**3. Q-Balls**

#### *3.1. Ansatz and Existence Conditions*

We begin by considering Lagrangian (6) where *φ*3 and *φ*∗<sup>3</sup> are replaced by |*φ*|<sup>3</sup> in order to recover the usual U(1)-symmetry needed to build Q-balls solutions.

The classical equations of motion in flat space-time with potential *<sup>U</sup>*(|*φ*|, *T*) = −*b*2(*T*) 2 |*φ*|<sup>2</sup> + *b*4(*T*)ln 1 − 6|*φ*|<sup>2</sup> + 8|*φ*|<sup>3</sup> − 3|*φ*|<sup>4</sup> read

$$
\partial\_{\mu}\partial^{\mu}\phi = \partial\_{\phi^{\*}}\mathcal{U} = -\frac{b\_{2}(T)}{2}\phi + 6b\_{4}(T)\frac{-\phi + 2|\phi|\phi - |\phi|^{2}\phi}{1 - 6|\phi|^{2} - 3|\phi|^{4} + 8|\phi|^{3}}\tag{10}
$$

plus the complex conjugated equation. We then perform the usual Q-ball ansatz on the scalar field :

$$\phi = \exp(i\omega t)\phi(r),\tag{11}$$

where *t* = *x*0 and where *r* = (*x*<sup>1</sup>)<sup>2</sup> + (*x*<sup>2</sup>)<sup>2</sup> + (*x*<sup>3</sup>)2. The solutions we will build can be characterized by their mas *M* and by a dimensionless conserved charge *Q*, respectively defined by

$$M = M\_{\text{phys}} \int d^3 \mathbf{x} \,\, T\_{00} \tag{12}$$

with *Mphys* = 1/*lphys* and

$$Q = 2\omega \int d^3 \mathbf{x} \, |\phi|^2. \tag{13}$$

The temporal component of the energy-momentum tensor represents the energy density, given by

$$T\_{00} = \omega^2 |\phi|^2 + \vec{\nabla}\phi \cdot \vec{\nabla}\phi^\* + \mathcal{U}(|\phi|). \tag{14}$$

The conserved charge *Q* finds its origin in the (artificially restored) U(1)-symmetry of the considered Lagrangian, leading to a conserved Noether current of the form *Jμ* = *i*(*φ∂μφ*∗ − *φ*<sup>∗</sup>*∂μφ*), *Q* being the space integral of *J*0. Axially symmetric solutions with *k* = 0 are spinning Q-balls whose angular momentum *J* is related to the charge *Q* according to *J* = *kQ* [27]. Here we focus on non-spinning Q-balls.

We have studied the equations for generic values of *ω* although it is clear that only the solutions corresponding to *ω* = 0 are physically relevant for the potential under consideration: The original potential is Z3-symmetric, not U(1). Note also that if *φ*(*r*) is a real solution of the equations of motion, e *ikπ* 3 with *k* ∈ Z is also a solution because of the system's symmetry.

The mass term of the potential plays a crucial role in the existence of the solutions. In a power expansion in |*φ*|,

$$
\hbar L(|\phi|, T) = m^2(T)|\phi|^2 + \text{\textquotedbl{}higher order\textquotedbl{}}\text{, with }m^2(T) = -\frac{b\_2(T)}{2} - \hbar b\_4(T). \tag{15}
$$

General results on Q-balls [27] state that the soliton exist for *ωmin* ≤ *ω* ≤ *ωmax* with

$$
\omega\_{\min} = \min\_{|\phi|} \frac{\mathcal{U}(|\phi|, T)}{|\phi|^2} \quad , \ \omega\_{\max} = m(T). \tag{16}
$$

In particular, if the potential *U*(|*φ*|) is negative in some interval of values of |*φ*|, the value *ω* = 0 belongs to the spectrum of the boson star. This turns out to be the case for *T* > *Tc*. The condition *m*(*T*)<sup>2</sup> > 0 also needs to be fulfilled; in terms temperature, this corresponds to *T*/*Tc* > 1.21. As a consequence, the general properties of Q-balls solutions sugges<sup>t</sup> that Q-ball solutions with zero frequency will exist for 1 < *T*/*Tc* < 1.21.

The parametrization (2) of the potential is not unique. In particular, a power expansion of the form

$$
\Delta I = -\frac{b\_2}{2} |\phi|^2 - \frac{b\_3}{6} (\phi^3 + \phi^{\*3}) + \frac{b\_4}{4} |\phi|^4 \tag{17}
$$

is often used in effective YM theories at finite *T*, the choice *b*2 = 6.75 − 1.95(*Tc*/*T*) + 2.63(*Tc*/*T*)<sup>2</sup> − 7.44(*Tc*/*T*)3, *b*3 = 0.75, *b*4 = 7.5 leading to a good agreemen<sup>t</sup> with lattice QCD data [32]. Using this alternative choice would not forbid the existence of Q-ball solutions: The mass term *b*2 is positive above *Tc* and the criterion (16) leads to *ωmin* < 0, so *ω* = 0 solitons are allowed also in this case.

#### *3.2. Numerical Results*

A numerical resolution of the equations (10) can now be performed. We use a collocation method for boundary-value ordinary differential equations, equipped with an adaptive mesh selection procedure [33]. The regularity of the solution at the origin implies *dφdr* (*r* = 0) = 0, the finiteness of the energy and the charge impose *φ*(∞) = 0. These are the boundary conditions.

We present in Figure 2 the spectrum of the Q-balls for *T*/*Tc* = 0.83, 1.01, 1.18. It can be observed that no Q-ball solution with *ω* = 0 can be found below *Tc*: It is a nice feature of our model that it does not lead to solutions modelling deconfined matter below *Tc*. In the range 1 < *T*/*Tc* < 1.21 suggested by the above analysis however, such solutions can be found. From now on, we concentrate on the latter *ω* = 0 solutions. Our results are summarized in Figures 3 and 4.

