Two-fermion bound states within the Bethe-Salpeter approach

Abstract

To solve the spinor-spinor Bethe-Salpeter equation in Euclidean space we propose a novel method related to the use of hyperspherical harmonics. We suggest an appropriate extension to form a new basis of spin-angular harmonics that is suitable for a representation of the vertex functions. We present a numerical algorithm to solve the Bethe-Salpeter equation and investigate in detail the properties of the solution for the scalar, pseudoscalar and vector meson exchange kernels including the stability of bound states. We also compare our results to the non relativistic ones and to the results given by light front dynamics.

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

“…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]. Our analysis, though in ladder approximation, is fully able to address a relevant issue for hadron physics, i.e.…”

“…Tables I (scalar coupling) and II (pseudoscalar coupling) show the comparison between the values of g 2 obtained within our approach, where the singularities have been singled out and analytically evaluated, and both (i) the calculations by Ref. [8], where a non trivial numerical treatment of the singular behaviors was introduced (without recognizing the possibility of a systematic analysis of the singularities as in [13]) and (ii) the available numerical results in Euclidean space [10], with a suitable number of digits.…”

Accounting for the analytical properties of the quark propagator from the Dyson-Schwinger equation

“…The general structure of vertex functions describing bound states of spinor particles has been investigated in detail, for example, in [24][25][26]. To release from the matrix structure, the vertex function Γ is expanded into functions which in turn are determined by angular momentum and parity of the corresponding meson known as the spin-angular harmonics [14,25,27]:…”

“…To release from the matrix structure, the vertex function Γ is expanded into functions which in turn are determined by angular momentum and parity of the corresponding meson known as the spin-angular harmonics [14,25,27]:…”

Schwinger–Dyson equation and NJL approximation in massive gauge theory with fermions

“…Actually, the works on light front dynamics indeed confirm the relation M H ≈ 2m [48] in the system with the exchange by massive vector bosons near to the criticality. We are aware of the investigations in the approach with BS equations only for the case M = 0 (see, for example, [49,2] and references therein), where this relation is confirmed as well.…”