Infrared finite cross sections at NNLO

In recent years there has been significant progress in the calculation of two-loop amplitudes [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. These amplitudes are needed for fully differential next-to-next-to-leading order (NNLO) calculations to improve the accuracy of theoretical predictions relevant to high-energy collider experiments. At the next-to-next-to-leading order level the ingredients for the third order term in the perrturbative expansion for quantities depending on n resolved “hard” partons are the already mentionend n-parton two-loop amplitudes, the (n + 1)parton one-loop amplitudes and the (n + 2) Born amplitudes. Taken separately, each one of these contributions is infrared divergent. Only the sum of all contributions is infrared finite. Here I review the state of the art for the cancellation of infrared divergences between these different contributions at NNLO. Infrared divergences occur already at next-toleading order. At NLO real and virtual corrections contribute. The virtual corrections contain the loop integrals and can have, in addition to ultraviolet divergences, infrared divergences. If loop amplitudes are calculated in dimensional regularisation, the IR divergences manifest themselves as explicit poles in the dimensional regularisation parameter ε = 2−D/2. These poles cancel with similar poles arising from amplitudes with additional partons but less internal loops, when integrated over phase space regions where two (or more) partons become “close” to each other. In general, the Kinoshita-Lee-Nauenberg theorem [ 14, 15] guarantees that any infrared-safe observable, when summed over all states degenerate according to some resolution criteria, will be finite. However, the cancellation occurs only after the integration