---
title: Tree-Level QED Processes
module: Quantum Electrodynamics
moduleNumber: 6
lessonNumber: 2
order: 602
summary: >
  The Feynman rules become numbers on the reference reactions of QED. Muon pair
  production $e^+e^-\to\mu^+\mu^-$ sets the scale with its $1+\cos^2\theta$
  distribution and $4\pi\alpha^2/3s$ total cross section, and its ratio to
  hadron production counts colors. Compton scattering gives the Klein-Nishina
  formula and the Thomson limit; Bhabha scattering shows $s$- and $t$-channel
  interference. Casimir's trick turns every spin-averaged square into a trace of
  gamma matrices.
topics: [Quantum Electrodynamics]
draft: false
sources:
  - book: Griffiths
    ref: "Ch. 7 §7.7–7.9 (Casimir's trick, the trace theorems, worked cross sections)"
  - book: Halzen & Martin
    ref: "Ch. 6 §6.3–6.6 (electron-muon, electron-electron, and Compton scattering)"
  - book: Thomson
    ref: "Ch. 6 (electron-positron annihilation and QED cross sections)"
  - book: Tong
    ref: "The Standard Model (Cambridge Part III), §5.3 (electroweak and QED processes)"
---

The Feynman rules of the [previous
lesson](/particle-physics/qed/feynman-rules-qed) produce an amplitude for any QED
process. This lesson evaluates the amplitude for the reactions that anchor the
subject. Muon pair production is the reference against which every $e^+e^-$ cross
section is measured; Compton scattering is the historical entry point and the
cleanest photon-electron process; Bhabha scattering shows two diagrams
interfering. Along the way the trace technology that converts a spinor amplitude
into a number — Casimir's trick — is set up once and reused.

Natural units and $\eta^{\mu\nu} = \operatorname{diag}(+,-,-,-)$ throughout, with
$\alpha = e^2/4\pi$. Electron and muon masses are neglected against the collision
energy except where a mass is the whole point.

## Casimir's trick and the trace theorems

An amplitude such as $\mathcal M \sim \bar u(p_3)\gamma^\mu u(p_1)$ depends on the
external spins. Detectors rarely resolve spin, so the measured cross section uses
the **spin-averaged square**: average over the initial spins, sum over the final
spins,

$$
\big\langle |\mathcal M|^2 \big\rangle
  = \frac{1}{(2s_1+1)(2s_2+1)}
    \sum_{\text{all spins}} |\mathcal M|^2 .
$$

The sums collapse by the **completeness relations** for the Dirac spinors,

$$
\sum_s u(p,s)\,\bar u(p,s) = \gamma\!\cdot\!p + m,
\qquad
\sum_s v(p,s)\,\bar v(p,s) = \gamma\!\cdot\!p - m .
$$

Because $|\mathcal M|^2 = \mathcal M \mathcal M^\ast$ and each factor is a spinor
sandwich, inserting the completeness relations turns every spin sum into a
**trace** over the four-dimensional Dirac indices. This is Casimir's trick: the
messy sum over spinor components becomes a trace of a product of gamma matrices and
slashed momenta, evaluated by algebraic identities rather than explicit
components.[^griffiths-trace] The identities used repeatedly are

$$
\operatorname{Tr}(\mathbb 1) = 4,
\qquad
\operatorname{Tr}(\gamma^\mu\gamma^\nu) = 4\,\eta^{\mu\nu},
$$

$$
\operatorname{Tr}(\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma)
  = 4\big(\eta^{\mu\nu}\eta^{\rho\sigma}
        - \eta^{\mu\rho}\eta^{\nu\sigma}
        + \eta^{\mu\sigma}\eta^{\nu\rho}\big),
$$

with the trace of any odd number of gamma matrices vanishing. These four lines
handle every tree-level QED cross section.

