Charge-Swapping Q-balls and Their Lifetimes
UUSTC-ICTS/PCFT-21-05
Charge-Swapping Q-balls and Their Lifetimes
Qi-Xin Xie a,b , Paul M. Saffin c and Shuang-Yong Zhou a,ba Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei,Anhui 230026, China b Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China c School of Physics and Astronomy, University Park, University of Nottingham,Nottingham NG7 2RD, United KingdomE-mail: [email protected] , [email protected] , [email protected] Abstract:
For scalar theories accommodating spherically symmetric Q-balls, there are also towers ofquasi-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 simulationswith efficient second order absorbing boundary conditions. We find that the evolution of a CSQtypically consists of 4 distinct stages: initial relaxation, first plateau (CSQ stage), fast decay andsecond plateau (oscillon stage). We chart the lifetimes of CSQs for different parameters of the initialconditions and of the potential, and show the attractor behavior and other properties of the CSQs. a r X i v : . [ h e p - t h ] J a n ontents A.1 Sommerfeld’s absorbing boundary condition 27A.2 Engquist-Majda’s absorbing boundary conditions 27
Non-perturbative configurations, such as solitons, together with their fascinating properties are integralparts of understanding field theories. Apart from topological defects, which are stable due to thepresence of topological charges, there also exist non-topological solitons, such as Q-balls [1, 2]. Q-ballsare spatially localized and stationary but non-static, and are stable due to the presence of Noethercharges. For example, Q-balls exist in U(1) symmetric scalar field theories with a potential that growsslower 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 localizedlump rather than dissipate to infinity. In other words, the Q-ball condensate is the energeticallypreferred state for such a system. The properties and dynamics of Q-balls have been extensivelystudied [3–24]. Crucially, they may play an important role in the early universe (see, e.g., [25–41]) andcan be candidates of dark matter (see, e.g., [42–50]). 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 configura-tions increase the gradient energy. Recently, non-spherically symmetric, composite Q-balls have beenidentified [53]. In the theories where the spherically symmetric Q-balls exist, there are also a tower ofcomposite Q-balls with different multipoles, within which both positive and negative charges co-existand swap with time (see Fig. 2 for a dipole CSQ and see Fig. 22 for higher multipole CSQs). Theyare thus dubbed charge-swapping Q-balls (CSQs) [53]. Nevertheless, their energy densities remainmostly spherically symmetric (see Figs. 3 and 22). They have been shown to exist in 2+1D and 3+1D, – 1 – nd can be prepared by simply placing positive and negative charge elementary Q-balls (or in generalsimply 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.As expected, CSQs are only quasi-stable and will ultimately decay. In this sense, althoughmuch more complex, they are spiritually like oscillons [54, 55] which are spatially localized and quasi-stationary and exist even in real scalar field theories. (Indeed, CSQs decay to oscillons, as will seein this paper.) Oscillons do not contain any Noether charges but nonetheless live for an extendedperiod of time (see, e.g., [56–75] for more details). While the existence of CSQs has been identified in[53], their detailed properties have not been sufficiently explored. In particular, Ref. [53] makes use ofperiodic boundary conditions, which are not suitable for extracting the lifetimes of the CSQs.In this paper, we shall investigate the properties and evolution of the CSQs in more detail, anddetermine their lifetimes. We will focus on dipole CSQs in 2+1D and 3+1D in the simplest φ potential,with quadrupole and octupole CSQs also briefly touched on. To be able to determine the lifetimesof CSQs, we shall utilize absorbing boundary conditions (ABCs). This is crucial as our method ofobtaining the CSQs simply involves superimposing elementary Q-balls, which is, of course, not theconfiguration of a CSQ. Thus, in the initial relaxing phase, the configuration radiates a substantialamount of energy (see, e.g., Fig. 8). With the previous periodic boundary conditions, this radiationwould travel back and echo around the periodic space, continuously perturbing the CSQ. So, while theperiodic boundary conditions of [53] are sufficient to determine the existence of CSQs, the lifetimes ofthe CSQs can not be reliably determined in that setup. Therefore, for an accurate determination ofthe 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 ABCscan be implemented at higher orders. For our applications, we find that the second order ABCs aresufficient to determine the lifetimes within a couple of percent, and Hidgon’s second order ABCs witha judicial choice of the c and c generally give the best absorbing effects, which are used to producemost of the results in the paper. Another way to damp radiation in the far field regions would be to adda Kreiss-Oliger term. However, in our simulations, we do not find the Kreiss-Oliger term significantlyincreases the accuracy, and therefore this artificial term is not adopted for the results presented in thepaper.