Soft Drop

TL;DR

This work introduces soft drop declustering, a jet grooming/tagging method controlled by $z_{ ext{cut}}$ and $\beta$, and develops a first-principles, resummed understanding of three observables on soft-dropped jets: the energy correlation function $C^{(\alpha)}_{1}$, the groomed jet radius $R_g$, and the jet energy drop $\Delta_E$. By exploring the full range of $\beta$, the authors show how grooming ($\beta>0$) preserves soft-collinear structure while tagging ($\beta\le0$) vetoes certain configurations, with the $\beta=0$ limit closely related to mMDT and exhibiting Sudakov safety. The analysis combines modified leading-logarithmic resummation, multiple-emission effects, and non-global logarithm considerations, and is validated against Pythia simulations, including non-perturbative effects. Applications to boosted $W$ tagging and pileup mitigation are demonstrated, highlighting practical benefits and guiding future exploration of event-wide implementations.

Abstract

We introduce a new jet substructure technique called "soft drop declustering", which recursively removes soft wide-angle radiation from a jet. The soft drop algorithm depends on two parameters--a soft threshold $z_\text{cut}$ and an angular exponent $β$--with the $β= 0$ limit corresponding roughly to the (modified) mass drop procedure. To gain an analytic understanding of soft drop and highlight the $β$ dependence, we perform resummed calculations for three observables on soft-dropped jets: the energy correlation functions, the groomed jet radius, and the energy loss due to soft drop. The $β= 0$ limit of the energy loss is particularly interesting, since it is not only "Sudakov safe" but also largely insensitive to the value of the strong coupling constant. While our calculations are strictly accurate only to modified leading-logarithmic order, we also include a discussion of higher-order effects such as multiple emissions and (the absence of) non-global logarithms. We compare our analytic results to parton shower simulations and find good agreement, and we also estimate the impact of non-perturbative effects such as hadronization and the underlying event. Finally, we demonstrate how soft drop can be used for tagging boosted W bosons, and we speculate on the potential advantages of using soft drop for pileup mitigation.

Figures (12)

• Figure 1: Phase space for emissions on the $(\log \frac{1}{z},\log \frac{R_0}{\theta})$ plane. In the strongly-ordered limit, emissions above the dashed line (Eq. (\ref{['eq:vetoline']})) are vetoed by the soft drop condition. For $\beta > 0$, soft emissions are vetoed while much of the soft-collinear region is maintained. For $\beta = 0$ (mMDT), both soft and soft-collinear emissions are vetoed. For $\beta < 0$, all (two-prong) singularities are regulated by the soft drop procedure.
• Figure 2: Phase space for emissions relevant for ${C^{(\alpha)}_{1}}$ in the $(\log \frac{1}{z},\log \frac{R_0}{\theta})$ plane. The soft dropped region is gray and the first emission satisfying the soft drop criteria is illustrated by the red dot. The leading emission for ${C^{(\alpha)}_{1}}$ is illustrated by the green dot with the forbidden emission region (the Sudakov exponent) shaded in pink.
• Figure 3: The energy correlation functions $C_1^{(\alpha=2)}$ for quark-initiated jets. Here we compare Pythia 8pythia8 (left), our MLL formula in Eq. (\ref{['all-order-ang-end']}) (right, dashed curves), and our MLL plus multiple-emissions formula in Eq. (\ref{['eq:multemis_c1']}) (right, solid curves). These $\alpha = 2$ curves correspond to the case of jet mass-squared (normalized to jet energy squared). We show both the ungroomed (plain jet) distribution, as well as groomed distributions from soft drop declustering with $z_\text{cut} = 0.1$ and various values of $\beta$. For $\beta = 2,1$, we see the expected Sudakov double logarithmic peaks, while $\beta = 0$ (mMDT) has only single logarithms and $\beta = -1$ cuts off at small values. The Pythia 8 distributions do not have hadronization effects, and the MLL distributions are evaluated by freezing $\alpha_s$ in the infrared.
• Figure 4: The energy correlation functions ${C^{(\alpha)}_{1}}$ with $\alpha=1.5,1,0.5$ (top to bottom) for quark-initiated jets. The plots on the left are obtained with Pythia 8, while the ones of the right are our MLL predictions (dashed) with multiple emissions included (solid).
• Figure 5: Phase space for emissions relevant for groomed jet radius $R_g$ in the $(\log \frac{1}{z},\log \frac{R_0}{\theta})$ plane. The soft dropped region is gray and the first emission satisfying the soft drop criteria is illustrated by the red dot. The forbidden emission region (the Sudakov exponent) is shaded in pink.