Feynman Rules for QED

The Dirac equation supplies free electrons and positrons; Maxwell's equations supply free photons. Quantum electrodynamics is the theory of what happens when the two couple, through the single interaction term that lets a photon be emitted or absorbed by a charged fermion. No exact solution exists, so every prediction is a perturbation series in the small coupling $\alpha \approx 1/137$. Feynman's achievement was to make each term in that series a picture, and to fix a dictionary that turns any picture into a definite complex number — the amplitude. This lesson states that dictionary for QED and shows how the amplitude becomes an observable rate. It assumes the Dirac spinors and the golden rule from the earlier modules.

Throughout, $\hslash =c=1$ and ${\eta }^{\mu \nu }=\mathrm{diag}(+,-,-,-)$. The electromagnetic coupling is written $e=\sqrt{4\pi \alpha }$, the positron charge in Heaviside-Lorentz units, so that $\alpha =e^2/4\pi$.¹

Perturbation theory as a sum of diagrams

The interaction between the Dirac field $\psi$ and the photon field $A_{\mu }$ is governed by the current-potential coupling

This is the entire content of QED beyond the free theory: one term, one coupling. Because $e$ is small, the transition amplitude for any process expands in powers of $e$, and each power corresponds to one factor of the interaction acting once. A term with $n$ factors of $e$ is drawn as a diagram with $n$ vertices, each vertex a point where a photon line meets a fermion line. The amplitude for a process is the sum over all topologically distinct diagrams with the correct external particles.

• Order counting. Every vertex contributes one factor of $e$, so a diagram with $V$ vertices scales as $e^V\sim {\alpha }^{V/2}$. A physical amplitude has an even number of vertices for the leading process; the cross section, being $|\mathcal{M}{|}^2$, then scales as ${\alpha }^V$. The lowest-order (tree) diagram dominates; each additional pair of vertices adds a loop and suppresses the contribution by $\alpha \approx 0.0073$.
• Tree vs loop. A tree diagram has no closed internal loops; its amplitude is finite and gives the leading prediction. A loop diagram contains a closed circuit of internal lines, carries an unconstrained internal momentum integrated over all values, and supplies the small quantum corrections treated in the later lessons on renormalization and $g-2$.

External lines: spinors and polarizations

Each particle entering or leaving a diagram contributes a wavefunction factor for its type and direction of travel. These are read directly off the Dirac and Maxwell plane-wave solutions.

• Incoming fermion (electron): a spinor $u(p,s)$.
• Outgoing fermion: the Dirac adjoint $\bar{u}(p,s)=u^{†}{\gamma }^0$.
• Incoming antifermion (positron): $\bar{v}(p,s)$.
• Outgoing antifermion: $v(p,s)$.
• Incoming photon: a polarization vector ${ϵ}_{\mu }(k,\lambda )$.
• Outgoing photon: the conjugate ${ϵ}_{\mu }^{\ast }(k,\lambda )$.

The pattern for antiparticles is the Feynman-Stückelberg rule that an outgoing antiparticle is bookkept as an incoming negative-energy particle, so $v$ and $\bar{v}$ swap roles relative to $u$ and $\bar{u}$. The spinors satisfy the momentum-space Dirac equations $(\gamma \cdot p-m)u=0$ and $(\gamma \cdot p+m)v=0$, where $\gamma \cdot p\equiv {\gamma }^{\mu }{p}_{\mu }$, and are normalized by $\bar{u}u=2m$, $\bar{v}v=-2m$. The photon polarization is transverse, $k^{\mu }{ϵ}_{\mu }=0$, with two physical states $\lambda =1,2$.

Propagators: internal lines

An internal line represents a virtual particle: one whose four-momentum $q$ is fixed by momentum conservation at the vertices but is off the mass shell, $q^2\ne m^2$. Its wavefunction is replaced by a propagator, the Fourier transform of the field's Green function, which is the amplitude to move from the emission vertex to the absorption vertex.

The fermion propagator for an internal line of momentum $q$ and mass $m$ is

and the photon propagator, in the Feynman gauge, is

The pole at $q^2=m^2$ is the imprint of the physical particle: as the virtual line approaches its mass shell the propagator blows up, which is why exchanged particles near resonance dominate a cross section. The $i\epsilon$ fixes the contour and encodes causal (Feynman) boundary conditions. The numerator $\gamma \cdot q+m$ carries the spin structure; for the massless photon the numerator $-{\eta }_{\mu \nu }$ carries the two transverse polarizations plus gauge artifacts that cancel against the conserved current ${\partial }_{\mu }{j}^{\mu }=0$.

