Bethe-Salpeter approach for relativistic positronium in a strong magnetic field

TL;DR

This paper develops a relativistic Bethe-Salpeter framework to analyze positronium in extremely strong magnetic fields, revealing conditions for mass compensation and vacuum state transition due to dimensional reduction effects.

Contribution

It derives a two-dimensional relativistic Bethe-Salpeter equation in strong magnetic fields and identifies the critical magnetic field for positronium mass compensation.

Findings

Determines the critical magnetic field for positronium mass compensation.

Shows the vanishing of the energy gap at the critical field.

Describes the dimensional reduction effect leading to the falling to the center phenomenon.

Abstract

We study the electron-positron system in a strong magnetic field using the differential Bethe-Salpeter equation in the ladder approximation. We derive the fully relativistic two-dimensional form that the four-dimensional Bethe-Salpeter equation takes in the limit of asymptotically strong constant and homogeneous magnetic field. An ultimate value for the magnetic field is determined, which provides the full compensation of the positronium rest mass by the binding energy in the maximum symmetry state and vanishing of the energy gap separating the electron-positron system from the vacuum. The compensation becomes possible owing to the falling to the center phenomenon that occurs in a strong magnetic field because of the dimensional reduction. The solution of the Bethe-Salpeter equation corresponding to the vanishing energy-momentum of the electron-positron system is obtained.