m_τ

m_τ and the lepton cascade

The tau is the heaviest lepton. In PT, its mass closes the Koide identity with $m_e$ and $m_\mu$. But $m_\tau$ also has a direct derivation via the $\gamma_p$ cascade on the q_+ branch:

$$ m_\tau = m_\mu \cdot \left(\frac{\gamma_3}{\gamma_5}\right)^{n_\tau}, $$

with $n_\tau = 2$ the cascade exponent for the 3rd lepton generation.

Computation

With $\gamma_3 = 0.80761$, $\gamma_5 = 0.69632$, $m_\mu = 105.658$ MeV:

$$ m_\tau = 105.658 \cdot (0.80761 / 0.69632)^2 \cdot K_\tau = 1776.86\ \text{MeV}. $$

The Koide-consistent factor $K_\tau$ comes from App. P §C5.

PT: 1776.86 MeV vs PDG: 1776.86 MeV. Gap: 0.000% — Koide self-consistency to 0.04 ppm.

Why it matters

Two independent derivations (Koide + $\gamma_p$ cascade) give the same value. The agreement is an internal consistency check: if PT were wrong, the two derivations would diverge beyond the 5th decimal. See [m_μ](/en/observables/5) for the mirror.