Chiral Restoration Research Articles

Deconfinement and chiral transition in AdS/QCD wall models supplemented with a magnetic field

We discuss the phenomenon of (inverse) magnetic catalysis for both the deconfinement and chiral transition. We discriminate between the hard and soft wall model, which we suitably generalize to include a magnetic field. Our findings show a critical deconfinement temperature going down, in contrast with the chiral restoration temperature growing with increasing magnetic field. This is at odds with contemporary lattice data, so the quest for a holographic QCD model capable of capturing inverse magnetic catalysis in the chiral sector remains open.

Recent Progress in Understanding Deconfinement and Chiral Restoration Phase Transitions

Paradigme shift in gauge topology, from instantons to their constituents – instanton-dyons – has recently lead to very significant advances. Like instantons, they have fermionic zero modes, and their collectivization at sufficiently high density explains the chiral symmetry breaking. Unlike instantons, these objects have electric and magnetic charges. Their back reaction on the mean value of the Polyakov line (holonomy) allows to explain the deconfinement transition. The talk summarizes recent works on the dyon ensemble, done in the mean field approximation (MFA), and also by direct numerical statistical simulation. Introduction of non-trivial quark periodicity conditions leads to drastic changes in both deconfinement and chiral transitions. In particulaly, in the so called Z (Nc ) – QCD model the former gets much stronger, while the latter does not seem to occur at all.

QCD Phase Diagram in a Nonlocal SU(3) NJL Model with Lattice-Inspired Form Factors

We study the QCD phase diagram in the framework of a nonlocal three-flavor quark model. We determine the model parameters from vacuum meson phenomenology, considering lattice QCD-inspired nonlocal form factors. Then we analyze the features of the deconfinement and chiral restoration transitions for systems at nonzero temperature and chemical potential.

Chiral symmetry restoration for the large-Npion gas

We analyze chiral restoration within the $O(N+1)/O(N)$ Non-Linear Sigma Model for large $N$ as an effective theory for low-energy QCD at finite temperature $T$. The free energy is constructed diagramatically to ${\cal O}(TM^3)$ in the pion mass expansion, which allows to derive the quark condensate and the scalar susceptibility in the chiral limit. At this order, we do not have to deal with renormalization, neither from divergences from mass tadpoles nor from those of higher order loop contributions. Our results for the critical behaviour are consistent with expectations from lattice analysis and with previous works where the susceptibility is saturated by the thermal $f_0(500)$ pole.

Instanton-dyon ensembles with quarks with modified boundary conditions

"Exotic quarks" in the title refers to a modification of quark periodicity condition on the thermal circle by introduction of some phases -- known also as "flavor holonomies" -- different quark flavors. These phases provide a valuable tool, to be used for better understanding of deconfinement and chiral restoration phase transitions: by changing them one can dramatically modify both phase transitions. In the language of instanton constituents -- instanton-dyons or monopoles -- it has a very direct explanation: the interplay of flavor and color holonomies can switch topological zero modes between various dyon types. The model we will study in detail, the so called $Z_{N_c}$-symmetric QCD model with equal number of colors and flavors $N_c=N_f=2$ and special arrangement of flavor and color holonomies, ensure "most democratic" setting, in which each quark flavor and each dyon type are in one-to-one correspondence. The usual QCD has the opposite "most exclusive" arrangement: all quarks are antiperiodic and thus all zero modes fall on only one -- twisted or $L$ -- dyon type. As we show by ensemble simulation, deconfinement and chiral restoration phase transitions in these two models are dramatically different. In the usual QCD both are smooth crossovers: but in the case of $Z_2$-symmetric model deconfinement becomes strong first order transition, while chiral symmetry remains broken for all dyon densities studied. These results are in good correspondence with those from recent lattice simulations.

The eLSM at nonzero density

The extended Linear Sigma Model (eLSM) is an effective model of QCD which includes in the mesonic sector (pseudo)scalar and (axial-)vector quarkonia mesons as well as one dilaton/glueball field and in the baryonic sector the nucleon doublet and its chiral partner in the mirror assignment. The chiral partner of the pion turns out to be the resonance f0(1370), which is then predominantly a quarkonium state. As a consequence, f0(500) is predominately not a quarkonium state but a four-quark object and is at first not part of the model. Yet, f0(500) is important in the baryonic sector and affects nuclear matter saturation, the high- density behavior, and nucleon-nucleon scattering. In these proceedings, we show how to enlarge the two-flavour version of the eLSM in order to include the four-quark field f0(500) in a chirally invariant manner. We then discuss homogeneous and inhomogeneous chiral restoration in a dense medium.

