Construction of infrared finite observables in N=4 super Yang-Mills theory

In this paper we give all the details of the calculation that we presented in our previous paper ArXiv:0908.0387 where the infrared structure of the MHV gluon amplitudes in the planar limit for ${\cal N}=4$ super Yang-Mills theory was considered in the next-to-leading order of perturbation theory. Explicit cancellation of the infrared divergencies in properly defined inclusive cross-sections is demonstrated first in a toy model example of "conformal QED" and then in the real ${\cal N}=4$ SYM theory. We give the full-length details both for the calculation of the real emission and for the diagrams with splitting in initial and final states. The finite parts for some inclusive differential cross-sections are presented in an analytical form. In general, contrary to the virtual corrections, they do not reveal any simple structure. An example of the finite part containing just the log functions is presented. The dependence of inclusive cross-section on the external scale related to the definition of asymptotic states is discussed