%Paper: 
%From: LANCE@SLACVM.SLAC.Stanford.EDU
%Date: Fri, 02 Sep 1994 15:00 -0800 (PST)


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% Recent Progress in One-Loop Multi-Parton Calculations,   %
% by Z. Bern, L. Dixon, D.C. Dunbar and D.A. Kosower       %
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%%%%%%%%%% espcrc2.tex %%%%%%%%%%
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% put your own definitions here:
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% declarations for front matter
\title{Elsevier instructions for the preparation of a
       2-column format camera-ready paper in \LaTeX}

\author{P. de Groot\address{Mathematics and Computer Science Division,
        Elsevier Science Publishers B.V., \\
        P.O. Box 103, 1000 AC Amsterdam, The Netherlands}%
        \thanks{Footnotes should appear on the first page only to
                indicate your present address (if different from your
                normal address), research grant, sponsoring agency, etc.
                These are obtained with the {\tt\ttbs thanks} command.}
        and
        X.-Y. Wang\address{Economics Department, University of Winchester, \\
        2 Finch Road, Winchester, Hampshire P3L T19, United Kingdom}}

\begin{document}

\begin{abstract}
These pages provide you with an example of the layout and style for
100\% reproduction which we wish you to adopt during the preparation of
your paper. This is the output from the \LaTeX{} document style you
requested.
\end{abstract}

% typeset front matter (including abstract)
\maketitle

\section{FORMAT}

Text should be produced within the dimensions shown on these pages:
each column 7.5 cm wide with 1 cm middle margin, total width of 16 cm
and a maximum length of 20.2 cm on first pages and 21 cm on second and
following pages. The \LaTeX{} document style uses the maximal stipulated
length apart from the following two exceptions (i) \LaTeX{} does not
begin a new section directly at the bottom of a page, but transfers the
heading to the top of the next page; (ii) \LaTeX{} never (well, hardly
ever) exceeds the length of the text area in order to complete a
section of text or a paragraph.

\subsection{Spacing}

We normally recommend the use of 1.0 (single) line spacing. However,
when typing complicated mathematical text \LaTeX{} automatically
increases the space between text lines in order to prevent sub- and
superscript fonts overlapping one another and making your printed
matter illegible.

\subsection{Fonts}

These instructions have been produced using a 10 point Computer Modern
Roman. Other recommended fonts are 10 point Times Roman, New Century
Schoolbook, Bookman Light and Palatino.

\section{PRINTOUT}

The most suitable printer is a laser printer. A dot matrix printer
should only be used if it possesses an 18 or 24 pin printhead
(``letter-quality'').

The printout submitted should be an original; a photocopy is not
acceptable. Please make use of good quality plain white A4 (or US
Letter) paper size. {\em The dimensions shown here should be strictly
adhered to: do not make changes to these dimensions, which are
determined by the document style}. The document style leaves at least
3~cm at the top of the page before the head, which contains the page
number.

Printers sometimes produce text which contains light and dark streaks,
or has considerable lighting variation either between left-hand and
right-hand margins or between text heads and bottoms. To achieve
optimal reproduction quality, the contrast of text lettering must be
uniform, sharp and dark over the whole page and throughout the article.

If corrections are made to the text, print completely new replacement
pages. The contrast on these pages should be consistent with the rest
of the paper as should text dimensions and font sizes.

\begin{table*}[hbt]
% space before first and after last column: 1.5pc
% space between columns: 3.0pc (twice the above)
\setlength{\tabcolsep}{1.5pc}
% -----------------------------------------------------
% adapted from TeX book, p. 241
\newlength{\digitwidth} \settowidth{\digitwidth}{\rm 0}
\catcode`?=\active \def?{\kern\digitwidth}
% -----------------------------------------------------
\caption{Biologically treated effluents (mg/l)}
\label{tab:effluents}
\begin{tabular*}{\textwidth}{@{}l@{\extracolsep{\fill}}rrrr}
\hline
                 & \multicolumn{2}{l}{Pilot plant}
                 & \multicolumn{2}{l}{Full scale plant} \\
\cline{2-3} \cline{4-5}
                 & \multicolumn{1}{r}{Influent}
                 & \multicolumn{1}{r}{Effluent}
                 & \multicolumn{1}{r}{Influent}
                 & \multicolumn{1}{r}{Effluent}         \\
\hline
Total cyanide    & $ 6.5$ & $0.35$ & $  2.0$ & $  0.30$ \\
Method-C cyanide & $ 4.1$ & $0.05$ &         & $  0.02$ \\
Thiocyanide      & $60.0$ & $1.0?$ & $ 50.0$ & $ <0.10$ \\
Ammonia          & $ 6.0$ & $0.50$ &         & $  0.10$ \\
Copper           & $ 1.0$ & $0.04$ & $  1.0$ & $  0.05$ \\
Suspended solids &        &        &         & $<10.0?$ \\
\hline
\multicolumn{5}{@{}p{120mm}}{Reprinted from: G.M. Ritcey,
                             Tailings Management,
                             Elsevier, Amsterdam, 1989, p. 635.}
\end{tabular*}
\end{table*}

