Dimensional reduction and catalysis of dynamical symmetry breaking by a magnetic field

dimensions is a strong catalyst of dynamical chiral symmetry breaking, leading to the generation of a fermion dynamical mass even at the weakest attractive interaction between fermions. The essence of this effect is the dimensional reduction

D\to D-2

in the dynamics of fermion pairing in a magnetic field. The effect is illustrated in the Nambu--Jona--Lasinio (NJL) model and QED. In the NJL model in a magnetic field, the low--energy effective action and the spectrum of long--wavelenth collective excitations are derived. In QED, it is shown that the dynamical mass of fermions (energy gap in the fermion spectrum) is

m_{\rm dyn}=C\sqrt{|eB|}\exp[-\frac{\pi}{2}(\frac{\pi}{2\alpha})~{1/2}]

, where

B

is the magnetic field, the constant

C

is of order one and

\alpha=e~2/4\pi

is the renormalized coupling constant. Possible applications of this effect and its extension to inhomogeneous field configurations are discussed.

fermion
quantum electrodynamics
Jona-Lasinio-Nambu model
magnetic field
dimension: 4
dimensional reduction
dimension: 3
dimension: 2
dynamical symmetry breaking
numerical calculations

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