$Q$-balls and charged $Q$-balls in a two-scalar field theory with generalized Henon-Heiles potential

We construct $Q$-ball solutions from a model consisting of one massive scalar field $\xi$ and one massive complex scalar field $\varphi$ interacting via the cubic couplings $g_1\xi {\varphi }^{*}\varphi +g_2{\xi }^3$, typical of Henon-Heiles-like potentials. Although being formally simple, these couplings allow for $Q$-balls. In one spatial dimension, analytical solutions exist, either with vanishing or nonvanishing $\varphi$. In three spatial dimensions, we numerically build $Q$-ball solutions and investigate their behaviors when changing the relatives values of $g_1$ and $g_2$. For $g_1<g_2$, two $Q$-balls with the same frequency exist, while $\omega =0$ can be reached when $g_1>g_2$. We then extend the former solutions by gauging the U(1) symmetry of $\varphi$ and show that charged $Q$-balls exist.

Figure 1

Effective potential (3.2) and typical allowed solutions (solid lines). The sketched solutions actually correspond to (3.7) (red), (3.8) (purple) and (3.10) (green). The displayed shape for $V$ has been obtained for $M=2.11166$, $m=1$, $g_1=2$, $g_2=1$, $\mathrm{\Omega }=0.8$, i.e., a configuration for which the three solutions may exist. The plot was made using geogebra software.

Figure 2

Left: The central values $F(0)$, $G(0)$, the mass and the Noether charge as function of $\omega$ for $D=3$, $M=1$, $g_1=1$. Right: The same data for $M=2$. For completeness we mention that the data used to generate this figure may be found at [22].

Figure 3

Profiles of the functions $F$, $G$ and the effective energy density $\epsilon$ for $D=3$, $M=2$, and $g_1=1$ for $\omega =0.9407$ (dashed lines) and for $\omega =0.999$ (solid lines).

Figure 4

Left: The central values $F(0)$ as function of $\omega$ for $D=3$, $M=2$ and several values of $g_1$. Right: The corresponding values of $G(0)$.

Figure 5

Left: The central values $F(0)$, $G(0)$ and the soliton mass as function of $\omega$ for $D=3$, $M=2$, $g_1=2.0$.

Figure 6

Left: The central values $F(0)$ as function of $\omega$ for $D=3$, $M=2$, and several values of $g_1$. Right: The corresponding values of $G(0)$.

Figure 7

Left: The central values $F(0)$, $G(0)$ and the mass as function of $\omega$ for $D=3$, $M=2$, $g_1=0.5$ (the Noether charge is roughly equal to the mass all along). a and b label the two different branches of solutions.

Figure 8

Left: Some parameters of charged $Q$-balls as functions of $\beta$ for $g_1=1$, $M=2$, and $e=0.1$. Right: Same for $g_1=2$.

Figure 9

Left: Central value $F(0)$ as functions of $\beta$ for $g_1=2$, $M=2$ and several values of $e$. Right: Same for the value $G(0)$.