A generalised-$k_t$ jet algorithm for Deep Inelastic Scattering

This paper introduces an inclusive generalised-$k_t$ jet algorithm for Deep Inelastic Scattering defined in the Breit frame, investigating its phenomenological applications for identifying the struck quark jet, assessing its sensitivity to non-perturbative effects, and comparing its performance with the Centauro algorithm.

The Gist

Imagine you are a detective trying to solve a crime that happened inside a tiny, invisible box. The "crime" is a high-speed collision between a particle of light (an electron) and a particle of matter (a proton). When they smash together, they shatter into a chaotic spray of new, smaller particles. Your job is to look at this messy spray and figure out exactly which piece came from the original "victim" (the quark inside the proton) and which pieces are just debris from the explosion.

This paper introduces a new, smarter way to sort that debris. Here is the breakdown in simple terms:

The Setting: The "Breit Frame"

Usually, when physicists look at these collisions, they see a messy, spinning mess. To make sense of it, they imagine moving to a special "camera angle" called the Breit frame.

• The Analogy: Imagine the proton is a train moving forward, and the electron is a bullet fired backward. In the Breit frame, we zoom in so that the train and the bullet are heading straight at each other like two cars in a head-on crash.
• The Result: After the crash, the "victim" (the struck quark) flies off in one direction (the "current hemisphere"), and the rest of the train (the "proton remnant") flies off in the other. The goal is to catch the victim's debris without accidentally grabbing the train's debris.

The Problem: Old Sorting Tools

For years, scientists have used different "jet algorithms" (sorting rules) to group these particles into clusters called "jets."

• Some tools are like sieves that only catch big rocks (hard particles).
• Some are like magnets that pull in everything nearby, regardless of size.
• The problem is that in this specific type of collision (Deep Inelastic Scattering), the old tools sometimes get confused. They might group the victim's debris with the train's debris, or they might miss the victim entirely because the debris is too soft or spread out.

The New Solution: The "Generalised-kT" Algorithm

The authors have created a new, flexible sorting tool called the Generalised-kT jet algorithm. Think of this tool as a smart, adjustable vacuum cleaner.

1. It's Tunable: The tool has a dial (called the parameter $p$) that changes how it behaves:

   • Setting $p=1$ (The "Soft-First" Mode): It acts like a vacuum that sucks up the light, fluffy dust (soft particles) first, then moves to the heavier rocks. This helps map out the shape of the debris cloud very precisely.
   • Setting $p=0$ (The "Angle-First" Mode): It ignores the weight of the particles and only cares about how close they are to each other. It groups things based purely on proximity.
   • Setting $p=-1$ (The "Hard-First" Mode): This is the "anti-kT" setting. It finds the biggest, heaviest rock first and then pulls everything else toward it. This creates very neat, circular clusters, like a perfect snowball.

2. The "Macrojet" Trick: One of the biggest challenges is knowing which cluster of debris belongs to the victim. The authors added a special rule to their tool: Find the cluster carrying the most "forward momentum."

   • The Analogy: Imagine the victim is a runner who was pushed forward. Even if they drop some items along the way, the group of items moving fastest in the forward direction is the one that belongs to them. The tool automatically picks this group (called the "macrojet") and ignores the stuff flying backward.

What They Found

The team tested their new tool against the old ones and a recently proposed tool called "Centauro."

• Neatness: The "Hard-First" (anti-kT) version creates the cleanest, most circular jets, making them easy to identify.
• Accuracy: The new tool is very good at separating the "victim's" debris from the "train's" debris. It avoids the mistake of accidentally sucking up the wrong pieces.
• Robustness: They tested the tool by simulating what happens when particles turn into real-world matter (a process called "hadronization"). They found that while the new tool is affected by this, it handles it much better than some older methods, keeping the data reliable.

Why It Matters

This new tool is like upgrading from a basic broom to a high-tech vacuum with a "find my lost keys" sensor. It allows scientists to look at the data from past experiments (like HERA) and future ones (like the Electron-Ion Collider) with much clearer eyes. By sorting the debris more accurately, they can better understand the fundamental rules of how matter holds together, specifically how the "glue" (gluons) inside the proton behaves.

In short: The paper gives physicists a new, customizable way to sort the messy aftermath of particle collisions, ensuring they can clearly see the "hero" of the crash (the struck quark) amidst the chaos.

Technical Summary

Technical Summary: A generalised-kt jet algorithm for Deep Inelastic Scattering

Problem Statement
Deep Inelastic Scattering (DIS) at lepton-hadron colliders, such as the former HERA and the forthcoming Electron–Ion Collider (EIC), provides a unique environment for investigating nucleon structure. While jet algorithms have been extensively developed for hadron-hadron collisions (notably the anti-$k_t$ algorithm at the LHC), the landscape for DIS lacks a single dominant algorithm. Existing DIS algorithms often struggle to balance the preservation of the natural hemisphere structure of the Breit frame with the need for robust clustering sequences that respect soft radiation and angular ordering. Furthermore, recent developments like the Centauro algorithm, while innovative, require comparison against a broader family of algorithms to assess their phenomenological utility, particularly regarding the identification of the struck quark jet and sensitivity to non-perturbative effects.

Methodology
The authors introduce an inclusive generalised-$k_t$ jet algorithm specifically formulated for DIS, implemented as a FastJet plugin (DISGenkt) within the fjcontrib package. The algorithm operates in the Breit frame, where the virtual photon and the incoming proton collide head-on, naturally separating the event into a "current hemisphere" (containing the struck quark) and a "target hemisphere" (containing the proton remnant).

