Charge-Swapping Q-balls in a Logarithmic Potential and Affleck-Dine condensate fragmentation

The physical models concerned are in an infinite space. However, in our numerical simulations, we can only have a finite grid. Previously, the logarithmic model has been studied with a periodic boundary condition Copeland:2014qra . While that approach is sufficient to establish the existence of CSQs as quasi-solitons, it would be insufficient to determine the long term stability of CSQs for two reasons. First, when we initially prepare the CSQs, we superimpose lumps of positive and negative charges (of which single Q-ball solutions would be one of the simplest choices) closely together and let the system relax to a CSQ. As CSQs are attractor solutions under the time evolution Copeland:2014qra ; Xie:2021glp , this is found to be an effective method to obtain CSQs. However, the relaxation process emits some amount of radiation. With a periodic boundary condition, the initial radiation would bounce back and forth in the box, which introduces destabilizing perturbations that are not in the physical system we are interested in. Second, as CSQs are quasi-stable, they emit further radiation as time goes by, which will also bounce within the box. Without reasonable methods to alleviate the unwanted radiation arising from the limitation of a lattice simulation, the long term stability of the CSQs can not be reliably established. This is especially true for the logarithmic potential, which, as we shall see, gives rise to extremely stable CSQs.

We shall make use of 2nd order Higdon’s absorbing boundary conditions higdon1994radiation ; higdon1986absorbing for our simulation grid. Higdon’s absorbing boundary conditions are designed based on the idea that a boundary condition that absorbs specific outgoing plane waves entirely will also absorb other plane waves partially and that we can use multiple of these conditions to improve the absorbing effect:

where $a$ is the location of the boundary and the $c_{j}$’s are tunable constants that should be adjusted for particular problems. For example, the differential operator labelled by $j$ in the above formula can exactly absorb plane waves moving in the $x^{i}$ direction with a phase velocity equal to $c_{j}$, with the $+$ ($-$) sign for waves moving to larger (smaller) $x^{i}$ direction. Although higher order Higdon’s methods have better absorbing effects, they are also more difficult to implement, due to the presence of higher order differential operators, and also computationally more expensive, and the second order Higdon’s absorbing boundary condition is already rather effective for our purposes:

which will be the boundary condition we use later.

To showcase the effectiveness of the 2nd order Higdon’s absorbing boundary condition, in Figure 3, we compare the initial evolution of $Q_{s}$, i.e., the total charge in the $y\geq 0$ half central region (see the text below Figure 2), for three different boundary conditions: 1) periodic boundary condition; 2) 2nd order Higdon’s absorbing boundary condition with $c_{1}=c_{2}=1$; 3) the reference simulation in which the lattice is chosen large enough such that the perturbations have not reached the boundary at the end of the test period, and which is taken to be the bona fide evolution. All the physical setup and other numeric parameters are chosen to be the same for the three evolutions. As we see in Figure 3, in the beginning, the three simulations are the same as the perturbations have not propagated to the boundary; soon after, we can see that the periodic boundary condition and absorbing boundary condition deviate from the reference case, with the absorbing boundary much better than the periodic boundary. Additionally, we can see that the deviations from the reference case modulate with time, especially for the periodic boundary, which is due to the fact that the initial burst of radiation bounces back and forth in the simulation box and the CSQ is affected the most when the bounced radiation hits and passes through it. Note that, thanks to the ultra-soft interactions of the logarithmic potential, the deviations from the reference case is relatively small during the initial test period, even for the periodic boundary condition. However, in the long run, without proper absorption of radiation, the perturbations can accumulate and significantly affect the physical configurations. In Figure 4, we see that, after a time evolution of $2\times 10^{7}$, while the CSQ configuration still holds up very well for the Higdon’s absorbing boundary condition, the periodic boundary condition introduces significant noise for the CSQ configuration just after a time evolution of $1\times 10^{5}$.