## Muon pair production

The process $e^+e^- \to \mu^+\mu^-$ proceeds through a single $s$-channel diagram:
the electron and positron annihilate into one virtual photon of invariant mass
$\sqrt s$, which materializes as a muon pair. There is no $t$-channel, because the
initial and final fermions are different species, which makes this the simplest
nontrivial QED reaction. The amplitude is

$$
\mathcal M = \frac{e^2}{s}\,
  \big[\bar v(p_2)\gamma^\mu u(p_1)\big]\,
  \big[\bar u(p_3)\gamma_\mu v(p_4)\big],
\qquad
s = (p_1 + p_2)^2 .
$$

Squaring and averaging over the four initial spin combinations, Casimir's trick
produces two traces, one for the electron current and one for the muon current.
Neglecting masses,

$$
\big\langle |\mathcal M|^2 \big\rangle
  = \frac{e^4}{4s^2}\,
    \operatorname{Tr}\!\big(\gamma\!\cdot\!p_2\gamma^\mu\gamma\!\cdot\!p_1\gamma^\nu\big)
    \operatorname{Tr}\!\big(\gamma\!\cdot\!p_3\gamma_\mu\gamma\!\cdot\!p_4\gamma_\nu\big)
  = \frac{2e^4}{s^2}\,(t^2 + u^2),
$$

with $t = (p_1-p_3)^2$ and $u = (p_1-p_4)^2$. In the center-of-momentum frame at
high energy, $t = -\tfrac{s}{2}(1-\cos\theta)$ and $u = -\tfrac{s}{2}(1+\cos\theta)$
with $\theta$ the muon scattering angle, so $t^2 + u^2 = \tfrac{s^2}{2}(1 +
\cos^2\theta)$ and

$$
\big\langle |\mathcal M|^2 \big\rangle = e^4\,(1 + \cos^2\theta).
$$

Feeding this to the golden rule gives the **differential cross section**

$$
\frac{\d\sigma}{\d\Omega}
  = \frac{\alpha^2}{4s}\,(1 + \cos^2\theta),
$$

and integrating over the solid angle, the **total cross section**

$$
\sigma(e^+e^- \to \mu^+\mu^-) = \frac{4\pi\alpha^2}{3s} .
$$

The $1 + \cos^2\theta$ shape is the signature of a spin-1 photon coupling to
spin-$\tfrac12$ fermions; it is forward-backward symmetric, as pure QED must be
(the parity-violating forward-backward asymmetry seen at high energy is the
interference with the $Z$, treated in the electroweak module). The $1/s$ fall-off
is pure dimensional analysis: a cross section has units of area, $\alpha$ is
dimensionless, and $s$ is the only scale.

$$
% caption: Muon pair production. Left: the single s-channel diagram, electron and
% positron annihilating into a virtual photon that produces the muon pair. Right:
% the angular distribution proportional to 1 plus cosine-squared theta, symmetric
% about ninety degrees, the fingerprint of a spin-one exchange between spin-half
% fermions.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- diagram (left) ---
  \draw[thick, ->] (-6.6,1.1) -- (-5.9,0.55);
  \draw[thick] (-5.9,0.55) -- (-5.3,0.1);
  \draw[thick, ->] (-6.6,-0.9) -- (-5.9,-0.35);
  \draw[thick] (-5.9,-0.35) -- (-5.3,0.1);
  \draw[acc, thick, dashed] (-5.3,0.1) -- (-3.7,0.1);
  \draw[thick, ->] (-3.7,0.1) -- (-3.0,0.65);
  \draw[thick] (-3.0,0.65) -- (-2.4,1.1);
  \draw[thick, ->] (-3.7,0.1) -- (-3.0,-0.45);
  \draw[thick] (-3.0,-0.45) -- (-2.4,-0.9);
  \fill (-5.3,0.1) circle (1.8pt);
  \fill (-3.7,0.1) circle (1.8pt);
  \node[left] at (-6.6,1.1) {electron};
  \node[left] at (-6.6,-0.9) {positron};
  \node[right] at (-2.4,1.1) {muon};
  \node[right] at (-2.4,-0.9) {antimuon};
  \node[acc] at (-4.5,0.45) {photon};
  % --- angular plot (right) ---
  \draw[->] (0.6,-1.3) -- (0.6,1.5) node[above] {cross section};
  \draw[->] (0.6,-1.3) -- (5.2,-1.3) node[right] {angle (deg)};
  % baseline for 1+cos^2: min value 1 at 90deg mapped to y=-0.3, max 2 at 0/180 mapped to y=1.1
  % theta from 0 (x=0.9) to pi (x=4.8); y = -1.3 + 0.7*(1+cos^2)
  \draw[very thick, domain=0:180, samples=60, variable=\t]
    plot ({0.9 + 3.9*\t/180}, {-1.3 + 0.7*(1 + (cos(\t))^2)});
  \node[below] at (0.9,-1.3) {0};
  \node[below] at (2.85,-1.3) {90};
  \node[below] at (4.8,-1.3) {180};
\end{tikzpicture}
$$