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 (oscillonstage). The presence of the initial relaxation stage is, as mentioned, because the CSQ is not preciselyprepared. Indeed, we find that the CSQ is an attractor solution, as a quasi-stable configuration shouldbe, and can be formed with relatively general initial configurations (see Figs. 19, 20 and 21). Aftersettling down, the CSQ stage is characterized by a mostly spherical and slightly oscillating energydensity profile (see Fig. 3) with charges swapping with time (see Fig. 2). The total energy and chargedecrease slowly with time, and the swapping period remains mostly constant (see Fig. 10), which is afew times the oscillating period of an elementary Q-ball. We find that long-lived dipole CSQs in 2+1Dcan be achieved in a diagonal strip of the parameter space of the initial Q-ball frequency ω and theinitial separation between them d (see Fig. 17). We also find the the lifetime of the dipole CSQ hasan exponential dependence on the φ coupling g (see Fig. 18). The CSQ stage is followed by a shortstage of fast decay of both energy and charges, the end result of which is, interestingly, an oscillonwith roughly half of the total energy (and with very small charge densities). In this oscillon stage, thecharge of the Q-ball components comprising the CSQ, Q s , decays exponentially. Although within the – 2 – ime limits of our simulations we have not seen the decay of the second oscillon plateau, the oscillonswill 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 butnevertheless 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 appearsthat the lifetimes converge to the same value for different g (see Fig. 26). In 3+1D, the lifetimes ofthe CSQs are much shorter (see Figs. 28 and 29), presumably because there are more possible decaymodes 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 arealso long-lived, even for the higher multipoles [80].The paper is organized as follows. In Section 2, we first introduce the fiducial field model weconsider in this paper, define a few quantities that will be used later, and review elementary Q-ballsand CSQs; then we specify the numerical implementations and introduce the Higdon’s ABCs thatare used to produce the results in the paper; additional ABCs, which are used to cross-check someof 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 evolutionare detailed, and various properties of the dipole CSQs are explored, in particular the lifetimes of thedipole CSQs are surveyed for different parameters ω , d and g . In Section 4, we briefly study highermultipole CSQs in 2+1D. In Section 5, we briefly study CSQs in 3+1D. The existence of Q-balls in a complex scalar field theory requires a potential with an attractive naturethat condenses field perturbations (or, loosely speaking, “particles” in the language of quantum fieldtheory), rather than dissipates them. The simplest Z symmetric ( ϕ → − ϕ ), polynomial potentialthat supports Q-balls is given by S = (cid:90) d d +1 ˜ x (cid:104) − (cid:12)(cid:12)(cid:12) ∂ϕ∂ ˜ x µ (cid:12)(cid:12)(cid:12) − V ( | ϕ | ) (cid:105) , with V ( | ϕ | ) = m | ϕ | − λ | ϕ | + ˜ g | ϕ | , (2.1)where d is the number of the spatial dimensions. Defining dimensionless variables x µ = m ˜ x µ , φ = λ / ϕ/m and g = ˜ gm /λ , the action can be re-written as S = λ − m − d (cid:90) d d +1 x (cid:104) − | ∂ µ φ | − V ( | φ | ) (cid:105) , with V ( | φ | ) = | φ | − | φ | + g | φ | , (2.2)which only contains one dimensionless free parameter g . In other words, we are expressing the co-ordinates in the units of the particle mass, m − , and the field value in the units of mλ − / . Unlessotherwise stated, in the following, we will consider the potential with g = 1 / V ( | φ | ) = | φ | − | φ | + 12 | φ | . (2.3)The energy density and conserved energy for a spatial volume are given respectively by H = | ˙ φ | + |∇ φ | + V, E = (cid:90) d d x H . (2.4)Since the action is invariant under a global symmetry φ → e iα φ ( α being constant), by Noether’stheorem, there is a conserved current and thus a conserved charge for this symmetry. The associated – 3 – r f (r) =0.74=0.78=0.82=0.86=0.90=0.94=0.98 Q E / Q Figure 1 : Profiles (left plot), total charges and ratios of total charge to total energy (rightplot) of elementary Q-balls with different frequencies for potential (2.3). The fact that
E/Q < m . charge density and conserved charge are given respectively by ρ = i ( φ ˙ φ ∗ − φ ∗ ˙ φ ) = − φ ˙ φ ∗ ) = 2( φ ˙ φ − ˙ φ φ ) , Q = (cid:90) d d xρ, (2.5)where we have defined φ = φ + iφ . (2.6)Numerically, we are simulating the system in a large but finite box, so the total energy E and totalcharge Q will refer to the total energy and charge in the simulation box.Note that for this potential the | φ | term is negative, which provides the attractive force mentionedabove, and thus Q-balls can form. Technically, this means that V / | φ | has a minimum away from | φ | = 0. Since V / | φ | at | φ | = 0 gives the mass of the particle in the free theory, V / | φ | having aminimum at | φ | (cid:54) = 0 means in the interacting theory there are condensate configurations where themass 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 leadinginteracting potential being shallower than the free quadratic potential, particles tend to condenserather than propagate away from each other.Elementary Q-balls are stable stationary solutions that are spherically symmetric. To obtain theradial profile of an elementary Q-ball, we take the following ansatz φ ( t, r ) = f ( r ) e iωt , (2.7)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 equationfor f ( r ) d fdr = 1 − dr dfdr − ω f + 12 ∂V∂f , (2.8)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 – 4 – t rest df (0) /dr = 0, moves with a time-dependent friction term and in the effective potential V eff = ω f − 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) (cid:54) = 0 and f ( ∞ ) = 0 without oscillations.If ω is greater than the perturbative mass of the free theory, f = 0 is a local minimum of V eff , sucha non-oscillating solution is impossible. To get an elementary Q-ball solution, the upper bound of thefrequency is given by ω = d V / d f | f =0 = V /f | f =0 = 1. If ω is smaller than the minimum of V /f at some f (cid:54) = 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 − = V /f | f = f .