On differential operators and unifying relations for $1$-loop Feynman integrands

Abstract: We generalize the unifying relations for tree amplitudes to the $1$-loop Feynman integrands. By employing the $1$-loop CHY formula, we construct differential operators which transmute the $1$-loop gravitational Feynman integrand to Feynman integrands for a wide range of theories, include Einstein-Yang-Mills theory, Einstein-Maxwell theory, pure Yang-Mills theory, Yang-Mills-scalar theory, Born-Infeld theory, Dirac-Born-Infeld theory, bi-adjoint scalar theory, non-linear sigma model, as well as special Galileon theory. The unified web at $1$-loop level is established. Under the well known unitarity cut, the $1$-loop level operators will factorize into two tree level operators. Such factorization is also discussed.