Charge-Swapping Q-balls and Their Lifetimes

For scalar theories accommodating spherically symmetric Q-balls, there are also towers of quasi-stable composite Q-balls, called charge swapping Q-balls (CSQs). We investigate the properties, particularly the lifetimes, of these long-lived CSQs in 2+1D and 3+1D using numerical simulations with efficient second order absorbing boundary conditions. We find that the evolution of a CSQ typically consists of 4 distinct stages: initial relaxation, first plateau (CSQ stage), fast decay and second plateau (oscillon stage). We chart the lifetimes of CSQs for different parameters of the initial conditions and of the potential, and show the attractor behavior and other properties of the CSQs.


Introduction and summary
Non-perturbative configurations, such as solitons, together with their fascinating properties are integral parts of understanding field theories. Apart from topological defects, which are stable due to the presence of topological charges, there also exist non-topological solitons, such as Q-balls [1,2]. Q-balls are spatially localized and stationary but non-static, and are stable due to the presence of Noether charges. For example, Q-balls exist in U(1) symmetric scalar field theories with a potential that grows slower than the quadratic term away from its minimum. The shallow potential creates some sort of "attractive forces between particles" in the theory, so particles prefer to condense to form a localized lump rather than dissipate to infinity. In other words, the Q-ball condensate is the energetically preferred state for such a system. The properties and dynamics of Q-balls have been extensively studied . Crucially, they may play an important role in the early universe (see, e.g., [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]) and can be candidates of dark matter (see, e.g., [42][43][44][45][46][47][48][49][50]). They can also be prepared in cold atom systems [51,52].
The most stable form of Q-balls are spherically symmetric, as other forms of spatial configurations increase the gradient energy. Recently, non-spherically symmetric, composite Q-balls have been identified [53]. In the theories where the spherically symmetric Q-balls exist, there are also a tower of composite Q-balls with different multipoles, within which both positive and negative charges co-exist and swap with time (see Fig. 2 for a dipole CSQ and see Fig. 22 for higher multipole CSQs). They are thus dubbed charge-swapping Q-balls (CSQs) [53]. Nevertheless, their energy densities remain mostly spherically symmetric (see Figs. 3 and 22). They have been shown to exist in 2+1D and 3+1D, and can be prepared by simply placing positive and negative charge elementary Q-balls (or in general simply lumps) tightly together such that their nonlinear cores overlap, and then the configurations, after initially emitting a burst of radiation, can quickly relax to CSQs.
In this paper, we shall investigate the properties and evolution of the CSQs in more detail, and determine their lifetimes. We will focus on dipole CSQs in 2+1D and 3+1D in the simplest φ 6 potential, with quadrupole and octupole CSQs also briefly touched on. To be able to determine the lifetimes of CSQs, we shall utilize absorbing boundary conditions (ABCs). This is crucial as our method of obtaining the CSQs simply involves superimposing elementary Q-balls, which is, of course, not the configuration of a CSQ. Thus, in the initial relaxing phase, the configuration radiates a substantial amount of energy (see, e.g., Fig. 8). With the previous periodic boundary conditions, this radiation would travel back and echo around the periodic space, continuously perturbing the CSQ. So, while the periodic boundary conditions of [53] are sufficient to determine the existence of CSQs, the lifetimes of the CSQs can not be reliably determined in that setup. Therefore, for an accurate determination of the lifetimes of CSQs, it is essential to employ effective ABCs.
We will survey the effectiveness of a few ABCs for our particular problem: Sommerfeld's ABCs [76], Engquist-Majda's ABCs [77] and Hidgon's ABCs [78,79]. Sommerfeld's ABCs are first order, designed to absorb spherically symmetric radiation, while Engquist-Majda's ABCs and Hidgon's ABCs can be implemented at higher orders. For our applications, we find that the second order ABCs are sufficient to determine the lifetimes within a couple of percent, and Hidgon's second order ABCs with a judicial choice of the c 1 and c 2 generally give the best absorbing effects, which are used to produce most of the results in the paper. Another way to damp radiation in the far field regions would be to add a Kreiss-Oliger term. However, in our simulations, we do not find the Kreiss-Oliger term significantly increases the accuracy, and therefore this artificial term is not adopted for the results presented in the paper.
