# Effects of a Finite Volume in the Phase Structure of QCD

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Two-Flavor Nambu–Jona–Lasinio (NJL) Model

## 3. Setting Parameters

#### 3.1. Multiple Reflection Expansion

#### 3.2. Boundary Conditions

## 4. Results

#### 4.1. Constituent Quark Mass

#### 4.1.1. Multiple Reflection Expansion

#### 4.1.2. Finite Volume Approximation by Boundary Conditions

#### 4.2. Phase Diagram

#### 4.2.1. Multiple Reflection Expansion

#### 4.2.2. Finite Volume Approximation by Boundary Conditions

#### 4.3. Crossover Zone

#### 4.3.1. Multiple Reflection Expansion

#### 4.3.2. Finite Volume Approximation by Boundary Conditions

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Multiple Reflection Expansion

#### Appendix A.1. Spherical Droplet Shape

#### Appendix A.2. Cubic Box

## Appendix B. Finite Volume Approximation by Boundary Conditions

#### Appendix B.1. Periodic Boundary Conditions PBC

#### Appendix B.2. Antiperiodic Boundary Conditions APBC

#### Appendix B.3. Stationary Wave Conditions SWC

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**Figure 1.**Constituent quark mass as a function of temperature at $\mu =0$: (

**a**) sphere-Dirichlet; (

**b**) sphere-Neumann; (

**c**) cube-Dirichlet.

**Figure 2.**Constituent quark mass as a function of temperature at $\mu =0$: (

**a**) APBC; (

**b**) PBC; (

**c**) SWC.

**Figure 3.**Phase diagram superposition for local criterion: (

**a**) sphere-Dirichlet; (

**b**) sphere-Neumann; (

**c**) cube-Dirichlet. The red dots indicate the location of the CEP.

**Figure 4.**Phase diagram superposition for global criterion: (

**a**) sphere-Dirichlet; (

**b**) sphere-Neumann; (

**c**) cube-Dirichlet. The red dots indicate the location of the CEP.

**Figure 5.**Phase diagram superposition for local criterion: (

**a**) APBC; (

**b**) PBC; (

**c**) SWC. The red dot indicates the location of the CEP.

**Figure 6.**Phase diagram superposition for global criterion: (

**a**) APBC; (

**b**) PBC; (

**c**) SWC. The red dot indicates the location of the CEP.

**Figure 7.**APBC phase diagram superposition for small volumes: (

**a**) local criterion; (

**b**) global criterion. The red dots indicate the location of the CEP.

**Figure 8.**Phase diagram superposition for $R=L=7$ fm. SN: sphere-Neumann; CD: cube-Dirichle; SD: sphere-Dirichlet; APBC: antiperiodic boundary conditions; PBC: periodic boundary conditions; SWC: stationary wave conditions. (

**a**) local criterion; (

**b**) global criterion. The red dots indicate the location of the CEP.

**Figure 9.**Inflection point criterion phase diagram for infinite volume systems: (

**a**) proper time regularization; (

**b**) ultraviolet cutoff. The red dot indicates the location of the CEP.

**Figure 10.**Inflection point criterion phase diagram for spherical geometry with Dirichlet boundary conditions: (

**a**) R = 30 fm; (

**b**) R = 7 fm; (

**c**) R = 4 fm. The red dot indicates the location of the CEP.

**Figure 11.**Inflection point criterion phase diagram for spherical geometry with Neumann boundary conditions: (

**a**) R = 30 fm; (

**b**) R = 7 fm; (

**c**) R = 4 fm. The red dots indicate the location of the CEP.

**Figure 12.**Inflection point criterion phase diagram for cubic geometry with Dirichlet boundary conditions: (

**a**) R = 30 fm; (

**b**) R = 7 fm; (

**c**) R = 4 fm. The red dot indicates the location of the CEP.

**Figure 13.**Inflection point criterion phase diagram for stationary wave conditions: (

**a**) L = 30 fm; (

**b**) L = 7 fm; (

**c**) L = 4 fm.

**Figure 14.**Inflection point criterion phase diagram for anti-periodic boundary conditions: (

**a**) L = 30 fm; (

**b**) L = 7 fm. (

**c**) L = 4 fm.

R [fm] | $\mathit{G}\times {10}^{-6}$ [MeV]${}^{-2}$ | ${\mathbf{\Lambda}}_{\mathit{UV}}$ [MeV] | ${\mathbf{\Lambda}}_{\mathit{IR}}$ [MeV] |
---|---|---|---|

30 | 5.471 | 629.054 | 11.844 |

20 | 5.472 | 629.031 | 17.766 |

7 | 5.503 | 628.331 | 50.760 |

4 | 5.642 | 625.402 | 88.830 |

2.5 | 6.168 | 615.737 | 142.128 |

R [fm] | $\mathit{G}\times {10}^{-6}$ [MeV]${}^{-2}$ | ${\mathbf{\Lambda}}_{\mathit{UV}}$ [MeV] | ${\mathbf{\Lambda}}_{\mathit{IR}}$ [MeV] |
---|---|---|---|

