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Tachyonic and parametric resonances for massive particle production in an intense plane wave background

Ekaterina Dmitrieva, Petr Satunin

TL;DR

The paper analyzes resonant production of massive particles in a toy model with a trilinear coupling $g\phi\chi^2$ driven by an intense plane-wave background $\phi$. It develops two complementary approaches: (i) a Heisenberg-equation treatment of χ in the external plane wave, and (ii) a boosted-rest-frame reduction to the Mathieu equation for a massive condensate. The results reveal mass-dependent stability: for a massless $\phi$ the instability manifests as tachyonic-like growth, while for a massive $\phi$ the instability is captured by the Mathieu equation with distinct narrow, broad, and tachyonic resonance regimes; a threshold amplitude/energy is identified to trigger resonance. The study clarifies the relative strengths and limitations of each method and highlights the role of Lorentz boosts in connecting plane-wave and condensate pictures, with implications for nonperturbative particle production in early-Universe and strong-field contexts.

Abstract

We investigate the stability of an intensive plane wave of a massless or light field $φ$ in a trilinear scalar model $gφχ^2$ due to the resonant production of massive $χ$ particles in a perturbatively forbidden regime. We apply two methods: first, we solve the Heisenberg equation for the quantum amplitudes of the field $χ$ in an external plane wave background, generalizing the solution of A.Arza. Second, for the light but massive $φ$ we perform the relativistic boost to the rest frame of $φ$, reducing the problem to the stability of the thoroughly investigated massive condensate. It turns out that the stability properties are significantly different for the cases of massless and light fields. In the first case, one should adopt the Heisenberg equation approach, since the alternative method cannot provide a comprehensive outcome. In the second case, the use of the Mathieu equation provides a more accurate solution, while for the massless case, this approach is not applicable.

Tachyonic and parametric resonances for massive particle production in an intense plane wave background

TL;DR

The paper analyzes resonant production of massive particles in a toy model with a trilinear coupling driven by an intense plane-wave background . It develops two complementary approaches: (i) a Heisenberg-equation treatment of χ in the external plane wave, and (ii) a boosted-rest-frame reduction to the Mathieu equation for a massive condensate. The results reveal mass-dependent stability: for a massless the instability manifests as tachyonic-like growth, while for a massive the instability is captured by the Mathieu equation with distinct narrow, broad, and tachyonic resonance regimes; a threshold amplitude/energy is identified to trigger resonance. The study clarifies the relative strengths and limitations of each method and highlights the role of Lorentz boosts in connecting plane-wave and condensate pictures, with implications for nonperturbative particle production in early-Universe and strong-field contexts.

Abstract

We investigate the stability of an intensive plane wave of a massless or light field in a trilinear scalar model due to the resonant production of massive particles in a perturbatively forbidden regime. We apply two methods: first, we solve the Heisenberg equation for the quantum amplitudes of the field in an external plane wave background, generalizing the solution of A.Arza. Second, for the light but massive we perform the relativistic boost to the rest frame of , reducing the problem to the stability of the thoroughly investigated massive condensate. It turns out that the stability properties are significantly different for the cases of massless and light fields. In the first case, one should adopt the Heisenberg equation approach, since the alternative method cannot provide a comprehensive outcome. In the second case, the use of the Mathieu equation provides a more accurate solution, while for the massless case, this approach is not applicable.

Paper Structure

This paper contains 17 sections, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Model
  3. The Heisenberg equation approach. Plane wave
  4. The solution using approximation
  5. The solution without approximation
  6. Problems of the Heisenberg equation approach
  7. Mathieu equation analysis for the instability in condenstae
  8. Narrow resonance
  9. Broad resonance
  10. Tachionic instability
  11. Comparison of two approaches for condensate
  12. The approach through the Heisenberg equation for condensate
  13. Conclusion
  14. Acknowledgements
  15. Solution for narrow resonance regime
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Dependence $\alpha_{\vec{\kappa}}(\mu_\chi)$ at fixed $\kappa=0.1, 0.2, 0.5$. The upper region related to the resonance.
  • Figure 2: Stability diagram of the Mathieu equation and stability bounds for the Heisenberg case (equation \ref{['eq_q_w_app']}) . Vertical axis: $A_k$, Horizontal: $q$. Grey area: stability regions of Mathieu equation. Blue solid line: Heisenberg instability bound, see eq. \ref{['eq_q_w_app']}. Black dashed line - line $A_k=2q$.
  • Figure 3: The resonant solutions. Upper line. Left panel: Narrow resonance, $A_k=1$, $q=0.1$. Right panel: Broad resonance. $A_k=100$, $q=50$. Lower line. Left panel: $N=2$ band resonance. $A_k=4$, $q=1$. Central panel: Tachyonic resonance. $A_k=1$, $q=1.5$. Right panel: destructive parametrical resonance in naively tachyonic unstable area. $A_k=3$, $q=2$