$$\varUpsilon $$ and $$\eta _b$$ mass shifts in nuclear matter

Abstract

By extending the previous works that two of the present authors were involved, we estimate for the first time the \(\varUpsilon \) and \(\eta _b\) 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 (\(J/\varPsi ,\varUpsilon \)) 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 \(\varUpsilon \) 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 \(\eta _b\), 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 \(\varUpsilon \) mass shift is made by comparing with the respectively corresponding \(DD, DD^*\), and \(D^*D^*\) meson loop contributions for the \(J/\varPsi \) mass shift. Based on the analysis for the \(\varUpsilon \), our prediction for the \(\eta _b\) mass shift is made on the same footing as that for the \(\varUpsilon \), namely including only the lowest order \(BB^*\) meson loop. The \(\varUpsilon \) 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 \(\varUpsilon BB\) coupling constant determined by the vector meson dominance model with the experimental data, while the \(\eta _b\) 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 \(\varUpsilon \) and \(\eta _b\) 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 \(J/\varPsi \) and \(\eta _c\) mass shifts. For the \(\eta _b\) 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 \(\varUpsilon \), \(-31\) to \(-38\) MeV for \(\eta _b\), and \(-8\) to \(-11\) MeV for \(\eta _b\) with a broken SU(5) symmetry, where the corresponding charm sector ones are, \(-5\) to \(-21\) for \(J/\varPsi \), \(-49\) to \(-87\) for \(\eta _c\), and \(-17\) to \(-51\) for \(\eta _c\) with a broken SU(4) symmetry.