A Complete Basis of Helicity Operators for Subleading Factorization

TL;DR

This work develops a complete SCET helicity-operator framework that extends factorization beyond leading power by constructing a complete, power-suppressed basis of hard scattering operators for back-to-back jet processes. It demonstrates how ultrasoft, transverse momentum insertions, and multiple collinear fields can be organized within a scalar helicity basis, enabling straightforward tree-level matching and clear symmetry constraints. The authors provide explicit operator bases up to ${ m O}( ext{λ}^2)$, perform tree-level matching for several subleading cases, and project results into the helicity basis to obtain Wilson coefficients, illustrating how angular momentum and CP symmetries constrain contributions to dijet observables. The framework is extensible to SCET$_{ m II}$, SCET$_+$, massive quarks, and evanescent operators, establishing a robust foundation for systematic power corrections in fixed-order and resummed QCD predictions for jet-related observables.

Abstract

Factorization theorems underly our ability to make predictions for many processes involving the strong interaction. Although typically formulated at leading power, the study of factorization at subleading power is of interest both for improving the precision of calculations, as well as for understanding the all orders structure of QCD. We use the SCET helicity operator formalism to construct a complete power suppressed basis of hard scattering operators for $e^+e^-\to$ dijets, $e^- p\to e^-$ jet, and constrained Drell-Yan, including the first two subleading orders in the amplitude level power expansion. We analyze the form of the hard, jet, and soft function contributions to the power suppressed cross section for $e^+e^-\to$ dijet event shapes, and give results for the lowest order matching to the contributing operators. These results will be useful for studies of power corrections both in fixed order and resummed perturbation theory.

Figures (2)

• Figure 1: Examples of the contributions to thrust, $\tau$, in the dijet limit, at leading power in a) and b), and subleading power in c) and d). There is an extra collinear gluon in a) from splitting, and in b) there is an extra gluon from soft emission. In c) the extra energetic gluon is collinear with the quark, but occurs without a nearly onshell parent propagator. Likewise in d) the extra soft emission amplitude is subleading.
• Figure 2: A illustration of the helicity selection rule for $e^+e^-\to$ dijets. In a) the collinear particles along the $n$ axis carry $|h=2|$, and have a vanishing projection onto the $J_{e\pm}$ current. In b), the collinear particles carry $|h=0|$ and therefore have a non-vanishing projection onto the $J_{e\pm}$ current.