Quantum Field Theory Course

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

Part IV, Chapter 2

Cross Sections & Decay Rates

From amplitudes to measurable quantities: connecting QFT to experiments

πŸ”—Course Connections

πŸ“šPrerequisites

QFT Part IV
Interaction Picture
S-matrix and scattering amplitudes
β†’
QFT Part III
Perturbation Theory
Computing amplitudes from Feynman diagrams
β†’
▢️

Video Lecture

Lecture 23: Cross Section and Decay Rate - MIT 8.323

Computing observable quantities from S-matrix elements (MIT QFT Course)

πŸ’‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

2.1 From Theory to Experiment

We've learned how to compute S-matrix elements Sfi = ⟨f|S|i⟩. But experiments don't measure amplitudes directly - they measure:

  • Cross sections Οƒ: Probability for scattering (measured in barns, 1b = 10-28mΒ²)
  • Decay rates Ξ“: Probability per unit time for a particle to decay (related to lifetime Ο„ = 1/Ξ“)

πŸ’‘What is a Cross Section?

Imagine shooting particles at a target. The cross section Οƒ is the effective target area per scattering center.

If you fire Nbeam particles at a target with n scattering centers, the number of scattering events is:

Nscatter = Nbeam Γ— n Γ— Οƒ

Larger Οƒ β†’ more likely to scatter!

2.2 Differential Cross Section

For 2 β†’ 2 scattering (a + b β†’ c + d), the differential cross section in the center-of-mass frame is:

$$\boxed{\frac{d\sigma}{d\Omega} = \frac{1}{64\pi^2 s}\frac{|\vec{p}_f|}{|\vec{p}_i|}|\mathcal{M}|^2}$$

where:

  • s = (pa + pb)Β² is the Mandelstam variable (total energy squared)
  • |pβƒ—i| is the initial momentum magnitude in CM frame
  • |pβƒ—f| is the final momentum magnitude in CM frame
  • M is the invariant amplitude (Lorentz scalar)
  • dΞ© is the solid angle element

For equal mass particles with |p⃗i| = |p⃗f| = |p⃗|, this simplifies to:

$$\frac{d\sigma}{d\Omega} = \frac{1}{64\pi^2 s}|\mathcal{M}|^2$$

2.3 Total Cross Section

Integrate over all angles:

$$\boxed{\sigma_{\text{tot}} = \int d\Omega \frac{d\sigma}{d\Omega}}$$

For spherically symmetric scattering (|M|Β² independent of angle):

$$\sigma_{\text{tot}} = \frac{4\pi}{64\pi^2 s}|\mathcal{M}|^2 = \frac{1}{16\pi s}|\mathcal{M}|^2$$

Example: φ⁴ Scattering

For φ⁴ theory with M = -Ξ» (tree level), we get:

$$\sigma_{\text{tot}} = \frac{\lambda^2}{16\pi s}$$

Cross section decreases as energy increases! This is typical for point-like interactions.

2.4 Phase Space Integrals

For n final particles, we must integrate over phase space:

$$d\Phi_n = \prod_{i=1}^n \frac{d^3p_i}{(2\pi)^3 2E_i}(2\pi)^4\delta^4\left(p_{\text{in}} - \sum_{j=1}^n p_j\right)$$

This is Lorentz-invariant phase space (LIPS). The factors ensure:

  • Lorentz invariance
  • Energy-momentum conservation (delta function)
  • On-shell particles: Ei = √(pβƒ—iΒ² + miΒ²)

2.5 Decay Rates

For a particle A decaying to n particles, the decay rate (or decay width) is:

$$\boxed{\Gamma = \frac{1}{2m_A}\int d\Phi_n |\mathcal{M}|^2}$$

The lifetime is Ο„ = 1/Ξ“. Particles with larger Ξ“ decay faster!

Two-Body Decay

For A β†’ B + C in the rest frame of A:

$$\Gamma = \frac{|\vec{p}|}{8\pi m_A^2}|\mathcal{M}|^2$$

where |p⃗| is the momentum of B (or C) in the rest frame:

$$|\vec{p}| = \frac{1}{2m_A}\sqrt{[m_A^2 - (m_B + m_C)^2][m_A^2 - (m_B - m_C)^2]}$$

2.6 Mandelstam Variables

For 2 β†’ 2 scattering, define three Lorentz-invariant variables:

\begin{align*} s &= (p_1 + p_2)^2 = (p_3 + p_4)^2 \quad \text{(total energy squared)} \\ t &= (p_1 - p_3)^2 = (p_2 - p_4)^2 \quad \text{(momentum transfer squared)} \\ u &= (p_1 - p_4)^2 = (p_2 - p_3)^2 \quad \text{(crossed channel)} \end{align*}

These satisfy:

$$s + t + u = \sum_{i=1}^4 m_i^2$$

Only two are independent! Different physical regions correspond to different scattering channels.

🎯 Key Takeaways

  • Differential cross section: dΟƒ/dΞ© = (1/64π²s)|M|Β²(|pβƒ—f|/|pβƒ—i|)
  • Total cross section: Integrate dΟƒ/dΞ© over all angles
  • Phase space: Lorentz-invariant measure for final states
  • Decay rate: Ξ“ = (1/2mA)∫dΞ¦n|M|Β²
  • Lifetime: Ο„ = 1/Ξ“ (shorter lifetime = larger decay rate)
  • Mandelstam variables: s, t, u describe kinematics
  • Next: QED - computing real scattering processes!