Fermion masses as mixing parameters in the SM

Flavor transitions via the charged current interactions are parametrized by a three dimensional and unitary transformation. This so called mixing matrix requires of four mixing parameters. Here we show that under the phenomenological observation of hierarchical fermion masses, $m_3 \gg m_2 \gg m_1$, a mixing parametrization can be built with its mixing parameters being the corresponding four independent mass ratios of each fermion sector, i.e., $m_u/m_c$, $m_c/m_t$, $m_d/m_s$, and $m_s/m_b$ and $m_e/m_μ$, $m_μ/m_τ$, $m_{ν1}/m_{ν2}$, and $m_{ν 2}/m_{ν3}$, for the quark and lepton sector, respectively.

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