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 thetotal 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 ( r ) cos ωt. (2.10)These are also localized lumps, very similar to the elementary Q-ball. Oscillons can be supportedby merely a real scalar field theory. For a complex field, if the initial configuration is real, the fieldremains real through out its time evolution. Indeed, for oscillons to exist, we also need a potentialthat 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, oscillonsare only quasi-stable solutions, although in 2+1D their life time can be long [66]. Recently, it has been observed that, apart from the elementary Q-balls, which have been proven tobe stable [2], there are also composite, quasi-stable solutions in the theory where elementary Q-ballsexist [53]. These composite Q-balls are not spherically symmetric. They can form when Q-balls andanti-Q-balls are placed closely together with their cores overlapping. Remarkably, the positive andnegative charges of the composite Q-balls swap with time, and thus they are dubbed charge-swappingQ-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 onthe negative y -axis, i.e. a system with reflection symmetry about x -axis. The initial relative phasedifference between the two Q-balls can be chosen to be zero. Because of this placement, the realcomponent of the scalar field is symmetric about the x -axis and y -axis, while the imaginary componentis 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 itwill quickly relax to a CSQ, as we shall see later. In fact, it is not essential to superpose exact Q-ballsand anti-Q-balls in the initial preparation; we may as well initially superpose oscillating lumps thatresemble Q-balls and anti-Q-balls. That is, CSQs are attractor solutions of the theory. See Figs. 2 and3 for the sequence of one charge-swapping period for the already relaxed dipole CSQ, and see Fig. 4 – 5 – igure 2 : Evolution sequence of the charge density of the dipole CSQ in one charge-swappingperiod T swap . The red color depicts positive charge density and the blue color depicts negativecharge density. T swap is usually a few times the oscillation period of the field, which is roughly T ≡ π/m = 2 π . Figure 3 : Evolution sequence of the energy density of the dipole CSQ in the same charge-swapping period T swap as Fig. 2, with the blue depicting lower density and the red depictinghigher density. for the evolution of the different charge integrals defined shortly below. On the other hand, the energydensity of a CSQ is mostly spherically symmetric. There are also more complex CSQs with moreQ-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 integrationsover various different regions of the dipole CSQ (the center of the CSQ is placed at the origin of thecoordinate system): • E and Q : the total energy and charge in the simulation box respectively • Q + : the positive charge over the whole simulation box – 6 –
040 2042 2044 2046 2048 t/T -2-1012 c ha r ge Q s Q up Q + Figure 4 : Evolutions of three different charge integrals Q s , Q up and Q + with time, theabsolute values of which are mostly the same. T = 2 π/m is roughly the oscillation period ofthe field φ . Figure 5 : Circular disk region used to evaluate E c and Q c and semi-circular disk region toevaluate energy E s and charge Q s . The radius of the semi-circle is 14, in units of 1 /m . (In3D, we will use a corresponding semi-ball with radius of 20.) • E up and Q up : the energy and charge obtained by integrating over the upper half space ( y > z > • E s and Q s : the energy and charge obtained by integrating over an upper semi-circular disk(2D) or an upper semi-ball (3D) with a radius of 14 (for 2D) or 20 (for 3D) around the CSQrespectively (see the black solid line in Fig. 5 for 2D) • E c and Q c : the energy and charge obtained by integrating over a circular disk (2D) or a ball – 7 –
10 20 30 40 50
MPI Processes T i m e / s S peed * s -3 TimeReciprocal of Time
Figure 6 : Times used to run simulations with a 256 grid for 145600 time steps on aworkstation with dual CPUs (two Intel Xeon Platinum 8280 CPUs) and shared memory.The “speed” is defined as the reciprocal of the time duration. (3D) with a radius of 14 (for 2D) or 20 (for 3D) around the CSQ respectively (see the thick,dashed line in Fig. 5 for 2D)While the existence of CSQs has been firmly shown in [53], the properties of CSQs are yet to bestudied 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, whichpropagate back to affect the CSQs in a periodic box. In this paper, we shall set up lattices that haveabsorbing boundary conditions that can absorb radiation effectively from the CSQs, which allows usto investigate CSQs in much more detail and determine the lifetimes of various CSQs. Our lattice code makes use of the open-source LATfield2 C++ library [81], which defines objects suchas 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. Wemodified the library to incorporate several absorbing boundary conditions, which will be introducedin Section 2.3 and Appendix A. We use a 4th order finite difference stencil for spatial derivativesand evolve in time with the classical Runge-Kutta 4th order method. The code has excellent parallelspeedups for many CPU cores with MPI as seen in Fig. 6.We shall superimpose elementary Q-ball solutions, both for φ (0 , x ) and ˙ φ (0 , x ), as the initialconfiguration and let it relax to obtain CSQs. The internal frequency ω of the elementary Q-ball andthe initial distance d between the elementary Q-balls are the free parameters we choose, and we willchart the lifetime of the CSQ in this two dimensional parameter space.The Courant-Friedrichs-Lewy (CFL) factor d t/ d x is set to be 0.1 both for 2D and 3D simulations.Unless otherwise stated, in 2D simulations, we will use a 512 lattice and the grid spacing is d x = 0 . lattice frequently used to check for convergence, and in 3D we will use a 256 latticeand the grid spacing is d x = 0 .
