Polynomial Structures in One-Loop Amplitudes

TL;DR

This work provides a constructive proof that the coefficients of master integrals in one-loop amplitudes are polynomial in the dimension-shift variable $u$ for massless propagators, enabling efficient dimensional reduction from $(4-2\epsilon)$-dimensional to dimensionally shifted bases. By refining triangle, bubble, box, and pentagon coefficients and isolating the $u$-dependence, the authors derive explicit expressions that separate box and pentagon contributions and establish precise degree bounds for the resulting polynomials. The dimensionally shifted basis and the derived polynomial structure yield practical routes for numerical implementations, with a detailed gluon five-point example illustrating the method's applicability. The work also clarifies connections to other contemporary approaches and discusses extensions to the massive case and numerical stability considerations. Overall, it advances robust, scalable one-loop amplitude calculations in a dimensional-regularization framework.

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.