Produced in pp, pep, hep, $^7Be$, $^8B$ processes. E < 20 MeV. Pure $\nu_e$

\(\nu_\tau\) are also produced in accelerators at much higher energy through charm decay: \(D_s \rightarrow \tau \nu_\tau\)

\(\nu_\mu\) dominate instead of \(\nu_e\) because of chiral structure ("wrong helicity" of the lepton)

Process: \(\nu_l + n \rightarrow p + l^-\)

To solve the solar neutrino problem: deficit of \(\nu_e\) in Homestake mine experiment (Ray Davis, inverse beta decay: \(\nu_e + Cl(n) \rightarrow Ar(p) + e^-\)

Primary channel: \(\nu_e\) elastic scattering (oxygen has large binding energy for inverse beta decay) \(\nu_e e^- \rightarrow \nu_e e^-\), (CC + NC).

50 kT water-cherenkov detector, surrounded by PMTs.

Aims to measure both \(\nu_e\) and the total solar neutrino flux.

10T deuteron (2.2 MeV binding energy; enough for inelastic scattering) shielded by pure water.

Conclusion: neutrino oscillates -> neutrino has mass -> weak (flavor) and mass eigenstates are different!

Unitary, 3 mixing angles, 1 phase. After decomposing to rotation matrix, have \(\theta_{12}, \theta_{23}, \theta_{13}\), and \(\delta_{CP}\)

\(P(\nu_e\rightarrow\nu_\mu) = \sin^2(2\theta)\sin^2\Big(\frac{\Delta m^2_{12}L}{4E_\nu}\Big)\) , = \(\sin^2(2\theta)\sin^2\Big(1.27\frac{\Delta m_{12}^2[eV^2]L[km]}{E_\nu[GeV]}\Big)\) where \(\Delta m_{12} = m_{\nu_1}^2 - m_{\nu_2}^2\)

1.27 comes from \(\hbar c = 0.197 [GeV] [fm]\)

To express a W vertex from flavor to mass states: \(-\frac{ig_w}{\sqrt{2}}(\bar{e}, \bar{\mu}, \bar{\tau})_i \gamma^\mu \frac{1}{2}(1-\gamma^5)U_{ij} (\nu_1, \nu_2, \nu_3)_j \)

If neutrino enters as adjoint spinor, use \(U_{ij}^*\)

Therefore, if a neutrino is produced, then e.g. \(\nu_e = \sum\limits_i U_{ei}^* \nu_i\)

One param \(\theta\); Suppose start with \(\nu_e\), write in terms of \(\nu_1, \nu_2\), apply time evolution, and convert back.

\(P(\nu_e\rightarrow \nu_e) \rightarrow 1 - 4|U_{e1}|^2|U_{e2}|^2\sin^2\Delta _{21} \) - \(4|U_{e1}|^2|U_{e3}|^2\sin^2\Delta_{31} \) - \(4|U_{e2}|^2|U_{e3}|^2\sin^2\Delta_{32}\)

\(\Delta_{ji} = \frac{\Delta m^2_{ji}L}{4E_\nu} = \frac{(m_j^2-m_i^2)L}{4E_\nu}\)

Only two of all 3 \(\Delta_{ji}\) are independent, e.g. \(\Delta_{31} = \Delta_{32} + \Delta_{21}\)

Note: MSW effect changes the oscillation when neutrino propagate in a medium with changing density. Net effect: enhanced \(\nu_e\) flux.

Measuring oscillation only gives the Abs(mass-squared difference), not the absolute value of mass or the hierarchy.

Measured: \(\Delta m_{21} = m_2^2 - m_1^2 = 8 \times 10^{-5} eV^2\), \(|\Delta m_{32}| = |m_3^2 - m_2^2| = 2 \times 10^{-3} eV^2 \)

Normal hierarchy: \(m_1 < m_2 < m_3\);
Inverted hierarchy: \(m_3 < m_1 < m_2\)

\(E_{\nu_e} > 0\) (just kinematic; also depending on nuclear binding energy)

\(E_{\nu_\mu} > 11 GeV\) (hard)

Process: \(\nu_l + e^- \rightarrow l^- + \nu_e\)

\(E_{\nu_e} > 0\) (just kinematic; also depending on nuclear binding energy)

Due to large threshold, usually only \(\nu_e\) participates in elastic CC.

