A muon neutrino produced in the atmosphere with energy E = 1 GeV travels a distance L through the Earth. The probability of it being detected as a muon neutrino is approximately P(nu_mu -> nu_mu) = 1 - sin^2(2*theta_23) * sin^2(1.27 * Delta m^2_{32} * L/E), where Delta m^2 is in eV^2, L in km, and E in GeV. What physical effect causes this disappearance?
AThe neutrino decays into lighter particles during propagation
BThe neutrino flavor state nu_mu is a superposition of mass eigenstates nu_2 and nu_3, which propagate with slightly different phases (because they have different masses); after traveling distance L, the phase difference causes the flavor composition to change — the muon neutrino has partially transformed into a tau neutrino
CThe neutrino interacts with matter in the Earth and changes flavor
DThe neutrino loses energy and falls below the detection threshold
Neutrino oscillation is a quantum interference effect. The flavor state |nu_mu> = cos(theta_23)|nu_2> + sin(theta_23)|nu_3> evolves as each mass eigenstate accumulates a phase proportional to m_i^2 * L / (2E). The resulting phase difference is proportional to Delta m^2 * L/E. After propagation, the overlap with |nu_mu> is reduced and the overlap with |nu_tau> is enhanced. The Super-Kamiokande experiment observed this effect in 1998 for atmospheric neutrinos, finding a strong zenith-angle-dependent deficit of upward-going muon neutrinos (which travel ~13,000 km through the Earth) compared to downward-going ones (~15 km), consistent with oscillation with Delta m^2_{32} ~ 2.5 x 10^{-3} eV^2.
Question 2 Short Answer
Neutrino oscillation experiments measure mass-squared differences (Delta m^2_{21} ~ 7.5 x 10^{-5} eV^2 and |Delta m^2_{32}| ~ 2.5 x 10^{-3} eV^2) but not the absolute mass scale. Why can't oscillation experiments determine the individual neutrino masses?
Think about your answer, then reveal below.
Model answer: The oscillation probability depends on the phase difference between mass eigenstates, which is proportional to (m_i^2 - m_j^2) * L / (2E) = Delta m^2_{ij} * L / (2E). An overall shift of all masses by the same amount does not change any Delta m^2 and therefore does not affect oscillation. For example, m_1 = 0.1 eV, m_2 = 0.1005 eV and m_1 = 1.0 eV, m_2 = 1.0005 eV give the same Delta m^2 = 10^{-4} eV^2. The absolute mass scale must be determined by other methods: beta decay endpoint measurements (KATRIN, current limit m_beta < 0.45 eV), neutrinoless double beta decay (sensitive to the Majorana mass), or cosmological constraints from the CMB and large-scale structure (sum of masses < ~0.12 eV from Planck).
The unknown absolute mass scale also means we don't know the 'mass ordering': whether m_3 is the heaviest (normal ordering) or lightest (inverted ordering). Determining the ordering is a primary goal of current experiments (JUNO, DUNE, Hyper-K) and has implications for neutrinoless double beta decay and cosmology.
Question 3 True / False
The solar neutrino problem -- a deficit of electron neutrinos from the Sun compared to theoretical predictions -- was resolved by the SNO experiment in 2001. SNO detected all three neutrino flavors and found that the total neutrino flux agreed with the solar model prediction.
TTrue
FFalse
Answer: True
Previous experiments (Homestake, Kamiokande, SAGE, GALLEX) detected only electron neutrinos and found about 1/3 to 1/2 of the expected flux. SNO used heavy water (D_2O), which enabled three detection channels: charged current (CC, sensitive only to nu_e), neutral current (NC, sensitive to all flavors equally), and elastic scattering (ES, sensitive to all flavors but enhanced for nu_e). The CC rate confirmed the deficit, but the NC rate showed the total flux agreed with the solar model. The missing electron neutrinos had oscillated into muon and tau neutrinos. This result, combined with the matter-enhanced oscillation (MSW effect) in the Sun, determined the solar oscillation parameters: Delta m^2_{21} ~ 7.5 x 10^{-5} eV^2 and theta_12 ~ 34 degrees.