The mean-field approximation based on effective interactions or density functionals plays a pivotal role in the description of finite quantum many-body systems that are too large to be treated by ab initio methods. Some examples are strongly interacting medium and heavy mass atomic nuclei and mesoscopic condensed matter systems. In this approach, the linear Schrodinger equation for the exact many-body wave function is mapped onto a non-linear density-dependent one-body potential problem. This approximation, not only provides computationally very simple solutions even for systems with many particles, but due to the non-linearity, it also allows for obtaining solutions that break essential symmetries of the system, often connected with phase transitions. In this way, additional correlations are subsumed in the system. However, the mean-field approach suffers from the drawback that the corresponding wave functions do not have sharp quantum numbers and, therefore, many results cannot be compared directly with experimental data. In this article, we discuss general group-theory techniques to restore the broken symmetries, and provide detailed expressions on the restoration of translational, rotational, spin, isospin, parity and gauge symmetries, where the latter corresponds to the restoration of the particle number. In order to avoid the numerical complexity of exact projection techniques, various approximation methods available in the literature are examined. Applications of the projection methods are presented for simple nuclear models, realistic calculations in relatively small configuration spaces, nuclear energy density functional theory, as well as in other mesoscopic systems. Further, unresolved problems in the application of the symmetry restoration methods to the energy density functional theories are highlighted in the present work.
A particle called η' meson is found to reduce its mass in less than 10^-22 second in high energy heavy ion collisions at RHIC. Apparently this is the fastest mass reduction ever observed. This indicates the restoration of a lost symmetry of strong interactions in a hot and dense, hadronic matter.
We shed light upon the eta' mass in nuclear matter in the context of partial restoration of chiral symmetry, pointing out that the U_{A}(1) anomaly effects causes the eta'-eta mass difference necessarily through the chiral symmetry breaking. As a consequence, it is expected that the eta' mass is reduced by order of 100 MeV in nuclear matter where partial restoration of chiral symmetry takes place. The discussion given here is based on Ref. [1].
Starting from the microscopic low-momentum nucleon-nucleon interaction V{low k}, we present the first systematic shell model study of magnetic moments and magnetic dipole transition strengths of the basic low-energy one-quadrupole phonon excitations in nearly-spherical nuclei. Studying in particular the even-even N=52 isotones from 92Zr to 100Cd, we find the predicted evolution of the predominantly proton-neutron non-symmetric state reveals a restoration of collective proton-neutron mixed-symmetry structure near mid-shell. This provides the first explanation for the existence of pronounced col…
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.
An alternative approach to symmetry restoration within Energy Density Functional, the Symmetry-Conserving EDF is discussed. In this approach, the energy is directly written in terms of the degrees of freedom encoded in the one-, two-... body density matrices of the state having good quantum numbers. The SC-EDF framework is illustrated within Projection After and Before variation applied to particle number restoration.
The effective restoration of the U_{A}(1) symmetry is revisited by implementing the functional renormalization group approach combining with the 2+1 flavor Polyakov-loop quark-meson model. A temperature-dependent 't Hooft term is taken to imitate the restoration of the U_{A}(1) symmetry. Order parameters, meson spectrum and mixing angles, especially the pressure and the entropy density of the system are calculated to explore the effects of different U_{A}(1) symmetry restoration patterns. We show then that the temperature for the restoration of the U_{A}(1) symmetry is much higher than that fo…
Symmetry restoration in the axially deformed proton-neutron quasiparticle random phase approximation for nuclear beta decay: The effect of angular-momentum projection
Sun Yat-sen University（中山大学）；University of North Carolina, Chapel Hill（北卡罗来纳大学教堂山分校）
Chiral symmetry restoration in a dense medium is to some extent a consequence of the nuclear pion cloud. These pions induce a mixing of the axial and vector current contributions in the axial and vector correlators. We discuss their influence on hadron masses and investigate the signal produced by the remaining contribution associated with chiral symmetry restoration. Using the quark-meson coupling model we find that the latter is associated with the reduction of hadron masses.
