#
Inverse Supersymmetry Breaking in S^{1} × R^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Twisted Sections and Non-Trivial Topology

#### 2.1. $N=1$, $d=4$ Supersymmetric Model

#### 2.2. Supersymmetric Effective Potential in ${S}^{1}\times {R}^{3}$

#### 2.3. Connection with Zeta Function

#### 2.4. Small L Expansion of the effective potential

## 3. Addition of Explicit Supersymmetry Breaking Terms

#### 3.1. A term of the form ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$

#### 3.2. Combination of the terms ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$ and ${g}_{3}{\chi}^{2}{\phi}_{{t}_{2}}^{2}$

#### 3.3. Combination of the terms $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{u}{\mathsf{\Psi}}_{u}\chi $ and ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$

#### 3.4. Combination of the terms $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{u}{\mathsf{\Psi}}_{u}\chi $, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$ and ${g}_{3}{\chi}^{2}{\phi}_{{t}_{2}}^{2}$

#### 3.5. Discussion

- Addition of fermion interactions with the untwisted scalar of the form
- *
- $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{u}{\mathsf{\Psi}}_{u}\chi $
- *
- $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{t}{\mathsf{\Psi}}_{t}\chi $
- *
- $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{u}{\mathsf{\Psi}}_{u}\chi $ and $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{t}{\mathsf{\Psi}}_{t}\chi $

- Addition of interactions of twisted fermions with twisted scalars of the form
- *
- $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{t}{\mathsf{\Psi}}_{t}\chi $, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$
- *
- $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{t}{\mathsf{\Psi}}_{t}\chi $, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{2}}^{2}$
- *
- $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{t}{\mathsf{\Psi}}_{t}\chi $, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{2}}^{2}$

## 4. Inverse Symmetry Breaking and Symmetry non-Restoration at Finite Temperature

## 5. Conclusions

## 6. Acknowledgments

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**Table 1.**${g}_{3}$ and the corresponding ${L}_{c}$ values, case ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$

${\mathit{g}}_{3}$ | ${\mathit{L}}_{\mathit{c}}^{-1}$ |
---|---|

0.1 | 32319 |

0.07 | 41352 |

0.05 | 50839 |

0.03 | 67994 |

0.01 | 121950 |

0.007 | 145900 |

0.005 | 173083 |

0.003 | 223990 |

0.001 | 388990 |

0.0005 | 557000 |

0.0001 | 1232000 |

0.00005 | 1740000 |

0.00001 | 3942000 |

**Table 2.**Plot of the ${L}_{c}-{g}_{3}$ (left) and fit with a continuous curve (right), case ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$

**Table 3.**Contour plots of the effective potential as a function of ${g}_{3}$ and ${L}_{c}$, case ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$

**Table 4.**${g}_{3}$ and the corresponding ${L}_{c}$ values for the ${g}_{3}{\chi}^{2}{\phi}_{{t}_{2}}^{2}$ case

${\mathit{g}}_{3}$ | ${\mathit{L}}_{\mathit{c}}^{-1}$ |
---|---|

0.1 | 32290 |

0.07 | 41320 |

0.01 | 121551 |

0.0001 | 1238950 |

0.00001 | 3922000 |

**Table 5.**${g}_{3}$ and the corresponding ${L}_{c}$ values, case ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{2}}^{2}$

${\mathit{g}}_{3}$ | ${\mathit{L}}_{\mathit{c}}^{-1}$ |
---|---|

0.1 | 32300 |

0.07 | 41359 |

0.05 | 51009 |

0.03 | 68100 |

0.01 | 121900 |

0.005 | 173500 |

0.001 | 388990 |

**Table 6.**Plot of the ${L}_{c}-{g}_{3}$ (right) and fit with a continuous curve (left), case ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{2}}^{2}$

**Table 7.**${g}_{3}$ and the corresponding ${L}_{c}$ values for the case $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{u}{\mathsf{\Psi}}_{u}\chi $, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$

${\mathit{g}}_{3}$ | ${\mathit{L}}_{\mathit{c}}^{-1}$ |
---|---|

0.1 | 32178 |

0.07 | 41205 |

0.05 | 50742 |

0.03 | 62250 |

0.01 | 84200 |

0.005 | 170000 |

0.001 | 381570 |

**Table 8.**Plot of the ${L}_{c}-{g}_{3}$ (right) and fit with a continuous curve (left), for the case $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{u}{\mathsf{\Psi}}_{u}\chi $, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$

**Table 9.**Contour plots of the effective potential as a function of ${g}_{3}$ and ${L}_{c}$, for the case $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{u}{\mathsf{\Psi}}_{u}\chi $, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$

**Table 10.**${g}_{3}$ and the corresponding ${L}_{c}$ values for the case $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{u}{\mathsf{\Psi}}_{u}\chi $, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{2}}^{2}$

${\mathit{g}}_{3}$ | ${\mathit{L}}_{\mathit{c}}^{-1}$ |
---|---|

0.1 | 32350 |

0.07 | 41900 |

0.05 | 50900 |

0.03 | 67990 |

0.01 | 121500 |

0.005 | 172000 |

0.001 | 300000 |

**Table 11.**Plot of the ${L}_{c}-{g}_{3}$ (right) and fit with a continuous curve (left), for the case $\frac{1}{2}{g}_{2}{\overline{\mathsf{\Psi}}}_{u}{\mathsf{\Psi}}_{u}\chi $, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{1}}^{2}$, ${g}_{3}{\chi}^{2}{\phi}_{{t}_{2}}^{2}$

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**MDPI and ACS Style**

Oikonomou, V.
Inverse Supersymmetry Breaking in S^{1} × R^{3}. *Symmetry* **2010**, *2*, 366-387.
https://doi.org/10.3390/sym2010366

**AMA Style**

Oikonomou V.
Inverse Supersymmetry Breaking in S^{1} × R^{3}. *Symmetry*. 2010; 2(1):366-387.
https://doi.org/10.3390/sym2010366

**Chicago/Turabian Style**

Oikonomou, Vasilis.
2010. "Inverse Supersymmetry Breaking in S^{1} × R^{3}" *Symmetry* 2, no. 1: 366-387.
https://doi.org/10.3390/sym2010366