## The R ratio and the color count

Replacing the outgoing muons with quarks, $e^+e^- \to q\bar q$, the same
$s$-channel diagram applies with the muon charge $-1$ replaced by the quark charge
$Q_q$ (in units of $e$). Each quark is produced in three colors, and each color
contributes an independent final state, so the hadronic cross section is the
muon-pair cross section scaled by $\sum_q Q_q^2$ summed over accessible flavors and
multiplied by the number of colors $N_c$. The **$R$ ratio**,

$$
R \equiv \frac{\sigma(e^+e^- \to \text{hadrons})}{\sigma(e^+e^- \to \mu^+\mu^-)}
  = N_c \sum_q Q_q^2,
$$

is therefore a direct count of colors. Below the charm threshold the open flavors
are $u,d,s$ with charges $+\tfrac23, -\tfrac13, -\tfrac13$, so

$$
R = N_c\left[\Big(\tfrac23\Big)^2 + \Big(\tfrac13\Big)^2 + \Big(\tfrac13\Big)^2\right]
  = N_c \cdot \tfrac23 = 2 \quad (N_c = 3),
$$

and the measured $R \approx 2$ in that energy range was among the first
quantitative confirmations that $N_c = 3$. Above each new quark threshold $R$ steps
up by $N_c Q_q^2$, and the size of each step measures the new quark's charge. The
reaction that fixes the QED scale doubles as a color meter.

## Compton scattering and the Klein-Nishina formula

Compton scattering $e^-\gamma \to e^-\gamma$ has two tree diagrams. In the
$s$-channel the electron absorbs the incoming photon, propagates as a virtual
electron, then emits the outgoing photon; in the $u$-channel the electron emits the
outgoing photon first and absorbs the incoming one after. Both are required by the
fact that the two photons are identical bosons, and the amplitude is their sum:

$$
\mathcal M = \mathcal M_s + \mathcal M_u,
$$

with internal electron propagators $i(\gamma\!\cdot\!q + m)/(q^2 - m^2)$ carrying
$q = p_e + k$ in the $s$-channel and $q = p_e - k'$ in the $u$-channel. Squaring,
averaging over electron spins and photon polarizations, and applying the trace
theorems yields the **Klein-Nishina formula** for the differential cross section in
the electron rest frame,

$$
\frac{\d\sigma}{\d\Omega}
  = \frac{\alpha^2}{2m^2}
    \left(\frac{\omega'}{\omega}\right)^{\!2}
    \left(\frac{\omega}{\omega'} + \frac{\omega'}{\omega} - \sin^2\theta\right),
$$

where $\omega$ and $\omega'$ are the incoming and scattered photon energies related
by the **Compton shift**

$$
\frac{\omega'}{\omega} = \frac{1}{1 + (\omega/m)(1 - \cos\theta)} .
$$

Two limits check the formula. At low photon energy $\omega \ll m$, the shift
becomes negligible, $\omega'/\omega \to 1$, and the bracket reduces to
$1 + \cos^2\theta$, recovering the classical **Thomson cross section**

$$
\sigma_{\text{Thomson}} = \frac{8\pi\alpha^2}{3m^2} = \frac{8\pi}{3}\,r_e^2,
\qquad
r_e = \frac{\alpha}{m},
$$

the elastic scattering of light by a free charge, independent of frequency. At high
energy $\omega \gg m$, the recoil suppresses $\omega'$, the forward peak sharpens,
and the total cross section falls as $(\ln\omega)/\omega$. The energy dependence of
the Compton cross section is thus a direct window on the quantum recoil that the
classical Thomson picture omits.

