Gauged Q-ball dark matter through a cosmological first-order phase transition

As a new type of dynamical dark matter mechanism, we discuss the stability of the gauged Q-ball dark matter and its production mechanism through a cosmological first-order phase transition. This work delves into the study of gauged Q-ball dark matter generated during the cosmic phase transition. We demonstrate detailed discussions on the stability of gauged Q-balls to rigorously constrain their charge and mass ranges. Additionally, employing analytic approximations and the mapping method, we provide qualitative insights into gauged Q-balls. We establish an upper limit on the gauge coupling constant and give the relic density of stable gauged Q-ball dark matter formed during a first-order phase transition. Furthermore, we discuss potential observational signatures or constraints of gauged Q-ball dark matter, including astronomical observations and gravitational wave signals.


Introduction
Exploring the nature of dark matter (DM) is one of the central issues in (astro)particle physics and cosmology [1].So far, there are no expected signals of conventional DM candidates like Weakly Interacting Massive Particles (WIMPs) in the DM direct detection and collider search experiments [2][3][4].Then simple WIMP scenarios are strongly disfavored.There are a number of ways to save WIMP scenarios.For example, DM may have substantial couplings only to the 3rd generation fermions [5][6][7], or dark sectors may consist of two or more stable DM species (see, for example, [8]).Or one can discard WIMP scenarios and consider other possibilities for DM productions and annihilations or decays in the early Universe.This status motivates us to study ultralight or ultra heavy DM candidate (for reviews, see Refs.[9,10]).
Solitons produced in the early Universe are natural candidates of heavy DM (see, for example, Ref. [11] for hidden sector monopole DM accompanied by stable spin-1 vector DM and massless dark radiation, and Ref. [12] for hunting for topological DM using atomic clocks).These solitons are specific field configurations which are classified into two classes, namely, the topological solitons and the non-topological solitons.Recently, as renaissance of the quark nuggets DM proposed by Witten [13], various new ideas on the non-topological soliton DM are proposed, where the DM relic density can be produced by the dynamical process of cosmological first-order phase transition (FOPT), such as the Q-ball DM [14][15][16][17].These new mechanisms can naturally avoid the unitarity problem for heavy DM [18].Dynamical DM mechanisms are specified by the DM penetration behavior into the bubble which depends on the DM mass and bubble wall velocity [19][20][21].Phase transitions in the early Universe can also be the source of primordial black holes [22][23][24].
There are extensive discussions on the non-topological solitons in a theory of complex scalar field with global U (1) symmetry, proposed in [25] and known as Q-balls [26].And it is natural to study the Q-balls in the gauged case [26][27][28][29][30][31][32][33][34], by promoting the global U (1) symmetry to the local gauge U (1) symmetry.For reviews of U (1) gauged Q-balls, see [35][36][37].Q-balls have been proposed as a potential DM candidate in supersymmetric theories [38,39].They can also explain the baryon asymmetry of the Universe [40].The gauged Q-ball DM in supersymmetry model has been studied in several papers [41,42].It is meaningful to search for other production mechanisms of Q-ball or gauged Q-ball DM without supersymmetry.In this paper, we study whether the U (1) gauged Q-balls produced during cosmic phase transition could be a viable DM candidate.If the gauged Q-balls can be stable under certain circumstances, we still need some mechanism to (1) produce the charge asymmetry (i.e.locally produce lots of particles with the same charge to form Q-ball) (2) and packet the same sign charge in the small size after overcoming the Coulomb repulsive interaction.For the first condition, the primordial charge asymmetry could be produced by some early Universe processes such as decays of heavier particles.Cosmological FOPT can naturally realize the second condition and can produce phase transition gravitational wave (GW) which can be detected by future GW experiments, such as LISA [43], TianQin [44,45], Taiji [46], BBO [47], DECIGO [48], and Ultimate-DECIGO [49].
In this work, for the first time, we study the natural production mechanism of gauged Q-ball DM through a cosmological FOPT.The paper is organised as follows.We describe the basic model that can produce the gauged Q-balls and the numerical solutions of the Q-ball profiles in section 2. Basic properties and the stable parameter space of gauged Qballs are discussed in section 3. Thin-wall approximation and the corresponding analytic evaluations are given in section 4. Phase transition dynamics in the Standard Model (SM) plus an extra singlet and the relic density of gauged Q-ball DM are elucidated in section 5. Signals and constraints of gauged Q-ball DM are given in section 6. Concise conclusions and discussions are given in section 7.
2 Gauged Q-ball 2.1 Friedberg-Lee-Sirlin-Maxwell model In this work, we adopt the Friedberg-Lee-Sirlin two-component model [30] 1 plus gauge component, which is called Friedberg-Lee-Sirlin-Maxwell (FLSM) model [29].This model and the corresponding stability of U (1) gauged Q-ball have been discussed in [27,29,[52][53][54].We begin our discussions with the following Lagrangian density where the potential V (ϕ, h) reads ϕ and h are the complex scalar field and (real) Higgs field respectively.D µ = ∂ µ +ig Ãµ and Ãµν = ∂ µ Ãν − ∂ ν Ãµ where Ãµ is a dark U (1) gauge field and g is the corresponding gauge coupling constant.Ãµ can be identified as the dark electromagnetic field.We fix the Higgs mass m h = 125 GeV and vacuum expectation value v 0 = 246 GeV at zero temperature then λ h = m 2 h /(2v 2 0 ) ≈ 0.13.The complex scalar ϕ gains mass through the portal coupling with the Higgs.In the true vacuum, m ϕ = λ ϕh 2 v 0 .We assume λ ϕh > 0, and thus the Lagrangian density is symmetric under the dark U (1) symmetry which remains unbroken when the Universe goes through the electroweak phase transition.The local U (1) gauge symmetry leads to the conserved current, and the corresponding conserved charge, Once the gauged Q-balls are formed in this FLSM model, one could consider a coherent configuration of ϕ, h, and Ãµ at a given charge Q.The lowest energy state will have no "magnetic field" so the space component Ãi = 0 [27,29].We assume spherical symmetry for the lowest energy configuration.Scaling away the physical dimensions, we introduce dimensionless field variables A, Φ, and H defined in the convention of Ref. [29].
where ρ ≡ √ 2λ h v 0 r = m h r.The Lagrangian, with the substitution of the field variables defined above, becomes where . By varying L with respect to A, Φ, and H, we find the equations of motion (EoM) for the three fields, ) The total energy is given by where And the total charge is given by (2.11) From Eqs. (2.7) and (2.11) we see that Therefore we get, for large ρ, A → λ h Q 2πρ or Ãt → gQ 4πr .Then Eq. (2.8) at large ρ becomes It has been shown in Ref. [36] for ν < k this equation at ρ → ∞ has the solution of the form, where C U is a constant and U (a, b, z) is the confluent hypergeometric function of the second kind.For We can see that in the limit g → 0 this form coincides with the nongauged global Qball, Φ(ρ The difference is caused by taking into account the dark electromagnetic potential A(ρ).For ω = m ϕ or ν = k, Eq. (2.13) takes the form of The solution to this equation reads Φ , where C K is a constant and K 1 (z) is the modified Bessel function of the second kind.For large ρ this solution has the form of Φ(ρ) ∼ ρ − 3 4 e − 2kg 2 Q π ρ which also differs from the global case, in which one expects Φ(ρ) ∼ 1 ρ for ν = k.For the case ν > k, it can be seen from Eq. (2.13) that the corresponding solutions for the complex scalar field are oscillatory at ρ → ∞, leading to unexpected infinite charge and energy.These solutions are unphysical and should be discarded.
It is convenient to write Eq. (2.7) in the form of (2.17) Suppose that A(0) > ν/α 2 .Eq. (2.17) then implies that ρ 2 ∂ ρ A is an increasing function of ρ such that ∂ ρ A > 0 and A(ρ) > ν/α 2 for all ρ > 0. This possibility is not acceptable, given that ν > 0 and A(ρ) → 0 at ρ → ∞.The only acceptable possibility is that A(0) ≤ ν/α 2 .Then ∂ ρ A < 0 and therefore A(ρ) is a monotonically decreasing function of ρ.We can then say that A(ρ) obeys the inequalities The energy integral Eq. (2.10) can be written in a different form.Demanding that the Lagrangian (2.6) is stationary at ϵ = 1 under the rescaling of the form ρ → ϵρ, leading to the relation dL(ρ → ϵρ)/dϵ| ϵ=1 = 0, this gives the relation: where in the third line we have integrated by part and use the fact that and A → 0 at large ρ.It can be seen that the energy for the free field solution when we neglect the variation of Φ and H takes the form The "dark electrostatic energy" is roughly proportional to R for charge uniformly distributed on scale R.

