Uj being a single-particle potential acting on a nucleon j. The definition of Uj is important for the convergence of the BBG expansion series, where Hi is treated as a perturbation. As the system is spatially uniform, Uj is constant in space, and unperturbed nucleon states are plane waves \p). For simplicity, we omit spin indices and express nucleon momenta p in units of h. In momentum representation, the unperturbed single-particle energy is h2p2
19The linked cluster theorem, formulated in terms of diagrams, which were latter named "Goldstone diagrams", and the equation governing the wave function of a nucleon pair in nuclear matter (Bethe-Goldstone equation) were all derived during the graduate studies of Goldstone at Trinity College of the Cambridge University (Cambridge, England).
where N = n or p. Because of the isotropy of the nuclear matter, which is assumed to be spin-unpolarized, single-particle energies are independent of momentum direction and nucleon spin. However, owing to a neutron excess, the single-particle potentials for neutrons and protons are different and the G-matrix is no longer charge symmetric (Gnn = Gpp). As we will see below, Un can be expressed in terms of the G-matrix, the central quantity of the BBG theory. Its calculation is equivalent to the summation of the "ladder diagrams" of the BBG expansion series. It is performed by solving the integral equation, which can be written in the operator form:
Here, QNN' is the two-particle exclusion-principle operator, which projects particle states outside the Fermi surface, and hNN' is the Hamiltonian operator acting on uncorrelated two-particle states,
QNN'\PlP2) = ©(Pi — PFN)©(p2 — Pfn') \P1P2) , (5.30)
while 2 is the starting energy parameter. In the low-density limit we get hNN'\P1P2) —► 2m(pI + p2)\PIP2) , Qnn' —► 1 . (5.32)
In this case the G-matrix equation transforms into the well known equation for the scattering T-matrix, which describes the NN scattering in vacuum (see, e.g., Messiah 1961, vol. II, Chapter XIX, §14). Passing to the momentum representation, we get
(P1P2IGW (z)|PlP2) = (pIpAVnn'\PlP2) dk\ dk2 (2n) Qnn' (ki, k2)
In G-matrix elements relevant for calculating the ground-state energy, the starting energy is the sum of single-particle energies of the initial \p1p2) state:
with single-particle potentials given in terms of the G-matrix via a Hartree-Fock expression
+2 / dp3(pip3\G nn(en (pi) + en(p3))\pip3)a , (5.35a)
+2 / dp3(pip3\Gpp(ep(pi) +ep(p3))\pip3)a • (5.35b) Jp
Here, for the sake of compactness, we use the notations
\pip2)a = \pip2) - \p2pi) , J^ dp = J (2^3 @(kFN - p) • (5.36)
As we are dealing with the spin-unpolarized system, we can use the spin-averaged G-matrix. The spin degeneracy gives a factor of two in front of the integrals.
The auxiliary single-particle potential term U(p), Eq. (5.27), crucial for the convergence of the linked-cluster expansion, deserves an additional comment. The choice of U (p) for states above the Fermi surface (p > pF) has been a subject of a long debate since the formulation of the BBG theory. Eventually, the so called continuous prescription for U has been regarded as the most advantageous. According to this prescription, no energy gap is introduced between the energies of occupied (p < pF) and empty (p > pF) momentum states.20 Such a choice turns out to be particularly suitable in view of the rapid convergence of the BBG expansion (see, e.g., Baldo et al. 2000,2001). The lowest-order BBG approximation for the energy density E (without nucleon rest energy contribution) is given by the Hartree-Fock expression, where the G-matrix acts as an effective interaction. This justifies the name "Brueckner-Hartree-Fock" (BHF) approximation, used by many authors. The BHF expression for E reads
E = EFFG(nn,np) + 2 J dpi j dp2(piP2\Gpp(ep(pi) + ep(p2))|pip2)a dpi / dp2(piP2|G nn(en (Pi) + en(P2))\pip2)a n
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