# Quantum Chromodynamics and the Hyperbolic Unitary Group SUh(3)

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## Abstract

**:**

## 1. Introduction

## 2. Pseudo-Euclidean Space ${\mathbb{E}}_{\mathbf{3},\mathbf{3}}$ as the Image of the Three-Dimensional Hyperbolic Space ${\mathbb{H}}^{\mathbf{3}}$

## 3. The Hyperbolic Color Group of Unitary Symmetry Uh^{3}(1) and Its Representation

## 4. Hyperbolic Unitary Group SUh(3) and Gluons

## 5. Rules to Determine the Permissible Types of Interactions between Color Quarks

## 6. Color Factors

## 7. Alternative Antiquark Representation

## 8. Permissible Types of Interactions between Gluon Pairs

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 8.**Process of exchange between color and different anti-color charges (scheme IIIa, $X\ne Y$).

**Figure 9.**Exchange processes between gluons of type (22).

Interaction | Scheme | Gluon | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{C}}_{\mathit{F}}$ for Antiquarks (13) | ${\mathit{C}}_{\mathit{F}}$ for Antiquarks (31) |
---|---|---|---|---|---|---|

R–R | IIIb | ${\Delta}_{3}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{3}$ | $\frac{1}{3}$ |

IIIb | ${\Delta}_{8}$ | $\frac{1}{\sqrt{6}}$ | $\frac{1}{\sqrt{6}}$ | |||

R–G | II | ${\Delta}_{R\tilde{G}}$ | 1 | 1 | $\frac{1}{3}$ | $\frac{1}{3}$ |

IIIb | ${\Delta}_{3}$ | $\frac{1}{\sqrt{2}}$ | $\frac{-1}{\sqrt{2}}$ | |||

IIIb | ${\Delta}_{8}$ | $\frac{1}{\sqrt{6}}$ | $\frac{1}{\sqrt{6}}$ | |||

R–B | II | ${\Delta}_{R\tilde{B}}$ | 1 | 1 | $\frac{1}{3}$ | $\frac{1}{3}$ |

IIIb | ${\Delta}_{8}$ | $\frac{1}{\sqrt{6}}$ | $\frac{-2}{\sqrt{6}}$ | |||

R–$\tilde{R}$ | I | ${\Delta}_{R\tilde{G}}$ | 1 | 1 | $\frac{4}{3}$ | $\frac{1}{3}$ |

I | ${\Delta}_{R\tilde{B}}$ | 1 | 1 | |||

IIIa | ${\Delta}_{3}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}$ | |||

IIIa | ${\Delta}_{8}$ | $\frac{1}{\sqrt{6}}$ | $\frac{1}{\sqrt{6}}$ | |||

R–$\tilde{G}$ | IIIa | ${\Delta}_{3}$ | $\frac{1}{\sqrt{2}}$ | $\frac{-1}{\sqrt{2}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ |

IIIa | ${\Delta}_{8}$ | $\frac{1}{\sqrt{6}}$ | $\frac{1}{\sqrt{6}}$ | |||

R–$\tilde{B}$ | IIIa | ${\Delta}_{8}$ | $\frac{1}{\sqrt{6}}$ | $\frac{-2}{\sqrt{6}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ |

B–B | IIIb | ${\Delta}_{8}$ | $\frac{-2}{\sqrt{6}}$ | $\frac{-2}{\sqrt{6}}$ | $\frac{1}{3}$ | $\frac{1}{3}$ |

B–$\tilde{B}$ | I | ${\Delta}_{B\tilde{R}}$ | 1 | 1 | $\frac{4}{3}$ | $\frac{1}{3}$ |

I | ${\Delta}_{B\tilde{G}}$ | 1 | 1 | |||

IIIa | ${\Delta}_{8}$ | $\frac{-2}{\sqrt{6}}$ | $\frac{-2}{\sqrt{6}}$ |

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**MDPI and ACS Style**

Popov, N.; Matveev, I.
Quantum Chromodynamics and the Hyperbolic Unitary Group SUh(3). *Axioms* **2023**, *12*, 1058.
https://doi.org/10.3390/axioms12111058

**AMA Style**

Popov N, Matveev I.
Quantum Chromodynamics and the Hyperbolic Unitary Group SUh(3). *Axioms*. 2023; 12(11):1058.
https://doi.org/10.3390/axioms12111058

**Chicago/Turabian Style**

Popov, Nikolay, and Ivan Matveev.
2023. "Quantum Chromodynamics and the Hyperbolic Unitary Group SUh(3)" *Axioms* 12, no. 11: 1058.
https://doi.org/10.3390/axioms12111058