Influence of virtualΔstates on the saturation properties of nuclear matter

The effect of virtual $\Delta \mathrm{}\left(1236\right)\mathrm{}$ states on the saturation properties of nuclear matter is studied within the framework of lowest-order Brueckner theory. The $\Delta$ is treated as a stable elementary particle. Transitions from the nucleon-nucleon ($\mathrm{NN}$) channel to the nucleon-$\Delta \left(N\Delta \right)$ channel are caused by a nonrelativistic potential obtained from the static limit of meson theory. The coupled-channel potentials are constrained to fit the $\mathrm{NN}$ phase shifts. Saturation curves are calculated for the couplings ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}\mathrm{}\left(\mathrm{NN}\right)\mathrm{}\leftrightarrows {}_{}{}^5{}_{}{}^{}D_0^{}{}_{}{}^{}\left(N\Delta \right)$ and ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}\mathrm{}\left(\mathrm{NN}\right)\mathrm{}\leftrightarrows {}_{}{}^5{}_{}{}^{}P_1^{}{}_{}{}^{}\left(N\Delta \right)$, and the effects of other $N\Delta$ couplings to nucleon-nucleon $P$ and $D$ waves are estimated. Calculations are done using both the Reid soft-core and Ueda-Green potentials for $\mathrm{NN}$ partial waves not coupled to the $N\Delta$ channel. The $N\Delta$ coupling does not change the usual tendency of the calculated saturation points to lie in a narrow band in the energy-density plane that does not contain the empirical saturation point. This result is illuminated by a rough approximation to the Pauli and dispersion effects. We have also used this approximation to estimate the loss of binding due to $N\Delta$ coupling in those channels not treated by detailed calculation. Combining all our results, we find that at the empirical density (1) the inclusion of $N\Delta$ coupling in nucleon-nucleon $S$, $P$, and $D$ waves reduced the binding energy by about 3.3, 3.2, and 0.8 MeV, respectively, and (2) each particle spends about 3.7% of its time as a $\Delta$. All these figures vary roughly quadratically with the $\pi N\Delta$ coupling constant and increase rapidly with density. The size of the shift in energy depends strongly on the suppression of the short-range part of the two-body wave function, but our approximate formulas indicate that the tendency of the calculated saturation points to remain in a narrow band is independent of the short-range behavior of the two-body interaction, i.e., it is model-independent.