We investigate the chiral phase transition in the strong coupling lattice QCD at finite temperature and density with finite coupling effects. We adopt one species of staggered fermion, and develop an analytic formulation based on strong coupling and cluster expansions. We derive the effective potential as a function of two order parameters, the chiral condensate sigma and the quark number density $ρ_q$, in a self-consistent treatment of the next-to-leading order (NLO) effective action terms. NLO contributions lead to modifications of quark mass, chemical potential and the quark wave function renormalization factor. While the ratio mu_c(T=0)/Tc(mu=0) is too small in the strong coupling limit, it is found to increase as beta=2Nc/g^2 increases. The critical point is found to move in the lower T direction as beta increases. Since the vector interaction induced by $ρ_q$ is shown to grow as beta, the present trend is consistent with the results in Nambu-Jona-Lasinio models. The interplay between two order parameters leads to the existence of partially chiral restored matter, where effective chemical potential is automatically adjusted to the quark excitation energy.
The outline of the two lectures given in "Dense Matter School 2015" is presented. After an overview on the relevance of the phase diagram to heavy-ion collisions and compact star phenomena, I show some basic formulae to discuss the QCD phase diagram in the mean field treatment of the Nambu-Jona-Lasinio model. Next, I introduce the strong-coupling lattice QCD, which is one of the promising methods to access the QCD phase diagram including the first order phase boundary. In the last part, I discuss the QCD phase diagram in asymmetric matter, which should be formed in compact star phenomena.
I report on the current status of QCD phase diagram at vanishing baryon density. I focus on the QCD phase diagram with three degenerate quark flavor using Highly Improved Staggered Quarks on $N_τ=6$ lattices. No evidence of a first order phase transition in the pion mass window of $80\lesssim m_π \lesssim 230 $MeV is found. The pion mass at the critical point where the chiral first order phase transition ends is estimated to be $m^c_π \lesssim 45$ MeV.
I present recent results from lattice simulations of SU(2) gauge theory with Nf=2 Wilson quark flavors, at non-zero quark chemical potential. The thermodynamic equation of state is discussed along with the nature of the high density matter which forms. It is conjectured that deconfinement may mean different things for bulk and Fermi surface phenomena.
We study the phase diagram of quark matter at finite temperature (T) and chemical potential (mu) in the strong coupling region of lattice QCD for color SU(3). Baryon has effects to extend the hadron phase to a larger mu direction relative to Tc at low temperatures in the strong coupling limit. With the 1/g^2 corrections, Tc is found to decrease rapidly as g decreases, and the shape of the phase diagram becomes closer to that expected in the real world.
The present knowledge of the QCD phase diagram based on simulations of lattice QCD is summarised. The main questions are whether there is a critical point in the QCD phase diagram and whether it is related to a chiral phase transition. It is shown that QCD at imaginary chemical potentials has a rich phase structure, which can be determined in a controlled way without sign problem and which severely constrains the phase structure at real chemical potentials.
We calculate the QCD cross-over temperature, the equation of state and fluctuations of conserved charges at finite density by analytical continuation from imaginary to real chemical potentials. Our calculations are based on new continuum extrapolated lattice simulations using the 4stout staggered actions with a lattice resolution up to $N_t=16$. The simulation parameters are tuned such that the strangeness neutrality is maintained, as it is in heavy ion collisions.
Neither the chiral limit nor finite baryon density can be simulated directly in lattice QCD, which severely limits our understanding of the QCD phase diagram. In this review I collect results for the phase structure in an extended parameter space of QCD, with varying numbers of flavours, quark masses, colours, lattice spacings, imaginary and isospin chemical potentials. Such studies help in understanding the underlying symmetries and degrees of freedom, and are beginning to provide a consistent picture constraining the possibilities for the physical phase diagram.
We report our recent results on the QCD phase diagram obtained from the lattice QCD simulation. The location of the phase boundary between hadronic and QGP phases in the two-flavor QCD phase diagram is investigated. The imaginary chemical potential approach is employed, which is based on Monte Carlo simulations of the QCD with imaginary chemical potential and analytic continuation to the real chemical potential region.
Two parameters that have a strong influence on the finite temperature QCD transition, and play an important role in various physical scenarios are the quark density and the external magnetic field. The effect of these parameters on the thermal properties of QCD is discussed, and an overview of the latest lattice results is given.
