Analytic Solutions

We discuss analytic solutions of MB representations for virtual loop and real phase space integrals. We present examples of how MB loop integrals can be transformed to nested sums over residues and derive analytic solutions for some one- and two-dimensional sums. The evaluation of real phase space angular integrals using MB representations and corresponding sums is also examined. We show how higher-dimensional MB integrals can be decoupled by changing MB variables and how MB integrals can be expressed via real Euler-type integrals. We also present a brief discussion on the symbolic evaluation of Euler-type integrals as a way of obtaining analytic solutions. Finally, we consider approximations of MB integrals in ratios of both kinematic parameters and masses present in propagators of FI, as well as by the method of expansion by regions.