Produce \(\bar{\nu}_e\) through \(\beta\) decay with known flux

Could only detect disappearance (kinematic threshold too high)!

\(e^++e^-\rightarrow \gamma \gamma\) (prompt photons)

neutron is captured later (e.g. by Gd), emitting another \(\gamma\) (delayed photon)

\(e^-\) are scattered by \(\gamma\) through Compton scattering, ionizing the liquid scintillator and produce scintillation light.

Sensitive to different params depending on the baseline

\(P(\bar{\nu_e} \rightarrow \bar{\nu_e}) = 1 - \cos^4(\theta_{13})\sin^2(2\theta_{12})\sin^2\Big(\frac{\Delta m^2_{21} L}{4E_\nu}\Big)\) - \(\sin^2(2\theta_{13})\sin^2\Big(\frac{\Delta m^2_{32} L}{4E_\nu}\Big)\)

Detect through inverse beta decay: \(\bar{\nu}_e + p \rightarrow n + e^+\)

Baseline: O(1km) sensitive to \(\theta_{13}, \Delta m^2_{32}\); O(100km) sensitive to \(\theta_{12}, \Delta m^2_{21}\)

Goal: measure \(\theta_{13}\) with six reactors + near/far detectors; distance ~ O(1km). \(\Delta m^2_{32}\) measured by other experiments.

Calculate event ratio in far/near detector (expect deficit due to oscillation). Then fit for \(\theta\).

Competed with RENO (Korea) and Double Chooz (France)

Result: \(\sin^2(2\theta_{13}) \sim 0.1\)

Same location as Super-K; detect \(\bar{\nu_e}\) from multiple reactors; flux-weighted distance ~O(180km). 1-kT liquid scintillator.

Could not resolve oscillation due to \(\Delta m^2_{32}\).

Result: \(\Delta m^2_{21} = 7.6 \times 10^{-5} eV^2\), \(\sin^2(2\theta_{12}) = 0.87\)

Measures oscillation as a function of \(L/E_\nu\)

Fermi lab -> \(\nu_\mu\) beam (peak at 3 GeV) -> Soudan mine; 735 km apart. ND: 1 km from the source, measuring neutrino flux and energy profile.

\(P(\nu_\mu \rightarrow \nu_\mu) \approx 1 - A \sin^2\Big(\frac{\Delta^2_{32} L}{4 E_\nu} \Big)\); sensitive to \(\theta_{23}\) and \(|\Delta m^2_{32}|\) ("atmospheric" neutrino oscillation)

With fixed baseline, measure oscillation as a function of \(E_\nu\)

Result: \(\sin^2(2\theta_{23}) > 0.9\), \(|\Delta m_{32}|^2 = 2.3 \times 10^{-3} eV^2\)

Channel: CC interaction; \(E_\nu = E_\mu + E_x\)

\(\delta_{CP}\) is not known yet

\(\theta_{12} =35^\circ, \theta_{23}=45^\circ, \theta_{13}=10^\circ\)

\(\Delta m^2_{21} = 7.6 \times 10 ^{-5} eV^2\), \(\Delta |m^2_{32}| = 2.3 \times 10^{-3} eV^2\)

NOTE: \(\Delta m^2_{21} >0\) is given by solar neutrino data; could also say it's by definition.

Study \(\nu_\mu \rightarrow \nu_e\) appearance

Search \(\nu_\mu \rightarrow \nu_\tau\)

Study \(\nu_\mu \rightarrow \nu_e\) appearance

Study \(\delta_{CP}, \theta_{13}, \Delta m^2_{13}\)