We find that the evolution of a CSQ can be divided into 4 distinct stages (see Figs. 8 and 9): (1) Initial relaxation, (2) First plateau (CSQ stage), (3) Fast decay and (4) Second plateau (oscillon stage). The presence of the initial relaxation stage is, as mentioned, because the CSQ is not precisely prepared. Indeed, we find that the CSQ is an attractor solution, as a quasi-stable configuration should be, and can be formed with relatively general initial configurations (see Figs. 19,20 and 21). After settling down, the CSQ stage is characterized by a mostly spherical and slightly oscillating energy density profile (see Fig. 3) with charges swapping with time (see Fig. 2). The total energy and charge decrease slowly with time, and the swapping period remains mostly constant (see Fig. 10), which is a few times the oscillating period of an elementary Q-ball. We find that long-lived dipole CSQs in 2+1D can be achieved in a diagonal strip of the parameter space of the initial Q-ball frequency ω and the initial separation between them d (see Fig. 17). We also find the the lifetime of the dipole CSQ has an exponential dependence on the φ 6 coupling g (see Fig. 18). The CSQ stage is followed by a short stage of fast decay of both energy and charges, the end result of which is, interestingly, an oscillon with roughly half of the total energy (and with very small charge densities). In this oscillon stage, the charge of the Q-ball components comprising the CSQ, Q s , decays exponentially. Although within the time limits of our simulations we have not seen the decay of the second oscillon plateau, the oscillons will ultimately decay as they are not absolutely stable.
For the higher multipole CSQs in 2+1D, their lifetimes are shorter than those of the dipoles but nevertheless remain at the same order, while their total energies are about twice those of the dipoles. On the other hand, the charges of the Q-ball components, Q s , decays faster than that of the dipoles, unless the coupling g is tuned to be a smaller value (see Fig. 25). Also, for high multipoles, it appears that the lifetimes converge to the same value for different g (see Fig. 26). In 3+1D, the lifetimes of the CSQs are much shorter (see Figs. 28 and 29), presumably because there are more possible decay modes with three spatial directions. However, this is also largely to do with the potential of the theory. For example, for the logarithmic potential, which is the fiducial example of [53], the 3+1D CSQs are also long-lived, even for the higher multipoles [80].
The paper is organized as follows. In Section 2, we first introduce the fiducial field model we consider in this paper, define a few quantities that will be used later, and review elementary Q-balls and CSQs; then we specify the numerical implementations and introduce the Higdon's ABCs that are used to produce the results in the paper; additional ABCs, which are used to cross-check some of the results, are introduced in the Appendix A. In Section 3, we investigate the evolution histories, lifetimes and attractor behavior of the dipole CSQs in 2+1D; the different stages of a CSQ evolution are detailed, and various properties of the dipole CSQs are explored, in particular the lifetimes of the dipole CSQs are surveyed for different parameters ω, d and g. In Section 4, we briefly study higher multipole CSQs in 2+1D. In Section 5, we briefly study CSQs in 3+1D.

Model and setup
The existence of Q-balls in a complex scalar field theory requires a potential with an attractive nature that condenses field perturbations (or, loosely speaking, "particles" in the language of quantum field theory), rather than dissipates them. The simplest Z 2 symmetric (ϕ → −ϕ), polynomial potential that supports Q-balls is given by where d is the number of the spatial dimensions. Defining dimensionless variables x µ = mx µ , φ = λ 1/2 ϕ/m and g =gm 2 /λ 2 , the action can be re-written as which only contains one dimensionless free parameter g. In other words, we are expressing the coordinates in the units of the particle mass, m −1 , and the field value in the units of mλ −1/2 . Unless otherwise stated, in the following, we will consider the potential with g = 1/2 as a fiducial model, The energy density and conserved energy for a spatial volume are given respectively by Since the action is invariant under a global symmetry φ → e iα φ (α being constant), by Noether's theorem, there is a conserved current and thus a conserved charge for this symmetry. The associated charge density and conserved charge are given respectively by Numerically, we are simulating the system in a large but finite box, so the total energy E and total charge Q will refer to the total energy and charge in the simulation box. Note that for this potential the |φ| 4 term is negative, which provides the attractive force mentioned above, and thus Q-balls can form. Technically, this means that V /|φ| 2 has a minimum away from |φ| = 0. Since V /|φ| 2 at |φ| = 0 gives the mass of the particle in the free theory, V /|φ| 2 having a minimum at |φ| = 0 means in the interacting theory there are condensate configurations where the mass of the particle is smaller than that of the free theory. The minimum of these configurations, i.e., elementary Q-balls, are the minima of the energy functional [2]. That is, because of the leading interacting potential being shallower than the free quadratic potential, particles tend to condense rather than propagate away from each other.
