Tree-Level QED Processes

The Feynman rules of the previous lesson 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 ${\eta }^{\mu \nu }=\mathrm{diag}(+,-,-,-)$ 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,

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

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.¹ The identities used repeatedly are

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

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,

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

Feeding this to the golden rule gives the differential cross section

and integrating over the solid angle, the total cross section

The $1+{\cos }^2\theta$ shape is the signature of a spin-1 photon coupling to spin-$\frac{1}{2}$ 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.

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,

is therefore a direct count of colors. Below the charm threshold the open flavors are $u,d,s$ with charges $+\frac{2}{3},-\frac{1}{3},-\frac{1}{3}$, so

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:

with internal electron propagators $i(\gamma \cdot q+m)/(q^2-m^2)$ carrying $q=p_e+k$ in the $s$-channel and $q=p_e-k^{\prime }$ 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,

where $\omega$ and ${\omega }^{\prime }$ are the incoming and scattered photon energies related by the Compton shift

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

the elastic scattering of light by a free charge, independent of frequency. At high energy $\omega \gg m$, the recoil suppresses ${\omega }^{\prime }$, 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.

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):

The squared amplitude has three pieces: $|{\mathcal{M}}_t{|}^2$, $|{\mathcal{M}}_s{|}^2$, and the interference $-2\mathrm{Re}({\mathcal{M}}_t{\mathcal{M}}_s^{\ast })$. 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.

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 $⟨|\mathcal{M}{|}^2⟩=e^4(1+{\cos }^2\theta )$, giving $\mathbf{d}\sigma /\mathbf{d}\Omega =({\alpha }^2/4s)(1+{\cos }^2\theta )$ 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.