Cross Sections

QFT predicts numbers measurable in experiments via two parallel observables: cross sections $\sigma$ for scattering processes and decay rates $\Gamma$ for unstable particles. Both are constructed in a uniform way from the invariant amplitude $\mathcal{M}$ produced by Feynman-rule calculations (see QED/historical.md §5.2 and QED Step 6).

Scope. This page focuses on cross sections. The companion observable — decay rates — shares the same kinematic ingredients but has its own master formula and conceptual status; that material lives in decay-rates.md. For the broader landscape — branching ratios, asymmetries, bound-state energies, $g-2$ and form factors, IR-safe QCD observables, masses, couplings, etc. — see observables.md, which places this page inside a wider inventory and explains how each observable type relates to the underlying $\mathcal{M}$ / correlator / vertex-function machinery.

This page collects the definitions, the formulas, and the kinematic ingredients (flux factor, Lorentz-invariant phase space) that the rest of the notes use without explanation. It complements QFT/preliminaries.md § S-Matrix and Cross Sections, which only sketches them in one line.

1. What Is a Cross Section?

1.1 What Is a Cross Section?

A cross section $\sigma$ is, in the original sense, the effective transverse area that one target particle presents to an incoming beam: a beam particle that crosses this area triggers a scattering reaction; one that misses it does not. It has dimensions of area — hence the name.

This is the picture Rutherford used in 1911 to interpret his alpha-on-gold-foil scattering experiments, and it is the historical origin of the term. We take it as the primary definition below; the per-particle probability (ii), the operational rate formula (iii), the quantum amplitude formula (iv), and the S-matrix expression (v) are then all derived from it (or from its quantum generalization), in the chain

$$\text{(i)}⟶\text{(ii)}⟶\text{(iii)}\text{(classical chain)};\text{(i)}\stackrel{\text{(iii)}}{\leftarrow }\text{(iv)}⟷\text{(v)}\text{(quantum side)}.$$

(i) Geometric / effective-area definition — primary

For a single target particle and a single beam particle approaching with impact parameter $b$, define ${\sigma }_{a\to f}$ as the area in the plane perpendicular to the beam such that the beam particle triggers reaction $a\to f$ if and only if it passes through that area:

$$\boxed{{\sigma }_{a\to f}\equiv \text{effective transverse area of the target for reaction }a\to f.}$$

Differential version: the probability per unit solid angle that the outgoing particle goes into direction $(\theta ,\varphi )$ defines the differential cross section $d\sigma /d\Omega$, with ${\sigma }_{\text{tot}}=\int d\sigma$.

For a hard sphere of radius $R$, this is exact and literal: any beam particle with impact parameter $b<R$ hits the sphere, any with $b\ge R$ misses, so ${\sigma }_{\text{tot}}=\pi R^2$. For other potentials (Coulomb, Yukawa, ...), the picture remains conceptually correct if you allow $\sigma$ to depend on the reaction (different final states correspond to different effective areas) and on the energy (slower particles spend more time near the target and have larger effective areas).

This was Rutherford's working concept and remains the cleanest mental picture. It also fixes the units of $\sigma$ (area) and motivates the unit name barn (§1.2 below).

(ii) Single-target (per-particle) reformulation — derived from (i)

Consider a single beam particle traversing a thin slab of target material of thickness $dz$, transverse area $A_{\perp }$, and target density $n_b$. The slab contains $N_b=n_b{A}_{\perp }dz$ target particles, each presenting effective area $\sigma$ to the beam (definition (i)). The total effective scattering area in the slab is

$$N_b\sigma =n_b\sigma A_{\perp }dz.$$

A single beam particle entering the slab at a random transverse position (within $A_{\perp }$) scatters iff it crosses one of these effective disks, so its scattering probability is

$$\boxed{d{P}_{\text{scatter}}=\frac{N_b\sigma }{A_{\perp }}=n_b\sigma dz.}$$

Both $A_{\perp }$ and the slab transverse geometry have cancelled — the probability depends only on $n_b$, $\sigma$, and $dz$.

Integrating along a path of length $L$, the fraction of beam particles that have scattered is $1-e^{-n_b\sigma L}$, and the mean free path is $\lambda =1/(n_b\sigma )$.

(Some texts take this as the primary definition of $\sigma$ — declare the cross section to be whatever number makes $d{P}_{\text{scatter}}=n_b\sigma dz$ hold for a thin slab. That gives the same $\sigma$ as (i), with the advantage of not requiring the literal "effective area" picture for soft potentials.)

(iii) Operational (rate) reformulation — derived from (ii)

For a beam of particles of type $a$ (number density $n_a$, velocity $v_a$) impinging on the slab, multiply the per-particle probability (ii) by the rate at which beam particles enter the slab. Throughout, $N_f$ denotes the cumulative number of scattering events of type $f$; $d{N}_f/dt$ is the event rate.

