Eur. Phys. J. A (2010) 46: 387-397
https://doi.org/10.1140/epja/i2010-11055-4

Regular Article - Theoretical Physics

Solving the Bethe-Salpeter equation for two fermions in Minkowski space

¹ Laboratoire de Physique Subatomique et Cosmologie, CNRS/IN2P3, 53 av. des Martyrs, 38026, Grenoble, France
² Lebedev Physical Institute, Leninsky Prospekt 53, 119991, Moscow, Russia

^* e-mail: karmanov@lpsc.in2p3.fr

Received: 16 July 2010
Accepted: 15 October 2010
Published online: 9 November 2010

Abstract

The method of solving the Bethe-Salpeter equation in Minkowski space, developed previously for spinless particles (V.A. Karmanov, J. Carbonell, Eur. Phys. J. A 27, 1 (2006)), is extended to a system of two fermions. The method is based on the Nakanishi integral representation of the amplitude and on projecting the equation on the light-front plane. The singularities in the projected two-fermion kernel are regularized without modifying the original BS amplitudes. The numerical solutions for the J = 0 bound state with the scalar, pseudoscalar and massless vector exchange kernels are found. The stability of the scalar and positronium states without vertex form factor is discussed. Binding energies are in close agreement with the Euclidean results. Corresponding amplitudes in Minkowski space are obtained.