It is well known that forward limits of tree-level amplitudes (and those trivalent diagrams they consist of) produce one-loop amplitudes and trivalent diagrams with propagators linear in the loop momentum. They naturally arise from one-loop worldsheet formulae, and an important open problem is how to recombine them into usual one-loop diagrams with quadratic propagators. In this talk we study a new collection of worldsheet functions: generalized one-loop Parke-Taylor factors with tensor numerators, which are conjectured to serve as a basis for one-loop worldsheet functions with this nice property. We present all-multiplicity, closed-form expressions for combinations of one-loop trivalent diagrams with quadratic propagators and tensor numerators to arbitrary rank (including possible tadpole contributions), produced by any pair of Parke-Taylor factors. What' s more, based on our new formulae, once the one-loop worldsheet functions for certain theories are expanded onto these Parke-Taylor basis, the Bern-Carrasco-Johansson numerotars can be read from the expansion directly, which unravels a new kind of double copy relations based on color-kinematics duality with quadratic propagators at one-loop level.