In the article "On chiral symmetry restoration at finite density in large-$N_c$ QCD" by Adhikari, Cohen, Ayyagari and Strother [Phys. Rev. C 83, 065201 (2011)] the description of dense nuclear matter by means of Skyrmions in hyperspherical unit cells is severely criticized. We point out that this criticism is based on invalid assumptions and therefore unwarranted.
We review the implementation of a q-deformed fermionic algebra in the Nambu--Jona-Lasinio model (NJL). The gap equations obtained from a deformed condensate as well as from the deformation of the NJL Hamiltonian are discussed. The effect of both temperature and deformation in the chiral symmetry restoration process as well as in the pion properties is studied.
The thermal and nonthermal pion production by sigma decay and its relation with chiral symmetry restoration in a hot and dense matter are investigated. The nonthermal decay into pions of sigma mesons which are popularly produced in chiral symmetric phase leads to a low-momentum pion enhancement as a possible signature of chiral phase transition at finite temperature and density.
Symmetry techniques based on group theory play a prominent role in the analysis of nuclear phenomena, and in particular in the understanding of observed regular patterns in nuclear spectra and selection rules for electromagnetic transitions. A variety of symmetry-based nuclear models have been developed in nuclear physics, providing efficient tools of choice to interpret nuclear spectroscopic data. This chapter provides a pedagogical introduction to the basic idea of symmetry-breaking mechanism and symmetry-restoration methods in modeling atomic nuclei.
We discuss medium modifications of the unitarized pion-pion interaction in the nuclear medium. We incorporate both the effects of chiral symmetry restoration and the influence of collective nuclear pionic modes, originating from the p-wave coupling of the pion to delta-hole configurations. We show how the resulting strong enhancement of the sigma-meson spectral function is related to large fluctuations of the condensate associated with the partial restoration of chiral symmetry.
The phase transition of chiral symmetry restoration in strange hadronic matter is studied in the chiral SU(3) quark mean field model. When the baryon density is larger than a critical density $ρ_c$, the minimal energy density of the system occurs at the point where the effective masses of nucleon, $Λ$ or $Ξ$ drop to zero. The physical quantities change discontinuously at this density and the system will be in the phase of chiral symmetry restoration. A rich phase structure of strange hadronic matter with different strangeness fraction $f_s$ is obtained.
We study chiral symmetry restoration and quark deconfinement beyond mean field approximation in a magnetized PNJL model. The feedback from mesons to quarks modifies the quark coupling constant and Polyakov potential. As a result, the separate critical temperatures for the two phase transitions at mean field level coincide and the magnetic catalysis becomes inverse magnetic catalysis, when the meson contribution is included.
Two recent experiments have demonstrated that the effective $ρ$-mass in nuclear medium, as extracted from the $^3He(γ, π^+ π^-)$ reaction, is substantially reduced. This has been advocated as an indication of partial restoration of chiral symmetry in nuclear matter. We show that even in the absence of chiral symmetry, effective mean field nuclear matter models can explain these findings quantitatively.
To accomodate recent RHIC data on eta prime multiplicity, we propose a minimal modification of the Witten-Veneziano relation at high temperature. This renders a significant drop of eta prime mass at high temperature signaling a restoration of the U(1)A, and the Goldstone character of etaprime.
We present a pfaffian formula for projection and symmetry restoration for wave functions of the general Bogoliubov form, including quasiparticle excited states and linear combinations of them. This solves a long-standing problem in calculating states of good symmetry, arising from the sign ambiguity of the commonly used determinant formula. A simple example is given of projecting good particle number and angular momentum from a Bogoliubov wave function in the Fock space of a single j-shell.
The temperature ($T$) dependence of the axion mass is predicted for $T$'s up to $\sim 2.3 \times$ the chiral restoration temperature of QCD. The axion is related to the $U_A(1)$ anomaly. The squared axion mass $m_a(T)^2$ is, modulo the presently undetermined scale of spontaneous breaking of Peccei-Quinn symmetry $f_a$ (squared), equal to QCD topological susceptibility $χ(T)$ for all $T$. We obtain $χ(T)$ by using quark condensates calculated in two effective Dyson-Schwinger models of nonperturbative QCD. They exhibit the correct chiral behavior, including the dynamical breaking of chiral symme…
We overview two interconnected topics: possible effective restoration of chiral symmetry in highly excited hadrons and possible existence of confined but chirally symmetric matter at low temperatures and high densities.
