Quantum Field Theory Course

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

Part IV, Chapter 2 | Page 3 of 3

Worked Examples & Applications

Computing cross sections step by step in concrete theories

Worked Example 1: φ⁴ Scattering at Tree Level

Consider the scalar field theory with interaction Lagrangian:

$$\mathcal{L}_{\text{int}} = -\frac{\lambda}{4!}\phi^4$$

Compute the total cross section for 2 → 2 scattering at tree level.

Solution

Step 1: Identify the Feynman diagram. At tree level, there is only one diagram: a single vertex connecting all four external legs. The vertex factor from the Feynman rules is:

$$-i\mathcal{M} = -i\lambda \quad \Rightarrow \quad \mathcal{M} = \lambda$$

The factor of 4! from the vertex rule exactly cancels the 1/4! in the Lagrangian when we account for the number of ways to contract 4 external legs with the vertex.

Step 2: Apply the cross-section formula. For identical particles with equal masses:

$$\frac{d\sigma}{d\Omega} = \frac{|\mathcal{M}|^2}{64\pi^2 s} = \frac{\lambda^2}{64\pi^2 s}$$

Step 3: Integrate over solid angle. Since |M|² is independent of the scattering angle, we get $\int d\Omega = 4\pi$. However, for identical final-state particles we must include a symmetry factor of 1/2:

$$\boxed{\sigma_{\text{tot}} = \frac{1}{2} \cdot \frac{4\pi \lambda^2}{64\pi^2 s} = \frac{\lambda^2}{32\pi s}}$$

Check: In natural units, [λ] = 0 (dimensionless in 4D), and [s] = energy², so [σ] = 1/energy² = length². Correct!

💡Energy Dependence of φ⁴ Cross Section

The cross section σ ~ 1/s decreases with energy. This is characteristic of a point-like interaction (no structure to resolve). At higher energies, the de Broglie wavelength shrinks and the effective target area decreases.

Compare: for processes mediated by massive particles (like W bosons), σ ~ 1/(s - M²)², showing a resonance peak at √s = M.

Worked Example 2: Yukawa Scattering

Consider two distinguishable fermions (mass m) interacting via exchange of a massive scalar (mass μ) with coupling g:

$$\mathcal{L}_{\text{int}} = -g\phi\bar{\psi}\psi$$

Compute the differential cross section for ψ₁ + ψ₂ → ψ₁ + ψ₂ at tree level.

Solution

Step 1: Draw the Feynman diagram. At tree level, there is one t-channel diagram: the scalar propagator connects the two fermion lines:

$$-i\mathcal{M} = (-ig)^2 \frac{i}{t - \mu^2} = \frac{-ig^2}{t - \mu^2}$$

where $t = (p_1 - p_3)^2$ is the Mandelstam variable (momentum transfer squared).

Step 2: Evaluate |M|² in the non-relativistic limit. For non-relativistic fermions, the spinor factors simplify and:

$$|\mathcal{M}|^2 = \frac{g^4}{(t - \mu^2)^2}$$

Step 3: Express t in terms of scattering angle. In the CM frame:

$$t = -2|\vec{p}|^2(1 - \cos\theta) = -4|\vec{p}|^2\sin^2\frac{\theta}{2}$$

Step 4: Differential cross section:

$$\boxed{\frac{d\sigma}{d\Omega} = \frac{g^4}{64\pi^2 s} \cdot \frac{1}{\left(4|\vec{p}|^2\sin^2\frac{\theta}{2} + \mu^2\right)^2}}$$

Step 5: Check limits. For μ → 0 (massless mediator), this becomes:

$$\frac{d\sigma}{d\Omega}\bigg|_{\mu=0} \propto \frac{1}{\sin^4(\theta/2)}$$

This is exactly the Rutherford scattering formula! Yukawa theory with a massless scalar reproduces Coulomb scattering, confirming the connection between virtual particle exchange and classical forces.

2.11 The Optical Theorem

Unitarity of the S-matrix ($S^\dagger S = 1$) leads to a powerful constraint. Writing $S = 1 + iT$:

$$(1 - iT^\dagger)(1 + iT) = 1 \quad \Rightarrow \quad -i(T - T^\dagger) = T^\dagger T$$

Taking the matrix element between identical initial and final states, and inserting a complete set of intermediate states:

$$2\,\text{Im}\,\mathcal{M}(a \to a) = \sum_f \int d\Phi_f \, |\mathcal{M}(a \to f)|^2$$

Using the cross-section formula, the right side is proportional to the total cross section. The optical theorem is:

$$\boxed{\sigma_{\text{tot}} = \frac{1}{2|\vec{p}_i|\sqrt{s}}\,\text{Im}\,\mathcal{M}(p_1, p_2 \to p_1, p_2)}$$

💡Meaning of the Optical Theorem

The total cross section (probability of anything happening) is determined by the imaginary part of the forward scattering amplitude (elastic scattering at zero angle). This connects total rates to forward amplitudes and is crucial for proving the finiteness of renormalized QFT.

At tree level, amplitudes are real, so Im(M) = 0. The optical theorem only becomes nontrivial at loop level, where intermediate particles go on-shell and contribute imaginary parts.

2.12 Spin Averaging and Summing

For processes with spinning particles (fermions, gauge bosons), the cross section depends on the spin states. If the initial beams are unpolarized, we average over initial spins:

$$\overline{|\mathcal{M}|^2} = \frac{1}{(2s_1+1)(2s_2+1)} \sum_{\text{spins}} |\mathcal{M}|^2$$

For spin-1/2 particles, (2s+1) = 2. If the detector doesn't measure final spins, we sum over final spins. The combined operation is:

$$\overline{|\mathcal{M}|^2} = \frac{1}{4} \sum_{\text{all spins}} |\mathcal{M}|^2$$

for two spin-1/2 particles in the initial state. The spin sums can be evaluated using the completeness relations:

\begin{align*} \sum_{s} u^s(p)\bar{u}^s(p) &= \not{p} + m \\[8pt] \sum_{s} v^s(p)\bar{v}^s(p) &= \not{p} - m \end{align*}

This converts the spin sum into a trace over gamma matrices:

$$\sum_{\text{spins}} |\mathcal{M}|^2 = \text{Tr}\left[(\not{p}_1 + m)\Gamma (\not{p}_2 + m)\bar{\Gamma}\right]$$

where Γ is the vertex structure. The essential trace identities are:

\begin{align*} \text{Tr}[\gamma^\mu \gamma^\nu] &= 4g^{\mu\nu} \\[4pt] \text{Tr}[\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma] &= 4(g^{\mu\nu}g^{\rho\sigma} - g^{\mu\rho}g^{\nu\sigma} + g^{\mu\sigma}g^{\nu\rho}) \\[4pt] \text{Tr}[\gamma^5 \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma] &= -4i\epsilon^{\mu\nu\rho\sigma} \\[4pt] \text{Tr}[\text{odd \# of } \gamma\text{'s}] &= 0 \end{align*}

Key Takeaways (Page 3)

  • • φ⁴ scattering: σ = λ²/(32πs), decreasing with energy (point-like interaction)
  • • Yukawa scattering reduces to Rutherford formula for massless mediator
  • • The optical theorem: σ_tot = Im M(forward)/(2|p⃗|√s) from unitarity
  • • Unpolarized cross sections require averaging over initial spins, summing over final
  • • Spin sums use completeness: Σ u ū = /p + m, converted to traces over gamma matrices
  • • Trace technology is the key computational tool for QED and QCD cross sections
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