How large is the contribution of excited mesons in coupled-channel effects?

We study the excited $B$ mesons’ contributions to the coupled-channel effects under the framework of the ${{}^{3}P}_0$ model for the bottomonium. Contrary to what has been widely accepted, the contributions of $P$ wave $B$ mesons are generally the largest, and to some extent, this result is independent of the potential parameters. We also push the calculation beyond $B(1P)$ and carefully analyze the contributions of $B(2S)$. A form factor is a key ingredient to suppress the contributions of $B(2S)$ for low lying bottomonia. However, this suppression mechanism is not efficient for highly excited bottomonia, such as $\mathrm{\Upsilon }(5S)$ and $\mathrm{\Upsilon }(6S)$. We give explanations why this difficulty happens to the ${{}^{3}P}_0$ model and suggest analyzing the flux-tube breaking model for the full calculation of coupled-channel effects.

Figure 1

Sketch of coupled-channel effects in the ${}^3{P}_0$ model for the bottomonium. $i$ and $f$, respectively, denote the initial and final states with the same $J^{PC}$ and $B\overline{B}$ stands for all possible $B$ meson pairs, including the excited $B$ mesons.

Figure 2

$-\mathrm{\Delta }M$ of different coupled-channels for $\mathrm{\Upsilon }(nS)$. The results with parameters in Table 1 and the results recalculated with the parameters of Ref. [40] are, respectively, represented by thick and thin curves. $\mathrm{\Delta }M(1S,1S)$, $\mathrm{\Delta }M(1S,1P)$, and $\mathrm{\Delta }M(1P,1P)$ are represented by black dot-dashed, red dashed, and blue solid curves, respectively.

Figure 3

$-\mathrm{\Delta }M$ of different coupled-channels for some selected representatives of the ${{}^{1}S}_0$, ${{}^{1}P}_1$, ${{}^{3}P}_J$, ${{}^{3}D}_1$ families. The results with parameters in Table 1, and the results recalculated with the parameters of Ref. [40] are, respectively, represented by thick and thin curves. $\mathrm{\Delta }M(1S,1S)$, $\mathrm{\Delta }M(1S,1P)$, and $\mathrm{\Delta }M(1P,1P)$ are represented by black dot-dashed, red dashed, and blue solid curves, respectively. Here, we omit the results of ${\eta }_b(nS)$, $h_b(nP)$, and ${\chi }_{bj}(nP)$ when $n\le 2$, because these results are quite similar to those of $\mathrm{\Upsilon }(1S)$ or $\mathrm{\Upsilon }(2S)$.

Figure 4

$R$ [defined in Eq. (13)] of the $\mathrm{\Upsilon }$ family. Results corresponding to $r=0$ (no form factors), $r=0.335\mathrm{fm}$ (taken from [49]), and $r=0.408\mathrm{fm}$ (our best fit) are denoted by black dot-dashed, blue dashed, and red solid curves, respectively.

Figure 5

$-\mathrm{\Delta }M$ of $\mathrm{\Upsilon }(5S)$ and $\mathrm{\Upsilon }(6S)$ with different form factors. Solid curves represent the denominator of Eq. (13), i.e., $\mathrm{\Delta }M(1S,1S)+M(1S,1P)+M(1P,1P)$, and dashed curves represent the numerator, i.e., $\mathrm{\Delta }M(2S,2S)+\mathrm{\Delta }M(2S,1P)+\mathrm{\Delta }M(2S,1S)$. $\mathrm{\Delta }M$ corresponding to $r=0$, $r=0.335\mathrm{fm}$, and $r=0.408\mathrm{fm}$ are denoted by black, blue, and red curves, respectively. The value of the $-\mathrm{\Delta }M$ cannot compare within different form factors because the ${}^3{P}_0$ coupling constants are not fit to reproduce experimental data.

Figure 6

Sketch of coupled-channel effects in the flux-tube breaking model. Red lines stand for the flux-tube. The regions outside the flux-tube are shaved off, i.e., the excited states’ contributions in the loops are suppressed.