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%% subsection 1.2 \it Difficulty of Brute-Force Calculations [slac-pub-7111-0-0-1-2 in slac-pub-7111-0-0-1: slac-pub-7111-0-0-1-3]
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\subsection{\usemenu{slac-pub-7111::context::slac-pub-7111-0-0-1-2}{\it Difficulty of Brute-Force Calculations}}\label{subsection::slac-pub-7111-0-0-1-2}
Gauge theories have an elegant construction based on the principle of
local gauge invariance. The QCD Lagrangian for massless quarks $q$ is
$$
{\cal L}_{QCD} = -{1\over4}\Tr(F_{\mu\nu}^2) -i \bar{q}\Dsl q \,,
\equn\label{QCDL}
$$
where the covariant derivative
$D_\mu = \partial_\mu - i g A_\mu/\sqrt{2}$
and field strength $F_{\mu\nu} = i\sqrt{2} [ D_\mu , D_\nu ]/g$
are given in terms of the matrix-valued gauge connection
$A_\mu = A_\mu^a T^a$.\footnote{
The normalization $\Tr(T^aT^b) = \delta^{ab}$ of the
fundamental-representation generators $T^a$ accounts for the
$\sqrt{2}$'s here; it serves to eliminate the $\sqrt{2}$'s from
the partial amplitudes defined below.}
%
Since ${\cal L}_{QCD}$ depends on a single coupling constant $g$,
all the interactions are dictated by gauge symmetry.
Unfortunately, the Feynman diagram expansion does not respect this
invariance, because the quantization procedure fixes the gauge
symmetry.
%
%For example, imposing the Lorentz gauge condition
%$\partial_\mu A^\mu=0$ leads to the additional term
%$-{1\over2\xi}\Tr(\partial_\mu A^\mu)^2$ (plus ghost terms),
%where $\xi$ is an arbitrary parameter, in the
%gauge-fixed Lagrangian from which Feynman rules are derived.
%
Individual diagrams are not gauge invariant,
and are often more complicated than the final sum over diagrams.
The non-abelian gluon self-interactions coming from the cubic and
quartic terms in \eqn{QCDL} have a complicated index structure and
momentum-dependence.
So while it is straightforward in principle to compute both tree and
loop amplitudes by drawing all Feynman diagrams and evaluating them,
%--- using standard reduction techniques for the loop integrals that
%are encountered ---
in practice this method becomes extremely
inefficient and cumbersome as the number of external legs grows.
For five or more external legs there are a large number of kinematic
variables, which allow the construction of complicated expressions.
Indeed, intermediate expressions tend to be vastly more
complicated than the final results, when the latter are represented in
an appropriate way.
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=FiveGluon.eps, width=1.in,clip=}
\end{center}
\vskip -.7 cm
\caption[]{
\label{FiveGluonFigure}
The five-gluon pentagon diagram.}
\end{figure}
%
%
%
As an example consider the five-gluon pentagon diagram, depicted
in \fig{FiveGluonFigure}, which would be encountered in a brute-force
computation of NLO corrections to three-jet production at
a hadron collider.
Each of the five non-abelian three-point vertices in the diagram is
given by
$$
V^{abc}_{\mu\nu\rho}(k,p,q) = f^{abc} \Bigl(
\eta_{\nu\rho}(p-q)_\mu
+ \eta_{\rho\mu} (q-k)_\nu + \eta_{\mu\nu} (k-p)_\rho \Bigr) \,,
\equn\label{FeynmanVertex}
$$
where $f^{abc}$ are the $SU(3)$ structure constants, $k$, $p$ and $q$
the momenta, and $\eta_{\mu\nu}$ the Minkowski metric. As the
non-abelian vertex has six terms, a rough estimate of the number of
terms is about $6^5$. Each term is associated with a loop integral
which evaluates to an expression on the order of a page in length.
This means that one is faced with about $10^4$ pages of algebra for
this single diagram. As bad as this brute-force approach might seem,
the situation is actually worse, because of the structure of the
results. After evaluating the integrals and summing over a few
hundred more diagrams one obtains expressions of the form $\sum_i {N_i
\over D_i}$, where the factors $N_i$ are polynomials in the gluon
polarization vectors and external momenta, and the $D_i$ (polynomials
in the external invariants) are produced when the loop integrals are
reduced to a standard set of functions. In general the $D_i$ contain
spurious kinematic singularities which cancel only after combining many
terms over a common denominator; this causes an explosion of
terms in the numerator.
In contrast to the complexity of intermediate expressions, the final
results can be strikingly simple. For example, the five gluon
amplitudes which we shall describe in
section~\docLink{slac-pub-7111-0-0-3.tcx}[SusyDecompositionSubsection]{3.2} are remarkably compact.
%%%%%%%%%%%%%%%%%%%%%%%%