Two-Pion-Exchange Three-Body Force in Nuclear Matter

The contribution of the two-pion-exchange three-nucleon force to the binding energy of nuclear matter is evaluated using the modified pion-mass method of Brown and Green and by the conventional potential approach. Using nonrelativistic formulas, the relation between the two methods is clarified. It is shown that the Brown-Green prescription, as given in their paper, is incorrect when a cutoff in the $\mathrm{NN}$ distance is applied and the modified pion mass is momentum dependent. When the correct prescription is used, the first-order contribution becomes sensitive to the pionic form factor chosen. Using the same form factor as used by Brown and Green, the contribution with an $\mathrm{NN}$ cutoff of 1 F is only about 0.1 MeV instead of 1.25-MeV repulsion per particle as quoted by them. Another form factor leads to about 0.5-MeV repulsion. The second-order contribution, including the cross terms involving the two-body one-pion-exchange potential and three-body forces, has been recalculated in the Brown-Green spirit after correcting for their earlier error. It is attractive, much larger than the first-order term, and is about the same for the two form factors used. The net effect of the two-pion-exchange three-body force is rather sensitive to the $\mathrm{NN}$ cutoff and is about a 4- to 8-MeV per particle attraction in nuclear matter, depending on the cutoff.