1. Beam particles entering the slab per unit time. The number of beam particles crossing the slab face per unit time is $J{A}_{\perp }=n_a{v}_{\text{rel}}{A}_{\perp }$, where $J=n_a{v}_{\text{rel}}$ is the incident flux (beam particles per unit area per unit time).

2. Multiply by per-particle probability. By linearity of expectation, the expected event rate is (attempts per unit time) × (probability per attempt):

   $$\frac{d{N}_f}{dt}=\underset{\text{attempts/sec}}{\underbrace{(n_a{v}_{\text{rel}}{A}_{\perp })}}\cdot \underset{\text{prob. per attempt, from (ii)}}{\underbrace{(n_b\sigma dz)}}=n_a{n}_b{v}_{\text{rel}}\sigma A_{\perp }dz.$$

   The dice analogy: "if you roll $N$ dice per second and each scatters with probability $p$, you get $Np$ events per second on average."

3. Divide by slab volume $dV=A_{\perp }dz$:

$$\boxed{\frac{d{N}_f}{dtdV}=n_a{n}_b{v}_{\text{rel}}{\sigma }_{a\to f}.}$$

The slab geometry has cancelled: $A_{\perp }$ and $dz$ both drop out. Equivalently, in terms of the incident flux $J=n_a{v}_{\text{rel}}$ and target areal density $N_b$ (per unit transverse area along the beam),

$$\frac{d{N}_f}{dt}=J{N}_b\sigma .$$

This is the experimentalist's working formula: measure $d{N}_f/dt$, divide by $J{N}_b$, and what comes out is $\sigma$. Many practical texts take this rate formula as the operational definition of $\sigma$ instead of (i); the two are equivalent.

Each ingredient has a clear physical role:

• $n_a$ — more beam particles per unit volume → more chances to scatter.
• $n_b$ — more targets per unit volume → more targets to scatter from.
• $v_{\text{rel}}$ — faster relative motion → more beam particles encounter each target per unit time.
• $\sigma$ — bigger effective area → each encounter is more likely to react.

Relativistic generalization

For relativistic beams the simple product $n_a{n}_b{v}_{\text{rel}}$ is not Lorentz-invariant, but the rate density $d{N}_f/(dtdV)$ is (events per spacetime volume). The Lorentz-invariant generalization replaces $n_a{n}_b{v}_{\text{rel}}$ with the Møller flux

$$\mathcal{F}=\sqrt{(v_a-v_b{)}^2-({\mathbf{v}}_a\times {\mathbf{v}}_b{)}^2}{n}_a{n}_b=\frac{1}{E_a{E}_b}\sqrt{(p_a\cdot p_b{)}^2-m_a^2{m}_b^2}{n}_a{n}_b,$$

which reduces to $n_a{n}_b|{\mathbf{v}}_a-{\mathbf{v}}_b|$ in the non-relativistic limit and gives the boxed formula in the lab frame with ${\mathbf{v}}_b=0$.

Worked example: numerical estimate

Suppose a 1 mA proton beam ($J\approx 6\times 10^{15}{\text{p/s/cm}}^2$ for a focused beam of $\sim 1{\text{mm}}^2$ cross-section) hits a 1 mm thick lead target ($n_b\approx 3.3\times 10^{22}{\text{nuclei/cm}}^3$). The geometric cross section of a Pb nucleus is $\sigma \sim \pi R_{\text{Pb}}^2\sim 0.5\text{b}=5\times 10^{-25}{\text{cm}}^2$. Then

• Target areal density: $N_b=n_{b}L\approx 3.3\times 10^{21}{\text{cm}}^{-2}$.
• Scattering probability per beam particle: $n_b\sigma L\approx 1.6\times 10^{-3}$.
• Event rate: $J{N}_b\sigma A_{\perp }\approx 6\times 10^{15}\cdot 3.3\times 10^{21}\cdot 5\times 10^{-25}\cdot 0.01{\text{s}}^{-1}\approx 10^{11}\text{events/s}.$

So at the geometric cross-section level, about $10^{11}$ scattering events per second occur. Cross sections for specific reactions (e.g. nuclear excitations, deep-inelastic scattering, particular final states) are typically much smaller — picobarn to femtobarn — yielding correspondingly fewer events.

