Vacuum instability in electric fields via AdS/CFT: Euler-Heisenberg Lagrangian and Planckian thermalization

Abstract

We analyze vacuum instability of strongly coupled gauge theories in a constant electric field using AdS/CFT correspondence. The model is the $ \mathcal{N}=2 $ 1-flavor supersymmetric large N c QCD in the strong ’t Hooft coupling limit. We calculate the Euler-Heisenberg effective Lagrangian $ \mathcal{L} $ (E), which encodes the nonlinear response and the quantum decay rate of the vacuum in a background electric field E, from the complex D-brane action in AdS/CFT. We find that the decay rate given by Im $ \mathcal{L} $ (E) becomes nonzero above a critical electric field set by the confining force between quarks. A large E expansion of Im $ \mathcal{L} $ (E) is found to coincide with that of the Schwinger effects in QED, replacing its electron mass by the confining force. Then, the time-dependent response of the system in a strong electric field is solved non-perturbatively, and we observe a universal thermalization at a shortest timescale “Planckian thermalization time” $ {\tau_{\mathrm{th}}}\sim \frac{\hbar }{{{k_B}T_{\mathrm{eff}}^{\infty }}}\sim \frac{\hbar }{{{k_B}}}{E^{{-{1 \left/ {2} \right.}}}} $ . Here, $ T_{\mathrm{eff}}^{\infty } $ is an effective temperature which quarks feel in the nonequilibrium state with nonzero electric current, calculated in AdS/CFT as a Hawking temperature. Stronger electric fields accelerate the thermalization, and for a realistic value of the electric field in RHIC experiment, we obtain τ th ~ 1 [fm/c], which is consistent with the believed value.