Adiabatic invariance of $\mathbf{\text{oscillons}}/I$-balls

Real scalar fields are known to fragment into spatially localized and long-lived solitons called oscillons or $I$-balls. We prove the adiabatic invariance of the $\text{oscillons}/I$-balls for a potential that allows periodic motion even in the presence of non-negligible spatial gradient energy. We show that such a potential is uniquely determined to be the quadratic one with a logarithmic correction, for which the $\text{oscillons}/I$-balls are absolutely stable. For slightly different forms of the scalar potential dominated by the quadratic one, the $\text{oscillons}/I$-balls are only quasistable, because the adiabatic charge is only approximately conserved. We check the conservation of the adiabatic charge of the $I$-balls in numerical simulation by slowly varying the coefficient of logarithmic corrections. This unambiguously shows that the longevity of $\text{oscillons}/I$-balls is due to the adiabatic invariance.

Figure 1

Numerical results of the $I$-ball for $D=1$. We have set $K=10^{-1}$ and ${\mathrm{\Phi }}_c=2M$. The top left panel shows snapshots of the spatial distribution of $\tilde{\rho }$ at $t=0$, 10, ${10}^2$, ${10}^3[1/m]$, and all the lines are overlapped, implying that $\tilde{\rho }$ is a constant of motion. The top right, bottom left, and bottom right panels show that the time evolutions of ${\tilde{\rho }}_c$, $R_{1/2}$, and ${\tilde{\mathrm{\Phi }}}_c$ in very good agreement with the analytic results (60), (61), and (62), respectively. The box size is $L=100/m$ and the grid number is $N=2048$.

Figure 2

Numerical results for $\alpha =10^{-2}$ and $K_0=10^{-1}$ in the case of $D=1$. The top panels show the snapshots of the spatial distribution of $\tilde{\rho }$ at $t=0$, ${10}^2$, ${10}^3$, ${10}^4[1/m]$ in linear and logarithmic scales. The bottom panels show the evolution of $R_{1/2}$ and ${\tilde{\mathrm{\Phi }}}_c$ from $t=10/m$ to ${10}^4[1/m]$. The red (green) line shows the numerical (analytical) result [see Eqs. (67) and (73)].

Figure 3

Numerical results for $\alpha =10^{-2}$ and $K_0=10^{-1}$ in the cases of $D=2$ and $D=3$. The left (right) panels show the results for $D=2(3)$ case. The top two panels show snapshots of the spatial distribution of $\rho$ at $t=0$, ${10}^2$, ${10}^3[1/m]$, where $N_x$, $N_y$, and $N_z$ represent the grid point number of the lattice. The middle (bottom) panels show the evolution of $R_{1/2}$ (${\tilde{\mathrm{\Phi }}}_c$) from $t=1/m$ to ${10}^3/m$ in a very good agreement with the analytical ones (67) and (73).

Figure 4

Same as Fig. 2 but for $\alpha =10^{-1}$. The $I$-ball deformation is no longer adiabatic, and it does not follow the analytic one.