Extracting quadratic propagators by refined graphic rule

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.