  In subleading powers of soft-collinear effective theory (SCET), the
Lagrangian contains couplings between soft quarks and hard-collinear quarks.
Matrix elements of the hard-collinear parts of these couplings are radiative
jet functions. In the position-space formulation of SCET, the Lagrangians are
constructed from operators that appear to be gauge invariant. Nevertheless, we
find violations of gauge invariance arise in the hard-collinear sector because
gauge transformations can shift the momentum of a hard-collinear quark field
from the hard-collinear sector to the soft sector, where the hard-collinear
fields, by definition, have no support. The violations of gauge invariance are
manifested in perturbation theory in the hard-collinear sector through the
absence of certain Feynman diagrams that would be present in full QCD. A
consequence of the absence of these diagrams is that the radiative jet
functions that follow directly from the position-space Lagrangians are not
gauge invariant, and we demonstrate this through explicit calculations in
lower-order perturbation theory. We obtain gauge-invariant Lagrangians by
adding to existing position-space Lagrangians terms that are proportional to
the soft-quark equation of motion. These gauge-invariant Lagrangians are valid
for nonzero, as well as zero, quark masses. We also remark briefly on the gauge
invariance of certain Lagrangians that have been constructed in the
label-momentum formulation of SCET.
