---
title: Cross Sections and the Golden Rule
module: Units and Kinematics
moduleNumber: 2
lessonNumber: 4
order: 204
summary: >
  The cross section measures how often a scattering happens and the decay width how
  fast a particle disintegrates. This lesson defines both, relates event rate to
  luminosity through $R=\mathcal L\,\sigma$ and lifetime to width through
  $\tau=\hbar/\Gamma$, and states Fermi's golden rule with Lorentz-invariant phase
  space, giving the master formulas that turn an amplitude $\mathcal M$ into a
  measurable rate for $1\to2$ decay and $2\to2$ scattering.
topics: [Units and Kinematics]
draft: false
sources:
  - book: Griffiths
    ref: "Ch. 6 — The Feynman Calculus, §6.1–6.3"
  - book: Halzen & Martin
    ref: "§4 — the golden rule and phase space"
  - book: Thomson
    ref: "§3 — decay rates and cross sections"
---

The kinematics of the previous lessons says what final states are allowed; it does
not say how often each occurs. That is the dynamics, encoded in two measurable
quantities: the cross section for a scattering reaction and the decay width for an
unstable particle. Both are computed from a single amplitude $\mathcal M$ through
Fermi's golden rule, with the kinematic content packaged in Lorentz-invariant phase
space. This lesson defines the observables and states the master formulas that the
QED, weak, and QCD modules evaluate.

## The cross section

The **cross section** $\sigma$ quantifies the likelihood of a reaction as an
effective target area presented by one particle to another. Consider a uniform beam
of particles striking a thin target. Let the beam deliver a **flux** $F$, the number
of incident particles per unit area per unit time, and let the target hold $N_t$
scattering centers. The reaction rate is

$$
R = F\, N_t\, \sigma,
$$

which defines $\sigma$ as the constant of proportionality with dimensions of area.
The cross section is not the geometric size of the particle; it is the area that
reproduces the observed rate, and it depends on the reaction, the energy, and the
force involved. A strong-interaction cross section is of order the geometric
nuclear area, tens of millibarns; a weak-interaction cross section at the same
energy is smaller by many orders of magnitude.

$$
% caption: The cross section as an effective area. A uniform flux F of beam
% particles crosses a thin target; each scattering center blocks an effective area
% sigma, and the reaction rate is the flux times the number of centers times that
% area.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % incoming flux arrows
  \foreach \y in {-1.2,-0.6,0,0.6,1.2}
    \draw[-{Stealth[length=2mm]}, black] (-3.6,\y) -- (-1.4,\y);
  \node[black, above, font=\scriptsize] at (-2.5,1.3) {beam $F$};
  % target slab
  \draw[black, thick, fill=black!5] (-1.0,-1.6) rectangle (0.4,1.6);
  \node[black, below, font=\scriptsize] at (-0.3,-1.7) {thin target};
  % scattering centers with effective area disks
  \foreach \y in {-1.0,-0.2,0.6,1.2}
    \fill[acc] (-0.3,\y) circle (2.2pt);
  \draw[acc, thick, fill=acc!12] (-0.3,0.6) circle (0.32);
  \node[acc, right, font=\scriptsize] at (2.0,0.6) {reaction area};
  \draw[-{Stealth[length=2mm]}, acc, thick] (2.0,0.6) -- (0.05,0.6);
  % scattered arrow
  \draw[-{Stealth[length=2mm]}, black] (0.4,0.6) -- (2.3,1.7);
\end{tikzpicture}
$$

## Luminosity and event rate

For a collider the flux-times-targets product is packaged into a single machine
parameter, the **luminosity** $\mathcal L$, with dimensions of inverse area per unit
time. The event rate for a process of cross section $\sigma$ is

$$
R = \mathcal L\, \sigma,
$$

and the total number of events collected over a run is the cross section times the
**integrated luminosity**,

$$
N = \sigma \int \mathcal L\, \d t.
$$

Integrated luminosity is quoted in inverse barns and their submultiples; an LHC
dataset of $150\ \text{fb}^{-1}$ multiplied by a picobarn-scale cross section yields
of order $10^{5}$ events. This factorization separates the machine, which delivers
$\int \mathcal L\, \d t$, from the physics, which supplies $\sigma$. The
accelerator module returns to luminosity as a design quantity; here it is the
bridge from a computed $\sigma$ to a counted number of events.

