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. Particularly, we establish the correspondence between refined graphs and bi-adjoint scalar (BS) Feynman diagrams with linear propagators. Using this correspondence and graph-based relations of Berends-Giele currents in BS theory, the nonlocal terms accompanied by refined graphs can either be canceled out or be collected into local ones. Once the locality has been achieved, the integrand with linear propagators can be directly arranged into that with quadratic propagators. Following this approach, we first convert the linear propagators in single-trace Yang-Mills-scalar (YMS) integrands (with a pure-scalar loop) into quadratic ones. This result is then 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.