Two-Loop g gg Splitting Amplitudes in QCD

Splitting amplitudes are universal functions governing the collinear behaviorof scattering amplitudes for massless particles. We compute the two-loop g ->gg splitting amplitudes in QCD, N=1, and N=4 super-Yang-Mills theories, whichdescribe the limits of two-loop n-point amplitudes where two gluon momentabecome parallel. They also represent an ingredient in a direct x-spacecomputation of DGLAP evolution kernels at next-to-next-to-leading order. Toobtain the splitting amplitudes, we use the unitarity sewing method. Incontrast to the usual light-cone gauge treatment, our calculation does not relyon the principal-value or Mandelstam-Leibbrandt prescriptions, even though theloop integrals contain some of the denominators typically encountered inlight-cone gauge. We reduce the integrals to a set of 13 master integrals usingintegration-by-parts and Lorentz invariance identities. The master integralsare computed with the aid of differential equations in the splitting momentumfraction z. The epsilon-poles of the splitting amplitudes are consistent with aformula due to Catani for the infrared singularities of two-loop scatteringamplitudes. This consistency essentially provides an inductive proof ofCatani's formula, as well as an ansatz for previously-unknown 1/epsilon poleterms having non-trivial color structure. Finite terms in the splittingamplitudes determine the collinear behavior of finite remainders in thisformula.Comment: 100 pages, 33 figures. Added remarks about leading-transcendentality argument of hep-th/0404092, and additional explanation of cut-reconstruction uniquenes