Regularization of zero-range effective interactions in finite nuclei

The problem of the divergences which arise in beyond mean-field calculations, when a zero-range effective interaction is employed, has not been much considered so far. Some of us have proposed, quite recently, a scheme to regularize a zero-range Skyrme-type force when it is employed to calculate the total energy, at second-order perturbation theory level, in uniform matter. Although this scheme looked promising, the extension for finite nuclei is not straightforward. We introduce such a procedure in the current paper, by proposing a regularization procedure that is similar, in spirit, to the one employed to extract the so-called $V_{\text{low-}k}$ from the bare force. Although this has been suggested already by B. G. Carlsson and collaborators, the novelty of our work consists of setting on equal footing uniform matter and finite nuclei; in particular, we show how the interactions that have been regularized in uniform matter behave when they are used in a finite nucleus with the corresponding cutoff. We also address the problem of the validity of the perturbative approach in finite nuclei for the total energy.

Figure 1

Diagrammatic representations of the first-order (Hartree-Fock) and second-order total energy, respectively, in the upper and lower parts of the figure. The labels outside (inside) parentheses are those used in the text for the case of uniform matter (finite nuclei).

Figure 2

Representation of matrix elements in the case of uniform matter. We compare the notation used in the present paper (left) with the one in which the transferred momentum $q$ appears (right).

Figure 3

Energy per particle at the HF level [cf. Eqs. (10) and (12)] for different values of the cutoff $\lambda$ on the relative momentum $\mathit{k}$.

Figure 4

Total HF energy as a function of a cutoff $\lambda$. The dashed (dot-dashed) curve corresponds to the result obtained with SkP with only $t_0$, $t_3$ terms (${\mathrm{SkP}}_{\Lambda }$). The thin dotted line corresponds to the result without any cutoff, obtained with SkP with only $t_0$, $t_3$ term, and is meant to guide the eye for the convergence of the dashed line. See the text for a more detailed discussion.

Figure 5

Total density profiles obtained with the renormalized ${\mathrm{SkP}}_{\Lambda }$ interactions. The thick black line refers to the bare interaction. The inset shows some detail of the region in which the density attains its largest values.

Figure 6

Total energy at second order as a function of the maximum particle energy (left panel) or as a function of the cutoff (right panel). All curves are obtained with renormalized interactions. See the text for a discussion.

Figure 7

The same as Fig. 6 in a three-dimensional representation, that is, total energy at second order as a function of both the maximum particle energy and the cutoff.