The transition between the hadronic phase and the quark gluon plasma phase at nonzero temperature and quark chemical potentials is studied within the large-Nc expansion of QCD.
We summarize our current understanding of the connection between the QCD phase line and the chemical freeze-out curve as deduced from thermal analyses of yields of particles produced in central collisions between relativistic nuclei.
We draw the three-flavor phase diagram as a function of light- and strange-quark masses for both zero and imaginary quark-number chemical potential, using the Polyakov-loop extended Nambu-Jona-Lasinio model with an effective four-quark vertex depending on the Polyakov loop. The model prediction is qualitatively consistent with 2+1 flavor lattice QCD prediction at zero chemical potential and with degenerate three-flavor lattice QCD prediction at imaginary chemical potential.
We extend to the clover improved lattice formulation of QCD the resummation of cactus diagrams, i.e. a certain class of tadpole-like gauge invariant diagrams. Cactus resummation yields an improved perturbative expansion. We apply it to the lattice renormalization of some two-fermion operators improving their one-loop perturbative estimates.
We show how to perform a resummation, to all orders in perturbation theory, of a certain class of gauge invariant diagrams in Lattice QCD. These diagrams are often largely responsible for lattice artifacts. Our resummation leads to an improved perturbative expansion. Applied to a number of cases of interest, this expansion yields results remarkably close to corresponding nonperturbative estimates.
The sign problem of QCD prevents standard lattice simulations to determine the phase diagram of strong interactions with a finite chemical potential directly. Complex Langevin simulations provide an alternative method to sample path integrals with complex weights. We report on our ongoing project to determine the phase diagram of QCD in the limit of heavy quarks (HDQCD) using Complex Langevin simulations.
I review recent lattice QCD results on a few selected topics which are relevant to the heavy ion physics community. Special emphasis is put on the QCD equation of state at vanishing and nonzero baryon chemical potential, the onset of deconfinement of open strange and charmed hadrons, the contribution from experimentally not yet observed hadrons to the thermodynamics of the hadronic gas and their influence on freeze-out conditions of strange and light-quark hadrons.
Some consequences of the analyticity of the free energy (pressure) of QCD at finite chemical potential are deduced. These include a method for numerical exploration of the full phase diagram by a novel use of simulations at imaginary chemical potential to extract Yang-Lee zeroes of the grand-canonical partition function. We make use of, and comment on, CPT symmetries, positivity of non-linear susceptibilities and the finiteness of screening lengths. We also comment on the structure of zeroes expected for the usual picture of the phases of QCD, following a discussion of the physics of imaginary…
We explore the phase diagram of two-flavor QCD at imaginary values of baryon and isospin chemical potentials analyzing the thermodynamic potential of QCD analytically and that of the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model numerically. QCD has no pion condensation at imaginary baryon and isospin chemical potentials, and therefore has discrete symmetries that are not present at real baryon and isospin chemical potentials. The PNJL model possesses all the discrete symmetries. The PNJL model can reproduce qualitatively lattice QCD data presented very lately.
We propose a method to study lattice QCD at non-vanishing temperature (T) and chemical potential (μ). We use n_f=2+1 dynamical staggered quarks with semi-realistic masses on L_t=4 lattices. The critical endpoint (E) of QCD on the Re(μ)-T plane is located. We calculate the pressure (p), the energy density (ε) and the baryon density (n_B) of QCD at non-vanishing T and μ.
I review recent new lattice QCD results on a few selected topics which are relevant to the heavy ion physics community. Special emphasis is put on the QCD phase diagram towards the chiral limit and at nonzero baryon density as well as the fate of quarkonia and heavy quark drag coefficients.
We review the potential method in lattice QCD, which has recently been proposed to extract nucleon-nucleon interactions via numerical simulations. We focus on the methodology of this approach by emphasizing the strategy of the potential method, the theoretical foundation behind it, and special numerical techniques. We compare the potential method with the standard finite volume method in lattice QCD, in order to make pros and cons of the approach clear. We also present several numerical results for the nucleon-nucleon potentials.
In the last years there has been remarkable progress in the comparison of experimental data on the shape of event-by-event distributions of conserved quantities and lattice thermodynamic predictions based on the grand canonical ensemble. In this talk we discuss how the QCD crossover temperature and the freezeout curve are extracted from the analysis of fluctuations. We report that one can also go further and locate the QCD critical point already from published data. We also list the systematics which must be brought under control in future.
