Compton Scattering — The Easiest Real QED Calculation

Compton scattering, $e^{-}+\gamma \to e^{-}+\gamma$, is the simplest physical QED process whose cross section can be computed end-to-end at tree level. It uses every piece of the QED machinery introduced in QED and QED/historical.md — Lagrangian, Feynman rules, LSZ, kinematics — but only at lowest order, with no loops, no renormalization, and no bound-state complications.

It is the right "first concrete application" to compare against the much more involved hydrogen calculation.

1. The Process

A photon of 4-momentum $k^{\mu }$ scatters off a free electron of 4-momentum $p^{\mu }$:

$$e^{-}(p,s)+\gamma (k,\lambda )\to e^{-}(p^{\prime },s^{\prime })+\gamma (k^{\prime },{\lambda }^{\prime }).$$

Initial and final states are asymptotic Fock-space vectors (see fock-space-inventory.md §3):

$$|i⟩=\sqrt{2{\omega }_{\mathbf{p}}2|\mathbf{k}|}{b}_{\mathbf{p},s}^{†}{a}_{\mathbf{k},\lambda }^{†}|0⟩,|f⟩=\sqrt{2{\omega }_{{\mathbf{p}}^{\prime }}2|{\mathbf{k}}^{\prime }|}{b}_{{\mathbf{p}}^{\prime },s^{\prime }}^{†}{a}_{{\mathbf{k}}^{\prime },{\lambda }^{\prime }}^{†}|0⟩.$$

Conservation of 4-momentum: $p+k=p^{\prime }+k^{\prime }$.

2. Feynman Diagrams

At lowest order in $e$ (i.e. $O(e^2)$), there are exactly two diagrams, related by exchange of the two photon vertices:

        s-channel                    u-channel

  γ(k)        γ(k')          γ(k')        γ(k)
   \           /              \             /
    \         /                \           /
     ●═══════●                  ●═════════●
    /  e(p+k) \                /  e(p-k')  \
   /           \              /             \
  e(p)         e(p')         e(p)           e(p')

• s-channel: electron absorbs $k$, propagates with momentum $p+k$, emits $k^{\prime }$.
• u-channel: electron emits $k^{\prime }$ first, propagates with momentum $p-k^{\prime }$, then absorbs $k$.

(There is no t-channel diagram because there is no photon self-coupling in QED — the photon doesn't interact directly with another photon at tree level.)

3. The Amplitude

Reading the diagrams off the QED Feynman rules (QED Step 6):

$$i\mathcal{M}=\bar{u}(p^{\prime },s^{\prime })[(-ie{\gamma }^{\mu })\frac{i(/p+/k+m)}{(p+k{)}^2-m^2}(-ie{\gamma }^{\nu })+(-ie{\gamma }^{\nu })\frac{i(/p-/k^{\prime }+m)}{(p-k^{\prime }{)}^2-m^2}(-ie{\gamma }^{\mu })]u(p,s){ϵ}_{\mu }^{\ast }(k^{\prime },{\lambda }^{\prime }){ϵ}_{\nu }(k,\lambda ).$$

Using $(p+k{)}^2-m^2=2p\cdot k$ and $(p-k^{\prime }{)}^2-m^2=-2p\cdot k^{\prime }$ (since $p^2=m^2$ and $k^2=k^{\prime 2}=0$), this simplifies to

$$\mathcal{M}=-e^2\bar{u}(p^{\prime },s^{\prime })\left[\frac{{\gamma }^{\mu }(/p+/k+m){\gamma }^{\nu }}{2p\cdot k}-\frac{{\gamma }^{\nu }(/p-/k^{\prime }+m){\gamma }^{\mu }}{2p\cdot k^{\prime }}\right]u(p,s){ϵ}_{\mu }^{\ast }(k^{\prime },{\lambda }^{\prime }){ϵ}_{\nu }(k,\lambda ).$$

This is the complete tree-level amplitude. No regularization, no counterterms, no infinities — it's a finite algebraic expression, written down in two lines from the Lagrangian.

