Regularization of the local pairing interaction

As discussed in Sec. 5, in many HFB applications, pairing interaction is often assumed to be in the form of the zero-range, density-dependent force. Calculations using the contact interaction are numerically simpler, but the pairing gap diverges when the dimension of the pairing-active space increases for a fixed strength of the interaction. In roots of this problem is the ultraviolet divergence of abnormal density for zero-range pairing interaction [80,81,54]:

$$\left.\tilde\rho(\mathbf{r}-\mathbf{x}/2,\mathbf{r}+\mathbf{... ...^2\vert\mathbf{x}\vert}\right\vert _{\mathbf{x}\rightarrow 0}.$$ (17)

Consequently, in practical calculations, one has to apply a cutoff procedure to truncate the pairing-active space of s.p. states [19,21,37], and the pairing strength has to be readjusted accordingly. 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 [21,82,83]. Such a sharp cut-off regularization 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 account for high energy effects. It has been shown that by an appropriate renormalization the pairing strength for each value of the cutoff energy, one practically eliminates the dependence of various observables on the cutoff parameter[21,54]. Figure 4 illustrates the procedure for the total energy in the tin isotopes. While for a fixed pairing strength total energies depend significantly on the cut-off energy (top), for a fixed pairing gap the changes obtained with renormalized interactions (bottom) are very small indeed.

Figure 4: Total energies in the tin isotopes calculated within the HFB+SkP$^\delta$ model[21]. Top panel shows the results for the fixed interaction strength $V_0$ and for several cut-off energies $\Delta E_{\mbox{\rm\scriptsize{max}}}$ added to the usual ${\ell}j$-dependent cut-off energy $E_{\mbox{\rm\scriptsize{max}}}$[19]. Bottom panel shows similar results when the values of $V_0$ are renormalized to keep the average neutron pairing gap in $^{120}$Sn the same for each $\Delta E_{\mbox{\rm\scriptsize{max}}}$.

The cutoff energy dependence of the pairing strength can also be handled by means of a regularization scheme by defining the regularized local abnormal density [84,85,80,81,86,87,54]:

$$\tilde\rho_r(\mathbf{r})=\lim_{\mathbf{x}\rightarrow 0} \lef... ...x}/2,\mathbf{r}+\mathbf{x}/2)-f(\mathbf{r},\mathbf{x})\right],$$ (18)

where $f$ is a regularization counterterm, which removes the divergence (17) at $\mathbf{x}=0$. For cutoff energies high enough, one can express $f$ through the s.p. Green's function at the Fermi level, $G(\mathbf{r}+\mathbf{x}/2,\mathbf{r}-\mathbf{x}/2)$, which also exhibits a $1/x$ divergence. In practical calculations, one cane use the Thomas-Fermi (TF) approximation for the local s.p. Green’s function; this approach has been used with success for a description of spherical and deformed nuclei [86,87,54,37]. As demonstrated in Ref.[54] the differences between pairing renormalization and regularization procedures are rather small.

A combination of the renormalization and regularization methods described above is the hybrid technique[57] based on the TF approximation to the non-resonant HFB continuum [88,89]. This approach is of great practical interest as it makes it possible to carry out calculations in wide pairing windows and very large coordinate spaces. In the hybrid method, the high-energy continuum above the cutoff energy $E_c$ is divided into the non-resonant part and deep-hole states. While deep-hole states have to be treated separately, the non-resonant continuum contribution to HFB densities and fields can be integrated out by means of the TF approximation. The choice of the cutoff $E_c$ is determined by positions of deep-hole levels[57]; this information can be obtained by solving the HF problem.