Oscillons in gapless theories

We show that large scale oscillons, i.e., quasiperiodic, long-living particlelike solutions, may exist in massless theories, too. Their existence is explained using an effective (smeared) mass threshold which takes into account nonlinear (finite) perturbations.

Figure 1

The massless potential (2) for various $\epsilon$.

Figure 2

Dynamics of oscillons in the massless deformation of the ${\varphi }^4$ model in ($1+1$) dimensions. Value of the field at the origin for $\epsilon =0.001$ (upper) and 0.003 (lower).

Figure 3

Dynamics of oscillons in the massless deformation of the ${\varphi }^4$ model in ($1+1$) dimensions. Upper: decay of the amplitude. Lower: evolution of the frequency of oscillons.

Figure 4

The smeared (finite perturbation) mass threshold: step function (full line) and Gaussian smearing (dashed-dotted line).

Figure 5

The effective frequency ${\omega }_{\mathrm{eff}}$.

Figure 6

The dependence of measured oscillation frequency on the amplitude of the oscillon in the massless model with $\epsilon =0.003$ (blue curve) and in the ${\varphi }^4$ theory.

Figure 7

Power spectrum of an oscillon in ($1+1$) dimensions in the massless deformed model with $\epsilon =0.001$.

Figure 8

Decay of the amplitude of oscillons in ($3+1$) dimensions. Upper: the ${\varphi }^4$ model, i.e., $\epsilon =0$. Lower: the massless model (1) with $\epsilon ={10}^{-5}$.

Figure 9

Dynamics and lifetime of the oscillon—envelope of $\varphi (0,t)$. Upper: the massless model (1) with $\epsilon ={10}^{-6}$ and different intial data parameters $r_0$. Lower: dynamics in different massless models and with $r_0=2.8496049$.

Figure 10

Dynamics and lifetime of the oscillon in ${\varphi }^4$ in ($3+1$) dimensions—envelope of $\varphi (0,t)$.

Figure 11

Comparison of the lifetime of the oscillon in ($3+1$) dimensions in ${\varphi }^4$ theory and massless deformed models (1).

Figure 12

Envelope of $\varphi (0,t)$ for different values of $\epsilon$ in $3+1d$ models for Gaussian initial conditions with $r_0=2.8496049$.

Figure 13

The effective mass threshold $m_{\mathrm{eff}}$ for the step function and Gaussian smearing functions for the ${\varphi }^4$ theory ($\epsilon =0$) and for the massless deformation with $\epsilon =0.001$.