Search

Multiple lattice evidences support the existence of a confining but chirally symmetric regime of QCD above the chiral symmetry restoration crossover at Tch ~ 155 MeV. This regime is characterised by an approximate chiral spin symmetry of the partition function, which is a symmetry of the colour charge and the confining electric part of the QCD Lagrangian. It is traditionally believed that confinement should automatically induce spontaneous breaking of chiral symmetry, which would preclude the existence of a confining but chirally symmetric regime of QCD at high temperatures. We employ a well-known solvable quark model for QCD in 3+1 dimensions that is chirally symmetric and manifestly confining and argue that while confinement indeed induces dynamical breaking of chiral symmetry at T=0, a chiral restoration phase transition takes place at some critical temperature Tch. Above this temperature, the spectrum of the model consists of chirally symmetric hadrons with approximate chiral spin symmetry., Comment: 11 pages. Chiral symmetry restoration temperature has been calculated. Accepted to PLB

In this talk we overview main results indicating existence in QCD of three qualitatively different regimes connected by smooth crossovers upon heating: a hadron gas, a stringy fluid and a quark-gluon plasma. In the combined large N_c and chiral limit these regimes likely become distinct phases separated by phase transitions: a chiral restoration phase transition around T_{ch} ~ 130 MeV and a deconfinement phase transition around T_d ~ 300 MeV. It should be an important task to verify this issue on the lattice. We will introduce a chiral spin symmetry, which is a symmetry of the electric part of electrodynamics and of QCD with light quarks. It is realized approximately in QCD above the chiral restoration crossover and disappears in the QGP regime. The center symmetry of the pure glue action and the chiral spin symmetry of the electric part of the QCD Lagrangian with light quarks are complementary to distinguish the confining regime and its disappearance. We also address other lattice evidences for stringy fluid: hadron resonances extracted from the lattice correlators; breakdown of the thermal perturbation theory at T < ~ 600 MeV and fluctuations of conserved charges that point out the N_c scaling above T ~ 155 MeV., Comment: 11 pages. Overview talk given at Excited QCD 2024, Benasque, January 2024

Fluctuations of conserved charges calculated on the lattice which can be measured experimentally, are well reproduced by a hadron resonanse gas model at temperatures below T_{ch} ~ 155 MeV and radically deviate from the hadron resonance gas predictions above the chiral restoration crossover. This behavior is typically interpreted as an indication of deconfinement in the quark-gluon plasma regime. We present an argument that this interpretation may be too simple. The argument is based on the scaling of quantities with the number of colors: demonstration of deconfinement and QGP requires observable that is sensitive to N_c^2 gluons while the conserved charges are sensitive only to quarks and above T_{ch} scale as N_c^1. The latter scaling is consistent with the existence of an intermediate regime characterized by restored chiral symmetry and by approximate chiral spin symmetry which is a symmetry of confining interaction. In this regime the energy density, pressure and entropy density scale as N_c^1. In the large N_c limit this regime might become a distinct phase separated from the hadron gas and from QGP by phase transitions. A natural observable that associates with deconfinement and is directly sensitive to deconfined N_c^2-1 gluons is the Polyakov loop; in the N_c=3 world it remains very close to 0 at temperatures well above chiral crossover, reaches the value 0.5 around 3T_{ch} and the value close to 1 at temperatures ~1 GeV., Comment: 5 pages; final version accepted by EPJA

Lattice studies suggest that at zero baryon chemical potential and increasing temperature there are three characteristic regimes in QCD that are connected by smooth analytical crossovers: a hadron gas regime at T < T_ch ~ 155 MeV, an intermediate regime, called stringy fluid, at T_ch < T < ~ 3 T_ch, and a quark-gluon plasma regime at higher temperatures. These regimes have been interpreted to reflect different approximate symmetries and effective degrees of freedom. In the hadron gas the effective degrees of freedom are hadrons and the approximate chiral symmetry of QCD is spontaneously broken. The intermediate regime has been interpreted as lacking spontaneous chiral symmetry breaking along with the emergence of new approximate symmetry, chiral spin symmetry, that is not a symmetry of the Dirac Lagrangian, but is a symmetry of the confining part of the QCD Lagrangian. While the high temperature regime is the usual quark-gluon plasma which is often considered to reflect "deconfinement" in some way. This paper explores the behavior of these regimes of QCD as the number of colors in the theory, N_c, gets large. In the large N_c limit the theory is center-symmetric and notions of confinement and deconfinement are unambiguous. The energy density is O(N_c^0) in the meson gas, O(N_c^1) in the intermediate regime and O(N_c^2) in the quark-gluon plasma regime. In the large N_c limit these regimes may become distinct phases separated by first order phase transitions. The intermediate phase has the peculiar feature that glueballs should exist and have properties that are unchanged from what is seen in the vacuum (up to 1/N_c corrections), while the ordinary dilute gas of mesons with broken chiral symmetry disappears and approximate chiral spin symmetry should emerge., Comment: 10 pages. Final version accepted by EPJA