Numerical results of the field configuration, energy, and charge
After qualitative analysis of the gauged Q-ball solution, we begin to numerically solve Eqs.(2.7), (2.8), and (2.9) with the following boundary conditions, The first boundary condition is necessary so that the terms do not become singular at ρ = 0, and the latter is necessary because the energy density E and charge density (ν − α 2 A)Φ 2 should be integrable over the infinite spatial volume and the integral should be finite.
It is convenient to introduce the dimensionless quantities for the total energy and charge of the gauged Q-ball, which can be calculated directly once the numerical solutions of the corresponding differential equations are found.We use the relaxation method [55,56] to solve coupled 2nd-order ordinary differential equations, Eqs.(2.7), (2.8) and (2.9), with boundary conditions, Eq. (2.22).The relaxation method solves the boundary value problems by updating the trial functions on the grid in an iterative way.As an example, we fix k = 3.5 and we scan over all of the solutions at a given value of α.The results can be seen in figure 1.The frequency ν firstly decreases in the direction of the arrow.We call this the "first branch" where the backreaction of gauge field is small.Then the solutions turn on the "second branch" as ν increases and the gauge field dominates.Contrary to the global Q-balls, the parameter ν does not uniquely determine the charge and energy of the gauged Q-balls.For the global case, the energy and charge increase as the ν approaches zero.However, in the case of gauged Q-balls, the ν is replaced by (ν − α 2 A).Then on the second branch where the gauge field A dominates, ν has to increase in order to satisfy (ν − α 2 A) > 0.
We choose four specific solutions P 1, P 2, P 3, P 4 in figure 1 for α = 1.0 which are marked by the purple triangles and the corresponding numerical profiles of A, Φ and H are shown in figure 2. We can see that as the value of A(0) of the gauged Q-ball becomes larger, the Higgs field value inside the Q-ball is closer to zero.Actually, when the value of A(0) increases, the radius, charge and energy also increase.In other words, the Higgs value is effectively zero inside for large gauged Q-balls.
The total charge of gauged Q-balls for different values of gauge coupling α is shown in the left panel of figure 3. It can be seen that the charge of the gauged Q-balls is also finite at a nonzero α whereas for the global Q-balls the charge is unbounded from above.In order to obtain gauged Q-balls with relatively large charge, the gauge coupling has to be small enough.
We can define two typical radii for the Higgs field H(ρ) and the Q-ball field Φ(ρ) respectively.The ρ ⋆ is defined by H(ρ ⋆ ) = 1/2 and ρ b is defined by Φ(ρ b ) = Φ(0)/2.These two radii are shown in the right panel of figure 3. We can see that ρ ⋆ is generally larger than ρ b .One may prefer to call ρ ⋆ the radius of the "Higgs ball".Hereafter, we use ρ ⋆ to represent the Q-ball radius which defines the charge and energy of gauged Q-balls.A(ρ)  Here we choose the marked points P 1, P 2, P 3, P 4 in figure 1 where k = 3.5 and α = 1.0.3 Basic properties of gauged Q-balls It is well known that for non-gauged global Q-balls the relations dE/dQ = ω holds.Here we will show that this also holds for gauged Q-balls in FLSM model.From Eq. (2.10), we have After integrating ∂ ρ Φ and ∂ ρ H by part and using Eqs.(2.8) and (2.9), we can get (3.2) In the third line we have integrated ∂ ρ A by part and used Eq.(2.7), then the integral vanishes.Finally,

Stability of gauged Q-balls
The stability of Q-balls is an important criterion to judge whether they can serve as the DM candidate.Unlike the global Q-balls, the stability of gauged Q-balls is still being discussed.In this subsection, we will systematically analyze four stability criteria of gauged Q-balls and show the viable parameter space of stable gauged Q-balls.