We derive exact constraints relating QCD at nonzero baryon chemical potential and temperature to QCD at nonzero isospin chemical potential and temperature, a theory which can be simulated by conventional methods. These results challenge the consistency of dynamical models of superconductivity and superfluidity in QCD.
Recent progress and the latest results on the bulk thermodynamic properties of QCD matter from lattice are reviewed. In particular, I will stress upon the fact that lattice techniques are now entering into precision era where they can provide us with new insights on even the microscopic degrees of freedom in different phases of QCD. I will discuss some instances, from the recent studies of topological fluctuations and screening masses. The progress towards understanding the effects of anomalous $U_A(1)$ symmetry on the chiral crossover transition and transport properties of QCD matter will als…
Monte Carlo methods cannot probe far into the QCD phase diagram with a real chemical potential, due to the famous sign problem. Complex Langevin simulations, using adaptive step-size scaling and gauge cooling, are suited for sampling path integrals with complex weights. We report here on tests of the deconfinement transition in pure Yang-Mills SU(3) simulations and present an update on the QCD phase diagram in the limit of heavy and dense quarks.
Complex Langevin simulations provide an alternative to sample path integrals with complex weights and therefore are suited to determine the phase diagram of QCD from first principles. Adaptive step-size scaling and gauge cooling are used to improve the convergence of our simulations. We present results for the phase diagram of QCD in the limit of heavy quarks and discuss the order of the phase transitions, which are studied by varying the spatial simulation volume.
The physics of the QCD phase diagram is discussed in view from heavy-ion collisions, compact astrophysical phenomena, lattice QCD and chiral effective models. We find that (T,mu_B) region probed in heavy-ion collisions and the black hole formation processes covers most of the critical point locations predicted in recent lattice QCD Monte-Carlo simulations and chiral effective models of QCD. We also discuss the consistency of the statistical model results and dynamical model description.
Simulations of QCD with a finite chemical potential typically lead to a severe sign problem, prohibiting any standard Monte Carlo approach. Complex Langevin simulations provide an alternative to sample path integrals with oscillating weight factors and therefore potentially enable the determination of the phase diagram of QCD. Here we present results for QCD in the limit of heavy quarks and show evidence that the phase diagram can be mapped out by direct simulation. We apply adaptive step-size scaling and adaptive gauge cooling to ensure the convergence of these simulations.
We present results on the QCD phase diagram for mu_B <= pi T. Our simulations are performed with an imaginary chemical potential mu_I for which the fermion determinant is positive. On an 8^3 x 4 lattice with 2 flavors of staggered quarks, we map out the phase diagram and identify the pseudo-critical temperature T_c(mu_I). For mu_I/T <= pi/3, this is an analytic function, whose Taylor expansion is found to converge rapidly, with truncation errors far smaller than statistical ones. The truncated series may then be continued to real mu, yielding the corresponding phase diagram for mu_B <~ 500 MeV…
We study the density of states method to explore the phase diagram of the chiral transition on the tempeature and quark chemical potential plane. Four quark flavours are used in the analysis. Though the method is quite expensive small lattices show an indication for a triple-point connecting three different phases on the phase diagram.
We study the finite temperature phase transition of four and two flavor staggered fermions with hypercubic smeared link actions. These preliminary studies suggest that the improved flavor symmetry of the fermionic action can have a significant effect on the finite temperature phase diagram.
The aim of this review is to summarize the contemporary understanding of the QCD phase diagram as a function of temperature and baryo-chemical potential. The focus is on recent theoretical developments due to lattice simulations of the phase diagram.
The first results on lattice at finite temperature $T$ and chemical potential $μ$ with 2+1 flavors of Wilson quarks are presented. We have studied the dependence of chiral phase transition and deconfinement phase transition on quark mass. Finite volume size analysis and Binder cumulants are used to determine the properties of phase transition. Phase diagram of QCD with 2+1 flavors of Wilson quarks are presented.
The phase diagram of lattice QCD in the strong coupling limit can be measured in the full $μ$-$T$ plane, also in the chiral limit. In particular, the phase diagram in the chiral limit features a tricritical point at some $(μ_c,T_c)$. This point may be related to the critical end point expected in the QCD phase diagram. We discuss the gauge corrections to the phase diagram at strong coupling and compare our findings with various possible scenarios in continuum QCD. We comment on the possibility that the tricritical point at strong coupling is connected to the tricritical point in the continuum,…
In this review we present the current state-of-the-art on complex Langevin simulations and their implications for the QCD phase diagram. After a short summary of the complex Langevin method, we present and discuss recent developments. Here we focus on the explicit computation of boundary terms, which provide an observable that can be used to check one of the criteria of correctness explicitly. We also present the method of Dynamic Stabilization and elaborate on recent results for fully dynamical QCD.
