Part IV, Chapter 5 | Page 1 of 4

Elementary QED Processes II

Moller scattering: identical fermions and the full derivation of the differential cross section

5.1 Moller Scattering: Identical Particles

Moller scattering, $e^-(p_1) + e^-(p_2) \to e^-(p_3) + e^-(p_4)$, is the first process where we encounter a fundamental subtlety of quantum field theory: the exchange of identical fermions. Because both incoming and outgoing particles are electrons, there are two Feynman diagrams that contribute at tree level — connected by swapping the two final-state electrons.

The Mandelstam variables for this process are:

$$s = (p_1 + p_2)^2, \quad t = (p_1 - p_3)^2, \quad u = (p_1 - p_4)^2$$

satisfying $s + t + u = 4m_e^2$. In the high-energy (ultrarelativistic) limit we take $m_e \to 0$, so $s + t + u = 0$.

💡Why Two Diagrams?

Imagine detecting an electron at angle $\theta$. It could be the "original" electron 1 that scattered to that angle (t-channel), or electron 2 that scattered there instead (u-channel). Since electrons are identical, quantum mechanics demands we add the amplitudes before squaring — with a crucial relative minus sign from Fermi statistics.

5.2 The t-Channel and u-Channel Amplitudes

t-channel diagram: Electron 1 emits a virtual photon and becomes electron 3; electron 2 absorbs it and becomes electron 4. The momentum transfer is$q = p_1 - p_3$ with $q^2 = t$.

Applying the QED Feynman rules (vertex factor $-ie\gamma^\mu$, photon propagator $-ig_{\mu\nu}/q^2$, external spinors):

$$i\mathcal{M}_t = \bar{u}(p_3)(-ie\gamma^\mu)u(p_1) \frac{-ig_{\mu\nu}}{(p_1 - p_3)^2} \bar{u}(p_4)(-ie\gamma^\nu)u(p_2)$$

Simplifying:

$$\mathcal{M}_t = -\frac{e^2}{t}\left[\bar{u}(p_3)\gamma^\mu u(p_1)\right]\left[\bar{u}(p_4)\gamma_\mu u(p_2)\right]$$

u-channel diagram: Now electron 1 scatters into electron 4, and electron 2 into electron 3. This is obtained from the t-channel by exchanging$p_3 \leftrightarrow p_4$ (equivalently, swapping the two outgoing electrons):

$$\mathcal{M}_u = -\frac{e^2}{u}\left[\bar{u}(p_4)\gamma^\mu u(p_1)\right]\left[\bar{u}(p_3)\gamma_\mu u(p_2)\right]$$

The total amplitude includes the relative minus sign from Fermi-Dirac statistics. When two identical fermion lines are exchanged, the amplitude picks up a factor of $(-1)$:

$$\boxed{\mathcal{M} = \mathcal{M}_t - \mathcal{M}_u = -e^2\left[\frac{\bar{u}_3\gamma^\mu u_1 \cdot \bar{u}_4\gamma_\mu u_2}{t} - \frac{\bar{u}_4\gamma^\mu u_1 \cdot \bar{u}_3\gamma_\mu u_2}{u}\right]}$$

💡The Fermion Minus Sign

This minus sign is the Feynman-diagram manifestation of the Pauli exclusion principle. In the path integral formalism, it arises because fermionic fields anticommute: exchanging two identical external fermion legs introduces a factor of $(-1)$. Without this sign, the forward-scattering amplitude would not have the correct analyticity properties, and the theory would violate unitarity.

5.3 Evaluating |M|$^2$: Spin Sums and Traces

To compute the unpolarized cross section, we average over initial spins and sum over final spins. The squared amplitude has three contributions:

$$\overline{|\mathcal{M}|^2} = \frac{1}{4}\sum_{\text{spins}}\left[|\mathcal{M}_t|^2 + |\mathcal{M}_u|^2 - 2\,\text{Re}(\mathcal{M}_t \mathcal{M}_u^*)\right]$$

The t-channel squared term

Using the completeness relation $\sum_s u^s(p)\bar{u}^s(p) = \not{p} + m$ and working in the massless limit:

$$\frac{1}{4}\sum_{\text{spins}}|\mathcal{M}_t|^2 = \frac{e^4}{t^2}\,\text{Tr}\!\left[\not{p}_3\gamma^\mu\not{p}_1\gamma^\nu\right]\,\text{Tr}\!\left[\not{p}_4\gamma_\mu\not{p}_2\gamma_\nu\right]$$