30 | 5.47056 | 629.063 | 4.655 |

20 | 5.47062 | 629.061 | 6.982 |

7 | 5.47257 | 629.018 | 19.949 |

4 | 5.48135 | 628.821 | 34.910 |

0.9 | 6.402 | 611.982 | 155.155 |

R [fm] | $\mathit{G}\times {10}^{-6}$ [MeV]${}^{-2}$ | ${\mathbf{\Lambda}}_{\mathit{UV}}$ [MeV] | ${\mathbf{\Lambda}}_{\mathit{IR}}$ [MeV] |
---|---|---|---|

30 | 5.47057 | 629.063 | 4.937 |

20 | 5.47064 | 629.061 | 7.406 |

7 | 5.473 | 629.009 | 21.159 |

4 | 5.483 | 628.775 | 37.027 |

1 | 6.268 | 614.112 | 148.110 |

Dirichlet | Neumann | ||
---|---|---|---|

R(L) [fm] | Sphere | Cube | Sphere |

∞ | 326.143 | 326.143 | 326.143 |

30 | 292.407 | 315.488 | 325.982 |

20 | 275.106 | 310.084 | 325.773 |

7 | 172.276 | 279.249 | 323.249 |

4 | 59.557 | 241.908 | 317.774 |

2.5 | 22.224 | - | - |

1 | - | 31.428 | - |

0.9 | - | - | 221.695 |

**Table 5.**Constituent quark mass [MeV] for finite volume approximation by boundary conditions at $\mu =0$ MeV.

R [fm] | APBC | PBC | SWC |
---|---|---|---|

∞ | 208.265 | 208.265 | 208.265 |

30 | 208.265 | 208.265 | 165.229 |

20 | 208.265 | 208.265 | 141.654 |

7 | 208.232 | 208.301 | 40.302 |

4 | 206.624 | 210.567 | 18.969 |

1 | 168.294 | 258.028 | 9.867 |

Dirichlet | Neumann | |||||
---|---|---|---|---|---|---|

Sphere | Cube | Sphere | ||||

R(L) [fm] | $\mathbf{\mu}$ | $\mathit{T}$ | $\mathbf{\mu}$ | $\mathit{T}$ | $\mathbf{\mu}$ | $\mathit{T}$ |

30 | 318 | 13 | 322 | 37 | 324 | 44 |

20 | 309 | 1 | 321 | 33 | 324 | 44 |

7 | - | - | 311 | 5 | 324 | 44 |

4 | - | - | - | - | 323 | 40 |

Dirichlet | Neumann | |||||
---|---|---|---|---|---|---|

Sphere | Cube | Sphere | ||||

R(L) [fm] | loc | glob | loc | glob | loc | glob |

∞ | 206 | 187 | 206 | 187 | 206 | 187 |

30 | 190 | 166 | 201 | 176 | 206 | 182 |

20 | 182 | 152 | 198 | 170 | 205 | 182 |

7 | 132 | 119 | 184 | 153 | 204 | 181 |

4 | 38 | 53 | 166 | 144 | 202 | 175 |

2.5 | 9 | 30 | - | - | - | - |

1 | - | - | 10 | 34 | - | - |

0.9 | - | - | - | - | 160 | 139 |

**Table 8.**Critical temperatures [MeV] and maximun reached chemical potential [MeV] for finite volume approximation by boundary conditions systems.

APBC | PBC | SWC | |||||||
---|---|---|---|---|---|---|---|---|---|

L [fm] | loc | glob | ${\mathbf{\mu}}_{\mathit{max}}$ | loc | glob | ${\mathbf{\mu}}_{\mathit{max}}$ | loc | glob | ${\mathbf{\mu}}_{\mathit{max}}$ |

∞ | 225 | 208 | 320 | 225 | 208 | 320 | 225 | 208 | 320 |

30 | 166 | 164 | 210 | 166 | 163 | 205 | 144 | 142 | 166 |

20 | 166 | 164 | 213 | 166 | 163 | 205 | 130 | 128 | 149 |

7 | 166 | 98 | 242 | 166 | 164 | 204 | 12 | 62 | 158 |

4 | 166 | 142 | 285 | 166 | 165 | 191 | - | 75 | 269 |

2 | 160 | 157 | 547 | 173 | 169 | 169 | - | - | 350 |

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Mata Carrizal, N.B.; Valbuena Ordóñez, E.; Garza Aguirre, A.J.; Betancourt Sotomayor, F.J.; Morones Ibarra, J.R.
Effects of a Finite Volume in the Phase Structure of QCD. *Universe* **2022**, *8*, 264.
https://doi.org/10.3390/universe8050264

**AMA Style**

Mata Carrizal NB, Valbuena Ordóñez E, Garza Aguirre AJ, Betancourt Sotomayor FJ, Morones Ibarra JR.
Effects of a Finite Volume in the Phase Structure of QCD. *Universe*. 2022; 8(5):264.
https://doi.org/10.3390/universe8050264

**Chicago/Turabian Style**

Mata Carrizal, Nallaly Berenice, Enrique Valbuena Ordóñez, Adrián Jacob Garza Aguirre, Francisco Javier Betancourt Sotomayor, and José Rubén Morones Ibarra.
2022. "Effects of a Finite Volume in the Phase Structure of QCD" *Universe* 8, no. 5: 264.
https://doi.org/10.3390/universe8050264