Quarks, Nuclei and Stars

The following sections are included:

• Contents
• Preface
• Foreword I
• Foreword II
• Foreword III
• Photos of Gerry Brown

Gerry Brown initiated some early studies on the coexistence of different nuclear shapes. The subject has continued to be of interest and is crucial for understanding nuclear fission. We now have a very good picture of the potential energy surface with respect to shape degrees of freedom in heavy nuclei, but the dynamics remain problematic. In contrast, the early studies on light nuclei were quite successful in describing the mixing between shapes. Perhaps a new approach in the spirit of the old calculations could better elucidate the character of the fission dynamics and explain phenomena that current theory does not model well.

The properties of the oxygen isotopes provide diverse examples of progress made in experiments and theory. This chain of isotopes has been studied from beyond the proton drip line in ¹²O to beyond the neutron drip line in ^25,26O. This short survey starts with the microscopic G matrix approach for ¹⁸O of Kuo and Brown in the 1960’s and shows how theory has evolved. The nuclear structure around the doubly-magic nucleus ²⁴O is particularly simple in terms of the nuclear shell model. The nuclear structure around the doubly-magic nucleus ¹⁶O exhibits the coexistence of single-particle and collective structure.

My father Geny Brown practiced science in a certain way during his life, but much of his approach to work and research was not clearly articulated. In this article I convey some of the lessons I learned growing up with him.

The formalism of next-to-leading order (NLO) Fermi Liquid Theory (FLT) is employed to calculate the thermal properties of symmetric nuclear and pure neutron matter in a relativistic many-body theory beyond the mean field level which includes two-loop (TL) effects. For all thermal variables, the semi-analytical NLO corrections reproduce results of the exact numerical calculations for entropies per baryon up to 2. This corresponds to excellent agreement down to subnuclear densities for temperatures up to 20 MeV. In addition to providing physical insights, a rapid evaluation of the equation of state (EOS) in the homogeneous phase of hot and dense matter is achieved through the use of the zero-temperature Landau effective mass function and its derivatives.

The understanding of the convergence properties of the shell-model effective Hamiltonian, within the framework of the many-body perturbation theory, is a long-standing problem. The infinite summation of a certain class of diagrams, the so-called “bubble diagrams,” may be provided calculating the Kirson–Babu–Brown induced interaction, and provides a valid instrument to study whether or not the finite summation of the perturbative series is well-grounded. Here, we perform an application of the calculation of the Kirson–Babu–Brown induced interaction to derive the shell-model effective Hamiltonian for p-shell nuclei starting from a modern nucleon–nucleon potential, obtained by way of the chiral perturbation theory. The outcome of our calculation is compared with a standard calculation of the shell-model Hamiltonian, where the core-polarization effects are calculated only up to third-order in perturbation theory. The results of the two calculations are very close to each other, evidencing that the perturbative approach to the derivation of the shell-model Hamiltonian is still a valid tool for nuclear structure studies.

The US Nuclear Science Advisory Committee (NSAC) recently recommended the construction of a high-luminosity, high-energy Electron Ion Collider (EIC), with polarized beams capable of colliding polarized electrons with polarized proton and light ion beams, and with any nucleus. The $\sqrt{s}$ range between 40 GeV and 140 GeV, and luminosity range from 10³³⁻³⁴ cm⁻² s⁻¹ were recommended. It is anticipated that under the current guidance from the DOE, the collider could become operational in the second half of the 2020’s. This paper summarizes its science and the scope of this over all project.

The Cosmic Microwave Background (CMB) yields information about our Universe at around 380,000 years after the Big Bang (BB). Due to the weak interaction of the neutrinos with matter, the Cosmic Neutrino Background (CNB) should give information about a much earlier time of our Universe, around one second after the BB. Probably, the most promising method to “see” the CNB is the capture of the electron neutrinos from the Background by Tritium, which then decays into ³He and an electron with the energy of the the Q-value = 18.562 keV plus the electron neutrino rest mass. The “KArlsruhe TRItium Neutrino” (KATRIN) experiment, which is in preparation, seems presently the most sensitive proposed method for measuring the electron antineutrino mass. At the same time, KATRIN can also look by the reaction ${\nu }_e(\sim 1.95\mathrm{\text{K}}){+}^3\mathrm{\text{H}}{\to }^3\mathrm{\text{He}}+e^{-}(Q=18.6\mathrm{\text{keV}}+m_{\nu e}{c}^2)$. The capture of the Cosmic Background Neutrinos (CNB) should show in the electron spectrum as a peak by the electron neutrino rest mass above $Q$. Here, the possibility to see the CNB with KATRIN is studied. A detection of the CNB by KATRIN seems not to be possible at the moment. But KATRIN should be able to determine an upper limit for the local electron neutrino density of the CNB.

