Particle Lifetimes and the Standard Model

Particle Lifetimes and the Standard Model

Particle lifetimes in the Standard Model span more than 40 orders of magnitude — from the proton's stability over cosmic timescales to the fleeting existence of the W boson. Understanding what sets these lifetimes reveals deep connections between symmetries, coupling strengths, and phase space.

The Width–Lifetime Relation

Every unstable particle has a decay width $\Gamma$ (in energy units) related to its lifetime $\tau$ by:

$\Gamma \cdot \tau = \hbar = 6.582119569 \times 10^{-25} \text{ GeV·s}$

A broad resonance ($\Gamma$ large) decays quickly ($\tau$ small). The natural linewidth of any state is just $\Gamma = \hbar/\tau$.

Lifetime Hierarchy

Relativistic Decay Length

A particle produced with momentum $p$ and mass $m$ travels (on average) a distance before decaying:

$L = \frac{p}{m}\, c\, \tau = \beta\gamma c\tau$

This is crucial for detector design. A charged pion with $p = 100$ GeV travels kilometers before decaying — it easily reaches the muon detectors. A B meson with $p \sim 50$ GeV travels only ~mm, requiring precision vertex detectors (silicon pixels) close to the interaction point.

Your Task

Implement:

• natural_width(tau_s) — decay width $\Gamma = \hbar/\tau$ in GeV
• decay_length_at_LHC(mass_GeV, tau_s, momentum_GeV) — relativistic decay length in metres
• compare_widths(Gamma1_GeV, Gamma2_GeV) — ratio of two decay widths

All constants must be defined inside each function body.