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%% subsection 1.1 \it Importance of Diagrammatic Calculations [slac-pub-7111-0-0-1-1 in slac-pub-7111-0-0-1: ^slac-pub-7111-0-0-1 >slac-pub-7111-0-0-1-2]
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\docLink{slac-pub-7111-0-0-0u1.tcx}[section::slac-pub-7111-0-0-0u1]{Table of Contents}%
\sectionLink{slac-pub-7111-0-0-1}{slac-pub-7111-0-0-1}{Above: 1. INTRODUCTION}%
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\subsection{\usemenu{slac-pub-7111::context::slac-pub-7111-0-0-1-1}{\it Importance of Diagrammatic Calculations}}\label{subsection::slac-pub-7111-0-0-1-1}
Gauge theories form the backbone of the Standard Model. The
weak-coupling perturbative expansion of gauge theory scattering
amplitudes, carried out by means of Feynman diagrams, has led to
theoretical predictions in remarkable agreement with high-energy
collider data~\cite{1}. This high-precision agreement
places strong bounds on new physics. In the strong-interaction sector
of the Standard Model --- described by quantum chromodynamics --- the
precision is not as great as in the electroweak sector. QCD is
asymptotically free, so the strong coupling constant $\alpha_s$
becomes weak at large momentum transfers, justifying a perturbative
expansion~\cite{2}. Physical quantities do depend on
nonperturbative, long-distance QCD, in the form of quantities such as
parton distribution and fragmentation functions, as well as on the
physics of hadronization. In many processes at modern colliders,
however, the dominant theoretical uncertainties are due to an
incomplete knowledge of the perturbation series, rather than to our
relative ignorance of nonperturbative aspects of scattering processes.
The situation is exacerbated by the slow approach to asymptopia
($\alpha_s$ is of order $0.1$ at the 100~GeV scale), and
by the presence of large logarithms of ratios of scales.
The leading-order (LO) term in the $\alpha_s$ expansion of a QCD
cross-section comes simply from squaring a tree-level scattering
amplitude. Efficient techniques for computing QCD tree amplitudes
have been available for some time now~\cite{3}, and the
results have provided a basic theoretical description of QCD processes
and thereby estimates of QCD backgrounds to new physics searches.
Unfortunately, higher order corrections, especially those enhanced by
logarithms, can be sizeable. The ultraviolet logarithms manifest
themselves in the residual renormalization-scale dependence of a
finite-order prediction. The renormalization scale $\mu_R$ is
introduced in order to define the coupling constant; renormalization
group invariance requires any physical quantity to be independent of
it. However, when a perturbative expansion is truncated at a finite
order, residual $\mu_R$-dependence appears, because the cancellation
takes place across different orders in $\alpha_s$. Calculations at
next-to-leading order (NLO) in $\alpha_s$ significantly reduce the
dependence on $\mu_R$ as compared to leading order. As an example,
\fig{ExptTheoryFigure} shows the comparison of the LO and NLO
theoretical predictions to the experimental measurement of a point in
the single-jet inclusive distribution. Note the good agreement
between NLO theory and experiment and the significant reduction of
theoretical uncertainties, compared to the LO calculation.
\def\yir{y_{\rm IR}}
Infrared logarithms
arise because jet processes involve more than one scale, at the very least
a scale characterizing the jet size in addition to the hard scale of the
short-distance scattering, and because of the infrared divergences of
perturbative QCD. These divergences transform the perturbation expansion
for such quantities from one in $\alpha_s$ alone to one in
$\alpha_s \log^2 \yir$ and $\alpha_s\log \yir$ in addition to $\alpha_s$,
where $\yir$ is a jet `resolution' parameter.
All three must be small for the perturbation expansion to be reliable; but
the first two cannot be calculated in an LO calculation. Only
in an NLO calculation are the corresponding terms determined
quantitatively, and only at this order can one establish the reliability of
the perturbative calculation.
Beyond the logarithmically-enhanced corrections, the ${\cal
O}(\alpha_s)$ corrections to most jet observables are larger than
non-perturbative power corrections and corrections due to quark
masses, and are thus the most important ones to calculate in order to
refine the precision of theoretical predictions.
