In this work we show how a double-cover (DC) extension of the Cachazo, He and
Yuan formalism (CHY) can be used to provide a new realization for the
factorization of the amplitudes involving gluons and scalar fields. First, we
propose a graphic representation for a color-ordered Yang-Mills (YM) and
special Yang-Mills-Scalar (YMS) amplitudes within the scattering equation
formalism. Using the DC prescription, we are able to obtain an algorithm
(integration-rules) which decomposes amplitudes in terms of three-point
building-blocks. It is important to remark that the pole structure of this
method is totally different to ordinary factorization (which is a consequence
of the scattering equations). Finally, as a byproduct, we show that the soft
limit in the CHY approach, at leading order, becomes trivial by using the
technology described in this paper.Comment: 50+7 pages and typos fixed. Some modifications were made to improve
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