When employing contact pairing interactions such as the density-dependent delta force resulting in the pairing functional (5), one has to apply a cut-off procedure and use a finite space of single-particle states [16]. When this space increases, the pairing energy diverges for any strength of the interaction; therefore, one has to readjust the pairing strength for each size of the single-particle space [17]. Such renormalization procedure is performed in the spirit of the effective field theory, whereupon contact interactions are used to describe low-energy phenomena while the coupling constants are readjusted for any given energy cut-off to take into account neglected high-energy effects. It has been shown that by carrying out renormalization for each value of the cut-off energy, one practically eliminates the dependence of the HFB results on the size of the single-particle space.

Recently, the subject of the contact pairing force has been addressed in Refs. [33,34,35,36,37] suggesting the renormalization procedure can be replaced by a regularization scheme which removes the cut-off energy dependence of the pairing strength. Differences between the HFB results emerging from the pairing renormalization and pairing regularization procedures have been analyzed in Ref. [38] for both spherical and deformed nuclei. Figure. 4 shows differences between the HFB-SkP results for the deformed Er nuclei obtained using pairing renormalization and regularization. While the regularization method is better theoretically motivated, it is seen that both methods give indeed very similar results.