On Power Suppressed Operators and Gauge Invariance in SCET

TL;DR

This study investigates gauge invariance at subleading power in SCET, showing that a field redefinition based on Wilson lines yields hat fields whose transformations remain governed by the leading-order gauge symmetry, enabling $O(\,\lambda^2)$ Lagrangians to be written with manifest invariance. It introduces a two-stage matching framework (SCET_I→SCET_II) to systematically construct power-suppressed soft-collinear operators, with jet functions arising from time-ordered products in SCET_I and constrained by power counting when matching to SCET_II. Through examples like heavy-to-light currents and factorization in B decays, the paper clarifies how SCET_I inputs determine the SCET_II operator basis and the suppression of soft-collinear interactions at leading subleading orders. The results illuminate how gauge invariance, RP invariance, and scale separation jointly shape the subleading operator structure, with implications for factorization and precision predictions in exclusive processes.

Abstract

The form of collinear gauge invariance for power suppressed operators in the soft-collinear effective theory is discussed. Using a field redefinition we show that it is possible to make any power suppressed ultrasoft-collinear operators invariant under the original leading order gauge transformations. Our manipulations avoid gauge fixing. The Lagrangians to O(lambda^2) are given in terms of these new fields. We then give a simple procedure for constructing power suppressed soft-collinear operators in SCET_II by using an intermediate theory SCET_I.

Figures (1)

• Figure 1: Examples of graphs contributing to the matching of the SCET$_{\rm I}$ T-products onto SCET$_{\rm II}$ operators in Eq. (\ref{['O12s']}). The dots denote the insertion of a ${\cal L}_{\xi q}^{(1)}$ and the circled crosses in the two diagrams are $J_{\xi\xi}^{(2,3)}$ operators respectively.