It has recently been suggested that the observed structure of parity doublets, seen in the spectrum of highly excited baryons, may be due to the effective restoration of chiral symmetry for these states. This chiral symmetry restoration high in the spectrum is consistent with the concept of quark-hadron duality. Moreover, if QCD dynamics implies the linear Regge trajectories for highly excited baryons, then the MacDowell symmetry requires the parity doubling to appear, which shows that linear Regge trajectories and chiral symmetry restoration are well compatible. On the contrary, in the low-en…
Recent topics on mesons in nuclei are discussed by especially emphasizing the role of the partial restoration of chiral symmetry in the nuclear medium. The spontaneously broken chiral symmetry in vacuum is considered to be incompletely restored in finite nuclear density systems with moderate reduction of the magnitude of the quark condensate. On the partial restoration of chiral symmetry, the wave function renormalization is important to be taken into account for the Nambu-Goldstone bosons. We also discuss the possible change of the meson properties in the nuclear medium and meson-nucleus syst…
This paper is devoted to the interplay between p-wave, s-wave pion-nucleon/ nucleus interaction and in-medium pion-pion interaction with special emphasis on the role of the nuclear pionic scalar density driving a large amount of chiral symmetry restoration. In particular we show that the $πNN$ coupling constant and the Goldberger-Treiman relation are preserved in the nuclear medium within certain conditions. We also discuss the related problem of the in-medium pion-pion strength function.
Recent developments on medium modifications of hadron properties in dense and hot hadronic matter are discussed. I will focus in particular on the behavior of spectral functions associated to collective scalar-isoscalar modes, kaons and vector mesons from ordinary nuclear matter to highly excited matter produced in relativistic heavy-ion collisions from SIS to SPS energies. Various theoretical approaches are presented in connection with the interpretation of experimental data. Special emphasis will be put on the role of chiral dynamics and chiral symmetry restoration. I will discuss in particu…
We study the spectrum of heavy-light mesons within a model with linear instantaneous confining potential. The single-quark Green function and spontaneous breaking of chiral symmetry are obtained from the Schwinger-Dyson (gap) equation. For the meson spectrum we derive a Bethe-Salpeter equation (BSE). We solve thiss equation numerically in the heavy-light limit and obtain effective restoration of chiral and $U(1)_A$ symmetries at large spins.
Physics of the low-lying and high-lying hadrons in the light flavor sector is reviewed. While the low-lying hadrons are strongly affected by the spontaneous breaking of chiral symmetry, in the high-lying hadrons the chiral symmetry is restored. A manifestation of the chiral symmetry restoration in excited hadrons is a persistence of the chiral multiplet structure in both baryon and meson spectra. Meson and baryon chiral multiplets are classified. A relation between the chiral symmetry restoration and the string picture of excited hadrons is discussed.
We explore the possibility to perform symmetry restoration with the variation after projection technique on a quantum computer followed by additional post-processing. The final goal is to develop configuration interaction techniques based on many-body trial states pre-optimized on a quantum computer. We show how the projection method used for symmetry restoration can prepare optimized states that could then be employed as initial states for quantum or hybrid quantum-classical algorithms. We use the quantum phase estimation and quantum Krylov approaches for the post-processing. The latter metho…
It is discussed how chiral symmetry restoration manifests itself through mixing of vector and axial-vector correlators. The vector correlator is directly accessible in relativistic heavy-ion collisions. Within models of the vector correlator its implications for low-mass dilepton spectra are reviewed.
