Mapping Gauged Q-Balls

Scalar field theories with particular U(1)-symmetric potentials contain non-topological soliton solutions called Q-balls. Promoting the U(1) to a gauge symmetry leads to the more complicated situation of gauged Q-balls. The soliton solutions to the resulting set of nonlinear differential equations have markedly different properties, such as a maximal possible size and charge. Despite these differences, we discover a relation that allows one to extract the properties of gauged Q-balls (such as the radius, charge, and energy) from the more easily obtained properties of global Q-balls. These results provide a new guide to understanding gauged Q-balls as well as providing simple and accurate analytical characterization of the Q-ball properties.


I. INTRODUCTION
Q-balls are stable nontopological solitons that can arise in theories involving complex scalars φ [1] (for a review see Ref. [2]). In the case of global Q-balls, φ carries a conserved global charge and the solitons are stabilized by a scalar potential that provides an attractive force [3]. Global Q-balls have been proposed as dark matter [4,5] due to their potential occurrence in supersymmetric models and provide in particular a simple realization of macroscopic dark matter [6,7].
The analytic construction of Q-balls requires solving a nonlinear differential equation. In certain potentials, the equation can be solved exactly [8][9][10][11]. For many other cases numerical solutions can be efficiently obtained via computer programs such as AnyBubble [12]. Recently, it was shown that almost all aspects of global Q-balls can be understood essentially analytically, even for potentials which are not exactly solvable [13]. Extremely accurate analytical expressions were obtained for global Q-ball properties such as radius, charge, and energy in some non-solvable scenarios which essentially obviate the need for numerical studies [13]. It appears that for all intents and purposes single-field global Q-balls are a solved problem.
The system's complexity increases if φ is charged under a local symmetry, which leads to gauged Q-balls [14][15][16]. Given the prevalence of gauge bosons in the Standard Model and its extensions, understanding gauged Q-balls is important phenomenologically. However, they are considerably more difficult to describe, both analytically and numerically. On the analytic side, no exactly solvable examples are known to us. Numerical studies are made difficult by the gauge field, which appears in the scalar potential as a field whose kinetic term has the opposite sign. This makes numerical studies (using, for example, the ever-popular shooting method) far more tedious to implement.
In this article we extend the methods of Ref. [13] to gauged Q-balls. In so doing we reveal a close connection between global Q-balls and gauged Q-balls. This enables us to use our understanding of global Q-balls to analytically calculate the properties of these gauged Q-balls-such as radius, charge, and energy. Furthermore, we find simple expressions for the scalar and gauge-field profiles that can be used to solve the differential equations efficiently using finite-element methods. This work paves the way for detailed phenomenological studies of these objects.
In the next section, we review global Q-balls and establish our notation. Section III introduces gauged Q-balls and analytical approximations for the scalar and gauge field profiles. In Sec. IV we present a method for solving the Q-ball differential equations using finite-element methods rather than the shooting method. The novel mapping between global and gauged solutions is given in Sec. V. The accuracy of our analytical predictions for the Q-ball profiles and observables such as energy, mass, and radius, are established in Sec. VI. We also derive quantities of interest such as the parametric regions of Q-ball stability before concluding in Sec. VII. A derivation of the Q-ball energy and alternative derivation of the mapping formula is given in Appendices A and B, respectively.