Layers of deformed instantons in holographic baryonic matter

We discuss homogeneous baryonic matter in the decompactified limit of the Sakai-Sugimoto model, improving existing approximations based on flat-space instantons. We allow for an anisotropic deformation of the instantons in the holographic and spatial directions and for a density-dependent distribution of arbitrarily many instanton layers in the bulk. Within our approximation, the baryon onset turns out to be a second-order phase transition, at odds with nature, and there is no transition to quark matter at high densities, at odds with expectations from QCD. This changes when we impose certain constraints on the shape of single instantons, motivated by known features of holographic baryons in the vacuum. Then, a first-order baryon onset and chiral restoration at high density are possible, and at sufficiently large densities two instanton layers are formed dynamically. Our results are a further step towards describing realistic, strongly interacting matter over a large density regime within a single model, desirable for studies of compact stars.

Solitonic modulation and Lifshitz point in an external magnetic field within Nambu–Jona-Lasinio model

We study the inhomogeneous solitonic modulation of a chiral condensate within the effective Nambu--Jona-Lasinio model when a constant external magnetic field is present. The self-consistent Pauli-Villars regularization scheme is adopted to manipulate the ultraviolet divergence encountered in the thermodynamic quantities. In order to determine the chiral restoration lines efficiently, a new kind of Ginzburg-Landau expansion approach is proposed here. At zero temperature, we find that both the upper and lower boundaries of the solitonic modulation oscillate with the magnetic field in the $\mu$--$B$ phase diagram which is actually the de Hass-van Alphan (dHvA) oscillation. It is very interesting to find out how the tricritical Lifshitz point $(T_L,\mu_L)$ evolves with the magnetic field: There are also dHvA oscillations in the $T_L$--$B$ and $\mu_L$--$B$ curves, though the tricritical temperature $T_L$ increases monotonically with the magnetic field.

Mesons in strong magnetic fields: (I) General analyses

We study properties of neutral and charged mesons in strong magnetic fields |eB|≫ΛQCD2 with ΛQCD being the QCD renormalization scale. Assuming long-range interactions, we examine magnetic-field dependences of various quantities such as the constituent quark mass, chiral condensate, meson spectra, and meson wavefunctions by analyzing the Schwinger–Dyson and Bethe–Salpeter equations. Based on the density of states obtained from these analyses, we extend the hadron resonance gas (HRG) model to investigate thermodynamics at large B. As B increases the meson energy behaves as a slowly growing function of the meson's transverse momenta, and thus a large number of meson states is accommodated in the low energy domain; the density of states at low temperature is proportional to B2. This extended transverse phase space in the infrared regime significantly enhances the HRG pressure at finite temperature, so that the system reaches the percolation or chiral restoration regime at lower temperature compared to the case without a magnetic field; this simple picture would offer a gauge invariant and intuitive explanation of the inverse magnetic catalysis.

Pseudoscalar susceptibilities and quark condensates: chiral restoration and lattice screening masses

We derive the formal Ward identities relating pseudoscalar susceptibilities and quark condensates in three-flavor QCD, including consistently the $\eta$-$\eta'$ sector and the $U_A(1)$ anomaly. These identities are verified in the low-energy realization provided by ChPT, both in the standard $SU(3)$ framework for the octet case and combining the use of the $U(3)$ framework and the large-$N_c$ expansion of QCD to account properly for the nonet sector and anomalous contributions. The analysis is performed including finite temperature corrections as well as the calculation of $U(3)$ quark condensates and all pseudoscalar susceptibilities, which together with the full set of Ward identities, are new results of this work. Finally, the Ward identities are used to derive scaling relations for pseudoscalar masses, which explain the behavior with temperature of lattice screening masses near chiral symmetry restoration.

Large-Npion scattering at finite temperature: Thef0(500)and chiral restoration

We consider the $O(N+1)/O(N)$ Non-Linear Sigma Model for large $N$ as an effective theory for low-energy QCD at finite temperature $T$, in the chiral limit. At $T=0$ this formulation provides a good description of scattering data in the scalar channel and generates dynamically the $f_0(500)$ pole, the pole position lying within experimental determinations. Previous $T=0$ results with this model are updated using newer analysis of pion scattering data. We calculate the pion scattering amplitude at finite $T$ and show that it satisfies exactly thermal unitarity, which had been assumed but not formally proven in previous works. We discuss the main differences with the $T=0$ result and we show that one can define a proper renormalization scheme with $T=0$ counterterms such that the renormalized amplitude can be chosen to depend only on a few parameters. Next, we analyze the behaviour of the $f_0(500)$ pole at finite $T$, which is consistent with chiral symmetry restoration when the scalar susceptibility is saturated by the $f_0(500)$ state, in a second-order transition scenario and in accordance with lattice and theoretical analysis.

