Session EM: Nuclear Theory III

Renormalization of the leading-order chiral nucleon-nucleon interaction and bulk properties of nuclear matter

It is well known [1] that the nucleon-nucleon (NN) interaction at leading order (LO) of chiral perturbation theory can be renormalized (i.e., cutoff independence can be achieved) when certain counter terms of next-to-leading (NLO) and next-to-next-to-leading order (NNLO) are promoted to LO. It is then of interest to investigate if also the predictions for few- and many-nucleon observables turn out to be cutoff independent when calculated from the renormalized LO NN interaction. As a first test, we have calculated the binding energy per nucleon in nuclear matter as a function of density and find saturation and cutoff independence of the results. \\[4pt] [1] A. Nogga \textit{et al.,} Phys. Rev. C \textbf{72}, 054006 (2005).   

Our recent progress in microscopic calculations of the equation of state

We are involved with a broad spectrum of studies aimed at improving our knowledge of nuclear matter, including its states far from equilibrium. Our ``ab initio'' approach is microscopic and relativistic, with the calculated nuclear matter properties being derived self- consistently from realistic nuclear forces. The isovector features of the equation of state, in particular, are still poorly understood. They have relevance for the physics of rare, short-lived nuclei and, on a dramatically different scale, the physics of neutron stars. In both cases, the crucial role is played by the symmetry energy, which determines the formation of the neutron skin in neutron-rich nuclei and the radius of a neutron star, a system 18 orders of magnitudes larger and 55 orders of magnitude heavier. We will report on our progress, which includes predictions of the energy per particle in hyperonic matter and, most recently, the effect of temperature on the single-particle properties.   

Neutron star and $\beta$-stable EOS with Brown-Rho scaled low-momentum NN interactions

Neutron star properties, such as its mass, radius, and moment of inertia, are calculated by solving the Tolman-Oppenheimer-Volkov equations using the ring-diagram equation of state (EOS) obtained from realistic low-momentum NN interactions $V_{low-k}$. Several NN potential models (CDBonn, Argonne, Nijmegen) have been employed to calculate the ring-diagram EOS where pphh ring diagrams are summed to all orders. The proton fraction for a $\beta$-stable neutron star is determined from the chemical potential condition $\mu_n-\mu_p=\mu_e$. The neutron star masses and radii given by the above potentials alone both tend to be about $30\%$ too small compared with accepted values. Our results are largely improved with the inclusion of medium corrections based on Brown-Rho scaling where the in-medium meson masses, particulaly those of $\omega$, $\rho$ and $\sigma$, are slightly decreased compared with their in-vacumn values. Initial results using such medium corrected $V_{low-k}$ are neutron star mass $M\sim 1.6 M_{sun}$ and radius $R\sim 8$ km. Effects from superconducting neutron EOS are discussed.   

Similarity renormalization group to the many-body problems

One of the major goals of nuclear structure theory is to explain many-body phenomena from nucleonic interactions. Since realistic nucleon-nucleon interactions have strong repulsion and tensor component at short distance, nuclear system is non-perturbative and even few-body problems are difficult to solve. Several methods based on renormalization group (RG) or unitary transformation can be used to treat the short-range correlation, the consequence of which nuclear many-body calculations converge rapidly. These methods, however, generate many-body forces which significantly affects the observable unless the induced forces are treated properly. To overcome this problem, one way is to keep the induced many-body forces explicitly. We propose an alternative way, In-medium similarity renormalization group (SRG), by extending the free-space SRG. We derive the flow equations for normal-ordered Hamiltonian assuming a core so that the dominant part of many-body correlations are incorporated into density dependent lower-body forces, driving the Hamiltonian more feasible form for the many-body calculations. In-medium SRG provides a new systematic and non-perturbative path from nucleonic interactions to the many-body calculations. We will show the newest results of the methods.   

Evolution of Nuclear Many-Body Forces with the Similarity Renormalization Group

The first practical method to evolve many-body nuclear forces to softened form using the Similarity Renormalization Group (SRG) in a harmonic oscillator basis is demonstrated. When applied to 4He calculations, the two- and three-body oscillator matrix elements yield rapid convergence of the ground-state energy with a small net contribution of the induced four-body force.   

