Lab Frame Decay Length

Lab Frame Decay Length

A particle produced in a collider is not at rest — it flies through the detector at high speed. This means its proper lifetime $\tau_0$ is dilated in the lab frame by the Lorentz factor $\gamma$.

Kinematics in Natural Units

In high-energy physics we work in natural units where $c = 1$. For a particle with energy $E$ and mass $m$:

$\gamma = \frac{E}{m}$

$\beta = \frac{p}{E} = \sqrt{1 - \frac{m^2}{E^2}}$

The lab-frame lifetime is stretched by time dilation:

$\tau_{\text{lab}} = \gamma \, \tau_0 = \frac{E}{m} \, \tau_0$

Mean Decay Length

The average distance a particle travels before decaying is:

$L = \gamma \beta c \tau_0 = \frac{p}{m} \cdot c \cdot \tau_0$

In SI-friendly mixed units, with energy/mass in GeV and $c = 2.998 \times 10^8$ m/s:

$L ;[\text{m}] = \frac{E_{\text{GeV}}}{m_{\text{GeV}}} \cdot \sqrt{1 - \left(\frac{m}{E}\right)^2} \cdot c \cdot \tau_0$

Example: Muons

A muon has mass $m_\mu = 0.10566$ GeV and proper lifetime $\tau_\mu = 2.197 \times 10^{-6}$ s. At $E = 1$ GeV, $\gamma \approx 9.46$, and the mean decay length is over 6 km — explaining why cosmic-ray muons survive from the upper atmosphere to sea level.

Your Task

Implement three functions. All constants must be defined inside each function body.

• lorentz_factor(E_GeV, m_GeV) — returns $E/m$
• beta(E_GeV, m_GeV) — returns $\sqrt{1-(m/E)^2}$
• mean_decay_length(E_GeV, m_GeV, tau_s) — returns $L$ in metres