$$
% caption: Compton scattering. The two tree diagrams: in the s-channel the electron
% first absorbs the incoming photon and later emits the outgoing one; in the
% u-channel the order is reversed. Both are summed because the photons are
% identical.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- s-channel ---
  \draw[thick, ->] (-6.6,-1.0) -- (-5.8,-1.0);
  \draw[thick] (-5.8,-1.0) -- (-5.0,-1.0);
  \draw[thick, dashed] (-6.4,0.9) -- (-5.0,-1.0);
  \draw[acc, thick, ->] (-5.0,-1.0) -- (-4.2,-1.0);
  \draw[acc, thick] (-4.2,-1.0) -- (-3.4,-1.0);
  \draw[thick, dashed] (-3.4,-1.0) -- (-2.0,0.9);
  \fill (-5.0,-1.0) circle (1.8pt);
  \fill (-3.4,-1.0) circle (1.8pt);
  \node[left] at (-6.6,-1.0) {electron};
  \node[left] at (-6.5,0.9) {photon};
  \node[right] at (-2.1,0.9) {photon};
  \node[acc] at (-4.2,-1.45) {virtual electron};
  \node at (-4.2,1.15) {$s$-channel};
  % --- u-channel ---
  \draw[thick, ->] (0.4,-1.0) -- (1.2,-1.0);
  \draw[thick] (1.2,-1.0) -- (2.0,-1.0);
  \draw[thick, dashed] (2.0,-1.0) -- (0.6,0.9);
  \draw[acc, thick, ->] (2.0,-1.0) -- (2.8,-1.0);
  \draw[acc, thick] (2.8,-1.0) -- (3.6,-1.0);
  \draw[thick, dashed] (5.0,0.9) -- (3.6,-1.0);
  \fill (2.0,-1.0) circle (1.8pt);
  \fill (3.6,-1.0) circle (1.8pt);
  \node[left] at (0.4,-1.0) {electron};
  \node[left] at (0.7,0.9) {photon};
  \node[right] at (5.0,0.9) {photon};
  \node at (2.8,1.15) {$u$-channel};
\end{tikzpicture}
$$

$$
% caption: The Klein-Nishina cross section normalized to the Thomson value, versus
% photon energy in units of the electron mass. At low energy it plateaus at the
% classical Thomson result; above the electron mass the quantum recoil suppresses
% it, and the total cross section falls roughly as the inverse energy.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->] (0,0) -- (6.4,0) node[right] {photon energy};
  \draw[->] (0,0) -- (0,3.2) node[above] {cross section ratio};
  % Thomson plateau reference
  \draw[dashed] (0,2.7) -- (6.0,2.7);
  \node[left] at (0,2.7) {$1$};
  % decreasing curve: plateau near 1 then ~1/omega falloff, via Bezier
  \draw[very thick]
    (0.15,2.68) .. controls (1.0,2.66) and (1.4,2.5) .. (2.0,2.0)
    .. controls (2.8,1.35) and (3.6,0.95) .. (4.4,0.7)
    .. controls (5.0,0.55) and (5.6,0.48) .. (6.0,0.45);
  \node at (1.0,2.95) {Thomson limit};
  \node[below] at (2.0,0) {$1$};
  \node[below] at (4.0,0) {$3$};
  \node[below] at (6.0,0) {$5$};
\end{tikzpicture}
$$

## Bhabha scattering and channel interference

Electron-positron elastic scattering $e^+e^- \to e^+e^-$, called **Bhabha
scattering**, has two diagrams because the same electron and positron appear in
both initial and final state. The $s$-channel is annihilation into a virtual photon
that recreates the pair; the $t$-channel is the photon exchanged between the
electron and positron as they pass, the direct Coulomb interaction. The amplitude
is the difference of the two (the relative minus sign is fixed by Fermi statistics
under exchange of the identical outgoing configurations):

$$
\mathcal M = \mathcal M_t - \mathcal M_s .
$$

The squared amplitude has three pieces: $|\mathcal M_t|^2$, $|\mathcal M_s|^2$, and
the **interference** $-2\operatorname{Re}(\mathcal M_t\mathcal M_s^\ast)$. The
$t$-channel piece carries $1/t^2$ and diverges as $\theta \to 0$ — the forward
Rutherford peak of Coulomb scattering — so Bhabha scattering is dominated by small
angles and is used at $e^+e^-$ colliders as the **luminosity monitor**: its rate is
calculable in pure QED, so counting forward Bhabha events calibrates the beam
luminosity. The $s$-channel and interference terms become important only at wide
angles and near a resonance, where the annihilation channel can go on shell.