Caveats

• Single-scattering assumption. The formula assumes each beam particle scatters at most once. For thick or dense targets ($n_b\sigma L\gtrsim 1$) one must use the exponential attenuation $1-e^{-n_b\sigma L}$ from (ii) instead, and account for multiple scattering.
• Reaction-specific $\sigma$. A given collision can produce many different final states, each with its own ${\sigma }_{a\to f}$. The total cross section ${\sigma }_{\text{tot}}={\sum }_f{\sigma }_{a\to f}$ counts any reaction.
• Coherent vs. incoherent. The formula assumes incoherent scattering off independent targets. Coherent processes (e.g. Bragg diffraction off a crystal) require summing amplitudes, not probabilities, and produce different angular distributions.

(iv) Quantum / amplitude reformulation — derived from (v); the working formula

For potentials (Coulomb, Yukawa, ...) where there is no literal "area", the geometric picture (i) is defined by the rate formula (iii) instead, and computed from the quantum-mechanical scattering amplitude $\mathcal{M}$. Extracting $\mathcal{M}$ from the off-diagonal $T$-matrix elements via $⟨f|T|i⟩=(2\pi {)}^4{\delta }^4(p_f-p_i){\mathcal{M}}_{fi}$, the cross section in (v) below takes the form:

$$d\sigma =\frac{1}{F}\overline{|\mathcal{M}{|}^2}d{\Pi }_n,$$

with the flux factor $F$ and Lorentz-invariant phase space $d{\Pi }_n$ defined in §2 below.

This is the working formula used throughout perturbative QFT — nearly all standard graduate textbooks (Peskin–Schroeder, Schwartz, Srednicki, Mandl–Shaw, Bjorken–Drell) take this as the quoted definition, with (v) shown only as motivation. The numerical agreement between this computed value and an experimental measurement via (iii) is what makes QFT predictive.

(iv) ↔ (iii): how the chains meet

The chain $(\text{i})\to (\text{ii})\to (\text{iii})$ is purely classical counting; the chain $(\text{v})\to (\text{iv})$ is purely quantum (extracting $\mathcal{M}$ from $T$). What links them is the observation that the quantum-side rate density turns out to have exactly the same functional form as (iii) — bilinear in $n_a$ and $n_b$, proportional to $v_{\text{rel}}$, with a kinematic factor of dimensions area. This is not a derivation of (iii) from (iv) (or vice versa); it is a non-trivial structural compatibility check that allows the quantum coefficient to be defined as a cross section.

Three steps make the compatibility manifest.

1. Transition probability per particle pair. The S-matrix element $⟨f|S-1|i⟩$ between asymptotic momentum eigenstates contains a momentum-conserving delta function $(2\pi {)}^4{\delta }^4(p_f-p_i)$. Squaring it produces $|(2\pi {)}^4{\delta }^4(p_f-p_i){|}^2\to VT\cdot (2\pi {)}^4{\delta }^4(p_f-p_i)$ in the long-time, large-volume limit (one delta function evaluates to the spacetime volume $VT$ at zero argument). With the standard relativistic normalization $⟨p|p^{\prime }⟩=2E_p(2\pi {)}^3{\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime })$ and ${\delta }^3(0)\to V/(2\pi {)}^3$, the transition probability per unit time per particle pair is

   $$\frac{d{P}_{i\to f}}{dt}=\frac{1}{4E_a{E}_{b}V}|{\mathcal{M}}_{fi}{|}^2(2\pi {)}^4{\delta }^4(p_f-p_i)d{\Pi }_n.$$

   The $V$'s from the delta-squaring and from the state norms partially cancel; one residual $1/V$ remains.

2. Sum over all pairs in the box. A box of volume $V$ contains $N_a=n_{a}V$ beam particles and $N_b=n_{b}V$ target particles, giving $N_a{N}_b=n_a{n}_b{V}^2$ pairs (assuming each pair scatters independently — the perturbative single-pair regime). Multiplying:

   $$\frac{d{N}_f}{dt}=N_a{N}_b\cdot \frac{d{P}_{i\to f}}{dt}=\frac{n_a{n}_{b}V}{4E_a{E}_b}|\mathcal{M}{|}^2(2\pi {)}^4{\delta }^4(p_f-p_i)d{\Pi }_n.$$

3. Divide by box volume to get the rate density. Dividing by $V$ and integrating over the final-state phase space (the delta function fixes one momentum):

   $$\boxed{\frac{d{N}_f}{dtdV}=\frac{n_a{n}_b}{4E_a{E}_b}\int |\mathcal{M}{|}^2(2\pi {)}^4{\delta }^4(p_f-p_i)d{\Pi }_n.}$$

   This is a derived result from the quantum side, with no input from (iii). What's striking about it: it is bilinear in $n_a$ and $n_b$, with a kinematic prefactor depending only on the two beam-particle energies and the matrix element. That bilinearity is not assumed — it follows from the standard normalization of momentum-eigenstate Fock states and from each beam particle scattering off one target at a time. (If the rate density had turned out to depend on $n_a^2$ or $n_b^{1.5}$, the cross-section interpretation would not be available.)

