Brown-Rho Scaling Research Articles

Systematics on the high-density nuclear equation of state from relativistic Hartree-Fock theory with Brown-Rho scaling

Using the Brown-Rho (BR) scaling law, several new relativistic Hartree-Fock (RHF) models with chiral limit are thoroughly investigated. The high-density nuclear equation of state (EOS) of RHF with the BR scaling become softer than one without the BR scaling. The EOS of RHF with BR scaling is consistent with the constraint extracted from collective flows and kaon production in heavy-ion collisions. It is found that, with a sizable strength parameter of the mass drop $x=0.126$, the symmetry energy is almost flat at density above 3 times the saturation density ${\ensuremath{\rho}}_{0}$, and even decreases slightly. This is caused from the fact that the decline of the potential part of symmetry energy is faster than the increase of the kinetic part of symmetry energy. The decrease of potential part is mainly because the neutron mass ${M}_{n}^{*}$ and the proton mass ${M}_{p}^{*}$ are close to each other when the density gradually approaches the critical density of chiral limit. For a mass drop of $x=0.092$, since the critical density of chiral limit is higher than one of $x=0.126$, the symmetry energy becomes flat at density above $5{\ensuremath{\rho}}_{0}$. While, since the critical density of chiral limit is very high for a small mass drop $x=0.053$, the symmetry energy of RHFs with $x=0.053$ always increases at the entire density domain of this work (below 6 ${\ensuremath{\rho}}_{0}$). The maximum mass of neutron star (NS) obtained with present models can satisfy $M=2.08\ifmmode\pm\else\textpm\fi{}0.07{M}_{\ensuremath{\bigodot}}$. However, the radius of 1.4 ${M}_{\ensuremath{\bigodot}}$ with the mass drop of $x=0.126$ will surpass the upper limit (13.7 km) extracted from the tidal deformability parameter of coalescence of a NS binary system and the radius of $\text{J0030}+0451$ being $12.{71}_{\ensuremath{-}1.19}^{+1.14}$ km. The radius of 1.4 ${M}_{\ensuremath{\bigodot}}$ with the mass drop of $x=0.092$ is 13.6 km, closing to 13.7 km and the radius of $\text{J0030}+0451$ being $12.{71}_{\ensuremath{-}1.19}^{+1.14}$ km. Therefore, the RHF model with BR scaling prefers the mass drop of $x\ensuremath{\lesssim}0.092$.

In search of a pristine signal for (scale-)chiral symmetry in nuclei

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 ([Formula: see text]) 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 [Formula: see text], 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.”

Dense baryonic matter in conformally-compensated hidden local symmetry: Vector manifestation and chiral symmetry restoration

We find that, when the dilaton is implemented as a (pseudo-)Nambu-Goldstone boson using a conformal compensator or "conformon" in a hidden gauge symmetric Lagrangian written to $O(p^4)$ from which baryons arise as solitons, namely, skyrmions, the vector manifestation and chiral symmetry restoration at high density predicted in hidden local symmetry theory --- which is consistent with Brown-Rho scaling --- are lost or sent to infinite density. It is shown that they can be restored if in medium the behavior of the $\omega$ field is taken to deviate from that of the $\rho$ meson in such a way that the flavor $U(2)$ symmetry is strongly broken at increasing density. The hitherto unexposed crucial role of the $\omega$ meson in the structure of elementary baryon and multibaryon systems is uncovered in this work. In the state of half-skyrmions to which the skyrmions transform at a density $n_{1/2}^{} \gtrsim n_0^{}$ (where $n_0^{}$ is the normal nuclear matter density), characterized by the vanishing (space averaged) quark condensate but nonzero pion decay constant, the nucleon mass remains more or less constant at a value $\gtrsim$ 60 \% of the vacuum value indicating a large component of the nucleon mass that is not associated with the spontaneous breaking of chiral symmetry. We discuss its connection to the chiral-invariant mass $m_0^{}$ that figures in the parity-doublet baryon model.

Core polarization, Brown–Rho scaling and a memory of Gerry's Princeton Years

Core-polarization (CP) and Brown-Rho (BR) scaling were among Gerry's most favorite topics. In this contribution, we will discuss some of the early history as well as more recent work associated with these two fascinating phenomena. I (TTSK) will begin with some recollections of Princeton, where I met Gerry for the first time in 1964 and worked as his postdoc. Core polarization was in fact the first topic he assigned to me. JWH started working with Gerry at Stony Brook from 2003 and was Gerry's last student in nuclear physics. We three had teamed up well, working closely on both CP and BR scaling, particularly on the latter's connection to the anomalously-long beta-decay lifetime of carbon-14. We shall here briefly review these topics, including a recently developed new Brown-Rho scaling based on a Skyrmion half-Skyrmion two-phase model.

