Hierarchies of the 4 Texture Zero Quark Mass Matrices and their equal spacing rule

TL;DR

This paper explores hierarchical regularities in quark mass matrices with four texture zeros, revealing an equal spacing rule and multiple states characterized by two parameters, offering new insights into quark mass structure.

Contribution

It introduces the equal spacing rule in 4 texture zero quark mass matrices and interprets the existence of multiple states labeled by (N,k).

Findings

Hierarchical regularity called equal spacing rule identified.

Quark mass matrices appear in multiple states labeled by (N,k).

Provides interpretation for these hierarchical patterns.

Abstract

We show that the parameters of the quark mass matrices M_u and M_d of a 4 texture zero model, exhibit an interesting hierarchical regularity we have called equal spacing rule, and that this regularity leads us to a further observation that the quark mass matrices (M_u, M_d) appear to exist in a plurality of states, each state labelled by two numbers (N,k). We give interpretation to these observations.