Eur. Phys. J. C (2019) 79:196
https://doi.org/10.1140/epjc/s10052-019-6717-4
Regular Article - Theoretical Physics
Radiative dynamical mass of planar charged fermion in a constant
homogeneous magnetic field
V.R. Khalilov
a
Faculty of Physics, M.V. Lomonosov Moscow State University, 119991 Moscow, Russia
Received: 29 November 2018 / Accepted: 22 February 2019 / Published online: 5 March 2019
© The Author(s) 2019
Abstract The effective Lagrangian and mass operator are
calculated for planar charged massive and massless fermions
in a constant external homogeneous magnetic field in the
one-loop approximation of the 2+1 dimensional quantum
electrodynamics (QED
2+1
). We obtain the renormalizable
effective Lagrangian and the fermion mass operator for a
charged fermion of mass m and then calculate these quan-
tities for the massless case. The radiative corrections to the
mass of charged massless fermion when it occupies the low-
est Landau level are found for the cases of the pure QED
2+1
as well as the so-called reduced QED
3+1
on a 2-brane. The
fermion masses were found can be generated dynamically in
an external magnetic field in the pure QED
2+1
if the charged
fermion has small bare mass m
0
and in the reduced QED
3+1
on a 2-brane even at m
0
= 0. The dynamical mass seems to
be likely to be revealed in monolayer graphene in the pres-
ence of constant homogeneous magnetic field (normal to the
graphene sample).
1 Introduction
Quantum systems of planar charged fermions in external
electromagnetic fields are interesting in view of possible
applications of the corresponding field-theory models to a
number of condensed-matter quantum effects such as, for
example, the quantum Hall effect [1] and high-temperature
superconductivity [2] as well as in connection with problems
of graphene (see, [3–6]). In graphene, the electron dynam-
ics near the Fermi surface can be described by the Dirac
equation in 2+1 dimensions for a zero-mass charged fermion
[4] though the case of massive charged fermions is also of
interest [7]. The field-theory models applied for study are the
pure 2+1 dimensional quantum electrodynamics (QED
2+1
)
as well as the so-called reduced QED
3+1
on a 2-brane. In
the latter model, fermions are confined to a plane, neverthe-
a
e-mail: khalilov@phys.msu.ru
less the electromagnetic interaction between them is three-
dimensional [8,9].
The radiative one-loop shift of an electron energy in the
ground state in a constant homogeneous magnetic field in
QED
2+1
was calculated in [10] and the one-loop electron
self-energy in the topologically massive QED
2+1
at finite
temperature and density was obtained in [11]. The effective
Lagrangian, the electron mass operator and the density of
vacuum electrons (induced by the background field) in an
external constant homogeneous magnetic field were derived
in the one-loop QED
2+1
approximation in [12].
Since the effective fine structure constant in graphene
is large, the QED
2+1
effects can be significant already in
the one-loop approximation. The polarization operator in
graphene in a strong constant homogeneous magnetic field
perpendicular to the graphene membrane has been obtained
in the one-loop approximation of the QED
2+1
in [9,13–15].
The effective potential and vacuum current in graphene in
a superposition of a constant homogeneous magnetic field
and an Aharonov–Bohm vortex was studied in [16]. We
also note that the induced vacuum current in the field of a
solenoid perpendicular to the graphene sample was investi-
gated in [17], and the vacuum polarization of massive and
massless fermions in an Aharonov–Bohm vortex in the one-
loop approximation of the QED
2+1
was studied in [18]. The
one-loop self-energy of a Dirac electron of mass m in a thin
medium simulating graphene in the presence of external mag-
netic field was investigated in the reduced QED
3+1
on a 2-
brane in [19], in which it was shown that the radiative mass
correction in the lowest Landau level does not vanish at the
limit m → 0.
In this work, we calculate the effective Lagrangian and
the mass operator of planar charged fermions in the pres-
ence of an external constant homogeneous magnetic field in
the one-loop approximation of the QED
2+1
. We also calcu-
late the radiative corrections to the mass of charged fermion
when it occupies the lowest Landau level for the cases of the
123
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