About two decades ago, Gerry Brown pushed me to study hadron physics using hadronic models based on the hidden local symmetry (HLS). Since then, I have been involving many works related to the HLS. In this contribution to Gerry’s memorial, I will summarize my recent analyses associated with the HLS. After a very brief summary of the HLS, I will show a way to determine parameters of the HLS model from holographic QCD models. This model provides a soliton solution which is identified with the nucleon. I will briefly introduce a model in which the solitons are put onto a crystal lattice for simulating nuclear matter. The results suggest existence of the chiral invariant mass of nucleon. I also briefly review my recent analyses on the nuclear matter by using a hadronic model including nucleon field together with the π, σ, ρ and ω mesons based on the parity doublet structure of nucleon combined with the HLS, in which the chiral invariant mass of the nucleon is naturally embedded. The result implies that studying effective masses of nucleons will give a clue for the chiral invariant mass. Finally, I will show our recent analysis on the medium modification of charmed meson masses using a chiral model based on the chiral doubling structure of charmed mesons. The result shows that the masses of the chiral partners of charmed mesons become closer in high density reflecting the particle chiral symmetry restoration in medium.

We review efforts to describe the approach to chiral symmetry restoration in neutron matter from the low-energy realization of QCD, chiral effective field theory.

In this paper, we present a simple discussion on the properties of compact stars using an EoS obtained in effective field theory anchored on scale and hidden-local symmetric Lagrangian endowed with topology change and a unequivocal prediction on the deformation of the compact star, that could be measured in gravitational waves. The objective is not to offer a superior or improved EoS for compact stars but to confront with a forthcoming astrophysical observable, the given model formulated in what is considered to be consistent with the premise of quantum chromodynamics (QCD). The model so obtained is found to satisfactorily describe the observation of a two-solar mass neutron star [P. B. Demorest et al., Nature 467 (2010) 1081, J. Antoniadis et al., Science 340 (2013) 1233232] with a minimum number of parameters. Specifically, the observable we are considering in this paper is the tidal deformability parameter λ (equivalently the Love number k₂), which affects gravitational wave forms at the late period of inspiral stage. The forthcoming aLIGO and aVirgo observations of gravitational waves from binary neutron star system will provide a valuable guidance for arriving at a better understanding of highly compressed baryonic matter.

For quark matter at finite baryon chemical potential, its density develops large fluctuations when it undergoes a first-order phase transition. Based on the Nambu–Jona–Lasinio (NJL) model, we have used the linear response theory to study the growth rate of density fluctuations and its dependence on the wavelength of unstable modes. Using the transport equation derived from the NJL model, we have also studied the time evolution of the unstable modes and the density fluctuations in a baryon-rich quark matter that is confined in a finite volume. Allowing the expansion of the quark matter, we have further studied the survivability of the density fluctuations as the density and temperature of the quark matter decrease. Possible experimental signatures of the density fluctuations have been suggested.

We solve the Dirac radial equation for a nucleon in a scalar Woods–Saxon potential well of depth $V_0$ and radius $r_0$. A sequence of values for the depth and radius are considered. For shallow potentials with $-1000\mathrm{\text{MeV}}\lesssim V_0<0$ the wave functions for the positive-energy states ${\mathrm{\Psi }}_{+}(r)$ are dominated by their nucleon component $g(r)$. But for deeper potentials with $V_0\lesssim -1500\mathrm{\text{MeV}}$ the ${\mathrm{\Psi }}_{+}(r)$s begin to have dominant anti-nucleon component $f(r)$. In particular, a special intruder state enters with wave function ${\mathrm{\Psi }}_{1/2}(r)$ and energy $E_{1/2}$. We have considered several $r_0$ values between 2 and 8fm. For $V_0\lesssim -2000\mathrm{\text{MeV}}$ and the above $r_0$ values, ${\mathrm{\Psi }}_{1/2}$ is the only bound positive-energy state and has its $g(r)$ closely equal to $-f(r)$, both having a narrow wave packet shape centered around $r_0$. The $E_{1/2}$ of this state is practically independent of $V_0$ for the above $V_0$ range and obeys closely the relation $E_{1/2}=\frac{\hslash c}{r_0}$.

Neutron stars are not only mines for clues to dense matter physics but may also be the auspicious sources of half of all nuclei heavier than A = 60 in the universe, including the auric isotopes. Although the cold dense matter above the nuclear saturation density cannot be directly explored in the laboratory, gilded constraints on the properties of matter from 1 to 10 times higher density can now be panned from neutron star observations. We show how upcoming observations, such as gravitational wave from mergers, precision timing of pulsars, neutrinos from neutron star birth and X-rays from bursts and thermal emissions, will provide the bullion from which further advances can be smelted.

