Υ and η${}_b$ mass shifts in nuclear matter

By extending the previous works that two of the present authors were involved, we estimate for the first time the Υ and η${}_b$ as well as B* meson mass shifts (scalar potentials) in symmetric nuclear matter. The main interest is, whether 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 (J/Ψ, Υ) and (η${}_c$, η${}_b$) meson group is usually assumed to have very similar properties based on the heavy charm and bottom quark masses. The estimate for the Υ is made using an SU(5) effective Lagrangian density and the anomalous coupling one, 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, we include only the BB meson loop contribution as our prediction. As for the η${}_b$, to be complete, we include the BB* and B*B* meson loop contributions in the self-energy for the analysis. 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. A detailed analysis on the BB, BB*, and B*B* meson loop contributions for the Υ mass shift is made by comparing with the respectively corresponding DD, DD*, and D*D* meson loop contributions for the J/Ψ mass shift. Based on the analysis for the Υ, our prediction for the η${}_b$ mass shift is made on the same footing as that for the Υ, namely including only the lowest order BB* meson loop. The Υ mass shift is predicted to be -16 to -22 MeV at the nuclear matter saturation density with the cutoff mass values in the range of 2000-6000 MeV using the Υ BB coupling constant determined by the vector meson dominance model with the experimental data, while the η${}_b$ mass shift is predicted to be -75 to -82 MeV with the SU(5) universal coupling constant determined by the /varUpsilon BB coupling constant for the same range of the cutoff mass values. Our results show an appreciable difference between the bottomonium-(nuclear matter) and charmonium-(nuclear matter) interaction strengths. We also study the Υ and η${}_b$ mass shifts in a heavy quark (heavy meson) symmetry limit, namely, by calculating their mass shifts using the same coupling constant value as that was used to estimate the J/Ψ and η${}_c$ mass shifts. For the η${}_b$ mass shift an SU(5) symmetry breaking case is also studied in this limit. Our predictions for these cases at nuclear matter saturation density are, -6 to -9 MeV for Υ, -31 to -38 MeV for η${}_b$, and -8 to -11 MeV for η${}_b$ with a broken SU(5) symmetry, where the corresponding charm sector ones are, -5 to -21 for J/Ψ, -49 to -87 for η${}_c$, and -17 to -51 for η${}_c$ with a broken SU(4) symmetry.