# Gravitating Bubbles of Gluon Plasma above Deconfinement Temperature

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## Abstract

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## 1. Introduction

_{3}-symmetric Lagrangian to Einstein gravity. To our knowledge, very few attempts to describe to interplay between gravity and confinement/deconfinement phase transition can be found in the literature. One can quote [21,22], respectively discussing the loss of simultaneity between chiral restoration and deconfinement in curved space, and the possible existence of deconfined regions near a black hole horizon. Here we go one step further by building “particle-like” solutions for our scalar field that are known to appear in pure $3+1$-dimensional Yang–Mills theory coupled to Einstein gravity, see the seminal paper [23]. Within our approach the Yang–Mills degrees of freedom are replaced by a complex scalar field, whose associated Q-balls, when coupled to gravity, are called boson stars—see the review [24] for more recent references. To our knowledge, such a problem has never been addressed at finite temperature although research devoted to “QCD boson stars” (at $T=0$) is currently ongoing [25]. We build gravitating solutions of static-boson-star-type, i.e., spherically symmetric localized configurations of the scalar field that lead to an asymptotically flat metric without singularity, see Section 5.

## 2. The Model

#### 2.1. Z_{3}-Symmetric Potential

_{3}−symmetry. We use the potential V of Ref. [26] which reads

_{3}-symmetric and not U(1)-symmetric as is often the case in Lagrangians based on a complex scalar field, with typical potentials of the form ${\left|\varphi \right|}^{6}-{2\left|\varphi \right|}^{4}+b{\left|\varphi \right|}^{2}$ [27]. A U(1)-symmetry can be recovered in the large-${N}_{c}$ limit of Z${}_{{N}_{c}}$-symmetric potentials, see [13,28].

#### 2.2. Coupling to Einstein Gravity

## 3. Q-Balls

#### 3.1. Ansatz and Existence Conditions

_{3}-symmetric, not U(1). Note also that if $\varphi \left(r\right)$ is a real solution of the equations of motion, ${\mathrm{e}}^{\frac{ik\pi}{3}}$ with $k\in \mathbb{Z}$ is also a solution because of the system’s symmetry.

#### 3.2. Numerical Results

#### 3.3. Symmetry Breaking

## 4. Q-Holes

## 5. Boson Stars

## 6. Summary and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Relation between $\mathsf{\omega}$ and $\mathsf{\Omega}\equiv \sqrt{{m}^{2}\left(T\right)-{\omega}^{2}}$ versus $\varphi \left(0\right)$ for three values of $T/{T}_{c}$ in flat space-time ($\alpha =0$) (solid lines). The dotted lines represent $\omega $ and ${g}_{tt}\left(0\right)$ in the case $T/{T}_{c}=1.01$ for gravitating solutions ($\alpha =1$).

**Figure 3.**Profiles of $\varphi \left(r\right)$ for several values of $T/{T}_{c}$. Distances are in units of ${l}_{phys}$.

**Figure 4.**Evolution of the mean radius $\langle R\rangle $ as a function of $T/{T}_{c}$, the insert contains $\varphi \left(0\right)$ and the mass (12) of the scalar field. Distances are in units of ${l}_{phys}$ and masses are in units of ${M}_{phys}$.

**Figure 5.**Evolution of the mean radius $<R>$ (black line, in units of ${l}_{phys}$) and of the gravitational mass ${M}_{G}$ (red lines, in units of ${M}_{phys}$) as function of ${T}_{c}/T$ for $\omega =0$ boson stars. The metric component ${g}_{00}=f\left(0\right)$ is represented by the solid (resp. dashed) blue lines for $\alpha =1$ (resp. $\alpha =0.1$). These values depend very weakly on $\alpha $ and the curves are mostly superimposed.

**Figure 6.**Profiles of the metric functions $f,l$ and of the scalar field $\varphi $ for $\alpha =1,\omega =0$ and two values of the temperature: $T/{T}_{c}=1.01$ (solid lines) $T/{T}_{c}=1.11$ (dashed lines). The radial variable r is in units of ${l}_{phys}$.

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Brihaye, Y.; Buisseret, F.
Gravitating Bubbles of Gluon Plasma above Deconfinement Temperature. *Symmetry* **2020**, *12*, 1668.
https://doi.org/10.3390/sym12101668

**AMA Style**

Brihaye Y, Buisseret F.
Gravitating Bubbles of Gluon Plasma above Deconfinement Temperature. *Symmetry*. 2020; 12(10):1668.
https://doi.org/10.3390/sym12101668

**Chicago/Turabian Style**

Brihaye, Yves, and Fabien Buisseret.
2020. "Gravitating Bubbles of Gluon Plasma above Deconfinement Temperature" *Symmetry* 12, no. 10: 1668.
https://doi.org/10.3390/sym12101668