# Radiative dynamical mass of planar charged fermion in a constant homogeneous magnetic field

## 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 quantities for the massless case. The radiative corrections to the mass of charged massless fermion when it occupies the lowest 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 presence 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, 4, 5, 6]). In graphene, the electron dynamics 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, nevertheless 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, 14, 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 investigated 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 magnetic 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 \(m\rightarrow 0\).

In this work, we calculate the effective Lagrangian and the mass operator of planar charged fermions in the presence of an external constant homogeneous magnetic field in the one-loop approximation of the QED\(_{2+1}\). We also calculate the radiative corrections to the mass of charged fermion when it occupies the lowest Landau level 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 presence of constant homogeneous magnetic field (normal to the graphene sample) though we also see that an one-loop result is not accurate certainly when the coupling constant is large (see, [19]).

We shall adopt the units where \(c=\hbar =1\).

## 2 Eigenfunctions, the Green’s function for the Dirac equation in a constant magnetic field in 2+1 dimensions. Effective Lagrangian

*m*and charge \(e=-e_0<0\) “minimally” interacting with the background electromagnetic field is written in the covariant form as

*eB*|

*n*label the solutions of Dirac Eq. (2) as well as the \(E_p\) eigenfunctions of operators (4) and (5) can also classifies by the eigenvalues \(\zeta =\pm 1\) of \(\sigma _3\). The eigenfunctions \(E_{p\zeta }(t, \mathbf{r})\) are given by

*s*is the “proper time”.

## 3 Mass operator of a charged fermion in a constant homogeneous magnetic field

*I*is the unit two-column matrix, and

It is worth while noting that the fermion propagator (i.e. the fermion Green’s function) in a constant homogeneous magnetic field in 2+1 dimensions (11) is derived with taking account of all Landau levels.

*s*on \(\pi /2\) (a Wick rotation). We shall calculate the mass correction for the (on mass-shell) level \(p_0=m,\quad n=0\). As a result of all these transformations, the mass operator can be written as

*z*in the limits \([0,\infty )\) to divide into two regions \([0, z_0]\) plus \([z_0,\infty )\) [33] (see, also, [19]) with \(z_0\sim 1\). Then, when integrating into region \([z_0,\infty )\) the exponentials can be neglected and integral between the limits 0 and \(z_0\) is small compared with that in the limits \([z_0,\infty )\) and it can be neglected. Having made the integrations over

*v*we obtained

*v*at \(z=1\) gives \(2\sqrt{2}\pi \).

*z*we find the mass corrections in the form

*c*and \(\hbar \) and rewrite Eq. (45) as follows

## 4 Resume

In this work we calculated the effective Lagrangian and the mass operator of planar charged fermions in an external constant homogeneous magnetic field in the one-loop approximation of the QED\(_{2+1}\). We also calculated the mass corrections of charged fermion in the lowest Landau level. The fermion mass can be generated dynamically in a constant homogeneous 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 \(m_0=0\). We believe the mass generation to be due to renormalization and to closely resemble to the dimensional transmutation phenomenon which occurs in massless relativistic field theories [38]. In our case the cutoff parameter, in fact, depending upon *eB* transmutes in an arbitrary mass. The dynamical mass seems to be likely to be revealed in monolayer graphene in the presence of constant homogeneous magnetic field (normal to the graphene sample).

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