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=ExptTheory.eps,clip=,width=3.3in}
\end{center}
\vskip -.75 cm
\caption[]{
\label{ExptTheoryFigure}
The inclusive cross section for single-jet production in $p \bar p$
collisions at $\sqrt{s} = 1.8$~TeV and jet transverse energy
$E_T=100$~GeV (using MRS\,D${}_0'$ structure functions \cite{4}),
showing the sensitivity of the LO result to the choice of
renormalization scale, $\mu_R$, and the reduced sensitivity at NLO. The CDF
data shown is extracted from ref.~\cite{5}; the band
shows statistical errors only.}
\end{figure}
Despite the need for higher-order QCD computations, at present no
quantities have been computed beyond next-to-next-to-lead\-ing order
(NNLO), and the only quantities that have been computed fully at NNLO
are totally inclusive quantities such as the total cross-section for
$e^+e^-$ annihilation into hadrons, and the QCD corrections to various
sum rules in deeply inelastic scattering \cite{6,7}. At
NLO, there are many complete calculations (in the form of computer
programs producing numerical results) for a variety of processes, but
at present results are still limited to where the basic process has
four external legs (counting electroweak vector bosons rather than
their decay products as external legs). The following are examples of
calculations which are relevant for current experiments but have not
yet been performed or assembled:
\begin{enumerate}
\item NLO corrections to three-jet production at hadron
colliders. These contributions would allow a measurement of $\alpha_s$
(via the three-jet to two-jet ratio) at the highest experimentally
available momentum transfers, as well as next-to-leading-order studies
of jet structure.
\item NLO corrections to $W$ + multi-jet production at hadron
colliders. These processes form a background to the $t$ quark signal
at Fermilab.
\item
NLO corrections to $e^+e^- \rightarrow 4$ jets. At the $Z$ resonance,
this is the lowest-order process in which the quark and gluon color
charges can be measured independently. It will also be useful for ruling
out the presence of light colored fermions (or scalars).
At LEP2 it is a background to threshold
production of $W$ pairs, when both $W$'s decay hadronically.
\item
NNLO corrections to $e^+e^-\rightarrow 3$ jets. These corrections are
the dominant uncertainty in a precision extraction of $\alpha_s$ from
hadronic event shapes at the $Z$ \cite{8}.
\end{enumerate}
In any of these processes, deviations of experimental results from the
theoretical predictions could indicate new physics.
Why do these higher-order QCD corrections remain uncalculated? NLO
corrections can be divided into real and virtual parts. (See
\fig{CrossSectionFigure}.) Real corrections arise from the emission
of one additional parton into the final state, and are straightforward
to compute from tree amplitudes with one more leg than the LO tree
amplitude. Virtual corrections arise from the interference of the LO
tree amplitude with a one-loop amplitude. Each contribution is
infrared divergent, but the divergences cancel in the sum, after
integrating the real contribution over ``unobserved'' partons in the
final state \cite{9}, and factorizing initial state
singularities into the definition of parton distributions in an
incoming hadron \cite{10}. The remaining finite
integrations are typically performed with a numerical program
\cite{11}.
While the numerical evaluation of NLO corrections can be non-trivial,
the major analytical bottleneck is simply the availability of one-loop
amplitudes, which enter into the virtual corrections. In particular,
one-loop amplitudes with more than four external legs (and all quarks
massless), which are required for the higher-order corrections listed
above, have only recently become available, thanks to the development
of new calculational techniques. The purpose of this review is to
provide an introduction to some of these techniques, together with
worked-out examples.
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=CrossSection.eps, width=2in,clip=}
\end{center}
\vskip -.9 cm
\caption[]{
\label{CrossSectionFigure}
In (a) the parton subprocesses required for the LO contribution to
two-jet production at hadron colliders are shown schematically.
In (b) the corresponding real and virtual NLO contributions are
shown.}
\end{figure}
%
%
Our emphasis will be on obtaining compact analytic results. In
general, it is preferable to have such results for matrix elements,
even though they are ultimately inserted into numerical programs for
computing cross-sections. Without compact results, numerical
instabilities can arise from the vanishing of spurious denominators in
the expression. With analytic forms it is also easier to compare
independent calculations, to understand better how to organize
calculations, and even to obtain results for an arbitrary number of
external legs \cite{12,13,14,15}.
%%%%%%%%%%%%%%%%%%%%%%%%