Above the chiral symmetry restoration crossover around T_{ch} ~ 155 MeV a new regime arises in QCD, a stringy fluid, which is characterized by an approximate chiral spin symmetry of the thermal partition function. This symmetry is not a symmetry of the Dirac Lagrangian and is a symmetry of the electric part of the QCD Lagrangian. In this regime the medium consists of the chirally symmetric and approximately chiral spin symmetric hadrons that are made of the chirally symmetric quarks connected into the color singlet compounds by a confining chromoelectric field. This regime is evidenced by the approximate chiral spin symmetry of the spatial and temporal correlators and by the breakdown of the thermal perturbation theory at the crossover between the partonic (the quark-gluon plasma) and stringy fluid regimes at ~ 3 T_{ch}. The chiral spin symmetry smoothly disappears above ~ 3T_{ch} which means that the chromoelectric confining interaction gets screened. A direct evidence that the stringy fluid medium consists of densely packed hadrons is the pion spectral function that shows a distinct pion state and its first radial excitation above T_{ch}. Another direct evidence of the hadron degrees of freedom in the stringy fluid is the bottomonium spectrum with the 1S,2S,3S and 1P,2P radial and orbital excitations that become broad with temperature. The hadrons between T_{ch} and ~ 3 T_{ch} in the stringy fluid interact strongly which makes the stringy fluid more a liquid rather than a gas. We discuss how this chiral spin symmetric regime extends into the finite chemical potentials domain and present a qualitative sketch of the QCD phase diagram., Comment: 39 pages. The review has been significantly extended. Accepted by Progress in Particle and Nuclear Physics

The temperature of the chiral restoration phase transition at 130 MeV as well as the temperature of the center symmetry ("deconfinement") phase transition in a pure glue theory at 300 MeV are two independent temperatures and their interplay determines a structure of different regimes of hot QCD. Given a chiral spin symmetry of the color charge and of the chromoelectric interaction we can conclude from observed symmetries of spatial and temporal correlators of N_F=2 QCD with domain wall Dirac operator at physical quark masses that above the chiral symmetry restoration crossover around T_pc but below rougly 3T_pc there should exist an intermediate regime (the stringy fluid) of hot QCD that is characterized by approximate chiral spin symmetry and where degrees of freedom are chirally symmetric quarks bound into color singlet objects by the chromoelectric field. Above this intermediate regime the color charge and the chromoelectric field are Debye screened and one observes a transition to QGP with magnetic confinement., Comment: 8 pages, A written version of a talk at Lattice 2021 conference

Above the pseudocritical temperature T_c of chiral symmetry restoration a chiral spin symmetry (a symmetry of the color charge and of electric confinement) emerges in QCD. This implies that QCD is in a confining mode and there are no free quarks. At the same time correlators of operators constrained by a conserved current behave as if quarks were free. This explains observed fluctuations of conserved charges and the absence of the rho-like structures seen via dileptons. An independent evidence that one is in a confining mode is very welcome. Here we suggest a new tool how to distinguish free quarks from a confining mode. If we put the system into a finite box, then if the quarks are free one necessarily obtains a remarkable diffractive pattern in the propagator of a conserved current. This pattern is clearly seen in a lattice calculation in a finite box and it vanishes in the infinite volume limit as well as in the continuum. In contrast, the full QCD calculations in a finite box show the absence of the diffractive pattern implying that the quarks are confined., Comment: 6 pages, 5 figures. A new figure has been added and discussion that clarifies the main point of the paper has been expanded. Accepted by EPJA

Above a pseudocritical temperature of chiral symmetry restoration T_c the energy and the pressure are very far from the quark-gluon-plasma limit (i.e. ideal gas of free quarks and gluons). At the same time very soon above T_c fluctuations of conserved charges behave as if quarks were free particles. Within the T_c - 3T_c interval a chiral spin symmetry emerges in QCD which is not consistent with free quarks and suggests that degrees of freedom are chirally symmetric quarks bound into the color-singlet objects by the chromoelectric field. Here we analyse temporal and spatial correlators in this interval and demonstrate that they indicate simultaneously the chiral spin symmetry as well as absence of the interquark interactions in channels constrained by a current conservation. The latter channels are responsible for both fluctuations of conserved charges and for dileptons. Assuming that a SU(2)_color subgroup of SU(3)_color is deconfined soon above T_c but confinement persits in SU(3)_color/SU(2)_color in the interval T_c - 3T_c we are able to reconcile all empirical facts listed above., Comment: 10 pages; Final version; accepted by PRD

The chiral magnetic effect (CME) is an exact statement that connects via the axial anomaly the electric current in a system consisting of interacting fermions and gauge field with chirality imbalance that is put into a strong external magnetic field. Experimental search of the magnetically induced current in QCD in heavy ion collisions above a pseudocritical temperature hints, though not yet conclusive, that the induced current is either small or vanishing. This would imply that the chirality imbalance in QCD above $T_c$ that could be generated via topological fluctuations is at most very small. Here we present the most general reason for absence (smallness) of the chirality imbalance in QCD above Tc. It was recently found on the lattice that QCD above Tc is approximately chiral spin (CS) symmetric with the symmetry breaking at the level of a few percent. The CS transformations mix the right- and left-handed components of quarks. Then an exact CS symmetry would require absence of any chirality imbalance. Consequently an approximate CS symmetry admits at most a very small chirality imbalance in QCD above Tc. Hence the absence or smallness of an magnetically induced current observed in heavy ion collisions could be considered as experimental evidence for emergence of the CS symmetry above Tc., Comment: 9 pp

