Two-loop corrections to the potential of a pointlike charge in a superstrong magnetic field
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EMENTARY PARTICLES AND FIELDS Theory
Two-Loop Corrections to the Potential of a Pointlike Charge in a Superstrong Magnetic Field S. I. Godunov* All-Russia Research Institute of Automatics (RRIA), Sushchevskaya ul. 22, Moscow, 127055 Russia Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117218 Russia Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia Received November 15, 2012; in final form, January, 24, 2013
Abstract—The potential of the pointlike charge in a superstrong homogeneous magnetic field B m2e /e3 ≈ 6 × 1015 G is considered. It is well known that Coulomb potential is significantly modified by taking into account vacuum polarization (calculated in one loop approximation). We consider electron selfenergy and correction to the vertex function at one loop, and show that these diagrams are not enhanced by magnetic field like eB. We calculate two-loop corrections to the vacuum polarization and find that these contributions are small. DOI: 10.1134/S1063778813070041
1. INTRODUCTION In classical electrodynamics, electromagnetic fields do not interact with one another, so that the resulting field is a superposition of electric and magnetic fields. All this changes upon taking into account radiative corrections. Loop diagrams usually make a small contribution (they determine, among other things, the Lamb shift of atomic levels and the anomalous magnetic moment of the electron), but, via the interaction with virtual electrons, an external field may enhance loop contributions. As was shown in [1, 2], a superstrong magnetic field (B m2e /e3 ≈ 6 × 1015 G)1) screens the Coulomb potential due to radiative corrections, thereby changing its form at distances in the range of l < 1/me (here, me is the electron mass). An analytic expression for the screened potential along the magnetic field was derived in [3]. An expression for the potential in a plane orthogonal to the magnetic field was obtained in [4], and higher order perturbative contributions were estimated there. In the present study, both of these issues are considered in detail. The screening arises in the one-loop approximation, and a dominant contribution to it comes from the vacuum-polarization diagram. Here, other oneloop contributions to the potential and corrections
from higher orders of perturbation theory are considered in addition to this diagram. The analysis described below covers not only the usual case of four dimensions (D = 4) but also two-dimensional (D = 2) electrodynamics, which is similar in many aspects to four-dimensional electrodynamics in a superstrong field (see [3, 5]), because, in the presence of a strong magnetic field, the motion of the particles becomes substantially two-dimensional.
2. POTENTIAL OF A POINTLIKE CHARGE IN TWO-DIMENSIONAL SPACETIME (D = 2)
2.1. Tree Level At the tree level, the interaction of two charges is described by the diagram in Fig. 1. In order to obtain an expression for the potential, one should use the transferred momentum k in the form (0, k ): Φ(k
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