Schroedinger's Wave Structure of Matter (WSM)

The puzzling electron is due to the belief that it is a discrete particle. Einstein deduced this structure was impossible since Nature does not allow the discrete particle. Clifford (1876) rejected discrete matter and suggested structures in `space'. Schroedinger, (1937) also eliminated discrete particles writing: What we observe as material bodies and forces are nothing but shapes and variations in the structure of space. Particles are just schaumkommen (appearances). He rejected wave-particle duality. Schroedinger's concept was developed by Milo Wolff and Geoff Haselhurst (SpaceAndMotion.com) using the Scalar Wave Equation to find spherical wave solutions in a 3D quantum space. This WSM, the origin of all the Natural Laws, contains all the electron's properties including the Schroedinger Equation. The origin of Newton's Law F=ma is no longer a puzzle; It originates from Mach's principle of inertia (1883) that depends on the space medium and the WSM. Carver Mead (1999) at CalTech used the WSM to design Intel micro-chips correcting errors of Maxwell's magnetic Equations. Applications of the WSM also describe matter at molecular dimensions: alloys, catalysts, biology and medicine, molecular computers and memories. See ``Schroedinger's Universe'' - at Amazon.com   

Neutron-deuteron scattering in configuration space II

A new computational method to solve the configuration-space Faddeev equations for the breakup scattering problem [1] has been applied to study the elastic \textit{nd} scattering above the deuteron threshold. To perform numerical calculations for arbitrary nuclear potentials and with arbitrary numbers of partial waves retained, we use the approach proposed in [2]. Calculations of the elastic differential cross section, nucleon and deuteron analyzing powers for lab energy 14.1 MeV were performed with the charge independent AV14 potential. To compute the observables, the maximum value for the total momentum $j$ of a nucleon pair was chosen equal to $3$ and all values of the conserved total three-nucleon angular momentum up to 13/2 with both signs of parity were taken into account. The results are compared with those of the Bochum group. \\[4pt] [1] V.M. Suslov and B. Vlahovic, Phys. Rev. C\textbf{69}, 044003 (2004). \\[0pt] [2] S.P. Merkuriev, C. Gignoux and A. Laverne, Ann. Phys. \textbf{99}, 30 (1976).   

Three-nucleon forces and neutron-rich nuclei

I will discuss the role of three-nucleon forces on neutron-rich nuclei, in particular for the evolution to and the location of the neutron dripline, as well as their impact on neutron-rich matter and the symmetry energy.   

The general relativistic harmonic oscillator

The relativistic harmonic oscillator has been solved analytically in two limits. One is the spin limit for which the scalar potential, V$_S $, is equal to the vector potential, V$_V $, plus a constant, and the other is the pseudospin limit in which the scalar potential is equal in magnitude but opposite in sign to the vector potential plus a constant [1,2]. Like the non-relativistic harmonic oscillator, each of these limits has a higher symmetry. For example, for the spherically symmetric oscillator, these limits have a SU(3) and pseudo-SU(3) symmetry respectively [3]. Atomic nuclei are close to the pseudospin limit. However, the analytic solutions in this limit are those of the Dirac ``negative'' energy states. In the exact pseudospin limit there are no bound Dirac valence states. For this reason we have started to investigate the general spherically symmetric relativistic harmonic oscillator for which V$_S =\frac{m}{2}\omega _S^2 $r$^2$ and V$_V =\frac{m}{2}\omega _V^2 $r$^2$. We report on the progress made in solving analytically the Dirac Hamiltonian with these potentials. \\[4pt] [1] Joseph N. Ginocchio, Phys. Rev. C \textbf{69}, 034318 (2004). \\[0pt] [2] Joseph N. Ginocchio, Phys. Rep. \textbf{414}, 165 (2005). \\[0pt] [3] Joseph N. Ginocchio, PRL \textbf{95, }252501 (2005).   

