RIKEN Accel. Prog. Rep. 48 (2015)

Ⅱ-6. Particle Physics

Magnetic instability in AdS/CFT : Schwinger effect andEuler-Heisenberg Lagrangian of Supersymmetric QCD†
K. Hashimoto,∗1,∗3 T. Oka,∗2 and A. Sonoda∗3
The renowned Schwinger effect1) creation process ofelectron-positron pairs in strong electric fields, is a big
challenge in the field of non-linear quantum field theory. Although the Schwinger limit E ∼ m2e has notbeen reached by the direct experiments such as strong
lasers, similar effective setups in materials are activelyinvestigated. Theoretical foundation of the Schwinger
effect was to evaluate the imaginary part of the effective action of QED under a constant electromagnetic field, the Euler-Heisenberg Lagrangian2) whichdates back to 1936. The Euler-Heisenberg Lagrangianis a generating function of nonlinear electromagnetic
responses of the vacuum. In its expression, the electricfield couples to the magnetic field in a complicated andnonlinear manner, and the total effective Lagrangian isa starting point in the research of strong fields in QED,including the non-perturbative Schwinger effect.
In our previous paper3) , two of the present authors
derived an Euler-Heisenberg Lagrangian for a supersymmetric QCD in the strong coupling limit, by using
the AdS/CFT correspondence but the Maxwell electricfield) is applied, a quark antiquark pair is created. The
nontrivial part is the gluon interaction at strong coupling in QCD. The quarks are confined, and between
the quark and the antiquark a confining force (a QCDstring) is present to bind them. If the electric field isstrong enough, the quarks are liberated. This truly
nonperturbative process is of importance, not only because it can be a realistic phenomenon occurring in theuniverse, but also because it may be a touchstone forunderstanding the quark confinement.
There are at least two cases in which the QCD
Schwinger effect may play an important role: First,
the heavy ion collision experiment, and second, magnetars (neutron stars with a very strong magnetic field).
In the heavy ion collisions, very strong electric fields
are generated by the the electric current induced byheavy ions passing by each other. Since the magnetic
field is time dependent, there appears strong electricfield as well, and it may be related to the formationof the quark gluon plasma. On the other hand, magnetars are known to be the most dense place in theuniverse, and the strong magnetic field accompanied
by some electric field can occur and affect the core
structure of the stars, possibly having a quark phaseinside. In these examples, the understanding of QCD
and Schwinger effect in strong electric and magnetic
†

∗1
∗2
∗3

fields can be tested by experiments/observations andserves as a playground at which we can test our theoretical knowledge on strongly coupled quantum field
theories.
The result of Ref.3) is summarized as follows; The
Euler-Heisenberg Lagrangian of strongly coupled N =2 supersymmetric QCD at large Nc limit was calculated in the presence of a constant electric field using
the AdS/CFT correspondence. Its imaginary part explicitly evaluated is found to agree with large electricfield expansion of the Schwinger effect of N = 2 supersymmetric QED (once the QCD string tension isreplaced by the electron mass). However, there, only
the electric field was considered. In this paper, we include the full dependence of the magnetic field, which
is important as is obvious from the physical situations
explained above.
Here we summarize the finding of the present paper:• We obtain the Euler-Heisenberg Lagrangian of theN = 2 supersymmetric QCD in a constant electromagnetic field, at strong coupling and large Nclimit.
• We evaluate the imaginary part of the EulerHeisenberg Lagrangian, and find that the rate of
the quark antiquark creation diverges at zero temperature for massless quarks.
• The divergent rate can be regularized, i.e., thevacuum is unstable but the lifetime becomes finite, by either introducing finite temperature or aquark mass.
• We compute the real part of the Euler-Heisenberg
Lagrangian, and show the disappearance ofCotton-Mouton effect in an expansion with theelectromagnetic field.
• The imaginary part of the Euler-Heisenberg Lagrangian for a small quark mass is shown to coincide with that of N = 2 supersymmetric QED, atthe leading order in electron mass. The agreement
is found also for the real parts responsible for theCotton-Mouton effect.
References1) J. S. Schwinger, Phys. Rev. 82, 664 (1951).
2) W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936)
3) K. Hashimoto and T. Oka, JHEP 1310, 116 (2013)[arXiv:1307.7423].

Condensed from the article in JHEP 1406 (2014) 085
RIKEN Nishina Center
Department of Applied Physics, University of Tokyo
Department of Physics, Osaka University- 148 -

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