An interesting feature of the model with a logarithmic potential Eq. (1) is that it has the Gaussian profile as its exact simple Q-ball solution bialynicki1975wave :

where $D$ is the number of the spatial dimensions and we have chosen the units according to Eq. (7). The $K$ parameter is chosen to be $-0.1$ through out this section. We shall use this Gaussian profile as the constituent charge lumps, but it is worth emphasizing that this is not necessary, and other charged lumps can also be used in the construction. To get the simplest dipole CSQ configuration, we can superimpose two opposite charge lumps initially close to each other in symmetric locations along the $y$-axis in the center of the simulation box, and then let the configuration relax to the CSQ. The length of the simulation box is at least three times (often five times) that of the central CSQ region where most of the energy density is located within, depending on the parameters of the initial lumps. That is, we adjust the box size when the initial lumps become noticeably larger. If unstated, simulations in the following are performed in 3+1D.

Our code in this section makes use of the LATfield2 package Daverio:2015ryl which provides a rudimentary framework for rapid development of parallel codes for classical lattice simulations, by defining objects such as lattice sites and fields on lattice. The Runge-Kutta 4th order method is used to evolve the system in time and spatial derivatives are approximated by 4th order finite differences. As mentioned, we use a 2nd order Higdon’s absorbing boundary condition to absorb out-going waves to reduce unphysical in-going reflection of waves, with the absorbing parameters chosen to be $c_{1}=c_{2}=1$. As the CSQ is prepared in the origin and along the $y$ axis, for a dipole CSQ, the field is symmetric with respect to the $x$-$y$ and $y$-$z$ plane, and its real (imaginary) component is symmetric (anti-symmetric) with respect to the $z$-$x$ plane. To speed up the simulations on top of the parallelization, we only simulate the first octant with positive $x,y,z$ and implement mirroring boundary conditions on the $x$-$y$, $y$-$z$ and $z$-$x$ planes; see Figure 5. This provides a speed-up of a factor of 8 for 3+1D simulations and a factor 4 for 2+1D simulations, where we simulate the first quadrant. Similar arrangements can also be made for higher multipole CSQs such as the quadrupole one.

A numerical subtlety when preparing the initial lumps is that the potential Eq. (2) contains a logarithm ${\rm ln}|\varphi|^{2}$. While the potential itself is not divergent at $|\varphi|=0$, thanks to the $|\varphi|^{2}$ factor in front, numerically, we may encounter problems, especially with the exponential profile Eq. (10), which gives rise to small numbers due to the $e^{K{|{\bf x}|^{2}}/2}$ factor with negative $K$ and large $|{\bf x}|$, exceeding the double precision. To overcome this, a small regulator $\varepsilon$ is added in, ${\rm ln}(|\varphi|^{2}+\varepsilon)$, when numerically evolving the equations of motion, with $\varepsilon$ chosen to be $1\times 10^{-60}$ times smaller than the maximum value of the $|\varphi|$ field. We have varied this ratio from $1\times 10^{-70}$ to $1\times 10^{-50}$ and found no noticeable differences in the simulation results.

As we have shown, CSQs may be formed in the early universe from fragmentation of some VEV in a preheating-like process. For this reason, we will focus on the 3+1D simulations in this subsection. As a fiducial example, we will investigate the properties of a dipole CSQ evolved from superimposing a Q-ball with $\omega=1.2$ and an anti-Q-ball with $\omega=-1.2$ (cf. Eq. (10)), with an initial separation between their centers equal to $2$. We will see that the properties of CSQs in the logarithmic potential are qualitatively similar to that in the $\varphi^{6}$ potential Xie:2021glp , but they are much more stable.

First, let us visualize the charge-swapping behavior of the CSQ in more details. In Figure 6, we plot the evolution sequence of the charge density of the dipole CSQ in about one charge-swapping period, that is, the process of the positive charge lump becoming negative and then turning back to being positive. In Figure 7, we also plot the evolution of various charges defined in the text below Figure 2. The purple line is the total charge of the whole region $Q$ and is always zero as expected, since the two initial constituent Q-balls have opposite charges. The red dot-dashed line, denoted by $Q^{up}$, is the total charge obtained by integrating over the up half simulation box, while the blue solid line, denoted by $Q_{s}$, which closely trails the red dot-dashed line, is the total charge obtained by integrating over the half spherical region defined in Figure 2. The dashed yellow line, denoted by $Q^{+}$, is the total charge obtained by summing all the positive charges in the simulation box. We see that most of the charges are within the CSQ, and the charge swapping period of the CSQ is much longer than the oscillation period of the field, the latter of which gives rise to the small wiggles on the large modulation that is mostly sinusoidal. We emphasize that while the charge density of the CSQ is dipolar and swapping, its energy density is approximately spherically symmetric.

