Quantum electrodynamics ($\mathrm{QED}$)

Quantum electrodynamics couples a Dirac field to an Abelian gauge field through $-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$.

The leading scattering amplitude is the sum of two tree-level Feynman diagrams, with the outgoing photons attached in either order to the fermion line. The Dirac propagator carries momentum $p-k_1$ or $p-k_2$ for incoming electron momentum $p$. Their sum obeys the two-photon fermion Ward identity, although individual diagrams need not. Crossing symmetry relates this amplitude to Compton scattering.

At tree level this QED process has two internal muon exchange diagrams, differing by the order of the two outgoing photon vertices. Their propagator momenta are $p_1-k_1$ and $p_1-k_2$ for incoming muon momentum $p_1$. The amplitude begins at order $e^2$ and obeys the two-photon fermion Ward identity. Identical final photons require the usual factorial correction when integration over the full final-state Lorentz-invariant phase space would otherwise count them twice.

In rationalized natural units, the fine-structure constant is $\alpha =e^2/(4\pi )$. It measures the electromagnetic interaction strength. Its use in a specified perturbative calculation requires a consistent coupling and renormalization scale convention.

Replace the derivative in the Dirac action by the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+iq{A}_{\mu }$. The interaction is then $-q{A}_{\mu }\bar{\Psi }{\gamma }^{\mu }\Psi$. With $\Psi \mapsto e^{-iq\alpha }\Psi$ and $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$, the covariant derivative transforms in the same way as the spinor, proving local gauge invariance.

The derivative of the local phase in $\Psi \mapsto e^{-iq\alpha }\Psi$ cancels the added gradient in the electromagnetic potential. Thus $D_{\mu }^{\prime }{\Psi }^{\prime }=e^{-iq\alpha }{D}_{\mu }\Psi$, and the charged Dirac equation maps solutions to solutions. The same cancellation makes its action gauge invariant.

At tree level, the Feynman diagram has an electron-positron current and a muon-antimuon current joined by one virtual photon. In the massless limit, the spin average and final-state spin sum give $\overline{|\mathcal{M}{|}^2}=e^4(1+{\cos }^2\theta )$, and the integrated relativistic scattering cross-section is $4\pi {\alpha }^2/(3s)$.

The QED Feynman rules combine the Dirac propagator, photon propagator, and vertex $-ie{\gamma }^{\mu }$ with external Dirac spinors and photon polarization vectors. In a covariant gauge they generate perturbative electron-photon scattering amplitudes.

Each elementary QED vertex has one photon leg, so counting these legs gives $V=2I_{\gamma }+E_{\gamma }$. An amplitude with two external photons has even $V$. In particular Compton scattering has no cubic-order elementary-vertex contribution. With four external legs and $V=4$, a connected diagram has four internal lines and one loop.

For massless quarks at photon-exchange tree level, the initial-spin-averaged, final-color-summed amplitude is $\overline{|\mathcal{M}{|}^2}=8N_c{e}^4{Q}_q^2[(p_1\cdot q_1)(p_2\cdot q_2)+(p_1\cdot q_2)(p_2\cdot q_1)]/s^2$. In the centre-of-mass frame this is $N_c{e}^4{Q}_q^2(1+{\cos }^2\theta )$. With $N_c=3$ and the standard two-body flux, its integrated cross-section is the displayed result.

Bhabha scattering is electron-positron elastic scattering $e^{-}{e}^{+}\to e^{-}{e}^{+}$. At tree level it receives an annihilation diagram in the $s$ channel and a photon-exchange diagram in the $t$ channel.

The Pauli term couples a fermion bilinear to electromagnetic field strength through $\bar{\psi }{\sigma }^{\mu \nu }{F}_{\mu \nu }\psi$. In four dimensions it is a dimension-five interaction.

Photon vacuum polarization is the one-particle-irreducible photon two-point function generated by charged-particle loops.

For $J_h^{\mu }={\sum }_f{Q}_f{\bar{q}}_f{\gamma }^{\mu }{q}_f$ without a factor of $e$, use $i\int e^{iqx}⟨T{J}_h^{\mu }(x)J_h^{\nu }(0)⟩d^{4}x=(q^{\mu }{q}^{\nu }-q^2{g}^{\mu \nu }){\Pi }_J(q^2)$. The inclusive tensor is $W_J^{\mu \nu }=2\pi {\rho }_J(s)(q^{\mu }{q}^{\nu }-s{g}^{\mu \nu })$. A massless free-quark calculation gives ${\rho }_J=N_c{\sum }_f{Q}_f^2/(12{\pi }^2)$; strong interactions alter this function.

