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Hierarchies of quark masses and the mixing matrix in the standard theory

We study the general dependence of mixing angles on heavy fermion masses when mass hierarchies exist among the fermions. For two generations and small Cabibbo angle, this angle is directly shown to scale like $μ_1/m_s \pm μ_2/m_c$, where $|μ_1| \ll m_s, |μ_2| \ll m_c$ are independent mass scales. For $n=3$ generations, we extend to the Yukawa matrices of $u$- and $d$-type quarks the property that the $2\times 2$ upper-left sub-matrix of the Cabibbo-Kobayashi-Maskawa matrix $K$ is a good approximation to the Cabibbo matrix $C$. Then, without any additional Ansatz concerning the existence of mass hierarchies or the smallness of the mixing angles, the moduli of its entries $K_{13},K_{23},K_{31},K_{32}$ are shown to scale like $[β_{13},β_{23},β_{31},β_{32}] \sqrt{{m_c}/{m_t}} \pm [δ_{13},δ_{23},δ_{31},δ_{32}] \sqrt{{m_s}/{m_b}}$, where the $β$'s and the $δ$'s are coefficients smaller than 10. This method, when used for two generations, gives a dependence on $m_s$ and $m_c$ ``weaker'' than the one obtained first, but which matches a well known behaviour for the Cabibbo angle: $θ_c \approx \sqrt{ε_d (m_d/m_s)} - \sqrt{ε_u(m_u/m_c)}$, with $ε_d,ε_u \leq 1$. The asymptotic behaviour in the case of three generations can also be strengthened into a $1/m_{b,t}$ behaviour by incorporating our knowledge about the hierarchies of quark masses and the smallness of the mixing angles.