The QED vertex

Every point where a photon line meets an electron line contributes the vertex factor

with $\mu$ the Lorentz index contracted into the attached photon line (through its propagator or polarization vector), and the gamma matrix sandwiched between the spinors running along the fermion line. The vertex is the derivative of the interaction $-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$ with respect to the three fields, so its structure — one gamma matrix, one power of $e$ — is dictated entirely by the form of ${\mathcal{L}}_{\text{int}}$.

Charge conservation at the vertex is automatic: the fermion number flowing in equals that flowing out, since a single fermion line passes through with a photon attached. A vertex therefore always has exactly two fermion ends and one photon end. This is why QED processes conserve the number of electrons minus positrons, and why a photon never couples to itself at tree level.

Assembling the amplitude

Reading a diagram into an amplitude $\mathcal{M}$ follows a fixed procedure. Label every line with a four-momentum, conserve momentum at each vertex, and write down the factors against the fermion arrows — that is, starting from the outgoing fermion (a $\bar{u}$ or $\bar{v}$) and moving backward along each fermion line to its incoming end. The gamma matrices, propagators, and spinors are multiplied in that order so the matrix indices contract correctly.

For a tree diagram there is no loop integral and no relative sign subtlety, so the recipe returns $\mathcal{M}$ as a product of spinors, gammas, and one propagator. The result is a Lorentz scalar: all indices are contracted, leaving a complex number for each choice of external spins and polarizations.

For the electron-muon diagram, the two currents are $\bar{u}(p_3){\gamma }^{\mu }u(p_1)$ (electron) and $\bar{u}(p_4){\gamma }^{\nu }u(p_2)$ (muon), the photon propagator supplies $-i{\eta }_{\mu \nu }/q^2$ with $q=p_1-p_3$, and each vertex a factor $ie$. Collecting,

The amplitude is the product of two conserved currents tied together by the photon propagator — the field-theory version of two charges interacting through the electromagnetic field.

From the amplitude to a rate

The amplitude enters an observable only through $|\mathcal{M}{|}^2$, averaged over initial spins and summed over final spins, fed into Fermi's golden rule. For a $2\to 2$ scattering in the center-of-momentum frame,

and for a $1\to 2$ decay of a particle of mass $M$,

with $⟨⟩$ the spin average and $|{\vec{p}}_f|$ the final-state momentum set by the kinematics. Evaluating $⟨|\mathcal{M}{|}^2⟩$ from the spinor expressions is the calculational core of QED, carried out by the trace technology of the next lesson. The Feynman rules have reduced the quantum field theory to a bookkeeping exercise: draw the diagrams, apply the dictionary, square, and integrate.

The same process, two channels

Momentum can flow through a diagram in more than one way, and each way is a separate diagram to be added into $\mathcal{M}$. The Mandelstam variables label the channels by which invariant the exchanged momentum carries. In an $s$-channel diagram the two incoming particles annihilate into a single virtual line of invariant mass $\sqrt{s}$, which then produces the final state; the propagator carries $q^2=s$. In a $t$-channel diagram one incoming particle emits the virtual line and continues, so the exchange carries the momentum transfer $q^2=t\le 0$. A $u$-channel is the same with the two final particles interchanged.

The distinction is physical, not cosmetic. An $s$-channel propagator $1/(s-m^2)$ becomes resonant when $\sqrt{s}$ hits the mass of a real intermediate particle, producing a Breit-Wigner peak in the cross section; a $t$-channel propagator $1/t$, with $t\le 0$, never resonates but peaks in the forward direction where $|t|$ is smallest. When the same external particles admit both, the two amplitudes add and interfere — the mechanism behind Bhabha scattering in the next lesson.

Summary

QED reduces to one interaction, $-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$, and one small parameter, $\alpha =e^2/4\pi$. Every amplitude is a sum of diagrams; every diagram translates by a fixed dictionary: external spinors $u,\bar{u},v,\bar{v}$ and polarizations ${ϵ}_{\mu }$; the propagators $i(\gamma \cdot q+m)/(q^2-m^2)$ and $-i{\eta }_{\mu \nu }/q^2$ for internal lines; the vertex $ie{\gamma }^{\mu }$. Reading a diagram against the fermion arrows yields $\mathcal{M}$; squaring it and applying the golden rule yields a rate. Each vertex costs one factor of $e$, so tree diagrams give the leading cross section of order ${\alpha }^2$ and loops add corrections one power of $\alpha$ at a time. The next lesson turns the dictionary on the reference reactions of QED and evaluates their cross sections.