Introduction to low-momentum effective interactions with Brown-Rho scaling and three-nucleon forces

Model-space effective interactions $V_{eff}$ derived from free-space nucleon-nucleon interactions $V_{NN}$ are reviewed. We employ a double decimation approach: first we extract a low-momentum interaction $V_{low-k}$ from $V_{NN}$ using a $T$-matrix equivalence decimation method. Then $V_{eff}$ is obtained from $V_{low-k}$ by way of a folded-diagram effective interaction method. For decimation momentum $\Lambda \simeq 2 fm^{-1}$, the $V_{low-k}$ interactions derived from different realistic $V_{NN}$ models are nearly model independent, and so are the resulting shell-model effective interactions. For nucleons in a low-density nuclear medium like valence nucleons near the nuclear surface, such effective interactions derived from free-space $V_{NN}$ are satisfactory in reproducing experimental nuclear properties. But it is not so for nucleons in a nuclear medium with density near or beyond nuclear matter saturation density. In this case it may be necessary to include the effects from Brown-Rho (BR) scaling of hadrons and/or three-nucleon forces $V_{3N}$, effectively changing the free-space $V_{NN}$ into a density-dependent one. The density-dependent effects from BR scaling and $V_{3N}$ are compared with those from empirical Skyrme effective interactions.