  We use the antenna subtraction method to isolate the double virtual infrared
singularities present in gluonic scattering amplitudes at
next-to-next-to-leading order. In previous papers, we derived the subtraction
terms that rendered (a) the double real radiation tree-level process finite in
the single and double unresolved regions of phase space and (b) the mixed
single real radiation one-loop process both finite and well behaved in the
unresolved regions of phase space. Here, we show how to construct the double
virtual subtraction term using antenna functions with both initial- and
final-state partons which remove the explicit infrared poles present in the
two-loop amplitude. As an explicit example, we write down the subtraction term
for the four-gluon two-loop process. The infrared poles are explicitly and
locally cancelled in all regions of phase space leaving a finite remainder that
can be safely evaluated numerically in four-dimensions.

High Energy Physics - Phenomenology