curve.
}
\end{figure*}
Contrary to our expectation,
the difference of the couplings exceeds 3\% even at very 
short distances, $1/r \simeq 100$~GeV.

Naturally the question arises: Why is there such a large discrepancy
between the potential constructed in momentum space and that
constructed in coordinate space?
To answer this question, let us examine a
relation connecting the effective coupling
in coordinate space, defined by Eq.~(\ref{cooreff}),
and the effective coupling in 
momentum space, defined from the momentum-space potential as
\bea
\alpha_{\rm V} ( q ) 
= \left( - 4 \pi C_F / q^2 \right)^{-1} V(q) .
\eea
The relation is derived from the renormalization group equation 
of $\alpha_{\rm V}(q)$ and 
exact to all orders. 
In
the asymptotic region where the couplings are small, 
it can be given in the form of an asymptotic series~\cite{jps},
which reads numerically
\bea
\bar{\alpha}_{\rm V}(1/r) = 
\alpha_{\rm V} + 1.225 \, \alpha_{\rm V}^3 
+ 5.596 \, \alpha_{\rm V}^4
+ 32.202 \, \alpha_{\rm V}^5 + \ldots
\nonumber \\
\label{rel}
\eea
for $n_f = 5$.
On the right-hand-side, 
$\alpha_{\rm V} = \alpha_{\rm V}(q = e^{-\gamma_E}/r)$.
All terms which are
written explicitly are determined from the known coefficients 
of the $\beta$ function, $\beta_0^V$, $\beta_1^V$, and $\beta_2^V$.
At present, we can use the above relation consistently only at 
$O(\alpha_{\rm V}^3)$ because we know the effective couplings
only up to the next-to-next-to-leading order corrections in
perturbative QCD,
i.e. we know the relation between 
$\alpha_{\rm V}$ and $\alpha\rms{\overline{MS}}$ only up to
${\cal O}(\alpha\rms{\overline{MS}}^3)$.
Due to this limitation, essentially, the effective coupling 
$\bar{\alpha}_{\rm V}$ defined from $V_{\rm SFHMN}$ 
is the right-hand-side of the above
equation truncated at the $O(\alpha_{\rm V}^3)$ term, 
while $\bar{\alpha}_{\rm V}$ defined from $V_{\rm JKPT}$ 
is the right-hand-side including all terms.
Numerically, the $O(\alpha_{\rm V}^4)$ term and the
$O(\alpha_{\rm V}^5)$ term
contribute as +1.4\% and +1.1\% corrections, respectively, for 
$\alpha_{\rm V} = 0.1379$ (corresponding to $1/r = 100$~GeV).
Therefore, these higher order
terms indeed explain the difference of the effective couplings
at small $r$.
Fig.~\ref{eff}(b) shows several curves derived from the above relation:
\begin{enumerate}
\item
The solid line is $\bar{\alpha}_{\rm V}(1/r)$ defined from
$V_{\rm JKPT}$.
\item
The dashed curve is 
$\alpha_{\rm V} + 1.225 \, \alpha_{\rm V}^3$, where
$\alpha_{\rm V} = \alpha_{\rm V}(q = e^{-\gamma_E}/r)$
is calculated using the perturbative prediction in momentum space.
This curve is essentially the same as 
$\bar{\alpha}_{\rm V}(1/r)$ defined from
$V_{\rm SFHMN}(r)$, since it is the next-to-next-to-leading order 
perturbative prediction for the coordinate-space coupling at
short distances.
\item
The dotted curve includes the next correction, 
$5.596 \, \alpha_{\rm V}^4$, which is in fact even larger than the 
${\cal O}(\alpha_{\rm V}^3)$ term below $1/r \sim 30$~GeV.
\item
The dash-dotted curve includes the ${\cal O}(\alpha_{\rm V}^5)$ term.
\end{enumerate}
We observe that the agreement of both sides of 
Eq.~(\ref{rel}) becomes better as
we include more terms at small $r$, while it becomes worse at large $r$ on
account of the asymptoticness of the series.
From the purely perturbative point of view, the discrepancy between our 
two potentials, $V_{\rm JKPT}$ and $V_{\rm SFHMN}$, in the asymptotic
region thus 
seems real, an indication of large higher order corrections.
When the third-order correction to the potential will be computed
in terms of $\alpha\rms{\overline{MS}}$
in the future, the ${\cal O}(\alpha_V^4)$ term will be treated consistently
and the difference will reduce by 1.4\% at $1/r \simeq 100$~GeV.

