A Z3-symmetric Quantum Chromodynamics

TL;DR

This work proposes a Z3-symmetric extension of QCD by encoding color into a 12-component coloured Dirac spinor built from six Pauli spinors, leading to a sixth-order Dirac-like equation with a $Z_3$-graded colour structure. The resulting propagator features six poles, including Lee–Wick-type complex poles, which suppress single-quark propagation, while carefully constructed ternary combinations of solutions can propagate freely, suggesting an algebraic form of confinement. In the massless limit, the Green function yields a natural quark-potential form $V(r) \sim \alpha/r + \beta r$ without assuming a priori potentials, indicating a link between the propagator structure and confinement physics. The paper also develops a $Z_3$-covering of the Lorentz group compatible with the colour Dirac framework and outlines next steps toward a fully second-quantized theory with potential ternary statistics and gauge interactions.

Abstract

We propose a description of colour triplets of quarks by entangled Z3-graded Lee-Wick type fields, one with real mass and the two remaining ones with mutually conjugate complex masses. This is obtained by attributing colour degrees of freedom to six Pauli spinors, three endowed with colours and three with anti-colours, which are unitrd into one 12-component generalized "coloured Dirac spinor". The so entangled triplet of quark fields satisfies a generalized Dirac equation, with generalized 12x1é gamma-matrices acting on 12-component coloured spinors. The sixth order dispersion relations lead to solutions suitably vanishing in asymptotic region, exhibiting the well established confinement property of coloured quarks' degrees of freedom. We show how one can construct certain cubic combinations of those solutions in a way that cansels the damping factors, producing freely propagating functions.