Quantum mechanical stability
The quantum mechanical stability is satisfied if This means that the gauged Q-balls are stable against decay to free scalar particles.It should be noted that if the Q-ball has decay channels into other fundamental scalar particles which have the mass m i that are smaller than m ϕ , we need to replace m ϕ by m i in Eq. (3.5).
The decay of global Q-balls or gauged Q-balls has been discussed in several works [40,[57][58][59].When the effective energy of DM particles inside the gauged Q-balls π/r ⋆ ∝ (ω − g Ãt ) is larger than the masses of decay products, the decay process is kinetically allowed.If the Q-ball radius r ⋆ is large enough or (ω − g Ãt ) is small, the gauged Q-balls do not decay into other daughter particles and thus are stable .
The ratio Ẽ/ Q is shown in the figure 4. As the frequency ν → k on the first branch, there is a region of parameter space where Ẽ/ Q > 1.It implies the existence of a minimal Q-ball charge Qs defined as Ẽ(Q s )/ Qs = 1 of the quantum mechanically stable gauged Q-ball.We can see from figure 4 that for the global Q-balls the Ẽ/ Q decreases with growing Q when Ẽ/ Q < 1.The branch of Ẽ/ Q > 1 corresponds to ν → k.So the global Q-balls are quantum mechanically stable as ν ≪ k and Q > Qs .One would wonder that the gauged Q-ball will destroy the quantum mechanical stability at large charge because the dark electrostatic energy is proportional to Q2 and thus the Ẽ/ Q is proportional to Q.However, we found the gauged Q-balls are always quantum mechanically stable on the second branch where the gauge field dominates because of the charge of the gauged Q-balls must be finite.

Stress stability
Now we investigate the effects of the electrostatic repulsion on the stability of the gauged Q-balls.In Ref. [53], the authors pointed out that, just like the hadrons, a necessary condition for stability of the configuration of gauged Q-balls is the balance of the internal forces, called von Laue condition [60,61] ∞ 0 drr 2 p(r) = 0 . (3.6) Here p(r) is the radial distribution of the pressure inside the Q-ball, which can be extracted from the energy-momentum tensor by using the following parametrization [62,63]: s(r) is the traceless part which yields the anisotropy of pressure (shear forces).This kind of stability has also been studied for global Q-balls [63,64].
One stronger local criterion is that the normal force per unit area acting on an infinitesimal area element at a distance r, must be directed outward [64,65], This is a necessary but not sufficient condition for stability.By using the rescaled parameters, we have the expressions of p(ρ) and s(ρ) in the FLSM model: ) from which we get (3.10)We briefly discuss the stress stability of global Q-balls by taking α → 0 in Eq. (3.10).For global Q-balls, we have then we can get (3.12)The EoM for Φ and H in the FLSM model for α = 0 read Substituting Eq. (3.13) into Eq.(3.12), we obtain This implies that F (ρ) is a decreasing function of ρ.Because F (ρ) must satisfy F (ρ) → 0 at large ρ, we find that F (ρ) is always positive.Then the global Q-balls will be always stable under stress stability criterion.Based on this, it can also be expected that the gauged Q-balls are also stable under stress stability on the first branch where the backreaction from gauge field is small enough.
For gauged Q-balls, we can first consider one extreme case.At the end point where (ν − α 2 A) → 0 on the second branch, the Φ is finite and Then, from Eq. (3.10), we can see that as H ≈ 0 inside gauged Q-balls in the large ball (thin-wall) limit.This implies the gauged Q-balls with maximal values of gauge filed A = ν/α 2 inevitably break the stress stability.
We also expect the gauged Q-balls are unstable under stress stability criterion on the second branch where the gauge field dominates.We prove this by using the numerical calculations.In the left panel of figure 5, we show the profile of F (ρ) for four marked points in figure 1 where α = 1.0.We can see that the F (ρ) has no nodes for P 1 and P 2 on the first branch.On the other hand, negative values exist for F (ρ) on the second branch.The gauged Q-balls on the second branch where the gauge field dominates are unstable.F (ρ) behaves similarly to Eq. (3.15) at ρ = 0 when (ν − α 2 A) → 0, which can also be seen from the P 4 of figure 5.It should be noted that the inequality (3.8) takes an approximation that Q-balls behave as a continuous media, which needs more discussions.And in the right panel we plot the F (ρ) at the transition point between the first and second branch for different values of gauge coupling α.The values of α are set to be 0.0001 − 0.7.It can be seen that they are all unstable under stress stability criterion.This implies that the solutions on the second branch are also unstable where the gauged field becomes larger [53].The conclusion does not change qualitatively for different values of α.
The stress-stability is the most stringent stability criterion in this work.We are not sure if the gauged Q-ball on the second branch would decay into free particles or smaller Q-balls.And it is meaningful to explore that how we can get rid of the strong constraints from stress-stability, in other words, how to get large gauged Q-balls with large gauge coupling.Maybe we can consider an example of two-scalar case where the two scalars ϕ and ψ possess opposite charge.If the electrostatic fields produced by the two scalars cancels with each other, which implies with g and g′ being the gauge couplings of ϕ and ψ respectively, this guarantees the electric neutrality of the interior of the gauged Q-balls [66,67].The gauged Q-balls will avoid the electric repulsion which leads to the stress-instability even when for large gauge coupling.
We expect these Q-balls can also be the DM candidate because they will behave as the ordinary global Q-balls.

Stability against fission
For nongauged global Q-balls the corresponding stability criterion against fission takes the form This clearly leads to when E(0) = 0.However, it may not hold everywhere for the gauged Q-balls due to the presence of the gauge potential.In Ref. [35] the authors gave a detailed discussion on the stability against fission of U (1) gauged Q-balls.They pointed out that we could not make any conclusion about the stability against fission for gauged Q-balls on the second branch.Nevertheless, it has been shown that the gauged Q-balls on the first branch are generally stable because the backreaction of the gauge field is generally small.