With H. A. Bethe, G. E. Brown worked on the merger rate of neutron star binaries for the gravitational wave detection. Their prediction has to be modified significantly due to the observations of $2M_{\odot }$ neutron stars and the detection of gravitational waves. There still, however, remains a possibility that neutron star-low mass black hole binaries are significant sources of gravitational waves for the ground-based detectors. In this paper, I review the evolution of neutron star binaries with super-Eddington accretion and discuss the future prospect.

The relevance of chiral symmetry in baryons is highlighted in three examples in the nucleon spectroscopy and structure. The first one is the importance of chiral dynamics in understanding the Roper resonance. The second one is the role of chiral symmetry in the lattice calculation of $\pi N\sigma$ term and strangeness. The third one is the role of chiral $U(1)$ anomaly in the anomalous Ward identity in evaluating the quark spin and the quark orbital angular momentum. Finally, the chiral effective theory for baryons is discussed.

We briefly review the formation of pion-mediated heavy-light exotic molecules with both charm and bottom, under the general structures of chiral and heavy quark symmetries. The charm isosinglet exotic molecules with $J^{PC}=1^{++}$ binds, which we identify as the reported neutral $X(3872)$. The bottom isotriplet exotic with $J^{PC}=1^{+-}$ binds, and is identified as a mixed state of the reported charged exotics $Z_b^{+}(10610)$ and $Z_b^{+}(10650)$. The bound bottom isosinglet molecule with $J^{PC}=1^{++}$ is a possible neutral $X_b(10532)$ to be observed.

Gerry Brown has played a major role in initiating and stimulating systematic work on the theories of nuclear forces. It is the purpose of this paper to review those theories — in his memory.

Dense compact objects like neutron stars or black holes have always been one of Gerry Brown’s favorite research topics. This is closely related to the effects of strangeness in nuclear physics. Here, we review the chiral Effective Field Theory approach to interactions involving nucleons and hyperons, the possible existence of strange dibaryons, the fate of hyperons in nuclear matter and the present status of three-body forces involving hyperons and nucleons.

We review the first fully relativistic description for the neutrinoless double-beta decay by the beyond-mean-field covariant density functional theory. The calculations of the nuclear transition matrix elements are based on a full relativistic transition operator and nuclear wave functions in which the dynamic effects of particle-number, angularmomentum, and parity conservations as well as shape fluctuations are incorporated. We find that the commonly used nonrelativistic approximation for the transition operator is justified within the mechanism of light-Majorana-neutrino exchange. We stress that reducing the discrepancies in the predicted nuclear matrix elements by different nuclear models is a major challenge in nuclear physics.

We report the discovery of a partial conservation law obeyed by a schematic Hamiltonian of two protons and two neutrons in a j shell. In our Hamiltonian, the interaction matrix element of two nucleons with combined angular momentum J is linear in J for even J and constant for odd J. It turns out that in some stationary states, the sum of the angular momenta J_p and J_n of the proton and neutron pairs is conserved. The energies of these states are given by a linear function of J_p + J_n. The systematics of their occurrence is described and explained.

The development of the chiral dynamics based description of nuclear electroweak currents is reviewed. Gerald E. (Gerry) Brown’s role in basing theoretical nuclear physics on chiral Lagrangians is emphasized. Illustrative examples of the successful description of electroweak observables of light nuclei obtained from chiral effective field theory are presented.

I describe the long-standing search for a “smoking-gun” signal for the manifestation of (scale-)chiral symmetry in nuclear interactions. It is prompted by Gerry Brown’s last unpublished note, reproduced verbatim below, on the preeminent role of pions and vector (ρ, ω) mesons in providing a simple and elegant description of strongly correlated nuclear interactions. In this note written in tribute to Gerry Brown, I first describe a case of an unambiguous signal in axial-charge transitions in nuclei and then combine his ideas with the more recent development on the role of hidden symmetries in nuclear physics. What transpires is the surprising conclusion that the Landau–Migdal fixed point interaction $G_0^{\prime }$, the nuclear tensor forces and Brown–Rho scaling, all encoded in scale-invariant hidden local symmetry, as Gerry put, “run the show and make all forces equal.”

Paradigm shift in gauge topology, from instantons to their constituents — instantondyons — has recently lead to very significant advances. Unlike instantons, these objects have three different set of charges, therefore allowing to explain and to tie together several important phenomena: confinement, chiral symmetry breaking and magnetic screening.

