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

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 in 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 meson (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 pseudoscalar and vector mesons. Results are in good agreement with experimental data. Our analysis is directly related to the future physics program at FAIR with respect to open charm degrees of freedom.

Figure 1

Positions of the few first poles (filled symbols) in the upper left hemisphere of the complex ${\tilde{k}}^2$ plane for various current quark masses (given in MeV). Relevant sections of the parabola, (13), corresponding to the meson bound-state mass $M$ are presented by solid curves for $M=1.1$, 1.5, 2.0, 2.5, 3.0, and $3.5$ GeV, from right to left. The pole positions for a 5 -MeV quark are represented by open symbols.

Figure 2

Bound-state masses (ground states) of a system $q{q}_x$ as a function of the bare mass $m_x$, for $m_q=0.005$ GeV (curve labeled “$ux$”) or $m_q=0.115$ GeV (curve labeled “$sx$”) or $m_q=1.0$ GeV (curve labeled “$cx$”). The effective parameters of the vertex-gluon kernel, (2), are $\omega =0.5$ GeV and $D=16{\mathrm{GeV}}^{-2}$. The masses of the pseudoscalar $\pi ,K,D,D_s$, and ${\eta }_c$ mesons according to [35] are indicated at the right. Dashed vertical lines represent the selected bare quark masses for $u$, $s$, and $c$.

Figure 3

Dynamically dressed quark mass $m_q\left({\tilde{k}}^2\right)=B\left({\tilde{k}}^2\right)/A\left({\tilde{k}}^2\right)$ from solutions of the Dyson-Schwinger equation along the real axis, i.e., for $\mathrm{Im}{\tilde{k}}^2=0$. (a) The dependence on $\mathrm{Re}{k}^2>0$, i.e., the behavior of the solution of the tDS equation in real Euclidean space. (b) Quark masses $m_q$ are calculated at the parabola vertex, $\mathrm{Re}{k}^2=-M^2/4$, and displayed as a function of $M$. A comparison of (a) and (b) illustrates the dynamical mass increase in the complex Euclidean plane. Solid curves represent the light $u$ and $d$ quarks; dashed curves, $s$ quarks; and dot-dashed curves, $c$ quarks.

Figure 4

Interaction domain of the tBS equation over the complex ${\tilde{k}}_{1,2}^2$ plane, which is the surface of the parabola $\mathrm{Im}{\tilde{k}}_{1,2}^2=\pm M\sqrt{M^2/4+\mathrm{Re}{\tilde{k}}_{1,2}^2}$ for $M=1.5$ GeV. The two self-conjugated poles for $u$ and $d$ quarks are represented by stars. Filled triangles represent the Gaussian mesh for $\mathrm{Re}{\tilde{k}}_{1,2}^2$ with 32 nodes (a) and 96 nodes (b) for the $d\tilde{k}$ integration, which actually extends to $+\infty$. For each value of the node $\mathrm{Re}{\tilde{k}}_i^2$ the angular integration $dcos{\chi }_k$ goes along the dashed vertical lines (only a few are exhibited) corresponding to $-1\le cos{\chi }_k\le +1$.