Method for Fitting and Deriving the CKM and PMNS Matrices from Underlying Wavefunctions

Abstract

The Cabibbo-Kobayashi-Maskawa (CKM) and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrices in the electroweak sector are now well-known experimentally. However, there have been only a few proposals to derive these matrices from an underlying theory. In this note, these matrices are derived from example scalar wavefunctions associated with a permutationally symmetric mass matrix with three states for each of the four fermion families. Such a mass matrix is consistent with an anomaly-free quantum field theory for the 4 fermion families. The derivation uses three-dimensional gaussian wavefunctions with specified widths and specified separations between the 3 wells implied by the theory. This approach first fits the diagonal elements of the matrices. A fourth element is then estimated from the properties of the scalar fields in the aforementioned model. Unitarity is then applied to compute the remaining matrix elements. The example calculations produce matrices that have a normalized root-mean-square error (RMSE) from the measured matrix of 6.09 × 10 −4 and 8.34 × 10 −3 for the CKM and PMNS matrices, respectively. The normalized RMSE for departure from unitarity is 9.18 × 10 −4 and 8.95 × 10 −3 for the two respective matrices. The results are within one standard deviation of almost all of the measured parameters for both matrices. The primary objective of this paper is to show that the matrices can be fit accurately in the context of at least one anomaly-free quantum field theory.