Solving the Bethe–Salpeter Equation in Euclidean Space

Abstract

Different approaches to solve the spinor-spinor Bethe-Salpeter (BS) equation in Euclidean space are considered. It is argued that the complete set of Dirac matrices is the most appropriate basis to define the partial amplitudes and to solve numerically the resulting system of equations with realistic interaction kernels. Other representations can be obtained by performing proper unitary transformations. A generalization of the iteration method for finding the energy spectrum of the BS equation is discussed and examples of concrete calculations are presented. Comparison of relativistic calculations with available experimental data and with corresponding non relativistic results together with an analysis of the role of Lorentz boost effects and relativistic corrections are presented.A novel method related to the use of hyperspherical harmonics is considered for a representation of the vertex functions suitable for numerical calculations.

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

“…Such a method is known as the exhausting (depletion) method and was reported in some details in Ref. [27]. It seems that a similar method has been recently employed in Ref.…”

“…Mmax n=1 (27) represents the sought solution in the form of a group of sets of partial wave components ϕ n α , specified on the integration mesh of the order N G . The matrix S is determined by the corresponding partial kernels (10), the Gaussian weights and the Jacobian of the mapping and is of the N × N dimension, where N = α max × M max × N G .…”

“…(3)- (5) the integral matrix form of the BS equation (1) can be reduced to a system of two-dimensional integral equations with respect to the partial vertices g α (p), cf. [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]:…”

Relative-energy dependence of the Bethe–Salpeter amplitude in two-dimensional massless QED

“…The analysis of that nontrivial aspect of a bound system has been undertaken in a couple of works [15][16][17][18][19][20][21] * Electronic address: t.radozycki@uksw.edu.pl both in euclidean [22][23][24][25][26][27][28] and in Minkowski space [28][29][30][31][32][33][34][35] (where it is more difficult), but a complete description of any B-S amplitude in its full complexity is missing, with one exception to be mentioned below. That has been mainly due to the lack of the appropriate model, which on one hand would constitute a nontrivial quantum field theory and on the other would allow for analytic and exact solutions.…”