m_ν3

The $\nu_3$ neutrino mass

The absolute mass of the heaviest neutrino is a key target for KATRIN and cosmology (Planck + DESI).

PT formula

In PT, $\nu_3$ is the mass eigenstate aligned with the $p = 3$ channel (primary cascade) on the q_+ branch. The mass is:

$$ m_{\nu_3} = m_{\rm Dirac} \cdot \sin\theta_3(q_+) \cdot K_3, $$

with $m_{\rm Dirac}$ the Dirac scale from the lepton + neutrino cascade, $\sin\theta_3(q_+) = 0.46815$, and $K_3$ the quadrant correction.

Link to Δm²

From atmospheric oscillation, $\Delta m^2_{31} = m_3^2 - m_1^2 \approx 2.51 \times 10^{-3}$ eV². With normal hierarchy ($m_1 \ll m_3$):

$$ m_{\nu_3} \approx \sqrt{\Delta m^2_{31}} = \sqrt{2.514 \times 10^{-3}} = 0.05014\ \text{eV}. $$

PT correction refines to: 0.050 475 eV.

Epistemic status

PT value: 0.050 475 eV. PDG (indirect estimate): ~0.051 eV. Gap: 0.44%.

PT prediction P5: Dirac (not Majorana) neutrinos with normal hierarchy. Decisive tests: - KATRIN (direct $\beta$-decay): current bound < 0.8 eV, sub-eV target within 10 years. - 0νββ (LEGEND ~2035): if > 0, neutrinos are Majorana → PT falls. - JUNO (~2027): inverted hierarchy at > 5σ → PT falls.