Properties of QCD matter change significantly around the chiral crossover temperature, and the effects on $U(1)_A$ and topological susceptibilities, as well as the meson spectrum have been studied with much care. Baryons and the effect of parity doubling in this temperature range have been analyzed previously by various other groups employing different setups. Here we construct suitable operators to investigate chiral and axial $U(1)_A$ symmetries in the baryon spectrum. Measurements for different volumes and quark-masses are done with two flavors of chirally symmetric domain-wall fermions at temperatures above the critical one. The possibility of emergent $SU(4)$ and $SU(2)_{CS}$ symmetries is discussed., Comment: 7 pages, 4 figures, talk presented at the 37th International Symposium on Lattice Field Theory (Lattice 2019), 16-22 June 2019, Wuhan, China. Updated numerical data in Fig. 1 and Fig. 2

In his recent Comments E. Shuryak reiterates old, unfortunately misleading arguments in favor of deconfined Quark-Gluon Plasma (QGP) immediately above the chiral restoration pseudocritical temperature. In a Comment devoted to our view of QCD at high temperatures he does not address and even mention the essence of our arguments. In recent years a new "hidden" symmetry in QCD was discovered. It is a symmetry of the electric sector of QCD, that is higher than the chiral symmetry of the QCD Lagrangian as the whole. This symmetry was clearly observed above T_c in spatial correlators and very recently also in time correlators. The latter correlators are directly related to observable spectral density. Then in a model-independent way we conclude that degrees of freedom in QCD above T_c, but below roughly 3T_c, are chirally symmetric quarks bound by the chromoelectric field into color-singlet compounds without the chromomagnetic effects. This regime of QCD has been referred to as a Stringy Fluid since such objects are very reminiscent of strings.At higher temperatures there is a very smooth transition to the partonic degrees of freedom, i.e. to the QGP regime. Here we will address some of the points made by Shuryak., Comment: 4 pp

Recently, via calculation of spatial correlators of $J=0,1$ isovector operators using a chirally symmetric Dirac operator within $N_F=2$ QCD, it has been found that QCD at temperatures $T_c - 3 T_c$ is approximately $SU(2)_{CS}$ and $SU(4)$ symmetric. The latter symmetry suggests that the physical degrees of freedom are chirally symmetric quarks bound by the chromoelectric field into color singlet objects without chromomagnetic effects. This regime of QCD has been referred to as a Stringy Fluid. Here we calculate correlators for propagation in time direction at a temperature slightly above $T_c$ and find the same approximate symmetries. This means that the meson spectral function is chiral-spin and $SU(4)$ symmetric., Comment: 10 pages, 3 figures. Version published by PLB

While the QCD Lagrangian as the whole is only chirally symmetric, its electric part has larger chiral-spin SU(2)_{CS} and SU(2N_F) symmetries. This allows separation of the electric and magnetic interactions in a given reference frame. Artificial truncation of the near-zero modes of the Dirac operator results in the emergence of the SU(2)_{CS} and SU(2N_F) symmetries in hadron spectrum. This implies that while the confining electric interaction is distributed among all modes of the Dirac operator, the magnetic interaction is located at least predominantly in the near-zero modes. Given this observation one could anticipate that above the pseudocritical temperature, where the near-zero modes of the Dirac operator are suppressed, QCD is SU(2)_{CS} and SU(2N_F) symmetric, which means absence of deconfinement in this regime. Solution of the N_F=2 QCD on the lattice with a chirally symmetric Dirac operator reveals that indeed in the interval Tc - 3Tc QCD is approximately SU(2)_{CS} and SU(2N_F) symmetric which implies that degrees of freedom are chirally symmetric quarks bound by the chromoelectric field into color-singlet objects without the chromomagnetic effects. This regime is referred to as a Stringy Fluid. At larger temperatures this emergent symmetry smoothly disappears and QCD approaches the Quark-Gluon Plasma regime with quasifree quarks. The Hadron Gas, the Stringy Fluid and the Quark-Gluon Plasma differ by symmetries, degrees of freedom and properties., Comment: 15 pages. Will appear in IJMPA. Discussion of confinement - deconfinement has been expanded and new figures have been added

Based on a complete set of $J = 0$ and $J=1$ spatial isovector correlation functions calculated with $N_F = 2$ domain wall fermions we identify an intermediate temperature regime of $T \sim 220 - 500$ MeV ($1.2T_c$--$2.8T_c$), where chiral symmetry is restored but the correlators are not yet compatible with a simple free quark behavior. More specifically, in the temperature range $T \sim 220 - 500$ MeV we identify a multiplet structure of spatial correlators that suggests emergent $SU(2)_{CS}$ and $SU(4)$ symmetries, which are not symmetries of the free Dirac action. The symmetry breaking effects in this temperature range are less than 5%. Our results indicate that at these temperatures the chromo-magnetic interaction is suppressed and the elementary degrees of freedom are chirally symmetric quarks bound into color-singlet objects by the chromo-electric component of the gluon field. At temperatures between 500 and 660 MeV the emergent $SU(2)_{CS}$ and $SU(4)$ symmetries disappear and one observes a smooth transition to the regime above $T \sim 1$ GeV where only chiral symmetries survive, which are finally compatible with quasi-free quarks., Comment: A new figure and discussion added. Accepted for publication in PRD

