U ( 1 ) gauged Q-balls and their properties

In the present manuscript, we briefly discuss the differences between ordinary Q-balls and U(1) gauged Q-balls. We focus on the differences in the existence of Q-ball solutions for appropriate values of the model parameters, as well as on the differences in classical stability of such Q-balls.

1 Ordinary Q-balls Q-balls are localized solutions in the theory of a complex scalar field, described by the action such that φ(t, � x) = e iωt f (r), Without loss of generality one can take f (r) to be real and such that f (r) > 0. Initially, such solutions were proposed in [1] and became popular after the well-known paper [2].The Q-ball charge is defined as whereas the Q-ball energy is defined as It is easy to show that the relation ⋆ e-mail: emin@ms2.inr.ac.ru ⋆⋆ e-mail: panin@ms2.inr.ac.ru ⋆⋆⋆ e-mail: smolyakov@theory.sinp.msu.ruholds for Q-balls.Since there exists the symmetry without loss of generality one can examine only the case ω ≥ 0. As a simple example of the Q-ball, we can consider a model with the scalar field potential where θ is the Heaviside step function (see Figure 1).For Q-balls, this piecewise scalar field potential was introduced in a more general form in [1].The corresponding solution for the scalar field takes the 1 2 The scalar field potential (7).
form [1] The solution for ω M = 0.8 is presented in Figure 2. The Q(ω) and E(Q) dependencies for these Q-balls are presented in Figure 3 (see [3] for a more detailed discussion of Q-balls in the model with the scalar field potential (7)).
The U(1) gauged Q-ball is a localized solution to the equations of motion for the fields, following from action (11), of the form Again, without loss of generality we take f (r) > 0. The charge of U(1) gauged Q-ball is defined as whereas its energy is defined as It is possible to show [6] that the relation holds for U(1) gauged Q-balls too.U(1) gauged Q-balls possess the following properties: • The sign of ω + eA 0 coincides with the sign of ω [5].
Thus, there exists the symmetry and without loss of generality one can examine only the case ω ≥ 0. After this brief introduction, we are ready to turn to examining the differences between ordinary Q-balls and U(1) gauged Q-balls.

Behavior of Q-balls at r → ∞
Let us consider the scalar field potential V(φ * φ) such that For f (r) → 0, the equation of motion for ordinary Q-balls reduces to In the case of U(1) gauged Q-balls the situation is completely different.Indeed, suppose that there exists a U(1) gauged Q-ball with the charge Q.Then, for f (r) → 0 the equation of motion for the scalar field reduces to Now we have two cases: where C is a constant and K 1 (b, z) is the modified Bessel function of the second kind, leading to for r → ∞.
We see that in the gauged case there may exist a solution possessing the finite charge and energy even for ω = M.This conclusion is supported by the results of numerical simulations (see [7] for details) for the scalar field potential (7) (Figures 4 and 5) and for the scalar field potential 6 and 7).
M 2 (thick lines).The thin lines stand for the nongauged case.The figure is taken from [7].Here The figure is taken from [7].

Classical stability of U(1) gauged Q-balls: theoretical considerations
Now we turn to examining another difference between ordinary and U(1) gauged Q-balls -the classical stability, i.e., stability of the Q-ball solution with respect to small perturbations of the fields.
It is known that ordinary Q-balls are stable with respect to small perturbations if the following conditions hold: 2. The operator where ∆ = ∂ i ∂ i , has only one negative eigenvalue.
The proof of the classical stability criterion based on the use of the energy functional of the system can be found in [8], the proof based on the use of the linearized equations of motion along the lines of the Vakhitov-Kolokolov method proposed in [9] can be found in [10].It should be noted that we are not aware of any explicit example of ordinary Q-ball, for which the operator L has more than one negative eigenvalues and which is classically unstable for dQ dω < 0. In order to examine the gauged case, we consider the perturbations above the U(1) gauged Q-ball solution in the form [10] A 0 (t, � x) = A 0 (r) + e γt a 0 (� x), (28) where γ is real, which passes through the linearized equations of motion.It is possible to show (see the detailed derivation in [10]) that U(1) gauged Q-balls are classically stable if the following conditions hold: 1. dQ dω < 0 (30) 2. The corresponding operator L has only one negative eigenvalue.
Here the operator L takes the form where I 3×3 is the 3 × 3 unit matrix and One can see that there always exist negative eigenvalues of the operator L, corresponding to the perturbations a i (because of the elements .Moreover, due to the fact that f (r) rapidly tends to zero at r → ∞, one expects that the operator ∆ 2 − e 2 f 2 − γ 2 2 has a negative continuous spectrum starting at − γ 2 2 .Thus, fulfillment of the condition dQ dω < 0 does not provide any information about possible stability regions of the U(1) gauged Q-ball, at least within the framework of the generalized Vakhitov-Kolokolov approach used in [10].At the moment it is not clear what is the origin of the problem described above.On the one hand, it is possible that the existence of the negative spectrum is just a technical artifact of the approach used in [10].On the other hand, it is possible, due to the repulsive nature of the electromagnetic field, that nonspherically symmetric perturbations destroy any U(1) gauged Q-ball.It is necessary to note that the question remains open as to whether it is possible to have only one negative eigenvalue of the operator L even in the case of spherically symmetric perturbations, in which the operator L takes a simpler form (see the detailed discussion in [10]).

