Internal quark symmetries and colour SU(3) entangled with Z_3-graded Lorentz algebra

TL;DR

This paper proposes a novel framework where internal quark symmetries, specifically color SU(3), are entangled with a Z_3-graded Lorentz algebra, leading to a new description of quark fields with extended components and mass structures.

Contribution

It introduces a Z_3-graded extension of Lorentz algebra to incorporate internal color symmetries, resulting in 12-component color Dirac equations and a master sextet for all Standard Model quarks.

Findings

Color multiplets described by 12-component Dirac equations.

Introduction of a Z_3-graded triplet of masses, including Lee-Wick type complex conjugates.

A unified 72-component master quark sextet for Standard Model quarks.

Abstract

In the current version of QCD the quarks are described by ordinary Dirac fields, organized in the following internal symmetry multiplets: the $SU(3)$ colour, the $SU(2)$ flavour, and broken $SU(3)$ providing the family triplets. \noindent In this paper we argue that internal and external (i.e. space-time) symmetries are entangled at least in the colour sector in order to introduce the spinorial quark fields in a way providing all the internal quark's degrees of freedom which do appear in the Standard Model. Because the $SU(3)$ colour algebra is endowed with natural $Z_3$-graded discrete automorphisms, in order to introduce entanglement the $Z_3$-graded version of Lorentz and Poincar\'e algebras with their realizations are considered. The colour multiplets of quarks are described by $12$-component colour Dirac equations, with a $Z_3$-graded triplet of masses (one real and a Lee-Wick complex conjugate pair). We argue that all quarks in the Standard Model can be described by the $72$-component master quark sextet of $12$-component coloured Dirac fields.