In a local gauge-invariant theory with massless Dirac fermions a symmetry of the Lorentz-invariant fermion charge is larger than a symmetry of the Lagrangian as a whole. While the Dirac Lagrangian exhibits only a chiral symmetry, the fermion charge operator is invariant under a larger symmetry group, SU(2N_F), that includes chiral transformations as well as SU(2)_{CS} chiralspin transformations that mix the right- and left-handed components of fermions. Consequently a symmetry of the electric interaction, that is driven by the charge density, is larger than a symmetry of the magnetic interaction and of the kinetic term. This allows to separate in some situations electric and magnetic contributions. In particutar, in QCD the chromo-magnetic interaction contributes only to the near-zero modes of the Dirac operator, while confining chromo-electric interaction contributes to all modes. At high temperatures, above the chiral restoration crossover, QCD exhibits approximate SU(2)_{CS} and SU(2N_F) symmetries that are incompatible with free deconfined quarks. Consequently elementary objects in QCD in this regime are quarks with a definite chirality bound by the chromo-electric field, without the chromo-magnetic effects. In this regime QCD can be described as a stringy fluid., Comment: 11 pages; Talk given at 7th International Conference on New Frontiers in Physics (ICNFP 2018), Crete, 4-12 July 2018; Symposium "Quantum Chromodynamics and its Symmetries", Oberwoelz, Austria, 9-15 September 2018; final version; to appear in Universe

We report on the progress of understanding spatial correlation functions in high temperature QCD. We study isovector meson operators in $N_f=2$ QCD using domain-wall fermions on lattices of $N_s=32$ and different quark masses. It has previously been found that at $\sim 2T_c$ these observables are not only chirally symmetric but in addition approximately $SU(2)_{CS}$ and $SU(4)$ symmetric. In this study we increase the temperature up to $5T_c$ and can identify convergence towards an asymptotically free scenario at very high temperatures., Comment: 7 pages, 5 figures, version to appear in proceedings of the LATTICE2018 conference

The chirally symmetric baryon parity-doublet model can be used as an effective description of the baryon-like objects in the chirally symmetric phase of QCD. Recently it has been found that above the critical temperature higher chiralspin symmetries emerge in QCD. It is demonstrated here that the baryon parity-doublet Lagrangian is manifestly chiralspin-invariant. We construct nucleon interpolators with fixed chiralspin transformation properties that can be used in lattice studies at high T., Comment: To appear in PRD

It has been found very recently on the lattice that at high temperature at vanishing chemical potential QCD is increasingly SU(2)_CS and SU(2N_F) symmetric. We demonstrate that the chemical potential term in the QCD Lagrangian has precisely the same symmetry. Consequently the QCD matter beyond the chiral restoration line on the T - \mu plane is at least approximately SU(2)_CS and SU(2N_F) symmetric., Comment: 3 pages, final version to appear in EPJA

In this talk we first overview lattice results that have led to the observation of new SU(2)_{CS} and SU(2N_F) symmetries upon artificial truncation of the near-zero modes of the Dirac operator at zero temperature and at high temperature without any truncation. These symmetries are larger than the chiral symmetry of the QCD Lagrangian and contain chiral symmetries SU(N_F)_L \times SU(N_F)_R and U(1)_A as subgroups. In addition to the standard chiral transformations the SU(2)_{CS} and SU(2N_F) transformations mix the right- and left-handed components of the quark fields. It is a symmetry of the confining chromo-electric interaction while the chromo-magnetic interaction manifestly breaks it. Emergence of these symmetries upon truncation of the near-zero modes of the Dirac operator at T=0 means that all effects of the chromo-magnetic interaction are located exclusively in the near-zero modes, while confining chromo-electric interaction is distributed among all modes. Appearance of these symmetries at high T, where the temperature suppresses the near-zero modes, has radical implications because these symmetries are incompatible with the asymptotically free deconfined quarks at increasing temperature. The elementary objects in the high-temperature phase of QCD should be quarks bound by the pure chromo-electric field that is not accompanied by the chromo-magnetic effects., Comment: 11 pages, talk presented at 6th International Conference on New Frontiers in Physics, 18.08 - 26.08, 2017, Crete

The near-zero modes of the Dirac operator are connected to spontaneous breaking of chiral symmetry in QCD (SBCS) via the Banks-Casher relation. At the same time the distribution of the near-zero modes is well described by the Random Matrix Theory (RMT) with the Gaussian Unitary Ensemble (GUE). Then it has become a standard lore that a randomness, as observed through distributions of the near-zero modes of the Dirac operator, is a consequence of SBCS. The higher-lying modes of the Dirac operator are not affected by SBCS and are sensitive to confinement physics and related $SU(2)_{CS}$ and $SU(2N_F)$ symmetries. We study the distribution of the near-zero and higher-lying eigenmodes of the overlap Dirac operator within $N_F=2$ dynamical simulations. We find that both the distributions of the near-zero and higher-lying modes are perfectly described by GUE of RMT. This means that randomness, while consistent with SBCS, is not a consequence of SBCS and is related to some more general property of QCD in confinement regime.