Expanding the covariant kinetic term of scalar electrodynamics gives a one-photon two-scalar vertex and a two-photon two-scalar seagull vertex. Their tree amplitudes and Ward identities depend on a consistent choice of charge-flow and gauge covariant derivative convention.

At order $e^2$, charged-scalar particle-antiparticle scattering has $s$ annihilation and $t$ photon exchange. For the convention $D=\partial +ieA$ and incoming momenta $p_1,p_2$, outgoing $p_3,p_4$, its amplitude isEqual external masses turn the numerators into $u-t$ and $s-u$. The scalar electrodynamics three-point vertex gives conserved currents, allowing the Coulomb-gauge propagator between conserved currents identity to replace each propagator by the covariant one. The seagull vertex has only two scalar legs and does not give a four-scalar tree at this order.

Expanding the covariant scalar kinetic term gives $ie{A}_{\mu }(\varphi {\partial }^{\mu }{\varphi }^{\ast }-{\varphi }^{\ast }{\partial }^{\mu }\varphi )$. With $D=\partial +ieA$, the vertex for an incoming charged particle of momentum $p$ and an outgoing one of momentum $p^{\prime }$ is $-ie(p+p^{\prime }{)}^{\mu }$. The antiparticle has the opposite charge-flow sign. The quadratic term $e^2{A}^2{\varphi }^{\ast }\varphi$ gives the separate seagull vertex $2i{e}^2{\eta }^{\mu \nu }$. On equal-mass external shells, contracting the three-point momentum with the photon momentum gives zero, the tree scalar quantum electrodynamics Ward identity.

The scalar-QED term $e^2{A}_{\mu }{A}^{\mu }{\varphi }^{\ast }\varphi$ produces a local two-photon two-scalar vertex, conventionally called the seagull vertex.

For an on-shell amplitude $\mathcal{M}={ϵ}_{\mu }(k){\mathcal{M}}^{\mu }$ with an external photon, gauge invariance implies $k_{\mu }{\mathcal{M}}^{\mu }=0$.

For tree Electron-Positron annihilation into three photons, all six orders along the fermion line must be summed. Write $D(r)=/r-m$ and $S(r)=D(r{)}^{-1}=(/r+m)/(r^2-m^2)$. Contracting an internal photon with its momentum gives $S(R)/qS(r)=S(R)-S(r)$ when $q=r-R$. Contracting an end photon removes its adjacent Dirac propagator by the external Dirac equation, with a minus sign at the electron end and a plus sign at the positron end. For either fixed order of the two uncontracted photons, the three insertion positions therefore telescope to zero. Summing their two orders proves the Ward identity. The mass terms in both remaining Dirac propagators are essential when $m\ne 0$. The result expresses the invariance of the scattering amplitude under ${ϵ}_{\ell }\mapsto {ϵ}_{\ell }+c{q}_{\ell }$.

Define the proper electromagnetic vertex with the electric charge removed, and define $S^{-1}$ without the propagator's overall factor $i$. The displayed off-shell identity follows by localizing the vector symmetry in a time-ordered correlation function, including the contact terms at the charged-field insertions, and amputating. Its on-shell form gives the Ward identity for physical photon amplitudes. At zero photon momentum it identifies the vertex with the momentum derivative of the inverse fermion propagator.

In a gauge-preserving regularization and consistent renormalization convention, the proper-vertex Ward-Takahashi identity in QED equates the vertex and electron wave-function renormalization constants. With ${\psi }_0=Z_2^{1/2}\psi$ and $A_0=Z_3^{1/2}A$, the vertex coefficient gives $e_0={\mu }_R^{ϵ}{Z}_{1}e/(Z_2\sqrt{Z_3})$, proving the displayed charge relation in $d=4-2ϵ$. Gauge invariance restricts counterterms but does not forbid charge running.

The two photon orders in a tree-level charged-fermion amplitude must be summed. Contracting one photon vertex with its momentum writes the contracted slash as a difference of inverse Dirac propagators. The external Dirac equations remove the on shell inverse factors; the two remaining terms have equal size and opposite sign. Thus the sum vanishes under the momentum replacement although individual diagrams generally do not. This implements the gauge equivalence of photon polarization vectors.

Localizing an infinitesimal symmetry inside a normalized path integral turns current conservation into a distributional identity. The variation of each inserted field contributes a delta function at that insertion. The displayed sign uses $\delta S=-\int ({\partial }_{\mu }{ϵ}^a)j_a^{\mu }$; the opposite current convention reverses the left side. Measure anomalies or noninvariant boundary states can add terms.

The soft photon theorem factorizes emission of a low-energy photon into the hard amplitude times ${\sum }_i{\eta }_i{Q}_i{p}_i\cdot ϵ/(p_i\cdot q)$. Its Ward identity is equivalent to conservation of total electric charge.