## Differential cross section

A total cross section counts all reactions; the angular distribution of the
products carries more information. The **differential cross section**
$\d\sigma/\d\Omega$ is the rate into a solid-angle element $\d\Omega$
about a direction $(\theta,\phi)$, normalized to the flux:

$$
\sigma = \int \frac{\d\sigma}{\d\Omega}\, \d\Omega
       = \int_0^{2\pi}\!\!\int_0^{\pi} \frac{\d\sigma}{\d\Omega}\,
         \sin\theta\, \d\theta\, \d\phi.
$$

Its shape tests the interaction: a $1 + \cos^2\theta$ distribution signals the
spin-1 photon exchange of $e^+e^- \to \mu^+\mu^-$, while a sharp forward peak at
small angle signals long-range $t$-channel exchange. Measuring
$\d\sigma/\d\Omega$ and comparing to the prediction is the standard test of a
theory at the amplitude level.

$$
% caption: A differential cross section as a polar distribution. The radial extent
% at each angle is the rate of scattering into that direction; the lobed shape shown
% is the 1 plus cosine-squared distribution of lepton-pair production, symmetric
% forward and backward.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, -{Stealth[length=2mm]}] (-3.4,0) -- (3.4,0)
    node[right, black, font=\scriptsize] {beam axis};
  \draw[black] (0,-2.1) -- (0,2.1);
  \fill[black!70] (0,0) circle (1.8pt);
  % 1+cos^2 lobes: r(theta) = 1+cos^2 theta, scaled
  \draw[acc, very thick]
    (2.4,0)
    .. controls (2.3,0.7) and (1.9,1.15) .. (1.3,1.3)
    .. controls (0.7,1.45) and (0.25,1.2) .. (0,1.2)
    .. controls (-0.25,1.2) and (-0.7,1.45) .. (-1.3,1.3)
    .. controls (-1.9,1.15) and (-2.3,0.7) .. (-2.4,0)
    .. controls (-2.3,-0.7) and (-1.9,-1.15) .. (-1.3,-1.3)
    .. controls (-0.7,-1.45) and (-0.25,-1.2) .. (0,-1.2)
    .. controls (0.25,-1.2) and (0.7,-1.45) .. (1.3,-1.3)
    .. controls (1.9,-1.15) and (2.3,-0.7) .. (2.4,0);
  \node[acc, right, font=\scriptsize] at (2.5,0.4) {rate into angle};
\end{tikzpicture}
$$

## Decay width, lifetime, and branching ratios

An unstable particle at rest decays with a constant probability per unit time. The
number surviving falls exponentially, $N(t) = N(0)\, e^{-\Gamma t}$, where the
**decay width** $\Gamma$ is that probability per unit time and has dimensions of
energy in natural units. The mean **lifetime** is its reciprocal,

$$
\tau = \frac{1}{\Gamma} = \frac{\hbar}{\Gamma},
$$

the second form restoring SI, so a width in GeV converts to a lifetime in seconds
through $\hbar = 6.582 \times 10^{-25}$ GeV·s. A particle with several decay modes
has a total width that is the sum of the **partial widths** $\Gamma_i$ of the
individual channels,

$$
\Gamma = \sum_i \Gamma_i,
$$

and the fraction going to channel $i$ is the **branching ratio**

$$
B_i = \frac{\Gamma_i}{\Gamma}, \qquad \sum_i B_i = 1.
$$

The width also appears as the energy spread of the state. A state of finite
lifetime is not a sharp energy eigenstate; its energy distribution is the
**Breit-Wigner** resonance

$$
|\mathcal A(E)|^2 \propto \frac{1}{(E - E_0)^2 + \Gamma^2/4},
$$

a peak centered at the rest mass $E_0$ with full width at half maximum equal to
$\Gamma$. The three descriptions — exponential decay in time, total width as a sum
of partial widths, and resonance width in energy — are one quantity $\Gamma$ viewed
three ways, connected by $\tau = \hbar/\Gamma$.

$$
% caption: The Breit-Wigner resonance. A state of lifetime tau shows an energy
% distribution peaked at its rest mass with a full width at half maximum equal to
% the decay width Gamma; a shorter lifetime is a broader peak, since width and
% lifetime are reciprocal.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, -{Stealth[length=2mm]}] (0,0) -- (7.0,0)
    node[right, black, font=\scriptsize] {energy};
  \draw[black, -{Stealth[length=2mm]}] (0,0) -- (0,3.5)
    node[above, black, font=\scriptsize] {rate};
  % Breit-Wigner peak centered at x=3.5, height 3, gamma controlling width
  \draw[acc, very thick]
    (0.3,0.32) .. controls (1.8,0.55) and (2.7,1.1) .. (3.1,2.4)
    .. controls (3.3,3.0) and (3.7,3.0) .. (3.9,2.4)
    .. controls (4.3,1.1) and (5.2,0.55) .. (6.7,0.32);
  % half-max line
  \draw[black, dashed] (2.75,1.5) -- (4.25,1.5);
  \node[black, above, font=\scriptsize] at (3.5,1.5) {full width};
  \draw[black, dashed] (3.5,0) -- (3.5,2.7);
  \node[black, below, font=\scriptsize] at (3.5,-0.12) {rest mass};
  \node[acc, right, font=\scriptsize] at (4.1,2.6) {resonance};
\end{tikzpicture}
$$

