Differential equations for two-loop four-point functions

At variance with fully inclusive quantities, which have been computed alrea dy at the two- or three-loop level, most exclusive observables are still kn own only at one loop, as further progress was hampered so far by the greate r computational problems encountered in the study of multi-leg amplitudes b eyond one loop. We show in this paper how the use of tools already employed in inclusive calculations can be suitably extended to the computation of l oop integrals appearing in the virtual corrections to exclusive observables , namely two-loop four-point functions with massless propagators and up to one off-shell leg. We find that multi-leg integrals, in addition to integra tion-by-parts identities, obey also identities resulting from Lorentz-invar iance. The combined set of these identities can be used to reduce the large number of integrals appearing in an actual calculation to a small number o f master integrals. We then write down explicitly the differential equation s in the external invariants fulfilled by these master integrals, and point out that the equations can be used as an efficient method of evaluating th e master integrals themselves. We outline strategies for the solution of th e differential equations, and demonstrate the application of the method on several examples. (C) 2000 Elsevier Science B.V. All rights reserved.