Complex Langevin simulations provide an alternative to sample path integrals with complex weights and therefore are suited to determine the phase diagram of QCD from first principles. We use our proposed method of Dynamic Stabilisation (DS) to ensure improved convergence to the right limit and present new systematic tests of this technique. We also show results on QCD in the limit of heavy quarks and an analysis of DS compared to known results from reweighting.
We demonstrate how to locate the critical endpoint of the QCD phase transition by means of simulations at imaginary μ. For the three flavor theory, we present numerical results for the pseudo-critical line as a function of chemical potential and bare quark mass, as well as the bare quark mass dependence of the endpoint.
We elucidate that the phase diagram of massless $N$-flavor QCD under $\mathbb{Z}_N$ flavor-twisted boundary condition (massless $\mathbb{Z}_N$-QCD) is constrained by an 't Hooft anomaly involving two-form gauge fields. As a consequence, massless $\mathbb{Z}_N$-QCD turns out to realize persistent order at any temperatures and quark chemical potentials, namely, the symmetric and gapped phase is strictly forbidden. This is the first result on the finite-$(T,μ)$ phase diagram in QCD-type theories based on anomaly matching related to center and discrete axial symmetries.
We present first results of a study of two flavour QCD with Wilson fermions at finite temperature. We have used tree level Symanzik improvement in both the gauge and fermion part of the action. In a first step we explore the phase diagramm on an $8^3 \times 4$ lattice, with particular emphasis on checking Aoki's conjecture with an improved action.
We present unambiguous evidence from lattice simulations of QCD with three degenerate quark species for two tricritical points in the (T,m) phase diagram at fixed imaginary μ/T=iπ/3 mod 2π/3, one in the light and one in the heavy mass regime. These represent the boundaries of the chiral and deconfinement critical lines continued to imaginary chemical potential, respectively. It is demonstrated that the shape of the deconfinement critical line for real chemical potentials is dictated by tricritical scaling and implies the weakening of the deconfinement transition with real chemical potential. T…
In recent years there has been much progress on the investigation of the QCD phase diagram with lattice QCD. This talk will focus on the developments in the last few years. Especially the addition of external influences and extended ranges of $T$ and $μ$ yield an increasing number of interesting results, a subset of which will be discussed. Many of these conditions are important for the understanding of both the QCD transition in the early universe and heavy ion collision experiments which are conducted for example at the LHC and RHIC. This offers many exciting opportunities for comparisons be…
Recent work by Armoni, Shifman, and Veneziano suggests a large-N equivalence between supersymmetric Yang-Mills Theory and one-flavor QCD. One consequence of this "orientifold projection" is that scalar and pseudoscalar mesons in one-flavor QCD should have degenerate mass since they lie within the same Wess-Zumino supermultiplet. We use lattice calculations to investigate the mass shifts caused by "double-hairpin" annihilation diagrams in quenched QCD to test for this degeneracy. Similar quark-antiquark annihilation processes are studied in the 2-dimensional CP(N-1) model with quenched fermions…
We discuss the phase diagram of QCD in the presence of a strong background magnetic field, providing numerical evidence, based on lattice simulations of QCD with $2+1$ flavours and physical quark masses, that the QCD crossover turns into a first order phase transition for large enough magnetic field, with a critical endpoint located between $eB=4$ GeV$^2$ (where we found an analytic crossover at a pseudo-critical temperature $T_c=(98\pm3)$ MeV) and $eB=9$ GeV$^2$ (where the measured critical temperature is $T_c=(63\pm5)$ MeV).
A general discussion is presented of the possible symmetries responsible for confinement of color and of their evidence in lattice simulations. The consequences on the phase diagram of $QCD$ are also analyzed.
We review our knowledge of the phase diagram of QCD as a function of temperature, chemical potential and quark masses. The presence of tricritical lines at imaginary chemical potential mu=i pi/3 T, with known scaling behaviour in their vicinity, puts constraints on this phase diagram, especially in the case of two light flavors. We show first results in our project to determine the finite-temperature behaviour in the two-flavour chiral limit.
