Local vertices, quadratic propagators and double-copy structure of one-loop integrands from forward limits

By worldsheet approach, $n$-point one-loop integrand can be expressed as a combination of $(n+2)$-point tree-level bi-adjoint scalar (BS) amplitudes under forward limit. The integrands constructed by this approach have two closely related features which differ from conventional Feynman diagrams. First, the denominators of loop propagators are linear functions of the loop momentum. Second, the local vertex expression is not manifest. In our previous work, a systematic approach was proposed to handle the nonlocal terms in the one-loop integrand of Yang-Mills-scalar (YMS) theory. Upon canceling the nonlocalities, quadratic propagator forms of both YMS and Yang-Mills (YM) integrands are naturally obtained. In this paper, we generalize the calculation to theories involving gravitons by introducing the one-loop double Yang-Mills-scalar (dYMS) integrands. The cancellation of the nonlocalities of the dYMS integrand in the forward limit coincides with the emergence of local multi-point vertices. We provide two equivalent methods for extracting the vertices and give the final expression of the dYMS integrand with quadratic propagators. In this formula, tree-level effective subcurrents are attached to the loop propagator line via local vertices. Each of the effective subcurrents exhibits double-copy structure, in the sense that it is expressed as a combination of tree-level BS subcurrents associated with two copies of kinematic coefficients. The quadratic propagator formulas for Einstein-Yang-Mills (EYM) and gravity (GR) integrands are further derived, by the help of the formula for dYMS. The extraction of local vertices in one-loop dYMS integrand also applies at tree-level, thus we have the corresponding expressions of tree-level dYMS, EYM, and GR amplitudes.