We study the properties of an instanton ensemble in three-flavor dense QCD which can be regarded as an instanton plasma weakly interacting by exchanging the eta' mesons. Based on this description, we explore the chiral phase transition induced by the instanton ensemble at high baryon density in analogy with the Berezinskii-Kosterlitz-Thouless transition. Using the renormalization group approach, we show that the instanton ensemble always behaves as a screened and unpaired plasma. We also demonstrate that the chiral condensate in dense QCD is proportional to the instanton density.
A Coulomb gas representation of dense-matter QCD is derived from a dual transformation of the low-energy effective Lagrangian. The point-like charges Q=+1,-1 of the gas are identified with the instantons and anti-instantons of such topological charges. An instanton repels another instanton with the same force as it attracts an anti-instanton, in contrast to the semiclassical interaction.
We revisit the instanton-dyon liquid model in the confined phase by using a non-linear Debye-Huckel (DH) resummation for the Coulomb interactions induced by the moduli, followed by a cluster expansion. The organization is shown to rapidly converge and yields center symmetry at high density. The dependence of these results on a finite vacuum angle are also discussed. We also formulate the hypernetted chain (HCN) resummation for the dense instanton-dyon liquid and use it to estimate the liquid pair correlation functions in the DH limit. At very low temperature, the dense limit interpolates betwe…
We discuss nuclear matter and the transition to quark matter in the decompactified limit of the Sakai-Sugimoto model. Nuclear matter is included through instantons on the flavor branes of the model. Our approximation is based on the flat-space solution, but we allow for a dynamical instanton width and deformation and compute the energetically preferred number of instanton layers in the bulk as a function of the baryon chemical potential. We determine the regions in parameter space where the binding energy of nuclear matter is like in QCD, and compute the phase diagram in the plane of temperatu…
We consider winding number transitions in the two dimensional O(3) non-linear sigma model, modified by a suitable conformal symmetry breaking term. We discuss the general properties of the relevant instanton solutions which dominate the transition amplitudes at finite energy, and find the solutions numerically. The Euclidean period of the solution increases with energy, contrary to the behavior found in the abelian Higgs model or simple one dimensional systems. This indicates that there is a sharp crossover from instanton dominated tunneling to sphaleron dominated thermal activation at a certa…
We present arguments suggesting that large size overlapping instantons are the driving mechanism of the confinement-deconfinement phase transition at nonzero chemical potential mu. The arguments are based on the picture that instantons at very large chemical potential in the weak coupling regime are localized configurations with finite size ρ\simμ^{-1}. At the same time, the same instantons at smaller chemical potential in the strong coupling regime are well represented by the so-called instanton-quarks with fractional topological charge 1/N_c. We estimate the critical chemical potential mu_c(…
We compute the first order chiral phase transition for an instanton motivated quark model with a local six-quark interaction. In order to compare different solutions of the gap equation we compute the bosonic effective action -- a two particle irreducible free energy functional. We find that the first order transition ends for a critical current quark mass, with continuous crossover for larger quark masses. Furthermore, we investigate different possible order parameters, including a color octet condensate. We also compare our formalism with mean field theory.
We derive an effective thermodynamic potential (Omega_eff) at finite temperature (T>0) and zero quark-chemical potential (mu_R=0), using the singular-gauge instanton solution and Matsubara formula for N_c=3 and N_f=2 in the chiral limit. The momentum-dependent constituent-quark mass is also obtained as a function of T, employing the Harrington-Shepard caloron solution in the large-N_c limit. In addition, we take into account the imaginary quark chemical potential mu_I = A_4, translated as the traced Polayakov-loop (Phi) as an order parameter for the Z(N_c) symmsetry, characterizing the confine…
Winding number transitions in the two dimensional softly broken O(3) nonlinear sigma model are studied at finite energy and temperature. New periodic instanton solutions which dominate the semiclassical transition amplitudes are found analytically at low energies, and numerically for all energies up to the sphaleron scale. The Euclidean period beta of these finite energy instantons increases with energy, contrary to the behavior found in the abelian Higgs model or simple one dimensional systems. This results in a sharp crossover from instanton dominated tunneling to sphaleron dominated thermal…
We have discussed the vacuum tunnelling in Friedberg-Lee model. The tunnelling coefficient is derived in the field configuration space by calculating the transition amplitude using the path integral under the stationary phase approximation and the dilute instanton gas approximation. By studying the tunnelling effect between the two degenerating vacuums at the critical temperature and chemical potential, we find that the system could be deconfined by tunnelling, which could possibly change the first order deconfinement phase transition to crossover.
