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 $$ \begin{equation} \langle pq|\tilde{v}|rs\rangle=\langle pq|\hat{v}|rs\rangle+\sum_{i\le F} \langle pqi|\hat{w}|rsi\rangle. \tag{19} \end{equation} $$ 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 \( \hat{w} \).