Lepton Sector & Neutrinos
Neutrino oscillations, PMNS matrix, and the mystery of neutrino mass
The Leptonic Generations
Like quarks, leptons come in three generations:
1st Generation
2nd Generation
3rd Generation
Key difference from quarks:
Leptons don't experience the strong force (color singlets). But neutrinos have revealed new physics beyond the SM through oscillations!
1. Charged Leptons
Masses from Yukawa Couplings
Charged leptons get mass like quarks:
ℒYukawaℓ = -ye L̄eL Φ eR - yμ L̄μL Φ μR - yτ L̄τL Φ τR + h.c.| Lepton | Mass | Yukawa y = √2 m/v | Lifetime |
|---|---|---|---|
| Electron (e) | 0.511 MeV | 2.9 × 10-6 | Stable |
| Muon (μ) | 105.7 MeV | 6.1 × 10-4 | 2.2 μs |
| Tau (τ) | 1.777 GeV | 1.0 × 10-2 | 290 fs |
Lepton number conservation:
In the SM, individual lepton numbers Le, Lμ, Lτ are conserved. This forbids processes like μ → eγ. But neutrino oscillations violate this! (See below)
Anomalous Magnetic Moment
The electron and muon magnetic moments are measured to incredible precision:
ae = (g-2)/2 = 0.001 159 652 180 73 (28) (experiment)Agrees with SM QED to 10-12 precision—best test of QED!
Muon g-2 anomaly:
Δaμ = aμexp - aμSM = (251 ± 59) × 10-114.2σ discrepancy (Fermilab 2021)! Possible sign of new physics (SUSY? Dark sector?)
2. Neutrino Oscillations
The Discovery
1998: Super-Kamiokande observed that atmospheric neutrinos oscillate between flavors. This proved neutrinos have mass, contradicting the minimal SM!
Key observation:
Neutrinos produced as definite flavor (νe, νμ, ντ) via weak interactions, but propagate as mass eigenstates (ν₁, ν₂, ν₃) with different masses.
|να⟩ = Σi Uαi |νi⟩U = PMNS matrix (analogous to CKM for quarks)
Oscillation Probability
The probability for να → νβ after distance L:
P(να → νβ) ≈ sin²(2θ) sin²(Δm² L / 4E)- • θ = mixing angle between flavors
- • Δm² = m2² - m1² (mass-squared difference)
- • L = baseline (distance traveled)
- • E = neutrino energy
3. The PMNS Matrix
The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix relates flavor and mass eigenstates:
UPMNS = ⎡ Ue1 Ue2 Ue3 ⎤
⎢ Uμ1 Uμ2 Uμ3 ⎥
⎣ Uτ1 Uτ2 Uτ3 ⎦Standard Parametrization:
Like CKM, parametrized by 3 angles + 1 phase (+ 2 Majorana phases if ν are Majorana):
U = ⎡ 1 0 0 ⎤ ⎡ c13 0 s13e-iδ ⎤ ⎡ c12 s12 0 ⎤
⎢ 0 c23 s23 ⎥ ⎢ 0 1 0 ⎥ ⎢-s12 c12 0 ⎥
⎣ 0 -s23 c23 ⎦ ⎣-s13eiδ 0 c13 ⎦ ⎣ 0 0 1 ⎦where cij = cos θij, sij = sin θij, δ = Dirac CP phase
Measured Values:
| Parameter | Value | Determined by |
|---|---|---|
| sin² θ12 | 0.304 ± 0.012 | Solar neutrinos |
| sin² θ23 | 0.573 ± 0.016 | Atmospheric ν |
| sin² θ13 | 0.0220 ± 0.0007 | Reactor experiments |
| δCP | ~230° (uncertain) | T2K, NOνA (2σ hint) |
| Δm21² | 7.5 × 10-5 eV² | Solar ν |
| |Δm32²| | 2.5 × 10-3 eV² | Atmospheric ν |
Key differences from CKM:
- • Large mixing angles: θ23 ~ 45°, θ12 ~ 33° (CKM has θC ~ 13°)
- • Mass hierarchy unknown: normal (m₁ < m₂ < m₃) or inverted (m₃ < m₁ < m₂)?
- • Absolute masses unknown: only Δm² measured, not mi individually
4. Neutrino Mass Mechanisms
Dirac vs Majorana
Two possibilities for neutrino mass:
Dirac Mass:
ℒDirac = -mD ν̄L νR + h.c.- • Requires νR (right-handed neutrino)
- • ν ≠ ν̄ (particle ≠ antiparticle)
- • Like charged fermions
- • Conserves lepton number
Majorana Mass:
ℒMaj = -(1/2)mM ν̄Lc νL + h.c.- • No νR needed!
- • ν = ν̄ (neutrino is its own antiparticle)
- • Violates lepton number by 2 units
- • Predicts 0νββ decay
The Seesaw Mechanism
Type-I Seesaw: Explains why neutrinos are so light compared to charged fermions.
Assume both Dirac mass mD ~ O(100 GeV) and heavy Majorana mass MR ~ 10¹⁵ GeV for νR:
mν ≈ mD² / MRExample: mD = 100 GeV, MR = 10¹⁴ GeV → mν ~ 0.1 eV ✓
This connects neutrino masses to GUT scale physics!
Neutrinoless Double Beta Decay
If neutrinos are Majorana, 0νββ decay is possible:
(A,Z) → (A,Z+2) + 2e- (no neutrinos!)Violates lepton number by 2. Current best limit: T1/2 > 10²⁶ years (GERDA, KamLAND-Zen). Discovery would prove Majorana nature!
5. Open Questions
Mass Hierarchy:
Normal (m₁ < m₂ < m₃) or inverted (m₃ < m₁ < m₂)? JUNO, DUNE will determine this.
Absolute Mass Scale:
What is mlightest? Tritium β-decay (KATRIN) bounds mνe < 0.8 eV. Cosmology (CMB+LSS) gives Σmν < 0.12 eV.
Dirac or Majorana?:
0νββ searches ongoing. If discovered → Majorana. If not found down to mββ ~ 1 meV → likely Dirac.
CP Violation in Leptons?:
Is δCP ≠ 0? T2K/NOνA see hints (~2-3σ). DUNE/HyperK will measure to 5σ. Could explain matter-antimatter asymmetry via leptogenesis!
Sterile Neutrinos?:
Anomalies in short-baseline experiments hint at 4th neutrino (~eV mass) that doesn't interact weakly. MicroBooNE, SBN program investigating.
Summary
- ✓ 3 charged leptons: e, μ, τ with masses from Yukawa couplings
- ✓ Neutrino oscillations discovered 1998: proof of BSM physics!
- ✓ PMNS matrix: 3 large mixing angles + CP phase δCP
- ✓ Mass differences known: Δm21² ~ 10-5 eV², |Δm32²| ~ 10-3 eV²
- ✓ Absolute masses unknown: mν < 1 eV, but hierarchy unclear
- ✓ Dirac vs Majorana: 0νββ decay experiments will decide
- ✓ Seesaw mechanism: connects mν to GUT/Planck scale physics
Further Resources
- • PDG Review - "Neutrino Masses, Mixing, and Oscillations"
- • Giunti & Kim - Fundamentals of Neutrino Physics (Oxford, 2007)
- • Mohapatra & Pal - Massive Neutrinos in Physics and Astrophysics
- • NuFIT Collaboration - http://www.nu-fit.org (global neutrino fits)