\section{TABLES AND ILLUSTRATIONS}

Tables should be made with \LaTeX; illustrations should be originals or
sharp prints. They should be arranged throughout the text and
preferably be included {\em on the same page as they are first
discussed}. They should have a self-contained caption and be positioned
in flush-left alignment with the text margin within the column. If they
do not fit into one column they may be placed across both columns
(using \verb-\begin{table*}- or \verb-\begin{figure*}- so that they
appear at the top of a page).

\subsection{Tables}

Tables should be presented in the form shown in
Table~\ref{tab:effluents}.  Their layout should be consistent
throughout.

Horizontal lines should be placed above and below table headings, above
the subheadings and at the end of the table above any notes. Vertical
lines should be avoided.

If a table is too long to fit onto one page, the table number and
headings should be repeated above the continuation of the table. For
this you have to reset the table counter with
\verb|\addtocounter{table}{-1}|. Alternatively, the table can be turned
by $90^\circ$ (`landscape mode') and spread over two consecutive pages
(first an even-numbered, then an odd-numbered one) created by means of
\verb|\begin{table}[h]| without a caption. To do this, you prepare the
table as a separate \LaTeX{} document and attach the tables to the
empty pages with a few spots of suitable glue.

\subsection{Line drawings}

Line drawings should be drawn in India ink on tracing paper with the
aid of a stencil or should be glossy prints of the same; computer
prepared drawings are also acceptable. They should be attached to your
manuscript page, correctly aligned, using suitable glue and {\em not
transparent tape}. When placing a figure at the top of a page, the top
of the figure should be at the same level as the bottom of the first
text line.

All notations and lettering should be no less than 2\,mm high. The use
of heavy black, bold lettering should be avoided as this will look
unpleasantly dark when printed.

\subsection{Black and white photographs}

Photographs must always be sharp originals ({\em not screened
versions\/}) and rich in contrast. They will undergo the same reduction
as the text and should be pasted on your page in the same way as line
drawings.

\subsection{Colour photographs}

Sharp originals ({\em not transparencies or slides\/}) should be
submitted close to the size expected in publication. Charges for the
processing and printing of colour will be passed on to the author(s) of
the paper. As costs involved are per page, care should be taken in the
selection of size and shape so that two or more illustrations may be
fitted together on one page. Please contact the Technical Editor in the
Camera-Ready Publications Department at Elsevier for a price quotation
and layout instructions before producing your paper in its final form.

\begin{figure}[htb]
\vspace{9pt}
\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\caption{Good sharp prints should be used and not (distorted) photocopies.}
\label{fig:largenenough}
\end{figure}
%
\begin{figure}[htb]
\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\caption{Remember to keep details clear and large enough.}
\label{fig:toosmall}
\end{figure}

\section{EQUATIONS}

Equations should be flush-left with the text margin; \LaTeX{} ensures
that the equation is preceded and followed by one line of white space.
\LaTeX{} provides the document-style option {\tt fleqn} to get the
flush-left effect.

\begin{equation}
H_{\alpha\beta}(\omega) = E_\alpha^{(0)}(\omega) \delta_{\alpha\beta} +
                          \langle \alpha | W_\pi | \beta \rangle
\end{equation}

You need not put in equation numbers, since this is taken care of
automatically. The equation numbers are always consecutive and are
printed in parentheses flush with the right-hand margin of the text and
level with the last line of the equation. For multi-line equations, use
the {\tt eqnarray} environment. For complex mathematics, use the
\AmS-\LaTeX{} package.

