Renormalization Group Methods for the Nuclear Many-Body Problem

The application of renormalization group (RG) methods to microscopic nuclear many-body calculations is discussed. We present the solution of the RG equations in the particle-hole channels for neutron matter and the application to S-wave pairing. Furthermore, we point out that the inclusion of tensor and spin-orbit forces leads to spin non-conserving effective interactions in nuclear matter. The solution to the many-body problem for systems of strongly interacting particles, such as finite nuclei and nuclear matter, can often be facilitated, by making a judicious use of the separation of lowand high-energy scales. One introduces a truncated Hilbert space (model space), where the particles are restricted to low energies. These are the so-called “slow” modes. The operators and the degrees of freedom in the truncated space must be renormalized to account for intermediate excitations to states outside the model space, the “fast” modes. The operators of interest include the effective interaction and various transition operators, e.g., the axial current. This procedure defines an effective theory, which is equivalent to the full theory at low energies. The RG method provides a systematic way to compute the effective operators of particles in the truncated space. There are several advantages of working in a truncated space. By integrating out the high-energy modes, the strong short-range repulsion of realistic nucleon-nucleon forces is tamed, and the resulting effective interaction is model independent, if it acts only at energies that are constrained by the scattering data [1, 2, 3, 4]. Due to the separation of scales, it can usually be achieved that the effective operators in the model space are energy-independent. For finite nuclei, the model space concept has been used for many years in the derivation of effective shell model interactions, where the truncated space contains the low-lying shells near the Fermi energy. 0 0.5 1 1.5 2 Diagonal Vlow k(k,k) [fm] vs. relative k [fm -1 ]