m_c

m_c — charm quark

Charm jumps to the 2nd upper generation. The cascade uses the anomalous-dimension ratio:

$$ m_c = m_s \cdot \left(\frac{\gamma_3}{\gamma_5}\right)^{n_c} \cdot R_c, $$

with $n_c = 5$ and $R_c$ electroweak correction (App. P §C8).

Computation

$\gamma_3 / \gamma_5 = 0.80761 / 0.69632 = 1.1599$. Power 5 = 2.108.

$$ m_c = 93.395 \cdot 2.108 \cdot 6.465 = 1272.4\ \text{MeV}. $$

PT: 1272.4 MeV vs PDG: 1270 ± 20 MeV. Gap: 0.19%.

The s → c jump

The gap $m_c / m_s \approx 13.6$ is smaller than $m_s / m_d \approx 20$ — the PT cascade predicts this non-monotonicity (passage from lower to upper generation via $\gamma_p$ rather than $\sin^2\theta_p$).