Elementary Q-balls are stable stationary solutions that are spherically symmetric. To obtain the radial profile of an elementary Q-ball, we take the following ansatz where ω is the Q-ball frequency or the angular velocity in the φ field space. We shall call it an anti-Q-ball if ω is negative. Substituting this ansatz to the equation of motion for φ * , we get an equation which is subject to the boundary conditions df (0)/dr = 0 and f (∞) = 0. Viewing r as "time", Eq. (2.8) along with its boundary conditions can be viewed as a problem where a point mass, initially at rest df (0)/dr = 0, moves with a time-dependent friction term and in the effective potential V eff = 1 2 ω 2 f 2 − 1 2 V , and eventually comes to stop in the infinite future at f (∞) = 0. For an appropriate ω, we can find a unique solution that interpolates between f (0) = 0 and f (∞) = 0 without oscillations. If ω 2 is greater than the perturbative mass of the free theory, f = 0 is a local minimum of V eff , such a non-oscillating solution is impossible. To get an elementary Q-ball solution, the upper bound of the frequency is given by ω 2 If ω 2 is smaller than the minimum of V /f 2 at some f 0 = 0, V eff does not have a maximum away from f = 0 that is greater than V eff (f = 0), such a Q-ball solution is again impossible, so the lower bound of the frequency is ω 2 − = V /f 2 | f =f0 . For our fiducial potential, we have ω − = 1/ √ 2. Solving Eq. (2.8) numerically by the shooting method, we can get the Q-ball profile and total charge for the corresponding ω; see Fig. 1. We see that the total charge Q is larger for smaller ω and the peak of f (r) is also higher except when ω is close to ω − . Also, for an elementary Q-ball, the ratio between the total energy and the total charge E/Q is, as expected, smaller than 1, meaning that the Q-ball will not decay into free particles.
Apart from the elementary Q-ball solutions, the theory (2.2) also admits nonlinear, quasi-stationary, real solutions called oscillons [54,55], which take the form φ = g(t, r), (2.9) where the imaginary part of the field vanishes. As a very crude approximation, g(t, r) goes like g(t, r) ∼ g 0 (r) cos ωt. (2.10) These are also localized lumps, very similar to the elementary Q-ball. Oscillons can be supported by merely a real scalar field theory. For a complex field, if the initial configuration is real, the field remains real through out its time evolution. Indeed, for oscillons to exist, we also need a potential that supports attractive forces, and thus a theory that supports Q-balls also has oscillon solutions. But different from the elementary Q-balls, without the protection of any exact symmetry, oscillons are only quasi-stable solutions, although in 2+1D their life time can be long [66].

Charge-swapping Q-balls (CSQs)
Recently, it has been observed that, apart from the elementary Q-balls, which have been proven to be stable [2], there are also composite, quasi-stable solutions in the theory where elementary Q-balls exist [53]. These composite Q-balls are not spherically symmetric. They can form when Q-balls and anti-Q-balls are placed closely together with their cores overlapping. Remarkably, the positive and negative charges of the composite Q-balls swap with time, and thus they are dubbed charge-swapping Q-balls (CSQs). The simplest CSQ can be prepared by superposing a Q-ball and an anti-Q-ball. For example, we can place a Q-ball on the positive y-axis and an anti-Q-ball with equal but negative charge on the negative y-axis, i.e. a system with reflection symmetry about x-axis. The initial relative phase difference between the two Q-balls can be chosen to be zero. Because of this placement, the real component of the scalar field is symmetric about the x-axis and y-axis, while the imaginary component is symmetric about the y-axis and antisymmetric about the x-axis. The charges will swap along the y axis as the system evolves. Of course, this superposed configuration is not the quasi-stable CSQ, but it will quickly relax to a CSQ, as we shall see later. In fact, it is not essential to superpose exact Q-balls and anti-Q-balls in the initial preparation; we may as well initially superpose oscillating lumps that resemble Q-balls and anti-Q-balls. That is, CSQs are attractor solutions of the theory. See   for the evolution of the different charge integrals defined shortly below. On the other hand, the energy density of a CSQ is mostly spherically symmetric. There are also more complex CSQs with more Q-balls and anti-Q-balls, as we shall see in Section 4. Their lifetime is shorter in 3+1D; see Section 5. We are focusing on the polynomial potential in this paper, but they also exist in other models [53].
For later convenience, we shall define a few energy and charge quantities denoting integrations over various different regions of the dipole CSQ (the center of the CSQ is placed at the origin of the coordinate system): • E and Q: the total energy and charge in the simulation box respectively   While the existence of CSQs has been firmly shown in [53], the properties of CSQs are yet to be studied in more detail. In particular, the periodic boundary conditions used in the simulations of [53] are not appropriate to determine the lifetimes of these CSQs, as CSQs radiate perturbations, which propagate back to affect the CSQs in a periodic box. In this paper, we shall set up lattices that have absorbing boundary conditions that can absorb radiation effectively from the CSQs, which allows us to investigate CSQs in much more detail and determine the lifetimes of various CSQs.

