Large gauged Q-balls with regular potential

In many models of gauged Q-balls, which were studied in the literature, there are upper limits for charge $Q$ (and size) of Q-balls due to repulsive Coulomb force. The only known model that allows large $Q$ without limitation is the V-shaped potential, $V\propto |\varphi |$, which is singular at $\varphi =0$. To make it clear whether the property of unlimited $Q$ is peculiar to the singular potential, we derive general conditions for potentials that allow Q-balls with unbounded $Q$. We find that large gauged $Q$-balls exist even for regular potentials. One of the simple models is $V=({\mu }^2/2){\varphi }^2[1+K\ln (\varphi /M{)}^2]$ with $K<0$. We investigate equilibrium solutions for this model systematically. As the electric charge $Q$ increases, the field configuration of the scalar field becomes shell-like; because the charge is concentrated on the surface, the Coulomb force does not destroy the Q-ball configuration. These properties are analogous to those in the V-shaped model. We also find that for each $K$ there is another sequence of unstable solutions, which is separated from the other sequence of the stable solutions. As $|K|$ increases, the two sequences approach; eventually at some point in $-1.07<K<-1.06$, the “recombination” of the two sequences takes place.

Figure 1

Interpretation of ordinary Q-balls by analogy with a particle motion in Newtonian mechanics.

Figure 2

Interpretation of gauged Q-balls by analogy with a particle motion in Newtonian mechanics. Examples of (a) monotonic solutions in the $V_4$ model and (b) nonmonotonic solutions in the $V_{\mathrm{V}}$ model.

Figure 3

The field configurations of $\tilde{\varphi }$ and $\tilde{\mathrm{\Omega }}$ for the $V_4$ model with ${\tilde{m}}^2=0.2$ and $\tilde{Q}=9$. The dashed and solid lines correspond to the ordinary and gauged Q-balls, respectively.

Figure 4

The field configurations of $\tilde{\varphi }$ and $\tilde{\mathrm{\Omega }}$ for the $V_{\mathrm{V}}$ model with $\tilde{Q}=120$. The dashed and solid lines correspond to the ordinary and gauged Q-balls, respectively.

Figure 5

(a) $\tilde{\mathrm{\Omega }}(0)\text{−}\tilde{\varphi }(0)$ and (b) $\tilde{Q}\text{−}\tilde{E}$ relations for the $V_{\mathrm{V}}$ model. The dashed line corresponds to the ordinary Q-balls. The dotted and black solid lines correspond to the gauged Q-balls with ${\tilde{r}}_{\max }=0$ and those with ${\tilde{r}}_{\max }\ne 0$, respectively. The blue solid line corresponds to the Q-shell solutions.

Figure 6

$\tilde{Q}\text{−}(\tilde{E}-3.7459\tilde{Q})$ relation for the $V_{\mathrm{V}}$ model near the points $B$, $C$, and $D$.

Figure 7

The field configurations of $\tilde{\varphi }$ for gauged Q-balls with $K=-1$ and $\tilde{Q}\simeq 1.7$, 11, and 103.

Figure 8

(a) $\tilde{\mathrm{\Omega }}(0)\text{−}\tilde{\varphi }(0)$ and (b) $\tilde{Q}\text{−}\tilde{E}$ relations for $K=-1$. The dashed lines correspond to ordinary Q-balls. The dotted and solid lines correspond to gauged Q-balls with ${\tilde{r}}_{\max }=0$ and those with ${\tilde{r}}_{\max }\ne 0$, respectively.

Figure 9

(a) $\tilde{\mathrm{\Omega }}(0)\text{−}\tilde{\varphi }(0)$ and (b) $\tilde{Q}\text{−}\tilde{E}$ relations for $K=-0.6$.

Figure 10

(a) $\tilde{\mathrm{\Omega }}(0)\text{−}\tilde{\varphi }(0)$ and (b) $\tilde{Q}\text{−}\tilde{E}$ relations for $K=-0.4$.

Figure 11

(a) $\tilde{\mathrm{\Omega }}(0)\text{−}\tilde{\varphi }(0)$ and (b) $\tilde{Q}\text{−}\tilde{E}$ relations for $K=-1.06$. The two sequences in red lines and in black lines are about to touch.

Figure 12

(a) $\tilde{\mathrm{\Omega }}(0)\text{−}\tilde{\varphi }(0)$ and (b) $\tilde{Q}\text{−}\tilde{E}$ relations with $K=-1.07$. The recombination of the two sequences happens.

Figure 13

(a) $\tilde{\mathrm{\Omega }}(0)\text{−}\tilde{\varphi }(0)$ and (b) $\tilde{Q}\text{−}\tilde{E}$ relations for $K=-1.07$ and $\tilde{Q}>500$. The dotted lines extend from Fig. 12.

Figure 14

Field distributions of $\tilde{\varphi }$ with $K=-1.07$ for (a) solutions $C\text{−}D\text{−}E$ and (b) solutions $E\text{−}F\text{−}G$.