Cross Sections and the Golden Rule

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

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.

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

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

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 $\int \mathcal{L}\mathbf{d}t$, 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$\mathbf{d}\sigma /\mathbf{d}\Omega$ is the rate into a solid-angle element $\mathbf{d}\Omega$ about a direction $(\theta ,\varphi )$, normalized to the flux:

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 $\mathbf{d}\sigma /\mathbf{d}\Omega$ and comparing to the prediction is the standard test of a theory at the amplitude level.

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,

the second form restoring SI, so a width in GeV converts to a lifetime in seconds through $\hslash =6.582\times 10^{-25}$ 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,

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

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

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 $\tau =\hslash /\Gamma$.

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

whose measure ${\mathbf{d}}^{3}p/2E$ 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,

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

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

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.

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

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.¹