---
title: "Reconstruct a collision and count its rate"
description: "Add four-momenta to reconstruct a mass, then connect selected event counts to cross section, luminosity and efficiency."
canonical_url: "https://sandboxphysics.com/stories/collider-qcd/four-momentum-and-rates/"
language: "en"
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  zh-CN: "https://sandboxphysics.com/zh/stories/collider-qcd/four-momentum-and-rates/"
last_modified: "2026-10-26"
---

[Collider physics from events to inference](https://sandboxphysics.com/stories/collider-qcd/)

FR03 · L1 · Chapter 1 of 8

# Reconstruct a collision and count its rate

Add four-momenta to reconstruct a mass, then connect selected event counts to cross section, luminosity and efficiency.

Before you beginVectors, the Pythagorean theorem and elementary probability. Natural units are introduced here.

Sandbox Physics · 2026-10-26

A short-lived particle can disappear before it reaches any detector layer. Its decay products can still reveal its mass: add their energies, add their momentum vectors, then ask what mass that combined four-momentum has. A second question follows naturally. Once we can recognise the process, how often does it happen?

## Start with what the detector measures

Take the beam direction as the positive longitudinal axis and the two perpendicular directions as the transverse plane. A reconstructed particle has energy $E$ and momentum components $p_x,p_y,p_z$. We use natural units, $c=1$, so energies, momenta and masses are quoted in GeV. A GeV is an energy unit; the usual factors of the speed of light are absorbed into the convention.

Four-momentum is the ordered set $p=(E,p_x,p_y,p_z)$. Its squared length uses a minus sign for each spatial component. This choice encodes the fact that different observers can disagree about energy and momentum but agree about mass.

$$
p^2=E^2-p_x^2-p_y^2-p_z^2=m^2.
$$

For several final-state particles, conservation of energy and momentum tells us to sum the four-vectors first. Squaring each particle separately would give their individual masses and lose the relative directions that contain the parent-mass information.

$$
M^2=\left(\sum_i E_i\right)^2-\left|\sum_i\mathbf p_i\right|^2.
$$

## A worked two-particle reconstruction

Suppose two approximately massless particles each have energy 50 GeV. They leave in opposite transverse directions, so their momenta cancel. The combined energy is 100 GeV and the invariant mass is 100 GeV. Now send the same energies out at a right angle. Some net momentum remains, and the reconstructed mass falls to about 70.7 GeV.

To see the angle explicitly, expand the square of the sum. Each individual mass vanishes and the momentum dot product is the product of the magnitudes and the cosine of the angle. The opening angle $\theta$ is measured between the two momentum vectors.

$$
M^2=2E_1E_2(1-\cos\theta),\qquad M(\theta=\pi/2)=\sqrt{5000}\ \mathrm{GeV}\approx70.7\ \mathrm{GeV}.
$$

An invariant mass is stable under a common Lorentz boost of the whole event. It changes when the measured energies or directions change. Detector resolution therefore broadens a narrow mass feature even when the underlying particle has a sharply defined mass.

## Turn a count into a rate

A cross section measures the probability of a specified reaction in units of area. Integrated luminosity describes how many collision opportunities the beams supplied, in inverse area. Their product is an expected event count. Acceptance is the fraction of the chosen physical final state that falls inside a stated geometric and kinematic region; efficiency is the probability that an event in that region is reconstructed and selected.

Let $\mathcal L$ be integrated luminosity, $\sigma$ the cross section, $A$ acceptance, $\epsilon$ selection efficiency and $B$ the expected background count. All factors except luminosity and cross section are dimensionless. A simple one-region counting model is

$$
\mathbb E[N]=\mathcal L\sigma A\epsilon+B,\qquad \widehat\sigma=\frac{N-B}{\mathcal L A\epsilon}.
$$

For a worked example, observe 120 selected events, estimate a background of 20, use an integrated luminosity of 10 inverse femtobarns, and take the product of acceptance and efficiency to be 0.5. The inferred cross section is 20 femtobarns. Multiplying back gives 100 expected selected signal events, which checks both the arithmetic and the units. These are invented classroom numbers, not a collider measurement.

A fiducial cross section defines the physical region as part of the result. That can reduce the extrapolation beyond the observed region. The analysis must still describe how reconstruction and migrations alter events within that definition. Recovering an unmeasured shape requires additional assumptions beyond an efficiency correction.

## Try the event reconstruction

Open the particle-collider Lab and inspect one event. Before changing the view, identify which objects belong in the parent candidate. Compare their directions with the reconstructed mass. The apparatus and decay samples are a teaching model; the reconstruction principle is the same four-vector sum used above.

Predict the effect of making a pair more nearly parallel at fixed energies. Calculate two opening-angle cases on paper and compare the sign of the change with the event geometry. A rotation of the camera should change the picture without changing the mass.

## Exercises with explanations

Exercise 1. Two massless particles have energies 40 and 90 GeV and travel in opposite directions. Find their invariant mass. Then explain why adding the energies alone gives a different answer.

Answer. The squared mass is four times the product of the energies, giving 120 GeV. Their total energy is 130 GeV, but their momenta do not cancel because their magnitudes differ. The remaining momentum accounts for the difference between energy and mass.

Exercise 2. Keep the selected signal count and luminosity in the worked rate example, but revise the acceptance-efficiency product from 0.5 to 0.4. Should the inferred cross section rise or fall?

Answer. It rises to 25 femtobarns. A less efficient detector must have received more physical signal events to leave the same selected count. This is a transfer question about the measurement model, not an extra statistical fluctuation in the observed sample.

Exercise 3. A mass peak doubles in height after the bin width is halved. What must you check before claiming twice as many particles?

Answer. Check whether the vertical axis is counts per bin or a density per unit mass, and compare the integrated yield over the same mass interval. A height depends on binning; the event count in a fixed interval has a direct counting interpretation.

## Try the chapter’s physical question

[Inspect a collision event](https://sandboxphysics.com/labs/particle-collider/)

## Sources and checked scope

1. [Particle Data Group, Kinematics (2024)](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-kinematics.pdf)

   Sections 49.1–49.2: four-vectors, scalar products and centre-of-mass energy. Natural units are stated on page 1.

2. [Particle Data Group, Quantum Chromodynamics (2024)](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-qcd.pdf)

   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.

[Next chapterWhy a proton beam has many collision energies](https://sandboxphysics.com/stories/collider-qcd/partons-and-factorization/)