The ultra-solfness of the logarithmic potential means that its relaxation process is quite different from that in the $\varphi^{6}$ potential Xie:2021glp . For the $\varphi^{6}$ case, after superimposing two charge lumps, the initial configuration quickly sheds away a significant amount of energy and settles down to the CSQ plateau. For the logarithmic case, however, we can barely see a relaxation process, and only a very small amount of energy ebbs away in the beginning. Also, during the relaxation, the lump in the $\varphi^{6}$ case does not have a well-defined swapping frequency. However, the lump in the logarithmic case starts to swap charges with a well-defined changing swapping frequency (within a relatively broad band) from the beginning, and the charge swapping frequency changes slowly for a relatively long time before settling down to a constant band. More interestingly, due to the ultra-solfness of the potential, the logarithmic CSQ is extremely stable. (This is of course when a CSQ can stably form from certain initial conditions, such as the ones we use in this subsection; outside the attractor basin of a CSQ, a swapping lump disintegrates very quickly; see Section 4.5.) Despite the acceleration of the simulation with parallelization and only evolving the octant, we have not captured the decay of the logarithmic dipole CSQs in 3+1D or in 2+1D. In Figure 8, we plot the evolution of energies and the charge-swapping period of the dipole CSQ. We see that in 3+1D the logarithmic CSQ’s lifetime is longer than

To achieve such a long simulation with a $256^{3}$ lattice for the first octant, we have used about $1.1\times 10^{5}$ CPU hours. In 2+1D, the logarithmic dipole CSQs have similar properties, and their lifetimes are found to be longer than

which is achieved by about $2\times 10^{4}$ CPU hours. Furthermore, we have also checked the stability of the quadrupole ones, and found that they are also stable to a similar time scales; Specifically, in 2+1D, the quadrupole logarithmic CSQ’s lifetime is longer than $2.3\times 10^{7}/m$, which takes $1.8\times 10^{4}$ CPU hours to simulate. We emphasize that for such long simulations to be reliable, it is essential to utilize absorbing boundary conditions to absorbing outgoing radiation at the boundary of the simulation box.

It is instructive to see the spectra of the oscillations of the CSQ. To this end, we pick two representative points, and plot the evolution of the scalar field at these points and their corresponding Fourier spectra in Figure 9. In the top left plot, the field evolution at the origin, i.e., the center of the CSQ, has been plotted, which has an oscillating real part with a frequency larger than the charge-swapping frequency and a vanishing imaginary part due to the dipole symmetry of the CSQ. The bottom left plot shows the Fourier spectrum of this point’s time evolution for the time from $t=0$ to $t\sim 2.7\times 10^{4}$. We see that the peaks of the spectrum are odd multiples of a base frequency $\omega\approx 1.16$, and decrease according to the power law of $e^{9.5}\omega^{-4.7}$. The reason why there are only odd multiples of the base frequency is related to the $\varphi\to-\varphi$ symmetry in the potential Salmi:2012ta . (For the regularized logarithmic potential, the leading nonlinear term in the equation of motion starts at $\varphi^{3}$ due to the $Z_{2}$ symmetry, which does not support even multiples of the base frequency; for a potential without the $Z_{2}$ symmetry, the leading nonlinear term in the equation of motion starts with $\varphi^{2}$, which supports all multiples of the base frequency.) The two plots in the right column show the evolution of the field and its Fourier spectrum at a point in the outer region of the CSQ. At this point, both the real and imaginary part of the field oscillate in a more complicated way, and the peaks of the spectrum become broader. Also, the dominant peak has two sub-peaks (see the inset of the bottom right plot in Figure 9), which arises because the real and imaginary part of the field have slightly different dominant frequencies, and the difference between the frequencies of the two sub-peaks is approximately the charge-swapping frequency of the CSQ. To see this, note that the field oscillation can be modeled to leading order by $\varphi_{r}\approx A_{r}(\mathbf{x})\cos\left(\omega_{r}t\right),~{}\varphi_{i}\approx A_{i}(\mathbf{x})\cos(\omega_{i}t+\phi_{0})$, where $\varphi_{r}$ and $\varphi_{i}$ are respectively the real and imaginary part of the scalar field, with $\omega_{r}$ and $\omega_{i}$ their dominant frequencies respectively (see Figure 9). As $\omega_{r}$ and $\omega_{i}$ are quite close to each other, we can approximate the 0-th component of the current with $j_{0}\approx A_{r}A_{i}(\omega_{r}+\omega_{i})\sin\left((\omega_{r}-\omega_{i})t-\phi_{0}\right)$, which means that the difference $\omega_{i}-\omega_{r}$ is approximately the charge-swapping frequency.

