A Pause for Reflection — Question 8

Question 8

A null rare-decay search constrains a branching fraction rather than proving zero

A search observes zero candidates with negligible background from $10^8$ parent decays and efficiency .25. For a Poisson signal, the probability of zero is $e^{-\mu}$.

1. Derive a one-sided 90 percent upper bound on the branching fraction.

2. If an additional invisible decay mode doubles the total width while the searched partial width stays fixed, find how the branching fraction and expected yield change.

Question 8 - Solution

A null rare-decay search constrains a branching fraction rather than proving zero

(a) Method and calculation

$$e^{-\mu_{90}}=.10\Rightarrow\mu_{90}=\ln10.$$$$\boxed{B_{90}=\ln10/(.25\times10^8)=9.21034\times10^{-8}}.$$

(b) Interpretation and further calculation

$$B=\Gamma_s/\Gamma_{\mathrm{total}},\quad\Gamma_{\mathrm{total}}\to2\Gamma_{\mathrm{total}}.$$$$\boxed{B\to B/2,\quad\mu=N\epsilon B\to\mu/2}.$$

Physical check

The bound depends on the statistical model and selection efficiency. No observed events does not establish an exactly forbidden channel.