Modern Physics · Advanced Topics

Neutrino Physics & Flavor Oscillations

Neutrinos are the most abundant matter particles in the universe yet barely interact. The discovery that they oscillate between flavors — and hence have mass — is the first confirmed physics beyond the Standard Model. Neutrino physics touches on nuclear reactions, cosmology, CP violation, and Majorana fermions.

Prerequisites

Particle physics (Ch. PP) \cdot Quantum mechanics (Ch. QM) \cdot Nuclear physics (Ch. Nuc) \cdot Speci

al relativity (Ch. SR)

Learning Goals

$Apply the two-flavor oscillation formula P(\nu_\alpha\to\nu_\beta) = sin^{2}(2\theta)sin^{2}(\Deltam^{2}L/4E) to solar and a$ tmospheric data.
Derive the MSW resonance condition and identify the electron density at which it occurs inside the Sun.
$Explain the seesaw mechanism and compute the required M_{R} to generate a light neutrino mas$ s of 0.05 eV.
Describe neutrinoless double beta decay and what its observation would prove about neutrino nature.
Distinguish Dirac and Majorana mass terms and outline how leptogenesis connects neutrino masses to the baryon asymmetry.

NU.1 Neutrinos in the Standard Model (and Beyond)

The SM has three massless left-handed neutrinos: ν_e, ν_μ, ν_τ (one per lepton family). Neutrinos interact only via the weak force (W±, Z) — cross section σ ~ G_F² s/π ≈ 10⁻⁴⁵ cm² at 1 MeV (10⁷ times smaller than electromagnetic). Mean free path in water: λ = 1/(nσ) ~ 10¹⁸ km — the Earth is transparent to low-energy neutrinos.

Solar neutrino problem (1968–2001): the Davis chlorine detector measured only 1/3 of the neutrino flux predicted by the Standard Solar Model. Resolution: electron neutrinos produced in the Sun oscillate to other flavors on their way to Earth — and the Davis detector was only sensitive to ν_e. Confirmed by SNO (2001): total flux of all flavors = predicted solar model flux. Nobel Prize 2015 to McDonald (SNO) and Kajita (Super-Kamiokande).

NU.2 Neutrino Oscillations

If neutrino mass eigenstates |ν₁⟩, |ν₂⟩, |ν₃⟩ differ from flavor eigenstates |ν_e⟩, |ν_μ⟩, |ν_τ⟩, mixing occurs via the PMNS matrix U:

|\nu_\alpha⟩ = \sum_i U*(\alpha,i) |\nu_i⟩ \qquad (PMNS mixing, \alpha = e,\mu,\tau; i = 1,2,3)

(NU.1)

For two-flavor mixing: U = [[cosθ, sinθ],[-sinθ, cosθ]]. A ν_α produced at t=0 evolves in time. The probability of finding ν_β at time t:

P(\nu_\alpha \to \nu_\beta) = sin^{2}(2\theta) sin^{2}(\Deltam^{2}L/(4E)) \qquad (oscillation formula, L = distance, E = energy)

(NU.2)

where Δm² = m₂² − m₁² in eV². The oscillation length: L_osc = 4πE/(Δm²) = 2.48 km × E[GeV]/(Δm²[eV²]).

Theorem NU.1 — MSW Effect

In matter, the effective potential for \nu_e differs from \nu_\mu,\tau due to coherent forward scat

tering: V_{CC} = \sqrt2 G_{F} n_{e} (charged current), V_{NC} = -G_{F} n_{n}/\sqrt2. The effective Hamiltonian

in the flavor basis: H_{eff} = (\Deltam^{2}/(4E))[[-cos2\theta + A, sin2\theta],[sin2\theta, cos2\theta - A]] where A =

2\sqrt2 G_{F} n_{e} E / \Deltam^{2}. Resonance condition A = cos2\theta: maximal mixing even for small vacuum \theta

This MSW (Mikheyev-Smirnov-Wolfenstein) effect explains why solar \nu_e are converted to \nu_\mu

_\tau inside the Sun

Example NU.1 — Atmospheric Neutrino Oscillation at Super-Kamiokande

$Super-Kamiokande (1998) observed a deficit of upward-going \nu_\mu vs. downward-going \nu_\mu (fro$ $m cosmic ray interactions). Use the two-flavor formula to find \Deltam^{2} and \theta$ .

Setup:

Cosmic ray pions decay: \pi \to \mu + \nu_\mu, \mu \to e + \nu_e + \nu_\mu. Expected ratio R = \nu_\mu/\nu_e \approx 2. Ob

served: R was lower and strongly L-dependent (upward = longer path through Earth, L ~ 10,0

00 km vs. downward L ~ 15 km). \nu_e shows no deficit; \nu_\mu shows strong deficit for upward-g

oing — \nu_\mu oscillating to \nu_\tau (\nu_\tau undetected at Super-K at that energy

L/E analysis:

Plot survival probability P(\nu_\mu \to \nu_\mu) = 1 - sin^{2}(2\theta)sin^{2}(1.27 \Deltam^{2}[eV^{2}] L[km]/E[GeV]) vs L

E. SK data: dip at L/E ~ 500 km/GeV. P_{min} \approx 0: maximal mixing sin^{2}(2\theta) \approx 1 \to \theta \approx 45°

Mass squared difference:

At minimum: 1.27 \Deltam^{2} \times 500 = \pi/2 \to \Deltam^{2} = \pi/(2 \times 1.27 \times 500) \approx 2.5\times10^{-3} eV^{2}. More precisely

from fit: \Deltam^{2}(23) \approx 2.4\times10^{-3} eV^{2}, sin^{2}(2\theta_23) > 0.99 (nearly maximal). Normal ordering: m_{3}

m_{2} > m_{1}; inverted: m_{3} < m_{1} < m_{2}

Implications:

