Electric field of a pointlike charge in a strong magnetic field and ground state of a hydrogenlike atom

Abstract

In an external constant magnetic field, so strong that the electron Larmour length is much shorter than its Compton length, we consider the modification of the Coulomb potential of a point charge owing to the vacuum polarization. We establish a short-range component of the static interaction in the Larmour scale, expressed as a Yukawa-like law, and reveal the corresponding "photon mass"parameter. The electrostatic force regains its long-range character in the Compton scale: the tail of the potential follows an anisotropic Coulomb law, decreasing away from the charge slower along the magnetic field and faster across. In the infinite-magnetic-field limit the potential is confined to an infinitely thin string passing though the charge parallel to the external field. This is the first evidence for dimensional reduction in the photon sector of quantum electrodynamics. The one-dimensional form of the potential on the string is derived that includes a δ-function centered in the charge. The nonrelativistic ground-state energy of a hydrogenlike atom is found with its use and shown not to be infinite in the infinite-field limit, contrary to what was commonly accepted before, when the vacuum polarization had been ignored. These results may be useful for studying properties of matter at the surface of extremely magnetized neutron stars.

Critical nucleus charge in a superstrong magnetic field

“…With the help of this formula the potential for 2 z = 0 with arbitrary ρ and for ρ = 0 with arbitrary z was calculated analytically [11,12]. It was found [7,8,12] that for z 1/m the equipotential lines are ellipses, though the potential in the mid-range distances, r = ρ 2 + z 2 1/m, was still unknown. In Fig.…”

“…With these approximations we follow [7, 8, 10-12, 15, 18] where the arguments in favour of this approximation can be found. The polarization operator enters the photon propagator leading to the modification of the pointlike charge potential (we consider elementary charge e): After integration over the angle in the plane transverse to the magnetic field the following result was obtained [7,8] (up to units and notations):…”

“…For example, in an external magnetic field vacuum polarization in one loop is strongly enhanced and proportional to magnetic field strength B. It was discovered by Shabad and Usov [7,8] (see also [9]) that this enhancement leads to the significant modification (screening) of the Coulomb potential for B m 2 /e 3 where m and e are electron mass and charge 1 . The potential was found numerically along the magnetic field on the line going through the charge and in the direction transverse to the magnetic field in the plane containing the charge.…”

“…Using formula (5) the analytical expressions for Φ (0, z) and Φ (ρ, 0) were found [11,12] (see also asymptotics in [7,8]):…”

Atmospheres and radiating surfaces of neutron stars

“…Such charge, in particular, screens the Coulomb interaction between an electron and an atomic nucleus at distances comparable to the Compton wavelength λ C = 2π /(m e c) = 2πα f a B . Shabad and Usov [317,318] noted that this screening affects the even atomic levels in superstrong magnetic fields, which squeeze the atom so that its size becomes comparable to λ C . As a result, instead of the unlimited growth of the binding energies of the tightly-bound states that is predicted by the nonrelativistic theory for unlimited increase of B, these energies ultimately level off.…”