Pairing Renormalization Procedure

Within the HFB theory, the energy cutoff can be applied either to the s.p. or to the quasiparticle spectrum. The first option is used when the HFB equations are solved within a restricted s.p. space. However, the s.p. energies play only an auxiliary role in the HFB method, and the cutoff applied to the quasiparticle spectrum is more justified. This is done by using the so-called equivalent s.p. spectrum [12]:

$$\bar{e}_{n}=(1-2P_{n})E_{n}+\mu,$$ (1)

where $E_{n}$ is the quasiparticle energy and $P_{n}$ denotes the norm of the lower component of the HFB wave function.

Due to the similarity between $\bar{e}_{n}$ and the s.p. energies, one takes into account only those quasiparticle states for which $\bar{e}_{n}$ is less than the assumed cutoff energy $\epsilon_{\mbox{\rm\scriptsize {cut}}}$.

It was shown [8] that for a fixed pairing strength the pairing energies depend significantly on the energy cutoff. Within the renormalization scheme employed in this work, we use the prescription of adjusting the pairing strength to obtain a fixed average neutron pairing gap [12],

$$\bar{\Delta}=-\frac{1}{N}\int d^3\mathbf{r}d^3\mathbf{r'}\sum... ...,\mathbf{r'}\sigma')\rho(\mathbf{r'}\sigma',\mathbf{r}\sigma),$$ (2)

in $^{120}$Sn equal to the experimental value of 1.245MeV. In Eq. (2) $N$ is the number of particles, $\rho$ is the particle density, and $\tilde h$ is the pairing Hamiltonian (see Appendix A).

Such a procedure almost eliminates the dependence of the HFB energy on the cutoff [8].