Classical stability
The problem of classical stability of U (1) gauged Q-balls is discussed in detail in Ref. [68].
The classical stability criterion was firstly derived in [69] for one-field Q-balls and was discussed for the model with two scalar fields in [30].The proof of Refs.[30,69] was based on examining the properties of the energy functional of the system.Instead, the examination of Ref. [68] is based on the Vakhitov-Kolokolov method [70,71] which utilized linearized EoM for the perturbations above the background solution.
We only consider the spherical perturbations on the gauged Q-ball.We adopt the following ansatz: ϕ(t, r) = e −iωt f (r) + e −iωt e γt (u(r) + il(r)) , Ãt (t, r) = Ãt (r) + e γt a 0 (r) , h(t, r) = h(r) + e γt σ(r) , where f (r), Ãt (r) and h(r) are the background solutions.u(r), l(r), a 0 (r) and σ(r) are the perturbations on the background.Then we obtain the linearized EoM as below, The boundary conditions are We want to obtain the parameter γ which depicts the growth of perturbations u, l, a 0 , σ.The classically unstable mode corresponds to γ > 0. This can be done by using the shooting method in Ref. [72].We introduce four basis solutions Ψ (i=1,2,3,4) (r) = (u (i) , l (i) , a (i) 0 , σ (i) ) which satisfy the Neumann boundary conditions dΨ (i) /dr| r=0 = 0 and Dirichlet boundary conditions Ψ (1) (0) = (1, 0, 0, 0), Ψ (2) (0) = (0, 1, 0, 0), Ψ (3) (0) = (0, 0, 1, 0) and Ψ (4) (0) = (0, 0, 0, 1).Then we integrate the Eqs.(3.19) numerically to find the values of Ψ (i) at large r = r ∞ .Now, recall that we are searching for a specific solution Ψ(r Eq. (3.22) has nontrivial solutions only if D c ≡ det Dc = 0.In figure 6 we plot Log|D c | as a function of Log 10 γ where we choose α 2 = 10 −5 and k = 3.5.We found that there is no classically unstable mode for ν = 3.45 as there is no solution for γ > 0. This also holds for other solutions of U (1) gauged Q-balls except for ν → k on the first branch.We found there is one unstable mode for ν = 3.49.Actually, in the limit ν → k, the contribution of gauged field can be neglected and one expects that the case is similar to the non-gauged Q-ball where dQ dω > 0 indicates there exists classically unstable mode.This is also discussed in Refs.[30,68].However, we usually do not have to worry about this because the region already has been excluded by the quantum instability.In Ref. [54], the authors have shown that the gauged Q-balls on the second branch with small gauge coupling are classically unstable with respect to axisymmetric perturbations.This enhances our confidence that the gauged Q-ball on the second branch is unstable.The region is almost covered by the stress stability criterion.
In summary, the parameter space of gauged Q-balls is shown in figure 7. The red line represents the region where the gauged Q-ball is dominated by the gauge field such that it is unstable under stress stability criterion.The green line represents the region where the gauged Q-ball is unstable under quantum stability criterion.We only plot the quantum mechanical stability and the stress stability criterion because they cover the space where the gauged Q-balls are classically unstable and are unstable against fission respectively.The gauged Q-balls are stable only in the region of ν ∈ [ν min , ν max ].The ν min corresponds to the maximal charge of gauged Q-balls and is close to the transition point between first and second branches which is marked by the purple point.In order to form gauged Q-balls of given charge at given gauge coupling, the charge has to be smaller than the maximal charge.It should also be noted that, if the dark gauge boson of gauged Q-ball kinetically mixes with SM photons or Z bosons, this would produce distinct experimental signatures which can constrain Q-ball charge and couplings.

Thin-wall approximation
As the radii of gauged Q-balls become large, the width of profile can be neglected and the gauged Q-balls can be depicted by the thin-wall approximation.The Higgs profile can be approximately viewed as a step function, the vacuum value is equal to zero and v 0 inside and outside the Q-balls, respectively.The derivative of the Higgs field only contributes to the surface term of gauged Q-balls which is negligible when the radius of the ball is large.
Then the problems are reduced to those or ones with two fields, ϕ and Ãµ .We will discuss the simplified piecewise model and show that it behaves similarly to the FLSM model.By using the mapping method introduced by Ref. [73], we give some semi-analytic results and some analytic evaluations of the maximal charge of gauged Q-balls.

Piecewise model
If the Higgs field is approximately h(ρ) = v 0 Θ(ρ − ρ ⋆ ), then we can approximately view the complex scalar moving in the piecewise parabolic potential [25,74,75].The Lagrangian density can be further approximated as where Θ(x) is the Heaviside step function.In our case, Note that v is chosen so that V (ϕ) is continuous at ϕ † ϕ = v 2 .This can be understood from the EoM of the Higgs field in the FLSM model, where we use the definition ϕ(r, t) = f (r)e −iωt and the prime denotes a derivative with respect to r.If the Higgs field is approximately a step function, we can neglect the derivatives, then we have and we could assume f (r) ≈ 0 outside the bubble where 0 outside and inside the Q-ball, respectively.The consistency between the piecewise model and the Friedberg-Lee-Sirlin model has been discussed in Ref. [75] and the classical stability of gauged Q-balls in piecewise model has been studied in Ref. [68,74].The EoM of gauged Q-balls in the piecewise model after the rescaling of Eq. (2.19) are These equations are easier to solve than the FLSM model by using undershooting/overshooting method.The gauged Q-ball energy and charge reads, Here, which has a different form from the FLSM model.
We solve Eqs.(4.5) and (4.6) numerically to get the profiles of the complex field and the gauge field.Then we can get the charge and energy of gauged Q-balls by substituting them into Eqs.(4.7).These numerical results are shown in figure 8.We can see that the piecewise model fits the FLSM model well when the gauged Q-ball radius is large.The distinctions appear at small ρ ⋆ at which the Higgs field value inside is not approximately zero.