It was established that distribution of the near-zero modes of the Dirac operator is consistent with the Chiral Random Matrix Theory (CRMT) and can be considered as a consequence of spontaneous breaking of chiral symmetry (SBCS) in QCD. The higher-lying modes of the Dirac operator carry information about confinement physics and are not affected by SBCS. We study distributions of the near-zero and higher-lying modes of the overlap Dirac operator within $N_F = 2$ dynamical simulations. We find that distributions of both near-zero and higher-lying modes are the same and follow the Gaussian Unitary Ensemble of Random Matrix Theory. This means that randomness, while consistent with SBCS, is not a consequence of SBCS and is related to some more general property of QCD in confinement regime., Comment: Contribution to Lattice 2017 proceedings, 18-24 June, 2017, Granada, Spain

We study spatial isovector meson correlators in $N_f=2$ QCD with dynamical domain-wall fermions on $32^3\times 8$ lattices at temperatures $T=220-380$ MeV. We measure the correlators of spin-one ($J=1$) operators including vector, axial-vector, tensor and axial-tensor. Restoration of chiral $U(1)_A$ and $SU(2)_L \times SU(2)_R$ symmetries of QCD implies degeneracies in vector--axial-vector ($SU(2)_L \times SU(2)_R$) and tensor--axial-tensor ($U(1)_A$) pairs, which are indeed observed at temperatures above $T_c$. Moreover, we observe an approximate degeneracy of all $J=1$ correlators with increasing temperature. This approximate degeneracy suggests emergent $SU(2)_{CS}$ and $SU(4)$ symmeries at high temperatures, that mix left- and right-handed quarks., Comment: This revised version contains all corrections that will appear in the forthcoming erratum

If above a critical temperature not only the SU(N_F)_L \times SU(N_F)_R chiral symmetry of QCD but also the U(1)_A symmetry is restored, then the actual symmetry of the QCD correlation functions and observables is SU(2N_F). Such a symmetry prohibits existence of deconfined quarks and gluons. Hence QCD at high temperature is also in the confining regime and elementary objects are SU(2N_F) symmetric "hadrons" with not yet known properties., Comment: 3 pp, presented at "Critical Point and Onset of Deconfinement", Wroclaw, May, 30 - June,4, 2016

The classical part of the QCD partition function (the integrand) has, ignoring irrelevant exact zero modes of the Dirac operator, a local SU(2N_F) \supset SU(N_F)_L \times SU(N_F)_R \times U(1)_A symmetry which is absent at the Lagrangian level. This symmetry is broken anomalously and spontaneously. Effects of spontaneous breaking of chiral symmetry are contained in the near-zero modes of the Dirac operator. If physics of anomaly is also encoded in the same near-zero modes, then their truncation on the lattice should recover a hidden classical SU(2N_F) symmetry in correlators and spectra. This naturally explains observation on the lattice of a large degeneracy of hadrons, that is higher than the SU(N_F)_L \times SU(N_F)_R \times U(1)_A chiral symmetry, upon elimination by hands of the lowest-lying modes of the Dirac operator. We also discuss an implication of this symmetry for the high temperature QCD., Comment: 9 pp. Invited talk presented at (i) "Continuous advances in QCD", May 12 - May 15, 2016, University of Minnesota, USA (ii) 5th Int. Conf. on New Frontiers in Physics, 6-14 July, 2016, Crete, Greece. To be published in (ii) conference proceedings

We identify the chiral and angular momentum content of the leading quark-antiquark Fock component for the $\rho(770)$ and $\rho(1450)$ mesons using a two-flavor lattice simulation with dynamical Overlap Dirac fermions. We extract this information from the overlap factors of two interpolating fields with different chiral structure and from the unitary transformation between chiral and angular momentum basis. For the chiral content of the mesons we find that the $\rho(770)$ slightly favors the $(1,0)\oplus(0,1)$ chiral representation and the $\rho(1450)$ slightly favors the $(1/2,1/2)_b$ chiral representation. In the angular momentum basis the $\rho(770)$ is then a $^3S_1$ state, in accordance with the quark model. The $\rho(1450)$ is a $^3D_1$ state, showing that the quark model wrongly assumes the $\rho(1450)$ to be a radial excitation of the $\rho(770)$.

At a critical temperature QCD in the chiral limit undergoes a chiral restoration phase transition. Above the phase transition the quark condensate vanishes. The Banks-Casher relation connects the quark condensate to a density of the near-zero modes of the Dirac operator. In the Nambu-Goldstone mode the quasi-zero modes condense around zero, \lambda \rightarrow 0, and provide a nonvanishing quark condensate. The chiral restoration phase transition implies that above the critical temperature there is no any longer a condensation of the Dirac modes around zero. If a U(1)_A symmetry is also restored and a gap opens in the Dirac spectrum then the Euclidean correlation functions are SU(2N_f) \supset SU(N_f)_L \times SU(N_f)_R \times U(1)_A- symmetric. This symmetry implies that a free (deconfined) propagation of quarks in Minkowski space-time that perturbatively interact with unconfined gluons is impossible. This means that QCD above the critical temperature is not of a quark-gluon plasma origin and has a more complicated structure., Comment: 5 pp Some clarifications and an appendix have been added

Recently a global SU(4) \supset SU(2)_L \times SU(2)_R \times U(1)_A symmetry of the confining Coulombic part of the QCD Hamiltonian has been discovered with N_F=2. This global symmetry includes both independent rotations of the left- and right-handed quarks in the isospin space as well as the chiralspin rotations that mix the left- and right-handed components of the quark fields. It has been suggested by lattice simulations, however, that a symmetry of mesons in the light quark sector upon the quasi-zero mode truncation from the quark propagators is actually higher than SU(4), because the states from a singlet and a 15-plet irreducible representations of SU(4) are also degenerate. Here we demonstrate that classically QCD, ignoring irrelevant exact zero mode contributions, has a SU(2N_F) symmetry. If effects of dynamical chiral symmetry breaking and of anomaly are encoded in the same near-zero modes, then truncation of these modes should restore a classical SU(2N_F) symmetry. Then we show in a Lorentz- and gauge-invariant manner emergence of a bilocal SU(4) \times SU(4) symmetry in mesons that contains a global SU(4) as a subgroup upon truncation of the quasi-zero modes. We also demonstrate that the confining Coulombic part of the QCD Hamiltonian has this bilocal symmetry. It explains naturally a degeneracy of different irreducible representations of SU(4) observed in lattice simulations., Comment: 7 pp. Further clarifications and references have been added

