Bhabha Scattering & Pair Production

4.13 Bhabha Scattering: e⁺e⁻ → e⁺e⁻

When the final-state particles are the same species as the initial state, a new diagram appears. Bhabha scattering ($e^+e^- \to e^+e^-$) has two tree-level diagrams:

The total amplitude is the sum (with a crucial relative minus sign from fermion statistics):

4.14 The Bhabha Cross Section

Squaring the amplitude gives three distinct contributions:

The s-channel squared term is identical to e⁺e⁻ → μ⁺μ⁻ (we already computed it):

The t-channel squared term: The trace calculation proceeds identically but with$s \leftrightarrow t$. The photon propagator has $1/t^2$ and the kinematics rearrange to give:

The interference term: This is the genuinely new calculation. Cross terms between the s and t channels produce a single trace over eight gamma matrices (rather than a product of two four-gamma traces). Using the trace identity for eight gamma matrices:

Combining all three pieces (in the massless limit):

Converting to the CM scattering angle $\theta$ using $t = -(s/2)(1-\cos\theta)$ and $u = -(s/2)(1+\cos\theta)$:

4.15 Interference Effects in Bhabha Scattering

The angular distribution of Bhabha scattering is dramatically different from $e^+e^- \to \mu^+\mu^-$due to the t-channel contribution:

• 1.Forward peak: The t-channel term has $1/t^2 \sim 1/\sin^4(\theta/2)$, producing a strong forward peak ($\theta \to 0$). This is Coulomb-like scattering.
• 2.Backward enhancement: The s-channel term contributes a $(1+\cos^2\theta)$pattern, enhancing the backward direction.
• 3.Destructive interference: The cross term is negative (note the minus sign), reducing the cross section at intermediate angles.

4.16 Pair Production from Crossing Symmetry

The process $\gamma\gamma \to e^+e^-$ is related to Compton scattering by crossing symmetry. The idea is that an incoming particle with momentum $p$ is equivalent to an outgoing antiparticle with momentum $-p$.

Starting from the Compton amplitude $\gamma(k) + e^-(p) \to \gamma(k') + e^-(p')$, we cross:

to obtain $\gamma(k_1) + \gamma(k_2) \to e^-(p_-) + e^+(p_+)$. The Compton Mandelstam variables transform as:

Applying this crossing to the Compton result, the pair production cross section in the CM frame (where $|\vec{k}_1| = |\vec{k}_2| = \omega$ and $\sqrt{s} = 2\omega$) is:

where $|\vec{p}| = \sqrt{\omega^2 - m^2}$ is the magnitude of the final-state electron 3-momentum. Note the threshold condition: pair production requires $\sqrt{s} \geq 2m$, i.e., $\omega \geq m$.

4.17 Properties of Pair Production

Near threshold ($\omega \to m$, or $\beta = |\vec{p}|/\omega \to 0$): the phase space factor $|\vec{p}|/|\vec{k}| = \beta$ suppresses the cross section, and:

High-energy limit ($s \gg m^2$, all masses negligible):

The logarithmic enhancement arises from the t and u channel propagators going nearly on-shell when the outgoing electron or positron is collinear with an incoming photon.

4.18 Pair Annihilation: e⁺e⁻ → γγ

The reverse process, $e^+e^- \to \gamma\gamma$, is obtained from pair production by time reversal (or equivalently, by another crossing). By detailed balance, at the same CM energy:

but the cross sections differ by the ratio of phase space factors and flux. In the non-relativistic limit ($v \to 0$), the annihilation cross section becomes:

The $1/v$ enhancement at low velocity is characteristic of s-wave annihilation. Multiplying by the relative velocity gives the annihilation rate$\sigma v \to \pi\alpha^2/m^2$, which is finite and well-behaved.