Many-body Green functions are a very efficient formulation of the many-body problem. We review the application of this method to nuclear physics problems. The formulas which can be derived are of general applicability, e.g., in self-consistent as well as in nonself-consistent calculations. With the help of the Landau renormalization, one obtains relations without any approximations. This allows to apply conservation laws which lead to important general relations. We investigate the one-body and two-body Green functions as well as the three-body Green function and discuss their connection to nuclear observables. The generalization to systems with pair correlations are also presented. Numerical examples are compared with experimental data.

Complexities of using an SU(4) and SU(3) basis in the 1p0f shell are discussed. SU(3) representations are detailed for 2–5 particles. Earlier calculations by McGrory are discussed with the view of using the SU(3) model to describe multi-particle multi-hole states.

We propose a method of multi-regulator functional renormalization group (MR-FRG) which is a novel formulation of functional renormalization group with multiple infrared (IR) regulators. It is applied to a two-component fermionic system with an attractive contact interaction to study crossover phenomena between the Bardeen–Cooper–Schrieffer (BCS) phase and the Bose–Einstein condensation (BEC) phase. To control both the fermionic one-particle excitations and the bosonic collective excitations, IR regulators are introduced, one for the fermionic two-point function and another for the four-fermion vertex. It is shown that the Nozières–Schmitt-Rink (NSR) theory, which is successful to capture qualitative features of the BCS–BEC crossover, can be derived from MR–FRG. Some aspects of MR-FRG to go beyond the NSR theory are also discussed.

The in-medium modifications of hadron properties such as masses and decay widths have been a major focus of the scientific work of Gerry Brown and the insights gained by him and his collaborators made them major drivers of this field for several decades. Their prediction of experimental signals in di-lepton pair production in relativistic heavy-ion collisions was instrumental in initiating large experimental campaigns which continue till today. In this chapter, we review recent results which elucidate the relation of hadronic spectral properties at finite temperature and density to the restoration of spontaneously broken chiral symmetry.

The phase diagram of quantum chromodynamics is conjectured to have a rich structure containing at least three forms of matter: hadronic nuclear matter, quarkyonic matter and quark–gluon plasma. We justify the origin of the quarkyonic phase transition in a chiral-quark model and describe its formulation in terms of Skyrme crystals.

The detection of galactic supernova (SN) neutrinos represents one of the future frontiers of low energy neutrino physics and astrophysics. The neutron coherence of neutral currents (NCs) allows quite large cross-sections in the case of neutron rich targets, which can be exploited in detecting earth and sky neutrinos by measuring nuclear recoils. They are relatively cheap and easy to maintain. These (NC) cross-sections are not dependent on flavor conversions and, thus, their measurement will provide useful information about the neutrino source. In particular, they will yield information about the primary neutrino fluxes and perhaps about the spectrum after flavor conversions in neutrino sphere. They might also provide some clues about the neutrino mass hierarchy. The advantages of large gaseous low threshold and high resolution time projection counters (TPC) detectors are discussed.

A nonzero electric dipole moment (EDM) of the neutron, proton, deuteron or helion, in fact, of any finite system necessarily involves the breaking of a symmetry, either by the presence of external fields (i.e., electric fields leading to the case of induced EDMs) or explicitly by the breaking of the discrete parity and time-reflection symmetries in the case of permanent EDMs. We discuss two theorems describing these phenomena and report about the cosmological motivation for an existence of CP breaking beyond what is generated by the Kobayashi–Maskawa mechanism in the Standard Model and what this might imply for the permanent EDMs of the nucleon and light nuclei by estimating a window of opportunity for physics beyond what is currently known. Recent — and in the case of the deuteron even unpublished — results for the relevant matrix elements of nuclear EDM operators are presented and the relevance for disentangling underlying New Physics sources is discussed.

Gerry Brown was a godfather of our hidden local symmetry (HLS) for the vector meson from the birth of the theory throughout his life. The HLS is originated from very nature of the nonlinear realization of the symmetry $G$ based on the manifold $G/H$, and thus is universal to any physics based on the nonlinear realization. Here, I focus on the Higgs Lagrangian of the Standard Model (SM), which is shown to be equivalent to the nonlinear sigma model based on $G/H=\mathrm{\text{SU}}{(2)}_L\times \mathrm{\text{SU}}{(2)}_R/\mathrm{\text{SU}}{(2)}_V$ with additional symmetry, the nonlinearly-realized scale symmetry. Then, the SM does have a dynamical gauge boson of the SU${(2)}_V$ HLS, “SM $\rho$ meson”, in addition to the Higgs as a pseudo-dilaton as well as the NG bosons to be absorbed in to the $W$ and $Z$. Based on the recent work done with Matsuzaki and Ohki, I discuss a novel possibility that the SM $\rho$ meson acquires kinetic term by the SM dynamics itself, which then stabilizes the skyrmion dormant in the SM as a viable candidate for the dark matter, what we call “dark SM skyrmion (DSMS)”.