One of the main goals of the low energy nuclear theory is to build a comprehensive microscopic framework, in which nuclear bulk properties, excitations, and low energy reactions can be described. For medium-mass and heavy nuclei, self-consistent methods based on the Density Functional Theory (DFT) [1,2] have already achieved a level of precision that allows for analysis of experimental data for a wide range of properties of nuclei throughout the chart of the nuclides. For example, the self-consistent Hartree-Fock-Bogoliubov (HFB) models based on the Skyrme energy functionals [3,4,5] are able nowadays to reproduce nuclear masses with an rms error of about 700keV [6,7]. The development of a universal nuclear density functional, however, still requires a better understanding and improved description of the density dependence, isospin effects, pairing force, many-body correlations, and symmetry restoration.
Nuclear pairing is an important ingredient of the nuclear density functional, and it becomes crucial for open shell nuclei, in particular weakly bound systems, where the effects of coupling to continuum become significant [8,9]. In this case, the BCS model is not adequate [8] and the fully self-consistent HFB approach must be used.
In most HFB applications, pairing interaction is assumed to be either in the form of the finite range Gogny force [10] or the zero-range, possibly density-dependent, delta force [11,8,9]. Gogny interaction in the pairing channel can be viewed as a regularized contact interaction, with regularization fixed through the finite range. The resulting pairing field is, however, nonlocal.
Calculations using the contact interaction are numerically simpler, but one has to apply a cutoff procedure within a given space of single-particle (s.p.) states [12,8]. When the dimension of this space increases, the pairing gap diverges for any given strength of the interaction. Therefore, the pairing strength has to be readjusted for each s.p. space. Thus the energy cutoff and the pairing strength together define the pairing interaction, and this definition can be understood as a phenomenological introduction of finite range [8,13]. Such a procedure is usually referred to as the renormalization of the contact pairing force. It 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 cutoff to take into account neglected high energy effects.
The renormalization procedure for the zero-range pairing interactions has been explored in Ref. [8] using the numerical solutions of the HFB equations. It has been shown that by renormalizing the pairing strength for each value of the cutoff energy one practically eliminates the dependence of the HFB energy on the cutoff parameter.
Recently, the issue of contact pairing force has been addressed in Refs. [14,15,16,17,18,19,20,21,22], suggesting that the renormalization procedure can be replaced by a regularization scheme which removes the cutoff energy dependence of the pairing strength. In subsequent papers, this regularization scheme has been applied to properties of the infinite nuclear matter [19], spherical nuclei [21,23], and trapped fermionic atoms [16,24].
In this study, we investigate the stability of the regularization scheme with respect to the cutoff energy for both spherical and deformed nuclei. Differences between the HFB results emerging from the pairing renormalization and pairing regularization procedures are analyzed.
The HFB and Skyrme HFB formalisms have been explained in great detail in many papers (see, e.g., Refs. [25,12]). The notation used in the present paper is consistent with that of Refs. [12,8,26]. This work is organized as follows. Sec. 2 gives a brief introduction to the pairing renormalization and regularization schemes. In Sect. 3 we explain the numerical framework used. The comparison between pairing regularization and renormalization techniques, studied for a large set of spherical and deformed nuclei, is discussed in Sec. 4. Finally, the summary and conclusions are given in Sec. 5.