Particles, Particles, Particles — Question 7

Question 7

Event yields determine a cross section only after efficiency correction

A collision sample has integrated luminosity $\mathcal L=50.0\,\mathrm{pb^{-1}}$. A selected channel has 500 candidates, estimated background 100, and signal efficiency .400. Treat luminosity, efficiency and background as exact for this exercise.

1. Infer the channel cross section.

2. Using Poisson uncertainty on the candidate count, estimate its standard uncertainty and the luminosity needed to halve that relative error at fixed rates.

Question 7 - Solution

Event yields determine a cross section only after efficiency correction

(a) Method and calculation

$$N_s=N_{\mathrm{obs}}-N_b=400,\quad N_s=\epsilon\mathcal L\sigma.$$$$\boxed{\sigma=400/[.400(50.0)]=20.0\,\mathrm{pb}}.$$

(b) Interpretation and further calculation

$$\boxed{\sigma_\sigma=\sqrt{500}/[.400(50.0)]=1.11803\,\mathrm{pb}}.$$$$\sigma_\sigma/\sigma\propto\mathcal L^{-1/2}\Rightarrow\boxed{\mathcal L\prime=4\mathcal L=200\,\mathrm{pb^{-1}}}.$$

Physical check

Background uncertainty would increase the error. An event count is not a cross section until acceptance, efficiency and exposure are accounted for.