Mapping gauged Q-balls
We make a further assumption that the complex scalar field can also be viewed as a step function, Φ(ρ) = Φ 0 (1 − Θ(ρ − ρ b )), where we denote Φ 0 = Φ(0) and ρ b is defined by Φ(ρ b ) = Φ 0 /2.Then the profile of gauged field is [27] The Q-ball radius is defined by Φ(ρ ⋆ ) = 1 2k .In Ref. [73], the authors propose a mapping between the gauged Q-ball and the global Q-ball.Specifically, where ν g is the value of frequency for the global Q-ball with the same ρ b .Then the profile of A is given by Eq. (4.8).This relation holds even for cases beyond the thinwall approximation.Interestingly, the global cases in the piecewise model have analytic solutions [74]: ρ⋆ sin(νgρ) ) and A(ρ) from Eq. (4.8), we have further semianalytic results for charge: .12) In the limit α → 0 and αΦ 0 ρ b → 0, because x coth x ∼ 1 + x2 3 + O(x 3 ) for x → 0. Then we have Q ∼ ν g Φ 2 0 ρ 3 b .And in this case, when ν g → 0, from Eq. (4.11) we have ν g ρ ⋆ ≃ π and sin(ν g ρ ⋆ ) ≃ ν g /k, which lead to From 2 sin(ν g ρ b ) = ν g ρ b , we have ν g ∼ C 1 ρ b , and which is consistent with the global Q-ball case.Using the same procedure that derives Eq. (2.20), we have analytic results for total energy: The second term comes from the integration over discontinuous (∂ ρ Φ) 2 by using the approximation of energy conservation [76].
In the first limit of αΦ 0 ρ b → 0 and large ρ b , the last term of Eq. (4.15) vanishes.We then have ν ≃ ν g , Φ 0 ≃ π 2νg ≃ ρ⋆ 2 and ρ ⋆ ≃ ρ b , then which is just the energy of global Q-ball.
In the opposite limit αΦ 0 ρ b → ∞, because A → Q/ρ, then from Eq. (4.8), Thus from Eq. (4.15) the energy of gauged Q-ball is The first term is the Coulomb energy and the second term is the potential energy difference between inside and outside of the Q-ball.The second term is proportional to ρ 2 b because in this case the Compton wavelength of the gauge field 1 gv 0 Φ 0 inside the Q-ball is much smaller than Q-ball radius, r b = ρ b /m h .So the Q ball is superconducting.The potential energy is therefore zero inside as well as outside of the Q ball and is nonzero only in the shell around ρ b [27].
For a given ρ ⋆ , we can solve the Eq.(4.11) to get the corresponding ν g .After using Φ 0 = 1 2k νgρ⋆ sin(νgρ⋆) , 2 sin(ν g ρ b ) = ν g ρ b and Eq.(4.9) we can get Φ 0 , ρ b and ν of gauged Qballs.Substituting them into Eqs.(4.12) and (4.15) gives the charge and energy of gauged Q-balls.These semi-analytic results are shown by the red lines in figure 8.We find the mapping works well for ρ ⋆ ≫ 1 on the first branch where the profiles of scalar fields ϕ and h can be safely viewed as step functions.However, the semi-analytic results of energy do not work well for the second branch because Φ 0 ̸ = 1 2k νgρ⋆ sin(νgρ⋆) in this case.The value of Φ 0 is lower because the gauge potential dominates.This can be seen in figure 1.At the transition point between the first and the second branches, the discrepancy between semi-analytic results and numerical results is about a factor of O(1).

Maximal charge and energy of gauged Q-balls: analytic approximations
We give analytic evaluations of maximal charge and maximal energy of gauged Q-balls for a given value of gauge coupling.Because the gauged Q-balls should be unstable under stress stability criterion on the second branch where the gauge potential dominates and the ν min is very close to the transition point, the maximal charge is approximately defined by dν dρ⋆ ρ⋆=ρmax = 0.In the limit at large Q-ball with ν g → 0, we have ρ b = C 1 /ν g and Φ 0 ≃ ρ⋆ 2 .Then, from (4.9) we get where we used ρ ⋆ ≃ π/ν g when ν g → 0. Then from dν dρ⋆ ρ⋆=ρmax = 0 when the charge is maximal, we have then we can obtain C 1 αρ 2 max 2π = C 2 with C 2 ≈ 1.08866 and αΦ 0 ρ b ≃ C 2 which is somewhere in the middle of cases of Eq. (4.16) and Eq.(4.18).We can also get the minimal frequency and the corresponding frequency for the global case ν gmin = π ρmax = πC 1 α 2C 2 .We can see that ν min ν gmin = C 2 coth(C 2 ) ≈ 1.367 which is independent of the Q-ball size.This implies that as the solutions are unstable on the second branch, the gauged Q-ball lives on the first branch and is close to the global case.Finally, the maximal charge is given by from which we can see the charge is unbounded from above in the global case where α = 0.The maximal energy reads This also gives us Ẽmax ∝ ρ 3 max ∝ Q3/4 max which is expected in the global case.The analytic results of Q max = 2π λ h Qmax and E max = 2πm ϕ λ h Ẽmax are shown in terms of the red lines in figure 9 and we can see that the maximal charge Q max fits well with the semi-analytic and numerical results.However, the analytic and semi-analytic E max is about 5-6 times larger than the numerical results due to the uncertainties of values of Φ 0 .

Gauged Q-ball DM from electroweak FOPT
In the above discussions, we have shown that the gauged Q-ball could be stable and hence be a DM candidate.In this section, we begin to discuss the detailed production mechanism of the gauged Q-ball DM from the electroweak FOPT in the early Universe.We consider the minimal Higgs extended model with a singlet scalar field which could trigger a strong FOPT [51,77,78].The discussions can also be applied to other FOPT models.The phase transition dynamics can be modified by introducing some new degrees of freedom beyond the standard model.

Electroweak FOPT
The electroweak FOPT dynamics is determined by the finite temperature effective potential V eff (h, T ) where h is the real component of the SM Higgs doublet as defined in Eq. (2.1), (5.1) The first term V tree (h) = λ h h 2 − v 2 0 2 /4 is the tree-level SM Higgs potential.V CW (h) is the one-loop quantum correction to the effective potential, i.e., Coleman-Weinberg potential [79].Using the on-shell renormalization scheme, we have where g i is the degree of freedom for each particle, F i = 1(0) for fermions(bosons), m i (h) are masses for = t, W, Z, h, ϕ.The finite-temperature correction term is given by where the integral with "−/+ " sign denotes the contribution of bosons/fermions.Π i is thermal masses of species i.Here, we use the daisy resummation scheme proposed by Dolan and Jackiw [80].It is worth noticing that only the scalar fields and the longitudinal components of the gauge fields have nonzero Π i .For the scalar fields where g and g ′ are the gauge coupling of SU (2) L and U (1) Y , respectively.For the longitudinal components of the gauge bosons, we have (5.5) Hence, for the longitudinal components of the gauge bosons, their physical masses are eigenvalues of the following matrix where

