The $gamma$ vibration is the most typical low-lying collective motion prevailing the nuclear chart. But only few one-phonon rotational bands in odd-$A$ nuclei have been known. Furthermore, two-phonon states, even the band head, have been observed in
a very limited number of nuclides not only of odd-$A$ but even-even. Among them, that in $^{105}$Mo is unique in that Coriolis effects are expected to be stronger than in $^{103}$Nb and $^{105}$Nb on which theoretical studies were reported. Then the purpose of the present work is to study $^{105}$Mo paying attention to rotational character change of the one-phonon and two-phonon bands. The particle-vibration coupling model based on the cranking model and the random-phase approximation is used to calculate the vibrational states in rotating odd-$A$ nuclei. The present model reproduces the observed yrast zero-phonon and one-phonon bands well. Emerging general features of the rotational character change from low spin to high spin are elucidated. In particular, the reason why the one-phonon band does not exhibit signature splitting is clarified. The calculated collectivity of the two-phonon states, however, is located higher than observed.
Multi-phonon excitations in atomic nuclei were observed very rarely although collective motions in quantum many-body systems are described as bosonic excitations. In particular, the first two-phonon $gamma$ vibrational ($2gamma$) excitation in odd-$A
$ nuclei was reported in 2006 and only a few have been known. Quite recently, conspicuously enhanced $B(E2)$s feeding $2gamma$ states were observed in $^{105}$Nb and conjectured that their parent states are candidates of $3gamma$ states. In the present work, the model space is enlarged from the present authors previous calculation for $^{103}$Nb. The purpose is twofold: One is to see how the description of $2gamma$ states is improved, and the other is to examine the existence of collective $3gamma$ states, and when they exist, study their collectivity through calculating interband $B(E2)$s. The particle-vibration coupling model based on the cranking model and the random-phase approximation is used to calculate the vibrational states in rotating odd-$A$ nuclei. Interband $B(E2)$s are calculated by adopting the method of the generalized intensity relation. The present calculation reproduces the observed spectra of $0gamma$ - $2gamma$ states well and gives collective $3gamma$ states with enhanced $B(E2)$s to $2gamma$ states in $^{103}$Nb and $^{105}$Nb. The most collective $3gamma$ state with the highest $K$ at zero rotation is thought to be the main component of the observed band.
One of the useful and practical methods for solving quantum-mechanical many-body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories
in nuclear physics have been formulated by use of the so called $hatQ$ box introduced by Kuo et.al. It has been one of the central problems how to calculate the $hatQ$ box accurately and efficiently. We first show that, introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With this transformed Hamiltonian, we next prove that the $hatQ$ box can be expressed in two ways: One is a form of continued fraction and the other is a simple series expansion up to second order with respect to renormalized vertices and propagators. This procedure ensures to derive an exact $hatQ$ box, if the calculation converges as the dimension of the Hilbert space tends to infinity. The $hatQ$ box given in this study corresponds to a non-perturbative solution for the energy-dependent effective interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. By applying the $hatZ$-box approach based on the $hatQ$ box proposed previously, we introduce a graphical method for solving the eigenvalue problem of the Hamiltonian. The present approach has a possibility of resolving many of the difficulties encountered in the effective-interaction theory.
Distribution of the two phonon $gamma$ vibrational collectivity in the rotating triaxial odd-$A$ nucleus, $^{103}$Nb, that is one of the three nuclides for which experimental data were reported recently, is calculated in the framework of the particle
vibration coupling model based on the cranked shell model plus random phase approximation. This framework was previously utilized for analyses of the zero and one phonon bands in other mass region and is applied to the two phonon band for the first time. In the present calculation, three sequences of two phonon bands share collectivity almost equally at finite rotation whereas the $K=Omega+4$ state is the purest at zero rotation.
We study the interplay of the chiral and the color superconducting phase transitions in an extended Nambu--Jona-Lasinio model with a multi-quark interaction that produces the nonlinear chiral-diquark coupling. We observe that this nonlinear coupling
adds up coherently with the omega^2 interaction to produce the chiral-color superconductivity coexistence phase or cancel each other depending on its sign. We discuss that large coexistence region in the phase diagram is consistent with the quark-diquark picture for the nucleon whereas its smallness is the prerequisite for the applicability of the Ginzburg-Landau approach.
