Reference

Formulas.

The core equations of neutrino physics — from Fermi's 1934 beta-decay theory and the PMNS oscillation framework to Freedman's CEvNS cross-section and the Schubart Master Equation for neutrinovoltaic conversion.

Foundations

Weak-interaction foundations

Fermi's 1934 four-fermion contact theory remains the low-energy effective description of charged-current weak interactions. The coupling constant G_F is the dimensional scale of every low-energy neutrino process.

Fermi's four-fermion Lagrangian

L_{F} = - \frac{G _{F}}{2} [\overset{ˉ}{ψ}_{p} γ^{μ} ψ_{n}] [\overset{ˉ}{ψ}_{e} γ_{μ} ψ_{ν}] + h.c.

Four-fermion contact term coupling the hadronic and leptonic currents. Reproduces the continuous beta-decay spectrum; superseded at high energies by intermediate W-boson exchange.

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Fermi coupling constant

G_{F} \approx 1.166 \times 1 0^{- 5} GeV^{- 2}

Extracted from muon lifetime with part-per-million precision. Related to the electroweak gauge coupling by G_F/√2 = g²/(8 M_W²).

Oscillation

Neutrino oscillation

Flavor eigenstates are linear combinations of the three mass eigenstates via the PMNS matrix. Propagating mass eigenstates accumulate phase differences, producing flavor oscillation along the trajectory.

Flavor decomposition

∣ ν_{α} ⟩ = i \sum U_{α i}^{*} ∣ ν_{i} ⟩

A flavor eigenstate is a superposition of mass eigenstates, weighted by complex-conjugated PMNS elements.

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PMNS mixing matrix (standard parametrization)

U = 100 0 c_{23} - s_{23} 0 s_{23} c_{23} c_{13} 0 - s_{13} e^{i δ} 010 s_{13} e^{- i δ} 0 c_{13} c_{12} - s_{12} 0 s_{12} c_{12} 0 001

Three mixing angles θ_{12}, θ_{13}, θ_{23} and one Dirac CP phase δ_CP. If neutrinos are Majorana, two additional phases enter but do not affect oscillation probabilities.

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Two-flavor vacuum oscillation probability

P (ν_{α} \to ν_{β}) = sin^{2} (2 θ) sin^{2} (\frac{Δ m ^{2} L}{4 E})

The leading behaviour in each dominant channel. Maximum probability at L/E such that the phase reaches π/2.

Three-flavor vacuum oscillation probability

P (ν_{α} \to ν_{β}) = δ_{α β} - 4 i > j \sum Re (U_{α i}^{*} U_{β i} U_{α j} U_{β j}^{*}) sin^{2} (\frac{Δ m _{ij}^{2} L}{4 E}) + 2 i > j \sum Im (U_{α i}^{*} U_{β i} U_{α j} U_{β j}^{*}) sin (\frac{Δ m _{ij}^{2} L}{2 E})

The complete three-flavor expression. The Im-term generates CP violation when all mixing angles are non-zero and masses non-degenerate.

Phase convention (practical units)

Δ_{ij} \equiv \frac{Δ m _{ij}^{2} L}{4 E} = 1.267 \cdot Δ m_{ij}^{2} [eV^{2}] \cdot \frac{L [ km ]}{E [ GeV ]}

Dimensionless kinematic phase in the units most commonly reported by experiments.

Matter effects

MSW matter effects

Electron neutrinos feel an additional charged-current potential from forward scattering on electrons in matter. This shifts the effective mass eigenvalues and mixing angles, producing resonant flavor conversion in the solar interior.

Matter potential (charged current)

V_{C C} = 2 G_{F} n_{e}

Shift of the νe energy due to forward scattering on electrons. About 10⁻¹² eV for solar-core densities.

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MSW resonance energy

E_{res} = \frac{Δ m ^{2} cos ( 2 θ )}{2 2 G _{F} n _{e}}

Energy at which the effective mixing angle in matter passes through maximal (45°). Solar electron neutrinos above this energy emerge as mass eigenstate ν₂.

Detection

Detection cross-sections

Neutrino detection rests on a small set of weak-interaction processes. Inverse beta decay dominates at MeV-range reactor energies; coherent elastic scattering dominates the low-energy total cross-section on heavy nuclei.

Inverse beta decay

\overset{ν}{ˉ}_{e} + p \to e^{+} + n, σ \approx 9.52 \times 1 0^{- 44} E_{e} p_{e} cm^{2}

The reactor-antineutrino detection process with E_e = E_ν − 1.293 MeV. The delayed-coincidence signature — prompt positron annihilation plus delayed neutron capture — has anchored reactor-neutrino detection since 1956.

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CEvNS differential cross-section

\frac{d σ}{d T} = \frac{G _{F}^{2} M}{2 π} [(G_{V} + G_{A})^{2} + (G_{V} - G_{A})^{2} (1 - \frac{T}{E _{ν}})^{2} - (G_{V}^{2} - G_{A}^{2}) \frac{M T}{E _{ν}^{2}}]

Freedman's 1974 coherent cross-section. T is the nuclear recoil kinetic energy, M the nuclear mass. At low momentum transfer, all nucleons scatter in phase.

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Weak nuclear charge

Q_{W} = (1 - 4 sin^{2} θ_{W}) Z - N \approx - N

Proton contribution nearly vanishes because (1 − 4sin²θ_W) ≈ 0.075. The neutron number dominates, giving the characteristic N² coherent enhancement.

Coherent N² scaling

σ_{CEvNS} \propto G_{F}^{2} N^{2} E_{ν}^{2}

Why the coherent cross-section on heavy nuclei exceeds incoherent neutrino-nucleon scattering by ~5000× and is the largest neutrino cross-section below ~50 MeV.

