An approach based on combined solutions of the Bethe-Salpeter (BS) and Dyson-Schwinger (DS) equations within the ladder-rainbow approximation in the presence of singularities is proposed to describe the meson spectrum as quark antiquark bound states.
We consistently implement into the BS equation the quark propagator functions from the DS equation, with and without pole-like singularities, and show that, by knowing the precise positions of the poles and their residues, one is able to develop reliable methods of obtaining finite interaction BS kernels and to solve the BS equation numerically. We show that, for bound states with masses $M < 1$ GeV, there are no singularities in the propagator functions when employing the infrared part of the Maris-Tandy kernel in truncated BS-DS equations. For $M >1 $ GeV, however, the propagator functions reveal pole-like structures. Consequently, for each type of mesons (unflavored, strange and charmed) we analyze the relevant intervals of $M$ where the pole-like singularities of the corresponding quark propagator influence the solution of the BS equation and develop a framework within which they can be consistently accounted for. The BS equation is solved for pseudo-scalar and vector mesons. Results are in a good agreement with experimental data. Our analysis is directly related to the future physics programme at FAIR with respect to open charm degrees of freedom.
In view of the mass spectrum of heavy mesons in vacuum the analytical properties of the solutions of the truncated Dyson-Schwinger equatio for the quark propagator within the rainbow approximation are analysed in some detail. In Euclidean space, the
quark propagator is not an analytical function possessing, in general, an infinite number of singularities (poles) which hamper to solve the Bethe-Salpeter equation. However, for light mesons (with masses M_{qbar q} <= 1 GeV) all singularities are located outside the region within which the Bethe-Salpeter equation is defined. With an increase of the considered meson masses this region enlarges and already at masses >= 1 GeV, the poles of propagators of u,d and s quarks fall within the integration domain of the Bethe-Salpeter equation. Nevertheless, it is established that for meson masses up to M_{qbar q}~=3 GeV only the first, mutually complex conjugated, poles contribute to the solution. We argue that, by knowing the position of the poles and their residues, a reliable parametrisation of the quark propagators can be found and used in numerical procedures of solving the Bethe-Salpeter equation. Our analysis is directly related to the future physics programme at FAIR with respect to open charm degrees of freedom.
The mass spectrum of heavy pseudoscalar mesons, described as quark-antiquark bound systems, is considered within the Bethe-Salpeter formalism with momentum dependent masses of the constituents. This dependence is found by solving the Schwinger-Dyson
equation for quark propagators in rainbow-ladder approximation. Such an approximation is known to provide both a fast convergence of numerical methods and accurate results for lightest mesons. However, as the meson mass increases, the method becomes less stable and special attention must be devoted to details of numerical means of solving the corresponding equations. We focus on the pseudoscalar sector and show that our numerical scheme describes fairly accurately the $pi$, $K$, $D$, $D_s$ and $eta_c$ ground states. Excited states are considered as well. Our calculations are directly related to the future physics programme at FAIR.
The mass spectrum of heavy pseudoscalar mesons, described as quark-antiquark bound systems, is considered within the Bethe-Salpeter formalism with momentum-dependent masses of the constituents. This dependence is found by solving the Schwinger-Dyson
equation for quark propagators in rainbow-ladder approximation. Such an approximation is known to provide both a fast convergence of numerical methods and accurate results for lightest mesons. However, as the meson mass increases, the method becomes less stable and special attention must be devoted to details of numerical means of solving the corresponding equations. We focus on the pseudoscalar sector and show that our numerical scheme describes fairly accurately the $pi$, $K$, $D$, $D_s$ and $eta_c$ ground states. Excited states are considered as well. Our calculations are directly related to the future physics programme at FAIR.
The production of pseudo scalar, Eeta, Eta-prime, and vector, Omega, Rho, Phi, mesons in NN collisions at threshold-near energies is analyzed within a covariant effective meson-nucleon theory. It is shown that a good description of cross sections and
angular distributions, for vector meson production, can be accomplished by considering meson and nucleon currents only, while for pseudo scalar production an inclusion of nucleon resonances is needed. The di-electron production from subsequent Dalitz decay of the produced mesons, $etato gamma gamma^* togamma e^+e^-$ and $omegato pigamma^*to pi e^+e^-$ is also considered and numerical results are presented for intermediate energies and kinematics of possible experiments with HADES, CLAS and KEK-PS. We argue that the transition form factor $omegato gamma^*pi$ as well as $etato gamma^*gamma$ can be defined in a fairly model independent way and the feasibility of an experimental access to transition form factors is discussed.
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 r
epresentation 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.