   Now compare with (iii): $d{N}_f/(dtdV)=n_a{n}_b{v}_{\text{rel}}\sigma$. The functional forms match. We define the cross section in the quantum framework as the coefficient that makes the matching work:

   $$\sigma \equiv \frac{1}{4E_a{E}_b{v}_{\text{rel}}}\int |\mathcal{M}{|}^2(2\pi {)}^4{\delta }^4(p_f-p_i)d{\Pi }_n.$$

   The combination $4E_a{E}_b{v}_{\text{rel}}$ in the denominator is exactly the lab-frame flux factor $F$ (which generalizes covariantly to the Lorentz-invariant Møller flux of §2.3). Absorbing the delta function into the phase-space integral $d{\Pi }_n$ recovers (iv) in its standard form $d\sigma =(1/F)\overline{|\mathcal{M}{|}^2}d{\Pi }_n$ ✓.

Two things to take away:

• The structural fact that the quantum rate density is bilinear in densities, proportional to $v_{\text{rel}}$, and has a kinematic factor of dimensions area is derived from the S-matrix formalism (steps 1–3). It is a non-trivial check; it could have failed for a non-perturbative or coherent process.
• The identification of that kinematic factor as the cross section $\sigma$ is a definition — one chosen so that the quantum framework reproduces the classical counting picture (iii). It is what makes the symbol $\sigma$ in (iv) refer to the same physical quantity as in (iii).

So the quantum chain (v) → (iv) and the classical chain (i) → (ii) → (iii) do not derive each other; they meet at this definitional matching, made possible by the structural compatibility shown in steps 1–3.

What is being postulated

The bridge above relies on two unstated assumptions, both of which are part of QFT's foundational postulate set rather than theorems derivable within the S-matrix formalism:

1. Born rule for scattering. $|⟨f|S|i⟩{|}^2$ is interpreted as the probability of a transition between asymptotic states. This is QM Postulate 3 applied to scattering — the same postulate, with $\hat{U}\to \hat{S}$ and the bookkeeping of delta-function-normalized states. The non-relativistic single-particle analog is Fermi's Golden Rule $\Gamma =(2\pi /\hslash )|V_{fi}{|}^2{\rho }_f$, which has the same (squared matrix element) × (final-state density) × (kinematic prefactor) structure as $d\sigma =(1/F)\overline{|\mathcal{M}{|}^2}d{\Pi }_n$. See QFT/remarks.md § Born Rule for Scattering for the full QM↔QFT comparison and the technical subtlety of non-normalizable asymptotic states. Without this postulate, $|\mathcal{M}{|}^2$ is just an algebraic object with no physical meaning.

2. Quantum-classical rate correspondence. The S-matrix transition rate per pair $|\mathcal{M}{|}^2\cdot d{\Pi }_n/F$ — computed in an idealized infinite-volume, infinite-time limit — equals the classical event rate per pair that an experimentalist measures with a real detector of finite size operating for finite time. This is a correspondence-principle postulate: it asserts that the asymptotic-state idealization captures what real apparatus does, modulo finite-size and resolution effects.

Without these two, the boxed quantum formula in step 3 would be a formal expression with no connection to laboratory measurements, and the matching with (iii) would not exist as anything more than dimensional coincidence. With them, (iv) becomes a prediction of the experimentally-measured (iii) — and the agreement of those two numbers, on which all of perturbative-QFT phenomenology depends, becomes the empirical content of the theory.

(v) S-matrix definition — quantum-foundational form

In axiomatic / S-matrix-based QFT, $\sigma$ is defined in terms of the S-matrix $S=1+iT$ acting between asymptotic in/out Fock states (see QFT/postulates.md Postulate 10). The cross section is what falls out of the squared modulus of $⟨f|S-1|i⟩$ after dividing by the natural normalization of momentum-eigenstate Fock states (which produces the flux factor $F$) and integrating over the final-state phase space:

$$d\sigma \propto |⟨f|S-1|i⟩{|}^2/F.$$

Factoring out the spacetime-translation delta function $⟨f|T|i⟩=(2\pi {)}^4{\delta }^4(p_f-p_i){\mathcal{M}}_{fi}$ and packaging the residue into the invariant amplitude $\mathcal{M}$ recovers (iv) as the practical form.

This is the form used in foundational / axiomatic texts (Weinberg QFT Vol. 1, Itzykson–Zuber, Streater–Wightman, Haag), where the S-matrix is taken as the primitive quantum object and (iv) is derived from it. The two are mathematically equivalent; the choice between them is a matter of which is taken as the definition vs. the working formula.

