Abstract:
We consider 1S0 pairing in infinite neutron matter and nuclear matter and show that in the lowest order approximation, where the pairing interaction is taken to be the bare nucleon-nucleon (NN) interaction in the 1S0 channel, the pairing interaction and the energy gap can be determined directly from the 1S0 phase shifts. This is due to the almost separable character of the NN interaction in this partial wave. Since the most recent NN interactions are charge-dependent, we have to solve coupled gap equations for proton-proton, neutron-neutron, and neutron-proton pairing in nuclear matter. The results, however, are found to be close to those obtained with charge-independent potentials.

Abstract:
An extended pairing plus QQ force model, which has been shown to successfully explain the nuclear binding energy and related quantities such as the symmetry energy, is applied to study the alpha-like four-nucleon correlations in 1f_{7/2} shell nuclei. The double difference of binding energies, which displays a characteristic behavior at $N \approx Z$, is interpreted in terms of the alpha-like correlations. Important roles of proton-neutron interactions forming the alpha-like correlated structure are discussed.

Abstract:
A simple model based on the group SO(5) suggests that both the like-particle and neutron-proton components of isovector pairing correlations in odd-A nuclei are Pauli blocked. The same effect emerges from Monte Carlo Shell-model calculations of proton-rich nuclei in the full fp shell. There are small differences between the two models in their representation of the effects of an odd nucleon on the competition between like-particle and neutron-proton pairing, but they can be understood and reduced by using a two-level version of the SO(5) model. On the other hand, in odd-odd nuclei with N not equal to Z SO(5) disagrees more severely with the shell model because it incorrectly predicts ground-state isospins. The shell model calculations for any fp-shell nuclei can be extended to finite temperature, where they show a decrease in blocking.

Abstract:
A detailed description of the nucleon direct Urca processes related to the neutron star cooling is given and how they are affected by the degrees of freedom of hyperons and hyperon direct Urca processes are presented. These results indicate that the appearance of hyperons can sharply suppress the neutrino emissivity of the nucleon direct Urca processes. However, the contribution of the hyperon direct Urca processes within the certain density (mass) range makes the total neutrino emissivity (luminosity) even exceed the corresponding value of the nucleon direct Urca processes in npe$\mu$ matter. Furthermore, when we only consider the nucleon direct Urca processes with the $^{1}S_{0}$ proton superfluidity in neutron star cooling, for example, it is hard to differentiate the theoretical cooling curves of 1.95$M_{\odot}$ for nphe$\mu$ matter from 2.03$M_{\odot}$ for npe$\mu$ matter in GM1 model. It is also difficult to rule out hyperons at the points of intersection for the two theoretical cooling curves of 2.0$M_{\odot}$ neutron star for npe$\mu$ and nphe$\mu$ matter in TM1 model.These results could be used to help prove appearing hyperons in PSR J1614-2230 and J0348+0432 from neutron star cooling perspective.

Abstract:
The neutron and proton odd-even mass differences are studied with Hartree-Fock+BCS (HFBCS) calculations with Skyrme interactions and an isospin dependent contact pairing interaction, which is recently derived from a microscopic nucleon-nucleon interaction. To this end, we perform HFBCS calculations for even and odd semi-magic Tin and Lead isotopes together with even and odd Z isotones with $N$= 50 and 82. The filling approximation is applied to the last unoccupied particle in odd nuclei. Comparisons with the experimental data show a clear manifestation of the isospin dependent pairing correlations in both proton and neutron pairing gaps.

Abstract:
We perform systematic calculations of pairing gaps in semi-magic nuclei across the nuclear chart using the Energy Density Functional method and a {\it non-empirical} pairing functional derived, without further approximation, at lowest order in the two-nucleon vacuum interaction, including the Coulomb force. The correlated single-particle motion is accounted for by the SLy4 semi-empirical functional. Rather unexpectedly, both neutron and proton pairing gaps thus generated are systematically close to experimental data. Such a result further suggests that missing effects, i.e. higher partial-waves of the NN interaction, the NNN interaction and the coupling to collective fluctuations, provide an overall contribution that is sub-leading as for generating pairing gaps in nuclei. We find that including the Coulomb interaction is essential as it reduces proton pairing gaps by up to 40%.

Abstract:
As a model which displays a picture of the symmetry energy as an energy of rotation in isospace of a Cooper pair condensate, a Hamiltonian with a pairing force and an interaction of isospins is analyzed in the Hartree-Bogolyubov (HB) plus Random Phase (RPA) approximation. The HB energy minus Lagrangian multiplier terms is shown to be locally minimized by a product of neutron and proton Bardeen-Cooper-Schrieffer states. Nambu-Goldstone RPA solutions appear due to global gauge invariance and isobaric invariance. In an idealized case of infinitely many equidistant single-nucleon levels, the symmetry energy is composed of contributions from the single-nucleon and isospin interaction energies and the RPA correlation energy. The contribution of the latter is dominated by a neutron-proton Nambu-Goldstone solution, which makes the total symmetry energy nearly proportional to T(T+1). Observations reported from Skyrme force calculations are discussed in the light of these results. Calculations with deformed Woods-Saxon single-nucleon levels give results similar to those of the idealized case, whereas a somewhat different behavior is found with spherical Woods-Saxon levels. The calculations with Woods-Saxon single-nucleon levels reproduce surprisingly well the empirical symmetry energy.

Abstract:
The present contribution reports the first systematic finite-nucleus calculations performed using the Energy Density Functional method and a non-empirical pairing functional derived from low-momentum interactions. As a first step, the effects of Coulomb and the three-body force are omitted while only the bare two-nucleon interaction at lowest order is considered. To cope with the finite-range and non-locality of the bare nuclear interaction, the 1S0 channel of Vlowk is mapped onto a convenient operator form. For the first time, neutron-neutron and proton-proton pairing correlations generated in finite nuclei by the direct term of the two-nucleon interaction are characterized in a systematic manner. Eventually, such predictions are compared to those obtained from empirical local functionals derived from density-dependent zero range interactions. The characteristics of the latter are analyzed in view of that comparison and a specific modification of their isovector density dependence is suggested to accommodate Coulomb effects and the isovector trend of neutron gaps at the same time.

Abstract:
Neutron-proton (np-) pairing is expected to play an important role in the N Z nuclei. In general, it can have isovector and isoscalar character. The existence of isovector np-pairing is well established. On the contrary, it is still debated whether there is an isoscalar np-pairing. The review of the situation with these two types of pairing with special emphasis on the isoscalar one is presented. It is concluded that there are no substantial evidences for the existence of isoscalar np-pairing.

Abstract:
We reexamine neutron-proton pairing as a phenomenon that should be explanable in a microscopic theory of nuclear binding energies. Empirically, there is an increased separation energy when both neutron and proton numbers are even or if they are both odd. The enhancement is present at some level in nearly all nuclei: the separation energy difference has the opposite sign in less than 1% of the cases in which sufficient data exist. We discuss the possible origin of the effect in the context of density functional theory (DFT) and its extensions. Neutron-proton pairing from the Hartree-Fock-Bogoliubov theory does not seem promising to explain the effect. We demonstrate that much of the increased binding in the odd-odd system might be understood as a recoupling energy. This suggests that the DFT should be extended by angular momentum projection to describe the effect.