4, with a 512 lattice to check for convergence. As we will see ( e.g. , inFigs. 16 and 15) that these numerical settings are sufficient for our purposes. – 8 –
50 100 150 200 t/T E c SommerfeldEM1EM2Hig1(2)Hig2(1,2.5)reference 0 50 100 150 200 t/T -0.1-0.0500.050.1 e rr o r o f E c t/T -14-70714 Q s
106 108 110 112 114 116 t/T -0.5-0.2500.250.5 e rr o r o f Q s Figure 7 : Comparisons of simulations with various absorbing boundary conditions (ABCs)in the initial relaxation phase. The left two plots are from the reference simulation, and theright two plots are deviations of simulations with various ABCs from the reference simulation.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 suchthat the radiation has not reached the boundaries in the run. Dashed lines represent first-order ABCs and solid lines represent second-order ones. We see that second order ABCsgenerally perform better than the first order ABCs, and the Hig2(1,2.5) ABC has the bestbehavior with regard to eliminating the initial relaxing radiation. T = 2 π/m is roughly theoscillation period of the field φ . As mentioned above, to determine the lifetime of a CSQ, we need to run the code for an extended periodof 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 duringthe initial relaxation phase, and this radiation travels back to interfere with the CSQs when usingperiodic boundary conditions. In reality we are interested in the lifetimes of CSQs in Minkowski spacewhere radiation should simply propagate to infinity. To use a finite computation region to solve aninfinite domain problem, we may make use of suitable absorbing boundary conditions (ABCs). Unlessthe problem is highly symmetric (say spherical symmetry) or the outgoing waves are very simple, – 9 –
BCs are usually not perfect. However, a good ABC should let the majority of outgoing waves gothrough the boundary transparently and only incur minor reflections. We will explore several ABCsin 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 inthe majority of the simulations in this paper, while the other ABCs are used for sanity checks. Wewill 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]: M (cid:89) j =1 (cid:18) ∂∂t ± c j ∂∂x i (cid:19) φ | x i = a = 0 , (2.11)where the + ( − ) sign is for a right (left) boundary, t and x i are Cartesian coordinates and c j , to bechosen by the user for specific problems, are the phase velocities of the normal outgoing plain wavesthat can be absorbed exactly. For the massive Klein-Gordon equation, c j should be chosen to be noless than 1. It is clear that c j should be chosen to annihilate the dominant wavenumbers near theboundary, which can be obtained by performing Fourier transforms of the field near the boundary. Arough guide is that c j is to be chosen to minimize the reflection rate between the amplitudes of theincoming waves and the outgoing waves R [ k i ] = M (cid:89) j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:112) ( k i ) + m + c j k i − (cid:112) ( k i ) + m − c j k i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.12)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 velocitiesare not c j , the Higdon ABC can still often absorb much of them. In practice, the most commonlyused Higdon ABCs are when M = 1 ,
2, which are simply given by (cid:18) ∂∂t ± c ∂∂x i (cid:19) φ | x i = a = 0 , (2.13) (cid:18) ∂ ∂t ± ( c + c ) ∂∂x i ∂t + c c ∂ ∂ ( x i ) (cid:19) φ | x i = a = 0 , (2.14)where the + ( − ) sign is for a right (left) boundary. Note that the 1st and 2nd order Higdon ABCwith c = 1 and c = c = 1 are actually equivalent to the 1st and 2nd order Engquist-Majda ABCrespectively, upon using the equations of motion. It is advised that, without prior knowledge of thespectrum of the outgoing waves, the default 2nd Higdon parameters can be chosen as c = c = 1 [78].As we find CSQs by superimposing elementary Q-balls and letting them relax, the initial relaxationphase of the CSQ evolutions produces the largest amount of radiation, and we shall choose our ABCsto maximize the absorption of the radiation at the boundary from this phase. We shall compare theabsorbing effects of different ABCs and select the best one among them. Since the relaxation phaseis short, we can actually simulate this phase with a sufficiently large lattice such that by the end ofthis phase the radiation has not reached the boundary yet. We compare the effects of different ABCswith this reference run, which allows us to pick the one with the smallest deviations. The ABCs thatwe have tested against the reference run are: • Sommerfeld : Sommerfeld boundary condition with v = 1, φ = 0 and no non-wavelike term h ( t ) • EM1 : Engquist-Majda’s 1st order condition, which is equivalent to Higdon’s 1st order conditionwith c = 1 – 10 – rrors Sommerfeld EM1 EM2 Hig1(2) Hig2(1,2.5) σ E c . × − . × − . × − . × − . × − σ Q s . × − . × − . × − . × − . × − Table 1 : Standard deviations of the various ABCs (see the text below Eq. (2.14)) against thereference simulation for energy E c ( t ) and charge Q s ( t ), which are defined in Section 2.1. Thereference simulation is set up such that the lattice is sufficiently large so that the radiationhas not reached the boundaries at the end of the run. The Hig2(1,2.5) ABC has the bestaccuracy 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 conditionwith c = c = 1 • Hig1(2) : Higdon’s 1st order condition with c = 2 • Hig2(1,2.5) : Higdon’s 2nd order condition with c = 1, c = 2 . grid for a runof 32769 steps. We use the fiducial model and prepare the consistent Q-balls with ω = ± .
84 andplace them together with a separation d = 2 .
8. We compute energy E c ( t ) and charge Q s ( t ), which aredefined in Section 2.1, as functions of time, and compute the standard deviations: σ E c = (cid:34) N (cid:88) i =1 ( E ABC c ( t i ) − E ref c ( t i )) N (cid:35) / , (2.15) σ Q s = (cid:34) N (cid:88) i =1 ( Q ABC s ( t i ) − Q ref s ( t i )) N (cid:35) / . (2.16)From Table 1 and Fig. 7, we see that all these ABCs, especially the second order ABCs, are relativelygood, and the Hig2(1,2.5) ABC appears to have the best accuracy in eliminating the outgoing radiationfrom the initial relaxation phase of the CSQ. The effectiveness of the Hig2(1,2.5) ABC can also beseen by looking at far-field regions of the CSQ, i.e. , at points far from the CSQ, in which case theruns 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 CSQsare probably most sensitive to the ABCs. Indeed, if we use different ABCs to find the lifetimes, weget 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 thatthe ABC absorbs the radiation better at the boundaries, on which basis we will use the Hig2(1,2.5)ABC as our default ABC. In this section, we will explore the properties of the simplest dipole CSQ in 2+1D, derived from afiguration with a Q-ball and an equal (opposite) charge anti-Q-ball. We will