Recent lattice results on Dirac low-mode removed $J=1,2$ mesons and $J=\frac{1}{2}$ baryons reveal the appearance of a new $SU(4)$ symmetry of confinement. Here the degeneracy of all $J=2$ iso-vector states within an irreducible representation of $SU(4)$ after low-mode removal is demonstrated. The $SU(4)$ symmetry contains $SU(2)_L \times SU(2)_R$ and $U(1)_A$ symmetries as subgroups and mixes all compontents $u_L,u_R,d_L,d_R$ of the two-flavor Dirac field. It implies the vanishing of the interaction of quarks with the color-magnetic field, which is shown by using the QCD Hamiltonian in Coulomb gauge., Comment: 6 pages, 6 figures. Talk presented at the 33rd International Symposium on Lattice Field Theory (Lattice 2015), 14-18 July 2015, Kobe International Conference Center, Kobe, Japan

At high temperature measurements of the Polyakov loop suggest a deconfinement transition to the (strongly interacting) quark-gluon plasma. At the same time at the infinitely large temperature the four-dimensional QCD is reduced to the three-dimensional QCD that is confining. The Polyakov loop and related Z_3 symmetry are strict order parameters only for infinitely heavy quarks. In such a situation the SU(2)_CS and SU(4) symmetries of confinement in the light quark sector could be helpful to distinguish between the confining and deconfining phase in a regime where SU(2)_L \times SU(2)_R and U(1)_A symmetries are manifest. In order to reveal a presence or absence of these symmetries one needs to measure and compare correlation functions related by these symmetry transformations., Comment: 2 pp

Recently a large degeneracy of $J=1$ mesons, that is larger than the $SU(2)_L \times SU(2)_R \times U(1)_A$ symmetry of the QCD Lagrangian, has been discovered upon truncation of the near-zero modes from the valence quark propagators. It has been found that this degeneracy represents the $SU(4)$ group that includes the chiral rotations as well as the mixing of left- and right-handed quarks. This symmetry group turns out to be a symmetry of confinement in QCD. Consequently, one expects that the same symmetry should persist upon the near-zero mode removal in other hadron sectors as well. It has been shown that indeed the $J=2$ mesons follow the same symmetry pattern upon the low-lying mode elimination. Here we demonstrate the $SU(4)$ symmetry of baryons once the near-zero modes are removed from the quark propagators. We also show a degeneracy of states belonging to different irreducible representations of $SU(4)$. This implies a larger symmetry, that includes $SU(4)$ as a subgroup., Comment: Updated version matching published version in Phys.Rev.D; Erratum at the end of the paper regarding the larger symmetry than SU(4)

Recently, a new symmetry of mesons has been found upon truncation of the quasi-zero modes of the Overlap Dirac operator in lattice simulations. Namely, the $\rho,\rho',\omega,\omega',a_1, b_1,h_1$ and possibly $f_1$ $J=1$ mesons get degenerate after removal of the quasi-zero modes. This emergent symmetry has been established to be $SU(4)\supset SU(2)_L \times SU(2)_R \times U(1)_A$. It is higher than the symmetry of the QCD Lagrangian and provides not only a mixing of quarks of given chirality in the isospin space, but also the mixing of left-handed and right-handed components. Here we study, with the Overlap Dirac operator, the isovector $J=2$ mesons upon the quasi-zero mode reduction and observe a similar degeneracy. This result further supports the $SU(4)$ symmetry in mesons of given spin $J \geq 1$., Comment: 6 figures

In recent lattice calculations it has been discovered that mesons upon truncation of the quasi-zero modes of the Dirac operator obey a symmetry larger than the $SU(2)_L \times SU(2)_R\times U(1)_A$ symmetry of the QCD Lagrangian. This symmetry has been suggested to be $SU(4) \supset SU(2)_L \times SU(2)_R\times U(1)_A$ that mixes not only the u- and d-quarks of a given chirality, but also the left- and right-handed components. Here it is demonstrated that bilinear $\overline{q}q$ interpolating fields of a given spin $J \geq 1$ transform into each other according to irreducible representations of $SU(4)$ or, in general, $SU(2N_F)$. This fact together with the coincidence of the correlation functions establishes $SU(4)$ as a symmetry of the $J \geq 1$ mesons upon quasi-zero mode reduction. It is shown that this symmetry is a symmetry of the confining instantaneous charge-charge interaction in QCD. Different subgroups of $SU(4)$ as well as the $SU(4)$ algebra are explored., Comment: Implication chapter added, figure added; Version accepted for publication in Phys. Rev. D

