Next-to-leading-order QCD corrections to jet cross sections and jet rates in deeply inelastic electron-proton scattering

Jet cross sections in deeply inelastic scattering in the case of transverse photon exchange for the production of (1+1) and (2+1) jets are calculated in next-to-leading-order QCD (here the "+1" stands for the target remnant jet, which is included in the jet definition). The jet definition scheme is based on a modified JADE cluster algorithm. The calculation of the (2+1) jet cross section is described in detail. Results for the virtual corrections as well as for the real initial- and final-state corrections are given explicitly. Numerical results are stated for jet cross sections as well as for the ratio $\frac{{\sigma }_{\mathrm{}(2+1)\mathrm{}\mathrm{jet}}}{{\sigma }_{\mathrm{tot}}}$ that can be expected at E665 and DESY HERA. Furthermore the scale ambiguity of the calculated jet cross sections is studied and different parton density parametrizations are compared.