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Mass

Mass measurements

Oscillation measures squared-mass differences; absolute scales require independent routes. The kinematic endpoint, the double-beta decay half-life, and cosmology each probe a different projection of the mass eigenvalues.

Measured squared-mass differences

Δ m_{21}^{2} \approx 7.4 \times 1 0^{- 5} eV^{2}, ∣Δ m_{31}^{2} ∣ \approx 2.5 \times 1 0^{- 3} eV^{2}

Solar-sector and atmospheric-sector splittings. Global-fit values as of 2024.

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Kinematic effective mass (tritium endpoint)

m_{ν_{e}}^{2, eff} = i \sum ∣ U_{e i} ∣^{2} m_{i}^{2}

The quantity measured by KATRIN and similar endpoint experiments. Current bound: m_νe < 0.45 eV (90% C.L.).

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Majorana effective mass (0νββ)

⟨ m_{β β} ⟩ = i \sum U_{e i}^{2} m_{i}

Neutrinoless double beta decay rate is proportional to this quantity squared. Current bound from KamLAND-Zen: ⟨m_ββ⟩ ≲ 40 meV.

Weinberg operator (dimension-5 Majorana mass generation)

L_{Weinberg} = - \frac{c _{α β}}{Λ} (\overset{ˉ}{L}_{α} \tilde{H}) (\tilde{H}^{†} L_{β}) + h.c.

The unique dimension-5 operator consistent with Standard Model gauge symmetries. After electroweak breaking, generates m_ν ~ c v²/Λ.

Seesaw relation (Type-I)

m_{ν} \sim \frac{m _{D}^{2}}{M}

For m_D ~ 100 GeV (electroweak scale) and m_ν ~ 0.1 eV, implies new-physics scale M ~ 10¹⁴ GeV near grand unification.

CP violation

Observable leptonic CP violation requires all three mixing angles non-zero, non-degenerate neutrino masses, and δ_CP distinct from 0 or π. The Jarlskog invariant packages the condition into a single convention-independent quantity.

Jarlskog invariant

J_{C P} = Im (U_{e 2} U_{μ 2}^{*} U_{e 3}^{*} U_{μ 3}) = \frac{1}{8} cos θ_{13} sin 2 θ_{12} sin 2 θ_{13} sin 2 θ_{23} sin δ_{C P}

Vanishes if any mixing angle is zero or if δ_CP ∈ {0, π}. Current best estimates give |J_CP| up to roughly 0.033 depending on δ_CP.

Cosmology

Neutrinos decoupled from the thermal plasma about one second after the Big Bang, leaving a relic population whose temperature and density are fixed by entropy conservation during subsequent e⁺e⁻ annihilation.

Cosmic neutrino background temperature

T_{ν} = (\frac{4}{11})^{1/3} T_{γ} \approx 1.95 K

Slightly colder than the CMB because photons were heated by e⁺e⁻ annihilation after neutrinos decoupled.

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Relic number density

n_{ν} = \frac{3}{11} n_{γ} \approx 336 cm^{- 3} (three flavors, ν + \overset{ν}{ˉ})

Relic neutrinos outnumber photons by roughly 3:1 per flavor, and outnumber baryons by a factor of 10⁹.

Effective number of relativistic species

N_{eff} = 3.044 (Standard Model prediction)

Slightly above 3 because of partial re-heating during e⁺e⁻ annihilation and finite-temperature QED corrections. Planck measurement: N_eff = 2.99 ± 0.17.

Applied research

The Schubart Master Equation

An engineering-integration framework that combines the CEvNS cross-section, cosmic-ray muon flux, ambient electromagnetic contributions, and thermal gradients into a single expression for device output power. Formulated by Holger Thorsten Schubart in collaboration with the Neutrino Energy Group. Physically grounded in the 2015 oscillation Nobel result, the 2017 COHERENT CEvNS observation, and 2025 precision flux measurements from JUNO.

Master Equation for neutrinovoltaic conversion

P (t) = η \cdot \int_{V} Φ_{eff} (r, t) \cdot σ_{eff} (E) d V

$P (t)$: Electrical output power as a function of time.
$η$: Dimensionless conversion efficiency of the material system.
$Φ_{eff} (r, t)$: Effective flux — weighted sum of neutrino, cosmic-ray muon, electromagnetic, and thermal contributions at position r and time t.
$σ_{eff} (E)$: Energy-dependent effective cross-section. For the neutrino channel this is the CEvNS cross-section.
$V$: Active material volume.

Full concept page → Holger Thorsten Schubart → Evidence thread →

Formulas.

Weak-interaction foundations

Fermi's four-fermion Lagrangian

Fermi coupling constant

Neutrino oscillation

Flavor decomposition

PMNS mixing matrix (standard parametrization)

Two-flavor vacuum oscillation probability

Three-flavor vacuum oscillation probability

Phase convention (practical units)

MSW matter effects

Matter potential (charged current)

MSW resonance energy

Detection cross-sections

Inverse beta decay

CEvNS differential cross-section

Weak nuclear charge

Coherent N² scaling

Mass measurements

Measured squared-mass differences

Kinematic effective mass (tritium endpoint)

Majorana effective mass (0νββ)

Weinberg operator (dimension-5 Majorana mass generation)

Seesaw relation (Type-I)

CP violation

Jarlskog invariant

Cosmology

Cosmic neutrino background temperature

Relic number density

Effective number of relativistic species

The Schubart Master Equation

Master Equation for neutrinovoltaic conversion

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