$Q$-ball superradiance: Analytical approach

TL;DR

The paper tackles the problem of waves scattering off a $Q$-ball and extracting energy via superradiance. It introduces an analytic framework that replaces the nonlinear $Q$-ball background with a discretized $(n+1)$-step profile, enabling perturbative perturbations to be solved in closed form as linear combinations of Bessel functions. The main contributions are (i) a detailed perturbative construction on piecewise backgrounds, (ii) explicit expressions for amplification factors in terms of outgoing particle numbers and energy flux, and (iii) thorough analysis of thin-wall and large-$Q$-ball limits, including large-$\omega$ asymptotics that explain the multi-peak spectra. The results provide both physical insight into peak structures and a fast, scalable method to evaluate amplification factors across a broad class of $Q$-balls, with potential applications in cosmology and beyond.

Abstract

It was recently discovered that waves scattering off a $Q$-ball can extract energy from it. We present an analytical treatment of this process by adopting a multi-step function approximation for the background field, which yields perturbative solutions expressed in terms of Bessel functions. For thin-wall $Q$-balls, the amplification factors reduce to simple sinusoidal functions, which explains the multi-peak structure of the spectrum and identifies the physical quantities that determine it. For instance, at high frequencies, the peak spacing is simply the inverse of the $Q$-ball size. The analytical solution further enables us to delineate the full range of possible amplification factors. For general $Q$-balls, this analytical framework also substantially improves the efficiency of evaluating the amplification factors.

Figures (8)

• Figure 1: Top left: The effective potential with $g = 1/3$. For varying $\omega_Q$, the potential lies within the region bounded by the red and dark-blue curves. The characteristic points for $\omega_Q = 0.7$ are $(f_{\rm min},f_z,f_0,f_{\rm max}) \approx (0.548,0.807,1.201,1.304)$ with $d=2$. The black dashed lines indicate the minimum and maximum values of $f_Q$ for varying $\omega_Q$, while the light-blue dashed line marks the corresponding $f_0$. Top right: Radial profiles of $Q$-balls for various $\omega_Q$ with $d=2$ and $g=1/3$. Bottom left: Radial profiles of $Q$-balls for various spatial dimensions $d$ with $\omega_Q=0.58$ and $g=1/3$. Bottom right: $(n+1)$-step function approximations of the background field with $g=1/3$, obtained from Eqs. \ref{['fq::app1']} and \ref{['fq::app2']}.
• Figure 2: Energy amplification factor $\mathcal{A}_{tt}^b$ and outgoing particle number $N^{out}_-$ with different parameters $\omega_Q , f_0 , r_*$ and $g$, from the full analytical results. The extrema of the outgoing particle number and the amplification factors almost coincide, and the value of the thin-wall location $r_*$ turns to be the key parameter that controls the number of peaks in the amplification factors.
• Figure 3: Amplification factors of energy $\mathcal{A}_{tt}^a,\mathcal{A}_{tt}^b$ and energy flux $\mathcal{A}_{rt}^a,\mathcal{A}_{rt}^b$ obtained from Eqs. \ref{['atta']}-\ref{['artb']} with $\omega_Q=0.52$ and $g=1/3$ for a thin-wall Q-ball ($n=1$). The top row shows solid lines for $N^{out}_\pm$ obtained from Eq. \ref{['app1']}, while the dotted lines indicate the numerical results, overlapping with the case where $N^{out}_\pm$ is obtained from Eq. \ref{['nout']}. In the bottom two rows, the left two columns present the approximate $N^{out}_\pm$ from Eq. \ref{['app1']}, which is valid for large $\omega$ or large $r_*$, whereas the right two columns display the $N^{out}_\pm$ obtained from Eq. \ref{['nout']}, which remain valid for all $\omega$ and $r_*$.
• Figure 4: Limits of the energy and energy flux amplification factors for $\omega_Q=0.52$ and $g=1/3$, with $f_0\in[f_z,f_{\rm max})$ and $r_*\in[3,50]$, for the thin-wall $Q$-ball ($n=1$). The red lines indicate the limits given by Eqs. \ref{['lim1']} and \ref{['lim2']}, while the blue lines correspond to those from Eqs. \ref{['app1']}. Solid lines represent the upper bounds, and dotted lines denote the lower bounds.
• Figure 5: Comparisons of the energy amplification factor in different approximations. ${\cal A}^{b(0)}_{\rm tt}$ is the full analytical result, ${\cal A}^{b(1)}_{\rm tt}$ is the large $r_*$ limit, and ${\cal A}^{b(2)}_{\rm tt}$ is given by Eq. \ref{['appn2']}.