Extensive data on strong interaction effects in pionic atoms are analyzed with a density-dependent isovector scattering amplitude suggested recently by Weise to result from a density dependence of the pion decay constant. Most of the so-called `missing s-wave repulsion' is removed when adopting this approach, thus indicating a partial chiral symmetry restoration in dense matter. The resulting potentials describe quite well also elastic scattering of 20 MeV pions on Ca. Further tests with elastic scattering are desirable.
Spectral enhancement near the 2m_π threshold in the I=J=0 channel in nuclei is shown to be a distinct signal of the partial restoration of chiral symmetry. The relevance of this phenomenon with the possible detection of 2π^{0} and 2γin hadron-nucleus and photo-nucleus reactions is discussed.
It is shown that in the large $N_c$ limit the light baryon spectrum exhibits the chiral restoration phase transition at high enough excitation energy. Such a phase transition is evidenced by the systematical parity doublets observed in the upper part of $N$ and $Δ$ spectra.
In-medium modification of the eta' mass is discussed in the context of partial restoration of chiral symmetry in nuclear medium. We emphasize that the U_A(1) anomaly effects causes the eta'-eta mass difference necessarily through the chiral symmetry breaking. As a consequence, the eta' mass is expected to be reduced by order of 100 MeV in nuclear matter where about 30% reduction of chiral symmetry takes place. The strong attraction relating to the eta' mass generation eventually implies that there should be also a strong attractive interaction in the scalar channel of the eta'-N two-body syste…
The in-medium mass of the $η^{\prime}$ meson is discussed in a context of partial restoration of chiral symmetry in nuclear medium. The $η^{\prime}$ mass is expected to be reduced by order of 100 MeV at the saturation density. The reduction is a consequence of the suppression of the anomaly effect on the $η^{\prime}$ mass induced by partial restoration of chiral symmetry. This strong attraction in $η^{\prime}$ nuclear systems does not accompany large absorption of $η^{\prime}$ into nuclear matter. This leads to the possibility of so narrow bound states of the $η^{\prime}$ meson in nuclei to be…
We discuss partial restoration of chiral symmetry in nuclear medium, which is realized as an effective reduction of the quark condensate in nuclear medium. We derive the in-medium Weinberg-Tomozawa relation. We also give a brief calculation of the in-medium quark condensate based on chiral perturbation theory. We conclude that the density dependence of the quark condensate beyond the linear density comes from the vertex correction of the chiral field insertion and nucleon correlations.
It is argued that an appearance of the near parity doublets in the upper part of the light baryon spectrum is an evidence for the chiral symmetry restoration in the regime where a typical momentum of quarks is around the chiral symmetry restoration scale. At high enough baryon excitation energy the nontrivial gap solution, which signals the chiral symmetry breaking regime, disappears and the chiral symmetry should be restored. Thus one observes a phase transition in the upper part of the light baryon spectrum. The average kinetic energy of the constituent quarks in this region is just around t…
Exploiting the recent lattice results for the infrared gluon propagator with light dynamical quarks, we solve the gap equation for the quark propagator. We thus model the chiral symmetry breaking mechanism with increasing number of flavours and study confinement (intimately tied with the analytic properties of QCD Schwinger functions) order parameters. We obtain, with this approach, clear signals of chiral symmetry restoration and deconfinement when the number of light quark flavors exceeds a critical value of $N_f^c \approx 8 \pm 1$, in agreement with the state-of-the-art direct lattice analy…
The chiral phase transition at high temperature and/or density is interpreted in terms of Georgi's vector limit. We discuss three cases as possible support for this scenario: Quark-number susceptibility, cool kaons in heavy-ion process and an instanton-molecule picture for chiral restoration. Both the notion of ``mended symmetry" and the Georgi vector limit are suggested to be relevant in nuclear physics of dense matter.
