Gauge invariance of radiative jet functions in the position-space formulation of SCET

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 gauge-invariant building blocks. Nevertheless, we find violations of gauge invariance in the radiative jet functions that we construct from these Lagrangians. We trace the violations of gauge invariance to Feynman diagrams that would be present in full QCD, but are missing in the radiative jet functions, owing to the requirement that the radiative jet functions contain only hard-collinear lines. We construct gauge-invariant Lagrangians for the interactions between soft quarks and hard-collinear quarks by making use of 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.