Requiring the ordinary electroweak vacuum with h
The phase transition is the process of symmetry breaking in the early Universe.Through a process of bubble nucleation, growth and merger, the Universe transits from a metastable state into a stable state.The critical temperature T c is defined by the time when the two minima of effective potential are degenerate, V eff (v(T c ), T c ) = V eff (0, T c ) with v(T c ) being the vacuum value in the true vacuum at T = T c .Bubbles begin to nucleate when the temperature drops to the nucleation temperature T n .The nucleation rate of bubbles is given by with S 3 (T ) being the action of the O(3) symmetric bounce solution [81].The nucleation temperature T n is typically defined by where H(T ) is the Hubble expansion rate, where M pl = 1.22×10 19 GeV is the Planck mass and g ⋆ is the number of relativistic degrees of freedom at temperature T .∆V eff (T ) is the potential energy difference between false and true vacuum ∆V eff (T ) = V eff (0, T ) − V eff (v(T ), T ).The potential difference between the inside and outside the bubbles will cause the bubbles to expand in the Universe so that the volume of the false vacuum diminishes with time.The probability of finding a point in the false vacuum reads, where I(T ) is the fraction of vacuum converted to the true vacuum [82,83], (5.12) where v w is the bubble wall velocity.In the radiation dominated Universe [83], The percolation temperature T p , is defined by I(T p ) = 0.34 [82].This means that 34% of the false vacuum has been converted to the true vacuum at T p .The percolation temperature T p is also the temperature at which the GW is produced from the FOPT [83][84][85][86][87].
We use the following definition of the phase transition strength at percolation temperature: where ρ r = π 2 g ⋆ T 4 /30 represents radiation energy density and ∆V eff is the potential difference between the false and the true vacua.The inverse time duration β at percolation temperature is defined as We use CosmoTransitions [88] to calculate the phase transition dynamics.In the left panel of figure 10, we show the three typical temperatures of the FOPT process in the minimal SM plus singlet model.And in the right panel of figure 10, we show the wash-out parameter v(T )/T at these different temperatures.

Bubble wall filtering during FOPT
As the particles gain mass inside the bubble, due to the energy conservation, only highenergy particles can pass through the bubble walls and the others are reflected.The condition of penetration in the bubble wall frame reads [19,20]: where p w z is the particle z-direction momentum in the bubble wall frame, ∆m is the mass difference between the true and false vacuum where m in i and m out i are the particle mass in the true vacuum and false vacuum respectively [89].We will set m out i to zero in this work.The particle flux coming from the false vacuum per unit area and unit time can be written as [19,20] where g i is the degree of freedom of the particle.p w is the magnitude of the threemomentum of the particles.f eq i is the equilibrium distribution outside the bubble in the bubble wall frame, where ∓ is for bosons and fermions respectively.γ w = 1/ 1 − v 2 w is the Lorentz boost factor.The particle number density inside the bubble n in i in the bubble center frame can be written as Assuming the particles are massless in the false vacuum, we can integrate Eq. (5.17) analytically and get [20,90] where we have used Maxwell-Boltzmann approximation of DM distribution.One can see that as v w → 1, Eq. (5.20) approaches n out i = g i T 3 /π 2 , which is approximately the equilibrium number density for Boltzmann distribution outside the bubble.The fraction of particles i that are trapped into the false vacuum is defined by In our case, complex scalars ϕ and ϕ † are trapped into the false vacuum due to the filtering effect.The symmetric part would annihilate away in terms of the process ϕ + ϕ † → h + h, then only the asymmetric part survives and composes the charge of gauged Q-balls.It can be easily seen that the penetrated particle number density is sensitive to the bubble wall velocity as it appears in the exponent of Eq. (5.20).The precise calculation of bubble wall velocity [91][92][93][94][95][96][97][98] is beyond the scope of this work and we set it as a free parameter.

