Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part IV, Chapter 5 | Page 2 of 4

Crossing Symmetry and Bhabha Scattering

Relating different physical processes by analytic continuation in Mandelstam variables

5.5 Crossing Symmetry: The Key Idea

One of the most powerful results in quantum field theory is crossing symmetry: different physical scattering processes are described by the same analytic function of Mandelstam variables, evaluated in different kinematic regions. An incoming particle with 4-momentum $p$ is equivalent to an outgoing antiparticle with momentum $-p$.

Consider the fundamental QED process $e^+e^- \to \mu^+\mu^-$ in the s-channel. The amplitude depends on $s$, $t$, $u$:

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

where the s-channel photon propagator gives $1/s = 1/(p_1 + p_2)^2$. The spin-averaged squared amplitude in the massless limit is:

$$\overline{|\mathcal{M}|^2}_{e^+e^- \to \mu^+\mu^-} = 2e^4\,\frac{t^2 + u^2}{s^2}$$

💡Crossing as a Road Map

Crossing symmetry is not an approximation — it is an exact consequence of the CPT theorem and the analytic structure of quantum field theory. Once you compute one process, you get a family of related processes for free by relabeling momenta and continuing Mandelstam variables to different kinematic regions.

5.6 From s-Channel to t-Channel

To obtain the amplitude for $e^-\mu^- \to e^-\mu^-$ (t-channel scattering) from $e^+e^- \to \mu^+\mu^-$ (s-channel annihilation), we "cross" the positron from the initial state to a final-state electron, and a final-state antimuon to an initial-state muon. The prescription is:

$$e^+e^- \to \mu^+\mu^- \quad\xrightarrow{\text{crossing}}\quad e^-\mu^- \to e^-\mu^-$$
$$p_{e^+} \to -p_{e^-}' \implies s \leftrightarrow t$$

Concretely, the s-channel annihilation amplitude with propagator $1/s$ becomes a t-channel exchange amplitude with propagator $1/t$:

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

The squared amplitude is obtained by the substitution $s \leftrightarrow t$:

$$\overline{|\mathcal{M}|^2}_{e^-\mu^- \to e^-\mu^-} = 2e^4\,\frac{s^2 + u^2}{t^2}$$

This is exactly the result we would obtain by direct Feynman diagram calculation! Crossing symmetry provides a powerful consistency check and labor-saving device.

The u-channel process

Similarly, crossing $s \leftrightarrow u$ gives $e^-\mu^+ \to e^+\mu^-$:

$$\overline{|\mathcal{M}|^2}_{e^-\mu^+ \to e^+\mu^-} = 2e^4\,\frac{t^2 + s^2}{u^2}$$

5.7 Bhabha Scattering: e$^+$e$^-$ → e$^+$e$^-$

Bhabha scattering (electron-positron elastic scattering) is the richest tree-level QED process because it receives contributions from two distinct topologies:

  • t-channel exchange: the electron and positron exchange a virtual photon, as in Moller scattering
  • s-channel annihilation: the $e^+e^-$ pair annihilates into a virtual photon, which then creates a new $e^+e^-$ pair

The t-channel amplitude:

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

The s-channel amplitude:

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

Here there is no relative minus sign between the two diagrams — the electron and positron are distinguishable particles (particle vs. antiparticle). The total amplitude is:

$$\boxed{\mathcal{M}_{\text{Bhabha}} = \mathcal{M}_t + \mathcal{M}_s}$$

The spin-averaged squared amplitude (massless limit):

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

💡Bhabha vs. Moller: Spot the Difference

Compare the Bhabha result $(t^2+u^2)/s^2 + (s^2+u^2)/t^2 + 2u^2/(st)$ with the Moller result $(s^2+u^2)/t^2 + (s^2+t^2)/u^2 + 2s^2/(tu)$. The Bhabha formula is obtained from Moller by the crossing $u \leftrightarrow s$: replace the u-channel (identical particle exchange) with the s-channel (annihilation). The variable that appears squared in the numerator of the interference term changes accordingly.

5.8 Connection to the CPT Theorem

Crossing symmetry is intimately connected to the CPT theorem, one of the deepest results in quantum field theory. The CPT theorem states that every Lorentz-invariant local quantum field theory with a Hermitian Hamiltonian is invariant under the combined operations of charge conjugation (C), parity (P), and time reversal (T).

The connection to crossing works as follows. Moving a particle from the initial state to the final state (with reversed momentum) is equivalent to replacing it with its antiparticle. Formally, if the S-matrix element for process A is:

$$\langle p_3, p_4 | S | p_1, p_2 \rangle$$

then crossing particle 2 gives:

$$\langle p_3, p_4, \bar{p}_2 | S | p_1 \rangle \quad \text{with } p_2 \to -p_2$$

This is guaranteed to be a valid analytic continuation of the original amplitude by the CPT theorem, which ensures that the amplitude is an analytic function of the Mandelstam variables in the appropriate domain. The key mathematical result is the LSZ reduction formula applied to crossed channels:

$$\mathcal{M}(s, t, u)\big|_{\text{physical region I}} \xrightarrow{\text{analytic continuation}} \mathcal{M}(s, t, u)\big|_{\text{physical region II}}$$

The physical regions are:

  • s-channel: $s > 0$, $t < 0$, $u < 0$ (annihilation/production)
  • t-channel: $t > 0$, $s < 0$, $u < 0$ (exchange scattering)
  • u-channel: $u > 0$, $s < 0$, $t < 0$ (crossed exchange)

💡CPT and Analyticity

The CPT theorem guarantees that the amplitude $\mathcal{M}(s,t,u)$ is a single analytic function defined throughout the complex $(s,t,u)$ plane (with branch cuts on the real axes). Different physical processes correspond to different sheets of this analytic function. This is why crossing symmetry is exact — it's not a perturbative statement but follows from the foundational axioms of quantum field theory: locality, Lorentz invariance, and unitarity.

Key Concepts (Page 2)

  • • Crossing symmetry: $s \leftrightarrow t$ relates $e^+e^- \to \mu^+\mu^-$ to $e^-\mu^- \to e^-\mu^-$
  • • An incoming particle with momentum p = outgoing antiparticle with momentum -p
  • • Bhabha scattering has both s-channel (annihilation) and t-channel (exchange) diagrams
  • • No relative minus sign in Bhabha (particle-antiparticle are distinguishable)
  • • Bhabha = crossed version of Moller: $u \leftrightarrow s$
  • • CPT theorem guarantees crossing as an exact, non-perturbative symmetry
← Page 1
Page 2 of 4
Chapter 5: Elementary QED Processes II
Next Page →