II. REVIEW OF GLOBAL Q-BALLS
The Lagrangian density for a complex scalar φ enjoys an explicit global U (1) symmetry φ → e iα φ. The conserved charge Q under this symmetry is φ number, normalized so that Q(φ) = 1. To preserve the U (1) symmetry, we require φ = 0 in the vacuum. We choose the potential energy to be zero in the vacuum by setting U (0) = 0 and enforce that the vacuum is a stable minimum of the potential by where m φ is the mass of the complex scalar. In this scenario, Coleman [1] showed that nontopological solitons, Q-balls, exist when the function U (|φ|)/|φ| 2 has a minimum at |φ| = φ 0 / √ 2 > 0 such that Spherical Q-ball solutions have the form for a constant ω 0 < ω G < m φ . We choose ω G to be positive, which results in a positive charge Q of the Q-ball. It is convenient to define the dimensionless quantities We can then write the Lagrangian as where a prime denotes a derivative with respect to ρ. The equation of motion for f is Q-ball solutions for f satisfy this nonlinear differential equation along with the boundary conditions As an explicit example, we consider the most generic U (1)-symmetric sextic potential studied in Ref. [13]. This can be parametrized as The differential equation of Eq. (7) then takes the form where κ 2 G ≡ Ω 2 G − Ω 2 0 . The solutions depend on the single parameter κ G ∈ (0, 1), which also determines the (dimensionless) Q-ball radius R * . 1 For small κ G , the Q-balls are large and the relation R * (κ G ) can be calculated analytically at leading order to be R * (κ G ) = 1/κ 2 G [13]. For these large Q-balls, the exact Q-ball profile is close to a step function f (ρ) 1 − Θ(ρ − R * ), this is the so-called thin wall limit [1]. As shown in Ref. [13], an even better profile for these thin-wall Q-balls around ρ ∼ R * 1 is This is called the transition profile, since it describes the rapid transition from the nearly constant f 1 inside the Q-ball to f 0 outside the Q-ball. The transition profile is actually a very good approximation to the full profile for all ρ and even works reasonably well for smaller Q-balls [13].
We also present here a new relation for R * (κ G ) which provides an approximation to the numerical result that is accurate to better than 2% in the region κ G < 0.84 (or R * 1.5) that leads to stable Q-balls (i.e. Q-balls with E < m φ Q). This relation can be used to produce extremely accurate expressions of the global Q-ball's energy and charge as a function of radius using the expressions in Ref. [13].

III. GAUGED Q-BALLS
Gauged Q-balls result from promoting the global U (1) symmetry to a local symmetry. The Lagrangian density is where D µ = ∂ µ − ieA µ is the gauge covariant derivative and F µν = ∂ µ A ν − ∂ ν A µ is the field-strength tensor. The parameter e is the gauge coupling normalized so that φ has charge one. After making the static charge ansatz [14] and defining dimensionless quantities we rewrite the Lagrangian as This has the form of two scalar fields under the influence of the potential However, it is important to notice that in this analogy the A field's kinetic term has the wrong sign.
The two equations of motion A are to be solved subject to the boundary conditions In the analogy of two fields moving in the potential V , ρ becomes a time coordinate and the terms with an explicit 1/ρ can be interpreted as time-dependent friction terms. As shown below, this analogy greatly aids our understanding of the Q-ball solutions.
The scalar frequency ω is restricted to the region ω 0 < ω ≤ m φ ; this is similar to the global Q-ball case, except that it is possible to have gauged Q-balls with ω = m φ (or κ = 1) [15], where no global Q-balls exist. In section VI we show that a stronger lower bound on ω exists.
The conserved charge Q is defined in the usual way as the integral over the time component of the scalar current [14] where the second line uses Eq. (18) and integration by parts. This implies that for large ρ, up to corrections that fall off faster than 1/ρ [14]. The gauged Q-ball energy E is obtained from the The second expression corrects a typo in Ref. [14] and is derived in App. A. The energy and charge also satisfy the non-trivial differential equation [17] dE This is a powerful relation among the Q-ball observables and, in particular, allows ω to be interpreted as the chemical potential.
For concreteness we restrict most of our discussion to the sextic scalar potential of Eq. (8), although we expect our results to be qualitatively applicable to a far larger class of potentials. Just like in the global case we only study ground-state Q-balls, which have no nodes; excited gauged Q-balls in the same potential have been discussed in Ref. [18].

IV. NUMERICAL METHODS
While the shooting method is quite successful for global Q-balls [1], the addition of the gauge field makes finding a solution using this method tedious, especially for large Q-balls. We avoid this by changing coordinates and solving the boundary value problem directly. A similar approach was employed in Ref. [19].
In order to enforce the boundary conditions at ρ = ∞ we switch to a compactified coordinate y, where a is a positive constant. The value of a makes no real difference in obtaining numerical solutions. However, choosing a much larger than the Q-ball radius ensures that the most drastic compactification effects occur outside the Q-ball. Clearly, y takes values y ∈ [0, a] and so we can require the conditions f (a) = 0 and A(a) = 0. The derivatives become so the boundary conditions at y = 0 are f (0) = 0 and A (0) = 0 where primes denote a derivative with respect to y. The set of equations can then be solved by finite element methods, using Mathematica's [20] routines for instance, and quickly converges to the exact solution if the initial guess is reasonably accurate. In the next section we present a method for finding analytical test functions for f and A that are close to the exact solutions. These can be successfully used as initial seed functions for this method.