All five describe the same object. (i) is taken as the classical foundational definition, following Rutherford's original geometric picture. (ii) is derived from (i) by counting targets in a thin slab. (iii) is then derived from (ii) by multiplying by the rate of beam particles entering the slab. On the quantum side, (v) is the most foundational form (S-matrix on asymptotic Fock states), and (iv) is the working formula derived from it by extracting $\mathcal{M}$. Numerical agreement between (iv)/(v) on the quantum side and the classical chain (i)→(ii)→(iii) on the experimental side is the empirical content of QFT predictions.

1.2 Units

Cross sections have units of area. Common units in particle physics:

The barn is roughly the geometric cross-section of a heavy nucleus ("as big as a barn door" — wartime Manhattan-Project lab humor).

In natural units ($\hslash =c=1$), cross sections have dimensions of $[\text{energy}{]}^{-2}$. The conversion is $1{\text{GeV}}^{-2}\approx 0.389\text{mb}$.

1.3 Differential vs. Total

• Differential cross section $d\sigma /d\Omega$ (or $d\sigma /dt$, $d\sigma /dE$, etc.) is the cross section per solid angle (or per Mandelstam $t$, etc.).
• Total cross section ${\sigma }_{\text{tot}}=\int d\sigma$ integrates over the full final-state phase space.

The differential form contains the angular / energy distribution of scattered particles; the total contains only the overall reaction rate.

1.4 Classical Analog

The cross section is not a quantum invention — it was defined and used in classical mechanics decades before QM existed. The QFT formula is a generalization of the classical one and reduces to it in appropriate limits.

Classical definition

For a beam of point particles incident on a fixed target, the operational definition (rate ∝ flux × density × $\sigma$) is literally identical to §1.1 above; only the computation differs between classical and quantum mechanics. Particles incident with impact parameter $b$ (perpendicular distance from the target's center to the beam line) and azimuthal angle $\varphi$ scatter to angle $\theta (b)$ determined by the classical equations of motion. Then

$$d\sigma =bdbd\varphi ,\frac{d\sigma }{d\Omega }=\frac{b}{\sin \theta }\left|\frac{db}{d\theta }\right|.$$

This is purely geometric — no $\hslash$, no $\mathcal{M}$.

The area formula

It's worth pausing to note what is exact and what is an approximation in the formula $d\sigma =bdbd\varphi$:

• The differential cross-section relation is exact only in the limit $db,d\varphi \to 0$. Equating the cross-sectional area $bdbd\varphi$ with $\frac{d\sigma }{d\Omega }d\Omega$ gives a true equality only when both differentials are infinitesimal. For finite patches, the Jacobian $\frac{d\sigma }{d\Omega }=\frac{b}{\sin \theta }\left|\frac{db}{d\theta }\right|$ varies across the patch, so the ratio (annulus area)/(solid angle) is approximate.

  • For hard-sphere scattering specifically, the Jacobian $b/\sin \theta \cdot |db/d\theta |=R^2/4$ is constant (independent of $b$), so the ratio is exact even for finite patches. This is a feature of the hard-sphere geometry, not a general fact.
  • For Coulomb scattering, the Jacobian diverges at $\theta \to 0$, so finite patches give noticeably different ratios.

• In the interactive visualizations above, the rendered annulus is a polygonal approximation of the smooth annular sector (triangulated mesh; a few subdivisions per edge). Bumping the subdivision counts would make it visually smooth, but does not change the algebraic content — both the area formula $bdbd\varphi$ and (for the hard sphere) the ratio $R^2/4$ are exact at any subdivision level.

Interactive 3D demo. An interactive three.js visualization of classical scattering — hard-sphere reflection and repulsive/attractive Coulomb trajectories — is available in cross-sections-3d.html. Open it in a browser; drag to rotate, scroll to zoom, and use the controls to switch modes, vary the impact-parameter range, and adjust the energy.

Spherical-coordinate visualization. A second three.js page, cross-sections-spherical.html, focuses specifically on the definition of $d\sigma /d\Omega$: the incoming annulus $dA=bdbd\varphi$ at the source plane is shown together with its image $d\Omega$ on a detection sphere surrounding the target, after hard-sphere reflection. Sliders control $b$, $\varphi$, $db$, $d\varphi$ — letting you watch how the two patches transform into each other.

   Classical scattering geometry
   ─────────────────────────────

                                                      ↗  outgoing ray
                                                     ╱      (at angle θ)
                                                    ╱
                          ●  target                ╱
                       ━━━┿━━━                    ╱  ← scattering angle θ
                          │ ↑                    ╱     measured from the
                          │ │                   ╱      incoming beam axis
                          │ │ b  (impact       ╱
                          │ │  parameter)     ╱
                          │ ↓                ╱
   ─────────────────●━━━━━┿━━━━━━━━━━━━━━━━━●──────→  beam axis  (θ = 0)
        incoming particle                  scattering
        (parallel ray, perp.                center
         distance b from axis)

   Key:  b = perpendicular distance from beam axis to the incoming-ray asymptote.
         φ = azimuthal angle of that ray around the beam axis  (not shown — out of page).
         θ = polar deflection angle measured from the beam axis to the outgoing ray.

