Numerical integration of one-loop Feynman diagrams for N-photon   amplitudes
  In the calculation of cross sections for infrared-safe observables in high energy collisions at next-to-leading order, one approach is to perform all of the integrations, including the virtual loop integration numerically. One would use a subtraction scheme that removes infrared and collinear divergences from the integrand in a style similar to that used for real emission graphs. Then one would perform the loop integration by Monte Carlo integration along with the integrations over final state momenta. In this paper, we have explored how one can perform the numerical integration. We have studied the N-photon scattering amplitude with a massless electron loop in order to have a case with a singular integrand that is not, however, so singular as to require the subtractions. We report results for N = 4, N = 5 with left-handed couplings, and N=6. 
Zoltan Nagy and Davison E. Soper