Our numerical analysis shows that in the limit *T*/*Tc* → 1.21 the scalar function *φ*(*r*) approaches uniformly the null function as expected by the existence criterion discussed before. The limit *T* → *Tc* reveals a peculiar behavior of the solitons: Their mean radius and mass increase considerably. In this limit the scalar function *φ*(*r*) is closer and closer to a nonzero constant solution, leading to the observed increase in mass and mean radius.

All the Q-balls we find have a mean radius smaller than 14× *lphys* = 10 fm and are lighter than 140 × *Mphys* = 38.9 GeV. Similar solutions were found in [14] with a simpler, power-law, *Z*3−symmetric potential of the form |*φ*|<sup>2</sup> − *a*(*φ*<sup>3</sup> + *φ*∗<sup>3</sup>) + *b*|*φ*|4. At *T* = 1.1 *Tc* they find a soliton with a typical size of 1.5−2 fm while we find a Q-ball with mean radius 1.43 fm and mass 200 MeV at the same temperature. Other results are obtained in [14] but in 2 + 1 dimensions so they cannot be compared to ours.

We have tried to construct radially excited solutions, i.e., solutions where the radial function presents one or more nodes, but so far, we cannot find any. The absence of solutions presenting nodes

for our model can be explained by the following argument. In general, the existence of node solutions is closely related to the shape of the effective potential

$$V\_{eff}(|\phi|) = \frac{\omega^2}{2} |\phi|^2 - \frac{1}{2} \mathcal{U}(|\phi|) \,. \tag{18}$$

Several conditions are necessary for node solutions to exist [27,34]: (i) *φ* = 0 should be a local maximum of *Veff* , (ii) the effective potential should admit local minima for both signs of *φ*. It would be challenging to have a generic proof of the absence of node solutions with our potential but an inspection of the potential *U*(|*φ*|) quickly reveals that no local minimum exist for *φ* < 0 when *φ* ∈ R (see Figure 1), so the condition (ii) cannot be fulfilled when *ω* = 0.

**Figure 2.** Relation between *ω* and Ω ≡ *m*<sup>2</sup>(*T*) − *ω*<sup>2</sup> versus *φ*(0) for three values of *T*/*Tc* in flat space-time (*α* = 0) (solid lines). The dotted lines represent *ω* and *gtt*(0) in the case *T*/*Tc* = 1.01 for gravitating solutions (*α* = 1).

**Figure 3.** Profiles of *φ*(*r*) for several values of *T*/*Tc*. Distances are in units of *lphys*.

The spectrum of the fundamental Q-ball appears quite different with the present potential than in more conventional U(1)-symmetric potentials. For example, one of us previously studied Q-balls with the SUSY-inspired potential *USUSY*(|*φ*|) ∼ 1 − exp −|*φ*|<sup>2</sup> *η*2 ), with *η* ∈ <sup>R</sup>+0 [35]. In contrast to our potential, solutions can be constructed for arbitrarily large values of the central density *φ*(0) with the latter potential, and radially excited solitons can be obtained. We considered an effective potential consisting of a linear superposition of our potential and the SUSY-potential. The latter is known to

admit node solutions: *Ueff* = cos(*y*) *USUSY* + sin(*y*) *U* with *y* ∈ [0, *<sup>π</sup>*/2]. It turns out that when we progressively deform the SUSY-potential into our potential (say with a fixed value *φ*(0) < 1) the zero-node solutions ge<sup>t</sup> continuously deformed and the frequency *ω* decreases with increasing the mixing parameter *y*. By contrast, for the one node solution, the *ω* quickly reaches *ω* = 1 and the solution becomes oscillating.

**Figure 4.** Evolution of the mean radius -*R* as a function of *T*/*Tc*, the insert contains *φ*(0) and the mass (12) of the scalar field. Distances are in units of *lphys*and masses are in units of *Mphys*.

#### *3.3. Symmetry Breaking*

An obvious outlook is to include the matter sector of QCD in our approach. As discussed in [12], an immediate effect of quarks is the breaking of *Z*3−symmetry in the potential and the simplest way to mimic that symmetry breaking is to add a term proportional to (*φ* + *φ*∗) or even (*φ* + *φ*∗)<sup>2</sup> to the potential. A more rigorous treatment of quark fields may be achieved by coupling *φ* to a NJL Lagrangian [15,36]. Within mean-field approximation for quarks, it has been show in [37] that complex-valued solutions of Q-ball-type still exist with broken *Z*3-symmetry. A question arising at this stage is therefore: May the *ω* = 0 Q-balls we constructed "survive" to such a symmetry breaking?

We propose to perform the substitution *U* → *Uβ* = *U* + *β* (*φ* + *φ*∗)<sup>2</sup> with *β* a real constant parameter. This ansatz breaks the *Z*3-symmetry while still allowing analytical calculations. In a power expansion in *φ* on the real axis, the *β*-term shifts the mass term: *m*<sup>2</sup>(*T*) → *m*<sup>2</sup>(*T*) + 4*β*. Graphical inspection of the modified potential shows that there is always an interval of *φ* values in which *Uβ* is negative above *Tc* if *β* < 0. Nevertheless a negative value of *β* lowers *m*<sup>2</sup> and therefore lowers the maximal temperature at which Q-balls may be expected (*m*<sup>2</sup> is indeed a decreasing function of *T*). If −0.5 < *β* < 0, there always exists an interval of temperatures above *Tc* for which the existence of Q-balls is guaranteed. We can thus safely assume that the solutions we find will not necessarily disappear in a more realistic theory including quarks.
