Part IV, Chapter 4 | Page 2 of 4

Compton Scattering

Photon-electron scattering: two diagrams, the Klein-Nishina formula, and the Thomson limit

4.8 Compton Scattering: Setup

Compton scattering is the process $\gamma(k) + e^-(p) \to \gamma(k') + e^-(p')$. At tree level, there are two Feynman diagrams:

s-channel (direct)

The electron absorbs the incoming photon, propagates as a virtual electron, then emits the outgoing photon.

$$e^-(p) + \gamma(k) \to e^*_{\text{virtual}} \to e^-(p') + \gamma(k')$$

u-channel (crossed)

The electron first emits the outgoing photon, propagates as a virtual electron, then absorbs the incoming photon.

$$e^-(p) \to e^*_{\text{virtual}} + \gamma(k') \to e^-(p') + \gamma(k)$$

💡Why s and u Channels?

There is no t-channel diagram because the photon does not carry electric charge — it cannot couple to the photon-electron-electron vertex by itself. The two diagrams correspond to the two time orderings of absorbing and emitting a photon. Both must be included and they interfere quantum mechanically.

4.9 Writing the Amplitude

Let $\epsilon^\mu(k)$ and $\epsilon'^\nu(k')$ be the polarization vectors of the incoming and outgoing photons. The s-channel amplitude is:

$$i\mathcal{M}_s = \bar{u}(p')(-ie\gamma^\nu)\epsilon'^*_\nu \frac{i(\not{p}+\not{k}+m)}{(p+k)^2 - m^2}(-ie\gamma^\mu)\epsilon_\mu\, u(p)$$

The u-channel amplitude (with the photon vertices exchanged):

$$i\mathcal{M}_u = \bar{u}(p')(-ie\gamma^\mu)\epsilon_\mu \frac{i(\not{p}-\not{k}'+m)}{(p-k')^2 - m^2}(-ie\gamma^\nu)\epsilon'^*_\nu\, u(p)$$

Using the on-shell condition $p^2 = m^2$ and $k^2 = k'^2 = 0$:

$$(p+k)^2 - m^2 = 2p\cdot k \equiv s - m^2, \qquad (p-k')^2 - m^2 = -2p\cdot k' \equiv u - m^2$$

The total amplitude is:

$$\boxed{\mathcal{M} = -e^2 \epsilon_\mu \epsilon'^*_\nu \bar{u}(p')\left[\frac{\gamma^\nu(\not{p}+\not{k}+m)\gamma^\mu}{2p\cdot k} + \frac{\gamma^\mu(\not{p}-\not{k}'+m)\gamma^\nu}{-2p\cdot k'}\right]u(p)}$$

4.10 Spin and Polarization Sums

For unpolarized scattering, we average over the 2 electron spin states and 2 photon polarizations for each photon, then sum over final states. The overall averaging factor is$1/(2 \cdot 2) = 1/4$.

The photon polarization sum (in Feynman gauge, where Ward identity guarantees the unphysical polarizations cancel) is:

$$\sum_{\lambda=1,2} \epsilon^\mu_\lambda(k)\epsilon^{*\nu}_\lambda(k) \to -g^{\mu\nu}$$

Combined with the electron spin sums, $|\mathcal{M}|^2$ becomes a product of traces involving the s-channel and u-channel propagator numerators. After a lengthy calculation using trace identities and the on-shell conditions, the result is:

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

Defining the Mandelstam-like variables $s' = 2p\cdot k$ and $u' = -2p\cdot k'$ (where$s' = s - m^2$ and $u' = u - m^2$), this can be written more compactly:

$$\boxed{\overline{|\mathcal{M}|^2} = -2e^4\left[\frac{u'}{s'} + \frac{s'}{u'} + 2m^2\left(\frac{1}{s'} + \frac{1}{u'}\right) + m^4\left(\frac{1}{s'} + \frac{1}{u'}\right)^2\right]}$$

4.11 The Klein-Nishina Formula

We evaluate the cross section in the lab frame where the electron is initially at rest:$p = (m, \vec{0})$. Let $\omega$ and $\omega'$ be the energies of the incoming and outgoing photons. From kinematics:

$$p\cdot k = m\omega, \qquad p\cdot k' = m\omega'$$

The Compton relation linking $\omega'$ to $\omega$ and the scattering angle $\theta$:

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

The differential cross section in the lab frame (accounting for the recoil of the electron via the phase space Jacobian) gives the Klein-Nishina formula (1929):

$$\boxed{\frac{d\sigma}{d\Omega}\bigg|_{\text{lab}} = \frac{\alpha^2}{2m^2}\left(\frac{\omega'}{\omega}\right)^2\left[\frac{\omega'}{\omega} + \frac{\omega}{\omega'} - \sin^2\theta\right]}$$

This was one of the earliest triumphs of quantum field theory, correctly predicting the energy dependence of photon-electron scattering that classical theory could not explain.

💡Physics of the Klein-Nishina Formula

At low energies ($\omega \ll m$), $\omega' \approx \omega$ and the formula reduces to Thomson scattering. At high energies ($\omega \gg m$), the cross section falls as$\sigma \sim (\alpha^2/m\omega)\ln(2\omega/m)$ — the photon "punches through" the electron more easily at higher energies. The $(\omega'/\omega)^2$ factor encodes quantum recoil, absent in classical electrodynamics.

4.12 The Thomson Limit

In the low-energy limit $\omega \ll m$, the electron barely recoils: $\omega' \approx \omega$. The Klein-Nishina formula becomes:

$$\frac{d\sigma}{d\Omega}\bigg|_{\text{Thomson}} = \frac{\alpha^2}{2m^2}(1 + \cos^2\theta) = \frac{r_e^2}{2}(1 + \cos^2\theta)$$

where $r_e = \alpha/m \approx 2.82 \times 10^{-13}$ cm is the classical electron radius. Integrating over angles:

$$\boxed{\sigma_{\text{Thomson}} = \frac{8\pi\alpha^2}{3m^2} = \frac{8\pi r_e^2}{3} \approx 0.665 \text{ barn}}$$

This is the Thomson cross section — the classical result for electromagnetic radiation scattering off a charged particle. Remarkably, the full quantum calculation reproduces the classical answer in the appropriate limit, as it must.

For the high-energy limit $\omega \gg m$, integrating the Klein-Nishina formula:

$$\sigma_{\text{Compton}} \approx \frac{\pi\alpha^2}{m\omega}\left[\ln\frac{2\omega}{m} + \frac{1}{2}\right] \qquad (\omega \gg m)$$

The cross section falls logarithmically at high energy — a purely quantum effect with no classical analog.

Key Concepts (Page 2)

• Compton scattering has two tree-level diagrams: s-channel and u-channel
• Photon polarization sums replace ε¹ε*² → -g¹² (Ward identity ensures gauge invariance)
• The Klein-Nishina formula (1929) gives the exact QED differential cross section in the lab frame
• Low-energy limit ($\omega \ll m$): Thomson scattering with $\sigma = \frac{8\pi\alpha^2}{3m^2} \approx 0.665 \text{ barn}$
• High-energy limit ($\omega \gg m$): cross section falls as $\sim \frac{\alpha^2}{m\omega} \ln\frac{2\omega}{m}$
• The Compton wavelength shift $\Delta\lambda = \frac{1}{m}(1 - \cos\theta)$ is a quantum recoil effect

← e⁺e⁻ → μ⁺μ⁻

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Chapter 4: Elementary QED Processes I

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