\begin{thebibliography}{9}
\bibitem{Scho70} S. Scholes, Discuss. Faraday Soc. No. 50 (1970) 222.
\bibitem{Mazu84} O.V. Mazurin and E.A. Porai-Koshits (eds.),
                 Phase Separation in Glass, North-Holland, Amsterdam, 1984.
\bibitem{Dimi75} Y. Dimitriev and E. Kashchieva,
                 J. Mater. Sci. 10 (1975) 1419.
\bibitem{Eato75} D.L. Eaton, Porous Glass Support Material,
                 US Patent No. 3 904 422 (1975).
\end{thebibliography}

References should be collected at the end of your paper. Do not begin
them on a new page unless this is absolutely necessary. They should be
prepared according to the sequential numeric system making sure that
all material mentioned is generally available to the reader. Use
\verb+\cite+ to refer to the entries in the bibliography so that your
accumulated list corresponds to the citations made in the text body.

Above we have listed some references according to the
sequential numeric system \cite{Scho70,Mazu84,Dimi75,Eato75}.
\end{document}

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%%%%%%%%%%%%% qcd94.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Recent Progress in One-Loop Multi-Parton Calculations,   %
% by Z. Bern, L. Dixon, D.C. Dunbar and D.A. Kosower       %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\documentstyle[twoside,fleqn,espcrc2]{article}

% put your own definitions here:
%   \newcommand{\cZ}{\cal{Z}}
%   \newtheorem{def}{Definition}[section]
%   ...
\newcommand{\spa}[3]{\left\langle#1\,#3\right\rangle}
\newcommand{\spb}[3]{\left[#1\,#3\right]}
%\newcommand{\ns}{n_s}
%\newcommand{\nf}{n_f}
\newcommand{\LP}{\left(}
\newcommand{\RP}{\right)}
\newcommand{\LB}{\left[}
\newcommand{\RB}{\right]}
\newcommand{\e}{\epsilon}
\newcommand{\cg}{c_\Gamma}
\newcommand{\hf}{\textstyle{1\over2}}
\newcommand{\Li}{\mathop{\rm Li}\nolimits}
\newcommand{\Ls}{\mathop{\rm Ls}\nolimits}
\newcommand{\Ll}{\mathop{\rm L}\nolimits}
\newcommand{\gluino}{{\tilde g}}
\newcommand{\qb}{{\bar q}}
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%
\newcommand{\ttbs}{\char'134}
\newcommand{\AmS}{{\protect\the\textfont2
  A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}

% add words to TeX's hyphenation exception list
\hyphenation{author another created financial paper re-commend-ed}

% declarations for front matter
\title{Recent Progress in One-Loop Multi-Parton Calculations${}^{*}$}%
        \thanks{Research supported by the US Department of Energy
                under grants DE-FG03-91ER40662 and DE-AC03-76SF00515,
                by the Alfred P. Sloan Foundation under grant BR-3222,
                by the US National Science Foundation under grant
                 by the {\it Commissariat \`a l'Energie
                Atomique\/} of France, and by NATO Collaborative
                Research Grants CRG--921322 and CRG--910285.
                Talk presented by L.D. at QCD94, July 7-13, 1994,
                Montpellier, France.}

\author{Z. Bern,\address{Department of Physics, UCLA, Los Angeles,
        CA 90024, USA}%
        \ L. Dixon,\address{Stanford Linear Accelerator Center,
        Stanford University, Stanford, CA 94309, USA}%
        \ D.C. Dunbar\address{University College of Swansea, UK}%
        \ and
        D.A. Kosower\address{Service de Physique Th\'eorique,
        Centre d'Etudes de Saclay,
        F-91191 Gif-sur-Yvette cedex, France}}%

\begin{document}

\begin{abstract}
We describe techniques that simplify the calculation
of one-loop QCD amplitudes with many external legs, which
are needed for next-to-leading-order (NLO) corrections to
multi-jet processes.  The constraints imposed by perturbative unitarity,
collinear singularities and a supersymmetry-inspired organization
of helicity amplitudes are particularly useful.
Certain sequences of one-loop helicity amplitudes
may be obtained for an arbitrary number of external gluons
using these techniques.
We also report on progress in completing the set of one-loop
helicity amplitudes required for NLO three-jet production at
hadron colliders, namely the amplitudes with two external quarks and
three gluons.
\end{abstract}

