Recently, the Cachazo-He-Yuan (CHY) approach has been extended to the loop level, but the resulting loop integrand has propagators that are linear in the loop momentum unlike Feynman’s. In this paper, we present a new technique that directly produces quadratic propagators identical to Feynman’s from the CHY approach. This paper focuses on the theory, but extensions to other theories are briefly discussed. In addition, our proposal has an interesting geometric meaning; we can interpret this new formula as a unitary cut on a higher genus Riemann surface.