Choose quantities that survive extra radiation

A detector may resolve one narrow energy deposit as two, or record a particle too soft to matter for the main event. An observable is infrared and collinear safe when its value approaches the original value as the extra energy tends to zero, or as a splitting becomes exactly parallel. This property lets a finite-resolution measurement connect to a perturbative calculation.

Two tests for the same event

Infrared refers here to low-energy radiation. Start with a fixed set of outgoing momenta and add a fixed momentum $q$ multiplied by a dimensionless scale $\varepsilon$ . Its energy vanishes as the scale tends to zero. A safe observable $O$ must approach its original value as that energy vanishes. A collinear test replaces a massless momentum with two parallel massless momenta sharing its energy, with fraction $z$ and complementary fraction $1-z$ .

O(p_1,\ldots,p_n,\varepsilon q)\longrightarrow O(p_1,\ldots,p_n)\quad(\varepsilon\to0),

O(\ldots,zp,(1-z)p,\ldots)\longrightarrow O(\ldots,p,\ldots).

A finite emission and an unresolved limit ask different questions

These are limits of the observable definition. A finite-energy emission can genuinely change an event; safety does not require every radiation pattern to look identical. In an actual momentum-conserving emission the other particles recoil too. The small perturbation tests isolate what the measurement definition does in the unresolved limit.

Counting particles fails the collinear test

Split a 100 GeV massless particle into exactly parallel particles of 40 and 60 GeV. Their four-vector sum is unchanged. Their multiplicity rises from one to two. Taking the opening angle closer to zero never removes that jump in the count.

Multiplicity can still be measured. Its interpretation depends on resolution, thresholds and hadron formation, and requires a suitable theoretical treatment. A jet momentum formed by adding four-vectors behaves differently: the parallel replacement gives exactly the same total momentum. This is the first useful contrast between a detector-level count and an energy-flow observable.

A threshold can create another edge. Adding an arbitrarily soft particle changes an unthresholded particle count by one. With a fixed nonzero threshold, the strict zero-energy addition no longer passes selection, but an unresolved collinear splitting can still move daughters below threshold. State the full observable, including selection, before testing it.

An event shape quantifies how pencil-like the energy flow is

Transverse thrust asks how much transverse momentum can be aligned with the best possible axis in the transverse plane. For each trial unit vector $\mathbf n_T$ , project every transverse momentum onto it, take the absolute value, add the projections, and divide by the sum of momentum magnitudes. Choose the axis giving the largest ratio.

T_\perp=\max_{|\mathbf n_T|=1}\frac{\sum_i|\mathbf p_{T,i}\cdot\mathbf n_T|}{\sum_i|\mathbf p_{T,i}|}.

The ratio is dimensionless. Two perfectly opposite particles give one. Energy distributed over several directions lowers it. A collinear splitting preserves both sums for every trial axis, so it also preserves their maximum. Adding a soft particle changes the numerator and denominator by quantities proportional to its momentum, giving a continuous zero-energy limit.

This definition is an ideal particle-level example over a specified complete set. Restricting rapidity coverage, choosing charged particles only or imposing thresholds creates a different measurement and can change the perturbative or nonperturbative treatment it needs.

A worked shape calculation

Put two opposite particles of transverse momentum 50 GeV along the horizontal direction. Add an opposite pair of 5 GeV particles vertically, so total transverse momentum still vanishes. In the first quadrant the thrust numerator is $100\cos\alpha+10\sin\alpha$ . Its largest value is the length of the vector with components 100 and 10, reached when $\tan\alpha=0.1$ .

T_\perp=\frac{\sqrt{100^2+10^2}}{110}\approx0.914.

Keeping the axis horizontal would give about 0.909. The small difference matters: the observable includes optimization over the axis. If each vertical particle becomes progressively softer, the thrust approaches one continuously. These momenta form a constructed analytic example; a shower calculation supplies a different source of events.

Compare safe and unsafe changes

In the jet-clustering Lab, begin with an unchanged hard event. Add a soft particle, then reduce its energy. Record the jet momentum and multiplicity separately. Next split a particle into a nearly collinear pair and shrink their opening angle. Predict which quantity returns to its original value before reading the result.

Keep the jet radius and momentum threshold fixed during each comparison. Keep a finite jet-boundary crossing separate from the zero-energy and zero-angle tests; they examine different changes to the event.

Exercises with answers

Exercise 1. A 40 GeV particle becomes two exactly parallel particles of 10 and 30 GeV. What happens to transverse thrust and to multiplicity?

Answer. Thrust is unchanged because each projection and the momentum-magnitude sum are additive for parallel momenta. Multiplicity increases by one. The two answers come from different observable definitions applied to the same replacement.

Exercise 2. Does infrared safety prove that detector noise has no effect?

Answer. No. It controls a limiting response to vanishing radiation. Finite noise, many soft contributions and thresholds can still produce a measurable change. Their size belongs in the detector and pileup model.

Exercise 3. Replace the vertical pair in the worked example by a pair of 10 GeV particles. Find the new thrust and explain the trend.

Answer. The maximizing numerator is $\sqrt{100^2+20^2}$ and the denominator is 120 GeV, giving about 0.850. More momentum now points away from the dominant horizontal direction, so the event is less pencil-like.

Sources and checked scope

Particle Data Group, Quantum Chromodynamics (2024)
Sections 9.2.2–9.2.3, especially Eq. (9.15): collinear factorization, scale dependence and infrared-safe observables. This is a fixed 2024 reference, not a claim about the latest PDF fit.
Cacciari, Salam & Soyez, The anti-kt jet clustering algorithm, arXiv:0802.1189v2
Definition (1a–b), Section 2.1 and Figure 1. The classroom calculation checks isolated-hard-particle soft capture; it does not regenerate the Herwig event or the full active-area plot.
Cacciari, Salam & Soyez, FastJet user manual, arXiv:1111.6097v1
Sections 3.1, 3.4 and 4: four-momentum representation, E-scheme recombination and sequential algorithms. The browser implementation is a small teaching implementation, not the FastJet package.