BCJ Relations for One-Loop QCD Integral Coefficients

Abstract

We present a set of one-loop integral coefficient relations in QCD. The unitarity method is useful for exposing one-loop amplitudes in terms of tree amplitudes. The coefficient relations are induced by tree-level BCJ amplitude relations. We provide examples for box, triangle, and bubble coefficients. These relations reduce the total number of independent coefficients needed to calculate one-loop QCD amplitudes.

Figures (13)

• Figure 1: The box, triangle, and bubble cuts. At each corner there are an arbitrary of external lines.
• Figure 2: The zero-mass, one-mass, two-mass-e, two-mass-h, three-mass, and four-mass box cuts are shown above. Two-mass-e and two-mass-h stand for 'easy' and 'hard'. The $K_i$ are sums of massless momenta and the $k_i$ is the momentum of a single external massless leg.
• Figure 3: The one-mass, two-mass, and three-mass triangle cuts are shown above.
• Figure 4: The needed two-mass bubble cut is shown above. The one-mass bubble cuts vanish.
• Figure 5: We consider two box cuts needed, which are identical up to a twisting of the $K_3$ leg. These two coefficients have the same loop solution, which allows for a tree level BCJ amplitude relation to be used to relate the two integral coefficients.