**4. Q-Holes**

Q-holes are configurations of the scalar field behaving like (11) but such that |*φ*(*r* = 0)| > 0 and |*φ*(*r* → ∞)| = *φc* > |*φ*(*r* = <sup>0</sup>)|, with *φc* a local minimum of the potential under study [20]. Are *ω* = 0 Q-holes solutions worth being constructed within the present approach?

A necessary condition for Q-holes to exist is that the second maximum of the effective potential (18) is lower than the maximum at the origin [38]. As illustrated in Figure 1, it can only happen below *Tc* in our model since *Veff*(|*φ*|) = −12*U*(|*φ*|) at vanishing *ω*. A configuration where *φ* is everywhere nonzero at a temperature below the deconfinement one is not physically relevant; although the problem is

interesting from a technical point of view we have thus to discard Q-hole solutions in the present work. The same conclusion holds for the parameterization (17).

#### **5. Boson Stars**

The action (8) leads to the Einstein equation

$$G\_{\mu\nu} = \frac{\alpha}{2} T\_{\mu\nu} \tag{19}$$

where the energy-momentum tensor is given by *<sup>T</sup>μν* = ( *Dμφ*)∗(*Dνφ*)+( *Dμφ*)( *Dνφ*)∗ − *<sup>g</sup>μν*(*<sup>D</sup>αφ*)∗(*Dαφ*) + *gμν* L. The metric defines *ds*<sup>2</sup> = *<sup>g</sup>μνdxμdx<sup>ν</sup>* and *<sup>D</sup>μ* is the covariant derivative. An estimation of this coupling constant *α* in the temperature range under study is 16 *π G Nl* 2 *phys T*<sup>4</sup> *c* = 3.58 × 10−38. It is so small that no significant change of the solutions can be observed by numerical investigation. To appreciate more clearly the influence of gravity on the system we will construct solutions with *α* = 0.01 and 1.

We search for boson-star solution. First, the ansatz (11) and the boundary conditions are kept for the scalar field. Second, we choose a spherically symmetric ansatz for the metric:

$$ds^2 = -f(r)dt^2 + \frac{l(r)}{f(r)}\left(dr^2 + r^2d\theta^2 + r^2\sin^2\varphi \, d\varphi^2\right),\tag{20}$$

with the boundary conditions *f*(*r* = + ∞) = *l*(*r* = + ∞) = 1 (asymptotically flat space-time) and *f* (*r* = 0) = *l* (*r* = 0) = 0 (no singularity at origin). The gravitational mass *MG* of these gravitating objects is defined as usual according to *f*(*r* → ∞) ∼ 1 − 2*MGGN r* . The explicit equations involving *φ*, *f* and *l* can be found in Appendix B of [34]; we do not recall them here for the sake of simplicity. The same numerical method as for Q-balls is used to construct boson-star solutions [33].

Even for such large values as *α* = 1, our results indicate that gravitating solutions with *ω* = 0 still exist on roughly the same interval of *T*/*Tc*, see Figure 5. However, the numerical analysis turns out to be tricky in the limit *T* → *Tc* likely because the local minimum of the potential disappears. Our results strongly sugges<sup>t</sup> that the gravitational mass and mean radius increase considerably in the limit *T* → *Tc* as shown by Figure 5, similarly as what is observed in the Q-ball case. As expected, the metric gets more deviated from the Minkowski metric in the central region of the soliton: for instance *g*00 1 (see the blue line of Figure 5) and one can expect an essential singularity of the metric to be formed at *Tc*.

The profiles of the solution corresponding to *T*/*Tc* = 1.01 are presented in Figure 6 for *α* = 1 (solid lines). This plot clearly demonstrates that the soliton splits the space into two distinct regions: an interior region where *φ* is practically constant and strongly curving space-time and a region with *φ* ∼ 0 where space-time is essentially Minkowski. These regions are separated by a "wall" of the scalar field. The profile of a solution at an intermediate temperature *T*/*Tc* = 1.11 is also shown in Figure 6; the same qualitative features are observed. The boson star finally presents different features for *T*/*Tc* → 1.21, i.e., the limit of vanishing *<sup>m</sup>*(*T*). In this limit the scalar field approaches uniformly the null function and the Minkowski space-time is approached.

The existence of boson-star configurations for *α* = 1 implies the existence of such solutions for much smaller, "realistic", values of the coupling constant around *Tc*, see Figure 5. Since potential (17) will also lead to Q-balls above *Tc*, we can state that boson stars also exist with that parameterization. We choose however not to perform full numerical computations since potential (2) leads to a more accurate modelling of QCD equation of state as computed on the lattice.

**Figure 5.** Evolution of the mean radius < *R* > (black line, in units of *lphys*) and of the gravitational mass *MG* (red lines, in units of *Mphys*) as function of *Tc*/*T* for *ω* = 0 boson stars. The metric component *g*00 = *f*(0) is represented by the solid (resp. dashed) blue lines for *α* = 1 (resp. *α* = 0.1). These values depend very weakly on *α* and the curves are mostly superimposed.

**Figure 6.** Profiles of the metric functions *f* , *l* and of the scalar field *φ* for *α* = 1, *ω* = 0 and two values of the temperature : *T*/*Tc* = 1.01 (solid lines) *T*/*Tc* = 1.11 (dashed lines). The radial variable *r* is in units of *lphys*.

#### **6. Summary and Outlook**

We have built Q-balls and boson stars from a model with a complex scalar field plus a temperature-dependent *Z*3-symmetric potential mimicking Yang–Mills theory at finite temperature. We have shown that static Q-balls only exist between 1 and 1.21 *Tc* with a mean radius smaller than 10 fm and that they cannot have radial nodes. The solutions we find are spherically symmetric and the scalar field is such that |*φ*(*r* = 0)| = 0 and |*φ*(*r* → ∞)| = 0; they can be interpreted as "bubbles" of deconfined gluonic matter. We also showed that Q-holes solutions should be discarded from a physical point of view since they are solutions modelling a deconfined phase, but that can only exist below *Tc* within our approach. Static boson stars exist in roughly the same temperature range as Q-balls. Their qualitative features are almost independent on the value of the matter-Einstein gravity coupling constant *α*.

To our knowledge, it is the first time that boson stars are constructed from an effective potential such as (2). Typical potentials used in boson-star-related studies are such that solutions exist for 0 < *ωmin* ≤ *ω* ≤ *ωmax*, see i.e., [35]. It is worth pointing out that the potential used here even allows the existence of static solutions with *ωmin* = 0.

Computation of the QCD equation of state in curved space-time shows that the latter may affect the phase-diagram of the theory by increasing the splitting between the critical points for chiral and deconfinement transitions [21]. We hope to present generalizations of our boson-star configurations to the case of a nontrivial quark field in a future work; they could shed new light on the interplay between confinement, chiral symmetry and gravity.

**Author Contributions:** Conceptualization, F.B. and Y.B.; software, Y.B.; validation, F.B. and Y.B.; formal analysis, F.B. and Y.B.; writing—original draft preparation, F.B. and Y.B.; writing—review and editing, F.B. and Y.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
