Far-from-equilibrium dynamics with broken symmetries from the 2PI-1/N expansion

TL;DR

We address far-from-equilibrium dynamics in an $O(N)$-invariant scalar quantum field theory with a nonvanishing field expectation value $\phi$ by developing a systematic $1/N$ expansion of the $2$PI effective action to next-to-leading order (NLO). The approach yields nonperturbative, causal evolution equations for the macroscopic field and two-point function, incorporating scattering and memory effects, and it is shown to be equivalent to an auxiliary-field formulation that uses a single diagram at NLO. The paper provides explicit forms for the $2PI$ action $\Gamma[\phi,G]$, the $NLO$ contributions via $\mathbf B$ and $\mathbf I$, and the complete set of evolution equations for spectral and statistical functions on the Schwinger-Keldysh contour, including a practical weak-coupling truncation. This framework enables controlled studies of nonequilibrium phenomena such as DCC formation and preheating beyond perturbation theory, with potential applications to heavy-ion physics and early-ununiverse cosmology, and it lays groundwork for numerical implementation on lattices with finite cutoffs.

Abstract

We derive the nonequilibrium real-time evolution of an O(N) - invariant scalar quantum field theory in the presence of a nonvanishing expectation value of the quantum field. Using a systematic 1/N expansion of the 2PI effective action to next-to-leading order, we obtain nonperturbative evolution equations which include scattering and memory effects. The equivalence of the direct method, which requires the resummation of an infinite number of skeleton diagrams, with the auxiliary-field formalism, which involves only one diagram at next-to-leading order, is shown.

Figures (8)

• Figure 1: Graphical representation of the $\phi$--dependent contributions for $\Gamma_2 \equiv 0$. The crosses denote field insertions $\sim \phi_a\phi_a$ for the left figure, which contributes at leading order, and $\sim \phi_a\phi_b$ for the right figure contributing at next-to-leading order.
• Figure 2: LO contribution to the $2PI$ effective action.
• Figure 3: NLO "double bubble" contribution.
• Figure 4: NLO $\phi$--independent contribution to the $2PI$ effective action. Higher loop diagrams in the infinite series can be obtained from the previous one by introducing another "rung" with two propagator lines at each vertex. The resummed series including the prefactors not displayed in the figure is given by the first term in Eq. (\ref{['NLOcont']}).
• Figure 5: NLO $\phi$--dependent contribution to the $2PI$ effective action. Each diagram in the infinite series can be obtained from the previous one by introducing another "rung" with two propagator lines at each vertex. The resummed series is given by the second term in Eq. (\ref{['NLOcont']}). The complete NLO contribution is given by the sum of the diagrams in Figs. \ref{['NLOfig']} and \ref{['NLOfigb']}.