We present the two-loop QCD helicity amplitudes for quark-quark and quarkantiquark scattering. These amplitudes are relevant for next-to-next-to-leading order corrections to (polarized) jet production at hadron colliders. We give the results in the 't Hooft-Veltman and four-dimensional helicity (FDH) variants of dimensional regularization and present the scheme dependence of the results. We verify that the finite remainder, after subtracting the divergences using Catani's formula, are in agreement with previous results. We also provide the amplitudes for gluino-gluino scattering in pure N = 1 supersymmetric Yang-Mills theory. We describe ambiguities in continuing the Dirac algebra to D dimensions, including ones which violate fermion helicity conservation. The finite remainders after subtracting the divergences using Catani's formula, which enter into physical quantities, are free of these ambiguities. We show that in the FDH scheme, for gluino-gluino scattering, the finite remainders satisfy the expected supersymmetry Ward identities.
JHEP09(2004)039logarithms [4]. There are also sizable uncertainties associated with the experimental input to the parton distribution functions [5]. Nevertheless, an exact next-to-next-to-leading order (NNLO) computation would be desirable. An important step has recently been accomplished with the computation of the three-loop splitting function by Moch, Vermaseren and Vogt [6]. There has also been some earlier work on global fits to the data [7] within an approximate next-to-next-to-leading order (NNLO) framework [8]. Once combined with the matrix elements in complete programs, this should considerably reduce the renormalization and factorization scale uncertainties in production rates. For a summary of the various expected improvements see, for example, ref. [9].Recent years have seen rapid progress in our ability to compute two-loop matrix elements, especially when there is dependence on more than a single kinematic variable [10]-[23]. Much of this progress has relied on new developments in loop integration [24]-[30] and in understanding the infrared divergences of the theory [31].An NNLO calculation of jet production requires six-point tree-level, one-loop fivepoint amplitudes and two-loop four-point amplitudes. The tree amplitudes for six external partons [32,33] and the one-loop amplitudes for five external partons [34] have been determined some time ago. Anastasiou, Glover, Oleari, and Tejeda-Yeomans have provided the NNLO interferences of the two-loop amplitudes with the tree amplitudes, for all QCD four-parton processes, summed over all external helicities and colors [15]. The helicity amplitudes for gg → gg were presented in ref. [18]. Theqq → gg and qg → qg helicity amplitudes were presented in refs. [21,22]. Recently, while preparing this paper, Glover presented the four-quark helicity amplitudes [23]. Here we present the same amplitudes using somewhat different methods, as well as the N = 1 supersymmetric version of these amplitudes. We also describe ambiguities in the...