Non-Global Logarithms, Factorization, and the Soft Substructure of Jets

TL;DR

This work tackles non-global logarithms (NGLs) in jet observables by proposing a factorization framework that uses multi-differential measurements to isolate in-jet and out-of-jet scales. Through soft-subjet factorization in SCET and the introduction of the dressed gluon approximation, the authors demonstrate RG-based resummation of NGLs and connect to the Banfi-Marchesini-Smye (BMS) equation. One- and two-dressed-gluon results reproduce leading NGL behavior with sub-percent accuracy over relevant LHC-like ranges and reveal the buffer-region dynamics near jet boundaries. The approach provides a systematic path toward all-orders NGL resummation for jet substructure and suggests avenues for subleading corrections and improved Monte Carlo implementations.

Abstract

An outstanding problem in QCD and jet physics is the factorization and resummation of logarithms that arise due to phase space constraints, so-called non-global logarithms (NGLs). In this paper, we show that NGLs can be factorized and resummed down to an unresolved infrared scale by making sufficiently many measurements on a jet or other restricted phase space region. Resummation is accomplished by renormalization group evolution of the objects in the factorization theorem and anomalous dimensions can be calculated to any perturbative accuracy and with any number of colors. To connect with the NGLs of more inclusive measurements, we present a novel perturbative expansion which is controlled by the volume of the allowed phase space for unresolved emissions. Arbitrary accuracy can be obtained by making more and more measurements so to resolve lower and lower scales. We find that even a minimal number of measurements produces agreement with Monte Carlo methods for leading-logarithmic resummation of NGLs at the sub-percent level over the full dynamical range relevant for the Large Hadron Collider. We also discuss other applications of our factorization theorem to soft jet dynamics and how to extend to higher-order accuracy.

Figures (11)

• Figure 1: (a) Illustration of the phase space configuration and dominant modes for a jet containing a hard core and a soft subjet. Here the gray radiation denotes global soft radiation $S_{n\bar{n} n_{sj}}$, and the green radiation denotes collinear radiation along the direction of the energetic jet axes, $J_n$ and $J_{\bar{n}}$. The soft subjet dynamics is described by soft jet modes, $J_{n_{sj}}$ shown in blue, and boundary soft modes shown in red, $S_{n_{sj} \bar{n}_{sj}}$. (b) Schematic of the ladder of factorization theorems defined by increasingly differential measurements made on the jet. With each additional measurement, the NGLs are pushed to the soft function at a lower unresolved scale. The $S'$, and $S"$ are schematic, typically being a product of multiple functions, but depend only on a single scale.
• Figure 2: (a) Illustration of the phase space for a jet on which $e_{2}^{(\alpha)}$ and $e_{3}^{(\alpha)}$ have been measured. Jets with a two-prong structure lie in the lower (red) region of phase space, where $e_{3}^{(\alpha)} \ll (e_{2}^{(\alpha)})^3$. The energy correlation functions parametrically separate the one-prong and two-prong regions of phase space. (b) Illustration of the phase space for a jet on which both $e_{2}^{(\alpha)}$ and $e_{2}^{(\beta)}$ have been measured, with $\alpha>\beta$, shown in gray. Jets dominated by soft radiation lie in the upper region of the phase space, where $e_{2}^{(\alpha)} \sim e_{2}^{(\beta)}$. Jets with two energetic collinear subjets populate the region $e_{2}^{(\alpha)} \sim {e_{2}^{(\beta)}}^{\alpha/\beta}$.
• Figure 3: The two distinct subjet configurations which exist in the two prong region of phase space. (a) Two energetic collinear subjets, which has been studied in Ref. Bauer:2011uc, and populates the region of phase space $e_{2}^{(\beta)} \sim \left ({e_{2}^{(\alpha)}}\right )^{\beta/\alpha}$. (b) Wide angle soft subjet, which populates the region of phase space $e_{2}^{(\alpha)} \sim e_{2}^{(\beta)}$.
• Figure 4: A schematic of the multi-stage matching procedure used to prove the factorization theorem of Eq. (\ref{['fact_inclusive_form_1']}) for the soft subjet region of phase space. As discussed in the text, the matching proceeds in three stages: matching to SCET, refactorizing the soft function to describe the soft jet production, and factorizing the boundary soft mode. The canonical scales of the modes in the final factorization theorem are shown on the right, ordered in virtuality. Here we have chosen an angular exponent $\alpha=2$ for concreteness.
• Figure 5: The diagrammatic structure of the cut diagrams of the soft jet production in the factorization equation \ref{['eq:N-eikonal_lines_soft']}. For concreteness, we have taken the soft jet to be created off of the $i, j$ eikonal lines of the parent $N$-jet factorization. The color matrices of each function are to be inserted at the indicated regions along the eikonal lines. Thus a $\mathbf{T}$ matrix of the soft jet production is inserted between the hard function $H_N$ and any global soft radiation. Note that the new soft jet eikonal line enters only into the color multipole function of the $i, j$ lines.