On the corner elements of the CKM and PMNS matrices

Abstract

Recent experiments show that the top-right corner element (Ue3) of the PMNS matrix is small but nonzero, and suggest further via unitarity that it is smaller than the bottom-left corner element (Uτ1). 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 mc/mt,ms/mb,mμ/mτ, and m2/m3, i) the corner elements are small but nonzero, ii) Vub < Vtd, Ue3 < Uτ1, iii) estimates result for the ratios Vub/Vtd and Ue3/Uτ1, and B) by inputting further the experimental values of Vus,Vtb and Ue2, Uμ3, iv) estimates result for the values of the corner elements themselves. All the inequalities and estimates obtained are consistent with present data within expectation for the approximations made.