We study the relativistic BCS-BEC crossover within a class of Nambu-Jona-Lasinio type models, including arbitrary Lorentz-scalar pairing channels. Using the mean-field approximation we investigate spectral properties of the collective bosonic excitations in the superfluid phase, with particular attention to the Nambu--Goldstone bosons of the broken symmetry. This is a first step towards a systematic improvement of the mean-field approximation by including the fluctuation effects. The general results are illustrated on pairing in dense two-flavor quark matter - the two-flavor color superconduct…
In this talk, we report our recent studies on an effective thermodynamic potential (Omega_eff) at finite temperature (T>0) and zero quark-chemical potential (mu_R=0), using the singular-gauge instanton solution and Matsubara formula for N_c=3 and N_f=2 in the chiral limit, i.e. m_q=0. The momentum-dependent constituent-quark mass is computed as a function of T, together with the Harrington-Shepard caloron solution in the large-N_c limit. In addition, we take into account the imaginary quark-chemical potential mu_I = A_4, indentified as the traced Polayakov-loop (Phi) as an order parameter for …
Dense QCD at zero temperature with a large number of colors is a crystal. We show that in the holographic dual description, the crystal is made out of pairs of dyons with $e=g=\pm 1$ charges in a salt-like arrangement. We argue that with increasing density the dyon masses and topological charges equalize, turning the salt-like configuration to a bcc of half-instantons. The latter is dual to a cubic crystal of half-skyrmions. We estimate the transition from an fcc crystal of instantons to a bcc crystal of dyons to about 3 times nuclear matter density with a dyon binding energy of about 180 MeV.
We study the QCD phase structure in the three-flavor Nambu-Jona-Lasinio model, incorporating the interplay between the chiral and diquark condensates induced by the axial anomaly. We demonstrate that for an appropriate range of parameters of the model, the interplay leads to the low temperature critical point in the phase structure predicted by a previous Ginzburg-Landau analysis. We also show that a Bose-Einstein condensate (BEC) of diquark molecules emerges in the intermediate density region, and as a result, a BEC-BCS crossover is realized with increasing quark chemical potential.
We investigate cold dense matter in the context of Sakai and Sugimoto's holographic model of QCD in the Wigner-Seitz approximation. In bulk, baryons are treated as instantons on S^3\times R^1 in each Wigner-Seitz cell. In holographic QCD, Skyrmions are instanton holonomies along the conformal direction. The high density phase is identified with a crystal of holographic Skyrmions with restored chiral symmetry at about 4 Mkk^3/pi^5. As the average density goes up, it approaches to uniform distribution while the chiral condensate approaches to p-wave over a cell. The chiral symmetry is effectivel…
Dense hadronic matter at low temperature is expected to be in crystal and at high density make a transition to a {\em chirally restored but color-confined} state which is a novel phase hitherto unexplored. This phase transition is predicted in both skyrmion matter in 4D and instanton matter in 5D, the former in the form of half-skyrmions and the latter in the form of half-instantons or dyons. We predict that when $K^-$'s are embedded in this half-skyrmion or half-instanton (dyonic) matter which may be reached not far above the normal density, there arises an enhanced attraction from the soft d…
In quantum chromodynamics (QCD), the role which topologically non-trivial configurations play in splitting the singlet pseudo-Goldstone meson, the $η^\prime$, from the octet is familiar. In addition, such configurations contribute to other processes which violate the axial $U(1)_A$ symmetry. While the nature of topological fluctuations in the confined phase is still unsettled, at temperatures well above that for the chiral phase transition, they can be described by a dilute gas of instantons. We show that instantons of arbitrary topological charge $Q$ generate anomalous interactions between $2…
We investigate chiral-restoration patterns of QCD for N_{c}=3 and N_{f}=2 at finite temperature (T) and zero quark-chemical potential beyond the chiral limit, indicating the explicit chiral-symmetry breaking. To this end, we employ the instanton-vacuum configuration for the flavor SU(2) sector and the Harrington-Shepard caloron for modifying relevant instanton parameters as functions of T. The meson-loop corrections (MLC), which correspond to 1/N_{c} corrections, are also taken into account to reproduce appropriate m_{q} dependences of chiral order parameters. We compute chiral condensate as a…
We study the nonperturbative QCD in terms of topological objects, i.e., QCD-monopoles and instantons. In the 't Hooft abelian gauge, QCD is reduced into an abelian gauge theory with QCD-monopoles, and the confinement mechanism can be understood with the dual superconductor picture. Regarding the flux-tube ring as the glueball, we study the glueball properties by combining the dual Ginzburg-Landau theory and the Nambu-Goto action. We find the strong correlations between instantons and monopole world-lines both in the continuum theory and in the lattice QCD. The nonperturbative QCD vacuum is fil…
Instantons lead to strong correlations between up and down quarks with spin zero and anti-symmetric color wave functions. In cold and dense matter, $n_b>n_c\simeq 1 fm^{-3}$ and $T<T_c\sim$ 50 MeV, these pairs Bose-condense, replacing the usual $< \bar qq >$ condensate and restoring chiral symmetry. At high density, the ground state is a color superconductor in which diquarks play the role of Cooper pairs. An interesting toy model is provided by QCD with two colors: it has a particle-anti-particle symmetry which relates $<\bar qq>$ and $< qq>$ condensates.