We may consider this difference of $\bar{\alpha}_{\rm V}(1/r)$ 
at short-distances as an estimate of
the higher order corrections on the basis of the following observations.
First, the two potentials are equal up to the 
next-to-next-to-leading order, and
there seems to be no reason {\it a priori} for considering one of the two
to be more favorable theoretically.
Secondly, if we apply the same method 
(the relation between $\bar{\alpha}_{\rm V}$ 
and $\alpha_{\rm V}$) to estimate the size of the already known 
${\cal O}(\alpha_s^3)$ correction, we obtain $\pi^2 \beta_0^2/3 = 193.4$,
which turns out to be a slight under-estimate of the true 
correction~\cite{MP97} $a_2 = 333.5$ ($n_f=5$).

Moreover,
the above 3\% uncertainty of $\bar{\alpha}_{\rm V}(1/r)$ at 
$1/r \simeq 100$~GeV
provides a certain criterion for the present theoretical uncertainty
of the $t\bar{t}$ cross section.
In fact, it would already limit the theoretical accuracy of 
$\bar{\alpha}_{\rm V}(1/r)$ at longer distances to be
not better than 3\%.
If we combine this with a naive estimate
$\sigma_{\rm peak} \propto \bar{\alpha}_{\rm V}^2$,
we expect a theoretical uncertainty of the peak cross section to be
$\delta \sigma_{\rm peak}/\sigma_{\rm peak} \simgt 6\%$.
Therefore, the large discrepancy of the cross sections which we have
seen turns out to be quite consistent with this estimated uncertainty.

One is tempted to include one more term of the series~(\ref{rel})
to define a (new) coordinate-space potential
despite our ignorance of the corresponding terms in the relation
between $\alpha_{\rm V}$ and $\alpha\rms{\overline{MS}}$, since 
this would apparently reduce the difference between the two 
effective couplings. 
In fact we did this exercise, but (to our surprise) it did not bring
the cross section closer to the one calculated
from the momentum-space potential $V_{\rm JKPT}$
in the peak region.  
This cross section calculated from the potential $V_{\rm new}(r)$,
which incorporates the ${\cal O}(\alpha_{\rm V}^4)$ term
of Eq.~(\ref{rel}),
is shown as a dotted curve in Fig.~\ref{fig1}.
This fact indicates that we are no longer able to improve the agreement
of the cross sections by including even higher order terms,
as we are confronting the problem of asymptoticness of the series.

Some indications can be obtained by looking into the nature of 
the perturbative expansion of each potential.
Within our present knowledge of the static QCD potential,
the perturbative series looks more convergent
for the momentum-space potential than for the coordinate-space
potential.
To see this, one may compare 
the $\beta$ functions of the effective
couplings (the V-scheme couplings) in both spaces~\cite{MP97}.
Numerically, the first three terms in the perturbative expansion 
read
\begin{itemize}
\item
(momentum-space coupling)
\bea
&&\mu^2 \frac{d \alpha_{\rm V}}{d \mu^2}
= - 0.6101 \, \alpha_{\rm V}^2
    - 0.2449 \, \alpha_{\rm V}^3
    - 1.198 \, \alpha_{\rm V}^4 + \ldots 
\nonumber \\
\eea
\item
(coordinate-space coupling)
\bea
&&\mu^2 \frac{d \bar{\alpha}_{\rm V}}{d \mu^2}
= - 0.6101 \, \bar{\alpha}_{\rm V}^2
    - 0.2449 \, \bar{\alpha}_{\rm V}^3
    - 1.945 \, \bar{\alpha}_{\rm V}^4 + \ldots 
\nonumber \\
\eea
\end{itemize}
for $n_f = 5$.
The first two coefficients are universal.  
The third coefficient depends on the scheme (the definition) of
the coupling.
As the third coefficients for the V-scheme couplings are quite
large, the third term of the $\beta$ function is comparable to
the second term already for $\alpha_V = 0.20$ and for
$\bar{\alpha}_V=0.13$, respectively.
The difference of the third coefficients between momentum space
and coordinate space originates from the
$\pi^2\beta_0^2/3$ term. 
Although the magnitude of the third coefficients is of the same order, 
in practice it makes a certain difference whether an apparent 
convergence is lost at
$\alpha_V = 0.20$ or $\bar{\alpha}_V = 0.13$
because there is a large scale difference between the two values.
This indicates a worse convergence in coordinate space than
in momentum space.

