The Leptons — Question 6

Question 6

A neutrino oscillation minimum measures mass-squared splitting

Use a two-flavor survival model $P_{\mu\mu}=1-\sin^2(2\theta)\sin^2(1.27\Delta m^2L/E)$, with $\Delta m^2$ in eV squared, L in km and E in GeV. Set $\sin^2(2\theta)=1$, $\Delta m^2=2.50\times10^{-3}\,\mathrm{eV^2}$ and E=1.00 GeV.

1. Find the first positive baseline giving minimum survival.

2. Find survival at half that baseline and explain why the experiment measures a mass-squared difference rather than an absolute mass.

Question 6 - Solution

A neutrino oscillation minimum measures mass-squared splitting

(a) Method and calculation

$$1.27\Delta m^2L_{\min}/E=\pi/2.$$$$\boxed{L_{\min}=\frac{\pi(1.00)}{2(1.27)(.00250)}=494.739\,\mathrm{km}}.$$

(b) Interpretation and further calculation

$$\boxed{P_{\mu\mu}(L_{\min}/2)=1-\sin^2(\pi/4)=.500}.$$$$\boxed{m_1^2\to m_1^2+C,\ m_2^2\to m_2^2+C}\quad\text{leaves }\Delta m^2\text{ unchanged}.$$

Physical check

This vacuum two-flavor approximation omits matter effects and the third flavor. Oscillations require distinct propagation phases.