V. MAPPING GLOBAL Q-BALLS TO GAUGED Q-BALLS
Much of the Q-ball profile can be understood by comparing it to the motion of a particle moving in the potential of Eq. (16) For constant A the potential in f ≥ 0 has three extrema, one at f = 0 and the other two at so the requirement that Eq. (31) is real implies Notice that this condition is trivially satisfied in the global case, i.e. for α → 0, but in the gauged case restricts αA to two possible regions: As shown below, the second inequality in Eq. (33) is not compatible with Q-ball solutions, leaving us with an upper bound on αA when Ω 0 ≥ 1/ √ 3.
As with the global case, we can determine the initial values of the fields by energy considerations.
Neglecting the friction terms, we can write the equations of motion as This means that the quantity is conserved as a function of ρ: Of course, when the friction is included this quantity is not conserved and we immediately find that This justifies our interpretation of the term on the right hand side of the equation as a friction.
For constant f , the potential for A has one extremum at Again, Eqs. (24) and (37) indicate that f and A affect the energy differently. The f profile behaves according to our usual intuition, but the A kinetic term has the opposite sign. Consequently, as Eq. (38) is a minimum in V , the dynamics of the system drive A uphill either toward A = 0 or A → ∞. If A is larger than A max it diverges as ρ → ∞, which clearly does not satisfy the Q-ball boundary conditions. This implies that for Q-ball solutions Ω−αA > 0, which has two consequences: First, because the right-hand side of the A equation of motion is always negative, A is monotonically decreasing for Q-ball solutions [14]. Second, as the system evolves the negative term under the square-root in Eq. (31) becomes smaller so the value of f + grows.
For some solutions, such as the one shown in Fig. 3, the "force" from the A gradient pushes f uphill toward this growing f + .
While the gauge field does affect the total Q-ball dynamics, it seems to play a relatively minor role when f transitions from near one to near zero. This observation suggests a relationship between the global Q-ball solutions and gauged Q-ball solutions. To explore this, we need analytic expressions for A and f . Beginning at the thin-wall limit, we approximate f by a step function, f (ρ) = 1−Θ(ρ−R * ) and then solve the equation of motion Eq. (18) for A. By demanding that A(ρ) and its derivative be continuous at ρ = R * one finds [14] A Remarkably, this result is a good approximation to the exact gauge field solution even beyond the thin-wall regime.
This result indicates that the derivative of αA is small if the Q-ball radius R * is large: which implies that αA is essentially constant over the transition. We can then refine our analysis of the scalar profile by solving the f equation of motion around ρ ∼ R * with a constant A: f Equation (42) is exactly the form of the equation for the global Q-ball Eq. (7) with the global value of Ω G given by Since the derivative of αA is small, it does not contribute significantly to the friction over the transition region. This means that the frictional effects over the transition are also nearly identical to the global case. Since the relation between Ω G and R * is determined by the friction, if the R * dependence of the global Q-ball parameter Ω G (R * ) is known, we can determine the R * dependence of the gauged Q-ball Ω(R * ) via