On chiral symmetry breaking, topology and confinement

We start with the relation between the chiral symmetry breaking and gauge field topology. New lattice result further enhance the notion of Zero Mode Zone, a very narrow strip of states with quasizero Dirac eigenvalues. Then we move to the issue of "origin of mass" and Brown-RHo scaling: a number of empirical facts contradicts to the idea that masses of quarks and such hadrons as $\rho,N$ decrease near $T_c$. We argue that while at $T=0$ the main contribution to the effective quark mass is chirally odd $m_{\snchi}$, near $T_c$ it rotates to chirally-even component $m_\chi$, because "infinite clusters" of topological solitons gets split into finite ones. Recent progress in understanding of topology require introduction of nonzero holonomy $<A_0>\neq 0$, which splits instantons into $N_c$ (anti)selfdual "instanton-dyons". Qualitative progress, as well as first numerical studios of the dyon ensemble are reported. New connections between chiral symmetry breaking and confinement are recently understood, since instanton-dyons generates holonomy potential with a minimum at confining value, if the ensemble is dense enough.

Liquid–gas phase transition in hot asymmetric nuclear matter with density-dependent relativistic mean-field models

The liquid–gas phase transition in hot asymmetric nuclear matter is studied within density-dependent relativistic mean-field models where the density dependence is introduced according to the Brown–Rho scaling and constrained by available data at low densities and empirical properties of nuclear matter. The critical temperature of the liquid–gas phase transition is obtained to be 15.7 MeV in symmetric nuclear matter falling on the lower edge of the small experimental error bars. In hot asymmetric matter, the boundary of the phase-coexistence region is found to be sensitive to the density dependence of the symmetry energy. The critical pressure and the area of phase-coexistence region increases clearly with the softening of the symmetry energy. The critical temperature of hot asymmetric matter separating the single-phase region from the two-phase region is analyzed to have a moderate sensitivity to the symmetry energy and is higher for the model possessing the softer symmetry energy.

Low-momentum interactions with Brown-Rho-Ericson scalings, and the density dependence of the nuclear symmetry energy

We have calculated the nuclear symmetry energy ${E}_{\mathrm{sym}}(\ensuremath{\rho})$ up to densities of $4\ensuremath{-}5{\ensuremath{\rho}}_{0}$ with the effects from the Brown-Rho (BR) and Ericson scalings with the in-medium mesons included. Using the ${V}_{\mathrm{low} k}$ low-momentum interaction with and without such scalings, the equations of state (EOSs) of symmetric and asymmetric nuclear matter have been calculated using a ring-diagram formalism where the particle-particle-hole-hole ring diagrams are included to all orders. The EOSs for symmetric nuclear matter and neutron matter obtained with linear BR scaling are both overly stiff compared with the empirical constraints of Danielewicz et al. [Science 298, 1592 (2002)]. In contrast, satisfactory results are obtained by either using the nonlinear Ericson scaling or by adding a Skyrme-type three-nucleon force (TNF) to the unscaled ${V}_{\mathrm{low} k}$ interaction. Our results for ${E}_{\mathrm{sym}}(\ensuremath{\rho})$ obtained with the nonlinear Ericson scaling are in good agreement with the empirical values of Tsang et al. [Phys. Rev. Lett. 102, 122701 (2009)] and Li et al. [Phys. Rev. C 72, 064611 (2005)], while those with the TNF are slightly below these values. For densities below the nuclear saturation density ${\ensuremath{\rho}}_{0}$, the results of the above calculations are nearly equivalent to each other and all are in satisfactory agreement with the empirical values.

Effects of Brown-Rho scalings in nuclear matter, neutron stars and finite nuclei

We have carried out calculations for nuclear matter, neutron stars and finite nuclei using NN potentials with and without the medium-dependent modifications based on the Brown-Rho (BR) scalings. Using the Vlow−k low-momentum interactions derived from such potentials, the equations of state (EOS) for symmetric and asymmetric nuclear matter, for densities up to ~ 5ρ0, are calculated using a RPA method where the particle-particle hole-hole ring diagrams are summed to all orders. The medium effects from both a linear BR scaling (BR1) and a non-linear one (BR2) are considered, and they both are essential for our EOSs to reproduce the nuclear matter saturation properties. For densities ρ below ρ0, results from BR1 and BR2 are close to each other. For higher densities, the EOS given by BR2 is more desirable and is well reproduced by that given by the interaction (Vlow−k+TBF) where Vlow−k is the unsealed low-momentum interaction and TBF is an empirical Skyrme three-body force. The moment of inertia of neutron stars is ~ 60 and ~ 25M⊙km2 respectively with and without the inclusion of the above BR2 medium effects. Effects from the BR scaling are important for the long half-life, ~ 5000yrs, of the 14C − 14N β-decay.

