Breit-Wigner Resonance

Breit-Wigner Resonance

When the centre-of-mass energy $\sqrt{s}$ passes through a particle's rest mass $M$, the cross section peaks sharply. This resonance is described by the non-relativistic Breit-Wigner formula:

$\sigma(E) = \frac{\sigma_{\text{peak}}}{1 + \left(\dfrac{2(E - M)}{\Gamma}\right)^2}$

where $\Gamma$ is the total decay width (inverse lifetime). The shape is a Lorentzian centred at $E = M$ with full width at half maximum equal to $\Gamma$.

Peak Cross Section for Z Production

For $e^+e^- \to Z \to$ hadrons, the peak cross section is:

$\sigma_{\text{peak}} = \frac{12\pi}{M_Z^2} \cdot \frac{\Gamma_{ee} \Gamma_{\text{had}}}{\Gamma_Z^2}$

In natural units this has dimensions of GeV$^{-2}$. Using the conversion $1 \text{ GeV}^{-2} = 3.894 \times 10^5 \text{ nb}$:

$\sigma_{\text{peak}} \approx 41.4 \text{ nb}$

This enormous cross section (compared to typical QED backgrounds $\sim 1$ nb) made the $Z$ lineshape measurement at LEP exquisitely precise.

Parameters used (PDG values)

Measuring $\Gamma$ from the Lineshape

The half-maximum points of the resonance occur at $E = M \pm \Gamma/2$. The full width $\Gamma = E_2 - E_1$ can be read directly from the energy scan.

Your Task

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

• breit_wigner(E_GeV, M_GeV, Gamma_GeV) — normalised Breit-Wigner (peak = 1)
• z_peak_cross_section_nb() — peak cross section for $e^+e^- \to Z \to$ hadrons in nb
• width_from_peak_shape(E1_GeV, E2_GeV, M_GeV) — extract $\Gamma$ from half-max energies