  Many search strategies for the Standard Model Higgs boson apply specific
selection criteria on hadronic jets observed in association with the Higgs
boson decay products, either in the form of a jet veto, or by defining event
samples according to jet multiplicity. To improve the theoretical description
of Higgs-boson-plus-jet production (and the closely related Higgs boson
transverse momentum distribution), we derive the two-loop QCD corrections to
the helicity amplitudes for the processes $H \to ggg$ and $H\to q \bar q g$ in
an effective theory with infinite top quark mass. The helicity amplitudes are
extracted from the coefficients appearing in the general tensorial structure
for each process. The coefficients are derived from the Feynman graph
amplitudes by means of projectors within the conventional dimensional
regularization scheme. The infrared pole structure of our result agrees with
the expectation from infrared factorization and the finite parts of the
amplitudes are expressed in terms of one- and two-dimensional harmonic
polylogarithms.
