Magnetic instability in AdS/CFT: Schwinger effect and Euler-Heisenberg Lagrangian of supersymmetric QCD

To reveal the Schwinger effect for quarks, i.e., pair creation process of quarks and antiquarks, we derive the vacuum decay rate at strong coupling using AdS/CFT correspondence. Magnetic fields, in addition to the electric field responsible for the pair creation, causes prominent effects on the rate, and is important also in experiments such as RHIC/LHC heavy ion collisions. In this paper, through the gravity dual we obtain the full Euler-Heisenberg Lagrangian of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} $\end{document} = 2 supersymmetric QCD and study the Schwinger mechanism with not only a constant electric field but also a constant magnetic field as external fields. We determine the quark mass and temperature dependence of the Lagrangian. In sharp contrast to the zero magnetic field case, we find that the imaginary part, and thus the vacuum decay rate, diverges in the massless zero-temperature limit. This may be related to a strong instability of the QCD vacuum in strong magnetic fields. The real part of the Lagrangian serves as a generating function for non-linear electro-magnetic responses, and is found such that the Cotton-Mouton effect vanishes. Interestingly, our results of the Schwinger/Cotton-Mouton effects coincide precisely with those of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} $\end{document} = 2 supersymmetric QED.