A significant difference between the spectrum plot in the logarithmic potential and that in the $\varphi^{6}$ case is that the spectra in the logarithmic case is much noisier. This may be easily understood in terms of the form of the potential. While the $\varphi^{6}$ case only has two nonlinear terms in the equation of motion, the regularized logarithmic potential leads to an equation of motion with an infinite number of nonlinear terms, and these nonlinear terms are large and alternating in signs so as to reproduce the ultra-soft logarithmic interaction. So compared to the $\varphi^{6}$ case, the higher order nonlinear terms in the logarithmic case can feed many higher order oscillation modes in the spectra, hence more “noise” near the peaks.

Figure 10 illustrates another important difference between the logarithmic case and the $\varphi^{6}$ case. In the model with the $\varphi^{6}$ potential, we observed Xie:2021glp that a unique dipole CSQ can be obtained with different initial lumps (simple Q-balls with different frequencies/energies or deformed Q-balls or Gaussian lumps) and with different initial separations. Extra energy in the initial configuration will just dissipate away in the initial relaxation process, leaving a CSQ with the same energy and the same charge swapping frequency. As we see in Figure 10, the situation is rather different in the case of a logarithmic potential, again due to the ultra-soft-ness of the interacting potential. In these plots, we superimpose together a simple Q-ball and a simple anti-Q-ball with different initial frequencies, and examine the resulting CSQ. From the right plot, we see that the resulting CSQ can have vastly different charge-swapping frequencies, depending on the initial frequencies of the Q-balls. This is an important feature of the logarithmic potential, and probably suggests that the attractors in the logarithmic case form a continuous attractor manifold.

In the left plot of Figure 10, we trace the dependence of the dominant frequency of Re($\varphi$) and Im($\varphi$) (at the same edge point as that in Figure 9) on the different initial frequencies of the constituent Q-balls, and find that the dependence is approximately linear. The slopes of the two linear scalings are very close, and the dominant frequency of Re($\varphi$) is always slightly smaller than the initial frequency of the constituent Q-balls, while that of Im($\varphi$) is always slightly larger. Of course, the linear scalings are not exact; otherwise the nonlinear dependence of charge-swapping frequency $\omega_{\rm swap}$ on the initial Q-ball frequencies (the right plot of Figure 10) can not be reproduced, as the difference between the dominant frequency of Re($\varphi$) and Im($\varphi$) is approximately the charge-swapping frequency.

In the last subsection, we have mainly focused on the simplest dipole CSQ. In generic circumstances, such as in the scenario of Section 3, a CSQ may have a much more complex multipolar structure. In this subsection, we will investigate the stability of more complex CSQs. We find that they are still very stable, at least for the simple ones among these complex CSQs. A new feature is that now there are also transit CSQs which are charge swapping for a relatively long time but lose energy faster than normal CSQs.

We may prepare more complex CSQs with unequal positive and negative charges. The simplest case is to superimpose a positive lump and a negative lump with an unequal charge, as in the left plot of Figure 11. In the right plot of Figure 11, on the other hand, we have superimposed four lumps with unequal charges. Both of these cases are still very stable, and we find that their lifetimes are at least $1.5\times 10^{5}/m$, each case taking $3.2\times 10^{4}$ CPU hours to simulate. These being shorter than the lifetimes in the previous subsection is due to our limitation of time and computational resources, and it is possible that these unequal complex CSQs are as stable as the dipole or quadrupole CSQs in the previous subsection.

