One-loop diagrams with quadratic propagators from the worldsheet

TL;DR

This paper introduces a new basis of worldsheet functions with quadratic propagators for one-loop amplitudes, providing explicit formulas and potential applications in physical theories.

Contribution

It develops a basis of one-loop worldsheet functions with quadratic propagators using generalized Parke-Taylor factors, including explicit all-multiplicity formulas.

Findings

Explicit formulas for one-loop trivalent diagrams with quadratic propagators.

Demonstration of the basis's completeness for arbitrary multiplicity.

Potential applications to simplifying one-loop amplitude computations.

Abstract

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 paper, 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. We also briefly comment on reducing worldsheet functions onto such a basis, and applications to one-loop amplitudes in physical theories.