Numerical setup
Our lattice code makes use of the open-source LATfield2 C++ library [81], which defines objects such as Lattice, Site and Field, allowing for fast and easy implementations of classical field simulations. However, LATfield2 uses periodic boundary conditions, which is not suitable for our purposes. We modified the library to incorporate several absorbing boundary conditions, which will be introduced in Section 2.3 and Appendix A. We use a 4th order finite difference stencil for spatial derivatives and evolve in time with the classical Runge-Kutta 4th order method. The code has excellent parallel speedups for many CPU cores with MPI as seen in Fig. 6.
We shall superimpose elementary Q-ball solutions, both for φ(0, x) andφ(0, x), as the initial configuration and let it relax to obtain CSQs. The internal frequency ω of the elementary Q-ball and the initial distance d between the elementary Q-balls are the free parameters we choose, and we will chart the lifetime of the CSQ in this two dimensional parameter space.
The Courant-Friedrichs-Lewy (CFL) factor dt/dx is set to be 0.1 both for 2D and 3D simulations. Unless otherwise stated, in 2D simulations, we will use a 512 2 lattice and the grid spacing is dx = 0.2, with a 1024 2 lattice frequently used to check for convergence, and in 3D we will use a 256 3 lattice and the grid spacing is dx = 0.4, with a 512 3 lattice to check for convergence. As we will see (e.g., in Figs. 16 and 15) that these numerical settings are sufficient for our purposes. The various ABCs are defined in the text below Eq. (2.14), and energy E c (t) and charge Q s (t) are defined in Section 2.1. The reference simulation is obtained by using a large lattice such that the radiation has not reached the boundaries in the run. Dashed lines represent firstorder ABCs and solid lines represent second-order ones. We see that second order ABCs generally perform better than the first order ABCs, and the Hig2(1,2.5) ABC has the best behavior with regard to eliminating the initial relaxing radiation. T 0 = 2π/m is roughly the oscillation period of the field φ.

Higdon's absorbing boundary conditions
As mentioned above, to determine the lifetime of a CSQ, we need to run the code for an extended period of time, and the periodic boundary conditions, while sufficient for determining the existence of CSQs [53], is unsuitable for this purpose. This is because CSQs emit radiation (or waves), especially during the initial relaxation phase, and this radiation travels back to interfere with the CSQs when using periodic boundary conditions. In reality we are interested in the lifetimes of CSQs in Minkowski space where radiation should simply propagate to infinity. To use a finite computation region to solve an infinite domain problem, we may make use of suitable absorbing boundary conditions (ABCs). Unless the problem is highly symmetric (say spherical symmetry) or the outgoing waves are very simple, ABCs are usually not perfect. However, a good ABC should let the majority of outgoing waves go through the boundary transparently and only incur minor reflections. We will explore several ABCs in this paper, namely Sommerfeld's ABC [76], Engquist-Majda's ABC [77] and Higdon's ABC [78,79]. We find that Higdon's 2nd order ABCs usually produce the best accuracy, which will be adopted in the majority of the simulations in this paper, while the other ABCs are used for sanity checks. We will introduce the Higdon ABCs below and the other ABCs are introduced in Appendix A. The Higdon ABCs are a set of easily implementable conditions at boundary x i = a [78]: where the + (−) sign is for a right (left) boundary, t and x i are Cartesian coordinates and c j , to be chosen by the user for specific problems, are the phase velocities of the normal outgoing plain waves that can be absorbed exactly. For the massive Klein-Gordon equation, c j should be chosen to be no less than 1. It is clear that c j should be chosen to annihilate the dominant wavenumbers near the boundary, which can be obtained by performing Fourier transforms of the field near the boundary. A rough guide is that c j is to be chosen to minimize the reflection rate between the amplitudes of the incoming waves and the outgoing waves for the dominant wavenumbers, where k i is the wavenumber and m is the perturbative mass of field. From the reflection rate formula, we can see that even for the outgoing waves whose phase velocities are not c j , the Higdon ABC can still often absorb much of them. In practice, the most commonly used Higdon ABCs are when M = 1, 2, which are simply given by where the + (−) sign is for a right (left) boundary. Note that the 1st and 2nd order Higdon ABC with c 1 = 1 and c 1 = c 2 = 1 are actually equivalent to the 1st and 2nd order Engquist-Majda ABC respectively, upon using the equations of motion. It is advised that, without prior knowledge of the spectrum of the outgoing waves, the default 2nd Higdon parameters can be chosen as c 1 = c 2 = 1 [78]. As we find CSQs by superimposing elementary Q-balls and letting them relax, the initial relaxation phase of the CSQ evolutions produces the largest amount of radiation, and we shall choose our ABCs to maximize the absorption of the radiation at the boundary from this phase. We shall compare the absorbing effects of different ABCs and select the best one among them. Since the relaxation phase is short, we can actually simulate this phase with a sufficiently large lattice such that by the end of this phase the radiation has not reached the boundary yet. We compare the effects of different ABCs with this reference run, which allows us to pick the one with the smallest deviations. The ABCs that we have tested against the reference run are: • Sommerfeld: Sommerfeld boundary condition with v = 1, φ 0 = 0 and no non-wavelike term h(t) • EM1: Engquist-Majda's 1st order condition, which is equivalent to Higdon's 1st order condition with c 1 = 1   14)) against the reference simulation for energy E c (t) and charge Q s (t), which are defined in Section 2.1. The reference simulation is set up such that the lattice is sufficiently large so that the radiation has not reached the boundaries at the end of the run. The Hig2(1,2.5) ABC has the best accuracy and is the default ABC used below unless stated otherwise.
