Complete One-Loop Amplitudes With Massless Propagators

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 (4−2ǫ)-dimensional master integrals; these formulas depend on an additional variable, u, which encodes the dimensional shift. Second, convert the u-dependent coefficients of (4 − 2ǫ)-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. The proof is constructive. As a byproduct, we produce simpler algebraic expressions for the scalar integral coefficients. In particular, we now separate the box and pentagon contributions explicitly.