4. The Spin- and Polarization-Averaged Squared Amplitude

For an unpolarized cross section, average over initial electron spins / photon polarizations and sum over final ones:

$$\overline{|\mathcal{M}{|}^2}=\frac{1}{4}\sum _{s,s^{\prime },\lambda ,{\lambda }^{\prime }}|\mathcal{M}{|}^2.$$

Standard trace technology — using ${\sum }_{s}u(p,s)\bar{u}(p,s)=/p+m$ and ${\sum }_{\lambda }{ϵ}_{\mu }(k,\lambda ){ϵ}_{\nu }^{\ast }(k,\lambda )=-{\eta }_{\mu \nu }$ (in Feynman gauge) — gives the classic result

$$\boxed{\overline{|\mathcal{M}{|}^2}=2e^4\left[\frac{p\cdot k^{\prime }}{p\cdot k}+\frac{p\cdot k}{p\cdot k^{\prime }}+2m^2\left(\frac{1}{p\cdot k}-\frac{1}{p\cdot k^{\prime }}\right)+m^4{\left(\frac{1}{p\cdot k}-\frac{1}{p\cdot k^{\prime }}\right)}^2\right].}$$

The trace evaluation is mechanical; it takes maybe two pages of algebra in any QFT textbook (e.g. Peskin–Schroeder §5.5, Schwartz §13.4).

5. The Klein–Nishina Cross Section

In the lab frame (electron initially at rest), with incoming photon energy $\omega$ and outgoing photon energy ${\omega }^{\prime }$ at angle $\theta$, momentum conservation gives the Compton wavelength shift:

$$\frac{1}{{\omega }^{\prime }}-\frac{1}{\omega }=\frac{1-\cos \theta }{m}.$$

Plugging into $\overline{|\mathcal{M}{|}^2}$ and folding in the standard $2\to 2$ phase-space factor (see QFT/cross-sections.md §2.5) yields the Klein–Nishina formula (1929):

$$\boxed{\frac{d\sigma }{d\Omega }=\frac{{\alpha }^2}{2m^2}{\left(\frac{{\omega }^{\prime }}{\omega }\right)}^2\left(\frac{\omega }{{\omega }^{\prime }}+\frac{{\omega }^{\prime }}{\omega }-{\sin }^2\theta \right).}$$

This is the central result. It gives the scattering cross section as an explicit closed-form function of incoming photon energy and scattering angle.

Limits

• Low-energy (Thomson) limit, $\omega \ll m$. Then ${\omega }^{\prime }/\omega \to 1$ and the formula reduces to

  $$\frac{d\sigma }{d\Omega }\to \frac{{\alpha }^2}{2m^2}(1+{\cos }^2\theta )=\frac{r_e^2}{2}(1+{\cos }^2\theta ),$$

  the classical Thomson scattering cross section, with $r_e=\alpha /m$ the classical electron radius. Total cross section ${\sigma }_{\text{Thomson}}=\frac{8\pi }{3}{r}_e^2\approx 6.65\times 10^{-29}{\mathrm{m}}^2$.

• High-energy limit, $\omega \gg m$. The cross section drops as $\sigma \sim \frac{\pi {\alpha }^2}{m\omega }\log (2\omega /m)$, a slow logarithmic falloff. Photons become more forward-peaked.

6. What Was — and Wasn't — Used

This is what makes Compton scattering the "easiest real QED application":

Compare with QED/hydrogen.md, which needs all of the above plus much more.

7. Why It's a Cleaner First Example Than Hydrogen

The scattering amplitude can be derived in a single sitting; the hydrogen spectrum requires the entire NRQED framework.

8. Historical and Experimental Status

• Klein and Nishina (1929) derived the formula immediately after Dirac's equation, before QED was fully formulated. Their derivation used the Dirac equation in an external EM field and a clever cancellation of negative-energy contributions; the modern QFT derivation (above) is conceptually cleaner but gives the same answer.
• Experimental tests: Compton's original 1923 X-ray scattering experiment confirmed the wavelength shift; Klein–Nishina's energy-dependent angular distribution has been verified for $\gamma$-rays from MeV to TeV energies (e.g. by the Compton Gamma Ray Observatory).
• Compton scattering is the dominant photon–matter interaction process in the energy range $\sim 100\mathrm{keV}$ to $\sim 10\mathrm{MeV}$, of central importance in medical imaging, $\gamma$-ray astronomy, and radiation shielding.

9. Take-Aways

• Compton scattering is the simplest real QED calculation: two diagrams, no loops, closed-form Klein–Nishina cross section.
• It exercises the full QED Feynman-rule machinery without any of the bound-state, loop, or renormalization complications of hydrogen.
• The Thomson and high-energy limits cleanly recover classical and ultra-relativistic regimes from the same formula.
• For pedagogical purposes, this is the right first place to land after learning the QED postulates; hydrogen is much later.