Particles, Particles, Particles — Question 10

Question 10

A rare-channel limit can be inferred even when no events are observed

A background-free search observes zero events. For Poisson mean mu, $P(0|\mu)=e^{-\mu}$. Define a one-sided 95 percent upper limit by $P(0|\mu_{95})=.0500$. Exposure is $100\,\mathrm{pb^{-1}}$ and efficiency .250.

1. Find the upper bound on expected signal events and cross section.

2. Find how that bound changes if exposure doubles with zero events again, and explain why zero observed events does not imply zero cross section.

Question 10 - Solution

A rare-channel limit can be inferred even when no events are observed

(a) Method and calculation

$$e^{-\mu_{95}}=.0500\Rightarrow\boxed{\mu_{95}=-\ln(.0500)=2.99573}.$$$$\boxed{\sigma_{95}=\mu_{95}/(\epsilon\mathcal L)=2.99573/25=.119829\,\mathrm{pb}}.$$

(b) Interpretation and further calculation

$$\boxed{\sigma_{95}\prime=.0599146\,\mathrm{pb}}\quad(\mathcal L\prime=200\,\mathrm{pb^{-1}}).$$$$\boxed{P(0|\mu)>0\text{ for every finite }\mu}\quad\text{so zero counts are not a proof of absence}.$$

Physical check

The confidence construction and zero-background assumption are specified. Uncertain efficiency or nonzero background requires a modified statistical treatment.