m_b

m_b — bottom quark

Bottom is the 3rd-generation down. The cascade jumps to the $p = 7$ channel:

$$ m_b = m_c \cdot \left(\frac{\gamma_5}{\gamma_7}\right)^{n_b} \cdot R_b, $$

with $n_b = 4$ and $R_b$ QCD correction (App. P §C8).

Computation

$\gamma_5 / \gamma_7 = 0.69632 / 0.59547 = 1.1693$. Power 4 = 1.872.

$$ m_b = 1272.4 \cdot 1.872 \cdot 1.748 = 4164.8\ \text{MeV}. $$

PT: 4164.8 MeV vs PDG: 4180 ± 30 MeV. Gap: 0.36%.

Convergence in B-mesons

$m_b$ determines B-meson lifetimes and is crucial for CKM physics (V_ub, V_cb). The 0.36% PT error propagates to V_xb, giving the ~0.1–0.15% uncertainty on those matrix elements.