VI. RESULTS
We can now use these results to construct an analytical estimate for the Q-ball profile. The mapping in Eq. (44) provides the radius of the gauged Q-ball given the known relationship Ω G (R * ) from the global Q-ball (Eq. 11). The scalar profile f (ρ) is taken to be the transition profile of global Q-balls (Eq. (10)); this is well motivated around ρ ∼ R * for large R * but happens to be a very good approximation for all other cases as well. Finally, the gauge profile A(ρ) is taken from Eq. (40). We can also use this analytical profile to find approximations for Q (via Eq. (20)) and E (via Eq. (24)); since the resulting expressions are lengthy we do not show them here.
These profiles serve as excellent seed functions for the numerical solution of the differential equations described in Sec. IV. Figure 4 shows a comparison between the numerical calculations and our analytical estimates for one choice of parameters. Note that the two solutions in Fig. 4 have the same potential parameters and scalar frequency ω, but differ in their Q-ball observables such as radius, charge, and energy. These two solutions correspond to the two solutions for R * obtained from the mapping in Eq. (44). As the plot illustrates, the analytical profiles for f and A match the numerical results remarkably well, especially for the large Q-balls (right panel).
We now discuss the Q-ball observables for the benchmark point Ω 0 = 5 and α = 1/100; we set results for E/m φ , Q, and (E/m φ Q). Overall, there is excellent agreement between the numerical and analytical results.
There are a number of features that restrict the allowable Q-ball solutions. First, we must have ω ≤ m φ (or κ ≤ 1) in order for the Q-ball solution to relax to zero for large ρ. This typically 2 implies a maximum Q-ball radius. Secondly, we must have E ≤ m φ Q so that the Q-ball is stable against decay to scalars. This constraint is most easily seen in the top right panel of Finally, we must impose the constraint of Eq. (33) that demands that the scalar potential have a second maximum away from f = 0. This puts an upper bound on the radius which, for this benchmark, is more restrictive than the maximal radius determined by the relation ω ≤ m φ . Using Eq. (33) with A = A(0) from our thin-wall expression Eq. (40), we can calculate this maximal radius R * max and impose this constraint on our analytical prediction shown in the figure, ending the solid line before κ = 1 . Since the thin-wall A(0) overestimates the true value, our maximal radius is slightly smaller than the true maximal radius (indicated by a rectangle in the plot), but the agreement is still good.
One interesting feature in the κ vs. R * plot is the existence of a minimum allowed value of κ. An analytic expression for this minimum value can be obtained; since this minimum value must be less than or equal to one for Q-balls to exist, we find the constraint In particular, this predicts that there are no gauged Q-balls with α > 0.58. Numerically, we find that the actual upper limit for α is 0.52, in quantitative agreement with the above mapping derivation.
Note that it was pointed out in Ref. [14] that for any scalar potential (and its implied attractive force) there must be an upper bound on the allowed gauge coupling (and its implied repulsive force) in order to form a stable Q-ball.
The lower panels of Fig. 5 show the behavior of Q and E as a function of R * . They inherit both a minimal and a maximal value from the corresponding radius. Our analytical predictions match the numerical results on the (phenomenologically interesting) stable Q-ball branch.
We compare the analytical and numerical data for several other benchmarks in Fig. 6. Our predictions show only small deviations with respect to the numerical results for all benchmarks.
This illustrates that the mapping in Eq. (44) holds qualitatively and quantitatively over the whole parameter space.
For these benchmarks, R * max is set by the condition κ = 1 rather than by Eq. (33). Using Eq. (44) and the large-R * relation κ G = 1/ √ R * [13] we find This equation cannot be solved analytically, but has the limiting cases: for Ω 0 α ,   Since both charge and energy grow with R * for large radii, this R * max also implies a maximal Qball charge and energy for a given set of potential parameters. This qualitative claim was made in Ref. [14], but here we provide easy-to-use quantitative predictions.
Using our analytical approximations together with numerical results, we can show that stable gauged Q-balls have R * 1.5, which is similar to the lower limit found for global Q-balls [13]. This matches the physical expectation that the introduction of a repulsive force to a Q-ball should not decrease the Q-ball radius.
We note that for Ω 0 = 0, the scalar profile is found to be essentially constant in the interior of thin-wall Q-balls (Fig. 4, right), and our approximations become more accurate, especially for small α, where the solutions approach the global Q-ball case. For larger Ω 0 , the solutions deviate from the global case (Fig. 2, left), but our results remain accurate. It would be interesting to explore the dependence on Ω 0 further; we leave this to future work.

VII. CONCLUSION
Global Q-balls are curious objects that arise in certain U (1)-symmetric scalar field theories and can be studied analytically and numerically with relative ease. Promoting the U (1) symmetry to a gauge symmetry complicates the discussion significantly and has eluded analytical descriptions outside of some limiting cases.
In this article we have exhibited a method to obtain essentially all properties of gauged Q-balls via a mapping from global Q-balls. Since the latter can be easily obtained numerically and often even analytically, this mapping allows for an excellent prediction of the gauged Q-ball properties without the need to solve the coupled, nonlinear differential equations. Our analytical expressions also make possible the solution of the differential equations by finite-element methods rather than the shooting method.
Finally, we stress that our analytical approximations are best in the thin-wall or large-radius limit. Smaller Q-balls show larger deviations, but these are also the Q-balls that are easiest to study numerically, providing good complementarity. Importantly, our analytical approximations also serve as good seed functions for numerical finite-element methods, significantly simplifying the numerical study of thick-wall gauged Q-balls.