On the other hand, if we prepare the initial configuration in the right plot of Figure 11 with slightly larger distances between the constituent lumps, we will end up with transit CSQs. Charges are still swapped within these transit states, but they continuously emit larger amounts of radiation and their total energies do not plateau as normal CSQs would do. With time, they evolve to an unequal CSQ with just one lump of positive charge and one lump of negative charge, just like the left plot of Figure 11. In fact, a transit CSQ decaying to an unequal CSQ with two lumps is usually what we encounter when we want to prepare more complex unequal CSQs via superposition of charge lumps. See Figure 12. While the superposition method is useful to obtain CSQs with fixed multipoles (dipole, quadruple, etc.), it does not seem to be reliable to construct more complex CSQs. On the other hand, as we see in Section 3, complex CSQs can actually arise naturally in a preheating-like setup, but in that case, we have little control over what kinds of complex CSQs are produced. We leave the search for alternative methods to construct complex CSQs for future work.

As the CSQs are quasi-stable configurations, they must have an attractor basin of formation. In this subsection, we investigate how easily logarithmic CSQs can form from colliding simple Q-balls. Since many salient dynamical properties of logarithmic CSQs in 3D are very similar to the ones in 2D, we will scan the parameter space in 2D for simplicity.

First, we consider the case where a Q-ball and an anti-Q-ball that have opposite frequencies $\omega$ and $-\omega$ and no relative phase are initially placed at a distance apart with a relative velocity, and then they collide head-on. (To get a Q-ball solution with a velocity, we can Lorentz boost the solution in Eq. (10).) In Figure 13 we plot the parameter space for this case where a CSQ can form. The simulation box is in a frame where the Q-ball and the anti-Q-ball have the same speed and opposite direction. The maximum initial Q-ball velocity (with respect to the simulation box) that can form a CSQ for the corresponding initial distance and frequency is plotted with different colors, and the white region is where a CSQ can not form for any velocity. We can see that whether a CSQ can form mainly depends on the initial distance between the center of the two Q-balls. The two Q-balls should be close so that their nonlinear cores have some overlap for the CSQ to form. Also, when the frequency $\omega$ or the distance is smaller, a large initial velocity is allowed before they can not form a CSQ. Notice that a Q-ball and an anti-Q-ball with appropriate phases attract each other when placed far apart, so a large initial distance is to some extent similar to a large initial velocity. The black data points (crosses, circles and boxes) on the right of Figure 13 are the special cases where a Q-ball and an anti-Q-ball, both of which have zero velocity initially, collide to form two CSQs moving away from each other. (The colored region on the left is when the Q-ball and anti-Q-ball collision results in only one CSQ.) In these cases, before two CSQs are formed and move away in opposite directions, the positive and negative charges can swap for a number of times. The crosses, circles and boxes correspond to the cases where the number of swaps is 5, 3 and 2 respectively. Just outside the boundary of the colored region, positive and negative charges can also swap for a number of times, but during these swaps all the energies in the lump are quickly radiated away.

For Q-balls that are initially far away or collide with large initial velocities, they will pass through each other in head-on collisions. However, it may be expected that non-head-on collisions are more probable in realistic situations, in which case CSQs can still form for large initial separations or velocities. In Figure 14 we chart the parameter space of CSQ formation for different initial velocities and impact parameters in non-head-on collisions. We see that in this velocity vs impact parameter plot it is the diagonal strip that supports CSQ formation, i.e., when the initial velocity is smaller, the impact parameter has to be larger, and vice versa. Not surprisingly, the results have many different end-states for the non-head-on collisions. When the initial conditions are unfavorable, the two balls just pass by each other. In the diagonal strip of the plot, the end state of the collision can be two or three CSQs, or one CSQ and two simple Q-balls, Fig. 15. As shown in the inset of Figure 14, we find that it is actually not rare to see three CSQs or one single CSQ as the end state. In the latter case, the positive and negative charges in the resulting single CSQ rotate around each other, much like the Tai Chi diagram.