Hidden Local Field Theory and Dileptons in Relativistic Heavy Ion Collisions

The notion of freedom is introduced based on the vector manifestation of hidden local symmetry and is used to suggest that the dileptons measured in relativistic heavy-ion collisions do not provide {\em direct} information on the spontaneous breaking of chiral symmetry and hence on the mechanism for mass generation of light-quark hadrons. We give arguments how the dileptons emitted from those vector mesons whose masses are shifted a la Brown-Rho (BR) scaling by the change in temperature -- as in heavy-ion collisions -- and/or density -- as in cold compressed matter -- could be strongly suppressed in the hadronic free region between the chiral restoration and the flash point at which the vector mesons recover $\gsim 90%$ of their free-space on-shell masses and the full strong coupling strength. It may seem ironical that the very mechanism i.e., the vector manifestation in hidden local symmetry, that is to make the mass drop and modify the spectral function is in turn responsible for the dilepton suppression. A possible falsification of this drastic prediction will be indicated. We also briefly discuss the potential role played by the holographic dimension intrinsic in gravity-gauge duality that provides a unified field theory description of hadrons -- both mesons and baryons -- under extreme conditions. Baryons arise as coherent states of pions and vector mesons of holographic QCD which define the ground state or the vacuum on which mesonic excitations could undergo BR scaling.

CHIRAL CONDENSATE IN NUCLEAR MATTER WITH VACUUM CORRECTIONS

Within the relativistic Hartree approach using a Lagrangian with density-dependent parameters according to the Brown–Rho scaling law, it is found that the vacuum corrections from the nucleon Dirac sea soften the equation of state and favor the chiral symmetry restoration at high densities.

QCD sum rules for ρ mesons in vacuum and in-medium, re-examined

An updated investigation of QCD sum rules for the first two moments of \ensuremath{\rho}-meson spectral functions, both in vacuum and in-medium, is performed with emphasis on the role of the scale related to spontaneous chiral symmetry breaking in QCD. It is demonstrated that these lowest moments of vector current spectral distributions do permit an accurate sum-rule analysis with controlled input including QCD condensates of the lowest dimensions, whereas higher moments are subject to uncertainties from higher dimensional condensates. Possible connections with Brown-Rho scaling are discussed. The factorization approximation for four-quark condensates is shown not to be applicable in any of the cases studied.

BROWN-RHO SCALING IN THE STRONG COUPLING LATTICE QCD

We examine the Brown-Rho scaling for meson masses in the strong coupling limit of lattice QCD with one species of staggered fermion. Analytical expression of meson masses is derived at finite temperature and chemical potential. We find that meson masses are approximately proportional to the equilibrium value of the chiral condensate, which evolves as a function of temperature and chemical potential.

Shell Model Description of theC14DatingβDecay with Brown-Rho-ScaledNNInteractions

We present shell model calculations for the beta decay of 14C to the 14N ground state, treating the states of the A=14 multiplet as two 0p holes in an 16O core. We employ low-momentum nucleon-nucleon (NN) interactions derived from the realistic Bonn-B potential and find that the Gamow-Teller (GT) matrix element is too large to describe the known lifetime. By using a modified version of this potential that incorporates the effects of Brown-Rho scaling medium modifications, we find that the GT matrix element vanishes for a nuclear density around 85% that of nuclear matter. We find that the splitting between the (J(pi),T)=(1(+),0) and (J(pi),T)=(0(+),1) states in 14N is improved using the medium-modified Bonn-B potential and that the transition strengths from excited states of 14C to the 14N ground state are compatible with recent experiments.

Neutron-skin thickness of finite nuclei in relativistic mean-field models with chiral limits

We study several structure properties of finite nuclei using relativistic mean-field Lagrangians constructed according to the Brown-Rho scaling due to the chiral symmetry restoration at high densities. The models are consistent with current experimental constraints for the equations of state of symmetric matter at both normal and supranormal densities and of asymmetric matter at subsaturation densities. It is shown that these models can successfully describe the binding energies and charge radii of finite nuclei. Compared to calculations with usual relativistic mean-field models, these models give a reduced thickness of neutron skin in $^{208}\mathrm{Pb}$ between 0.17 fm and 0.21 fm. The reduction of the predicted neutron skin thickness is found to be due to not only the softening of the symmetry energy but also the scaling property of $\ensuremath{\rho}$ meson required by the partial restoration of chiral symmetry.

Equation of state of isospin-asymmetric nuclear matter in relativistic mean-field models with chiral limits

Using in-medium hadron properties according to the Brown–Rho scaling due to the chiral symmetry restoration at high densities and considering naturalness of the coupling constants, we have newly constructed several relativistic mean-field Lagrangians with chiral limits. The model parameters are adjusted such that the symmetric part of the resulting equation of state at supra-normal densities is consistent with that required by the collective flow data from high energy heavy-ion reactions, while the resulting density dependence of the symmetry energy at sub-saturation densities agrees with that extracted from the recent isospin diffusion data from intermediate energy heavy-ion reactions. The resulting equations of state have the special feature of being soft at intermediate densities but stiff at high densities naturally. With these constrained equations of state, it is found that the radius of a 1.4 M ⊙ canonical neutron star is in the range of 11.9 km ⩽ R ⩽ 13.1 km , and the maximum neutron star mass is around 2.0 M ⊙ close to the recent observations.