see that they have distinct – 11 – t/T ene r g y EE c (1) Initial relaxation(2) First plateau(CSQ stage) (4) Second plateau(Oscillon stage)(3) Fast decay Figure 8 : Evolution of total energy E and energy E c , which are defined in Section 2.1. t isin the units of the oscillation period of the field T , which is 2 π in the units of 1 /m . Inset: anenlarged view of the initial relaxation phase where there are noticeable differences between E and E c . The constituent elementary Q-balls are initially placed d = 2 . ω = ± . t/T -6 -4 -2 en v e l ope o f Q s Figure 9 : Envelope of the Q s evolution, where Q s is defined in Section 2.1. Q s oscillatesquickly according to the charge swapping frequency T swap within the envelope. All the settingsare identical to Fig. 8. T = 2 π/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 areattractor solutions. We will also chart the lifetimes of these CSQs for different initial conditions. As we mentioned above, we prepare the dipole CSQ by superimposing an elementary Q-ball and anequal charge elementary anti-Q-ball with a short separation. An elementary Q-ball or anti-Q-ball istotally specified by ω in Eq. (2.7), that is, f ( r ) is determined by ω . For a typical CSQ prepared in this – 12 – t/T T s w ap Figure 10 : Charge-swapping period T swap . T = 2 π/m = 2 π is roughly the oscillation periodof the field φ . All the settings are identical to Fig. 8. t/T pe r i od period of Re( (0,0))period of a Q-ball with same energy Figure 11 : Evolution of the period of the real part of the field φ at the origin ( x = 0 , y = 0).All the settings are identical to Fig. 8. The orange dashed line is the oscillation period of anelementary Q-ball with the same energy as the CSQ. Note that if we prepare the CSQ withelementary Q-balls with different frequencies, the period of the resulting CSQ remains thesame. way, there are four distinct stages of its evolution, as can be seen in Figs. 8, 9, 10 and 11: (1) Initialrelaxation, (2)
First plateau (CSQ stage), (3)
Fast decay and (4)
Second plateau (oscillon stage). Inthe following, we will discuss these four stages separately. (1) Initial relaxation
As we set up the initial configuration by simply superimposing two elementary Q-balls, whichwe emphasize are not the quasi-stable CSQs, this stage is characterized by fast decrease of energyand 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 – 13 – bout a hundred oscillations, after which the initial configuration settles down to become a CSQ. Theradiation of energy from the initial lump can be easily seen in the inset of Fig. 8, where the energyin 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 ahundred oscillations for this fiducial model. During this transition stage, the swapping period of thelump T swap , which is defined at zero points of Q s by summing the time durations to the previous andnext zero point, decreases quickly to the plateau of the CSQ (see Fig. 10), and the oscillating periodof the real part of the field φ at the origin, which is defined at zero points of Re φ ( t, ,
0) by summingthe time durations to the previous and next zero point, quickly settles to a value that is slightly largerthan the intrinsic period of the elementary Q-ball with the same energy(see Fig. 11). From the trendsin 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 slowlywith time; see Figs. 8 and 9. While the average value of the swapping period of the CSQ , T swap , alsoremains mostly unchanged, its value oscillates noticeably around the average; see Fig. 10. This stagelasts for an extended period of time, the length of which depends on the oscillation frequency ω ofthe constituent elementary Q-balls and the initial separation d between them, the parameter space ofwhich will be charted in Section 3.2. This stage is usually what we refer to as the CSQ, and we willrefer 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, theoscillation period of the CSQ, defined as the period of Re φ ( x = 0 , y = 0), is greater than that ofthe elementary Q-ball. On the other hand, if an elementary Q-ball and a dipole CSQ have the sameoscillation 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) andpoint ( x = 6 . , y = 6 . ω = 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 apotential with the Z 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 notprepared to be exactly in anti-phase, these portrait rectangles will be rotated around the origin byan angle proportional to the initial phase misalignment. For an elementary Q-ball, the correspondingphase portrait is a circle at every field point, and for an oscillon the corresponding phase portrait is aline at every point, so the CSQ may be considered a hybrid between the two, whose phase portraitsare different at different points. (3) Fast decay In this short stage, the energy and charge of the CSQ decrease dramatically in a short period oftime and the Q-ball swapping period starts to increase unboundedly, which may be taken as the end ofthe life of the CSQ. The significant changes during the decay process can also be seen from the powerspectrum of the field ˜Φ k = k (cid:82) d k θ | Φ( k ) | where k = | k | and Φ( k ) = (cid:82) d xφ ( x ) e − i k · x . From the leftplot 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 tothe higher wave-numbers, and the power of the first peak at low wave-numbers decreases substantially. – 14 –
008 2012 2016 t/T -0.02-0.0100.010.02 ( . , . ) Re( )Im( ) t/T -1-0.500.51 R e (( , ) -4 -2 | ( . , . ) | -2 | ( , ) | =0.82 Figure 12 : The field values at ( x = 6 . , y = 6 .
4) and ( x = 0 , y = 0) and their temporalFourier transforms in the CSQ stage. Φ ω ( x, y ) is the Fourier transform of φ ( t, x, y ). Dashedlines in the two bottom plots show the odd multiples of ω = 0 .
82, which match the spectralpeaks. -1 0 1
Re( (0,1.6)) -1-0.500.51 I m (( , . )) -0.5 0 0.5 Re( (0,3.2)) -0.500.5 I m (( , . )) -0.04 -0.02 0 0.02 0.04 Re( (0,8.4)) -0.04-0.0200.020.04 I m (( , . )) Figure 13 : Phase portraits on three points on the y axis in the CSQ stage. (4) Second plateau (oscillon stage) In this stage, the energy becomes quasi-stable again at a second plateau, which is around half ofthe first plateau. After dropping a couple of orders of magnitudes in the fast decay stage, the charge inthis stage decays exponentially with time. That is, in this stage, the charge has been mostly radiatedaway, and the remnant of the fast decay of the CSQ is essentially an oscillon, which only oscillates – 15 – k before decayafter decay k -3 before decayafter decay Figure 14 : Power spectrum of φ ( x ) over the whole lattice (left plot) and over a smallrectangular area outside the CSQ along the positive x -axis (right plot). Typical resultsbefore the decay (orange line, t ≈ T ) and after the decay (green line, t ≈ T ) areshown. The constituent elementary Q-balls with ω = ± .