Based on the 1960 idea of Lipkin, the minimization of energy of a symmetry-restored mean-field state is equivalent to the minimization of a corrected energy of a symmetry-broken state with the Peierls-Yoccoz mass. It is interesting to note that the "unphysical" Peierls-Yoccoz mass, and not the true mass, appears in the Lipkin projected energy. The Peierls-Yoccoz mass can be easily calculated from the energy and overlap kernels, which allows for a systematic, albeit approximate, restoration of translational symmetry within the energy-density formalism. Analogous methods can also be implemented …
Partial restoration in nuclear matter of the chiral symmetry of QCD is discussed together with some of its possible signals. Estimates of corrections to the leading, linear dependence of the quark condensate are found to be small, implying a significant reduction of that condensate in matter. The importance of the pion cloud for the scalar quark density of a single nucleon indicates a close connection between chiral symmetry restoration and the attractive two-pion exchange interaction between nucleons. This force is sufficiently long-ranged that nucleons in nuclear matter will feel a significa…
We consider the parity doublet model for nucleonic and delta matter to investigate the structure of neutron stars. We show that it is possible to reconcile the multi-messenger astronomy constraints within a purely hadronic equation of state (EOS), which accounts for the self-consistent treatment of the chiral symmetry restoration in the baryonic sector. We demonstrate that the characteristics of the EOS required by the astrophysical constraints do not necessarily imply the existence of a hadron-quark phase transition in the stellar core.
The pion production by sigma decay and its relation with chiral symmetry restoration in a hot and dense matter are investigated in the framework of the Nambu-Jona-Lasinio model. The decay rate for the process sigma -> 2pion to the lowest order in a 1/N_c expansion is calculated as a function of temperature T and chemical potential mu. The thermal and nonthermal enhancements of pions generated by the decay before and after the freeze-out present only in the crossover region of the chiral symmetry transition. The strongest nonthermal enhancement is located in the vicinity of the endpoint of the …
We discuss here some aspects related to the symmetries of a quantum many-body problem when trying to treat it on a quantum computer. Several features related to symmetry conservation, symmetry breaking, and possible symmetry restoration are reviewed. After briefly discussing some of the standard symmetries relevant for many-particle systems, we discuss the advantage of encoding some symmetries directly in quantum ansätze, especially to reduce the quantum register size. It is, however, well-known that the use of symmetry-breaking states can also be a unique way to incorporate specific internal …
The sigma-meson, which is at best a broad resonance in the vacuum, may suffer a substantial red-shift and appear as a soft and narrow collective mode at finite temperature and baryon densities. The physics behind this softening phenomena and its relation to the partial restoration of chiral symmetry are discussed. Possible experiments in laboratory to test the idea are also summarized.
We propose a simple explanation for the connection between chiral symmetry restoration and deconfinement in QCD at high temperature. In the Higgs description of the QCD vacuum both spontaneous chiral symmetry breaking and effective gluon masses are generated by the condensate of a color octet quark-antiquark pair. The transition to the high temperature state proceeds by the melting of this condensate. Quarks and gluons become (approximately) massless at the same critical temperature. For instanton-dominated effective multiquark interactions and three light quarks with equal mass we find a firs…
Chiral symmetry restoration at nonzero temperature is studied in the framework of the O(4) linear sigma model and the U(N_f)_r x U(N_f)_l linear sigma model with N_f=2,3, and 4 quark flavors. We investigate the temperature dependence of the masses of the scalar and pseudoscalar mesons, and the non-strange, strange, and charm condensates within the Hartree approximation as derived from the Cornwall-Jackiw-Tomboulis formalism. We find that the masses of the non-strange and strange mesons at nonzero temperature depend sensitively on the particular symmetry of the model and the number of light qua…
Chiral symmetry imposes some constrains on hadron correlation functions in dense medium. These constraints imply a certain general structure of the two- and three-point correlation functions of chiral partners and lead to the effect of mixing arising from the interaction of the long-ranged nuclear pions with corresponding interpolating currents. It reflects the phenomena of partial restoration of chiral symmetry. We consider soft pion corrections for quark condensates and several possible pairs of chiral partners for both mesons and nucleons and analyse their mixing patterns. We explored both …
Isospin symmetry of atomic nuclei is explicitly broken by the charge-dependent interactions, primarily the Coulomb force. Within the nuclear density functional theory, isospin is also broken spontaneously. We propose a projection scheme rooted in a mean field theory, that allows the consistent treatment of isospin breaking in both ground and exited nuclear states. We demonstrate that this scheme is essentially free from spurious divergences plaguing particle-number and angular-momentum restoration approaches. Applications of the new technique include excited high-spin states in medium-mass N=Z…
We investigate the effect of chiral phase transition on the pentaquark $Θ^+$ mass and width at one-loop level of $NΘ^+K$ coupling at finite temperature and density. The behavior of the mass, especially the width in hadronic medium is dominated by the characteristics of chiral symmetry restoration at high temperature and high density. The mass and width shifts of positive-parity $Θ^+$ are much larger than that of negative-parity one, which may be helpful to determine the parity of $Θ^+$ in high energy nuclear collisions.