Geometrically, all incoming particles whose impact parameter lies in the annulus $b\to b+db$ scatter into the same conical region $\theta \to \theta +d\theta$ around the beam axis (modulo the $d\varphi$ azimuthal range):

   Annulus → solid angle  (top view, beam coming out of page)
   ────────────────────────────────────────────────────────

           area  dA = b db dφ                  solid angle  dΩ
                ╭─────╮                            ╭───╮
              ╭─┤ b+db├─╮                          │   │
             ╭──┤     ├──╮       scattering        │ θ │
             │  │  ●  │  │   ─────────────→        │   │  →  (cone of
             ╰──┤     ├──╯                         │   │      opening
              ╰─┤  b  ├─╯                          ╰───╯      angle θ)
                ╰─────╯

Two textbook examples

• Hard sphere of radius $R$. Geometry gives $b=R\cos (\theta /2)$, hence

  $$\frac{d\sigma }{d\Omega }=\frac{R^2}{4},{\sigma }_{\text{tot}}=\pi R^2.$$

  The total cross section equals the geometric cross-sectional area — the source of the "effective area" intuition. Classical and quantum results agree at energies where the de Broglie wavelength is much smaller than $R$.

     Hard-sphere reflection
     ──────────────────────
  
                              outgoing
                                  ╲
                                   ╲   θ      angle of scattering
                                    ╲
                                     ●━━━━━ surface
                                  α ╱│
                                   ╱ │       α = angle of incidence
                                  ╱  │ R         (incidence = reflection)
                                 ╱   │
                                ╱    │       Geometry:
                               ╱     │         sin α = b/R
            b →  ─────────────●──────●         θ = π - 2α
                              │   center  ⟹  b = R cos(θ/2)
                              │           ⟹  dσ/dΩ = R²/4
              incoming                    ⟹  σ_tot = π R²
              parallel ray
  

• Rutherford scattering. A point charge $Z_{1}e$ scattering off a fixed point charge $Z_{2}e$ with kinetic energy $E$:

  $$\frac{d\sigma }{d\Omega }={\left(\frac{Z_1{Z}_2{e}^2}{4E{\sin }^2(\theta /2)}\right)}^2.$$

  Remarkably, the non-relativistic Born-approximation quantum-mechanical calculation gives exactly the same answer, and tree-level QED (Mott) reduces to it in the non-relativistic limit. Rutherford scattering is one of the few cases where classical, NRQM, and QED results all coincide — a useful consistency check.

     Rutherford (Coulomb) scattering — hyperbolic trajectory
     ───────────────────────────────────────────────────────
  
                                                       ↗ outgoing
                                                      ╱  asymptote
                                                     ╱
                                                    ╱
                                                   ╱
                                                  ╱
                                              ⌒╱  θ
                                            ⌒ ╱
                                          ⌒  ╱
                                       ⌒    ╱
                                    ⌒      ╱
                                ⌒         ╱
              ⊕  Z₂e          ⌒         ╱     scattering angle θ
              ●═══════════════════════ ╱      determined by the
              │                       ╱       hyperbolic orbit
              │ b                    ╱
   ───────────●─────────────────────╱──────→  beam axis
                ────────────────⌒                                     ─⌒  ← incoming asymptote
                              (parallel to beam axis at distance b)
  
     Solving the orbit equation gives  cot(θ/2) = (2 E b)/(Z₁ Z₂ e²),
     and inverting yields the Rutherford formula above.
     The cross section diverges at small θ because the Coulomb
     potential has infinite range (1/r tail).
  

How the QFT formula reduces to the classical one

The QFT master formula $d\sigma =(1/F)\overline{|\mathcal{M}{|}^2}d{\Pi }_n$ reduces to $d\sigma =bdbd\varphi$ in the WKB / eikonal limit, where:

• Wavelengths are small compared to the target ($\hslash /p\ll R$).
• The wavepacket localization is much smaller than the impact-parameter resolution.

In this regime the amplitude $\mathcal{M}$ is dominated by stationary-phase paths, which are exactly the classical trajectories. The squared amplitude reproduces the geometric Jacobian $b/\sin \theta |db/d\theta |$.

These are one formula in three regimes, not three different formulas. The meaning of $\sigma$ as "effective area" is unchanged across all of them.

