Symmetric texture-zero mass matrices with eigenvalues quark mass

TL;DR

This paper establishes conditions for symmetric texture-zero quark mass matrices to have positive, real eigenvalues, classifies all such matrices, and analytically solves for the CKM mixing matrix in specific zero-texture cases.

Contribution

It provides a comprehensive classification of symmetric texture-zero mass matrices with positive eigenvalues and analytically derives the CKM matrix for selected zero-texture scenarios.

Findings

Conditions for real, positive eigenvalues of texture-zero matrices are derived.

All symmetric texture-zero matrices are classified systematically.

Analytic solutions for the CKM matrix in four-zero and related cases are obtained.

Abstract

Working within the context of texture-zeros mechanism for fermionic mass matrices, we provide necessary and sufficient conditions on the characteristic polynomial coefficients such that it has real, simple and positive roots. We translate these conditions in terms of invariants from congruent matrices. Then, all symmetric texture-zero matrices are counted and classified. Next we apply the result from the first part to analyze the three, two and one zero texture matrices in a systematic way. Finally we solve analytically the $V_{ckm}$ mixing matrix for the four zero sets; we also analyze the $V_{ckm}$ for a particular case of four zero, four zero-perturbed and three zero sets.