Cross Section Units and Luminosity

Cross Section Units and Luminosity

In particle physics experiments the interaction probability between two particles is quantified by the cross section $\sigma$ — a measure of the effective area for a given process.

The Barn and its Submultiples

The natural unit for cross sections is the barn:

$1 \text{ b} = 10^{-28} \text{ m}^2 = 10^{-24} \text{ cm}^2$

Modern collider physics works with much smaller cross sections:

Conversion from Natural Units

In natural units (where $\hbar = c = 1$), cross sections are computed in GeV$^{-2}$. The conversion to physical units uses:

$1 \text{ GeV}^{-2} = 0.3894 \text{ mb} = 3.894 \times 10^8 \text{ pb}$

Luminosity and Event Rates

The instantaneous luminosity $\mathcal{L}$ [cm$^{-2}$s$^{-1}$] relates the event rate $R$ to the cross section:

$R = \mathcal{L} \cdot \sigma$

The integrated luminosity $\mathcal{L}_{\text{int}}$ [fb$^{-1}$] gives the total number of events:

$N_{\text{events}} = \mathcal{L}_{\text{int}} \cdot \sigma$

At the LHC, Run 2 (2015–2018) delivered about 140 fb$^{-1}$ of integrated luminosity at 13 TeV. Key benchmark cross sections:

• $pp \to H$ (Higgs production via gluon fusion): $\sigma \approx 55$ pb at 13 TeV
• $pp \to Z$: $\sigma \approx 60$ nb $= 60{,}000$ pb

Your Task

Implement:

• GeV2_to_pb(sigma_GeV2) — convert cross section from GeV$^{-2}$ to pb
• events_expected(sigma_pb, luminosity_fb_inv) — number of events (1 fb$^{-1}$ = 1000 pb$^{-1}$)
• luminosity_from_rate(event_rate_per_s, sigma_cm2) — instantaneous luminosity in cm$^{-2}$s$^{-1}$

All constants must be defined inside each function body.