86 are initially placed at d = 4 . t/T E c default size1.5 times2 times t/T E c CFL=0.4CFL=0.2CFL=0.1
Figure 15 : Convergence study with different physical box sizes (left) and different Courant-Friedrichs-Lewy (CFL) factors (right). In the left plot, the blue solid line is the physical boxsize we use for most of our simulations, which is 102.4 in each spatial direction, and the redand orange lines are 1 . x kept at 0.2. along one linear direction in the phase space of Re φ and Im φ . An important goal of this paper is to determine the lifespans of CSQs. In this subsection, we shallsurvey the lifespans of the dipole CSQs in 2D.Calculating the CSQ lifetimes is prone to accumulated numerical errors, as we need to run thecode 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 – 16 – t/T E c Figure 16 : Convergence study with different spatial resolutions. The lifetime decreasesslightly as the resolution increases. We use a 512 lattice to survey the lifespans of the CSQ,and the uncertainty due to the lattice resolution is within 1%. The physical box size is 102 . and thus d x = 0 . sufficient for our purposes. Also, from the right plot, we see that our choice of the CFL factor 0 . x , 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 theCSQ stage by how long the charge-swapping period oscillates around a steady value, then the differencebetween the lifetimes of the 512 and 1024 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 anygiven time, the differences between energies of the three resolutions always remain fourth order, asexpected for the finite difference method we use.) Using the linear Richardson extrapolation, even fora second order convergence, we see that the difference between the extrapolated value and the 512 run is about 0.55%. In the following, we will survey the lifetimes in the parameter space of ω and d with a 512 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 wejust 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 recognizablefeature 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 thelifetime 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 initialseparation between them. We see that they can form in the diagonal strip of the parameter space,roughly between the lines of d + 60 ω = (50 . , . E c and Q s , and their lifetimes are typically ofthe order of a few thousands of swapping periods, or tens of thousands in terms of 1 /m . The longestlifetime is observed at when ω (cid:39) . d (cid:39) .
4. Of course, not every set of parameters leads to the – 17 – .75 0.8 0.85 0.9 0.95 of constituent Q-balls i n i t i a l d i s t an c e d Figure 17 : Lifespans of a dipole CSQ with different initial parameters. The parameterspace is spanned by the internal frequency ω of constituent Q-balls and the initial distance d between the Q-ball centers. Lifetimes are shown in units of T = 2 π/m . The white regions arewhere CSQ can not form. The two dashed guiding lines are d +60 ω = 50 . d +60 ω = 56 . formation of a CSQ. When trying to prepare a CSQ with initial elementary Q-balls placed too closeto each other, they will violently repel each other and scatter. On the other hand, if they are placedtoo 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 initialdistance d is not the most efficient way. In our fiducial model (Eq. (2.2)), the only theory parameter isthe coupling of the φ term g . In Fig. 18, we show how the lifetime will vary with g . We see that thelifetime increases at least exponentially when g is increased. There are also some other effects causedby 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 significantlyaffect the lifetime of CSQs. It mainly changes the relative magnitude of the real and imagery part ofthe field φ . For this reason, in the above we set the initial relative phase of the two constituent Q-ballsto be zero. As quasi-stable solitons, CSQs must be attractor solutions, that is, their formation should be insensitiveto initial conditions: if there are favorable but otherwise quite generic initial conditions, they can formspontaneously. Arguably, this is what makes them relevant in many physical circumstances. In thissection, 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 evolvesto the same trajectories after the short relaxation. This is why in Fig. 17 we see that once CSQsare formed, their lifetimes are usually quite similar to each other. On the other hand, if the initialconstituent Q-balls are (slightly) too close to each other, for 0 . (cid:46) ω (cid:46) .
92, the envelope of Q s willsteadily increase to that of the quasi-stable CSQ, which leads to longer lifetimes; see Fig. 20. – 18 – .42 0.46 0.5 0.54 0.58 g li f e t i m e / T data pointexponential fit Figure 18 : Dependence of the dipole CSQ lifetime on the coupling constant g . The greendashed line is the exponential fit a e b g with a = 3 . × − , b = 51 .
5. The data pointsare obtained by superposing two Gaussian lumps with an oscillating frequency ω = 0 .
84 andwith the same peak amplitude and width as those of an elementary Q-ball with ω = 0 .
84 and g = 0 .
5, for more equal comparison. We vary distance d to get the maximum lifetime for agiven g . t/T E c =0.88,d=1.6=0.84,d=4.0=0.80,d=6.4=0.76,d=8.8 t/T -4 -2 en v e l ope o f Q s =0.88,d=1.6=0.84,d=4.0=0.80,d=6.4=0.76,d=8.8 Figure 19 : Attractor behaviors of the dipole CSQ in the E c and envelope of Q s plot. Theinitial configurations are constructed by superimposing elementary Q-balls. We have so far only superimposed elementary Q-balls to prepare the CSQs. The attractor natureof the CSQs means that one should also be able to use other configurations to prepare CSQs. Forexample, we can prepare the CSQs with oscillating Gaussian lumps φ ( t, x ) = Ae − ( x − x ) /σ e iωt , (3.1)or deformed Q-balls φ ( t, x ) = Λ f ( | x − x | ) e iωt , (3.2) – 19 – t/T -4 -2 en v e l ope o f c ha r ge Figure 20 : Evolution of the envelope of Q s for the case where ω = 0 . d = 0 .
1. In this case,the initial amplitude of Q s is smaller than the steady value of the CSQ, and the amplitude of Q s will increase from 0 .
29 to 0 .
