A see-saw scenario of an A4A_4 flavour symmetric standard model

A see-saw scenario for an $A_4$ flavour symmetric standard model is presented. The latter, compared with the standard model, has an extended field content adopting now an additional $A_4$ symmetry structure (along with the standard model symmetry). As before, the see-saw mechanism can be realized in several models of different types depending on different ways of neutrino mass generation corresponding to the introduction of new (heavy in general) fields with different symmetry structures. In the present paper, a general description of all these see-saw types is made with a more detailed investigation on type-I models, while for type-II and type-III models a similar strategy can be followed. As within the original see-saw mechanism, the symmetry structure of the standard model fields decides the number and the symmetry structure of the new fields. In a model considered here, the scalar sector consists of three standard-model-Higgs-like iso-doublets ($SU_L(2)$-doublets) forming together an $A_4$-triplet, and three iso-singlets transforming as three singlets (1,$1^{'}$ and $1^{''}$) of $A_4$. In the lepton sector, the three left-handed lepton iso-doublets form an $A_4$-triplet, while the three right-handed charged leptons are either $A_4$-singlets in one version of the model, or components of an $A_4$-triplet in another version. To generate neutrino masses through, say, the type-I see-saw mechanism, it is natural to add four right-handed neutrino multiplets, including one $A_4$-triplet and three $A_4$-singlets. For an interpretation, the model is applied to deriving some physics quantities such as neutrinoless double beta decay effective mass $|\langle m_{ee}\rangle|$, CP violation phase $\delta_{CP}$ and Jarlskog parameter $J_{CP}$, which can be verified experimentally.Comment: LaTeX, 31 pages, 12 figures, 6 tables. V3: some parts modifie

Energies of the ground state and first excited 0+0^{+} state in an exactly solvable pairing model

Several approximations are tested by calculating the ground-state energy and the energy of the first excited $0^{+}$ state using an exactly solvable model with two symmetric levels interacting via a pairing force. They are the BCS approximation (BCS), Lipkin - Nogami (LN) method, random-phase approximation (RPA), quasiparticle RPA (QRPA), the renormalized RPA (RRPA), and renormalized QRPA (RQRPA). It is shown that, in the strong-coupling regime, the QRPA which neglects the scattering term of the model Hamiltonian offers the best fit to the exact solutions. A recipe is proposed using the RRPA and RQRPA in combination with the pairing gap given by the LN method. Applying this recipe, it is shown that the normal-superfluid phase transition is avoided, and a reasonably good description for both of the ground-state energy and the energy of the first excited $0^{+}$ state is achieved.Comment: 18 pages, 4 figure

Superfluid-normal phase transition in finite systems and its effect on damping of hot giant resonances

Thermal fluctuations of quasiparticle number are included making use of the secondary Bogolyubov's transformation, which turns quasiparticles operators into modified-quasiparticle ones. This restores the unitarity relation for the generalized single-particle density operator, which is violated within the Hartree-Fock-Bogolyubov (HFB) theory at finite temperature. The resulting theory is called the modified HFB (MHFB) theory, whose limit of a constant pairing interaction yields the modified BCS (MBCS) theory. Within the MBCS theory, the pairing gap never collapses at finite temperature T as it does within the BCS theory, but decreases monotonously with increasing T. It is demonstrated that this non-vanishing thermal pairing is the reason why the width of the giant dipole resonance (GDR) does not increase with T up to T around 1 MeV. At higher T, when the thermal pairing is small, the GDR width starts to increase with T. The calculations within the phonon-damping model yield the results in good agreement with the most recent experimental systematic for the GDR width as a function of T. A similar effect, which causes a small GDR width at low T, is also seen after thermal pairing is included in the thermal fluctuation model.Comment: Invited lecture at the Predeal international summer school in nuclear physics on ``Collective motion and phase transitions in nuclear systems'', 28 August - 9 September, 2006, Predeal, Romania; 18 pages, 3 figures; to be published by World Scientific in the proceedings of this schoo