Charge of Q-ball in the electroweak FOPT
We can define T ⋆ as the temperature at which the false vacuum or old phase remnants can still form an infinitely connected "cluster", just like the definition of percolation temperature [17].The T ⋆ satisfies p(T ⋆ ) = 1 − p(T p ) = 0.29 which corresponds to I(T ⋆ ) = 1.24.T ⋆ is the temperature when Q-balls start to form.Below the temperature T ⋆ , the false vacuum remnants formed during FOPT may further fragment into smaller pieces.Ultimately, these pieces would shrink into Q-balls if there exists a non-zero primordial charge asymmetry.We can define the critical radius, r c , at which the remnant shrinks to an insignificant size before another true vacuum bubble forms within it [14].This means [17] Γ (T ⋆ ) 4π 3 r 3 c ∆t ∼ 1 .
( 5.22) where ∆t = r c /v w is the time cost for shrinking.The number density of the remnants n ⋆ Q can be expressed as: since the condition n ⋆ Q 4π 3 r 3 c = p(T ⋆ ) ≃ 0.29.The formation of Q-balls requires a nonzero conserved primordial charge which comes from the primordial DM asymmetry η ϕ = (n ϕ −n ϕ † )/s with entropy density s = 2π 2 g ⋆ T 3 /45.If the DM asymmetry is produced by thermal freeze-out in the early Universe, it is bounded from above by the equilibrium value, where we have used n eq ϕ (T ) = 2ζ(3)T 3 /π 2 with ζ(3) = 1.20206 being the value of Riemann zeta function ζ(s) at s = 3.In order to overcome this constraint, we assume the DM is produced by some non-thermal processes like decay.In this work, we do not specify the origin of primordial charge of the complex scalar ϕ.In the early Universe at higher temperature, new physical processes beyond the standard model may have occurred.The would-be Q-ball DM particles ϕ may have their own conserved charge and be created in asymmetric decays of heavier particles [99].For example, heavy Majorana neutrino could decay into a light fermion and a scalar, like N → χ + ϕ and N → χ + ϕ † where χ is a fermion [100].Assuming the process is CP-violating, the decay rates of these two channels differentiate from each other at loop level (this is similar to the process of leptogenesis.).So the asymmetry between ϕ and ϕ † appears and is retained until the phase transition in this work.The large DM asymmetry can be discussed in a similar manner to the large lepton asymmetry in the leptogenesis scenarios.Recent measurement of 4 He abundance coming from the EMPRESS experiment suggests a large degeneracy parameter of the electron neutrino [101], ξ e = 0.05 +0.03 −0.02 . (5.25) Since the neutrino oscillations among three flavors, the neutrinos with three flavors have the same amount of asymmetry, and then the total lepton asymmetry reads, where g BBN = 10.75 is the relativistic degree of freedom at the epoch of big bang nucleosynthesis.The large lepton asymmetry may come from the low-scale leptogenesis [102,103], Affleck-Dine mechanism [104] or L-ball decay [105].
In a remnant, the trapped Q-charge is given by In figure 11, we show the charge of the gauged Q-ball DM formed during the FOPT for different values of bubble wall velocities.We have chosen η ϕ = η L .When λ ϕh is larger, both the Γ(T ⋆ ) and the Q-ball number density are suppressed, so that the charge is larger at a given η ϕ .Since n Q /s and Q does not change in the adiabatic Universe, at present they are with s 0 = 2891.2cm −3 being the entropy density in current time [106].
We take the approximation T ⋆ ≈ T p then the bounce action can be approximated written as [ In reality, the mass or charge of Q-balls should have a distribution, which has been discussed in detail in Ref. [108].The false vacuum size distribution is given by Eq. ( 15) of Ref. [108].However, it has been shown in Eq. ( 24) of Ref. [108] that the average size of Q-balls during a FOPT is still approximately equal to v w /β, just as in the monochromatic case.Based on this, for simplicity we can still use the average value instead of the distribution of false vacuum.In order to get stable gauged Q-ball with a given value of charge, we must impose Q max > Q, so the gauge coupling g has to satisfy g < 1.28 × 10 −18 1 T ⋆ 100 GeV where ⟨σ anni v rel ⟩ is the annihilation cross section.
67 where H 0 is the Hubble constant today [109].The cross section of process ϕ+ϕ † ↔ h+h reads [110].In our parameter space, where the λ ϕh is around 7, the relic abundance from freeze-out is approximately Ω freeze−out h 2 100 ≈ 2.35 × 10 −4 .So we can omit the DM produced from thermal freeze-out.
The DM relic density also receives the contribution from penetrated asymmetric components of DM particles which is given by the excess of ϕ over ϕ † , The relic density of Q-balls at present is where ρ c = 3H 2 0 M 2 pl /(8π) is the critical energy density.We have found that, the gauged Q-ball is generally a mixed state of Eq. (4.16) and Eq.(4.18) as 0 ≤ αB 0 ρ b ≤ C 2 .So we can write down the energy of gauged Q-ball at a given charge, where V 0 = λ h 4 v 4 0 is the potential difference between inside and outside of the gauged Qballs at zero temperature.The first term on the right-side is the zero-point energy of the scalar particles, the second term is the vacuum volume energy inside the Q-ball and the third term is the electrostatic self energy.By minimizing this expression respect to r ⋆ , we obtain, . (5.35) In the limit of zero gauge coupling, which is just the energy of global Q-ball.By using we finally arrive at the expression: (5.37) the Γ(T ⋆ ) can also be expanded by using Γ(T ⋆ ) ≈ T 4 ⋆ e −S 3 (T⋆)/T⋆ and Eq.(5.28), but we keep the expression here to give more accurate results.
Although the expression hitherto is general, we apply these to the minimal SM plus singlet model.We choose λ ϕh = 6.8 for which T n = 71.65 GeV and T p = 68.9GeV.The value of this portal coupling satisfies the validity of the perturbative analysis which indicates the portal coupling should be roughly smaller than 10 [111].The number density of gauged Q-balls at production is n ⋆ Q ≃ 3.0 × 10 −31 GeV −3 .In this case, the value of v(T p )/T p is approximately 3.5 and there are still 50% of the DM particles trapping inside the false vacuum even at v w = 0.6.However, it should be noted that in this case, the contribution from penetrated asymmetric DM will dominate, as can be seen in Eq. (5.32).This can be avoided in two ways.One way is to increase the phase transition strength and the corresponding v(T )/T so there are little DM particles penetrating into the true vacuum, resulting F trap ϕ ≈ 1.This can be achieved by introducing new freedoms beyond the standard model or considering dark FOPT instead of electroweak FOPT.The other way is model dependent: one can introduce new decay channels that penetrated ϕ could decay into dark radiation or SM leptons which can account for the lepton asymmetry.The decay process does not destroy the stability of Q-balls as long as π/r ⋆ < m d where m d is the mass of decay products.In this work, we focus on the gauged Q-ball DM so we do not consider the penetrated asymmetric DM in detail.We show the gauged Q-ball DM relic density in figure 12.The colored stars represent the values for the gauged Q-ball with maximal charge.The gray region represents the region where the DM is overproduced.We can see that due to the finiteness of the charge, the gauged Q-balls can explain the whole DM at v w = 0.01 only when the rescaled gauge coupling α ≲ 10 −16 .The DM relic density is slightly enhanced due to the extra electrostatic energy.
In table 1 we choose four benchmark points that satisfy the correct DM relic density and show the corresponding F trap ϕ and η ϕ /η L .The condition of a strong FOPT leads to obvious deviation of triple Higgs coupling, which might be detected by the loop-induced e + e − → hZ process [112,113] at future lepton colliders, such as FCC-ee, CEPC, and ILC.One can define the δσ Zh as the fractional change in Zh production relative to the SM prediction at one loop.We list the corresponding δσ Zh for the four benchmark points in table 1.We can see that the value of the required DM asymmetry η ϕ is close to the lepton asymmetry η L .This prompts us to speculate that they have the same origin.Actually, if the DM asymmetry comes from the process N → χ + ϕ and N → χ + ϕ † .We could assume the dark fermion χ is long-lived and decay suddenly into leptons after electroweak phase transition in order to avoid the electroweak sphaleron process.Then we expect the η ϕ and η L is at the same order.The detailed model building is beyond the scope of this work and we leave this in our future studies. .(6.2)

• Sound wave
The contribution from sound waves could be more significant.The formula GW spectrum from sound waves is [125]: Υ sw is the suppression factor of the short period of the sound wave, where τ sw H p ≈ (8π) v w (H p /β) 3κ v α p /(4 + 4α p ) .(6.6)