Effect of Gluon and Pion Exchanges on Hyperons

A new version of the quark-meson coupling model, which involves not only the gluon-quark interaction but also the pion-quark coupling based on chiral symmetry, is applied to hyperons in a nuclear medium. Our aim is to study the effects of one-gluon exchange (OGE) and pion-cloud on the mass of hyperon ($\Lambda$, $\Sigma$ and $\Xi$) in a nuclear matter. To describe a nuclear matter, we add the intermediate attractive and repulsive forces by introducing the $\sigma$, which is {\it not} the chiral partner of the $\pi$ meson, and the $\omega$ mesons. We determine the model parameters by fitting the nuclear saturation condition at normal density. As a consequence, we find that the effect of the gluon and pion exchanges provides the hyperfine splitting in the hadron spectra, and the hyperfine interaction due to the gluon exchange plays an important role in the in-medium baryon spectra. In contrast, the pion-cloud effect is relatively small. At the quark mean-field level, the $\Lambda$ feels more attractive force than the $\Sigma$ or $\Xi$ in matter.   

Advances in Energy and Integral Area Calculations

A transform function, and new additional theorem, allows work and energy, to be calculated for any constant, or time-varying force function, in integral form, as a function of time. In addition, the work W = \`{o} f(x) dx, and corresponding change in kinetic energy DK, of an object or particle, with a time-varying mass m(t), can be determined using the transform function. The individual work, and change in kinetic energy, can be calculated for the vector component(s), which make-up a resultant force function, when more than one force vector component simultaneously do work on an object or particle, along the same axis, using the transform function, and new additional theorem.   

A method to compute the QRPA

We introduce the finite amplitude method (FAM) for the QRPA. This method allows to build fully self consistent QRPA codes; since the FAM method is not limited to spherically symmetric systems it is helpful in the solution of the deformed QRPA problem where the construction of the matrices is a difficult task in itself. All that is needed to write a QRPA code with the FAM method is a HFB code; the residual fields ( $\delta h(\omega)$, $\delta h^{\dagger}(\omega)$, $\delta \Delta (\omega)$ and $\delta \Delta^{\dagger}(\omega)$), which usually are the difficult part to be calculated, are computed with a numerical derivation which requires the quasi-particle amplitudes previously obtained with the HFB code and the QRPA amplitudes. The FAM method is not involved in the diagonalization of the QRPA matrices, a task which can be solved via iterative methods (like the Conjugate Gradient Method).\\[4pt] [1] T. Nakatsukasa, T. Inakura and K. Yabana: Phys. \ Rev. \textbf{C 76} 024318 (2007)   

The covering SU(3) group over anisotropic harmonic oscillators

We propose new non-linear boson transformation by which all the anisotropic oscillator states can be embedded in the SU(3) bases. We start from the oscillator Hamiltonian without spin- orbit interaction, and suppose that three oscillator frequencies have an integral rational ratio $a:b:c$. In order to construct a SU(3)-invariant expression, we express the harmonic oscillator boson operator $c_k$ ($k=x,y,z$), in terms of a $m$-fold product of new bosons $s_m$ ($m=a,b,c$), by requiring $s_{m}^{\dag}s_m=mc_{k}^{\dag}c_k$. The general form of the new bosons $s_m$, for any positive integer $m$, is given by $c_{k}=[ m \prod_{r=1}^{m-1}({\hat n}_m +r) ]^{-1/2}(s_{m}) ^m$, with ${\hat n}_{m}=s_m^{\dag}s_m$. Applying the analogy of Elliott's group operators, we obtain a similar set of group operators from new bosons $s_a$, $s_b$ and $s_c$, i.e., ${\tilde Q}_q$ for $q=0, \pm 1$ and $\pm 2$, and ${\tilde \ell}_k$ for $k=a,b$ and $c$. Then, the commutation relations among these 8 operators are closed, and they commute with $H$. Together with Casimir operator and two operators which have diagonal form in number operators, i.e., ${\tilde Q}_{0}$, and ${\tilde Q}_{2}+ {\tilde Q}_{-2}$, we can classify the single-particle states in $N_{\rm sh}$, and find the new magic numbers for the triaxially deformed field.