Where the classical picture breaks

The classical analog is misleading or absent in several genuinely quantum cases:

• Diffraction at long wavelength. When $\hslash /p\gtrsim R$, the cross section is no longer geometric. Hard-sphere scattering at low energy gives ${\sigma }_{\text{tot}}=4\pi R^2$ — a factor of 4 larger than the classical $\pi R^2$, due to wave diffraction.
• Resonances and threshold cusps. Breit–Wigner peaks and channel-opening cusps have no classical analog.
• Polarization-dependent processes. Spin / polarization don't exist classically (in the relevant sense), so cross sections involving them have no classical limit.
• Identical-particle interference. Møller and Bhabha scattering have antisymmetric amplitudes producing interference patterns absent in classical-particle scattering.
• Production processes. $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$ creates new particles — fundamentally a quantum process with no classical analog.

In all these cases the QFT formula still applies and gives finite cross sections; there's just nothing classical to compare against.

(For the analogous discussion of decay rates and their weaker classical analog, see decay-rates.md §4.1.)

2. The Master Formula

2.1 General $2\to n$ Cross Section

For $a(p_1)+b(p_2)\to f_1(p_3)+f_2(p_4)+\cdots +f_n(p_{n+2})$,

$$\boxed{d\sigma =\frac{1}{F}\overline{|\mathcal{M}{|}^2}d{\Pi }_n,}$$

with three ingredients explained below: the flux factor $F$, the squared invariant amplitude $\overline{|\mathcal{M}{|}^2}$, and the Lorentz-invariant phase space (LIPS) $d{\Pi }_n$.

2.2 The Invariant Amplitude $\mathcal{M}$

The S-matrix element factorizes as

$$⟨f|iT|i⟩=(2\pi {)}^4{\delta }^4(\sum p_{\text{in}}-\sum p_{\text{out}})i{\mathcal{M}}_{fi}.$$

${\mathcal{M}}_{fi}$ is what Feynman-rule calculations directly produce. The overline $\overline{|\mathcal{M}{|}^2}$ denotes averaging over initial spins / polarizations and summing over final ones, appropriate for unpolarized beams and detectors:

$$\overline{|\mathcal{M}{|}^2}=\frac{1}{(\text{init. dof})}\sum _{\begin{array}{c}\text{init. spins}\\ \text{final spins}\end{array}}|\mathcal{M}{|}^2.$$

For example, $e^{-}+\gamma \to e^{-}+\gamma$ has 2 initial electron spins × 2 photon polarizations = 4 initial states, so the prefactor is $1/4$ — exactly as in QED/compton.md §4.

2.3 The Flux Factor $F$

$F$ accounts for the relative motion of the incoming beam and target. The Lorentz-invariant form is

$$F=4\sqrt{(p_1\cdot p_2{)}^2-m_1^2{m}_2^2}.$$

Equivalent expressions in common frames:

• Lab frame (target at rest, $p_2=(m_2,0)$): $F=4m_2|{\mathbf{p}}_1|$.
• CM frame (total 3-momentum zero): $F=4|{\mathbf{p}}_1^{\text{cm}}|\sqrt{s}$, where $\sqrt{s}$ is the CM energy.

The factor of 4 is a convention tied to the normalization $⟨p|p^{\prime }⟩=2E_p(2\pi {)}^3{\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime })$ used throughout the notes (see fock-space-inventory.md §3).

2.4 Lorentz-Invariant Phase Space

For $n$ final-state particles,

$$d{\Pi }_n=(2\pi {)}^4{\delta }^4(\sum p_{\text{in}}-\sum p_{\text{out}})\prod _{f=1}^n\frac{d^3{p}_f}{(2\pi {)}^{3}2E_f}.$$

Each final-state particle gets a Lorentz-invariant measure $d^{3}p/[(2\pi {)}^{3}2E]$; the overall delta function imposes 4-momentum conservation. The "$2E$" denominator is the same normalization factor that appears in the asymptotic states (see §2.3 above).

For commonly encountered cases:

• $2\to 2$: after using the delta function and integrating over one final-state magnitude, $d{\Pi }_2=\frac{1}{8\pi }\frac{|{\mathbf{p}}_3^{\text{cm}}|}{\sqrt{s}}\frac{d{\Omega }^{\text{cm}}}{4\pi }\cdot 4\pi$ — equivalently the classic $2\to 2$ CM-frame formula $d{\Pi }_2=\frac{1}{32{\pi }^2}\frac{|{\mathbf{p}}_3^{\text{cm}}|}{\sqrt{s}}d{\Omega }^{\text{cm}}$.
• $2\to 3$: more involved — requires Dalitz-plot variables or a numerical integration.

