Oscillons from $Q$-balls through renormalization

TL;DR

Oscillons are long-lived, localized excitations in nonlinear field theories. The authors use a renormalization-group perturbation expansion (RGPE) to derive a universal Q-ball equation that matches the integrable complex sine-Gordon model, encapsulated by $∂^2 Ψ + Ψ = 2 Ψ |Ψ|^2$. They show that generic (1+1)-dimensional oscillons emerge from this universal Q-ball dynamics, and that modulated (excited) oscillons are bound states of two Q-balls, captured by a two-Q-ball solution in the sine-Gordon limit. The approximate integrability explains oscillon longevity and suggests extensions to higher dimensions and quantum oscillons, with distinct universality classes arising when higher-order terms alter the effective Q-ball equation.

Abstract

Using a renormalization-inspired perturbation expansion we show that oscillons in a generic field theory in (1+1) dimensions arise as dressed $Q$-balls of a universal (up to the leading nonlinear order) complex field theory. This theory reveals a close similarity to the integrable complex sine-Gordon model which possesses exact multi-$Q$-balls. We show that excited oscillons, with characteristic modulations of their amplitude, are two-oscillons bound states generated from a two $Q$-ball solution.

Figures (3)

• Figure 1: Comparison between the unmodulated oscillon with (blue) a solution to full nonlinear equations of motion and renormalized solution (orange) for the single Q-ball solution given via Eq. (\ref{['eq:qball']}) inserted into Eq. (\ref{['eq:renormsol']}), which is supplied as initial conditions. Upper: $\phi^3$ theory and $\lambda = 0.2$. Lower: double well $\phi^4$ theory and $\lambda = 0.15$. The oscillons are small and with negligible modulations. The FFHL approximation is indistinguishable from the orange curve.
• Figure 2: Comparison between the modulated oscillon with (blue) a solution to full nonlinear equations of motion and renormalized solution (orange) for the two Q-ball solution given via Eq. (\ref{['eq:twoQ']}) inserted into Eq. (\ref{['eq:renormsol']}), which is supplied as initial conditions. Upper: $\phi^3$ theory and $\lambda_1=0.3$, $\lambda_2=-0.05$. Lower: the double well $\phi^4$ theory and $\lambda_1=0.15$, $\lambda_2=-0.05$. The two Q-ball solution captures the characteristic amplitude modulation.
• Figure 3: Comparison for two Q-ball solution supplied as initial configuration. For better clarity, we plot $|\partial^2 \phi+\phi|$ as a function of $x$ and $t$ for both the numerical solution (left panel) and the renormalized solution (right panel). Upper: $\phi^3$ theory for $\lambda_1=0.3$, $\lambda_2=-0.1$ (left) and $\lambda_1=0.25$, $\lambda_2=-0.15$ (right). Lower: the double well $\phi^4$ theory for $\lambda_1=0.05$, $\lambda_2=-0.15$ (left) and $\lambda_1=0.1$, $\lambda_2=-0.15$ (right). As we see, the two Q-ball solution captures the characteristic amplitude modulation.