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

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.

arXiv comment33 pages, 7 figures. [v2] Major revision. Soft-quark equation of motion used to reformulate the Lagrangians. [v3] conceptual clarifications, version published in Physical Review D

References in corpus (14)

• An effective field theory for collinear and soft gluons: heavy to light decays
• Soft-Collinear Factorization in Effective Field Theory
• Summing Sudakov logarithms in B -> X_s + gamma in effective field theory
• Invariant Operators in Collinear Effective Theory
• Hard Scattering Factorization from Effective Field Theory
• Soft-collinear effective theory and heavy-to-light currents beyond leading power
• Multipole-expanded soft-collinear effective theory with non-abelian gauge symmetry
• Reduced Effective Lagrangians
• A Complete Basis for Power Suppressed Collinear-Ultrasoft Operators
• On Power Suppressed Operators and Gauge Invariance in SCET
• Threshold factorization of the Drell-Yan process at next-to-leading power
• Subleading Power Factorization with Radiative Functions
• Two-Loop Radiative Jet Function for Exclusive $B$-Meson and Higgs Decays
• On the gauge-invariance of SCET beyond leading power

Cited by in corpus (1)

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