THERMAL DILEPTON PRODUCTION FROM DROPPING ρ IN THE VECTOR MANIFESTATION

We study the pion electromagnetic form factor and the dilepton production rate in hot matter based on the vector manifestation (VM) of chiral symmetry in which the massless vector meson becomes the chiral partner of the pion, giving a theoretical framework of the dropping ρ à la Brown–Rho scaling. The VM predicts a strong violation of the vector dominance (VD) near the phase transition point associated with the dropping ρ. We present the effect of the VD violation to the dilepton production rate and make a comparison to the one predicted by assuming the VD together with the dropping ρ.

Equation of state for hot and dense matter: [formula omitted] model with scaled hadron masses and couplings

The proposed earlier relativistic mean-field model with hadron masses and coupling constants depending on the σ-meson field is generalized to finite temperatures. Within this approach we simulate the in-medium behavior of the hadron masses motivated by the Brown–Rho scaling. The high-lying baryon resonances and boson excitations as well as excitations of the σ, ω and ρ fields interacting via mean fields are incorporated into this scheme. Thermodynamic properties of hot and dense hadronic matter are elaborated with the constructed equation of state. Even at zero baryon density, effective masses of σ – ω – ρ – N excitations abruptly drop down for T ≳ 170 MeV and reach zero at a critical temperature T = T c σ ∼ 210 MeV . Below T c σ (at T ∼ 190 MeV ) the specific heat gets a peak like at crossover. We demonstrate that our EoS can be matched with that computed on the lattice for high temperatures provided the baryon resonance couplings with nucleon are partially suppressed. In this case the quark liquid would masquerade as the hadron one. The model is applied to the description of heavy ion collisions in a broad collision energy range. It might be especially helpful for studying phase diagram in the region near possible phase transitions.

Nuclear matter with Brown–Rho-scaled Fermi liquid interactions

We present a description of symmetric nuclear matter within the framework of Landau Fermi liquid theory. The low momentum nucleon–nucleon interaction V low - k is used to calculate the effective interaction between quasiparticles on the Fermi surface, from which we extract the quasiparticle effective mass, the nuclear compression modulus, the symmetry energy, and the anomalous orbital gyromagnetic ratio. The exchange of density, spin, and isospin collective excitations is included through the Babu–Brown induced interaction, and it is found that in the absence of three-body forces the self-consistent solution to the Babu–Brown equations is in poor agreement with the empirical values for the nuclear observables. This is improved by lowering the nucleon and meson masses according to Brown–Rho scaling, essentially by including a scalar tadpole contribution to the meson and nucleon masses, as well as by scaling g A . We suggest that modifying the masses of the exchanged mesons is equivalent to introducing a short-range three-body force, and the net result is that the Brown–Rho double decimation [G.E. Brown, M. Rho, Phys. Rep. 396 (2004) 1] is accomplished all at once.

Thermal dilepton production from droppingρbased on the vector manifestation

We study the pion electromagnetic form factor and the dilepton production rate in hot matter using the hidden local symmetry theory as an effective field theory for pions and $\ensuremath{\rho}$ mesons. In this framework, the chiral symmetry restoration is realized as the vector manifestation (VM) in which the massless vector meson becomes the chiral partner of the pion, giving a theoretical support to the dropping $\ensuremath{\rho}$ \`a la Brown-Rho scaling. In the VM, the vector dominance (VD) is strongly violated near the phase transition point associated with the dropping $\ensuremath{\rho}$. We show that the effect of the violation of the VD substantially suppresses the dilepton production rate compared with the one predicted by assuming the VD together with the dropping $\ensuremath{\rho}$.

Influence of In-Medium Properties of Vector Mesons on Dilepton Production in Heavy Ion Collisions at SIS Energies

Dileptons represent a unique probe for nuclear matter under extreme conditions reached in heavy ion collisions. They allow the study of meson properties, like mass and decay width, at various density and temperature regimes. Up to now, in the Tubingen model for dilepton production, modification of meson properties in nuclear medium has been accounted for by allowing a density dependence of the mass (Brown-Rho scaling) together with an ad hoc dependence of the meson decay widths on the same variable. We use the extended vector meson dominance (eVMD) model to extract meson properties in nuclear matter by computing the in-medium meson spectral functions. Dilepton spectra for C+C at 1.0 and 2.0 AGeV are calculated and compared with previous results.