53 before the decay of the CSQ. This leads to a longer lifetime. t/T E c =1=1.1=0.9Gaussian t/T -4 -2 en v e l ope o f Q s =1=1.1=0.9Gaussian Figure 21 : Evolution of energy E c and charge Q s for deformed CSQs. The deformationis done by multiplying the profile f ( r ) of the constituent elementary Q-ball by a factor ofΛ, Λ = 1 being the un-deformed elementary Q-ball. Different cases are shifted to matchthe fast decay stages. We see that different initial configurations are attracted to the samequasi-stable CSQ configuration. where A, σ,
Λ are constants and f ( | x − x | ) is the elementary Q-ball profile. We find that if they aresufficiently close to the elementary Q-balls, CSQs can form. Indeed, these configurations evolve tothe same trajectory as those prepared with elementary Q-balls, although their lifespans are slightlyshorter; see Fig. 21.Of course, if they are too different from Q-balls, CSQs can not form. Take the deformed Q-ballsabove for example. If Λ is too big, the two deformed Q-balls will repel without forming a CSQ, whileif Λ is too small, the configuration will quickly shed away energy and charge and then decay to anoscillon without having the CSQ plateau. – 20 – igure 22 : Evolution sequences of the charge and energy density of a quadrupole and oc-tupole CSQs in 2+1D. In the first and third rows, the red color depicts positive chargesand the blue color depicts negative charges. The corresponding energy density sequences aredisplayed in the second and fourth rows, with the blue depicting lower density and the reddepicting higher density. Here we have focused on the evolution of E c and Q s for different initial conditions, and wedemonstrated an attractor behavior. One can also verify that similar attractor behavior can be foundin the evolution of the charge-swapping frequencies, field-oscillating frequencies inside the CSQs andso on. The dipole CSQs are the simplest ones one can construct. As shown in [53], there are also morecomplex, higher multipole CSQs. In this section, we shall only briefly touch on these CSQs. In – 21 – t/T E c dipole CSQquadrupole CSQoctupole CSQ Figure 23 : Energy evolution of different multipole CSQs. The initial distances betweenconstituent Q-balls and the coordinate origin are: 2 for the dipole CSQ, 4 for the quadrupoleCSQ and 8 for the octupole CSQ, in units of 1 /m . The frequencies of constituent Q-balls areall chosen to be ω = 0 . particular, we will look at the basic features and lifetimes of the quadrupole and octupole CSQs withequal 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 relevantnumbers 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 withrespect 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 constructhigher multipole CSQs with charges swapped between the opposite parts. We see from Fig. 22 thatthe energy density of higher multipole CSQs are mostly spherically symmetric. The charge swappingperiods for the quadrupole and octupole CSQ are about 29 /m , while in the dipole case it is about36 /m .The evolution of energy E c of the quadrupole and octupole CSQ are plotted, together with thedipole CSQ, in Fig. 23. Similar to the dipole case, the energy initially drops rapidly before coming to aplateau, 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 thelifetime of the dipole CSQ. Also, we see that the first E c plateau of the quadrupole and octupole CSQis about twice that of the first E c of the dipole CSQ, which interestingly is around the level of thesecond plateau of the quadrupole and octupole CSQ. Fig. 24 shows how the lifetime of CSQ changeswith different initial Q-ball frequency ω and distance from the origin d . Roughly speaking, along thediagonal strip of Fig. 17, higher multipole CSQs share similar lifespans, while away from the diagonalstrip (i.e., the case of ω = 0 . , d = 4 . g = 1 /
2, the charges of the quadrupole and octupole CSQ decreasemuch faster than those in the dipole CSQ, so even in the first plateau the charges in these highermultipole CSQs are very small at late times, due to the fast exponential decay. Nevertheless, fully – 22 – t/T E c =0.80,d=4.5=0.84,d=4.5=0.84,d=3=0.88,d=3 t/T E c =0.80,d=7.5=0.84,d=7=0.88,d=6.5 Figure 24 : Lifetimes of quadrupole (left) and octupole (right) CSQs with different initial ω and d . Lifetimes are shown in units of T = 2 π/m . t/T -4 -2 en v e l ope o f Q s g=0.456g=0.454g=0.452 Figure 25 : Charge plateau of a quadrupole CSQ. The other parameters are set to be d = 3and ω = 0 . fledged higher multipole CSQs do exist for different g or with different forms of the potential [80]. Forexample, in Fig. 25, we see that for a smaller g the first charge plateau does form at least for quadrupoleCSQs. As we have seen in Section 3.2, this is surprising as reducing g increases the amplitude of thecharge densities, thus increasing the charge to energy ratio. Note that here Q s is defined only in thespirit of Fig. 5: It is defined as the same charge in one sector of the CSQ, that is, a quadrant for thequadrupole Q-balls CSQ.In Fig. 26, we show how the CSQ lifetime changes with coupling g and its multipole. Note that forsmaller g the higher multipole CSQs actually have longer lifetimes, and it appears that the lifetimesconverge to the same value at high multipoles for different g . (Note that here the lifetime is directlyextracted from the energy E c plateaus.) – 23 – multipoles of CSQ li f e t i m e / T g=0.50g=0.48g=0.46E c ~25E c ~52 Figure 26 : Lifetimes of CSQs for different couplings ( g ) and multipoles. Lifetimes areshown in units of T = 2 π/m . Q E / Q Figure 27 : Dependence of total charge Q and energy-charge ratio E/Q on frequency ω ofan elementary Q-ball in 3+1D in the potential (2.3). In this section, we shall briefly investigate CSQs and their lifetimes in 3+1D. We shall still constructthe CSQs with elementary Q-balls. See Fig. 27 for the dependence of the total charge and the ratiobetween the total energy and the total charge on the oscillating frequency for 3+1D elementary Q-ballsin the potential (2.3). From Fig. 27, we see that Q-balls become unstable when frequency ω (cid:38) .
