Recursive box and vertex integrations for the one-loop hexagon reduction in the physical region

We provide a technique which makes an efficient numerical evaluation feasible for the ndimensional triangle functions and n+2-dimensional box functions, resulting from a reduction of the one-loop hexagon integral. At the level of the three- and four-point functions, a dimensional recursion of an adaptive numerical integration with extrapolation is effective for the corresponding low-dimensional integrals, even to integrate through threshold singularities present in cases of physical kinematics. An important reason for the reduction to these levels (in lieu of resorting to analytical formulas at a higher level), is that infrared divergences are made transparent so they can be separated. We give results for various sets of kinematic configurations, showing the feasibility and accuracy of the approach. Thus a bridge is provided between the computations of the reduction and direct numerical integration at a lower level.