On the Corner Elements of the CKM and PMNS Matrices

TL;DR

This paper proposes a model with a rotating universal rank-one mass matrix to explain the small but nonzero corner elements of the CKM and PMNS matrices, matching experimental data.

Contribution

It introduces a rotating rank-one mass matrix framework that predicts the hierarchy and values of corner elements in quark and lepton mixing matrices.

Findings

Predicts small but nonzero corner elements consistent with data

Estimates ratios of corner elements matching experimental bounds

Provides a unified explanation for CKM and PMNS matrix structures

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) $V_{ub} < V_{td}$, $U_{e 3} < U_{\tau 1}$, (iii) estimates result for the ratios $V_{ub}/V_{td}$ and $U_{e 3}/U_{\tau 1}$, 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.