In the large N limit cold dense nuclear matter must be in a lattice phase. This applies also to holographic models of hadron physics. In a class of such models, like the generalized Sakai-Sugimoto model, baryons take the form of instantons of the effective flavor gauge theory that resides on probe flavor branes. In this paper we study the phase structure of baryonic crystals by analyzing discrete periodic configurations of such instantons. We find that instanton configurations exhibit a series of "popcorn" transitions upon increasing the density. Through these transitions normal (3D) lattices …
We demonstrate that the QCD topological susceptibility nonperturbatively gets a significant contribution signaled by flavor-nonuniversal quark condensates at around the pseudo-critical temperature of the chiral crossover. It implies a remarkable flavor breaking in the axial anomaly as well as the QCD theta vacuum in high temperature QCD, which are almost flavor universal in the vacuum. A nontrivial flavor breaking is triggered by nonperturbative thermal loop corrections at around the chiral crossover, which is different from the trivial flavor violation just scaled by the quark mass ratio, obs…
We investigate the contribution that instantons make to the QCD chiral condensate for Nf= 0, 1 and 2 quark flavours. We use a simplified model: the instantons are a (weighted) gas, the fermionic integrations are restricted to the subspace spanned by the would-be zero-modes and only the dimensional and chiral features of the Dirac operator are retained. Previous work along these lines found a power-like divergence of the Dirac spectral density at zero eigenvalue. By changing the various components of the model, we show that a power divergence appears to be a generic feature of instanton mixing …
QCD-like theories possess a positively definite fermion determinant at finite baryon chemical potential $μ_{\text B}$ and the lattice simulation can be successfully performed. While the chiral perturbation theories are sufficient to describe the Bose condensate at low density, to describe the crossover from Bose-Einstein condensation (BEC) to BCS superfluidity at moderate density we should use some fermionic effective model of QCD, such as the Nambu-Jona-Lasinio model. In this paper, using two-color two-flavor QCD as an example, we examine how the Nambu-Jona-Lasinio model describes the weakly …
The singlet coupling to the topological charge density in the instanton vacuum, causes the instantons and antiinstantons to be screened over distances of the order of 1/2 fm. Dilute instanton systems behave as a free gas, while dense instanton systems behave as a plasma. The free gas behaviour is favored by a density of 1 fm$^{-4}$. Owing to the Higgs mechanism, the $η$' mass is heavy (1100 MeV). The vacuum topological susceptibility is small (0.07 fm$^{-4}$) and consistent with the QCD Ward identity. In the chiral limit, the singlet screening vanishes, leading to a four dimensional plasma sta…
"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 betwe…
Recently, it was argued that the thermal deconfinement transition in pure Yang-Mills theory is continuously connected to a quantum phase transition in softly-broken N=1 SYM theory on R^3 x S^1. The transition is semiclassically calculable at small S^1 size L, occurs as the soft mass m_soft and L vary, and is driven by a competition between perturbative effects and nonperturbative topological molecules. These are correlated instanton--anti-instanton tunneling events, whose constituents are monopole-instantons "bound" by attractive long-range forces. The mechanism driving the transition is unive…
Nontrivial topological structures of non-Abelian gauge fields were discovered in the 1970's. Instanton solutions, describing vacuum tunneling through topological barriers, have fermionic zero modes which are at the origin of $^\prime$t Hooft effective Lagrangian. In the 1980's instanton ensembles have been used to explain chiral symmetry breaking. In the 1990's a large set of numerical simulations were performed deriving Euclidean correlation functions. The special role of scalar diquarks in nucleons, and color superconductivity in dense quark matter have been elucidated. In these lectures, we…
We conjecture that the confinement- deconfinement phase transition in QCD at large number of colors $N$ and $N_f\ll N$ at $T\neq 0$ and $μ\neq 0$ is triggered by the drastic change in $θ$ behavior. The conjecture is motivated by the holographic model of QCD where confinement -deconfinement phase transition indeed happens precisely at $T=T_c$ where $θ$ dependence experiences a sudden change in behavior. The conjecture is also supported by quantum field theory arguments when the instanton calculations (which trigger the $θ$ dependence) are under complete theoretical control for $T>T_c$, suddenly…
We investigate the low-lying eigenmodes of the Dirac matrix with the aim to gain more insight into the temperature dependence of the anomalous $U_A(1)$ symmetry. We use the overlap operator to probe dynamical QCD configurations generated with (2+1)-flavors of highly improved staggered quarks. We find no evidence of a gap opening up in the infrared region of the eigenvalue spectrum even at $1.5\,T_c$, $T_c$ being the chiral crossover temperature. Instead, we observe an accumulation of near-zero eigenmodes. We argue that these near-zero eigenmodes are primarily responsible for the anomalous brea…
The problem of color superconductivity in dense QCD is reconsidered in the improved rainbow approximation to the Schwinger-Dyson equation. The effect of the unscreened magnetic modes of gluons on the value of the color condensate is studied. In particular, it is shown that, at sufficiently large values of the chemical potential, these modes lead to the enhancement of the superconducting order parameter. The interplay between the instanton induced interaction and the one-gluon induced one in color superconductivity is discussed.