Each trace is evaluated using the standard trace identity:

$$\text{Tr}\!\left[\gamma^\alpha\gamma^\beta\gamma^\gamma\gamma^\delta\right] = 4\left(g^{\alpha\beta}g^{\gamma\delta} - g^{\alpha\gamma}g^{\beta\delta} + g^{\alpha\delta}g^{\beta\gamma}\right)$$

Applying this to the first trace:

$$\text{Tr}\!\left[\not{p}_3\gamma^\mu\not{p}_1\gamma^\nu\right] = 4\left(p_3^\mu p_1^\nu - g^{\mu\nu}(p_3 \cdot p_1) + p_3^\nu p_1^\mu\right)$$

Contracting the two traces and expressing everything in Mandelstam variables ($p_1 \cdot p_2 = s/2$, $p_1 \cdot p_3 = -t/2$,$p_1 \cdot p_4 = -u/2$ in the massless limit):

$$\frac{1}{4}\sum_{\text{spins}}|\mathcal{M}_t|^2 = 2e^4\,\frac{s^2 + u^2}{t^2}$$

The u-channel squared term

By the exchange $t \leftrightarrow u$ (which corresponds to $p_3 \leftrightarrow p_4$):

$$\frac{1}{4}\sum_{\text{spins}}|\mathcal{M}_u|^2 = 2e^4\,\frac{s^2 + t^2}{u^2}$$

The interference term

The cross term involves a single trace over all four fermion propagators (since the spinor indices connect differently). After careful evaluation:

$$-\frac{2}{4}\sum_{\text{spins}}\text{Re}(\mathcal{M}_t\mathcal{M}_u^*) = 2e^4\,\frac{2s^2}{tu}$$

5.4 The Moller Cross Section

Combining all three contributions, the spin-averaged squared amplitude is:

$$\boxed{\overline{|\mathcal{M}|^2} = 2e^4\left[\frac{s^2 + u^2}{t^2} + \frac{s^2 + t^2}{u^2} + \frac{2s^2}{tu}\right]}$$

Using $e^2 = 4\pi\alpha$ and the 2-body cross section formula$d\sigma/d\Omega = |\mathcal{M}|^2/(64\pi^2 s)$:

$$\frac{d\sigma}{d\Omega}\bigg|_{\text{CM}} = \frac{\alpha^2}{2s}\left[\frac{s^2 + u^2}{t^2} + \frac{s^2 + t^2}{u^2} + \frac{2s^2}{tu}\right]$$

In terms of the CM scattering angle $\theta$, with $t = -(s/2)(1 - \cos\theta)$and $u = -(s/2)(1 + \cos\theta)$:

$$\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{2s}\left[\frac{(3 + \cos^2\theta)^2}{\sin^4\theta}\right]$$

Note the $1/\sin^4\theta$ behavior, which diverges both at$\theta \to 0$ (forward scattering, $t \to 0$) and$\theta \to \pi$ (backward scattering, $u \to 0$). This is a signature of massless photon exchange — the Coulomb divergence. For identical particles, the backward divergence is just as physical as the forward one.

💡Forward-Backward Symmetry

The Moller cross section is symmetric under $\theta \to \pi - \theta$(equivalently $t \leftrightarrow u$). This is a direct consequence of the identical nature of the two electrons: swapping "which electron went where" is physically indistinguishable. The interference term $2s^2/(tu)$ ensures this symmetry is exact.

Key Concepts (Page 1)

• Identical fermions require antisymmetrization: $\mathcal{M} = \mathcal{M}_t - \mathcal{M}_u$
• The relative minus sign is the Feynman-diagram form of Fermi-Dirac statistics
• Spin sums convert spinor products into traces via $\sum_s u\bar{u} = \not{p} + m$
• Three contributions to $|\mathcal{M}|^2$: t-squared, u-squared, and interference
• The interference term $\sim s^2/(tu)$ enforces forward-backward symmetry
• Coulomb divergences at $\theta = 0$ and $\theta = \pi$ from massless photon exchange

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