Compact Q-balls and Q-shells within a $CP^N$ Skyrme-Faddeev type model

Abstract

While $CP^N$ models with analytic potentials are known to support finite-energy compact Q-ball and Q-shell solutions, their behavior in more complex Lagrangian frameworks remains a subject of active research. This work explores these non-topological structures within an extended Skyrme-Faddeev-type model that incorporates quartic derivative terms. In this context, harmonic time dependence and the presence of quartic terms constitute two independent stabilization mechanisms that allow the configurations to circumvent Derrick's scaling argument. We investigate the necessary conditions for the existence of these solutions and analyze the influence of quartic terms on the properties of the resulting compactons, specifically examining the $E(Q)$ relationship between energy and Noether charge. Our findings provide valuable insights into the stability and characteristics of compact boson stars within $CP^N$ models featuring higher-order derivative terms.

Figures (20)

• Figure 1: The coupling constant space delimited by the conditions $\beta e^2+2\gamma e^2\le 0$, $2\beta e^2+2\gamma e^2-1\ge 0$, and $\beta e^2-4\ge 0$, ensuring a non-negative contribution from the quartic terms.
• Figure 2: Radial solutions for the $CP^1$ model are characterized by the value of $a_0$. The solutions fall into three categories: (a) those with $a_2>0$, (b) those with $a_2<0$ and (c) the compacton case, where the radial function and its derivative simultaneously vanish at the external radius, i.e., $f(R_{\rm out})=0=f'(R_{\rm out})$.
• Figure 3: The dependence of the coefficient $a_2$ on the parameter $a_0$ is shown for increasing values of $\omega$ for the $CP^1$ case.
• Figure 4: The sign of the coefficient $a_2$ is shown as a function of the parameters $a_0$ and $\omega$ for the $CP^1$ case with $\beta e^2=5.0$ and $\gamma e^2=-2.5$. The gray color indicates regions where ${\rm sgn}(a_2)=-1$. The horizontal lines that correspond to the values of the parameter $\omega$ used to generate Fig. \ref{['fig:CP1grid']}, specifically $\omega={0.624}$, $\omega={0.854}$, $\omega={0.906}$ and $\omega={0.94}$.
• Figure 5: $CP^1$ Q-balls. (a) Radial profiles and energy density for a solution with a minimal contribution from quartic terms. (b) Radial profiles and energy density for a solution with a significant contribution from quartic terms, demonstrating a shallow local minimum at the origin. (c) The top panel shows the coefficient $a_2$ as a function of $a_0$ in the region where $a_0<a^{(-)}_0$. The bottom panel shows the position of the left vertical asymptote, $a^{(-)}_0$, as a function of $\omega$.