---
title: "Jet Clustering Microscope Interactive Lab | Sandbox Physics"
description: "Can one very soft particle change what we call a jet? A jet momentum changes smoothly as extra radiation becomes softer or a splitting becomes collinear. Counting individual particles jumps with every added particle, so that count has no such limit."
canonical_url: "https://sandboxphysics.com/labs/jet-clustering/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/jet-clustering/"
last_modified: "2026-10-27"
---

FR03 · QCD jets

# Jet Clustering Microscope

**Can one very soft particle change what we call a jet?** A jet momentum changes smoothly as extra radiation becomes softer or a splitting becomes collinear. Counting individual particles jumps with every added particle, so that count has no such limit.

Drag the gold particle between sprays and compare three clustering algorithms. Split a hard particle into two and watch which quantities approach the original event.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Synthetic massless particle lists are clustered by inclusive generalized sequential recombination with four-vector addition. This is a finite event calculation; showering, hadronization and detector response are left out. Dashed ellipses show radius guides around the final axes, not measured jet boundaries.

Leading jet Second jet Drag the soft particle Dots: particles · diamonds: summed axes · dashed: radius guide

A CONTROLLED SOFT LIMIT

### The jet momentum changes continuously

The scan keeps the event and probe direction fixed. A gold marker shows the current setting.

### Counting particles reacts differently

Any added particle changes the count by one, however small its momentum becomes. At exactly zero opening, splitting changes the count while leaving the clustered four-vector unchanged.

Distance used by the algorithm

$$
\begin{aligned}d_{ij}&=\min(p_{T,i}^{2p},p_{T,j}^{2p})\\&\quad\cdot\frac{\Delta R_{ij}^2}{R^2}\\d_{iB}&=p_{T,i}^{2p}\end{aligned}
$$

$$
\begin{aligned}\Delta R_{ij}^2&=\Delta y_{ij}^2+\Delta\phi_{ij}^2\\\Delta y_{ij}&=y_i-y_j\end{aligned}
$$

The exponent is minus one for anti-kT, zero for Cambridge/Aachen and one for kT. Merge the closest pair or declare a jet when its beam distance is smallest.

[Cacciari, Salam & Soyez (2008), Eq. 1 and §2.1](https://arxiv.org/abs/0802.1189) · [FastJet 3.4 manual, §3.4 and §4](https://fastjet.fr/repo/fastjet-doc-3.4.0.pdf)

[From a collision to jets and inference: eight connected chapters](https://sandboxphysics.com/stories/collider-qcd/)

Start with four-momentum, work through clustering, response matrices and likelihoods, then check extrapolation against a public jet spectrum.

Physics tutorial

## Why a jet must survive soft radiation

**Background** A quark or gluon reaches a detector as a spray of particles. Clustering assigns particles to jets so that their total momenta can be compared with a short-distance calculation.

**Why it matters** A perturbative prediction sums over unresolved soft emissions and collinear splittings. The observable needs a stable limit when those changes become arbitrarily small.

Start with the essentials

- **Focus question**: Which measurements change smoothly when a particle becomes arbitrarily soft?
- **One-sentence intuition**: Jet four-momenta have a soft and collinear limit. A raw particle count changes by one for each extra particle.

### Core mathematical model

#### Generalized sequential-recombination distance

$$
d_{ij}=\min(p_{T,i}^{2p},p_{T,j}^{2p})\frac{\Delta y_{ij}^{2}+\Delta\phi_{ij}^{2}}{R^{2}},\qquad d_{iB}=p_{T,i}^{2p}
$$

Compare all pair and beam distances. Merge the closest pair, or remove a final jet when its beam distance is smallest. The exponent chooses anti-kT, Cambridge/Aachen or kT.

#### E-scheme recombination

$$
p_J^\mu=\sum_{i\in J}p_i^\mu,\qquad m_J^2=E_J^2-\lVert\mathbf p_J\rVert^2
$$

The merged object generally has mass. Its rapidity is recomputed from energy and longitudinal momentum; averaging input directions would give another recombination scheme.

### Common difficulties

#### Safety allows a finite response

**Typical misconception** An infrared-safe jet never changes when any soft particle is added.

**Better mental model** Safety is a limiting statement: the hard observable approaches its original value as the added momentum tends to zero. Finite soft radiation contributes finite momentum.

#### Finite opening prepares another parent

**Typical misconception** A massless on-shell particle can split at a finite angle while conserving its original four-vector.

**Better mental model** Two non-collinear massless daughters have a positive invariant mass. This preset changes the prepared event at finite angle; the exact collinear limit is the conservation comparison.

### Run the experiment

1. 01

   #### Move a soft particle

   Drag the gold ring across the two sprays, then lower its momentum toward zero.

   **What to observe:** Its jet assignment can switch at a boundary. Away from a degenerate event, its effect on a hard jet vanishes with its momentum.

2. 02

   #### Split a hard particle

   Choose the splitting preset and reduce the opening angle all the way to zero.

   **What to observe:** The two daughters become exactly collinear and their total four-vector returns to the original. The particle count stays one higher.

3. 03

   #### Compare ordering rules

   Choose the nearby-hard-cores preset and switch algorithms. Increase the radius to join the sprays.

   **What to observe:** All three algorithms are infrared and collinear safe. Their finite-radiation assignments and clustering histories can still differ.

Physical conclusion

Use a quantity with a controlled soft and collinear limit before comparing a jet calculation with data.

## Related experiments and articles

Collider physics · Reading path

### Collider physics from events to inference

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Choose quantities that survive extra radiation**](https://sandboxphysics.com/stories/collider-qcd/radiation-and-safe-observables/)
2. [Article **Build jets by following the nearest pair**](https://sandboxphysics.com/stories/collider-qcd/sequential-jet-clustering/)
3. [Article **Recalculate a published benchmark and test a public jet spectrum**](https://sandboxphysics.com/stories/collider-qcd/reading-and-reproducing-a-paper/)
4. [Lab **Collider Signals and Systematic Errors** Fit a signal and its competing background](https://sandboxphysics.com/labs/collider-likelihood/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)