Branching Ratios and Partial Widths

Branching Ratios and Partial Widths

An unstable particle can decay via multiple channels. Each channel $i$ has a partial width $\Gamma_i$ — the rate of decay into that mode. The total decay width is:

$\Gamma_{\text{total}} = \sum_i \Gamma_i$

The mean lifetime is $\tau = \hbar / \Gamma_{\text{total}}$. Each decay mode's probability is its branching ratio:

$\text{BR}_i = \frac{\Gamma_i}{\Gamma_{\text{total}}}$

The Z Boson

The $Z$ boson ($M_Z \approx 91.2$ GeV, $\Gamma_Z \approx 2.4952$ GeV) decays to:

Counting Neutrino Generations

The invisible width $\Gamma_{\text{inv}} = \Gamma_Z - \Gamma_{\text{visible}}$ is carried by neutrinos. Since we know $\Gamma(Z \to \nu_e \bar{\nu}_e) \approx 167.17$ MeV from the Standard Model, the number of light neutrino generations is:

$N_\nu = \frac{\Gamma_{\text{inv}}}{\Gamma(Z \to \nu\bar{\nu})} \approx 3$

This LEP measurement proved there are exactly three neutrino families.

Your Task

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

• branching_ratio(partial_width_GeV, total_width_GeV)
• partial_width(total_width_GeV, branching_ratio)
• number_of_neutrino_generations(Gamma_inv_GeV, Gamma_Z_to_nunu_GeV) — returns an integer