On the Corner Elements of the CKM and PMNS Matrices

Recent experiments show that the top-right corner element ($U_{e3}$) of the PMNS, like that ($V_{ub}$) of the CKM, matrix is small but nonzero, and suggest further via unitarity that it is smaller than the bottom-left corner element ($U_{τ1}$), again as in the CKM case ($V_{ub} < V_{td}$). An attempt in explaining these facts would seem an excellent test for any model of the mixing phenomenon. Here, it is shown that if to the assumption of a universal rank-one mass matrix, long favoured by phenomenologists, one adds that this matrix rotates with scale, then it follows that (A) by inputting the mass ratios $m_c/m_t, m_s/m_b, m_μ/m_τ$, and $m_2/m_3$, (i) the corner elements are small but nonzero, (ii) $V_{ub} < V_{td}$, $U_{e 3} < U_{τ1}$, (iii) estimates result for the ratios $V_{ub}/V_{td}$ and $U_{e 3}/U_{τ1}$, and (B) by inputting further the experimental values of $V_{us}, V_{tb}$ and $U_{e2},U_{μ3}$, (iv) estimates result for the values of the corner elements themselves. All the inequalities and estimates obtained are consistent with present data to within expectation for the approximations made.

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