• EM2: Engquist-Majda's 2nd order condition, which is equivalent to Higdon's 2nd order condition with c 1 = c 2 = 1 • Hig1 (2): Higdon's 1st order condition with c 1 = 2 • Hig2(1,2.5): Higdon's 2nd order condition with c 1 = 1, c 2 = 2.5 (this is the best ABC for our problem, and we will mostly use this method in the following, with other methods used for sanity checks.) Specifically, we test the different ABCs against the reference simulation in a 256 2 grid for a run of 32769 steps. We use the fiducial model and prepare the consistent Q-balls with ω = ±0.84 and place them together with a separation d = 2.8. We compute energy E c (t) and charge Q s (t), which are defined in Section 2.1, as functions of time, and compute the standard deviations: (2.16) From Table 1 and Fig. 7, we see that all these ABCs, especially the second order ABCs, are relatively good, and the Hig2(1,2.5) ABC appears to have the best accuracy in eliminating the outgoing radiation from the initial relaxation phase of the CSQ. The effectiveness of the Hig2(1,2.5) ABC can also be seen by looking at far-field regions of the CSQ, i.e., at points far from the CSQ, in which case the runs with the Hig2(1,2.5) ABC match the reference run much better than the other ABCs. Among the properties of CSQs that will be explored in the later sections, the lifetimes of CSQs are probably most sensitive to the ABCs. Indeed, if we use different ABCs to find the lifetimes, we get slightly different results. However, the differences are small, about a couple of percent. Generally, the Hig2(1,2.5) ABC leads to the longest lifetimes. We will assume that a longer lifetime means that the ABC absorbs the radiation better at the boundaries, on which basis we will use the Hig2(1,2.5) ABC as our default ABC.

Dipole CSQs in 2+1D
In this section, we will explore the properties of the simplest dipole CSQ in 2+1D, derived from a figuration with a Q-ball and an equal (opposite) charge anti-Q-ball. We will see that they have distinct  Figure 9: Envelope of the Q s evolution, where Q s is defined in Section 2.1. Q s oscillates quickly according to the charge swapping frequency T swap within the envelope. All the settings are identical to Fig. 8. T 0 = 2π/m is roughly the oscillation period of the field φ.
stages of evolution and can be formed from various different initial setups, confirming that they are attractor solutions. We will also chart the lifetimes of these CSQs for different initial conditions.   (4) Second plateau (oscillon stage). In the following, we will discuss these four stages separately.

Stages of a CSQ evolution
(1) Initial relaxation As we set up the initial configuration by simply superimposing two elementary Q-balls, which we emphasize are not the quasi-stable CSQs, this stage is characterized by fast decrease of energy and charge from the initial lump. For example, in Figs. 8 and 9, we see that, for a typical CSQ, about a third of the energy and about three quarters of the charge of the initial lump are shed during about a hundred oscillations, after which the initial configuration settles down to become a CSQ. The radiation of energy from the initial lump can be easily seen in the inset of Fig. 8, where the energy in the central circle E c initially decreases much faster than that of the total energy in the box E, and upon absorption at the boundaries the difference between E c and E diminishes within about a hundred oscillations for this fiducial model. During this transition stage, the swapping period of the lump T swap , which is defined at zero points of Q s by summing the time durations to the previous and next zero point, decreases quickly to the plateau of the CSQ (see Fig. 10), and the oscillating period of the real part of the field φ at the origin, which is defined at zero points of Reφ(t, 0, 0) by summing the time durations to the previous and next zero point, quickly settles to a value that is slightly larger than the intrinsic period of the elementary Q-ball with the same energy(see Fig. 11). From the trends in this initial stage of these figures, we might have expected that the CSQs are attractor solutions. We will look at this in more detail in the next subsection.
