# Asymptotically Flat, Spherical, Self-Interacting Scalar, Dirac and Proca Stars

^{*}

## Abstract

**:**

## 1. Introduction and Motivation

#### 1.1. General Remarks

#### 1.2. Conventions

**spin 0:**$\Phi $ is a complex scalar field, which is equivalent to a model with two real scalar fields, ${\Phi}^{R},{\Phi}^{I}$, via $\Phi ={\Phi}^{R}+i{\Phi}^{I}$.**spin 1/2:**${\Psi}^{\left(A\right)}$ are massive spinors, with four complex components, the index $\left(A\right)$ corresponding to the number of copies of the Lagrangian. For a spherically symmetric configuration, one should consider (at least) two spinors ($A=1,2$), with equal mass $\mu $. A model with a single spinor necessarily possesses a nonzero angular momentum density and it cannot be spherically symmetric.**spin 1:**$\mathcal{A}$ is a complex 4-potential, with the field strength $\mathcal{F}=d\mathcal{A}$. Again, the model can be described in terms of two real vector fields, $\mathcal{A}={\mathcal{A}}^{R}+i{\mathcal{A}}^{I}$.

## 2. The General Framework

#### 2.1. The Action and Field Equations

#### 2.2. The Ansätze and Explicit Equations

#### 2.2.1. The Metric and Matter Fields

#### 2.2.2. The Explicit Equations

#### 2.3. Units and Scaling Symmetries

## 3. The Probe Limit: Flat Spacetime Solutions

#### 3.1. Deser’s Argument and Virial-Type Identities

- scalar field:

- Dirac field:

- Proca field:

#### 3.2. Numerical Results

#### 3.2.1. General Remarks

#### 3.2.2. Solutions with a Sextic Self-Interaction Term, $\beta >0$

#### 3.2.3. Solutions without a Sextic Self-Interaction Term, $\beta =0$

## 4. Including the Gravity Effects

#### 4.1. The Boundary Conditions

#### 4.2. Virial Identities

- scalar field:

- Dirac field:

- Proca field:

#### 4.3. General Features

## 5. Other Aspects

#### 5.1. No Hair Results

**Scalar case****Dirac case****Proca case**A no hair theorem has been proven in [30] for a massive, non-selfinterating Proca field. In Appendix B, we generalize it for an arbitrary Proca potential $U\left({\mathcal{A}}^{2}\right)$.

#### 5.2. The Issue of Particle Numbers: Bosons vs. Fermions

## 6. Further Remarks. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. The Dirac Field: Conventions

## Appendix B. Self-Interacting Proca Field: A No Hair Result

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**Figure 1.**(

**Left**): the radial profile of a typical non-gravitating scalar soliton. (

**Right**): the mass and Noether charge are shown $vs.$ the scalar field frequency for the fundamental family of non-gravitating scalar solitons.

**Figure 2.**Same as Figure 1 for Dirac stars. Note that the single particle condition, $Q=1$, is not imposed here.

**Figure 3.**Same as Figure 1 for Proca stars.

**Figure 4.**ADM mass and Noether charge of the gravitating scalar boson stars $vs.$ the scalar field frequency for families of solutions with three different values of the coupling constant $\alpha $. The solutions in the right panel do not possess a sextic self-interacting term.

**Figure 5.**The same as Figure 4 for Dirac stars. The single particle condition, $Q=1$, is not imposed here.

**Figure 6.**Same as Figure 4 for Proca stars.

**Figure 7.**(

**Left panel**) Soliton mass $vs.$ the mass of the elementary quanta of the field, for non-gravitating solutions of the Dirac equations with three different values of $\beta $. (

**Right panel**) The same for the gravitating solutions with several values of the coupling constant $\alpha $ and $\beta =0$. The single particle condition, $Q=1$, is imposed here.

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Herdeiro, C.A.R.; Radu, E.
Asymptotically Flat, Spherical, Self-Interacting Scalar, Dirac and Proca Stars. *Symmetry* **2020**, *12*, 2032.
https://doi.org/10.3390/sym12122032

**AMA Style**

Herdeiro CAR, Radu E.
Asymptotically Flat, Spherical, Self-Interacting Scalar, Dirac and Proca Stars. *Symmetry*. 2020; 12(12):2032.
https://doi.org/10.3390/sym12122032

**Chicago/Turabian Style**

Herdeiro, Carlos A. R., and Eugen Radu.
2020. "Asymptotically Flat, Spherical, Self-Interacting Scalar, Dirac and Proca Stars" *Symmetry* 12, no. 12: 2032.
https://doi.org/10.3390/sym12122032