**1. Introduction**

A fascinating feature of Yang–Mills theory is the existence of a deconfinement temperature, *Tc*, above which free color charges (free gluons) may propagate without being confined into color singlets [1,2]. This deconfined phase can be thought as a "gluon plasma", in analogy with the celebrated quark-gluonplasmaexperimentallycreatedfirstatRHIC[3,4],i.e.,thedeconfinedphaseoffullQCD.

In pure SU(*Nc*) Yang–Mills theory, deconfinement might be driven by the breaking of a global center symmetry, i.e., <sup>Z</sup>*Nc* symmetry [5,6], exemplified by the behavior of the Polyakov loop at finite temperature *T*. The Polyakov loop is defined as *<sup>L</sup>*(*<sup>T</sup>*,*y*) = 1*Nc* Tr*cP* e*i g* - 1/*T* 0 *dτA*0(*<sup>τ</sup>*,*y*), with *A*0 the temporal component of the Yang–Mills field and *y* the spatial coordinates. *P* is the path-ordering, *g* is the strong coupling constant and units where *h*¯ = *c* = *kB* = 1 are used. Since gauge transformations belonging to the center of the gauge group only cause *<sup>L</sup>*(*<sup>T</sup>*,*y*) to be multiplied by an overall factor, the Polyakov loop is such that the spatial average -*L* = 0 (= 0) when the <sup>Z</sup>*Nc* symmetry is present (broken), hence when the theory is in a (de)confined phase [2,7]. Hence -*L* is commonly seen as an order parameter for Yang–Mills theory at finite *T*, although it may not be the best physical candidate. As shown in [8], distinct <sup>Z</sup>*Nc* phases above *Tc* do not actually label different physical states. Moreover, the value −*T* ln -*<sup>L</sup>*, giving the free energy of a static color source in the heath bath, may take unphysical complex values above *Tc*. The real number |-*L*| may be a better candidate, with a free energy given by −*T* ln |-*<sup>L</sup>*|. It is also argued in [8] that the correlator *L*(*<sup>T</sup>*,*<sup>y</sup>*)*L*(*<sup>T</sup>*,-0)† at large distances may be a proper order parameter. Another relevant order parameter for pure Yang–Mills theory has finally been formulated in [9]: It is the spatial 't Hooft loop *V*(*C*) [10], *C* being a closed spatial contour. *V*(*C*) shows a perimeter law in the confined phase and an area law in the deconfined phase: *V*(*C*) ∼ exp(−*<sup>m</sup> P*(*C*)) and *V*(*C*) ∼ exp(−*<sup>α</sup> S*(*C*)) respectively, with *P* and *S* the perimeter and the area of the closed contour.

In view of the above results we can safely assume that there exists one scalar field *φ* playing the role of an order parameter. Moreover, there should exist a Z *Nc* temperature-dependent potential *<sup>V</sup>*(*φ*, *φ*<sup>∗</sup>, *<sup>T</sup>*), whose global minimum is different at *T* < *Tc* and *T* > *Tc*, reproducing the equation of state computed in lattice QCD [11] in the mean-field approximation—recall that the pressure reads *p* = − min*φ*,*φ*<sup>∗</sup> *V*. In the case *Nc* = 3, the *Z*3-symmetry should be present through terms in *φ*3 + *φ*∗<sup>3</sup> at the lowest-order in a power expansion of *V* [12]: The explicit form we will use is given in Section 2. In the following we go beyond mean-field theory and treat *φ* as a complex, position-dependent, scalar field: This dynamical field mimics the behavior of pure Yang–Mills theory at finite *T*. The value of |*φ*| being related to the phase of the gluonic matter, we can summarize the aim of our study as follows: We search for configurations describing localized regions of (de)confined gluonic matter, either in flat or curved 750/9.

On one hand we have already shown the existence of nontrivial solutions for *φ*(*y*), vanishing at infinity, at the deconfinement temperature in flat space-time and at large *Nc* [13]. On the other hand, nontrivial static configurations in *Z*3-symmetric potentials have already been found in [14,15]. Most of the effort in the field has been devoted to study the temporal evolution of such solutions in close relation with thermalization issues of experimentally observed quark-gluon-plasma [14,16–19]. In this work we search for spherically symmetric static Q-ball solutions with a focus on conditions constraining their existence: temperature range, radial nodes, etc. Less standard solitons as Q-holes [20], never studied up to now within that framework, are also discussed. Q-balls and Q-holes are discussed in Sections 3 and 5.

