Model calculation of effective three-body forces

We propose a scheme for extracting an effective three-body interaction originating from a two-nucleon interaction. This is based on the $\stackrel{\wedge }{Q}$-box method of Kuo and collaborators, where folded diagrams are obtained by differentiating a sum of nonfolded diagrams with respect to the starting energy. To gain insight we have studied several examples using the Lipkin model where the perturbative approach can be compared with exact results. Numerically the three-body interactions can be significant and in a matrix example good accuracy was not obtained simultaneously for both eigenvalues with two-body interactions alone.

Figure 1

Two-body diagrams discussed in the text.

Figure 2

Three-body diagrams discussed in the text.

Figure 3

Third-order (a) and fourth-order [(b) and (c)] three-body diagrams discussed in the text. Diagram (b) is a folded diagram drawn in unfolded form for clarity.

Figure 4

Fourth-order three-body diagrams discussed in the text. Diagram (c) is a folded diagram drawn in unfolded form for clarity, and the dot represents a one-body insertion.

Figure 5

Comparison of the eigenvalues (in units of ξ) of the effective interaction with n folds to the exact result. For the solid lines two- and three-body interactions are included, whereas for the dashed lines only two-body interactions are considered.

Figure 6

A three-body diagram discussed in the text.

Figure 7

Eigenvalues (in units of ξ) obtained with the two-body effective interaction (dashed curves) compared with the exact values (solid curves), as a function of $w/\xi$.