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.

Jets above 10 GeV $2$

Leading transverse momentum $99.3\,\mathrm{GeV}$

Leading jet mass $16.7\,\mathrm{GeV}$

Change in particle count $+1$

ONE EVENT, DIFFERENT QUESTIONS

What survives a soft change?

Clustering algorithm

Jet radius

R

0.700

Gold particle momentum

p_{T,s}

3.00\,\mathrm{GeV}

Daughter opening angle

\delta\phi

0\,\mathrm{rad}

Drag the gold ring in the event. Arrow keys move it too; hold Shift for smaller steps.

Model and angular conventions

The beam defines the longitudinal axis. The map uses true rapidity and periodic azimuth in radians; natural units set the speed of light to one. Particle positions on this map are momentum directions.

y=\frac12\ln\frac{E+p_z}{E-p_z}

\phi=\operatorname{atan2}(p_y,p_x)

At zero opening angle, the two daughters exactly conserve the original four-vector. A finite opening prepares a different event with a different parent virtuality.

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 jetSecond jetDrag the soft particleDots: 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 · FastJet 3.4 manual, §3.4 and §4

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 misconceptionAn infrared-safe jet never changes when any soft particle is added.

Better mental modelSafety 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 misconceptionA massless on-shell particle can split at a finite angle while conserving its original four-vector.

Better mental modelTwo 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

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.
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.
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.

Jet Clustering Microscope

The jet momentum changes continuously

Counting particles reacts differently

Why a jet must survive soft radiation

Core mathematical model

Generalized sequential-recombination distance

E-scheme recombination

Common difficulties

Safety allows a finite response

Finite opening prepares another parent

Run the experiment

Move a soft particle

Split a hard particle

Compare ordering rules

Related experiments and articles

Collider physics from events to inference