Renormalization out of equilibrium in a superrenormalizable theory

We discuss the renormalization of the initial value problem in nonequilibrium quantum field theory within a simple, yet instructive, example and show how to obtain a renormalized time evolution for the two-point functions of a scalar field and its conjugate momentum at all times. The scheme we propose is applicable to systems that are initially far from equilibrium and compatible with nonsecular approximation schemes which capture thermalization. It is based on Kadanoff-Baym equations for non-Gaussian initial states, complemented by usual vacuum counterterms. We explicitly demonstrate how various cutoff-dependent effects peculiar to nonequilibrium systems, including time-dependent divergences or initial-time singularities, are avoided by taking an initial non-Gaussian three-point vacuum correlation into account.

Figure 1

Closed time path $\mathcal{C}$ used for the computation of expectation values. The time path starts and ends at the initial time $t_{\mathrm{init}}\equiv 0$, and $t_{\max }$ should be chosen larger than the largest time for which correlation functions are computed. The boundaries of the time path are denoted by $0_{+}$ and $0_{-}$, respectively.

Figure 2

One-loop contributions to the $\phi$ self-energy $\mathrm{\Pi }(x,y)$. The box represents the initial three-point correlation $i{\alpha }_3$. The first and second diagrams contribute to ${\mathrm{\Pi }}_{F/\rho }$ and ${\mathrm{\Pi }}_{F/\rho }^{\lambda \alpha }$ respectively, while the other two are $\propto {\delta }_{s/a}(x_0)$ and not needed in what follows.

Figure 3

Time-evolution of the statistical propagator $F_p(t,t)$ and the momentum-momentum correlator ${\partial }_t{\partial }_{t^{\prime }}{F}_p(t,t^{\prime })-{\partial }_t{\partial }_{t^{\prime }}{F}_p^{\text{vac}}(t,t^{\prime }){|}_{t=t^{\prime }}$ (for $p=0$) at equal times for three different values of the cutoff $\mathrm{\Lambda }/m_{\phi }=10$, 25, 100, and three different initial conditions (G1) (dashed), (G2) (dotted) and (NG) (solid). We used ${\mathrm{\Delta }}_{p=0}^{(0)}=n^{in}/m_{\phi },{\mathrm{\Delta }}_{p=0}^{(1)}=0,{\mathrm{\Delta }}_{p=0}^{(2)}=n^{in}{m}_{\phi }$ with $n^{in}=0.5$, $\gamma /m_{\phi }=0.28$, $T_{\chi }=m_{\chi }=0$.

Figure 4

Time-evolution of the unequal-time correlation function ${\partial }_t{F}_p(t,0)$ for the same set of initial conditions and values of $\mathrm{\Lambda }$ as in Fig. 3. Note that both Gaussian initial conditions (G1) and (G2) coincide for this correlator (dashed lines), and approach the continuum limit shown by the grey dot-dashed line for $\mathrm{\Lambda }\to \infty$. For the non-Gaussian initial condition (NG) the (solid) lines for the three values $\mathrm{\Lambda }/m_{\phi }=100$, 25, 10 lie almost on top of each other.