First Subleading Power Resummation for Event Shapes

TL;DR

The paper develops the first all-orders leading-logarithmic resummation of subleading-power (NLP) corrections for an event shape (thrust) in a gauge theory, using soft-collinear effective theory. It identifies a new class of universal subleading operators, the $\theta$-jet and $\theta$-soft functions, which arise from cross-section level mixing and close into a $2\times2$ RG system driven by cusp anomalous dimensions. By solving the NLP RG mixing equations (including running coupling) and applying them to $H\to gg$ in pure glue QCD, it derives analytic NLP LL results and demonstrates explicit exponentiation into Sudakov-like factors, with fixed-order checks through ${\cal O}(\alpha_s^3)$ confirming the structure. The work also establishes RG-consistency constraints that relate NLP anomalous dimensions and constrain the form of the mixing functions, providing a general framework for systematic NLP resummation of collider observables. Overall, this work lays out a path toward controlled NLP resummation for event shapes and related observables in high-energy processes.

Abstract

We derive and analytically solve renormalization group (RG) equations of gauge invariant non-local Wilson line operators which resum logarithms for event shape observables $τ$ at subleading power in the $τ\ll 1$ expansion. These equations involve a class of universal jet and soft functions arising through operator mixing, which we call $θ$-jet and $θ$-soft functions. An illustrative example involving these operators is introduced which captures the generic features of subleading power resummation, allowing us to derive the structure of the RG to all orders in $α_s$, and provide field theory definitions of all ingredients. As a simple application, we use this to obtain an analytic leading logarithmic result for the subleading power resummed thrust spectrum for $H\to gg$ in pure glue QCD. This resummation determines the nature of the double logarithmic series at subleading power, which we find is still governed by the cusp anomalous dimension. We check our result by performing an analytic calculation up to ${\cal O}(α_s^3)$. Consistency of the subleading power RG relates subleading power anomalous dimensions, constrains the form of the $θ$-soft and $θ$-jet functions, and implies an exponentiation of higher order loop corrections in the subleading power collinear limit. Our results provide a path for carrying out systematic resummation at subleading power for collider observables.

Figures (2)

• Figure 1: Two different routings for the soft momentum. In a) the additional soft momentum is routed into the collinear sectors. In b) the additional momentum is routed in through the hard scattering vertex, simplifying the large momentum routed into the collinear sectors.
• Figure 2: Plots of the LP and NLP fixed order and resummed predictions for thrust in pure glue $H\to gg$, with and without running coupling. In a) we show $d\sigma/d\tau$ and in b) we show $\tau d\sigma/ d\tau$. Resummation at LP cures a $1/\tau$ divergence, while resummation at NLP overturns a much weaker logarithmic divergence, leading to a broader shoulder.