# A Brief Review of Chiral Chemical Potential and Its Physical Effects

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## Abstract

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## 1. Introduction

#### 1.1. The ${U}_{a}(1)$ Problem, Chiral Anomaly, and Instantons

#### 1.2. $\theta $-Vacuum and Strong CP Problem

#### 1.3. Event by Event P and CP Violation and Chiral Chemical Potential

## 2. The Effects of Chiral Imbalance

#### 2.1. The Charge Separation Effect (CSE) and Chiral Magnetic Effect (CME)

#### 2.2. The Effects of Chiral Chemical Potential on QCD Phase Structure

#### 2.3. The Effects of Chiral Chemical Potential on Quark Stars

## 3. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The illustration of the charge separation effect and chiral magnetic effect. The blue arrows and the red arrows respectively denote the spin and the momentum of quarks. (1) At the beginning, the u and d quarks are all in the lowest Landau level and can only move along the direction of the magnetic field. (2) The quarks interact with a nontrivial gauge configuration with ${q}_{w}$. Assuming ${q}_{w}=-1$, this gauge configuration can convert the chiralities of quarks from left-hand to right-hand. It will lead to the chiral imbalance between the left- and right-hand quarks. (3) In the presence of a strong magnetic, the u quarks (or d quark) with different chiralities move in different directions. Due to the chiral imbalance, the total net charge moving along the direction of the magnetic is $Q=2e$. In addition, it will result in a charge difference between two domain walls perpendicular to the magnetic field. Reprinted from [28], with permission from Elsevier.

**Figure 2.**Charge separation effect—the regions inside the domain walls with $\theta \ne 0$, outside with $\theta =0$. The domain walls is charged in the case of a strong magnetic field, with the surface charge density ∼$\frac{{e}^{2}\theta B}{2{\pi}^{2}}$. Reprinted from [36], with permission from Elsevier.

**Figure 3.**Chiral magnetic effect—in the case of a strong magnetic field, in the region with $\dot{\theta}\ne 0$, an electric current $\mathbf{J}\sim \frac{{e}^{2}\dot{\theta}\mathbf{B}}{2{\pi}^{2}}$ is induced. Reprinted from [36], with permission from Elsevier.

**Figure 4.**The mass–radius relation of quark star with different chiral chemical potential at $\alpha =0.9$ and ${B}^{\frac{1}{4}}$ = 90 MeV.

**Table 1.**The maximum mass ${M}_{max}$ of quark star with different ${\mu}_{5}$ are shown. In addition, the tidal deformabilities of the 1.4-solar-mass stars are also listed.

${\mathit{\mu}}_{5}$ (MeV) | ${\mathit{M}}_{\mathit{max}}({\mathit{M}}_{\odot})$ | ${\mathbf{\Lambda}}_{1.4}$ |
---|---|---|

0 | 1.96 | 236 |

10 | 1.79 | 221 |

20 | 1.71 | 204 |

30 | 1.61 | 150 |

40 | 1.53 | 104 |

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**MDPI and ACS Style**

Yang, L.-K.; Luo, X.-F.; Segovia, J.; Zong, H.-S.
A Brief Review of Chiral Chemical Potential and Its Physical Effects. *Symmetry* **2020**, *12*, 2095.
https://doi.org/10.3390/sym12122095

**AMA Style**

Yang L-K, Luo X-F, Segovia J, Zong H-S.
A Brief Review of Chiral Chemical Potential and Its Physical Effects. *Symmetry*. 2020; 12(12):2095.
https://doi.org/10.3390/sym12122095

**Chicago/Turabian Style**

Yang, Li-Kang, Xiao-Feng Luo, Jorge Segovia, and Hong-Shi Zong.
2020. "A Brief Review of Chiral Chemical Potential and Its Physical Effects" *Symmetry* 12, no. 12: 2095.
https://doi.org/10.3390/sym12122095