## Fermi's golden rule and phase space

Both observables come from the same master formula. A transition rate is the
product of a dynamical factor, the squared amplitude $|\mathcal M|^2$, and a
kinematic factor, the density of available final states. This is **Fermi's golden
rule**. The kinematic factor is written in Lorentz-invariant form. For each
final-state particle of four-momentum $p_f$, the invariant phase-space element is

$$
\widetilde{\d p_f}
  = \frac{\d^3 p_f}{(2\pi)^3\, 2 E_f},
$$

whose measure $\d^3 p / 2E$ is Lorentz invariant because it is the
mass-shell restriction of the four-dimensional integral. The **Lorentz-invariant
phase space** for an $n$-body final state combines these with overall
four-momentum conservation,

$$
\d\Pi_n = (2\pi)^4\, \delta^4\!\Big(P - \sum_f p_f\Big)
  \prod_{f=1}^{n} \frac{\d^3 p_f}{(2\pi)^3\, 2 E_f}.
$$

The golden rule then gives the rate as $|\mathcal M|^2$ integrated against this
phase space, with a flux or normalization prefactor set by the initial state. Two
cases carry the course.

**Decay $1 \to 2$.** For a particle of mass $m_1$ decaying into two bodies, the
phase-space integral is elementary because the daughter momentum $p^\ast$ is fixed
by the masses (previous lesson). The width is

$$
\Gamma = \frac{S\, |p^\ast|}{8\pi\, m_1^2}\, \overline{|\mathcal M|^2},
$$

where $\overline{|\mathcal M|^2}$ is the amplitude squared, averaged over initial
and summed over final spins, and $S$ is a statistical factor of $1/j!$ for each
group of $j$ identical final particles. Everything kinematic is in the single
factor $|p^\ast|/m_1^2$; the dynamics is entirely in $\mathcal M$.

**Scattering $2 \to 2$.** For $a + b \to c + d$ in the CM frame, with initial and
final CM momenta $|p_i|$ and $|p_f|$, the differential cross section is

$$
\frac{\d\sigma}{\d\Omega}
  = \frac{1}{64\pi^2\, s}\, \frac{|p_f|}{|p_i|}\, \overline{|\mathcal M|^2},
$$

with $s$ the squared CM energy. The prefactor $1/(64\pi^2 s)$ and the momentum
ratio $|p_f|/|p_i|$ are pure kinematics; the physics is the amplitude. For elastic
scattering $|p_f| = |p_i|$ and the ratio is one.

$$
% caption: Two-body phase space in the center-of-momentum frame. The two products
% leave back-to-back on a sphere of radius p-star fixed by the invariant energy; the
% only free variable is the direction, so the phase-space integral reduces to an
% integral over solid angle.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, thick] (0,0) circle (1.9);
  \draw[black, dashed] (0,0) ellipse (1.9 and 0.62);
  \fill[black!70] (0,0) circle (1.8pt);
  \draw[-{Stealth[length=2.6mm]}, acc, very thick] (0,0) -- (1.55,1.1);
  \draw[-{Stealth[length=2.6mm]}, acc, very thick] (0,0) -- (-1.55,-1.1);
  \node[acc, right, font=\scriptsize] at (1.55,1.1) {product};
  \node[acc, left, font=\scriptsize] at (-1.55,-1.1) {product};
  \node[black, right, font=\scriptsize] at (0.55,0.55) {radius};
  \node[black, below, font=\scriptsize] at (0,-2.1) {direction free, magnitude set};
\end{tikzpicture}
$$

## The event budget

Combining the pieces gives the chain from theory to a counted signal. The predicted
number of events of a given process in a dataset is

$$
N = \Big(\int \mathcal L\, \d t\Big) \times \sigma \times B \times \epsilon,
$$

the integrated luminosity from the machine, the production cross section from the
amplitude, the branching ratio into the observed final state, and the detector
efficiency. Each factor is computed or measured separately, and a discovery
requires the product to exceed the fluctuation of the background — the statistics
developed in the experimental module. The cross section and width defined here are
the two theoretical inputs; the rest of the course computes $\mathcal M$ for the
electromagnetic, weak, and strong interactions and feeds it through the golden
rule.[^gold]

[^gold]: Griffiths, _Introduction to Elementary Particles_, Ch. 6, §6.1–6.3, derives Fermi's golden rule, Lorentz-invariant phase space, and the $1\to2$ and $2\to2$ master formulas in the normalization used here; Halzen & Martin, _Quarks and Leptons_, §4, and Thomson, _Modern Particle Physics_, §3, give the same with the flux and phase-space conventions. Widths, branching ratios, and luminosities follow the Particle Data Group, _Review of Particle Physics_, [pdg.lbl.gov](https://pdg.lbl.gov).