THE EUROPEAN PHYSICAL JOURNAL A Short Note Momentum distribution of relativistic nuclei with Hartree-Fock mesonic correlations

The impact of Hartree-Fock correlations on the nuclear momentum distribution is studied in a fully relativistic one-boson-exchange model. Hartree-Fock equations are exactly solved to first order in the coupling constants. The renormalization of the Dirac spinors in the medium is shown to affect the momentum distribution, as opposed to what happens in the non-relativistic case. The unitarity of the model is shown to be preserved by the present renormalization procedure. PACS. 21.60.Jz Hartree-Fock and random-phase approximations – 21.65.+f Nuclear matter – 24.10.Jv Relativistic models – 11.10.Gh Renormalization It is well known that in a non-relativistic framework the momentum distribution of nuclear matter is not affected by the Hartree-Fock (HF) field. This arises because, due to general invariance principles [1], the non-relativistic self-energy cannot depend on spin in an infinite system: hence the single-nucleon wave functions are not modified and only the energy-momentum relation is affected by the medium. Of course, correlations in the nuclear wave function beyond the mean-field approximation are very important already at the non-relativistic level [2–4]. Due to such correlations, the momentum distribution is reduced for momenta below kF and the states above kF acquire small but finite occupation probabilities. Although the momentum distribution is not an observable, it is also true that over the years electron scattering reactions have frequently been expressed in terms of momentum densities. In recent work [5,6] we have evaluated the impact of mesonic correlations and meson exchange currents (MEC) on the electroweak response functions within a fully relativistic, gauge-invariant model. We have shown that the consistency of the theory necessarily implies the inclusion in the calculation of Hartree-Fock self-energy insertions. In order to deal properly with the divergencies associated with these diagrams, not only the a e-mail: [email protected] energy but also the nucleon wave functions must be renormalized by the medium. As a consequence, in a relativistic HF framework, the momentum distribution is also modified for k < kF, since now the Dirac four-spinors describing the nucleons display new features and the self-energy becomes spin dependent. The aim of this paper is to quantify this genuine relativistic effect in a one-boson-exchange model for the NN interaction, while the corresponding observable consequences on the response functions were analyzed in depth in [5,6]. An unambiguous treatment of relativistic HartreeFock does not exist in the literature, since the presence of the Dirac sea requires (at least) the specification of a prescription to take it into account [7]. The approach we use is equivalent to that used in [7,8], where the nucleon proper self-energy is calculated in terms of positive-energy spinors only. This approximation is valid in the first iteration of a fully self-consistent calculation to which we confine ourselves in this work. This procedure was shown in [7] to reproduce the non-relativistic HF equations in the limit M → ∞, and it reduces the relativistic Hartree approximation to the mean-field theory. The proton momentum distribution of nuclear matter in the independent-particle approximation is