Subleading terms in the collinear limit of Yang–Mills amplitudes

For two massless particles i and j, the collinear limit is a special kinematic configuration in which the particles propagate with parallel four-momentum vectors, with the total momentum P   distributed as pi=xPpi=xP and pj=(1−x)Ppj=(1−x)P, so that sij≡(pi+pj)2=P2=0sij≡(pi+pj)2=P2=0. In Yang–Mills theory, if i and j are among N gauge bosons participating in a scattering process, it is well known that the partial amplitudes associated to the (single trace) group factors with adjacent i and j   are singular in the collinear limit and factorize at the leading order into (N−1)(N−1)-particle amplitudes times the universal, x-dependent Altarelli–Parisi factors. We give a precise definition of the collinear limit and show that at the tree level, the subleading, non-singular terms are related to the amplitudes with a single graviton inserted instead of two collinear gauge bosons. To that end, we argue that in one-graviton Einstein–Yang–Mills amplitudes, the graviton with momentum P can be replaced by a pair of collinear gauge bosons carrying arbitrary momentum fractions xP   and (1−x)P(1−x)P.