We compute the O(α 3 s ) virtual QCD corrections to the γ * → qqg matrix element arising from the interference of the two-loop with the tree-level amplitude and from the self-interference of the one-loop amplitude. The calculation is performed by reducing all loop integrals appearing in the two-loop amplitude to a small set of known master integrals. Infrared and ultraviolet divergences are both regularized using conventional dimensional regularization, and the ultraviolet renormalization is performed in the MS scheme. The infrared pole structure of the matrix elements agrees with the prediction made by Catani using an infrared factorization formula. The analytic result for the finite terms of both matrix elements is expressed in terms of one-and two-dimensional harmonic polylogarithms.Among jet observables, the three-jet production rate in electron-positron annihilation plays an outstanding role. The initial experimental observation of three-jet events at PETRA [1], in agreement with the theoretical prediction [2], provided first evidence for the gluon, and thus strong support for the theory of Quantum Chromodynamics (QCD). Subsequently the three-jet rate and related event shape observables were used for the precise determination of the QCD coupling constant α s (see [3] for a review). Especially at LEP, three-jet observables were measured to a very high precision and the error on the extraction of α s from these data is dominated by the uncertainty inherent in the theoretical next-to-leading order (NLO) calculation [4][5][6][7][8] of the jet observables. The planned TESLA [9] linear e + e − collider will allow precision QCD studies at even higher energies than at LEP. Given the projected luminosity of TESLA, one again expects the experimental errors to be well below the uncertainty of the NLO calculation.Related to e + e − → 3 jets by crossing symmetry are (2+1)-jet production in deep inelastic ep scattering and vector-boson-plus-jet production at hadron colliders. The experimental data from HERA on ep → (2 + 1) jets and related event shape observables have already reached a level of precision demanding predictions beyond the present NLO accuracy; a further improvement on these data is expected soon from the HERA high luminosity programme. Similarly, vector-boson production at large transverse momentum is a classic test of QCD in hadron-hadron collisions and demands the theoretical prediction to be as precise as possible. In this case, it is also an important background in searches for new physics at the Tevatron and the LHC.Besides its phenomenological importance, the three-jet rate has also served as a theoretical testing ground for the development of new techniques for higher order calculations in QCD: both the subtraction [4] and the phase-space slicing [5] methods for the extraction of infrared singularities from NLO real radiation processes were developed in the context of the first three-jet calculations. The systematic formulation of phase-space slicing [7] as well as the dipole subtraction [8] method w...

We compute the two-loop QCD helicity amplitudes for the process e + e − → qqg. The amplitudes are extracted in a scheme-independent manner from the coefficients appearing in the general tensorial structure for this process. The tensor coefficients are derived from the Feynman graph amplitudes by means of projectors, within the conventional dimensional regularization scheme. The actual calculation of the loop integrals is then performed by reducing all of them to a small set of known master integrals. The infrared pole structure of the renormalized helicity amplitudes agrees with the prediction made by Catani using an infrared factorization formula. We use this formula to structure our results for the finite part into terms arising from the expansion of the pole coefficients and a genuine finite remainder, which is independent of the scheme used to define the helicity amplitudes. The analytic result for the finite parts of the amplitudes is expressed in terms of one-and two-dimensional harmonic polylogarithms.

Differential cross sections for the scattering of electrons by a short-range potential in the presence of an intense, linearly polarized, low-frequency field are obtained in three different ways: within the Kroll-Watson approximation, within the impulse approximation and through an exact Floquet calculation. Examples are given where the cross section for small-angle scattering with emission or absorption of photons is larger by several orders of magnitude than predicted by the Kroll-Watson approximation. The enhancement can be ascribed to off-shell contributions and is correctly reproduced by the impulse approximation. However, no such enhancement is found in model calculations for helium. The accuracy of the impulse approximation in the vicinity of a resonance is also studied.