(2) First plateau (CSQ stage) In this stage, the energy and the charge of the CSQ have reached a plateau, decreasing very slowly with time; see Figs. 8 and 9. While the average value of the swapping period of the CSQ , T swap , also remains mostly unchanged, its value oscillates noticeably around the average; see Fig. 10. This stage lasts for an extended period of time, the length of which depends on the oscillation frequency ω of the constituent elementary Q-balls and the initial separation d between them, the parameter space of which will be charted in Section 3.2. This stage is usually what we refer to as the CSQ, and we will refer to the duration of this stage as the lifetime of a CSQ.
In Fig. 11, we see that if an elementary Q-ball and a dipole CSQ have the same energy, the oscillation period of the CSQ, defined as the period of Reφ(x = 0, y = 0), is greater than that of the elementary Q-ball. On the other hand, if an elementary Q-ball and a dipole CSQ have the same oscillation frequency, the energy of the elementary Q-ball is greater than that of the CSQ. In Fig. 12, we plot the evolution of the field values and their Fourier transforms at point (x = 0, y = 0) and point (x = 6.4, y = 6.4). We see that most power in the spectra occurs around the first few odd times of the base frequency ω 0 = 0.82. This is similar to the case of oscillons, the real scalar "cousin" of a Q-ball, where again only odd times of the base frequencies are significant in the spectra with a potential with the Z 2 symmetry [66]. In Fig. 13, we plot the phase portraits of various points on the y axis within the CSQ, which carve out near-rectangle areas. When the elementary Q-balls are not prepared to be exactly in anti-phase, these portrait rectangles will be rotated around the origin by an angle proportional to the initial phase misalignment. For an elementary Q-ball, the corresponding phase portrait is a circle at every field point, and for an oscillon the corresponding phase portrait is a line at every point, so the CSQ may be considered a hybrid between the two, whose phase portraits are different at different points.

(3) Fast decay
In this short stage, the energy and charge of the CSQ decrease dramatically in a short period of time and the Q-ball swapping period starts to increase unboundedly, which may be taken as the end of the life of the CSQ. The significant changes during the decay process can also be seen from the power spectrum of the fieldΦ k = k dk θ |Φ(k)| 2 where k = |k| and Φ(k) = d 2 xφ(x)e −ik·x . From the left plot of Fig. 14, we can see the CSQ power spectrum decreases significantly during the decay, while, from the right plot, we see that the second peak of the power spectrum of the outgoing waves shifts to the higher wave-numbers, and the power of the first peak at low wave-numbers decreases substantially.    In this stage, the energy becomes quasi-stable again at a second plateau, which is around half of the first plateau. After dropping a couple of orders of magnitudes in the fast decay stage, the charge in this stage decays exponentially with time. That is, in this stage, the charge has been mostly radiated away, and the remnant of the fast decay of the CSQ is essentially an oscillon, which only oscillates

Lifetimes of CSQs
An important goal of this paper is to determine the lifespans of CSQs. In this subsection, we shall survey the lifespans of the dipole CSQs in 2D.
Calculating the CSQ lifetimes is prone to accumulated numerical errors, as we need to run the code for very long time. So it is essential that we get all the numerical errors under sufficient control. Let us first check how the lifetimes vary with different numerical setups. From the left plot of Fig. 15, we see that, thanks to the effective absorbing boundary condition, a box size of around 100/m is sufficient for our purposes. Also, from the right plot, we see that our choice of the CFL factor 0.1 is also very accurate to compute the lifetime. Our main numerical uncertainties for the lifetimes of the CSQs are from the lattice resolution dx, or the number of lattice points for a fixed physical box size, which however are also under good control, as one may see in Fig. 16. If we define the lifetime of the CSQ stage by how long the charge-swapping period oscillates around a steady value, then the difference between the lifetimes of the 512 2 and 1024 2 runs is about 0.39% in Fig. 16. With the three resolutions, we find the convergence rate for the lifetime, due to the long evolution, is about second order. (At any given time, the differences between energies of the three resolutions always remain fourth order, as expected for the finite difference method we use.) Using the linear Richardson extrapolation, even for a second order convergence, we see that the difference between the extrapolated value and the 512 2 run is about 0.55%. In the following, we will survey the lifetimes in the parameter space of ω and d with a 512 2 lattice, and thus the accuracy of this survey is expected to be around 1%.
