At variance with fully inclusive quantities, which have been computed already at the two-or three-loop level, most exclusive observables are still known only at one-loop, as further progress was hampered so far by the greater computational problems encountered in the study of multi-leg amplitudes beyond 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 loop 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 integration-by-parts identities, obey also identities resulting from Lorentz-invariance. 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 of master integrals. We then write down explicitly the differential equations in the external invariants fulfilled by these master integrals, and point out that the equations can be used as an efficient method of evaluating the master integrals themselves. We outline strategies for the solution of the differential equations, and demonstrate the application of the method on several examples.

The distribution of the spin of the nucleon among its constituents can be parametrized in the form of polarized parton distribution functions for quarks and gluons. Using all available data on the polarized structure function g 1 (x, Q 2 ), we determine these distributions both at leading and next-to-leading order in perturbation theory. We suggest three different, equally possible scenarios for the polarized gluon distribution, which is found to be only loosely constrained by current experimental data. We examine various possibilities of measuring polarized parton distributions at future experiments.

The two-loop QCD corrections to vector boson pair production at hadron colliders involve a new class of Feynman integrals: two-loop four-point functions with two off-shell external legs. We describe their reduction to a small set of master integrals by solving linear relations among them. We then use differential equations in the external invariants to compute all master integrals that are relevant to planar Feynman amplitudes. Our results are expressed analytically in terms of generalized harmonic polylogarithms. The calculation relies heavily on techniques that exploit the algebraic structure of these functions, which we describe in detail.

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