#
Four-Fermion Interaction Model on ℳ^{D−1} ⊗ S^{1}

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Basic Model: Four-Fermion Interaction Model on ${\mathcal{M}}^{\mathit{D}}$

## 3. Four-Fermion Interaction Model on ${\mathcal{M}}^{\mathit{D}\mathbf{-}\mathbf{1}}\mathbf{\otimes}{\mathit{S}}^{\mathbf{1}}$

## 4. Casimir Effect

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NJL | Nambu–Jona-Lasinio |

GN | Gross–Neveu |

## Appendix A. Effective Potential on $\mathcal{M}$^{D−1} ⊗ S^{1}

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**Figure 2.**Dynamically generated mass, ${m}_{0}$, (solid lines) and the mass scale, ${m}_{1}$, (dashed lines).

**Figure 3.**Behavior of the effective potential on ${\mathcal{M}}^{D-1}\otimes {S}^{1}$ at $L{m}_{0}=2.5$ for ${\lambda}_{r}>{\lambda}_{cr}$.

**Figure 4.**Behavior of the effective potential on ${\mathcal{M}}^{D-1}\otimes {S}^{1}$ at $L{m}_{1}=2.5$ for ${\lambda}_{r}<{\lambda}_{cr}$.

**Figure 9.**Phase structure on $\delta -L$ plane. The chiral symmetry is broken above the lines for ${\lambda}_{r}>{\lambda}_{cr}$ and below the lines for ${\lambda}_{r}<{\lambda}_{cr}$.

**Figure 10.**Phase structure on $D-L$ plane. The chiral symmetry is broken above the lines for ${\lambda}_{r}>{\lambda}_{cr}$ and below the lines for ${\lambda}_{r}<{\lambda}_{cr}$.

**Figure 13.**Boundary between the repulsive (left side of the lines) and attractive (right side of the lines) force.

D | 2 | 2.5 | 3 | 3.5 |
---|---|---|---|---|

$\delta $ | 0.42265 | 0.44575 | 0.46166 | 0.47280 |

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**MDPI and ACS Style**

Inagaki, T.; Matsuo, Y.; Shimoji, H.
Four-Fermion Interaction Model on ℳ^{D−1} ⊗ *S*^{1}. *Symmetry* **2019**, *11*, 451.
https://doi.org/10.3390/sym11040451

**AMA Style**

Inagaki T, Matsuo Y, Shimoji H.
Four-Fermion Interaction Model on ℳ^{D−1} ⊗ *S*^{1}. *Symmetry*. 2019; 11(4):451.
https://doi.org/10.3390/sym11040451

**Chicago/Turabian Style**

Inagaki, Tomohiro, Yamato Matsuo, and Hiromu Shimoji.
2019. "Four-Fermion Interaction Model on ℳ^{D−1} ⊗ *S*^{1}" *Symmetry* 11, no. 4: 451.
https://doi.org/10.3390/sym11040451