• Turbulence
The formula of the GW spectrum from turbulence is [126]: κ turb represents the efficiency of vacuum energy being converted into turbulent flow: where the ε is set to 0.1.
The total contribution to the GW spectra can be calculated by summing these individual contributions: Ω GW h 2 100 = Ω co h 2 100 + Ω sw h 2 100 + Ω turb h 2 100 .(6.11) We show the GW spectra Ω GW h 2 100 for four benchmark points in figure 14.These four benchmark points are shown in Table 1.We choose λ ϕh = 6.8 and 7.0 and for each value of λ ϕh we choose two bubble wall velocities v w = 0.1, 0.6.The colored regions represent the sensitivity curves for future GW detectors LISA [43] and TianQin [44,45] with the signalto-noise ratio (SNR) about 5. We can see that the LISA and TianQin could detect this new DM mechanism when the bubble wall velocity is relatively large.Taiji [46], BBO [47], DECIGO [48], Ultimate-DECIGO [49] could also detect this new DM mechanism by GW signals.1.The future GW detectors LISA [43], TianQin [44,45] could detect BP 2 and BP 4 whose bubble wall velocities are relatively large.

Discussion and conclusion
In this work, we have systematically discussed the gauged Q-ball DM formed during a FOPT.We have investigated the stability of gauged Q-balls, including quantum stability, stress stability, fission stability, and classical stability.Different from the global Q-balls, the gauge interaction restricts the size of the stable gauged Q-balls.For a given value of the gauge coupling, the stable gauged Q-balls can only be realized in the region of charge Q s < Q < Q max .The upper limit Q max and lower limit Q s mainly come from the stress stability and quantum stability criterion respectively.By using the thin-wall approximation, we show that the piecewise model can describe the basic properties of gauged Q-balls in FLSM model well.Based on this, we further give an approximately analytic evaluation of Q max by using the mapping method.We find the maximal charge is approximately Q max ∝ g−2 where g is the gauge coupling of the dark U (1) symmetry.The constraint on the value of gauge coupling g is given by Eq. (5.30) if the gauged Q-balls are produced by a FOPT.We discuss the relic density of gauged Q-ball DM formed during an electroweak FOPT.Even in the minimal electroweak FOPT model (SM plus singlet), the gauged Q-balls can comprise all the observed DM.And we have found that in order to satisfy the relic abundance of DM, the original DM asymmetry surprisingly coincides with the observed large lepton number asymmetry.The average charge and mass of gauged Q-ball DM can be varied by modifying the phase transition dynamics or the primordial DM asymmetry.Besides, we give combined constraints on the gauged Q-ball DM from DM direct detection (Mica, XENON1T, Ohya), and astronomical observations (CMB, neutron stars, white dwarfs).The formation process of gauged Q-ball DM during a FOPT also produces phase transition GW signals which could be detected by future GW experiments such as LISA, TianQin, and Taiji.The phase transition dynamics in the early Universe provide new formation mechanisms of various soliton DM or dynamical DM.For example, it is reasonable to discuss other species of soliton DM formed during FOPT, such as gauged Fermi-ball DM.The configuration of the Fermi-ball is different from the Q-ball because of the extra Fermi-gas degeneracy pressure.We leave these in our future works.

Figure 1 .
Figure 1.Values of the three fields at the Q-ball center and the total energy Ẽ for different values of gauge coupling α.The arrows represent the evolution of the frequency ν.We choose k = 3.5 which corresponds to λ ϕh ≈ 6.Four specific solutions P 1, P 2, P 3, P 4 for α = 1.0 are marked by the triangles and the corresponding profiles are shown in figure 2.

Figure 2 .
Figure 2. Profiles of the dark gauge field, complex scalar field, Higgs field of the gauged Q-ball.Here we choose the marked points P 1, P 2, P 3, P 4 in figure1where k = 3.5 and α = 1.0.

Figure 3 .
Figure 3. Left: total charge Q for different values of α for U (1) gauged Q-balls.Right: the two typical radii for the Q-ball field Φ(ρ) and Higgs field H(ρ).

40 QFigure 4 .
Figure 4.The energy over charge Ẽ/ Q for different values of α.The Qs represents the value of charge which satisfies Ẽ/ Q = 1.

Figure 5 .
Figure 5. Left: the profile of F (ρ) defined by Eq. (3.10) for the four marked points in figure 1 where α = 1.0.Right: F (ρ) at the transition point between the first and second branches for different values of gauge couplings α.

Figure 7 .
Figure 7. Viable space of ν of gauged Q-balls for α = 0.7.The red line represents the region where the gauged Q-ball is dominated by gauge field such that it is unstable under stress stability criterion.The green line represents the region where the gauged Q-ball is unstable under quantum stability criterion.The purple point labels the transition point.Gauged Q-balls in the blue region between ν min and ν max are stable.

Figure 8 .
Figure 8. Charge and Energy as functions of Gauged Q-ball radius.The blue lines are the numerical results in piecewise model; red lines are the semi-analytic results Eq. (4.12) and Eq.(4.15), black lines come from the numerical results in FLSM model.

5 Figure 9 .
Figure 9. Maximal charge Q max = 2π λ h Qmax (left panel) and energy E max = 2πm ϕ λ h Ẽmax (right panel) of gauged Q-ball for different α and k.The red lines are analytic evaluations Eq. (4.22) and (4.24); the blue dotted lines are the semi-analytic results and the black dashed lines represent the numerical results for piecewise model.

Figure 10 .
Figure 10.Phase transition parameters as functions of λ ϕh .Left: the critical, nucleation and percolation temperatures for different values of λ ϕh .For the percolation temperature we choose v w = 0.1.Right: the wash-out parameters v(T )/T for various temperatures as functions of λ ϕh .

100 Ω Q h 2 100 > 0 Figure 12 .
Figure 12.Q-ball relic density as function of DM asymmetry η ϕ /η L in the SM plus singlet model where λ ϕh = 6.8.We choose v w = 0.01.The star represents the value for the gauged Q-ball with maximal charge.The gray region represents the region where the DM is overproduced.

sw h 2 100 ≃2. 65 × 10 − 6 Υ sw H p β κ v α p 1 5
where κ v reflects the fraction of vacuum energy that transfers into the fluid's bulk motion.The peak frequency of sound wave processes is:f sw ≃ 1.9 × 10 −Hz