$$
% caption: Bhabha scattering. The t-channel diagram is the direct photon exchange
% between electron and positron, dominant at small angles as the Coulomb peak; the
% s-channel diagram is annihilation into a virtual photon that recreates the pair.
% The two amplitudes interfere.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- t-channel ---
  \draw[thick, ->] (-6.6,1.1) -- (-5.8,1.1);
  \draw[thick] (-5.8,1.1) -- (-5.0,1.1);
  \draw[thick, ->] (-5.0,1.1) -- (-4.2,1.1);
  \draw[thick] (-4.2,1.1) -- (-3.4,1.1);
  \draw[thick, ->] (-6.6,-1.1) -- (-5.8,-1.1);
  \draw[thick] (-5.8,-1.1) -- (-5.0,-1.1);
  \draw[thick, ->] (-5.0,-1.1) -- (-4.2,-1.1);
  \draw[thick] (-4.2,-1.1) -- (-3.4,-1.1);
  \draw[acc, thick, dashed] (-5.0,1.1) -- (-5.0,-1.1);
  \fill (-5.0,1.1) circle (1.8pt);
  \fill (-5.0,-1.1) circle (1.8pt);
  \node[left] at (-6.6,1.1) {electron};
  \node[left] at (-6.6,-1.1) {positron};
  \node at (-4.2,1.55) {$t$-channel};
  % --- s-channel ---
  \draw[thick, ->] (0.4,1.1) -- (1.2,0.6);
  \draw[thick] (1.2,0.6) -- (1.8,0.1);
  \draw[thick, ->] (0.4,-1.1) -- (1.2,-0.6);
  \draw[thick] (1.2,-0.6) -- (1.8,0.1);
  \draw[acc, thick, dashed] (1.8,0.1) -- (3.4,0.1);
  \draw[thick, ->] (3.4,0.1) -- (4.1,0.6);
  \draw[thick] (4.1,0.6) -- (4.7,1.1);
  \draw[thick, ->] (3.4,0.1) -- (4.1,-0.4);
  \draw[thick] (4.1,-0.4) -- (4.7,-1.1);
  \fill (1.8,0.1) circle (1.8pt);
  \fill (3.4,0.1) circle (1.8pt);
  \node[left] at (0.4,1.1) {electron};
  \node[left] at (0.4,-1.1) {positron};
  \node at (2.6,1.55) {$s$-channel};
\end{tikzpicture}
$$

## Summary

Casimir's trick — insert $\sum u\bar u = \gamma\!\cdot\!p + m$, take the trace, apply the
four trace theorems — reduces every tree-level QED cross section to gamma-matrix
algebra. Muon pair production has the single $s$-channel amplitude with
$\langle|\mathcal M|^2\rangle = e^4(1+\cos^2\theta)$, giving
$\d\sigma/\d\Omega = (\alpha^2/4s)(1+\cos^2\theta)$ and
$\sigma = 4\pi\alpha^2/3s$; its ratio to hadron production, $R = N_c\sum_q Q_q^2$,
counts three colors. Compton scattering sums an $s$- and a $u$-channel diagram into
the Klein-Nishina formula, which reduces to the Thomson cross section
$8\pi\alpha^2/3m^2$ at low energy. Bhabha scattering adds $s$- and $t$-channel
diagrams whose interference and forward Coulomb peak make it the collider
luminosity monitor. Every result is finite at tree level; the divergences that
appear when loops are added are the subject of the
[next lesson](/particle-physics/qed/renormalization-running-coupling).

[^griffiths-trace]: Casimir's trick, the completeness relations $\sum u\bar u = \gamma\!\cdot\!p + m$, the trace theorems, and the worked cross sections for $e^+e^-\to\mu^+\mu^-$, Compton, and Bhabha scattering are in Griffiths, §7.7–7.9. Halzen & Martin, Ch. 6 §6.3–6.6, works the same processes with the trace technology; Thomson, Ch. 6, gives the $e^+e^-$ annihilation cross sections and the $R$ ratio. The Klein-Nishina formula and the Thomson limit follow the same references. Measured values of $R$ and the fine-structure constant are from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).