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%% section 4 Top quark production at the TEVATRON [slacpub7176004 in slacpub7176004: slacpub7176005]
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\section{\usemenu{slacpub7176::context::slacpub7176004}{Top quark production at the TEVATRON}}\label{section::slacpub7176004}
In the light of our discussion, let us consider recent results for the
resummed top quark production
cross section, which we summarize
in Table~1. We concentrate on the comparison of two new calculations
\cite{13,14}. Both assumed $m_t =175$ GeV and used the same parametrisations
for the structure functions. Hence the difference is entirely due to
different resummation prescriptions. The difference in central values
is of order 15\%, compared to $\sim 10\%$ renormalization
scale dependence and $\sim 5\%$ due to uncertainty in the structure
functions. Apparently, resummation causes the largest ambiguity.
Note that the resummed cross section of Ref.~14 practically coincides
with the strict $O(\alpha^2)$ result. Thus, in Ref.~14 the resummation
of $\ln N$ terms has a negligible effect,
while the resummation in Ref.~13 produces a $1015\%$ enhancement.
Since both procedures sum all leading logarithms
(in the sense of Eq.~\docLink{slacpub7176001.tcx}[LLA]{2}),
the difference is entirely due to terms with
less powers of logarithms which are beyond the accuracy
of the resummation in the strict sence. Unless we
can prefer one particular resummation procedure, the difference
would have to be considered as the present theoretical uncertainty.
Our discussion of DrellYan production suggests the criterium
that resummation procedures should not introduce power corrections
(factorial divergence in large orders) which are not already
present in finite order approximations. From this point of view,
we are led to prefer the prescription used in Ref.~14, which starts
from Eq.~\docLink{slacpub7176003.tcx}[form2]{15}.
\begin{table}[t]
\caption{Resummed cross section for the $t\bar t$ production at the
TEVATRON. $m_t=175\,$GeV, MRSA' parton distributions for the
central value quoted. \label{tab:exp}}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{ccc}
\hline
& & \\
Ref. & $\sigma_{t\bar t}$, pb & Uncertainty
\\ & & \\ \hline
LSN \cite{15} & 4.95 & 4.53  5.66 \\
BC \cite{13} & 5.32 & 4.93  5.40 \\
CMNT \cite{14} & 4.75 & 4.25  5.00 \\
\hline
\end{tabular}
\end{center}
\end{table}
The major numerical difference between Ref.~13 and Ref.~14
comes from a different procedure to implement the inverse Mellin
transformation from moment space to momentum space.
The subtle problem here is to which extent one can avoid contributions
of very large moments
$N\ge Q/\Lambda_{QCD}$, which strictly speaking can not be
treated by shortdistance methods.
This problem is somewhat similar
to the problem of analytically
continuing perturbative QCD predictions from Euclidian in Minkowski
space, relevant, for example, in connection with the $\tau$lepton
hadronic width.
The particular way of performing the analytic continuation
becomes important when one uses a {\em resummed} coupling constant,
and the guiding principle
proves to be to avoid the region $Q<\Lambda_{QCD}$
in the complex $Q$plane, where no shortdistance treatment is
possible.
If the region $Q< \Lambda_{QCD}$ is not avoided, one may introduce
spurious $1/Q^2$ power corrections to the decay width \cite{7},
which are absent
in the OPE. Similarly, Catani {\em et al.} find \cite{14} that the
inverse Mellin transform of the resummed cross section in moment space
has to be done by exact numerical integration in the complex $N$ plane,
avoiding the region ${\rm Re}\,N\to \infty$ where the IR singularity in
the running coupling becomes important. Failure to avoid this region
may result in spurious effects of order $(\Lambda_{QCD}/Q)^{\sim 0.3}$.
Within the approach of Ref.~13 this problem is somewhat masked by
using a resummation formula similar to Eq.~\docLink{slacpub7176003.tcx}[form11]{7}, in which
the sensitivity to the IR behavior of the coupling is of order $1/Q$
for {\em any} $N$. As explained above, this IR sensitivity is an
artifact of truncating the anomalous dimensions to finite order.
One also notes that applying the prescription of Ref.~13 to
DrellYan production requires to introduce a phenomenological
$1/Q$correction, that mainly seems to cancel the $1/Q$effects
generated by the resummation prescription \cite{16}.
The difference between various resummation
procedures should also be perceptible in high$p_\perp$ jet production
at the TEVATRON, and a theoretical understanding of this difference
might be one aspect in understanding the apparent excess of
large$p_\perp$ jets seen by CDF.
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