$\Upsilon$ and $\eta_{b}$ mass shifts in nuclear matter and the nucleus bound states

Abstract

The $\Upsilon$ and $\eta_b$ as well as $B^*$ meson mass shifts (scalar potentials) are estimated for the first time in symmetric nuclear matter. The main interest is, whether or not the strengths of the bottomonium-nuclear matter and charmonium-nuclear matter interactions are similar or very different, in the range of a few tens of MeV at the nuclear matter saturation density. This is because, each ($\Upsilon,J/\Psi$) and ($\eta_c,\eta_b$) meson group is usually assumed to have very similar properties based on the heavy charm and bottom quark masses. The estimate for the $\Upsilon$ is made using an SU(5) effective Lagrangian density, by studying the $BB$, $BB^*$, and $B^*B^*$ meson loop contributions for the self-energy in free space and in nuclear medium. As a result, only the $BB$ meson loop contribution is included as our minimal prediction. As for the $\eta_b$, is included only the $BB^*$ meson loop contribution in the self-energy, to be consistent with the minimal prediction for the $\Upsilon$. The in-medium masses of the $B$ and $B^{*}$ mesons appearing in the self-energy loops are calculated by the quark-meson coupling model. Form factors are used to regularize the loop integrals with a wide range of the cutoff mass values. The results suggest that both $\Upsilon$ and $\eta_b$ should form bound states with a variety of nuclei considered in this study, for which the $\Upsilon$-nucleus and $\eta_b$-nucleus bound state energies are calculated. The results also show an appreciable difference between the bottomonium-nuclear matter and charmonium-nuclear matter interaction strengths. Are also studied the $\Upsilon$ and $\eta_b$ mass shifts in a heavy quark (heavy meson) symmetry limit. In addition, an initial study was done to investigate the influence of the choice of the form factor on our predictions.