Building Blocks for Subleading Helicity Operators

TL;DR

This work extends on-shell helicity methods by building a complete, scalar helicity-building-block basis to describe subleading power operators in Soft-Collinear Effective Theory (SCET). By introducing collinear and ultrasoft helicity blocks, including $\,\mathcal{B}_{i\pm}^a$, $\chi_{i\pm}$, and a hierarchy of helicity currents, the authors provide a framework to construct hard-scattering operators at any order in the SCET power expansion with manifest gauge invariance and transparent color structure via ultrasoft Wilson lines from the BPS field redefinition. A central contribution is the angular momentum conservation constraint, $|h_{n_i}^{tot}| \leq \sum_{j: \hat{n}_j \neq \hat{n}_i} |h_{n_j}^{tot}|$, which significantly reduces the allowed helicity configurations at subleading power and guides operator enumeration. The paper illustrates the method by building an explicit ${\cal O}(\lambda^2)$ basis for back-to-back $q\bar{q}gg$ channels, showing how angular-momentum rules and BPS-induced Wilson lines streamline the operator basis and color algebra. The results enable systematic subleading power factorization and resummation in collider observables, with a companion paper promising a complete two-direction basis to $\mathcal{O}(\lambda)$ and $\mathcal{O}(\lambda^2)$ and broader channel coverage.

Abstract

On-shell helicity methods provide powerful tools for determining scattering amplitudes, which have a one-to-one correspondence with leading power helicity operators in the Soft-Collinear Effective Theory (SCET) away from singular regions of phase space. We show that helicity based operators are also useful for enumerating power suppressed SCET operators, which encode subleading amplitude information about singular limits. In particular, we present a complete set of scalar helicity building blocks that are valid for constructing operators at any order in the SCET power expansion. We also describe an interesting angular momentum selection rule that restricts how these building blocks can be assembled.

Figures (2)

• Figure 1: Example of scattering amplitudes with energetic particles in four distinct regions of phase space, 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 schematic illustration of the helicity selection rule with two axes, as relevant for the case of $e^+e^-\to$ dijets. In a) the $n$-collinear sector carries $|h|=2$, and therefore has a vanishing projection onto the $J_{e\pm}$ current. In b), the collinear sector carries $|h|=0$ and has a non-vanishing projection onto the $J_{e\pm}$ current.