Classical stability of U(1) gauged Q-balls: numerical simulations
In order to examine the classical stability regions of U(1) gauged Q-balls, the numerical simulations in the spherically symmetric case were performed (see the details in [10]).Here we present the results for two different scalar field potentials.The first one is the potential (7).We see from Figure 8 (7).The longdashed line stands for the nongauged Q-balls.The solid line stands for the classically stable gauged Q-balls.The short-dashed line stands for the classically unstable gauged Q-balls.The figure is taken from [10].
that the inequality dQ dω > 0 holds for classically unstable solutions, which formally is in agreement with the classical stability criterion.Meanwhile, the inequality dQ dω > 0 also holds for gauged Qballs on the upper branch of the Q(ω) dependence, which appear to be stable at least with respect to spherically symmetric perturbations.Although formally this fact does not contradict the classical stability criterion (in the case of ordinary Q-balls the classical stability criterion does not guarantee that there exist unstable modes in the case dQ dω > 0, it just states that there are no classical instabilities if dQ dω < 0), this situation differs from the case of ordinary Q-balls, for which usually there exist spherically symmetric unstable modes for Q-balls with dQ dω > 0.
The second scalar field potential has the form In the nongauged case this scalar field potential was proposed in [11].An interesting feature of this model is that the linearized equations of motion for perturbations can be solved exactly in the nongauged case [12].The dQ dω < 0 stability criterion is valid for ordinary Q-balls in this model -Q-balls with dQ dω < 0 are classically stable (and Q-balls with dQ dω > 0 are classically unstable).The results of the numerical simulation in the gauged case can be found in Figure 9.One can easily see from Figure 9 that there are classically unstable U(1) gauged Q-balls with dQ dω < 0 (for example, the Q-ball marked by the dot in Figure 9).This differs considerably from the case of ordinary Q-balls.An example of time evolution of the perturbed U(1) gauged Q-ball is shown in Figure 10, where the scalar field profile is presented at different moments of time.
In the present manuscript, we briefly discussed the differences between ordinary and U(1) gauged Q-balls.It is shown that U(1) gauged Q-balls may exist for the values of the parameters (ω = M), for which ordinary Q-balls do not exist.It is also shown that the classical stability criterion for ordinary Q-balls cannot be applied to U(1) gauged Q-balls.An explicit example of classically unstable U(1) gauged Q-ball with dQ dω < 0 is presented.These results demonstrate that addition of the gauge field changes drastically the properties of Q-balls.
The work was supported by the Grant 16-12-10494 of the Russian Science Foundation.

Figure 8 .
Figure8.The stability and instability regions of Q-balls in the model with the scalar field potential(7).The longdashed line stands for the nongauged Q-balls.The solid line stands for the classically stable gauged Q-balls.The short-dashed line stands for the classically unstable gauged Q-balls.The figure is taken from[10].

Figure 9 .
Figure 9. Ordinary (left plot) and U(1) gauged Q-balls for e/βµ = 1.1 (right plot) for the logarithmic scalar field potential.The solid lines stand for the classically stable Q-balls.The short-dashed lines stand for the classically unstable Q-balls.The right plot is taken from [10].

Figure 10 .
Figure 10.The scalar field profile of the classically unstable gauged Q-ball at different moments of time.The initial solution (at µt = 0) is marked by the dot in the upper right corner of Figure 9.The figure is taken from [10].