Order statistics for multijet events

TL;DR

The paper introduces rank-ordered jet rapidities, i.e. the order statistics of the event's rapidities $Y_{(1)}<...<Y_{(N)}$, as simple yet powerful observables to probe high-energy QCD at the LHC. By mapping jet rapidities to order-statistic distributions $f_{(l)}(y)$ derived from a parent density $f(y)$, the authors show that different dynamics—BFKL-like vs DGLAP-like showers—imprint distinctive shapes on the rank histograms, even when inclusive densities are similar. Using LL BFKLex and LL DGLAP-based generators (PYTHIA8 and HERWIG7) at 8 and 13 TeV, they find robust, observable differences in inner ranks (and edge vs central behavior) that persist under variations of jet radius, PDFs, and MPI/hadronization, and remain when enforcing large rapidity separations. The proposed approach offers a practically accessible, data-driven handle on high-energy radiation patterns and sets the stage for future extensions to NLL BFKL and Run 2/3 data analyses, potentially with joint rank distributions or lower $p_T$ thresholds.

Abstract

We show that rank-ordered jet rapidity distributions - a direct application of order statistics - provide a simple yet powerful probe of high-energy (small-x) QCD dynamics at the LHC. In inclusive dijet topologies at s^(1/2) = 8 and 13 TeV, with realistic jet selections, we compare a BFKL-based Monte Carlo (BFKLex) to two general-purpose event generators based on collinear factorization and DGLAP parton showers, PYTHIA8 (pT-ordered) and HERWIG7 (angular-ordered). Even when two underlying dynamics happen to give similar inclusive jet rapidity distributions, such observables are too coarse to discriminate their underlying rapidity point processes, whereas the rank-ordered distributions remain sensitive to the differences in how rapidity space is filled. For fixed multiplicity (N=3) and for the second-most-forward/backward jets across multiplicities, BFKLex populates the rapidity interval more democratically, whereas the general-purpose event generators exhibit comparatively stronger edge enhancement for N=3 and narrower, more centrally concentrated distributions for the second-most ranks. These shape differences are stable under variations of jet radius, proton PDFs, and MPI/hadronization settings, and persist when requiring large rapidity separation between the outer jets. Rank-ordered rapidities thus compress genuinely exclusive information about the multi-jet final state into one-dimensional, normalized histograms that are directly measurable with existing dijet and Mueller-Navelet selections and provide a new handle on high-energy radiation patterns.

Figures (6)

• Figure 1: 8 TeV, fixed multiplicity $N{=}3$. Normalized to unit area rapidity distributions for Jet1, Jet2 and Jet3 (ordered by rapidity) from BFKLex, PYTHIA8 and HERWIG7. For all three ranks we see a clear separation: BFKLex fills the inter--tag interval more democratically, while the DGLAP showers exhibit comparatively stronger edge enhancement (Jet1, Jet3) and narrower central distribution (Jet2).
• Figure 2: 8 TeV, across multiplicities. Normalized rapidity distributions for the most--backward (MB) and most--forward (MF) jets. The three generators give very similar shapes, dominated by PDFs and acceptance.
• Figure 3: 8 TeV, across multiplicities. Normalized rapidity distributions for the second--most--backward (SMB) and second--most--forward (SMF) jets. These inner ranks show the largest separation between BFKLex and the DGLAP showers: PYTHIA8 and HERWIG7 exhibit more central weight, while BFKLex remains broader with a much more pronounced left (backward jet) and right (forward jet) skewness.
• Figure 4: 13 TeV, fixed multiplicity $N{=}3$. Normalized rapidity distributions for Jet1, Jet2 and Jet3. The central rank again shows a clear separation between BFKLex and the DGLAP showers.
• Figure 5: 13 TeV, across multiplicities. Normalized rapidity distributions for the MB and MF jets. As at 8 TeV, the three generators give very similar shapes.