92 astheir
E/Q > – 24 –
10 315 320 325 t/T -100-50050100 c ha r ge Q s Q up Q + Figure 28 : Charge swapping patterns of a dipole CSQ in 3+1D. Q s , Q up and Q + are definedanalogously to those in Section 2.1; see Fig. 5. t/T ene r g y EE c (4) Second plateau(Oscillon stage)(1) Initial relaxation(2) First plateau(CSQ stage)(3) Fast decay t/T -2 en v e l ope o f Q s Figure 29 : Evolution of energy E and E c and envelope of charge Q s for a dipole CSQ in3+1D. The initial constituents are elementary Q-balls with frequency ω = ± .
84 and spacing d = 10. (1) Initial relaxation, (2) First plateau (CSQ stage), (3) Fast decay and (4) Second plateau (oscillonstage). 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-stableCSQs. However, it is also to do with the potential we are using. For the logarithmic potential, forexample, the 3+1D CSQs are also very long lived [80], even for the quadrupole or octupole CSQs. InFig. 30, we see that CSQs in 3+1D are also attractor solutions, as expected, and their lifetimes canbe 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 asin Fig. 31, unlike the planar octupole configuration in 2+1D. In Fig. 32, we find a very short CSQstage for the octupole CSQ, which is close to the oscillon plateau. If we were to construct the planaroctupole CSQ in 3+1D, we would not be able to identity the first CSQ plateau in its evolution. – 25 –
500 1000 1500 2000 t/T E c =0.76,d=18=0.80,d=14=0.84,d=10 t/T -2 en v e l ope o f Q s =0.76,d=18=0.80,d=14=0.84,d=10 Figure 30 : Attractor behavior of the dipole CSQ in 3+1D. The initial configurations aresuperimposed elementary Q-balls. Initial configurations close to ω = 0 . , d = 14 have longerlifetimes. Figure 31 : Octupole CSQ in 3+1D. t/T E c t/T -1000-50005001000 en v e l ope o f Q s Figure 32 : Energy (left) and charge (right) evolution of the octupole CSQ of Fig. 31. Theparameters in the potential are chosen as g = 0 . , ω = 0 . , d = 3.– 26 – cknowledgments We would like to thank Zachariah Etienne, Xiao-Xiao Kou and Chi Tian for helpful discussions. PMSacknowledges support from STFC grant ST/P000703/1. SYZ acknowledges support from the start-ing grants from University of Science and Technology of China under grant No. KY2030000089 andGG2030040375, and is also supported by National Natural Science Foundation of China under grantNo. 11947301, 12075233 and 12047502, and supported by the Fundamental Research Funds for theCentral Universities under grant No. WK2030000036.
APPENDIX
A Other absorbing boundary conditions
Here we describe other ABCs we have explored and used to cross-check the validity of some of ourresults. The Sommerfeld ABC is widely used in numerically relativity [76], while the Engquist-MajdaABCs have been previously used to study stability of oscillons in spherically symmetry [66].
A.1 Sommerfeld’s absorbing boundary condition
The Sommerfeld boundary condition assumes that the outgoing waves have a spherical form φ = φ + u ( r − vt ) r ( d − / + h ( t ) r n , (A.1)where d is the number of the spatial dimensions, φ is the field value at the spatial infinity, v is thewave velocity at spatial infinity, u ( r − vt ) is an out-going spherical perturbation and h ( t ) simulatesthe non-wavelike behavior that has an n -th power law decay. The Sommerfeld ABC that absorbs sucha wave is given by ∂ t φ + v∂ r φ + ( d − v r ( φ − φ ) = h (cid:48) r n , (A.2)and can be re-cast in Cartesian coordinates as ∂ t φ + vx i r ∂ i φ + ( d − v r ( φ − φ ) = h (cid:48) r n , (A.3)with r replaced by ( δ ij x i x j ) / . Setting v = 1 and φ = 0, in the large r limit, Eq. (A.2) is just the1st order Higdon ABC with the addition of a non-wavelike term. Practically, h (cid:48) is taken as only afunction of t , and its value at fixed t is evaluated at the outermost layer adjacent to the boundary byEq. (A.3), which is then used to solve the boundary conditions at the boundary. A.2 Engquist-Majda’s absorbing boundary conditions
The Engquist-Majda ABCs [77] are designed to absorb plane waves φ = φ e i ( ωt + k j x j ) . At boundary x i = a , such a wave is annihilated by the local operator (cid:18) ∂∂x i − ik i (cid:19) φ | x i = a = ∂∂x i − i sign( k i ) (cid:115) ω − (cid:88) j (cid:54) = i k j k j − m φ | x i = a = 0 , (A.4) – 27 – hich uses both the Fourier space and real space coordinates. To extract boundary conditions in realspace in terms of differential operators, we expand the square root around (cid:80) j (cid:54) = i k j k j + m = 0 by aTaylor series (Alternatively, one may also use a Pade expansion [77].). The first two orders are givenby (cid:18) ∂∂x i − sign( k i ) iω (cid:19) φ | x i = a = 0 , (A.5) − iω ∂∂x i + sign( k i ) ( iω ) − (cid:88) j (cid:54) = i ( ik j )( ik j ) + 12 m φ | x i = a = 0 . (A.6)Transforming the Fourier space coordinates to real space coordinates, we have (cid:18) ∂∂t ± ∂∂x i (cid:19) φ | x i = a = 0 , (A.7) ∂∂x i ∂t ∓ ∂ ∂t − (cid:88) j (cid:54) = i ∂ ∂x j ∂x j + 12 m φ | x i = a = 0 , (A.8)where the + ( − ) sign is for the right (left) boundary. The 1st order ABC above is just that ofthe 1st order Higdon ABC with c = 1, and, as mentioned, the 2nd order ABC above is just thatof the 2nd order Higdon ABC with c = c = 1, upon using the linearized Klein-Gordon equationof motion. Engquist and Majda also generalized their absorbing boundary conditions for generalcurvilinear coordinates [77]. References [1] R. Friedberg, T. Lee and A. Sirlin,
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