It is interesting to examine the level of uncertainties within
the momentum-space approach or the coordinate-space 
approach by itself.
\begin{figure*}
 \begin{center}
 \epsfxsize=8cm \mbox{\epsfbox{sigmaasrun1.eps}}
 \end{center}
 \caption[]{\label{scalevar}
Comparison of the cross sections for different choices of the scale.
The upper three curves are for the momentum-space approach:
$\mu = q$ (solid), $\mu = \sqrt{2}q$ (dotted), and $\mu = q/\sqrt{2}$
(dashed line).
The lower three curve are for the coordinate-space approach:
$\mu = \mu_2 = {\rm exp}(-\gamma_E)/r$ (solid), 
$\mu = \sqrt{2}\mu_2$ (dotted), 
and $\mu = \mu_2/\sqrt{2}$
(dashed line).
}
\end{figure*}
Fig.~\ref{scalevar}
shows how the cross section changes when we vary the 
scale by a factor of 2 in each approach: from
$\mu = q/\sqrt{2}$ to $\mu = \sqrt{2}q$ in Eq.~(28) of \cite{MP97} in
the momentum-space approach (upper three curves), and
from $\mu = \mu_2/\sqrt{2}$ to 
$\mu =\sqrt{2}\mu_2$ in Eq.~(44) of \cite{MP97} in the
coordinate-space approach (lower three curves).
For the momentum-space approach,
the variation of the cross section is 2.2\% at the peak
and around 0.6\% for c.m.\ energies above threshold.
Meanwhile in the coordinate-space approach,
the variation of the cross section amounts to 0.7\% at the peak
and 0.9\% at larger c.m.\ energies.
These results may be regarded as an internal consistency check for each
approach.

So far we have examined the difference between the momentum-space
potential and the coordinate-space potential in detail, and we have
taken this difference as an estimate of the theoretical uncertainty
of the QCD potential.
This estimation may, however, be somewhat misleading.
One might well argue that the difference is an artifact of our 
inadequate use of the
perturbative expansion in describing the potentials related by
Fourier transformation.
For illustration, let us consider a hypothetical case
where we know $\alpha_{\rm V}(q)$ exactly.
In this case, if we calculate $\bar{\alpha}_{\rm V}(1/r)$
via (numerical) Fourier transformation of the momentum-space potential, 
in principle we can calculate $\bar{\alpha}_{\rm V}(1/r)$ to any desired
accuracy by investing more time.
On the other hand, if we calculate $\bar{\alpha}_{\rm V}(1/r)$ using
the series on the right-hand-side of Eq.~(\ref{rel}) naively,
there is a limitation in the achievable accuracy 
because the series is only asymptotic.
Certainly such a limited accuracy does not reflect any theoretical uncertainty
of $\bar{\alpha}_{\rm V}(1/r)$.
This nature should not be confused with our claim.
We claim that there is a limitation within perturbative QCD in relating
$\alpha_{\rm V}(q)$ or $\bar{\alpha}_{\rm V}(1/r)$ to 
$\alpha\rms{\overline{MS}}(M_Z^2)$,
and estimate theoretical uncertainties in this relation using the
difference of the potentials in the two spaces.

Let us comment on the relation between this work and the
work~\cite{ht} which has been completed very recently.
Ref.~\cite{ht} presents a fixed-order calculation (without log resummations),
which includes the full second-order corrections 
to the $t\bar{t}$ threshold cross sections.
While we include corresponding 
corrections only to the static QCD potential in our analyses, we also employ
the renormalization-group improvement and thus resum large logarithms
in the potential.
In this sense the two works are complementary to each other.
Both works give the common conclusion that the second-order corrections
to the $t\bar{t}$ threshold cross section are large and may indicate
a poor convergence of perturbative QCD, although they are
based on qualitatively different arguments.
The full set of 
the fixed-order ${\cal O}(\alpha_s^2)$ corrections to the cross section near
threshold~\cite{ht} are larger in size and even more scale
dependent than the corrections to the potential alone.
The theoretical uncertainty may therefore even be larger than indicated by the
study in our paper.

\section*{\bf Summary}
\begin{itemize}
\item
There is a difference between the 
potential constructed
in momentum space and that constructed in coordinate space even at
a fairly short-distance, $1/r \sim 100$~GeV.
The difference can be understood within the framework of perturbative QCD.
We already know that there is a large correction at ${\cal O}(\alpha_s^4)$
in the relation between the two potentials, although
a consistent treatment is not possible
until the full ${\cal O}(\alpha_s^4)$ corrections to the
QCD potential are calculated.

