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%% subsection 2.2 \it Spinor Helicity Formalism [slac-pub-7111-0-0-2-2 in slac-pub-7111-0-0-2: slac-pub-7111-0-0-2-3]
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\subsection{\usemenu{slac-pub-7111::context::slac-pub-7111-0-0-2-2}{\it Spinor Helicity Formalism}}\label{subsection::slac-pub-7111-0-0-2-2}
\label{SpinorHelicitySubsection}
In explicit calculations, it is very convenient to adopt a
helicity (circular polarization) basis for external gluons.
The spinor helicity formalism~\cite{16} expresses
the positive- and negative-helicity polarization vectors
in terms of massless Weyl spinors $\vert k^{\pm} \rangle$,
$$
\pol^{+}_\mu (k;q) = {\sandmm{q}.{\gamma_\mu}.k
\over \sqrt2 \spa{q}.k}\, ,\hskip 1cm
\pol^{-}_\mu (k;q) = {\sandpp{q}.{\gamma_\mu}.k
\over \sqrt{2} \spb{k}.q} \, ,
\equn
\label{PolarizationVector}
$$
where $q$ is an arbitrary null `reference' momentum which drops out of
the final gauge-invariant amplitudes. (Changing $q$ is equivalent to
performing a gauge transformation on the external legs.)
We use the compact notation
$$
\langle k_i^{-} \vert k_j^{+} \rangle \equiv \langle ij \rangle \, ,
\hskip 1 cm
\langle k_i^{+} \vert k_j^{-} \rangle \equiv [ij] \, .
\equn
$$
These spinor products are crossing-symmetric, antisymmetric in their
arguments, and satisfy
$$
\spa{i}.j \spb{j}.i = 2 k_i \cdot k_j \equiv s_{ij} \, .
\equn
$$
Helicity amplitudes can be given a manifestly crossing symmetric
representation, with the convention that a helicity label corresponds
to an outgoing particle; the helicity of an incoming particle is
reversed. As we shall discuss in Section \docLink{slac-pub-7111-0-0-5.tcx}[FactorizationSection]{5}, in the
collinear limit where $k_i$ and $k_j$ become parallel, helicity
amplitudes have a square-root singular behavior, $\sim
{1\over\sqrt{s_{ij}}} \sim {1\over \spa{i}.{j}} \sim {1\over
\spb{i}.{j}}$, whose magnitude and phase are captured concisely by the
spinor products. This helps explain why spinor products provide an
extremely compact representation of amplitudes.
In performing calculations, the Schouten identity is useful,
$$
\spa{i}.{j}\spa{k}.{l}
= \spa{i}.{l}\spa{k}.{j} + \spa{i}.{k}\spa{j}.{l} \,.
\equn\label{SchoutenIdentity}
$$
A more complete discussion, including further identities and numerical
representations of the spinor products, can be found in
refs.~\cite{16,3,21}.
%
To maximize the benefit of the spinor helicity formalism for loop
amplitudes we must choose a compatible regularization scheme. In
conventional dimensional regularization \cite{29}, the
polarization vectors are $(4-2\eps)$-dimension\-al; this is
incompatible with the spinor helicity method, which assumes
four-dimensional polarizations. To avoid this problem, we modify the
regularization scheme so all helicity states are four-dimensional and
only the loop momentum is continued to $(4-2\eps)$ dimensions. This
is the four-dimensional-helicity (FDH) scheme \cite{28}, which has
been shown to be equivalent \cite{30} to an appropriate
helicity formulation of Siegel's dimensional-reduction scheme
\cite{31} at one-loop. The conversion between schemes has been
given in ref.~\cite{30}, so there is no loss of
generality in choosing the FDH scheme.
%%%%%%%%%%%%%%%%%%%%%%%%