Modified Coulomb Law in a Strongly Magnetized Vacuum

Abstract

We study electric potential of a charge placed in a strong magnetic field B ≫ B0 = 4.4 · 10 13 G, as modified by the vacuum polarization. In such field the electron Larmour radius is much less than its Compton length. At the Larmour distances a scaling law occurs, with the potential determined by a magnetic-field-independent function. The scaling regime implies short-range interaction, expressed by Yukawa law. The electromagnetic interaction regains its long-range character at distances larger than the Compton length, the potential decreasing across B faster than along. Correction to the nonrelativistic ground-state energy of a hydrogenlike atom is found. In the limit B = ∞, the modified potential becomes the Dirac δ-function plus a regular background. With this potential the ground-state energy is finite -the best pronounced effect of the vacuum polarization.PACS numbers: 03.50, 77.22.Ch, 97.60.Gb There is now compelling evidence that many compact astronomical objects (soft gamma-ray repeaters, anomalous X-ray pulsars, and some radio pulsars) identified with neutron stars have surface magnetic fields as high as ∼ 10 14 − 10 15 G [1]. More strong magnetic fields (B ∼ 10 16 − 10 17 G) are predicted to exist at the surface of cosmological gamma-ray bursters if they are rotationpowered neutron stars similar to radio pulsars [2]. All these fields, however, are much smaller than the maximum value inherent in quantum electrodynamics [3].Vacuum in an external magnetic field B behaves as an anisotropic dielectric medium with spatial and frequency dispersion, (e.g., [4]). These properties may become important, provided that the field strength achieves the characteristic value B 0 = m 2 /e ≃ 4.4×10 13 G, where m is the electron mass and e is its charge. [Henceforth, we set = c = 1 and refer to the Heaviside-Lorentz system of units.] Although much work has been devoted to study of electromagnetic wave propagation in the magnetized vacuum, problems of electro-and magneto-statics in this medium did not attract sufficient attention, save Refs. [5,6], where corrections to the Coulomb law were found when these are small: for B/B 0 ≪ 1 in [5], or at large distances from the source for 1 ≪ B/B 0 ≪ 3παIn this Letter, we find that for sufficiently large b ≡ B/B 0 ≫ 1 the electric field produced by a pointlike charge at rest may be significantly modified by the vacuum polarization, the modification being determined by the characteristic factor αb. The modified Coulomb potential in the close vicinity of its charge, characterized by the Larmour length L B = (eB) −1/2 = (1/m √ b), goes steeper than the standard one, following a Yukawa law, whereas it obeys a long-range "anisotropic Coulomb law" far from the source, at distances characterized by the electron Compton length m(Details of the corresponding derivations can be found in the accompanying preprint [7]). The short-range part of the modified potential tends to the Dirac δ-function in the limit b → ∞. The modification of the Coulomb law should affect, first of all, the field of...

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