Parametric Resonance in Quantum Field Theory

We present the first study of parametric resonance in quantum field theory from a complete next-to-leading order calculation in a $1/N$ expansion of the two-particle irreducible effective action, which includes scattering and memory effects. We present a complete numerical solution for an $O(N)$-symmetric scalar theory and provide an approximate analytic description of the nonlinear dynamics in the entire amplification range. We find that the classical resonant amplification at early times is followed by a collective amplification regime with explosive particle production in a broad momentum range, which is not accessible in a leading-order calculation.

Figure 1

The dots indicate that each diagram is obtained from the previous one by adding another “rung” with two full propagator lines at each vertex. The crosses denote field insertions.

Figure 2

Total energy (solid line) and classical-field energy (dotted line) as a function of time for $\lambda ={10}^{-6}$. The dashed line represents the fluctuation part, showing a transition from a classical-field to a fluctuation dominated regime.

Figure 3

Effective particle number density for the transverse modes as a function of time for various momenta $p\le 5p_0$. At early times, modes with $p\simeq p_0$ are exponentially amplified with a rate $2{\gamma }_0$. Because of nonlinearities, one observes subsequently an enhanced growth with rate $6{\gamma }_0$ for a broad momentum range.

Figure 4

Same as in Fig. 3, for the longitudinal modes. Nonlinear source effects trigger an exponential growth with rate $4{\gamma }_0$ for $p\lesssim 2p_0$. The thick line corresponds to a mode in the parametric resonance band, and the long-dashed line for a similar one outside the band. The resonant amplification is quickly dominated by source-induced particle production.

Figure 5

The rescaled field $\sigma$ as a function of time for $\lambda ={10}^{-6}$ (top panel) and $\lambda =10$ (bottom panel).