Spectroscopic factors: constraints from elastic scattering observables and ground-state data*

The independent-particle shell model (IPSM) provides a simple description of the nucleus in which particles fully occupy orbits associated with corresponding single-particle energies. However, knockout or transfer reactions demonstrate that these orbits exhibit features that include substantial fragmentation and suggest partial occupation. For valence hole orbits the largest contribution comes from the transition to the hole-like state near or at the corresponding Fermi energy. This contribution is often referred to as the spectroscopic factor for this transition. The reaction that appears most suitable to establish the normalization of these transition amplitudes, or overlap functions, is the $(e,e'p)$ reaction. 

A Green's function description of the nucleon in the nucleus also identifies this spectroscopic factor as the energy derivative of the nucleon self-energy used to solve the Dyson equation at the corresponding energy. We explore this link between the self-energy and spectroscopic factor through the calculation of the $^{40}$Ca$(e,e'p)$$^{39}$K cross sections employing the nonlocal implementation of the Dispersive Optical Model (DOM). The DOM self-energy is constrained by all available elastic nucleon scattering observables and ground-state information. Once determined, the DOM provides both the bound and scattering states needed for a distorted wave impulse approximation (DWIA) calculation of the $(e,e'p)$ cross section as well as the corresponding spectroscopic factor. Spectroscopic factors of 0.7 and 0.6, calculated from the DOM, for the 0d${}_{3/2}$ and 1s${}_{1/2}$ orbitals, respectively, reproduce the measured cross sections at several outgoing proton energies in a completely satisfactory manner, lending conclusive support for the DWIA interpretation of this reaction.
Other implications of the DOM will also be presented related to results for $^{48}$Ca and $^{208}$Pb.