## Modification due to a three-body force

The two-body matrix elements are now modified in order to account for medium-modified contributions from the three-body interaction, resulting in $$$$\langle pq|\tilde{v}|rs\rangle=\langle pq|\hat{v}|rs\rangle+\sum_{i\le F} \langle pqi|\hat{w}|rsi\rangle. \tag{19}$$$$ In Eq. (19), the effective two-body interaction $$\tilde{v}$$ can contain both a standard two-nucleon interaction and a density dependent contribution stemming from a three-body interaction $$\hat{w}$$.