Effects of a dressed quark-gluon vertex in pseudoscalar heavy-light mesons

Using a simple model in the context of the Dyson-Schwinger-Bethe-Salpeter approach, we investigate the effects of a dressed-quark-gluon vertex on pseudoscalar meson masses. In particular, we focus on the unequal-mass case and investigate heavy-light meson masses; in addition, we study the premise of the effective treatment of heavy quarks in our approach.

Figure 1

Dressing functions $A(s)$ and $M(s)$ for different values of $n$, corresponding to a quark mass $m_u=0.01\mathrm{GeV}$.

Figure 2

$n$-dependence of various quantities: dressing functions $M(s)$ and meson masses, all in GeV. Left panel: $M(s=0)$ for all four quark masses; center panel: Calculated mass function $M(s)$ at $s=-m_H^2/4$. For the unequal-mass case the mass function of the heavier quark is presented. The experimental values are used for the meson masses $m_H$ (see text). The labels $H$ of the subplots denote $\pi$, $K$, $D$, $B$; right panel: Pseudoscalar bound-state masses for the equal-mass case calculated from the BSE for different quark flavors.

Figure 3

Relative differences $R_n$ of dressing functions $M(0)$ for $n$ loops compared to the case $n\to \infty$, for all four quark masses.

Figure 4

Bound-state mass for the kaon as a function of $n$ and $\eta$, given in GeV. Even $n$ are depicted by dashed lines, odd ones by dotted lines, and the fully summed result by a solid line.

Figure 5

Bound-state masses for $\pi$ meson, ${\eta }_c$, and flavored pseudoscalar ground states as a function of $n$, given in GeV. The dependence on $\eta$ is illustrated via the error bars. Calculated results are given by blue dots; experimental data are represented by horizontal lines.

Figure 6

Quark propagator dressing functions $A(s)$ and $M(s)$ as functions of $n$, where $n=0$ and $n=\infty$ are represented by the dashed and thick solid lines, respectively. Upper left and right panels: $A(s)$ and $M(s)$ for $m_s=0.166\mathrm{GeV}$; middle left and right panels: $A(s)$ and $M(s)$ for $m_c=1.33\mathrm{GeV}$; lower left and right panels: $A(s)$ and $M(s)$ for $m_b=4.62\mathrm{GeV}$.

Figure 7

Comparison of $A(s)$ for the four quark masses, plotted on the same domain for each $n$. Left upper panel: $n=0$; right upper panel: $n=1$; Left middle panel: $n=2$; right middle panel: $n=3$; Left lower panel: $n=4$; right lower panel: $n=\infty$.

Figure 8

Comparison of the relative difference of our calculated $A(s)$ to an Ansatz $A(s)=1$ for the four quark masses, plotted on the same domain for each $n$. In particular, we plot $A(s)-1$. Left upper panel: $n=0$; right upper panel: $n=1$; Left middle panel: $n=2$; right middle panel: $n=3$; Left lower panel: $n=4$; right lower panel: $n=\infty$.

Figure 9

Comparison of the relative difference of our calculated $M(s)$ to an Ansatz $M(s)=\text{const}=M(s=0)$ for the four quark masses, plotted on the same domain for each $n$. In particular, we plot $[M(s)-M(0)]/M(0)$. Left upper panel: $n=0$; right upper panel: $n=1$; Left middle panel: $n=2$; right middle panel: $n=3$; Left lower panel: $n=4$; right lower panel: $n=\infty$.

Figure 10

Meson bound-state masses as functions of $n$ and $\eta$, given in GeV. Even $n$ are depicted by dashed lines, odd ones by dotted lines, and the fully summed result by a solid line. If no solution is found, no surface is plotted at the corresponding $n$. Left upper panel: pion; right upper panel: $D$; Left upper center panel: $B$; right upper center panel: $D_s$; Left lower center panel: $B_s$; right lower center panel: $B_c$ Left lower panel: ${\eta }_c$; right lower panel: ${\eta }_b$.