We study the interplay between chiral and diquark condensates within the framework of the Ginzburg-Landau free energy, and classify possible phase structures of two and three-flavor massless QCD. The QCD axial anomaly acts as an external field applied to the chiral condensate in a color superconductor and leads to a crossover between the broken chiral symmetry and the color superconducting phase, and, in particular, to a new critical point in the QCD phase diagram.
We investigate the interplay between the chiral and diquark condensates on the basis of the Ginzburg-Landau potential with QCD symmetry. We demonstrate that the axial anomaly drives a new critical point at low temperature in the QCD phase diagram and leads to a smooth crossover between the hadronic and color superconducting phases.
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 …
In the instanton ensemble the fermionic zero modes collectivize and break chiral symmetry. Recent studies of resulting zero mode zone confirm its very small width and overall importance for lattice simulations. Confinement however has been related with completely different topological objects, the magnetic monopoles. Instanton constituents -- instanton dyons, discovered at nonzero holonomy by Pierre van Baal and others -- are able to explain both confinement and chiral symmetry breaking. The talk summarizes recent works deriving the instanton-dyon mutual interactions, and statistical studies o…
We discuss the violation of quark-flavor symmetry at high temperatures, induced from nonperturbative thermal loop corrections and axial anomaly, based on a three-flavor linear-sigma model including an axial-anomaly induced-flavor breaking term. We employ a nonperturbative analysis following the Cornwall-Jackiw-Tomboulis formalism, and show that the model undergoes a chiral crossover with a pseudo-critical temperature, consistently with lattice observations. We find following features regarding the flavor breaking eminent around and above the pseudo-critical temperature: i) up-and down-quark co…
We consider the model of holographic QCD with asymptotic freedom and gluon condensation in its vacuum. It consists of the color D4-branes and D0-branes as a background and the flavor D8-branes as a probe. By taking a specific field theory limit, the effective coupling decreases to vanish in UV region. We then introduce the uniformly distributed baryons in terms of the baryon vertices and study the density dependence of chiral condensate, which is evaluated using the worldsheet instanton method. In the confined phase, the chiral condensate as a function of density monotonically decreases in hig…
It was predicted qualitatively in ref.[1] that skyrmion matter at low density is stable in an inhomogeneous phase where skyrmions condensate into lumps while the remaining space is mostly empty. The aim of this paper is to proof quantitatively this prediction. In order to construct an inhomogeneous medium we distort the original FCC crystal to produce a phase of planar structures made of skyrmions. We implement mathematically these planar structures by means of the 't Hooft instanton solution using the Atiyah-Manton ansatz. The results of our calculation of the average density and energy confi…
Lectures discuss applications of semiclassical methods based on instantons to vacuum structure (chiral symmetry breaking), hadronic structure (various correlation functions), and high energy collisions of hadrons and heavy ions. We discuss phase diagram of hot/dense hadronic matter, and in the last lecture review current puzzles/lesons from the first RHIC run in summer of 2000.
