# Emergent Dynamics of Five-Colour QCD Due to Dimensional Frustration

## Abstract

**:**

## 1. Introduction

## 2. The Cho–Faddeev–Niemi–Shabanov Decomposition

## 3. The Vacuum State of Five-Colour QCD

## 4. The Emergence of Massless and Massive Photons

## 5. Matter Field Representations

## 6. The Origin of Mass

## 7. Conclusions

## Acknowledgements

## References

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**Table 1.**Candidate parallel components for vacuum condensate. The column on the left is for parallel vectors, the column on the right is for antiparallel vectors. $\Delta \mathcal{H}$ should be multiplied by ${H}^{2}\frac{11}{96{\pi}^{2}}$.

${\mathbf{H}}^{\left(\mathit{i}\right)}=+{\mathbf{H}}^{\left(\mathit{j}\right)}$ | $\Delta \mathcal{H}$ | ${\mathbf{H}}^{\left(\mathit{i}\right)}=-{\mathbf{H}}^{\left(\mathit{j}\right)}$ | $\Delta \mathcal{H}$ |
---|---|---|---|

${\mathbf{H}}^{\left(1\right)}=+{\mathbf{H}}^{\left(2\right)}$ | $1.06381$ | ${\mathbf{H}}^{\left(1\right)}=-{\mathbf{H}}^{\left(2\right)}$ | $1.06381$ |

${\mathbf{H}}^{\left(1\right)}=+{\mathbf{H}}^{\left(3\right)}$ | $0.857072$ | ${\mathbf{H}}^{\left(1\right)}=-{\mathbf{H}}^{\left(3\right)}$ | $0.857072$ |

${\mathbf{H}}^{\left(1\right)}=+{\mathbf{H}}^{\left(4\right)}$ | $0.715651$ | ${\mathbf{H}}^{\left(1\right)}=-{\mathbf{H}}^{\left(4\right)}$ | $0.715651$ |

${\mathbf{H}}^{\left(2\right)}=+{\mathbf{H}}^{\left(3\right)}$ | $1.01655$ | ${\mathbf{H}}^{\left(2\right)}=-{\mathbf{H}}^{\left(3\right)}$ | $0.656584$ |

${\mathbf{H}}^{\left(2\right)}=+{\mathbf{H}}^{\left(4\right)}$ | $0.882589$ | ${\mathbf{H}}^{\left(2\right)}=-{\mathbf{H}}^{\left(4\right)}$ | $0.577976$ |

${\mathbf{H}}^{\left(3\right)}=+{\mathbf{H}}^{\left(4\right)}$ | $1.00042$ | ${\mathbf{H}}^{\left(3\right)}=-{\mathbf{H}}^{\left(4\right)}$ | $0.540983$ |

**Table 2.**The antisymmetric flux, the number of flux quanta, and the restricted flux for each root and weight.

AS flux | Nb flux quanta | Restricted flux | |
---|---|---|---|

??XX | 0 | 0 | 0 |

??OX | $1/6$ | 9 | 0.0119 |

XXXO | $\left(4+\sqrt{15}\right)/16$ | 28 | 0.0105 |

??OO | $\left(8-\sqrt{15}\right)/48$ | 5 | 0 |

down ($rbg$) | $\left(4+\sqrt{15}\right)/80$ | 5 | 0.0124 |

$u\phantom{\rule{-0.166667em}{0ex}}v$ | $\left(8-\sqrt{15}\right)/80$ | 3 | 0 |

$i\phantom{\rule{-0.166667em}{0ex}}r$ | $1/10$ | 5 | 0.0140 |

$u\phantom{\rule{-0.166667em}{0ex}}v$-shadow | $\left(8+10\sqrt{\frac{3}{5}}\right)/120$ | 7 | 0.0108 |

$i\phantom{\rule{-0.166667em}{0ex}}r$-shadow | $\left(32+30\sqrt{\frac{3}{5}}\right)/120$ | 26 | 0.0132 |

up | $\left(4+\sqrt{15}\right)/40$ | 11 | 0.0077 |

${e}^{\pm}$ ($u\phantom{\rule{-0.166667em}{0ex}}v\phantom{\rule{0.166667em}{0ex}}i\phantom{\rule{-0.166667em}{0ex}}r$) | $3\left(4+\sqrt{15}\right)/80$ | 17 | 0.0029 |

Particle | mass/${\mathit{m}}_{\mathit{e}}$ |
---|---|

${e}^{\pm}$ | 1 |

down ($rbg$) | 18.3 |

$u\phantom{\rule{-0.166667em}{0ex}}v$ | 0 |

$i\phantom{\rule{-0.166667em}{0ex}}r$ | 23.3 |

$u\phantom{\rule{-0.166667em}{0ex}}v$-shadow | 13.9 |

$i\phantom{\rule{-0.166667em}{0ex}}r$-shadow | 20.7 |

up | 6.96 |

??XX | 0 |

??OX | 16.8 |

XXXO | 13.2 |

??OO | 0 |

**Table 4.**Predicted and experimentally determined [42] mass ratios.

Experimental | Predicted | |
---|---|---|

${m}_{u}/{m}_{e}$ | 2.94 - 6.46 | 6.96 |

${m}_{d}/{m}_{e}$ | 6.85 - 11.7 | 18.3 |

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Walker, M.L.
Emergent Dynamics of Five-Colour QCD Due to Dimensional Frustration. *Symmetry* **2010**, *2*, 1322-1337.
https://doi.org/10.3390/sym2031322

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Walker ML.
Emergent Dynamics of Five-Colour QCD Due to Dimensional Frustration. *Symmetry*. 2010; 2(3):1322-1337.
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**Chicago/Turabian Style**

Walker, Michael Luke.
2010. "Emergent Dynamics of Five-Colour QCD Due to Dimensional Frustration" *Symmetry* 2, no. 3: 1322-1337.
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