  One-loop integrands in Cachazo-He-Yuan (CHY) formula, which is based on the
forward limit of tree-level amplitudes, involves linear propagators that are
different from quadratic ones in traditional Feynman diagrams. In this paper,
we provide a general approach to converting linear propagators in one-loop CHY
formula into quadratic propagators, by refined graphic rule stemming from the
recursive expansion of tree-level Einstein-Yang-Mills amplitudes. We establish
the correspondence between refined graphs with cut lines and the bi-adjoint
scalar (BS) Feynman diagrams with linear propagators. Using this correspondence
and helpful properties of Berends-Giele currents in BS theory, we convert the
linear propagators in single-trace Yang-Mills-scalar (YMS) integrands (with a
pure-scalar loop) into quadratic ones. This result is further demonstrated to
match the traditional one-loop Feynman diagrams. The discussions on
single-trace YMS integrands are generalized to multi-trace YMS and Yang-Mills
integrands.
