Neutrino Physics

8.1 Neutrino Oscillations

Neutrinos are produced and detected in flavour eigenstates $(\nu_e, \nu_\mu, \nu_\tau)$ But propagate As mass eigenstates $(\nu_1, \nu_2, \nu_3)$ related by the PMNS mixing matrix $U$ :

$|\nu_\alpha\rangle = \sum_i U_{\alpha i}^* |\nu_i\rangle$

As a neutrino of flavour $\alpha$ propagates, the mass eigenstates acquire different phases: $\exp(-im_i^2 L/(2E))$ Leading to oscillations.

Two-flavour oscillation probability:

$P(\nu_\alpha \to \nu_\beta) = \sin^2(2\theta)\sin^2\left(\frac{\Delta m^2 L}{4E}\right)$

Where $\Delta m^2 = m_2^2 - m_1^2$ , $\theta$ is the mixing angle, $L$ is the distance, and $E$ is the Energy.

Evidence: The Solar Neutrino Problem (deficit of $\nu_e$ from the Sun, resolved by $\nu_e \to \nu_\mu, \nu_\tau$ oscillations) and atmospheric neutrino oscillations (Super-Kamiokande, 1998).

8.2 Neutrino Masses

Neutrino oscillations imply that neutrinos have mass, but the masses are extremely small: $\sum m_\nu \lt 0.12$ eV (Planck 2018).

In the Standard Model, neutrinos are massless. Their masses require physics beyond the Standard Model, most commonly via the seesaw mechanism:

$m_\nu \sim \frac{m_D^2}{M}$

Where $m_D$ is a Dirac mass and $M \gg m_D$ is the mass of a heavy right-handed neutrino.

Example 8.1: Atmospheric neutrino oscillation calculation

Atmospheric neutrinos are produced when cosmic rays strike the upper atmosphere, creating Pions that decay: $\pi^+ \to \mu^+ + \nu_\mu$ Followed by $\mu^+ \to e^+ + \bar{\nu}_\mu + \nu_e$ .

Super-Kamiokande (1998) observed that upward-going muon neutrinos (travelling through the Earth, $L \sim 10^4$ km) were significantly depleted relative to downward-going ones ( $L \sim 10$ km), while electron neutrinos showed no such deficit.

Using the two-flavour formula with the atmospheric parameters $\Delta m^2_{32} \approx 2.5 \times 10^{-3}$ eV $^2$ and $\sin^2(2\theta_{23}) \approx 1$ (maximal mixing):

For upward-going $\nu_\mu$ with $E = 1$ GeV and $L = 10\,000$ km:

$\frac{\Delta m^2 L}{4E} = \frac{2.5 \times 10^{-3}\;\mathrm{eV}^2 \times 10^4\;\mathrm{km}{4 \times 1\;\mathrm{GeV}}}$

Converting to natural units ( $\hbar c = 1.973 \times 10^{-7}$ eV $\cdot$ M): $L = 10^7$ m, so $L/E = 10^7 / 10^9 = 10^{-2}$ eV $^{-1}$ .

$\frac{\Delta m^2 L}{4E} = \frac{2.5 \times 10^{-3} \times 10^{-2}}{4} = 6.25 \times 10^{-6}\;\mathrm{eV}^2\cdot\mathrm{eV}^{-1}$

Wait --- we need to be more careful with units. Using the practical formula:

$\frac{\Delta m^2 [\mathrm{eV}^2] \cdot L [\mathrm{km}]}{4E [\mathrm{GeV}]} = \frac{2.5 \times 10^{-3} \times 10^4}{4 \times 1} = \frac{25}{4} = 6.25\;\mathrm{rad}$

$P(\nu_\mu \to \nu_\mu) = 1 - \sin^2(2\theta_{23})\sin^2(6.25) = 1 - 1 \times \sin^2(6.25) \approx 1 - 0.018 \approx 0.98$

Hmm, this gives almost no oscillation. Let me reconsider. Actually:

$P(\nu_\mu \to \nu_\tau) = \sin^2(2\theta)\sin^2\left(\frac{\Delta m^2 L}{4E}\right) = \sin^2(6.25) \approx 0.018$

This seems small. But at $E = 0.5$ GeV:

$\frac{\Delta m^2 L}{4E} = \frac{25}{2} = 12.5\;\mathrm{rad}$

$\sin^2(12.5) \approx \sin^2(0.35) \approx 0.12$

And at the first oscillation maximum, $L/E = 2\pi/(\Delta m^2) = 2\pi/(2.5 \times 10^{-3}) \approx 2513$ km/GeV. For $E = 1$ GeV, $L_{\mathrm{osc} \approx 2513}$ km, which is comparable to the Earth”s diameter ( $\sim 12\,700$ km). The observed deficit is an average over many oscillations and energies, Giving roughly $\langle P\rangle \approx 1/2$ for maximal mixing, consistent with the Super-Kamiokande observation of approximately half the expected upward-going $\nu_\mu$ flux.