\Deltam^{2}(23) = 2.4\times10^{-3} eV^{2} implies m_{3} \ge \sqrt\Deltam^{2} \approx 0.05 eV (at least one mass eigenstate \ge 50 meV

Solar mixing: \Deltam^{2}(12) = 7.5\times10^{-5} eV^{2}, \theta_{12} \approx 34°. Absolute masses unknown; upper bound from

osmology: \summ_\nu < 0.12 eV (Planck 2018). KATRIN experiment: m(\nu_e) < 0.45 eV directly

NU.3 Neutrino Mass Mechanisms

The SM gives massless neutrinos (no right-handed component). To add mass:

Dirac mass: add right-handed ν_R (sterile). Yukawa coupling y: m_D = yv/√2 (v = Higgs VEV = 246 GeV). To get m_ν ~ 0.1 eV: y ~ 4×10⁻¹³ — unnaturally small.

Majorana mass: if ν = ν̄ (its own antiparticle). Forbidden for charged particles (charge conservation). Allowed for neutral neutrinos. Majorana mass term: m_M ν_c ν (Lorentz invariant for neutral fermion).

m_\nu ~ m_{D}^{2}/M_{R} \qquad (seesaw type I: heavy M_{R} \to naturally light m_\nu)

(NU.3)

The seesaw mechanism: a heavy right-handed Majorana neutrino (mass M_R ~ 10¹⁵ GeV, GUT scale) generates naturally light left-handed neutrinos. Baryogenesis via leptogenesis: CP-violating decay of heavy N_R in early universe → lepton asymmetry → converted to baryon asymmetry via sphaleron processes.

Neutrinoless double beta decay (0νββ): N → N+2 + 2e⁻ (no neutrinos). Only possible if ν is Majorana. Half-life: T_(1/2) ≥ 10²⁶ yr (KamLAND-Zen). Discovery would confirm Majorana nature and measure |m_ββ| = |Σ U^2_(ei) m_i|.

NU.4 Neutrino Sources and Detectors

Solar neutrinos: pp chain dominates (pp → d + e⁺ + ν_e, E < 0.42 MeV); ⁸B neutrinos (E ~ 14 MeV) detected by SNO. Total flux: ~6×10¹⁰ cm⁻²s⁻¹.Reactor antineutrinos (ν̄_e): from β-decay of fission products. KamLAND: 180 km average baseline, measured θ₁₂ and Δm²(12). Daya Bay, RENO: short baseline (1-2 km), measured θ₁₃ = 8.5° (2012).

IceCube: cubic-kilometer detector at South Pole. Cherenkov light from ν interactions. TeV-PeV astrophysical neutrinos detected (2013): diffuse flux consistent with E^(-2.5) spectrum. First sources: Seyfert galaxy NGC 1068 (2022), blazar TXS 0506+056.DUNE: long-baseline experiment (Fermilab to Homestake, 1300 km). Goals: CP violation in neutrino sector (δ_CP), mass ordering, proton decay.

Definition NU.1 — Common Traps

Flavor states are not mass states: oscillations happen because propagation phases differ.
Oscillation needs nonzero mass differences: absolute mass scale is a separate question.
Matter changes mixing: the MSW effect can enhance flavor conversion.
Neutrinos are hard to detect because weak interactions are weak: huge fluxes still produce few events.

Exercises — NU.1–NU.4 Neutrino Physics

Calculate the \nu_e survival probability for solar ^{8}B neutrinos (E = 10 MeV, L = 1.5\times10^{8} km

using two-flavor vacuum oscillation with \Deltam^{2}(12) = 7.5\times10^{-5} eV^{2} and sin^{2}(2\theta) = 0.855

Straightforward

Derive

the MSW resonance condition in matter. At what electron density does resonance occur for ^{8}

neutrinos (E = 10 MeV)? Is this density present in the Sun

cm⁻³

Intermediate

Explain the type-I seesaw mechanism. What right-handed neutrino mass M_{R} is needed to gene

rate m_\nu ~ 0.05 eV with Dirac mass m_{D} ~ m_\tau? Outline how leptogenesis generates the baryo

n asymmetry.

Intermediate

Explain neutrinoless double beta decay. What is the effective Majorana mass m_\beta\beta? What do

current experimental bounds (KamLAND-Zen: T\frac{1}{2} > 2.3\times10^{26} yr) imply for the neutrino mass hi

rarchy?

Challenging

Key Takeaways

$Neutrino oscillations: P(\nu_\alpha\to\nu_\beta) = sin^{2}(2\theta)sin^{2}(1.267 \Deltam^{2} L/E). Require nonzero masses \to$ BSM physics.
$Atmospheric: \Deltam^{2}(23) \approx 2.4\times10^{-3} eV^{2}, \theta_{23} \approx 45°. Solar: \Deltam^{2}(12) = 7.5\times10^{-5} eV^{2}, \theta_{12} = 34°$ . $\theta_{13} = 8.5°$
$MSW effect: matter potential for \nu_e from W exchange. Resonance A=cos2\theta \to level crossing \to$ adiabatic conversion.
$Seesaw: m_\nu = m_{D}^{2}/M_{R}. GUT-scale M_{R} ~ 10^{15} GeV gives naturally m_\nu ~ 0.1 eV$ .
Maj $orana vs Dirac: neutrinoless double beta decay tests Majorana nature. Current: T\frac{1}{2} > 10^{2}$ $r \to m_\beta\beta < 36-156 meV$ .
$Absolute masses: cosmological bound \summ_\nu < 0.12 eV (Planck). KATRIN direct: m_\nu_e &lt$ ; 0.45 eV.