Relativistic Kinematics & Feynman Diagrams
Relativistic Kinematics & Feynman Diagrams
Particle collisions are analyzed using 4-momenta in special relativity. Feynman diagrams provide an intuitive pictorial representation of quantum field theory perturbation theory, encoding propagators and vertices systematically.
Definition
4-momentum: \(p^\mu = (E/c, \mathbf{p})\) with \(p^2 = m^2c^2\). Invariant mass \(s = (p_1+p_2)^2\) determines the available energy in a collision. At the LHC: \(\sqrt{s} = 13\) TeV.
Key Result
Each Feynman diagram corresponds to a term in the perturbation series. Internal lines are propagators \(1/(p^2-m^2)\); each vertex contributes a coupling constant \(g\). Amplitude squared gives cross-section.
Example 1
At LEP (\(e^+e^-\) at 91 GeV), production cross-section peaks sharply at the Z boson mass. The 2 MeV width corresponds to a lifetime \(\tau \sim \hbar/\Gamma \approx 3\times10^{-25}\) s.
Example 2
Compton scattering \(e^-\gamma\to e^-\gamma\): two Feynman diagrams contribute at leading order (t-channel and u-channel electron propagators), and their sum gives the Klein-Nishina formula.
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Practice
- Draw the Feynman diagram for muon decay \(\mu^-\to e^-\bar{\nu}_e\nu_\mu\).
- Compute the invariant mass of two photons with energies 500 MeV traveling at 90° to each other.
- What is the optical theorem, and what does it relate?
- Explain crossing symmetry in Feynman diagram calculations.
Show Answer Key
1. Muon decay $\mu^- \to e^-\bar{\nu}_e\nu_\mu$: a muon emits a virtual $W^-$ boson, which decays to $e^-\bar{\nu}_e$. The $\nu_\mu$ continues forward. Vertex 1: $\mu^- \to \nu_\mu + W^-$. Vertex 2: $W^- \to e^- + \bar{\nu}_e$. The $W$ is internal (virtual, off-shell).
2. Two photons: $p_1 = (E_1/c, \mathbf{p}_1)$, $p_2 = (E_2/c, \mathbf{p}_2)$ with $E_1=E_2=500$ MeV, angle $\theta=90°$. Invariant mass: $M^2c^4 = (p_1+p_2)^2c^2 = 2E_1E_2(1-\cos\theta) = 2(500)(500)(1-0) = 500000$ MeV². $Mc^2 = \sqrt{500000} \approx 707$ MeV.
3. The optical theorem relates the imaginary part of the forward scattering amplitude to the total cross section: $\text{Im}\,f(0) = \frac{k}{4\pi}\sigma_{\text{tot}}$. It follows from unitarity of the S-matrix ($S^\dagger S = I$) and conservation of probability. It connects elastic and inelastic scattering: the total cross section includes all possible final states.
4. Crossing symmetry: a Feynman diagram for $A+B\to C+D$ can be analytically continued to describe $A+\bar{C}\to\bar{B}+D$ (by crossing particle $B$ to the initial state as $\bar{B}$ and $C$ to the final state). The scattering amplitudes are related by replacing $s\leftrightarrow t$ (Mandelstam variables). This reduces the number of independent calculations needed.