8.3 Key Relationships

Parameter	Value (best fit)	Experiment	Role
$\Delta m^2_{21}$	$7.5 \times 10^{-5}$ eV $^2$	Solar (SNO, Borexino)	Drives solar $\nu_e \to \nu_{\mu,\tau}$
$\Delta m^2_{32}$	$2.5 \times 10^{-3}$ eV $^2$	Atmospheric (Super-K)	Drives $\nu_\mu \to \nu_\tau$ oscillations
$\sin^2(2\theta_{12})$	0.86	Solar (SNO)	Solar mixing angle
$\sin^2(2\theta_{23})$	1.0 (maximal)	Atmospheric (Super-K)	Atmospheric mixing angle
$\sin^2(2\theta_{13})$	0.092	Reactor (Daya Bay, RENO, Double Chooz)	Non-zero, enables CP violation

8.4 Common Pitfalls

Confusing flavour and mass eigenstates. Neutrinos are produced and detected as flavour eigenstates but propagate as mass eigenstates. Fix: The PMNS matrix $U$ relates the two bases; oscillations arise from phase differences between mass components.
Assuming all oscillations average to zero. While fast oscillations average over energy and baseline, the survival probability for solar $\nu_e$ is $\approx 0.55$ , not $0.5$ , due to the MSW matter effect in the Sun. Fix: Matter effects modify the effective mixing angle at high densities.
Forgetting the practical unit conversion. When using the oscillation formula $\Delta m^2 L / (4E)$ with $L$ in km and $E$ in GeV, the result is in radians directly: $\Delta m^2 [\mathrm{eV}^2] L [\mathrm{km}] / (4E [\mathrm{GeV}])$ . Fix: Use $L/E$ in km/GeV for quick estimates.
Thinking neutrinos are massless in the Standard Model. While the SM predicts massless neutrinos, oscillations prove they have mass. Fix: The seesaw mechanism extends the SM with heavy right-handed neutrinos.

8.5 Applications

Solar neutrino spectroscopy: Precise measurement of solar neutrino fluxes (pp, $^7$ Be, $^8$ B) tests solar models and constrains the MSW effect transition between vacuum and matter-dominated oscillations.
Reactor neutrino monitoring: Antineutrino detectors at nuclear reactors (Daya Bay, Double Chooz) measure $\theta_{13}$ and can monitor reactor power and fuel composition for non-proliferation.
Supernova neutrinos: Core-collapse supernovae release $\sim 99\%$ of their gravitational binding energy as neutrinos. Detecting these (SN 1987A, $\sim 20$ events) tests models of stellar death and neutron star formation.
Neutrino telescopes: IceCube and KM3NeT detect high-energy astrophysical neutrinos from blazars, gamma-ray bursts, and possibly dark matter annihilation.
Cosmology: The sum of neutrino masses $\sum m_\nu < 0.12$ eV (Planck + BAO) affects structure formation; future surveys (Euclid, DESI) will tighten constraints and potentially determine the mass hierarchy.

8.6 Summary Table

Neutrino source	Typical energy	Baseline $L$	Oscillation probed	Key experiment
Solar	0.1-10 MeV	$1.5 \times 10^8$ km	$\Delta m^2_{21}$	SNO, Borexino, Super-K
Atmospheric	0.1-100 GeV	10-10 $^4$ km	$\Delta m^2_{32}$	Super-Kamiokande
Reactor	1-10 MeV	0.1-100 km	$\theta_{13}$ , $\Delta m^2_{21}$	Daya Bay, RENO, KamLAND
Accelerator	0.1-10 GeV	100-1000 km	$\delta_{\rm CP}$	T2K, NO $\nu$ A, DUNE

8.7 Open Questions

Mass hierarchy: Is the ordering $m_1 < m_2 < m_3$ (normal) or $m_3 < m_1 < m_2$ (inverted)? Future experiments like JUNO and DUNE aim to resolve this via matter effects in oscillation probabilities.
CP violation in the lepton sector: The phase $\delta_{\rm CP}$ in the PMNS matrix determines whether neutrinos and antineutrinos oscillate differently. A non-zero $\delta_{\rm CP}$ could help explain the matter-antimatter asymmetry via leptogenesis.
Dirac vs. Majorana nature: Are neutrinos their own antiparticles? Neutrinoless double-beta decay ( $0\nu\beta\beta$ ) experiments search for this; a positive signal would prove the Majorana nature and fix the absolute mass scale.
Absolute mass scale: Oscillations only measure mass-squared differences. KATRIN measures the electron neutrino mass via tritium beta decay, currently constraining $m_{\nu_e} < 0.8$ eV.

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