We also need to adopt a reasonable measure to define the lifetime of a CSQ. One way, as we just used above, is to extract the length of the plateau of the CSQ stage in the T swap evolution plot (such as in Fig. 16). One can similarly define the lifetime from other plots, but the easily recognizable feature of the swapping period plateau in Fig. 10 arguably provides the best measure for this purpose. After all, those objects have been dubbed charge-swapping Q-balls. So, for definiteness, we define the lifetime of a CSQ as the length of the thick bar in Fig. 10. Fig. 17 is the survey of the lifespans of the dipole CSQ in 2+1D in the parameter space of ω and d, where ω is the oscillating frequency of the constituent elementary Q-balls and d is the initial separation between them. We see that they can form in the diagonal strip of the parameter space, roughly between the lines of d + 60ω = (50.8, 56.8). That is, if the constituent Q-ball has lower frequencies, the separation between the constituents should be greater. When CSQs do form, as we see in Fig. 19, they evolve to some universal values of E c and Q s , and their lifetimes are typically of the order of a few thousands of swapping periods, or tens of thousands in terms of 1/m. The longest lifetime is observed at when ω formation of a CSQ. When trying to prepare a CSQ with initial elementary Q-balls placed too close to each other, they will violently repel each other and scatter. On the other hand, if they are placed too far away, they will attract and pass through each other and scatter.
If one wants to prolong the lifetimes of CSQs, changing the Q-ball frequency ω and the initial distance d is not the most efficient way. In our fiducial model (Eq. (2.2)), the only theory parameter is the coupling of the φ 6 term g. In Fig. 18, we show how the lifetime will vary with g. We see that the lifetime increases at least exponentially when g is increased. There are also some other effects caused by increasing g: 1) the plateau CSQ energy E c decreases slightly; 2) the CSQ charge Q s decreases; 3) the CSQ swapping period increases.
We also find that the initial relative phase of the two constituent Q-balls does not significantly affect the lifetime of CSQs. It mainly changes the relative magnitude of the real and imagery part of the field φ. For this reason, in the above we set the initial relative phase of the two constituent Q-balls to be zero.

Attractor behaviors of CSQs
As quasi-stable solitons, CSQs must be attractor solutions, that is, their formation should be insensitive to initial conditions: if there are favorable but otherwise quite generic initial conditions, they can form spontaneously. Arguably, this is what makes them relevant in many physical circumstances. In this section, we will explore this aspect of the dipole CSQs.
In Fig. 19, we see that for different initial ω and d the CSQ energy E c and charge Q s evolves to the same trajectories after the short relaxation. This is why in Fig. 17 we see that once CSQs are formed, their lifetimes are usually quite similar to each other. On the other hand, if the initial constituent Q-balls are (slightly) too close to each other, for 0.86 ω 0.92, the envelope of Q s will steadily increase to that of the quasi-stable CSQ, which leads to longer lifetimes; see   Figure 19: Attractor behaviors of the dipole CSQ in the E c and envelope of Q s plot. The initial configurations are constructed by superimposing elementary Q-balls.
We have so far only superimposed elementary Q-balls to prepare the CSQs. The attractor nature of the CSQs means that one should also be able to use other configurations to prepare CSQs. For example, we can prepare the CSQs with oscillating Gaussian lumps or deformed Q-balls Of course, if they are too different from Q-balls, CSQs can not form. Take the deformed Q-balls above for example. If Λ is too big, the two deformed Q-balls will repel without forming a CSQ, while if Λ is too small, the configuration will quickly shed away energy and charge and then decay to an oscillon without having the CSQ plateau. Figure 22: Evolution sequences of the charge and energy density of a quadrupole and octupole CSQs in 2+1D. In the first and third rows, the red color depicts positive charges and the blue color depicts negative charges. The corresponding energy density sequences are displayed in the second and fourth rows, with the blue depicting lower density and the red depicting higher density.
Here we have focused on the evolution of E c and Q s for different initial conditions, and we demonstrated an attractor behavior. One can also verify that similar attractor behavior can be found in the evolution of the charge-swapping frequencies, field-oscillating frequencies inside the CSQs and so on.

Higher multipole CSQs in 2+1D
The dipole CSQs are the simplest ones one can construct. As shown in [53], there are also more complex, higher multipole CSQs. In this section, we shall only briefly touch on these CSQs. In particular, we will look at the basic features and lifetimes of the quadrupole and octupole CSQs with equal charges, as shown in Fig. 22. There are many possible configurations for higher multiple CSQs, a full characterization of which is certainly interesting but is beyond the scope of this paper. Higher multiple CSQs can be prepared analogously as the dipole case. In Fig. 22, we place relevant numbers of equal and opposite Q-balls next to each other with no phase differences, and then we see, with time, the positive charge will turn negative and the negative charge will turn positive. That is, the charges are swapping with neighboring charges, rather than swapping with the opposite part with respect to the origin, which are the same kind of charges in the quadrupole and octupole CSQs case. This is due to the limitation of 2 spatial dimensions, and we will see that in 3+1D, we can construct higher multipole CSQs with charges swapped between the opposite parts. We see from Fig. 22 that the energy density of higher multipole CSQs are mostly spherically symmetric. The charge swapping periods for the quadrupole and octupole CSQ are about 29/m, while in the dipole case it is about 36/m. The evolution of energy E c of the quadrupole and octupole CSQ are plotted, together with the dipole CSQ, in Fig. 23. Similar to the dipole case, the energy initially drops rapidly before coming to a plateau, which lasts for thousands of oscillation periods, and then quickly decays to a second plateau. The quadrupole and octupole CSQ share almost the same duration of lifetime, which is less than the lifetime of the dipole CSQ. Also, we see that the first E c plateau of the quadrupole and octupole CSQ is about twice that of the first E c of the dipole CSQ, which interestingly is around the level of the second plateau of the quadrupole and octupole CSQ. Fig. 24 shows how the lifetime of CSQ changes with different initial Q-ball frequency ω and distance from the origin d. Roughly speaking, along the diagonal strip of Fig. 17, higher multipole CSQs share similar lifespans, while away from the diagonal strip (i.e., the case of ω = 0.84, d = 4.5), the lifespans can be much shorter.
