Perspectives on Quark Mass and Mixing Relations

Abstract

Two down-up perspectives on quark mass-mixing relations are reviewed. The modified Fritzsch path relates $V_{cb}$ and $V_{ub}$ to $M_u$, but has trouble with the low $V_{cb} \simeq 0.04$ value. The modified Wolfenstein path focuses on the change in $V_{ub}$ from $λ^3 { (ca. 1983)} \leadsto λ^4 { (ca. 1994)}$. The relations $m_d/m_s\sim m_s/m_b \sim δ\sim λ^2 \simeq | V_{cb}| \simeq | V_{us}|^2$ and $m_u/m_c\sim m_c/m_t \sim δ^2 \sim λ^4 \sim | V_{ub}|$ suggest a clean separation of the origin of $V_{KM}$: $| V_{us}| \equiv λ$ and $| V_{cb}| \equiv δ$ arise from $M_d$ while $V_{ub} \equiv Bλ^4 e^{-iϕ}$ comes from $M_u$. Five to six parameters might suffice for ten mass-mixing parameters, with $δ$ seemingly the more sensible expansion parameter, while $λ\simeq \sqrt{m_d/m_s}$ is tied empirically to $(M_d)_{11} = 0$. The approximate relations suggest a near weak scale origin of flavor.