% typeset front matter (including abstract)
\maketitle

\section{INTRODUCTION}


High-$p_T$ events in
hadronic collisions and high-energy $e^+e^-$ annihilations
often produce a large number of jets.
Quantitative QCD predictions for such multi-jet rates
are important, but require at least
next-to-leading-order (NLO) calculations.
NLO corrections in turn require one-loop amplitudes with many
external partons.
The analytic complexity of one-loop calculations grows very
rapidly with the number of external legs.
As a result, complete NLO results are currently available only
for processes with up to four external partons or vector bosons,
such as $e^+e^-\ \to\ 3\ {\rm jets}$, and $p\bar{p}$ (or $pp$)
production of inclusive jets, di-jets, and $(W,Z)\ +\ 1\ {\rm jet}$.
To go further requires one-loop QCD amplitudes with five or more
external partons or vector bosons.
Here we briefly describe some very useful tools for carrying out
such calculations, and summarize recent progress.

\section{ORGANIZATIONAL TOOLS}

The basic principle in calculating a complicated one-loop amplitude
is to break it up into as many simpler, yet physical, pieces as
possible.
Traditional Feynman diagrams are not a good way to do this since
individual diagrams are gauge-variant, and hence unphysical.
Instead one should use the quantum numbers of the external particles,
namely their {\it helicities} $(\pm)$ and {\it color} quantum-numbers,
to decompose the amplitude into
{\it color-ordered helicity sub-amplitudes},
or {\it sub-amplitudes} for short.
The helicity~\cite{SpinorHelicity} and
color~\cite{TreeColor} decompositions of multi-parton
tree-level amplitudes have been essential to their
efficient calculation, and the same is true at one loop.
The analytic properties of the one-loop sub-amplitudes  ---
namely their collinear behavior (poles) and their unitarity
properties (cuts) --- are simpler than those
of the full amplitude, yet they provide powerful constraints.
They are simpler mainly because they only involve ``color-adjacent''
kinematic invariants such as $(k_i+k_{i+1})^2$, where $k_i$ and
$k_{i+1}$ are adjacent momenta with respect to the color ordering.
Consequently the sub-amplitudes themselves tend to be simple;
analytic expressions for each one-loop five-parton sub-amplitude takes
up less than a page~\cite{FiveGluon,KSTFivePt},
whereas the color-summed cross-section would fill hundreds of pages.
Thus one should calculate sub-amplitudes analytically,
and carry out the squaring of amplitudes and the sum over colors
and helicities numerically at the very end.

The analytic properties of the sub-amplitudes can be
exploited to simplify their calculation.
Consider the constraints from perturbative unitarity.
It is well known that the cuts (absorptive parts) of a loop amplitude are
much easier to calculate than the full amplitude, because they are
given by phase-space integrals of products of tree
amplitudes~\cite{Cutting}.
The phase-space integrals can be performed to obtain
integral functions with the correct cuts,
omitting the need to do a dispersion integral~\cite{SusyFour}.
Thus the full amplitude is easily reconstructed from the
various cuts, up to additive ``polynomial'' terms (lacking
branch cuts).
It may be possible to determine the polynomial terms recursively
using their collinear singularities
(there is still a uniqueness question here).
If a certain power-counting criterion holds ---
the degree of the loop-momentum polynomial for each diagram
should be two fewer than the maximum possible in gauge theory
--- then there are actually no
polynomial ambiguities and the sub-amplitudes can be completely
reconstructed from the cuts~\cite{SusyFour,SusyOne}.
Supersymmetric amplitudes provide examples satisfying the criterion;
infinite sequences of ``maximally helicity violating''
$n$-gluon supersymmetric amplitudes can be efficiently calculated
via their cuts~\cite{SusyFour,SusyOne}.