We show the universality of phase diagrams in QCD and QCD-like theories through the large-N_c equivalence. The whole phase diagrams are identical between QCD at finite isospin chemical potential and SO(2N_c) and Sp(2N_c) gauge theories at finite baryon chemical potential. In the chiral limit, they are also identical to that of QCD at finite chiral chemical potential. Outside the pion or diquark condensed phase in these theories, the universality of phase diagrams is applicable to QCD at finite baryon chemical potential. We further argue that the universality may work approximately even for N_c…
QCD, the theory of the strong interactions, involves quarks interacting with non-Abelian gluon fields. This theory has many features that are difficult to impossible to see in conventional diagrammatic perturbation theory. This includes quark confinement, mass generation, and chiral symmetry breaking. This paper is a colloquium level overview of the framework for understanding how these effects come about.
Theta vacuum effects on the QCD phase structure in the μ-T plane are studied by using the Polyakov-loop extended Nambu-Jona-Lasinio model and its extension, where μis the quark chemical potential and T is temperature, respectively. As the parameter θof the theta vacuum increases, the chiral transition becomes stronger. For large θ, it eventually becomes first order even at zero μ.
The strong coupling limit of lattice QCD with staggered fermions has been studied for decades, both via Monte Carlo and via mean field theory. In this model, the finite density sign problem can be made mild and the full phase diagram can be obtained, even in the chiral limit. It is however desirable to understand the effect of a finite lattice gauge coupling $β$ on the phase diagram in the $μ-T$ plane in order to understand how it evolves into the phase diagram of continuum QCD. Here we discuss how to construct a partition function for non-zero lattice coupling, exact to $\mathcal{O}(β)$, and …
The thermodynamics of QCD with sufficiently heavy dynamical quarks can be described by a three-dimensional Polyakov loop effective theory, obtained after a truncated character and hopping expansion. We investigate the resulting phase diagram for low temperatures by mean field methods. Taking into account chemical potentials for both baryon number and isospin, we obtain clear signals for a liquid-gas type transition to baryon matter at $μ_I=0$ and a Bose-Einstein condensation transition at $μ_B=0$, as well as for their connection when both chemical potentials are non-zero.
We investigate the phase diagram of QCD with heavy quarks at finite temperature and chemical potential in the context of background field methods. In particular, we use a massive extension of the Landau-DeWitt gauge which is motivated by previous studies of the deconfinement phase transition in pure Yang-Mills theories. We show that a simple one-loop calculation is able to capture the richness of the phase diagram in the heavy quark region, both at real and imaginary chemical potential. Moreover, dimensionless ratios of quantities directly measurable in numerical simulations are in good agreem…
We study the possible significance of four-quark states in the iso-singlet scalar mesons ($J^{PC}=0^{++}$, $I=0$) by performing two-flavor full lattice QCD simulations on an $8^3 \times 16$ lattice using the improved gauge action and the clover-improved Wilson quark action. In particular, we evaluate the propagators of molecular and tetra-quark operators together with singly disconnected diagrams. In the computation of the singly disconnected diagrams we employ the $Z_2$-noise method with the truncated eigenmode approach. We show that the quark loops given by the disconnected diagrams play an …
A comprehensive number of integrals emerging in one-loop computations in a gauge perturbation theory on the lattice with Wilson fermions at $r=1$ is computed using the Burgio--Caracciolo--Pelissetto algorithm and the FORM package. An explicit analytical form of the recursion relations needed for such computations is presented.
We evaluate properties of neutral mesons in Nf=2 dynamical simulations of TMQCD at maximal twist. The pion is explored - establishing the size of the isospin splitting (an order a^2 effect). We investigate the eta' (the Nf=2 flavour singlet pseudoscalar meson) and neutral rho and scalar mesons. We show that disconnected diagrams can be evaluated very efficiently in TMQCD using variance reduction methods.
We study lattice QCD with four flavors of staggered quarks. In the limit of infinite gauge coupling, "dual" variables can be introduced, which render the finite-density sign problem mild and allow a full determination of the $μ-T$ phase diagram by Monte Carlo simulations, also in the chiral limit. However, the continuum limit coincides with the weak coupling limit. We propose a strong-coupling expansion approach towards the continuum limit. We show first results, including the phase diagram and its chiral critical point, from this expansion truncated to next-to-leading order.
