UV and IR rational terms in two-loop amplitudes: first insights

In this proceeding, we review the construction of UV rational terms at two loops, and present the first insights of the cancellation mechanism of two-loop IR rational terms with an example in massive QED. The rational terms are crucial ingredients that are needed for the four-dimensional calculation of scattering amplitudes, which is suitable for automation in a numerical framework. Within automated one-loop tools, the numerator of D dimensionally regularised loop amplitudes are usually constructed through numerical algorithms in D=4 dimensions, while the contributions of the missing (D-4) dimensional parts can be reconstructed a posteriori by means of rational counterterms. At two loops with four-dimensional numerators, we show that the missing (D-4) part can also be reconstructed through the process-independent two-loop rational terms in any renormalisable theories. This opens the door to the development of an efficient two-loop amplitudes automation in the future.