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

TL;DR

This paper investigates gauge invariance issues in the position-space formulation of SCET, revealing violations in the hard-collinear sector and proposing gauge-invariant Lagrangians by adding soft-quark equation of motion terms.

Contribution

It identifies gauge invariance violations in the hard-collinear sector of position-space SCET and constructs gauge-invariant Lagrangians by incorporating soft-quark equations of motion.

Findings

Violations of gauge invariance arise in the hard-collinear sector.

Explicit calculations show radiative jet functions are not gauge invariant.

Gauge-invariant Lagrangians are obtained by adding soft-quark EOM terms.

Abstract

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.