Quantitative studies of the homogeneous Bethe-Salpeter equation in Minkowski space

Abstract

The Bethe-Salpeter Equation for a bound system, composed by two massive scalars exchanging a massive scalar, is quantitatively investigated in ladder approximation, within the Nakanishi integral representation approach. For the S-wave case, numerical solutions with a form inspired by the Nakanishi integral representation, have been calculated. The needed Nakanishi weight functions have been evaluated by solving two different eigenequations, obtained directly from the Bethe-Salpeter equation applying the Light-Front projection technique. A remarkable agreement, in particular for the eigenvalues, has been achieved, numerically confirming that the Nakanishi uniqueness theorem for the weight functions, demonstrated in the context of the perturbative analysis of the multi-leg transition amplitudes and playing a basic role in suggesting one of the two adopted eigenequations, can be extended to a non perturbative realm. The detailed, quantitative studies are completed by presenting both probabilities and Light-Front momentum distributions for the valence component of the bound state.

Transition electromagnetic form factor and current conservation in the Bethe-Salpeter approach

“…[5,6] the BS amplitude is represented as an integral over a weight function g -the so called Nakanishi transform [7] -which satisfies a nonsingular equation. A modification of this method, based on the light-front projection of the BS amplitude, was developed in [8][9][10][11][12] and used to find the bound state Minkowski BS amplitude. The elastic EM form factor was also calculated in [13].…”

Advances in solving the two-fermion homogeneous Bethe-Salpeter equation in Minkowski space

“…In this Letter, we first present the formally exact integration of the singularities that prevent a straightforward application of the NIR for solving the two-fermion ladder BSE in Minkowski space, as it was accomplished in the case of two-scalar systems [4][5][6][7]. Then, after exactly transforming BSE in a coupled eigen-equation system, we compare our eigenvalues with both (i) the ones still obtained in Minkowski space [8], but introducing an auxiliary smoothing function, and (ii) outcomes in Euclidean space [10].…”

“…[2] and references therein), has allowed to make substantial steps forward in obtaining accurate numerical solutions of the actual Bethe-Salpeter equation (BSE). With massiveboson exchanges, it has been investigated: (i) two-scalar bound and zero-energy states [3][4][5][6][7] as well as two-fermion ground states [8], with a ladder kernel, governing, as wellknown, the tail of the momentum distributions; (ii) a two-scalar system, with a cross-ladder kernel [9].…”

Solving Bethe-Salpeter scattering state equation in Minkowski space

“…The interest in solving the Bethe-Salpeter (BS) equation [1] in its natural Minkowski-space formulation has increased in the recent years [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. There are several reasons for dealing with Minkowski solutions.…”

“…A modified method to the one developed in [4,5] aimed to compute the scattering states was proposed in [9]. It has already been successfully tested for the bound states [15].…”