  Pfaffian diagrams are formulated to represent gluon amplitudes computed from
the Cachazo-He-Yuan (CHY) formula. They may be regarded as a systematic
regrouping of Feynman diagrams after internal momenta are expanded and products
of vertex factors are evaluated. This reprocessing enables gluon amplitudes
expressed in Pfaffian diagrams to contain less terms. For example, there are 19
terms for the four-point amplitude in Pfaffian diagrams, and 35 terms in
Feynman diagrams. Gauge invariance is simpler and more explicit in Pfaffian
diagrams, in that subset of diagrams with the same root configuration are
already gauge invariant in all lines but two. In getting to these results,
several technical difficulties must be overcome. Double poles must be converted
to simple poles, integrations must be carried out directly and formulated into
simple rules, and the three \M constant lines must be suitably chosen to
minimize the number of terms present.
