# Spatial QCD theory of the dressing of quarks and the origin of three generations

## TL;DR

This paper presents a spatial QCD-based theory describing how bare quarks become dressed, revealing three distinct solutions that correspond to the three quark generations, and links their properties to fundamental constants and gravity.

## Contribution

It introduces an exact operator solution to QCD equations for single quark dressing, resulting in three soliton-like solutions representing quark generations.

## Key findings

- Three distinct soliton solutions correspond to three quark generations.
- Radii of quark generations are proportional to inverse energy scale E.
- Masses of dressed quarks relate to gravitational and cosmological constants.

## Abstract

The dressing of bare massless quarks is described with a spatial theory based on the self-consistent solution of the QCD field equations. After quantization these equations are expressed in terms of quark and gluon creation and annihilation operators and admit a surprisingly elegant exact (operator) solution which eliminates any multi-quark admixtures in the state vector. Hence, this theory is uniquely and exclusively suited to describe the dressing of single bare quarks. After factorizing out the operators, a finite set of coupled non-linear differential equations results for the reduced c-number quark and gluon fields. These yield three distinct solitonlike solutions, corresponding to the three observed quark generations. Physically each solution represents a quark absolutely confined by the gluon potentials it generates. The radii of the three generations are given by (2.0428..., 1 , 1)/E, while the binding energies are linked to the SU(3) structure constants and given by E - (32/9, 16/9, 1)E, where E sets the energy scale of the system. To stabilize the system general relativity is required, putting E in the Planck domain. After the introduction of a vacuum term characterized by the cosmological constant the dressed quark mass can be expressed in terms of the gravitational and cosmological constant, nevertheless lying in the MeV range. After the inclusion of the other gauge interactions this theory might well serve as a theoretical laboratory for quantitative tests of the unification of general relativity and QFT in a constrained Planck scale environment.