Evidence for a Continuous Hadron--Quark Transition in Cold Dense Matter
Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Science（中国科学院紫金山天文台暗物质与空间天文重点实验室）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所交叉理论数学科学中心（iTHEMS））；Research Center for Nuclear Physics, The University of Osaka（大阪大学核物理研究中心）
Using a general Ginzburg-Landau effective Lagrangian, we study the topological structure and low-lying collective modes of dense QCD having both chiral and diquark condensates, for two and three massless flavors. As we found earlier, the QCD axial anomaly acts as an external field applied to the chiral condensate in a color superconductor and, as a new critical point emerges, leads to a crossover between the broken chiral symmetry and color superconducting phases. At intermediate densities where both chiral and diquark condensates are present, we derive a generalized Gell-Mann- Oakes-Renner re…
The emergence and importance of hidden local symmetry in the structure of hadrons under extreme conditions is discussed. The topics covered are the potentially important role played by an infinite tower of vector mesons encoded in holographic dual QCD (or AdS/QCD) in chiral dynamics of mesons and baryons, in the vector dominance and its violation in EW response functions and the presence of the vector manifestation fixed point and its influence on the properties of hadrons in hot temperature (i.e., in relativistic heavy ion collisions) and in dense matter (i.e., in compact stars). Also discuss…
We conjecture that the confinement- deconfinement phase transition in QCD at large number of colors N and N_f\ll N at T\neq 0 and μ\neq 0 is triggered by the drastic change in θbehavior. The conjecture is motivated by the holographic model of QCD where confinement -deconfinement phase transition indeed happens precisely at the value of temperature T=T_c where θdependence experiences a sudden change in behavior[1]. The conjecture is also supported by quantum field theory arguments when the instanton calculations (which trigger the θdependence) are under complete theoretical control for T>T_c, s…
We investigate properties of the net baryon number fluctuations near chiral crossover in a hot and dense medium of strongly interacting quarks. The chirally invariant quark-antiquark interactions are modeled by an effective quark-meson Lagrangian. To preserve remnants of criticality in the O(4) universality class, we apply the functional renormalization group method to describe thermodynamics near chiral crossover. Our studies are focused on the influence of the momentum cuts on the critical behavior of different cumulants of the net quark number fluctuations. We use the momentum scale depende…
Trace anomaly, effective degrees of freedom, and chemical potential effects near the QCD crossover
National Technical University of Ukraine（乌克兰国立技术大学）；Institute for Nuclear Research of the NAS of Ukraine（乌克兰国家科学院核物理研究所）；Taras Shevchenko National University of Kyiv（基辅塔拉斯舍甫琴科国立大学）
Dense $\mathrm{QC_2D_2}$ with uniform matrix product states
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所跨学科理论与数学科学中心）
This talk consists of two very different parts: the first one deals with non-perturbative QCD and physics of the chiral restoration, the second with rather low-key (and still unfinished) work aiming at obtaining EOS and other properties of hot/dense hadronic matter from data on heavy ion collisions. The microscopic mechanism for chiral restoration phase transition is a transition from randomly placed tunneling events (instantons) at low T to a set of strongly correlated tunneling-anti-tunneling events (known as instanton - anti-instanton molecules) at high T. Many features of the transition ca…
We start with the relation between the chiral symmetry breaking and gauge field topology. New lattice result further enhance the notion of Zero Mode Zone, a very narrow strip of states with quasizero Dirac eigenvalues. Then we move to the issue of "origin of mass" and Brown-RHo scaling: a number of empirical facts contradicts to the idea that masses of quarks and such hadrons as $ρ,N$ decrease near $T_c$. We argue that while at $T=0$ the main contribution to the effective quark mass is chirally odd $m_{\snchi}$, near $T_c$ it rotates to chirally-even component $m_χ$, because "infinite clusters…
We calculate the electromagnetic Pauli form factor of quark induced by the nontrivial topological fluctuations of QCD vacuum called instantons. It is shown that such contribution is significant. We discuss the possible implications of our result in the photon-hadron reactions and in the dynamics of quark-photon interactions in the dense/hot quark matter.
We obtain the equation of state (EoS) for two-color QCD at low temperature and high density from the lattice Monte Carlo simulation. We find that the velocity of sound exceeds the relativistic limit ($c_s^2/c^2=1/3$) after BEC-BCS crossover in the superfluid phase. Such an excess of the sound velocity is previously unknown from any lattice calculations for QCD-like theories. This finding might have a possible relevance to the EoS of neutron star matter revealed by recent measurements of neutron star masses and radii.
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.
In this talk, we report our recent studies for the chiral magnetic effect, which signals the P- and CP-violations at the heavy ion collisions. We compute the electric current and its correlations, induced by the external magnetic field, inside the hot and dense QCD matter created in the heavy ion collisions. For this purpose, we employ the instanton-liquid model, modified by the Harrington-Shepard caloron at finite T. We observe that the chiral magnetic effect current and its correlations increase with respect to the magnetic field, whereas they decrease as functions of T, due to the diluting …
