Hidden self-energy contributions of collinear functions in SCET

Abstract

The LSZ reduction formula requires one to identify and amputate complete propagators on external legs of a Green's function and to evaluate complete two-point functions in the mass-shell limit. Motivated by these requirements, we analyze quark self-energy contributions on external legs in soft-collinear effective theory (SCET). We examine an operator basis that follows directly from full quantum chromodynamics (QCD) (upon application of the SCET equations of motion to express small Dirac components in terms of large Dirac components). We find that, for this basis, the self-energy contributions can be identified from their diagrammatic topologies, as in full QCD. However, for an alternative operator basis that is obtained from the direct-QCD basis by an application of Wilson-line identities, interactions are shifted from a covariant derivative to a Wilson line. Consequently, some self-energy contributions are hidden in diagrams involving Wilson lines, making their identification subtle. We find that the hidden self-energy contributions to the two-point function are ill-defined in the mass-shell limit, making their computation problematic. We introduce a generalization of the LSZ formula that allows one to make different choices for the complete propagator and that compensates for those choices through the factor that arises from the on-shell residue of the two-point function. We use this generalization to explore, in both SCET operator bases, various options for using the LSZ formula to construct the $S$-matrix.