Direct calculation of sub-amplitudes can also be simplified using a
decomposition based on the internal (spin) quantum numbers of
particles going around the loop.  For example, in a one-loop $n$-gluon
amplitude the contribution of a gluon propagating around the loop
would traditionally lead to a large amount
of algebra, due to the complicated non-abelian self-interaction vertex.
However, it is possible to rewrite the gluon self-interaction (using
either background-field gauge~\cite{Background} or a string-based
approach~\cite{StringBased}) so that the
gluon in the loop looks like a scalar in the loop,
plus ``a little bit more''.
One can rewrite~\cite{FiveGluon},
\begin{eqnarray}
  \hbox{gluon}\ &=&\ \hbox{scalar}  \nonumber \\
  && \hskip -18mm
  -4 \times [ \hbox{fermion}\ +\ \hbox{scalar} ] \nonumber \\
  && \hskip -18mm
  +\ [ \hbox{gluon}\ +\ 4 \times \hbox{fermion}
                      \ +\ 3 \times \hbox{scalar} ]\ ,
\end{eqnarray}
where all entries correspond to two-component fields circulating
in the loop (gluons, Weyl fermions, complex scalars).
The ``little bit more'' represented by the last two lines is
supersymmetric
(the contribution of 4 $N=1$ chiral multiplets, and of an $N=4$
super-Yang-Mills theory), and is calculable via its cuts.
In this way the gluon computation can be traded for the easier
scalar case.

Recursive techniques, which were first applied at tree-level
by Berends and Giele~\cite{RecursiveA,RecursiveB},
are now beginning to show promise at loop-level.
Mahlon~\cite{MahlonB} has recently obtained two infinite sequences
of one-loop $n$-gluon amplitudes by ``sewing up'' recursively determined
off-shell tree amplitudes.

\section{EXPLICIT RESULTS}

What are the practical consequences of these tools so far?
At the one-loop five-parton level, they have been used to
calculate the full set of helicity amplitudes
for five external gluons~\cite{FiveGluon} and for two quarks
and three gluons (which are almost complete~\cite{qqggg}).
In contrast, the calculation of the four-quark one-gluon
amplitudes used (of the above tools) only the sub-amplitude
decomposition~\cite{KSTFivePt}.

Here we present one of the $\bar{q}qggg$ partial amplitudes,
$A_{5;1}(1_{\bar{q}}^-,2_q^+,3^-,4^+,5^+)$, which is the coefficient of
the color structure $N_c\,(T^{a_3}T^{a_4}T^{a_5})_{i_2}^{~\bar{i_1}}$ for
the indicated helicity assignments; particles 3,4,5 are gluons.
For the gauge group $SU(N_c)$ with $n_f$ flavors of quark and $n_s$
flavors of scalar quarks, $A_{5;1}$ can be decomposed into
{\it primitive amplitudes},
from which all factors of $N_c,n_f,n_s$ have been extracted, as:
\begin{eqnarray}
  && \hskip -8mm
   A_{5;1}(1_{\bar{q}}^-,2_q^+;3^-,4^+,5^+) \nonumber \\
 &=&  \left( 1+{1\over N_c^2} \right)
 \,A_5^L(1_{\bar{q}}^-,2_q^+,3^-,4^+,5^+) \nonumber \\
 && \hskip -8mm
 -{1\over N_c^2}\,{\spa3.2\over\spa3.1}
   A_{5;1}^\susy(1^-,2^+,3^-,4^+,5^+) \nonumber \\
 && \hskip -8mm
 -\left( {\nf\over N_c}+{1\over N_c^2}\right)\,
   A_5^f(1_{\bar{q}}^-,2_q^+;3^-,4^+,5^+) \nonumber \\
 && \hskip -8mm
 +\left({\ns-\nf\over N_c}-{1\over N_c^2}\right)\,
   A_5^s(1_{\bar{q}}^-,2_q^+;3^-,4^+,5^+),
\end{eqnarray}
where $A_{5;1}^\susy(1^-,2^+,3^-,4^+,5^+)$ is the pure super-Yang-Mills
amplitude for five external gluons, taken from ref.~\cite{FiveGluon}.
In the remaining components, the poles in $\e$ in
$D=4-2\e$ dimensional reduction are separated out into
``$V$'' pieces by
\begin{equation}
A_5^x = \cg\left( V^x A^\tree_5 + i F^x \right),
\hskip 7mm x= L, f, s\;,
\end{equation}
where
\begin{equation}
  c_\Gamma = {1\over(4\pi)^{2-\e}}
  {\Gamma(1+\e)\Gamma^2(1-\e)\over\Gamma(1-2\e)}\;,
\end{equation}
\begin{equation}
A_5^\tree =
i \, {{\spa1.3}^3\spa2.3 \over \spa1.2\spa2.3\spa3.4\spa4.5\spa5.1}\;,
\end{equation}
and
\begin{eqnarray}
  V^L &=&
  -{1\over\e^2}\sum_{j=2}^5 \LP {\mu^2\over -s_{j,j+1}}\RP^\e
   \nonumber \\
 &&  +\sum_{j=1}^5 \ln\LP{-s_{j,j+1}\over -s_{j+1,j+2}}\RP\,
                        \ln\LP{-s_{j+2,j-2}\over -s_{j-2,j-1}}\RP
   \nonumber \\
&& +{5\over6}\pi^2
   -{3\over2\e}\LP{\mu^2\over-s_{34}}\RP^\e
   + \ln\LP {-s_{51}\over -s_{12}}\RP-3 \nonumber \\
 V^f &=& V^s = 0.
\end{eqnarray}

