Polynomial Structures in One-Loop Amplitudes

TL;DR

This paper proves that coefficients of master integrals in one-loop scattering amplitudes depend polynomially on a variable u for massless propagators, enabling explicit algebraic expressions with separated box and pentagon contributions.

Contribution

It provides a constructive proof of polynomial dependence of master integral coefficients on u, facilitating algebraic expressions with explicit separation of contributions.

Findings

Polynomial dependence of coefficients is proven for massless propagators.

Explicit algebraic formulas with separated contributions are derived.

The approach simplifies the calculation of one-loop amplitudes.

Abstract

A general one-loop scattering amplitude may be expanded in terms of master integrals. The coefficients of the master integrals can be obtained from tree-level input in a two-step process. First, use known formulas to write the coefficients of (4-2epsilon)-dimensional master integrals; these formulas depend on an additional variable, u, which encodes the dimensional shift. Second, convert the u-dependent coefficients of (4-2epsilon)-dimensional master integrals to explicit coefficients of dimensionally shifted master integrals. This procedure requires the initial formulas for coefficients to have polynomial dependence on u. Here, we give a proof of this property in the case of massless propagators. The proof is constructive. Thus, as a byproduct, we produce different algebraic expressions for the scalar integral coefficients, in which the polynomial property is apparent. In these formulas, the box and pentagon contributions are separated explicitly.