Oscillons from $Q$-balls

TL;DR

This work shows that oscillon solutions in (1+1) dimensions can be generated from $Q$-ball seeds through Renormalization Group Perturbation Expansion, revealing a universality across many scalar theories and explaining amplitude modulations as two-$Q$-ball bound states. By mapping the leading nonlinear dynamics to a universal $Q$-ball equation, closely approximated by the integrable complex sine-Gordon theory, the authors derive analytic seeds for both unmodulated and modulated oscillons and demonstrate remarkable agreement with numerical solutions across several models. Higher-order RG corrections introduce model-dependent terms that lift universality, with sine-Gordon serving as a consistency check via the breather expansion. The results unify oscillon physics with $Q$-ball dynamics, justify the two-DoF origin of modulations, and open avenues toward higher-dimensional generalizations and quantum extensions.

Abstract

Using Renormalization Group Theory we show that oscillons in (1+1)-dimensions can be obtained, at the leading nonlinear order, from $Q$-balls of universal complex field theories. For potentials with a nonzero cubic or quartic term the universal $Q$-ball theory is well approximated by the integrable complex sine-Gordon model. This allows us to generalize the usual perturbative expansion by Fodor et. al. beyond the simplest unmodulated oscillon case. Concretely, we explain the characteristic amplitude modulations of excited oscillons as an effect of formation of a two-$Q$-ball (two-oscillon) bound state.

Figures (18)

• Figure 1: Comparison among the exact (numerical) solution of Eq. \ref{['eq:quartosc']}, bare solution $y_{\mathrm{B}}$, Eq. \ref{['eq:sol1']}, and renormalized solution $y_{\mathrm{R}}$, Eq. \ref{['eq:sol2']} for $y_0 = 1$ which corresponds to $R \approx 0.486$. This choice illustrates that the match is fairly good even for a large initial amplitude. Note that for times larger than $1/R^2 \approx 4.2$ the bare solution deviates significantly from the true solution, as expected.
• Figure 2: Field theoretical potentials in models of oscillons that are discussed in our examples. Blue: $\phi^3$; orange: the inverse $\phi^4$; green: the double well $\phi^4$; red: the exotic $\phi^6$.
• Figure 3: Comparison between numerically found oscillon (blue) and renormalized solution (orange) for the single Q-ball solution in $\phi^3$ theory. We plot the value of the field at origin $\phi(x=0,t)$. Upper: $\lambda = 0.2$; Lower: $\lambda= 0.3$; Bottom: $\lambda= 0.6$.
• Figure 4: Comparison between numerically found oscillon (blue) and renormalized solution (orange) for the single Q-ball solution in $\phi^3$ theory. Upper: $\lambda=0.2$ and we plot the field profiles at $t=0, 1.8, 3.1$ and $t=51$. Lower: $\lambda=0.6$ and $t=0, 3,3.5$ and $t=17$.
• Figure 5: Comparison between numerically found oscillon (blue) and renormalized solution (orange) for the single Q-ball solution in the inverse $\phi^4$ theory. We plot the value of the field at origin $\phi(x=0,t)$. Upper: $\lambda = 0.2$; Lower: $\lambda= 0.5$; Bottom: $\lambda= 0.6$.