The finite terms possess all the analytic complexity, and are given by
\begin{eqnarray*}
F^L &=& F^s  \\
&& \hskip -8mm
- {\spa1.2\spa2.3\spa3.4\spa4.1{\spb2.4}^3
        \over\spa4.5\spa5.1}
    {\Ls_2\LP {-s_{23}\over -s_{51}},\,{-s_{34}\over -s_{51}}\RP
         \over s_{51}^3}  \\
&& \hskip -8mm
    - {\spa1.2\spa2.3{\spa3.5}^2{\spb2.5}^3
       \over\spa3.4\spa4.5}
    {\Ls_2\LP {-s_{12}\over -s_{34}},\,{-s_{51}\over -s_{34}}\RP
         \over s_{34}^3}  \\
&&\hskip -8mm
    - 2 {\spa1.3\spa2.3\spa4.1{\spb2.4}^2\over\spa4.5\spa5.1}
    {\Ls_1\LP {-s_{23}\over -s_{51}},\,{-s_{34}\over -s_{51}}\RP
         \over s_{51}^2}  \\
&&\hskip -8mm
    - 2 {\spa1.3\spa2.3\spa3.5{\spb2.5}^2\over\spa3.4\spa4.5}
    {\Ls_1\LP {-s_{12}\over -s_{34}},\,{-s_{51}\over -s_{34}}\RP
         \over s_{34}^2}  \\
%%
&&\hskip -8mm
   - { {\spa1.3}^2\spb2.4\over\spa4.5\spa5.1}
    {\Ls_0\LP {-s_{23}\over -s_{51}},\,{-s_{34}\over -s_{51}}\RP
         \over s_{51}}  \\
&&\hskip -8mm
    - { {\spa1.3}^2\spa3.5\spb2.5\over\spa3.4\spa4.5\spa5.1}
    {\Ls_0\LP {-s_{12}\over -s_{34}},\,{-s_{51}\over -s_{34}}\RP
         \over s_{34}}  \\
%%
  &&\hskip -8mm
 - \LP {\spa1.3{\spa2.3}^2{\spb2.5}^2\spa1.5 \over \spa1.2\spa3.4\spa4.5}
    +{1\over2}{{\spa1.3}^2\spb1.2\spa2.3\spb2.5 \over \spa3.4\spa4.5}\RP
      \\
  && \times
      {\Ll_1\LP {-s_{34}\over -s_{51}}\RP \over s_{51}^2}  \\
  &&\hskip -8mm
   + {1\over2} {\spa1.3\spa1.4\spa2.3{\spb2.4}^2\over\spa4.5\spa5.1}
       {\Ll_1\LP {-s_{23}\over -s_{51}}\RP \over s_{51}^2}  \\
  &&\hskip -8mm
    - {1\over2} {\spa1.3\spa1.5\spa3.4{\spb4.5}^2\over\spa1.2\spa4.5}
    {\Ll_1\LP {-s_{12}\over -s_{34}}\RP \over s_{34}^2}  \\
  &&\hskip -8mm
   + {1\over2} { {\spa1.3}^2\spb2.4
                \over\spa4.5\spa5.1}
      {\Ll_0\LP {-s_{34}\over -s_{51}}\RP \over s_{51}}  \\
  &&\hskip -8mm
   -\LB 2{ {\spa1.3}^2\spb4.5\over\spa1.2\spa4.5}
        +{ {\spa1.3}^2\spa3.5\spb2.5\over\spa3.4\spa4.5\spa5.1}\RB
   { \Ll_0\LP{-s_{12}\over -s_{34}}\RP \over s_{34} }  \\
%%
  &&\hskip -8mm
    +{1\over2} {\spa1.4\spb2.4^2\spb4.5\over
      \spa4.5 \spb2.3\spb3.4 s_{51}}
    -  {\spa1.3\spa2.3\spb2.5\spb4.5 \over
        \spa1.2 s_{34} \spa4.5\spb5.1}  \\
  &&\hskip -8mm
    -  {1\over2} {{\spa1.3}^2\spb1.2\spa2.3\spb2.5 \over
      s_{34} \spa3.4\spa4.5 s_{51}} \ ,
\end{eqnarray*}
%%%%
\begin{eqnarray*}
F^s &=& {1\over 3} \Biggl[
   { \spa1.5 \spb2.5 \spa3.4 \spa3.5 {\spb4.5}^2 \over \spa4.5 }
    { 2\, \Ll_2\LP{-s_{12}\over -s_{34}}\RP \over s_{34}^3 }
     \\