However, for the model with g = 1/2, the charges of the quadrupole and octupole CSQ decrease much faster than those in the dipole CSQ, so even in the first plateau the charges in these higher multipole CSQs are very small at late times, due to the fast exponential decay. Nevertheless, fully    fledged higher multipole CSQs do exist for different g or with different forms of the potential [80]. For example, in Fig. 25, we see that for a smaller g the first charge plateau does form at least for quadrupole CSQs. As we have seen in Section 3.2, this is surprising as reducing g increases the amplitude of the charge densities, thus increasing the charge to energy ratio. Note that here Q s is defined only in the spirit of Fig. 5: It is defined as the same charge in one sector of the CSQ, that is, a quadrant for the quadrupole Q-balls CSQ.
In Fig. 26, we show how the CSQ lifetime changes with coupling g and its multipole. Note that for smaller g the higher multipole CSQs actually have longer lifetimes, and it appears that the lifetimes converge to the same value at high multipoles for different g. (Note that here the lifetime is directly extracted from the energy E c plateaus.)

CSQs in 3+1D
In this section, we shall briefly investigate CSQs and their lifetimes in 3+1D. We shall still construct the CSQs with elementary Q-balls. See Fig. 27 for the dependence of the total charge and the ratio between the total energy and the total charge on the oscillating frequency for 3+1D elementary Q-balls in the potential (2.3). From Fig. 27, we see that Q-balls become unstable when frequency ω 0.92 as their E/Q > 1.
To construct dipole CSQs in 3+1D, we still superimpose an elementary Q-ball and an elementary anti-Q-ball closely to each other so that their nonlinear cores overlap, and then let the configuration relax. In Fig. 28, we see that the charge swapping patterns of a dipole CSQ in 3+1D is very similar to the 2+1D counterpart (cf. Fig. 4), although the charge swapping period in 3+1D is about 1.5 times that in 2+1D for the dipole case. In Fig. 29, we see that the time evolution still has four stages: (1) Initial relaxation, (2) First plateau (CSQ stage), (3) Fast decay and (4) Second plateau (oscillon stage). The big difference is that the lifetimes of the CSQ stages are much shorter than those in 2+1D, which is in line with the fact that in 3+1D there are more "channels" to decay for the quasi-stable CSQs. However, it is also to do with the potential we are using. For the logarithmic potential, for example, the 3+1D CSQs are also very long lived [80], even for the quadrupole or octupole CSQs. In Fig. 30, we see that CSQs in 3+1D are also attractor solutions, as expected, and their lifetimes can be significantly prolonged by tuning initial ω and d.
There are also higher multipole CSQs in 3+1D, although their lifetimes are even shorter. In 3+1D, while the quadrupole CSQ still has a planar configuration, the octupole CSQ could be arranged as in Fig. 31, unlike the planar octupole configuration in 2+1D. In Fig. 32, we find a very short CSQ stage for the octupole CSQ, which is close to the oscillon plateau. If we were to construct the planar octupole CSQ in 3+1D, we would not be able to identity the first CSQ plateau in its evolution.   which uses both the Fourier space and real space coordinates. To extract boundary conditions in real space in terms of differential operators, we expand the square root around j =i k j k j + m 2 = 0 by a Taylor series (Alternatively, one may also use a Pade expansion [77].). The first two orders are given by Transforming the Fourier space coordinates to real space coordinates, we have where the + (−) sign is for the right (left) boundary. The 1st order ABC above is just that of the 1st order Higdon ABC with c 1 = 1, and, as mentioned, the 2nd order ABC above is just that of the 2nd order Higdon ABC with c 1 = c 2 = 1, upon using the linearized Klein-Gordon equation of motion. Engquist and Majda also generalized their absorbing boundary conditions for general curvilinear coordinates [77].