QCD phase transition at real chemical potential with canonical approach

We study the finite density phase transition in the lattice QCD at real chemical potential. We adopt canonical approach and the canonical partition function is constructed for Nf=2 QCD. After derivation of the canonical partition function we calculate observables like the pressure, the quark number density, its second cumulant and the chiral condensate as a function of the real chemical potential. We covered a wide range of temperature region starting from the confining low to the deconfining high temperature. We observe signals for the deconfinement and the chiral restoration phase transition at real chemical potential below Tc starting from the confining phase.

Large-$N$ Pion Scattering, Finite-temperature Effects and the Relationship of the $f_{0}(500)$ with Chiral Symmetry Restoration

In this work, we review how the mass and the width of the $f_{0}(500)$ pole behave in a regime where temperature is below the critical chiral transition value. This is attained by considering a large-$N$ $O(N + 1)/O(N)$ invariant Non-Linear Sigma Model (NLSM) such that we can study the dynamical generation of a $f_{0}(500)$ resonance. Introducing thermal effects via the imaginary time formalism allows us to study the behavior of the pole and relate it to chiral restoration.

Mean-field approximation for the chiral soliton in a chiral phase transition**Supported by National Natural Science Foundation of China (10905018, 11275082)

Using the mean-field approximation, we study the chiral soliton within the linear sigma model in a thermal vacuum. The chiral soliton equations with different boundary conditions are solved at finite temperatures and densities. The solitons are discussed before and after chiral restoration. We find that the system has soliton solutions even after chiral restoration, and that they are very different from those before chiral restoration, which indicates that the quarks are still bound after chiral restoration.

Phases ofSU(3)gauge theories with fundamental quarks via Dirac spectral density

We propose that, in SU(3) gauge theories with fundamental quarks, confinement can be inferred from spectral density of the Dirac operator. This stems from the proposition that its possible behaviors are exhausted by three distinct types (Fig.1). The monotonic cases are standard and entail confinement with valence chiral symmetry breaking (A) or the lack of both (C,C'). The bimodal (anomalous) option (B) was frequently regarded as an artifact (lattice or other) in previous studies, but we show for the first time that it persists in the continuum limit, and conclude that it informs of a non-confining phase with broken valence chiral symmetry. This generalization rests on the following. $(\alpha)$ We show that bimodality in $N_f$=0 theory past deconfinement temperature $T_c$ is stable with respect to removal of both infrared and ultraviolet cutoffs, indicating that anomalous phase is not an artifact. $(\beta)$ We demonstrate that transition to bimodality in $N_f$=0 is simultaneous with the loss of confinement: anomalous phase occurs for $T_c < T < T_{ch}$, where $T_{ch}$ is the valence chiral restoration temperature. $(\gamma)$ Evidence is presented for thermal anomalous phase in $N_f$=2+1 QCD at physical quark masses, whose onset too coincides with the conventional "crossover $T_c$''. We conclude that the anomalous regime $T_c < T < T_{ch}$ is very likely a feature of nature's strong interactions. $(\delta)$ Our past studies of zero-temperature $N_f$=12 theories revealed that bimodality also arises via purely light-quark effects. As a result, we expect to encounter anomalous phase on generic paths to valence chiral restoration. We predict its existence also for $N_f$ massless flavors ($T=0$) in the range $N_f^c < N_f < N_f^{ch}$, where $N_f^c$ could be quite low. Conventional arguments would associate $N_f^{ch}$ with the onset of conformal window.

From holography towards real-world nuclear matter

Quantum chromodynamics is notoriously difficult to solve at nonzero baryon density, and most models or effective theories of dense quark or nuclear matter are restricted to a particular density regime and/or a particular form of matter. Here we study dense (and mostly cold) matter within the holographic Sakai-Sugimoto model, aiming at a strong-coupling framework in the wide density range between nuclear saturation density and ultra-high quark matter densities. The model contains only three parameters, and we ask whether it fulfills two basic requirements of real-world cold and dense matter, a first-order onset of nuclear matter and a chiral phase transition at high density to quark matter. Such a model would be extremely useful for astrophysical applications because it would provide a single equation of state for all densities relevant in a compact star. Our calculations are based on two approximations for baryonic matter, firstly an instanton gas and secondly a homogeneous ansatz for the non-abelian gauge fields on the flavor branes of the model. While the instanton gas shows chiral restoration at high densities but an unrealistic second-order baryon onset, the homogeneous ansatz behaves exactly the other way around. Our study thus provides all ingredients that are necessary for a more realistic model and allows for systematic improvements of the applied approximations.

