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%% subsection 2.3 \it Parity and Charge Conjugation [slac-pub-7111-0-0-2-3 in slac-pub-7111-0-0-2: slac-pub-7111-0-0-2-4]
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\subsection{\usemenu{slac-pub-7111::context::slac-pub-7111-0-0-2-3}{\it Parity and Charge Conjugation}}\label{subsection::slac-pub-7111-0-0-2-3}
\label{PCSubsection}
The reader might worry that the color and helicity decompositions
will lead to a huge proliferation in the number of primitive or partial
amplitudes that have to be computed.
In fact, this does not happen,
thanks to the group theory relations mentioned above, plus the
discrete symmetries of parity and charge conjugation.
Parity simultaneously reverses all helicities in an amplitude;
\eqn{PolarizationVector} shows that it is implemented by
the exchange $\spa{i}.{j} \leftrightarrow \spb{j}.{i}$.
Charge conjugation is related to the antisymmetry of the color-ordered
rules; for pure-glue partial amplitudes it takes the form of a
reflection identity,
$$
A_n^\tree(1,2,\ldots,n)\ =\ (-1)^n\ A_n^\tree(n,\ldots,2,1) \,.
\equn\label{reflectionid}
$$
For amplitudes with external quarks, it allows one to exchange a
quark and anti-quark, or equivalently to flip the helicity on a
quark line.
As an example,
with the use of parity and cyclic ($Z_5$) symmetry, we
can reduce the five-gluon amplitude at tree level to a combination
of just four independent partial amplitudes:
$$
\eqalign{
& A_5^\tree(1^+,2^+,3^+,4^+,5^+)\,,\qquad
A_5^\tree(1^-,2^+,3^+,4^+,5^+)\,, \cr
& A_5^\tree(1^-,2^-,3^+,4^+,5^+)\,,\qquad
A_5^\tree(1^-,2^+,3^-,4^+,5^+) \,. \cr}
\equn\label{treefiveg}
$$
Furthermore, the first two partial amplitudes here vanish (see below),
and there is a group theory ($U(1)$ decoupling) relation between the
last two \cite{3,21}, so there is only one
independent nonvanishing object to calculate. At one loop there are
four independent objects --- \eqn{treefiveg} with $A_5^\tree$ replaced
by $A_{5;1}$ --- but only the last two contribute to the NLO
cross-section, due to the tree-level vanishings. The explicit
expression for $A_{5;1}(1^-,2^-,3^+,4^+,5^+)$ is given in
section~\docLink{slac-pub-7111-0-0-3.tcx}[SusyDecompositionSubsection]{3.2}.
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