Effects of a dressed quark-gluon vertex in vector heavy-light mesons and theory average of the $B_c^{*}$ meson mass

We extend earlier investigations of heavy-light pseudoscalar mesons to the vector case, using a simple model in the context of the Dyson-Schwinger-Bethe-Salpeter approach. We investigate the effects of a dressed quark-gluon vertex in a systematic fashion and illustrate and attempt to quantify corrections beyond the phenomenologically very useful and successful rainbow-ladder truncation. In particular we investigate the dressed quark-photon vertex in such a setup and make a prediction for the experimentally as yet unknown mass of the $B_c^{*}$, which we obtain at 6.334 GeV well in line with predictions from other approaches. Furthermore, we combine a comprehensive set of results from the theoretical literature. The theoretical average for the mass of the $B_c^{*}$ meson is $6.336\pm 0.002\mathrm{GeV}$.

Figure 1

Bound-state masses for $\rho$ meson, $\phi$, $J/\mathrm{\Psi }$, $\mathrm{\Upsilon }$, and all different flavored vector 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 [112].

Figure 2

Relative mass differences $\mathrm{\Delta }{m}_H^{\mathrm{rel}n}$ to the fully dressed result for $\rho$ meson, $\phi$, $J/\mathrm{\Psi }$, $\mathrm{\Upsilon }$, and all different flavored vector ground states as a function of $n$ analogous to Fig. 1. Note that by the definition (14) one obtains $\mathrm{\Delta }{m}_H^{\mathrm{rel}\infty }=0$ in each case.

Figure 3

Left panel: $B_c^{*}-B_c$ mass splitting as a function of $n$ and $\eta$ in our scheme. Right panel: $B_c^{*}-B_c$ mass splitting as a function of $n$ in our scheme for fixed $\eta =0.75$ (blue circles); error bars indicate variations with respect to $\eta$.

Figure 4

$B_c^{*}$ mass values including error bars (blue data points) taken from the references as listed on the lower axis in the figure, corresponding to Table 2. Years of appearance are given on the upper axis for each data point. The horizontal red line is the average; its error is about the size of the thickness of the line. The value and error of the average are provided in the insert.

Figure 5

Inverse of the transversal component $1/f_1$ of the quark-photon vertex for different $n$ in the MN model studied herein together with a sophisticated (Maris-Tandy, MT) model [11] result in RL truncation for comparison (see text). Left panel: Detailed view in the region around $Q^2=0$. Right panel: Large-scale view without error bars on the MN curves to emphasize the asymptotic behavior.

Figure 6

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: $\rho$. Right upper panel: $\phi$. Left lower panel: $J/\mathrm{\Psi }$. Right lower panel: $\mathrm{\Upsilon }$.

Figure 7

Same as Fig. 6. Left upper panel: $K^{*}$. Right upper panel: $D^{*}$. Left center panel: $B^{*}$. Right center panel: $D_s^{*}$. Left lower panel: $B_s^{*}$. Right lower panel: $B_c^{*}$.