m_u

m_u — up quark, q_- branch

Up is the lightest +2/3 quark. On the q_- branch (geometry), its mass comes from a cascade off the lepton scale:

$$ m_u = m_e \cdot \sin^2\theta_3(q_-)^{-2} \cdot \sin^2\theta_5(q_-) \cdot R_u, $$

with $R_u$ the isospin correction (u/d split).

Values at μ* = 15

• $\sin^2\theta_3(q_-) = 0.12489$
• $\sin^2\theta_5(q_-) = 0.11181$
• $m_e = 0.511$ MeV
• $R_u \approx 5.28$ (App. P §C8)

$$ m_u = 0.511 \cdot (0.12489)^{-2} \cdot 0.11181 \cdot 5.28 \approx 2.156\ \text{MeV}. $$

PT: 2.156 MeV vs PDG: 2.16 ± 0.05 MeV. Gap: 0.17%.

u-d-c-... hierarchy

The 6 quark masses follow a geometric cascade on q_- with combinatorial exponents (App. P §C8 closes the catalogue at 24 exponents). See also [m_d](/en/observables/8), [m_s](/en/observables/9).