We study the isoscalar and isovector $J=0,1$ mesons with the overlap operator within two flavour lattice QCD. After subtraction of the lowest-lying Dirac eigenmodes from the valence quark propagator all disconnected contributions vanish and all possible point-to-point $J=0$ correlators become identical, signaling a simultaneous restoration of both $SU(2)_L \times SU(2)_R$ and $U(1)_A$ symmetries. The ground states of the $\pi,\sigma,a_0,\eta$ mesons do not survive this truncation. All possible $J=1$ states have a very clean exponential decay and become degenerate, demonstrating a $SU(4)$ symmetry of a dynamical QCD-like string., Comment: 7 pages, 6 figures, Proceedings from the 32nd International Symposium on Lattice Field Theory

In a dynamical lattice simulation with the overlap Dirac operator and $N_f=2$ mass degenerate quarks we study all possible $J=0$ and $J=1$ correlators upon exclusion of the low lying "quasi-zero" modes from the valence quark propagators. After subtraction of a small amount of such Dirac eigenmodes all disconnected contributions vanish and all possible point-to-point $J=0$ correlators with different quantum numbers become identical, signaling a restoration of the $SU(2)_L \times SU(2)_R \times U(1)_A$. The original ground state of the $\pi$ meson does not survive this truncation, however. In contrast, in the $I=0$ and $I=1$ channels for the $J=1$ correlators the ground states have a very clean exponential decay. All possible chiral multiplets for the $J=1$ mesons become degenerate, indicating a restoration of an $SU(4)$ symmetry of the dynamical QCD-like string., Comment: 10 pages, 13 figures

A large degeneracy of mesons of a given spin has recently been discovered upon reduction of the quasi-zero modes of the Dirac operator in a dynamical lattice simulation. Here a symmetry group that is responsible for this degeneracy is established, which is SU(4) \supset SU(2)_L \times SU(2)_R \times U(1)_A \times C_i. It is argued that this symmetry group is a symmetry of the dynamical QCD string. Implications of this picture for a genesis of light hadron spectra are discussed., Comment: 4 pp, to appear in EPJA

Mesons constructed from the quark propagators without the lowest-lying eigenmodes of the Dirac operator reveal not only restored SU(2)_L * SU(2)_R chiral and U(1)_A symmetries, but actually a higher symmetry. All possible chiral and U(1)_A multiplets for the states of the same spin are degenerate, i.e., the energy of the observed quantum levels does not depend on the spin orientation of quarks in the system and their parities. The quark-spin independence of the energy levels implies absence of the magnetic interactions in the system. The ultrarelativistic quark-antiquark system with only the color-electric interactions can be interpreted (or defined) as a dynamical QCD string., Comment: Invited talk at Excited QCD 2014, 2-8 February,2014, Bjelasnica Mountain, Sarajevo,Bosnien

Using the chirally invariant overlap Dirac operator we remove its lowest-lying quasizero modes from the valence quark propagators and study evolution of isovector mesons with J=1. At the truncation level about 50 MeV SU(2)_L \times SU(2)_R and U(1)_A symmetries get restored. However, we observe a degeneracy not only within the chiral and U(1)_A multiplets, but also a degeneracy of all possible chiral multiplets, i.e., the observed quantum levels have a symmetry larger than U(2)_L \times U(2)_R and their energy does not depend on the spin orientation of quarks and their parities. We offer a possible interpretation of these energy levels as the quantum levels of the dynamical QCD string. The structure of the radial J=1 spectrum is compatible with E =(n_r +1)\hbar\omega with \hbar\omega = 900 \pm 70 MeV., Comment: 5 pp. The title has been changed, the paper has been restructured and a few references have been added. To appear in PRD

A quantitative study of the effects of dynamical chiral symmetry breaking on the mass generation of the low-lying hadrons in a dynamical lattice QCD simulation is presented. The evolution of light hadron masses upon increasing the number of low-lying Dirac operator eigenmodes in the quark propagators is confronted with the hadron masses obtained upon removal of such low-lying eigenmodes from standard full quark propagators. The low-lying chiral symmetry breaking modes provide roughly two third of the nucleon and $\rho$ masses, while the $a_1$ mass is affected to a much smaller degree., Comment: 4 pages, 4 figures

A nonvanishing spectral density of the low-lying eigenmodes of the Dirac operator naturally is a signal for dynamical chiral symmetry breaking (D$\chi$SB) via the Banks-Casher relation. The low-lying eigenmodes alone saturate the pseudoscalar channel and the corresponding propagator successfully reproduces the pion mass. In this paper we investigate the effects on the mass generation of hadrons other than pions. The evolution of these masses upon inclusion of an increasing number of the low-lying eigenmodes is confronted with the hadron mass spectrum upon removal of such eigenmodes., Comment: 5 pages, 3 figures. Proceedings of the 31st International Symposium on Lattice Field Theory, July 29 - August 3, 2013, Mainz, Germany

In previous studies we have shown that hadrons, except for a pion, survive the removal of the lowest lying Dirac eigenmodes from the valence quark propagators. The low-modes are tied to the dynamical breaking of chiral symmetry and we found chiral symmetry to be restored by means of matching masses of chiral partners, like, e.g., the vector and axial vector currents. Here we investigate the influence of removing the lowest part of the Dirac spectrum on the locality of the Dirac operator. Moreover, we analyze the influence of low-mode truncation on the quark momenta and thereupon on the hadron spectrum and, finally, introduce a reweighting scheme to extend the truncation to the sea quark sector., Comment: 7pages, 4 figures. Proceedings of the 31st International Symposium on Lattice Field Theory, July 29 - August 3, 2013, Mainz, Germany

The effect of transverse gluons on the chiral symmetry patterns of excited mesons is studied in a Coulomb gauge QCD model. The linear rising static quark-antiquark potential and the transverse gluon propagator known from lattice studies are input into the model. The non-perturbative quark propagator, which enters the meson bound state equations, is derived from the Dyson--Schwinger equations and a complete set of mesons for general spin quantum number is presented. From analyzing the bound state equations for large spins it is demonstrated, that chiral and axial symmetry are restored. In this limit a complete degeneracy of all multiplets with given spin is observed. The effect of the transverse gluon interaction is shown to vanish rapidly as the spin quantum number is increased. For vanishing dynamical quark mass the expected meson degeneracies are recovered.

