Four-Hole-Line Diagrams in Nuclear Matter

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Abstract
The Brueckner-Goldstone diagrams with four independent hole lines, which give the third term in the expansion for the ground-state energy of infinite nuclear matter, are enumerated. These diagrams are grouped in a natural way into 16 distinct classes. Only one of these classes (the four-body clusters) involves the solution of a four-body equation. Six classes require the solution of the three-body Bethe-Faddeev equations, and nine classes can be evaluated in terms of two-body matrix elements alone. Exact formal expresions are given for the contribution to the energy from each class of diagrams. In these expressions, all exchange diagrams are included, and all energy denominators are clearly defined. Numerical estimates are made for each class of diagrams, assuming the two-body interaction to be the Reid soft-core potential. The sum of all contributions is attractive and is about 0.6-1.6 MeV. Most of the uncertainty in this result is caused by omission of the tensor force in certain diagrams. The implications of these results for the convergence of the energy expansion are discussed.