On the Corner Elements of the CKM and PMNS Matrices

Abstract

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 ($U_{\tau 1}$), 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 $m_c/m_t,m_s/m_b,m_{\mu }/m_{\tau }$, and $m_2/m_3$, (i) the corner elements are small but nonzero, (ii) $V_{ub}<V_{td}$, $U_{e3}<U_{\tau 1}$, (iii) estimates result for the ratios $V_{ub}/V_{td}$ and $U_{e3}/U_{\tau 1}$, and (B) by inputting further the experimental values of $V_{us},V_{tb}$ and $U_{e2},U_{\mu 3}$, (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.

Comments: 9 pages, 2 figures, updated with new experimental data and more details