The Leptons — Question 7

Question 7

Averaging oscillation phases reveals the mixing amplitude

A broad energy spectrum averages $\sin^2\phi$ to 1/2 in the supplied two-flavor survival law. The measured average survival is $.700\pm.020$, with standard uncertainty. Assume no other losses.

1. Infer $\sin^2(2\theta)$ and its uncertainty.

2. Find the two possible mixing angles between 0 and 90 degrees and explain the degeneracy.

Question 7 - Solution

Averaging oscillation phases reveals the mixing amplitude

(a) Method and calculation

$$\langle P\rangle=1-\tfrac12\sin^2(2\theta)\Rightarrow\boxed{\sin^2(2\theta)=2(1-.700)=.600}.$$$$\boxed{\sigma_{\sin^2(2\theta)}=2(.020)=.040}.$$

(b) Interpretation and further calculation

$$2\theta=\sin^{-1}\sqrt{.600}\ \text{or}\ 180^\circ-\sin^{-1}\sqrt{.600}.$$$$\boxed{\theta=25.3842^\circ\ \text{or}\ 64.6158^\circ}.$$

Physical check

The survival probability is invariant under theta changing to 90 degrees minus theta. A phase-averaged measurement loses oscillation-length information.