&& \hskip -8mm
 - { \spa1.3 \spa1.5 \spa3.4 {\spb4.5}^2
    \over \spa1.2 \spa4.5 }
    { \Ll_1\LP{-s_{12}\over -s_{34}}\RP \over s_{34}^2 }  \\
&& \hskip -8mm
- { \spa1.3 \spb2.4 \spb4.5 \over \spa1.2 \spb1.2 \spb3.4 \spa4.5 }
 + { {\spb2.4}^2 \spb2.5 \over \spb1.2 \spb2.3 \spb3.4 \spa4.5 }
                     \Biggr]\ ,
\end{eqnarray*}
%%%%
\begin{eqnarray}
F^f &=& - {{\spa1.3}^2\spb4.5 \over \spa1.2\spa4.5}
   { \Ll_0\LP{-s_{12}\over -s_{34}}\RP \over s_{34} } \ ,
\end{eqnarray}
where the logarithms and dilogarithms (${\rm Li}_2$) are contained in
\begin{eqnarray*}
  \Ll_0(r) &=& {\ln(r)\over 1-r}\,,\hskip 2mm
  \Ll_1(r) = {\Ll_0(r)+1\over 1-r}\,,  \\
  \Ll_2(r) &=& {\ln(r)-\hf(r-1/r)\over (1-r)^3}\,, \\
\hskip -6mm
  \Ls_0(r_1,r_2) &=&  { \Li_2(1-r_1) + \Li_2(1-r_2)
   + \ln r_1\,\ln r_2 - {\pi^2\over6} \over (1-r_1-r_2) } \;,
  \\
  \Ls_1(r_1,r_2) &=& { \Ls_0(r_1,r_2) + \Ll_0(r_1)+\Ll_0(r_2)
    \over (1-r_1-r_2) }\;,  \\
  \Ls_2(r_1,r_2) &=& { \Ls_1(r_1,r_2) +\LP\Ll_1(r_1)+\Ll_1(r_2)\RP/2
    \over (1-r_1-r_2) }\;.  \\
\end{eqnarray*}

Once the full set of $\bar{q}qggg$ helicity amplitudes
are available (roughly speaking, six expressions of the above
type are required), numerical programs can be constructed for
NLO three-jet production at hadron colliders.  There are various
general formalisms available~\cite{KunsztSingular,GG}
for combining $(n+1)$-parton tree
contributions and $n$-parton loop contributions into a NLO
correction; the one of Giele, Glover, and Kosower is convenient
because it is in a color-ordered framework which meshes well with
a color-ordered decomposition of the amplitudes.

\section{FUTURE PROSPECTS}

Projecting into the future, it seems that recently
developed tools --- especially the combination of unitarity, collinear
singularities and recursive techniques --- will make the
calculation of one-loop six-parton and perhaps even seven-parton
amplitudes quite practical within the next couple of years.
Indeed, we expect that the bottleneck in getting NLO results out will
shift from the analytical to the numerical end of the process.
On the analytical side, the emphasis should shift
(perhaps fairly soon) to two-loop multi-parton calculations,
which are needed for next-to-next-to-leading (NNLO) results,
such as the NNLO correction to $e^+e^-$ annihilation to 3 jets,
a result which could significantly reduce the theoretical error in
determining $\alpha_s$ at the $Z$ pole.

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\end{document}


