Analytical properties of the quark propagator from a truncated Dyson-Schwinger equation in complex Euclidean space

In view of the mass spectrum of heavy mesons in vacuum, the analytical properties of the solutions of the truncated Dyson-Schwinger equation for the quark propagator within the rainbow approximation are analyzed 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 solving the Bethe-Salpeter equation. However, for light mesons (with masses $M_{q\overline{q}}\le 1$ 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 $\ge$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 $M_{q\overline{q}}\simeq 3$ 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 parametrization 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 program at FAIR with respect to open charm degrees of freedom.

Figure 1

The Euclidean space, where numerical solutions of the tBS and tDS equations are sought. The quark inverse propagator part, $\Pi \left(p\right)=p^2{A}^2\left(p\right)+B^2\left(p\right)$, where $p^2=-M_{q\overline{q}}^2/4+k^2\pm i{M}_{q\overline{q}}kcos\chi$, entering the tBS equation is defined within the colored areas of the corresponding parabolas ($-1\le cos\chi \le 1$). Left (right) panel: the integration domain for $M_{q\overline{q}}=140$ MeV ($M_{q\overline{q}}=2$ GeV). The integration domain is restricted to $k_{\max }\le 3$ GeV/$c$; for larger values of $k$ all the partial Bethe-Salpeter amplitudes are already negligible small (see, e.g., Ref. [25]). For light mesons (left panel, $m_q=0.005$ GeV) there are no singularities in the colored tBS integration domain. In the right panel we display the first six self-conjugated poles of propagators of light, $u$, $d$ quarks (asterisks) and the first four poles for the $s$ quarks (open stars) which enter the solution of the tBS equation. The pole positions are quoted in Table 3. Axes are in units of ${(\mathrm{GeV}/\text{c})}^2$.

Figure 2

The solution of the tDS equations (8) and (9) with the IR part of the interaction kernel (4) only as a function of the spacelike ($p^2>0$) real Euclidean momentum $p^2$. Solid curves correspond to $u$ and $d$ quarks with bare mass $m_q=5$ MeV, dashed curves depict results for $s$ quarks with bare mass $m_q=115$ MeV, and dash-dotted curves are for $c$ quarks with $m_q=1$ GeV.

Figure 3

The solution of the tDS equations (8) and (9) in the complex Euclidean space Re $p^2>0$ along rays ${\varphi }_p=$ const for two values of ${\varphi }_p$ (black solid curves: ${\varphi }_p={32}^o$; red dashed curves: ${\varphi }_p={34}^o$) for $u$, $d$ quarks with $m_q=5$ MeV.

Figure 4

The functios ${\sigma }_s$ and ${\sigma }_v$ for real values of $p^2$. The solid curves are for the numerically determined solutions, while dashed curves depict the approximation (11) with $i=1,2$. For $m_q=1$ GeV.

Figure 5

The force lines of the inverse propagator part $\Pi \left(p^2\right)$ for $u$, $d$ quarks. The self-conjugated nature of the singularities located within the “turbulent” regions is evident.

Figure 6

Positions of few first poles in the upper hemisphere of the complex $p^2$ plane, labeled in correspondence to Table 2, for $u$, $d$ (asterisks), and $s$ (open stars) quarks. The relevant sections of the parabola (5) corresponding to the meson bound-state mass $M_{q\overline{q}}$ are presented for $M_{q\overline{q}}=1,1.5,2,2.5$, and 3 GeV, from right to left. To emphasize the dependence of the pole positions on the bare quark mass $m_q$, the “lowest” poles for the fiducial values $m_q=0.5$ GeV (open circle), $m_q=0.9$ GeV (triangle), and $m_q=1$ GeV (open square) are displayed as well. As an illustration of the behavior on the real axis, the fourth pole for $m_q=0.5$ GeV is presented (circle). The tendency is that, with increasing bare quark mass, the corresponding pole is shifted towards larger values of $\text{Im}{p}^2$ and $|\text{Re}{p}^2|$. The area to the left after the axis break is the “unrevealed terrain,” here further singularities could be located.