The algorithm is governed by two primary parameters:

1. The exponent $p$: Controls the transverse-momentum dependence of the clustering, defining a family of algorithms analogous to those in $e^+e^-$ and $pp$ collisions:
   • $p=0$: Angular-ordered (Cambridge/Aachen) variant.
   • $p=1$: $k_t$ variant.
   • $p=-1$: Anti-$k_t$ variant.
2. The jet radius $R$: Defines the angular scale of clustering.

The distance measures are defined as:
$$d_{ij} = \frac{\min(E_i^{2p}, E_j^{2p})(1 - \cos \theta_{ij})}{1 - \cos R}$$
$$d_{iB} = E_i^{2p}(1 - \cos \theta_{iB})$$
where $\theta_{iB}$ is the angle between pseudojet $i$ and the proton beam. Crucially, the beam distance $d_{iB}$ is sensitive to the angular separation from the beam, allowing for the early promotion of remnant-hemisphere particles to jets, thereby preserving the separation between hemispheres.

To identify the "macrojet" (the jet most likely to contain the fragments of the struck quark), the authors propose a prescription based on the light-cone fraction $z_{jet}$, defined via a Sudakov decomposition. The macrojet is selected as the one maximizing $z_{jet}$. This approach is highlighted as infrared and collinear safe, unlike selecting the jet with the most negative rapidity, which can be disrupted by soft emissions.

Key Contributions

• Algorithm Formulation: The paper presents a generalised-$k_t$ algorithm for DIS that extends the angular-ordered variant previously introduced in Ref. [8] to the full $p$-family. It combines the structure of generalised-$k_t$ algorithms with a beam distance measure sensitive to the proton beam direction in the Breit frame.
• Macrojet Identification: A robust, infrared-safe method for identifying the struck-quark jet is provided, utilizing the light-cone momentum fraction $z_{jet}$ rather than rapidity alone.
• Phenomenological Comparison: The authors perform a comprehensive comparison between the new generalised-$k_t$ variants ($p=0, 1, -1$) and the recently proposed Centauro algorithm using Pythia 8.3 event generation.
• Implementation: The algorithm is made publicly available as a FastJet plugin, facilitating immediate use in re-analyses of HERA data and future EIC studies.

Results
The study analyzes reconstructed jets and macrojet observables (Breit-frame energy, $z_{jet}$, and transverse momentum $q_T$) across different jet radii ($R_0 = 2/3$ and $R_0 = 1$):

• Jet Boundaries: The anti-$k_t$ variant produces perfectly conical jets in the $(\theta, \phi)$ plane, whereas $k_t$, Cambridge/Aachen (C/A), and Centauro algorithms yield irregular, jagged boundaries.
• Macrojet Identification: Selecting the macrojet via $z_{jet}$ successfully suppresses the soft tail observed when using rapidity-based selection, ensuring infrared safety.
• Sensitivity to $R_0$:
  • The anti-$k_t$ variant is stable against variations in $R_0$ regarding constituent selection but risks absorbing remnant-hemisphere particles into the current-hemisphere macrojet if $R_0$ is too large.
  • The $k_t$ and C/A variants show significant sensitivity to $R_0$, where small changes can alter which particles are clustered into the macrojet.
  • The Centauro algorithm produces smaller, more isolated jets and tends to suppress clustering in the target hemisphere more effectively than the generalised-$k_t$ variants.
• Distributions:
  • In the soft region ($2E_{jet}/Q < 1$), Centauro produces a significantly softer macrojet distribution compared to the generalised-$k_t$ algorithms.
  • In the hard region, the generalised-$k_t$ variants (especially anti-$k_t$ and $k_t$) shift the peak of the energy distribution slightly higher compared to C/A, while Centauro shifts it lower for $R_0=2/3$.
  • The $z_{jet}$ distribution reveals three distinct regions (peak, dijet, multi-jet). The generalised-$k_t$ algorithms behave similarly in the peak and dijet regions, while Centauro shows a smeared peak towards lower $z_{jet}$ values.
• Non-Perturbative Effects: The study assesses the impact of hadronisation and beam remnants. Hadronisation induces large distortions in the soft region of the energy spectrum and shifts the $z_{jet}$ peak to lower values. However, the normalisation of the generalised-$k_t$ algorithms in the hard region is surprisingly insensitive to hadronisation, unlike Centauro which shows large corrections. The $q_T$ observable is found to be relatively insensitive to hadronisation but sensitive to beam remnants, making it a potential tool for studying parton transverse momentum distributions.

Significance and Claims
The paper claims that the proposed generalised-$k_t$ algorithm offers a flexible and theoretically sound tool for DIS jet physics. Its primary significance lies in its ability to provide a clean separation between target and current hemisphere jets through the $z_{jet}$ macrojet definition. The authors suggest that these algorithms are well-suited for re-analysing historical HERA data to provide complementary constraints on the strong coupling constant $\alpha_s$ and to constrain non-perturbative parameters (hadronisation and intrinsic $k_t$) in event generators.

The authors remain modest regarding the immediate impact on $\alpha_s$ extraction, noting that such an extraction is unlikely to be competitive with other methods on its own. Instead, they emphasize the value of studying non-perturbative physics in DIS, particularly the ability to study jets in bins of $Q$ to probe the energy dependence of non-perturbative parameters. The work serves as a foundation for future studies and practical applications in the context of the upcoming EIC.