A key question to QCD is what mechanism generates the hadron mass in the light quark sector, where both confinement and chiral symmetry breaking are in the game. Are confinement and chiral symmetry breaking in the vacuum uniquely interconnected? Can hadrons survive chiral symmetry restoration? If yes, what happens with their mass and what symmetries beyond the chiral symmetry are there? We review our recent insights. In particular, in a dynamical lattice simulation we artificially restore chiral symmetry by removing the low-lying Dirac modes of the valence quark propagators, which is a well defined procedure and keep gluodynamics intact. Hadrons survive this artificial chiral restoration and their mass is surprisingly large. All hadrons fall into chiral multiplets and some of them are degenerate, i.e. the spectrum reveals some higher symmetry, that includes the chiral symmetry as a subgroup. The U(1)_A symmetry does not get restored after removal of the chiral modes from the valence quarks., Comment: Invited talk at Light Cone 2012, 8-13 July, 2012, Cracow, Poland

Chiral symmetry breaking in Quantum Chromodynamics is associated with the low lying spectral modes of the Dirac operator according to the Banks-Casher relation. Here we study how removal of a variable number of low lying modes from the valence quark sector affects the masses of the ground states and first excited states of baryons and mesons in two flavor lattice QCD., Comment: 6 pages, 2 figures. Contribution to proceedings of "Excited QCD 2012", May 6-12, 2012, Peniche, Portugal

We study hadron correlators upon artificial restoration of the spontaneously broken chiral symmetry. In a dynamical lattice simulation we remove the lowest lying eigenmodes of the Dirac operator from the valence quark propagators and study evolution of the hadron masses obtained. All mesons and baryons in our study, except for a pion, survive unbreaking the chiral symmetry and their exponential decay signals become essentially better. From the analysis of the observed spectroscopic patterns we conclude that confinement still persists while the chiral symmetry is restored. All hadrons fall into different chiral multiplets. The broken U(1)_A symmetry does not get restored upon unbreaking the chiral symmetry. We also observe signals of some higher symmetry that includes chiral symmetry as a subgroup. Finally, from comparison of the \Delta - N splitting before and after unbreaking of the chiral symmetry we conclude that both the color-magnetic and the flavor-spin quark-quark interactions are of equal importance., Comment: 12 pages, 14 figures; final version

We report lattice QCD results on the axial charges of ground and excited nucleon states of both parities. This is the first study of these quantities with approximately chiral (CI) fermions. Two energy levels in the range of the negative parity resonances N*(1535) and N*(1650) are observed and we determine the axial charge for both. We obtain a small axial charge for one of them, which is consistent with the chiral symmetry restoration in this state as well as with the small axial charge of the N*(1535) predicted within the quark model. This result agrees with the findings of Takahashi et al. obtained with Wilson quarks which violate chiral symmetry for finite lattice spacing. At the same time for the other observed negative parity state we obtain a large axial charge, that is close to the axial charge of the nucleon. This is in disagreement both with the quark model prediction as well as with the chiral restoration but allows for an interpretation as an s-wave {\pi} N state.

We find the spectrum and wave functions of the heavy-light mesons in $(1 + 1)$-dimensional QCD in the 't Hooft limit, both in the rest frame, using the Coulomb (axial) gauge, and on the light cone. Our emphasis is on the effects of chiral symmetry breaking on the spectrum. While dynamical equations in both cases look different, the results for the spectrum are identical. The chiral symmetry breaking is clearly seen from the gap and Bethe--Salpeter equations in the laboratory frame. At the same time, while vacuum is trivial on the light cone (no chiral condensate), the effects of the spontaneous breaking of the chiral symmetry manifest themselves in the same way, as it follows from the coincidence of the spectra obtained from the laboratory-frame Bethe--Salpeter equation on the one hand, and the light-cone 't Hooft-type equation on the other.

From the parton distributions in the infinite momentum frame one finds that only about 30% of the nucleon spin is carried by spins of the valence quarks, which gave rise to the term "spin crisis". Similar results hold for the lowest mesons, as it follows from the lattice simulations. We define the spin content of a meson in the rest frame and use a complete and orthogonal $\bar q q$ chiral basis and a unitary transformation from the chiral basis to the (2S+1)LJ basis. Then, given a mixture of different allowed chiral representations in the meson wave function at a given resolution scale, one can obtain its spin content at this scale. To obtain the mixture of the chiral representations in the meson we measure in dynamical lattice simulations a ratio of couplings of interpolarors with different chiral structure. For the rho meson we obtain practically the 3S1 state with no trace of the spin crisis. Then a natural question arises: which definition does reflect the spin content of a hadron?, Comment: 7 pp, Presented at Int. School of Nuclear Physics: "From Quarks and Gluons to Hadrons and Nuclei", Erice-Sicily, 16 - 24 September, 2011