Energies of the giant Gamow-Teller and analog resonances - $E_{\rm G}$ and $E_{\rm A}$, are presented, calculated using the microscopic theory of finite Fermi system. The calculated differences $ΔE_{\rm G-A}=E_{\rm G}-E_{\rm A}$ go to zero in heavier nuclei indicating the restoration of Wigner SU(4)-symmetry. The calculated $ΔE_{\rm G-A}$ values are in good agreement with the experimental data. The average deviation is 0.30 MeV for the 33 considered nuclei for which experimental data is available. The $ΔE_{\rm G-A}$ values were calculated for heavy and superheavy nuclei up to the mass number $…
In this paper we propose a general framework for deriving the effective interactions for many-body systems. We show how the selfconsistent interaction can be constructed on the quantum level for both local and nonlocal potentials. In particular the relation between the selfconsistent and symmetry restoring interaction is derived. The difference between the two is related to the coupling of the perturbing field to the kinetic energy operator.
In this work, we calculate the matrix elements of $0νββ$-decay of $^{150}$Nd with the deformed pn-QRPA method. We adopted the approach introduced by Rodin, Faessler [Phys. Rev. C84, 014322 (2011)] and Simkovic {\it et. al.} [Phys. Rev. C87,045501(2013)] to restore the isospin symmetry by enforcing $M^{2ν}_F=0$. We found that with this restoration, the Fermi matrix elements are reduced by about 30\% while the more important Gamow-Teller matrix elements remains the same. The results of an enlarged model space is also presented, which changes the matrix elements by less than 10\%.
The contribution of nucleons to the quark condensate in nuclear matter includes a piece of first order in $m_π$, arising from the contribution of low-momentum virtual pions to the $πN$ sigma commutator. Chiral symmetry requires that no term of this order appears in the $NN$ interaction. The mass of a nucleon in matter thus cannot depend in any simple way on the quark condensate alone. More generally, pieces of the quark condensate that arise from low-momentum pions should not be associated with partial restoration of chiral symmetry.
The effect of altered hadron masses is studied for its effect with regard to final-state hadronic observables. It is shown that the final phase space densities of pions and kaons, which can be inferred experimentally, are sensitive to in-medium properties of the excited matter at earlier stages of the collision, but that the sensitivity is significantly moderated by interactions that change the effective numbers of pions and kaons during the latter part of the collision.
We analyze the effects of chiral symmetry restoration in hadronic matter, including the lowest-lying baryonic resonance $Δ$ based on the parity doublet model. We study the role of $Δ$ and its chiral partner on the equation of state (EoS) of dense matter under neutron star (NS) conditions of $β$-equilibrium and charge neutrality. We find that the softening of the EoS driven by the early onset of $Δ$ matter due to partial restoration of chiral symmetry allows accommodating the modern multi-messenger astrophysical constraints on the mass, radius, and tidal deformability. The softening above the s…
We study the nuclear modification of the scalar QCD susceptibility, calculated as the derivative of the quark condensate with respect to the quark mass. We show that it has two origins. One is the low lying nuclear excitations. At normal nuclear density this part is constrained by the nuclear incompressibility. The other part arises from the individual nucleon response and it is dominated by the pion cloud contribution. Numerically the first contribution dominates. The resulting increase in magnitude of the scalar susceptibility at normal density is such that it becomes close to the pseudoscal…
