τ_N (intuitively)

Generate Jet — Presets

Formula

$\tau_N = \frac{1}{d_0} \sum_k p_{T,k}\,\min\!\big(\Delta R_{1,k},\,\Delta R_{2,k},\,\dots,\,\Delta R_{N,k}\big)$

with $d_0 = R \sum_k p_{T,k}$ and $\Delta R_{a,k}=\sqrt{(\eta_a-\eta_k)^2+(\phi_a-\phi_k)^2}$

Subjet axes

N =

Yellow ✕ = subjet axis. Drag to move. Particles are colored by which axis is nearest. Axes start at the optimal positions.

Live values

$\tau_N$ (custom)—

$\tau_1$—

$\tau_2$—

$\tau_3$—

$\tau_2/\tau_1$—

$\tau_3/\tau_2$—

$d_0$—

particles—

What is each piece doing?

Particles ($k$). A jet is a spray of particles; each has a transverse momentum $p_{T,k}$ and a position in the $(\eta,\phi)$ plane. Bigger circle = harder particle.

Axes. Pick $N$ candidate subjet axes. For every particle, find the nearest axis in $\Delta R$. That's the $\min(\Delta R_{1,k},\dots,\Delta R_{N,k})$ inside the sum.

Weighted distance. Multiply that nearest distance by the particle's $p_T$. Hard particles dominate — soft junk barely matters.

Normalization. Divide by $d_0 = R\sum p_T$ so $\tau_N$ is dimensionless and bounded above by ~1 (with jet radius $R$).

Why does it work?

$\tau_N$ is a "jet's distortion when forced into $N$ blobs." If the jet really has $N$ hard prongs, you can place $N$ axes so every hard particle sits right on one — sum is small, $\tau_N \to 0$. If the jet is one-prong but you force $N=2$, the second axis only catches soft stuff far away — $\tau_2$ stays close to $\tau_1$.

This is why ratios are the discriminators: $\tau_2/\tau_1$ small ⇒ 2-prong (W/Z); $\tau_3/\tau_2$ small ⇒ 3-prong (top).

Try it: Pick the W preset. With $N=1$ a single axis can't cover both prongs — $\tau_1$ is large. Switch to $N=2$ and minimize — both axes lock onto the prongs, $\tau_2$ collapses, ratio $\tau_2/\tau_1$ becomes small. Now try the same on the QCD preset: even at $N=2$ the second axis has nothing to catch, ratio stays close to 1.