Transmutation operators and expansions for one-loop Feynman integrands

In this paper, the connections among one-loop Feynman integrands of a wide range of theories are further investigated. The work includes two parts. First, we construct a new class of differential operators which transmute the one-loop gravitational Feynman integrands to Einstein-Yang-Mills and Yang-Mills Feynman integrands, thus linking these theories together. Second, by using one-loop level transmutation relations, together with some general conditions such as gauge and Lorentz invariance, we derive the expansions of Feynman integrands of certain theories to those of other theories. In particular, we find the Feynman integrands of all theories under consideration can be expanded to integrands of bi-adjoint scalar theory. The unified web for expansions is established, including gravity, Einstein-Yang-Mills theory, Yang-Mills-scalar theory, Dirac-Born-Infeld theory, extended Dirac-Born-Infeld theory, special Galileon theory, and nonlinear sigma model. The systematic rules for evaluating coefficients in the expansions are provided, and the duality between transmutation relations and expansions is shown.