Finally, we couple our Z3-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*

Let us model SU(3) Yang–Mills theory at finite temperature by an effective Lagrangian based on a complex scalar field *φ* plus Z3−symmetry. We use the potential *V* of Ref. [26] which reads

$$\mathcal{U}(\phi, \phi^\*, T) = \frac{V(\phi, \phi^\*, T)}{T^4},\tag{1}$$

with

$$\mathcal{U}(\phi, \phi^\*, T) = -\frac{b\_2(T)}{2} |\phi|^2 + b\_4(T) \ln\left[1 - 6|\phi|^2 + 4(\phi^3 + \phi^{\*3}) - 3|\phi|^4\right],\tag{2}$$

and

$$b\_2(T) = 3.51 - 2.47 \frac{T\_c}{T} + 15.22 \left(\frac{T\_c}{T}\right)^2, \quad b\_4(T) = -1.75 \left(\frac{T\_c}{T}\right)^3. \tag{3}$$

We have retained the above parametrization because it to an optimal agreemen<sup>t</sup> with the equation of state of pure SU(3) Yang–Mills theory computed in lattice QCD [11], and also with the full

*Nf* = 2 lattice QCD equation of state at zero and nonzero chemical potential when coupled to a Nambu–Jona–Lasinio (NJL) model [26]. We note that the latter reference explicitly identifies the scalar field with the Polyakov loop. In fact, their result is more general since *φ* can be regarded as the actual order parameter of the model, not necessarily the Polyakov loop.

Potential (1) is displayed in Figure 1 for the values (3) of the parameters and for several temperatures. The change in minimum is clearly seen above and below *Tc*: A (non)vanishing value for |*φ*| gives the minimum of *U* in the (de)confined phase. We notice that potential (1) is only Z3-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 |*φ*|<sup>6</sup> − 2|*φ*|<sup>4</sup> + *b*|*φ*|<sup>2</sup> [27]. A U(1)-symmetry can be recovered in the large-*Nc* limit of <sup>Z</sup>*Nc* -symmetric potentials, see [13,28].

**Figure 1.** The potential *<sup>U</sup>*(*φ*, *φ*<sup>∗</sup>, *T*) versus *φ* for various temperatures. *U* is given by Equations (2) and (3), and the plot is restricted to *φ* ∈ R for the sake of clarity.

According to the suggestion of e.g., Ref. [29], we choose for *φ* a kinetic part of the form *<sup>N</sup>*2*c <sup>T</sup>*<sup>2</sup>*∂μφ∂μφ*<sup>∗</sup>/*λ*, which has both the correct energy dimensions and the expected *Nc*-scaling when the gauge group SU(*Nc*) is chosen. *λ* is the 't Hooft coupling. Minkowski metric has signature (+ − −−). In our SU(3) case, recalling that *αs* = *<sup>λ</sup>*/(<sup>12</sup>*π*), we can write our Lagrangian as

$$\mathcal{L}\_{\text{phys}} = \frac{3T^2}{4\pi\alpha\_s} \partial\_{\mu}\phi \partial^{\mu}\phi^\* - T^4 \mathcal{U}(\phi, \phi^\*, T), \tag{4}$$

where *φ* = *φ*(*y<sup>μ</sup>*), *yμ* are the space-time coordinates. It is convenient to further define dimensionless variables *xμ* related to the original (physical) ones by

$$l\_{\mu}y^{\mu} = l\_{\text{phys}}x^{\mu}, \quad \text{with} \quad l\_{\text{phys}} = \frac{\sqrt{3}}{T\sqrt{4\pi\alpha\_{s}}},\tag{5}$$

so that the above Lagrangian can be replaced by the dimensionless one

$$\mathcal{L} = \frac{\mathcal{L}\_{\text{phys}}}{T^4} = \partial\_{\mu} \phi \partial^{\mu} \phi^{\*} - \mathcal{U}(\phi, \phi^{\*}, T), \tag{6}$$

where *φ* = *φ*(*x<sup>μ</sup>*) and where *T* is expressed in units of *Tc*.

It is worth estimating the physical length used in the model. First, a typical value for the deconfinement temperature in pure gauge QCD is *Tc* = 0.3 GeV [30]. Second, a way to estimate *αs* is to note that the short-range part of the static interaction between a quark and an antiquark scales as <sup>−</sup>(4/3)*<sup>α</sup>s*/*<sup>r</sup>*, at least from *T* = 0 to *Tc*. Lattice studies, performed at *Nc* = 3, favor *αs* = 0.2 up to *T* = *Tc* [31], which is the value we retain here. We are then in position to estimate that, at *T* = *Tc*,

$$d\_{\rm phys} = 3.6 \,\text{GeV}^{-1} = 0.72 \,\text{fm}.\tag{7}$$

#### *2.2. Coupling to Einstein Gravity*

The coupling of the above Lagrangian to gravity can be performed by minimally coupling the scalar field to Einstein gravity: The action reads

$$S = \int d^4x \sqrt{-g} \left(\frac{R}{a} + \mathcal{L}\right),\tag{8}$$

with the effective coupling constant

$$
\kappa = 16\pi G\_N l\_{phys}^2 T^4. \tag{9}
$$

The replacement of the partial derivatives by covariant ones in (6) must be performed.