\item
The above difference at short distances would limit the theoretical
accuracy of the QCD potential at longer distances and thus
provides a criterion for our present theoretical uncertainty
of the $t\bar{t}$ cross section, 
$\delta \sigma_{\rm peak}/\sigma_{\rm peak} \simgt 6\%$.

\item
In addition, it seems that
we are confronting the problem of the asymptoticness of the
perturbative series in the calculation of 
the $t\bar{t}$ cross section, as the 
top quarks do not probe a region which is sufficiently deep
in the potential.
We may not be able to improve our 
theoretical precision
even if the higher order corrections are calculated in perturbative QCD.

\item
We may, however, discuss which of the two approaches gives a more
favorable result theoretically.
Up to the second-order corrections, the perturbative series looks 
better convergent
for the momentum-space potential than for the coordinate-space potential.

\end{itemize}


\def\app#1#2#3{{\it Acta~Phys.~Polonica~}{\bf B #1} (#2) #3}
\def\apa#1#2#3{{\it Acta Physica Austriaca~}{\bf#1} (#2) #3}
\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\sovnp#1#2#3{{\it Sov.~J.~Nucl.~Phys.~}{\bf #1} (#2) #3}
\def\yadfiz#1#2#3{{\it Yad.~Fiz.~}{\bf #1} (#2) #3}
\def\jetp#1#2#3{{\it JETP~Lett.~}{\bf #1} (#2) #3}
\def\zpc#1#2#3{{\it Z.~Phys.~}{\bf C #1} (#2) #3}

\begin{thebibliography}{99}

\bibitem{prd} 
  M. Je\.zabek, J.H. K\"uhn, M.~Peter, Y.~Sumino and T.~Teubner, 
{\it Phys.~Rev.} {\bf D58} (1998) 014006.

\bibitem{sp} M.J. Strassler and M.E. Peskin, 
{\it Phys.~Rev.} {\bf D43} (1991) 1500.

\bibitem{fk} V.S. Fadin and V.A. Khoze, {\it JETP~Lett.}
{\bf 46} (1987) 525;
{\it Sov.~J.~Nucl.~Phys.} {\bf 48} (1988) 309.

\bibitem{sfhmn} Y. Sumino, K. Fujii, K. Hagiwara, H. Murayama and
C.-K. Ng, {\it Phys.~Rev.} {\bf D47} (1993) 56.

\bibitem{JKT92}
  M. Je\.zabek, J.H. K\"uhn and T. Teubner, 
  {\it Z. Phys.} {\bf C56} (1992) 653.

\bibitem{JT93}
  M. Je\.zabek and T. Teubner, {\it Z. Phys.} {\bf C59} (1993) 669.

\bibitem{ps}
  M. Peter and Y. Sumino, University of Karlsruhe Report 
 No.~TTP97-27, hep-ph/9708223, to appear in
  {\it Phys. Rev.} {\bf D}.

\bibitem{hoang}
A.H.~Hoang, {\it Phys.~Rev.} {\bf D56} (1997) 5851.

\bibitem{hl}
A.H.~Hoang, P.~Labelle and S.M.~Zebarjad, {\it Phys. Rev. Lett.} {\bf 79} 
 (1997) 3387.

\bibitem{adkins} 
G.~Adkins, R.N.~Fall and P.M.~Mitrikov, 
{\it Phys. Rev. Lett.} {\bf 79} (1997) 3383.

\bibitem{MP97}
  M. Peter, {\it Phys. Rev. Lett.} {\bf 78} (1997) 602;
            {\it Nucl. Phys.} {\bf B501} (1997) 471.

\bibitem{cm}
    A. Czarnecki and  K. Melnikov, University of Karlsruhe Report 
 No.~TTP97-54, hep-ph/9712222, and references therein. 

\bibitem{ht}
A.~Hoang and T.~Teubner, University of California, San Diego Report No.~UCSD/PTH 98-01, DESY 98-008, hep-ph/9801397.

\bibitem{jps}
  M. Je\.zabek, M. Peter and Y. Sumino, 
{\it Phys. Lett.} {\bf B428} (1998) 352.

\bibitem{Rich79}
  J.L. Richardson, {\it Phys. Lett.} {\bf B82} (1979) 272.

\bibitem{jezabek}
  M.~Je\.zabek, 
  {\it Nucl. Phys.} (Proc.\ Suppl.) {\bf 37B} (1994) 197, and
  reference therein.

\end{thebibliography}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                     END OF LTWOL2e.TEX                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