2.5 The $2\to 2$ Differential Cross Section

Putting flux and phase space together for the most common case ($2\to 2$ scattering in the CM frame):

$$\boxed{{\left(\frac{d\sigma }{d\Omega }\right)}_{\text{CM}}=\frac{1}{64{\pi }^{2}s}\frac{|{\mathbf{p}}_3^{\text{cm}}|}{|{\mathbf{p}}_1^{\text{cm}}|}\overline{|\mathcal{M}{|}^2}.}$$

For elastic scattering ($|{\mathbf{p}}_3^{\text{cm}}|=|{\mathbf{p}}_1^{\text{cm}}|$) the kinematic ratio is unity and one is left with $\frac{d\sigma }{d\Omega }=\frac{|\overline{\mathcal{M}}{|}^2}{64{\pi }^{2}s}$.

This is the formula applied (with a frame-conversion to the lab frame) to derive the Klein–Nishina cross section in QED/compton.md §5.

3. Decay Rates

Decay rates use the parallel master formula

$$d\Gamma =\frac{1}{2M}\overline{|\mathcal{M}{|}^2}d{\Pi }_n,$$

with the same $\overline{|\mathcal{M}{|}^2}$ and $d{\Pi }_n$ as above and the flux factor $F$ replaced by the rest-frame normalization $2M$ of the decaying particle. The full treatment — master formula, lifetime, branching ratios, the muon-lifetime worked example, classical analog, and connection to complex propagator poles — lives in decay-rates.md. Cross-references from the rest of the notes that previously pointed here for $d\Gamma$ should now point there.

4. Practical Conversions

A few rules of thumb for going between formulas and numbers.

4.1 Useful Numerical Factors

4.2 Mandelstam Variables

For a $2\to 2$ process $1+2\to 3+4$, the Lorentz-invariant kinematic variables are

$$s=(p_1+p_2{)}^2,t=(p_1-p_3{)}^2,u=(p_1-p_4{)}^2,$$

satisfying $s+t+u=m_1^2+m_2^2+m_3^2+m_4^2$. Differential cross sections are often quoted as $d\sigma /dt$, which has the advantage of being manifestly Lorentz-invariant.

The relation to the CM scattering angle ${\theta }^{\text{cm}}$ for elastic scattering is $t=-2|{\mathbf{p}}^{\text{cm}}{|}^2(1-\cos {\theta }^{\text{cm}})$.

4.3 Converting Between Frames

Since $\sigma$ is Lorentz-invariant, total cross sections are the same in any frame. But differential cross sections are not: $d\sigma /d\Omega$ in the lab frame differs from $d\sigma /d{\Omega }^{\text{cm}}$ by the Jacobian of the angle transformation, which is computable from kinematics.

Frame-invariant differential forms ($d\sigma /dt$, $d\sigma /dy$ in rapidity, etc.) are preferred when the choice of frame is irrelevant.

4.4 Optical Theorem

A useful consistency check: the optical theorem relates the imaginary part of the forward scattering amplitude to the total cross section,

$${\sigma }_{\text{tot}}=\frac{1}{2|{\mathbf{p}}_1^{\text{cm}}|\sqrt{s}}\mathrm{Im}\mathcal{M}(p_1,p_2\to p_1,p_2),$$

a direct consequence of $S^{†}S=1$ (unitarity). It provides a non-trivial constraint on perturbative calculations — every loop-level imaginary part must correspond to a contribution to a physical cross section.

5. Where Cross Sections Appear in the Notes

6. Take-Aways

• A cross section $\sigma$ has units of area; it is the proportionality between scattering rate and (flux × target density). The barn ($10^{-24}{\mathrm{cm}}^2$) is the standard unit.
• The master formula is $d\sigma =(1/F)\overline{|\mathcal{M}{|}^2}d{\Pi }_n$ — flux factor, squared amplitude, Lorentz-invariant phase space.
• The parallel decay-rate formula is $d\Gamma =(1/2M)\overline{|\mathcal{M}{|}^2}d{\Pi }_n$, treated in decay-rates.md.
• Total cross sections are Lorentz-invariant; differential ones are not (use $d\sigma /dt$ for invariance).
• The optical theorem ties total cross sections to forward-amplitude imaginary parts via S-matrix unitarity.
• A measured cross section is itself an inferred quantity, obtained from event counts via $\sigma =N/(\mathcal{L}\cdot \mathcal{A}\cdot ϵ)$ — luminosity, acceptance, efficiency. The taxonomy of what is directly measured by detectors vs. what is fit-extracted is in direct-vs-inferred.md.

These formulas are the universal interface between "what QFT computes" ($\mathcal{M}$) and "what experiments measure" ($\sigma$, $\Gamma$, $\tau$, $\mathrm{BR}$).