One-loop diagrams with quadratic propagators from the worldsheet

It is well known that forward limits of tree-level amplitudes (and those trivalent diagrams they consist of) produce one-loop and with propagators linear in the loop momentum. They naturally arise from worldsheet formulae, an important open problem how to recombine them into usual quadratic propagators. In this paper, we study a new collection functions: generalized Parke-Taylor factors tensor numerators, which are conjectured serve as basis for functions nice property. We present all-multiplicity, closed-form expressions combinations numerators arbitrary rank (including possible tadpole contributions), produced by any pair factors. also briefly comment on reducing onto such basis, applications physical theories.