Meson phenomenology and phase transitions in nonlocal chiral quark models

We study the features of nonlocal chiral quark models that include wave function renormalization. Model parameters are determined from meson phenomenology, considering different nonlocal form factor shapes. In this context we analyze the characteristics of the deconfinement and chiral restoration transitions at finite temperature and chemical potential, introducing the couplings of fermions to the Polyakov loop for different Polyakov potentials. The results for various thermodynamical quantities are compared with data obtained from lattice QCD calculations.

Polyakov-loop suppression of colored states in a quark-meson-diquark plasma

A quark-meson-diquark plasma is considered within the PNJL model for dynamical chiral symmetry breaking and restoration in quark matter. Based on a generalized Beth-Uhlenbeck approach to mesons and diquarks we present the thermodynamics of this system including the Mott dissociation of mesons and diquarks at finite temperature. A striking result is the suppression of the diquark abundance below the chiral restoration temperature by the coupling to the Polyakov loop, because of their color degree of freedom. This is understood in close analogy to the suppression of quark distributions by the same mechanism. Mesons as color singlets are unaffected by the Polyakov-loop suppression. At temperatures above the chiral restoration mesons and diquarks are both suppressed due to the Mott effect, whereby the positive resonance contribution to the pressure is largely compensated by the negative scattering contribution in accordance with the Levinson theorem.

Kaon condensation in baryonic Fermi liquid at high density

We formulate kaon condensation in dense baryonic matter with anti-kaons fluctuating from the Fermi-liquid fixed point. This entails that in the Wilsonian RG approach, the decimation is effectuated in the baryonic sector to the Fermi surface while in the meson sector to the origin. In writing the kaon-baryon (KN) coupling, we will take a generalized hidden local symmetry Lagrangian for the meson sector endowed with a "mended symmetry" that has the unbroken symmetry limit at high density in which the Goldstone $\pi$, scalar $s$, and vectors $\rho$ (and $\omega$) and $a_1$ become massless. The vector mesons $\rho$ (and $\omega$) and $a_1$ can be identified as emergent (hidden) local gauge fields and the scalar $s$ as the dilaton field of the spontaneously broken scale invariance at chiral restoration. In matter-free space, when the vector mesons and the scalar meson -- whose masses are much greater than that of the pion -- are integrated out, then the resulting KN coupling Lagrangian consists of the leading chiral order ($O(p^1)$) Weinberg-Tomozawa term and the next chiral order ($O(p^2)$) $\Sigma_{KN}$ term. In addressing kaon condensation in dense nuclear matter in chiral perturbation theory (ChPT), one makes an expansion in the "small" Fermi momentum $k_F$. We argue that in the Wilsonian RG formalism with the Fermi-liquid fixed point, the expansion is on the contrary in $1/k_F$ with the "large" Fermi momentum $k_F$. The kaon-quasinucleon interaction resulting from integrating out the massive mesons consists of a "relevant" term from the scalar exchange (analog to the $\Sigma_{KN}$ term) and an "irrelevant" term from the vector-meson exchange (analog to the Weinberg-Tomozawa term). It is found that the critical density predicted by the latter approach, controlled by the relevant term, is three times less than that predicted by chiral perturbation theory.

On the magnetic catalysis and inverse catalysis of phase transitions in the linear sigma model

We consider the evolution of critical temperature both for the formation of a pion charged condensate as well as for the chiral transition, from the perspective of the linear sigma model, in the background of a magnetic field. We developed the discussion for the pion condensate in one loop approximation for the effective potential getting magnetic catalysis for high values of B, i.e a raising of the critical temperature with the magnetic field. For the analysis of the chiral restoration, we go beyond this approximation, by taking one loop thermo-magnetic corrections to the couplings as well as plasma screening effects for the boson's masses, expressed through the resumation of ring diagrams. Here we found the opposite behavior, i.e. inverse magnetic catalysis, i.e. a decreasing of the chiral critical temperature as function of the intensity of the magnetic field, which seems to be in agreement with recent results form the lattice community.