One-loop same helicity YM amplitudes from BG currents

We propose and prove a new formula for the one-loop all same helicity Yang-Mills amplitudes. These amplitudes are seen to arise as a sum of products of two tree-level Berends-Giele currents connected by an effective propagator. To make sense of the propagators one needs to introduce the so-called region, or dual momenta. The formula is proven by observing that it readily implies the correct collinear limit properties. The only non-trivial part of the proof is establishing that our formula for the amplitude is invariant under shifts of the region momenta.