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A remarkable progress has been made in the understanding of the hot and dense QCD matter using lattice gauge theory. The issues which are very well understood as well as those which require both conceptual and algorithmic advances are highlighted. The recent lattice results on QCD thermodynamics which are important in the context of the heavy ion experiments are reviewed. Instances of greater synergy between the lattice theory and the experiments in the recent years are discussed where lattice results could be directly used as benchmarks for experiments, and results from the experiments would be a crucial input for lattice computations.

A large part of the visible matter in our universe is made out of protons and neutrons, collectively called hadrons. Hadrons are made out of more fundamental particles called quarks and gluons. The quantum theory for these particles is quantum chromodynamics (QCD). QCD is a strongly interacting theory, and the strength of interaction becomes vanishingly small only at asymptotically high energies. Due to this reason, the quarks and gluons are not visible directly in our world and remain confined within the hadrons. Lattice gauge theory has emerged as the most successful nonperturbative tool to study QCD, with very precise lattice results available for hadron masses and decay constants which are in excellent agreement with the experimental values [

It is expected that at high enough temperatures that existed in the early universe, the hadrons would melt into a quark gluon plasma (QGP) phase. Signatures of such a phase have been seen during the last decade in the heavy ion collision experiments at the relativistic heavy ion collider (RHIC), in Brookhaven National Laboratory. This is particularly exciting for the lattice theory community which has been predicting such a phase transition since a long time [

In this paper, I have selected the most recent results from lattice QCD thermodynamics that are relevant for the heavy ion phenomenology. I have tried to review the necessary background but not attempted to provide a comprehensive account of the development of the subject throughout these years. I have divided this paper into two major sections. The first section deals with QCD at finite temperature and zero baryon density, where lattice methods are very robust. I have given a basic introduction to the lattice techniques, and how the continuum limit is taken, which is essential to relate the lattice data with the real world experiments. I have discussed the current understanding we have of the nature of QCD phase transition as a function of quark masses, inferred from lattice studies. Subsequently, the different aspects of the hot QCD medium for physical quark masses are discussed; the EoS, the nature and the temperature of transition, and the behaviour of various thermodynamic observables in the different phases. In the study of thermodynamics, the contribution of the lighter

The second section is about lattice QCD at finite density, where there is an inherent short coming of the lattice algorithms due to the so-called sign problem. A brief overview of the different methods used, and those being developed by the lattice practitioners to circumvent this problem, is given. It is an active field of research, with a lot of understanding of the origin and the severity of this problem gained in recent years, which is motivating the search for its possible cure. In the regime where the density of baryons is not too large, which is being probed by the experiments at RHIC, lattice techniques have been used successfully to produce some interesting results. One such important proposal in the recent time is the first principles determination of the chemical freezeout curve using experimental data on the electric charge fluctuations. This and the lattice results on the fluctuations of different quantum numbers in the hot medium and the EoS at finite baryon density are discussed in detail. An important feature of the QCD phase diagram is the possible presence of a critical end-point for the chiral first order transition. Since critical end-point search is one of the main objectives at RHIC, I have reviewed the current lattice results on this topic. The presence of the critical end-point is still not conclusively proven from lattice studies. It is a very challenging problem and I mention the further work in progress to address this problem effectively. Fermions with exact chiral symmetry on the lattice are important in this context. I have discussed the recent successful development to construct fermion operators that have exact chiral symmetry even at finite density which would be relevant for future studies on the critical end-point. The signatures of the critical end-point could be detected in the experiments if the critical region is not separated from the freezeout curve. It is thus crucial to estimate the curvature of the critical line from first principles, and I devote an entire subsection to discuss the lattice results on this topic.

I apologize for my inability to include all the pioneering works that have firmly established this subject and also to review the extensive set of interesting contemporary works. For a comprehensive review of the current activity in lattice thermodynamics, at finite temperature and density, I refer to the excellent review talks of the lattice conference, 2012 [

The starting point of any thermodynamic study is the partition function. The QCD partition function for

With the choice of a suitable gauge and the fermion operators on the lattice, different physical observables are measured on statistically independent configurations generated using suitable Monte-Carlo algorithms. To make connection with the continuum physics, one needs to take the

To characterize different phases, one needs to define a suitable order parameter which depends on the symmetries of the theory. In the limit of infinitely heavy quark masses, QCD is just a pure gauge theory with an exact order parameter; the expectation value of the Polyakov loop is given as

Based on effective field theories with same symmetries as QCD, using universality arguments and renormalization group inspired techniques, a schematic diagram of different phases of QCD as a function of quark mass is summarized in the famous “Columbia plot” [

The present status of the Columbia plot.

In the following subsections, the lattice results for the QCD EoS for physical quark masses are discussed, which is an input for the hydrodynamics of the QGP medium. The current results on the pseudocritical temperature, the entropy density, and the speed of sound are also shown. All the results are for

The Equation of State (EoS) is the relation between the pressure and energy density of a system in thermal equilibrium. For estimating the QCD EoS, the most frequently used method by the lattice practitioners is the integral method [

The results for the trace anomaly are available for different lattice discretizations of the fermions. For staggered quarks, there are two sets of results, one from the HotQCD collaboration using HISQ discretization [

The results for the trace anomaly using the HISQ action for low (a) and high (b) temperatures for lattice sizes with temporal extent

The latest data with the stout smeared fermions (a), from [

There are lattice results for the EoS using alternative fermion discretizations, the Wilson fermions. The WHOT-QCD collaboration has results for

The results for the pressure, energy density, and the trace anomaly with clover-improved Wilson fermions on a

We recall that the QCD transition, from a phase of color singlet states to a phase of colored quantum states, is an analytic crossover, for physical quark masses. This is fairly well established by now from lattice studies using two different approaches. One approach is to monitor the behaviour of the thermodynamic observables in the transition region for physical values of quark masses while the other is to map out the chiral critical line as a function of light quark mass [

The results for the subtracted chiral condensate (a) and the renormalized Polyakov loop (b) from the HotQCD collaboration, from [

The results for the EoS and the pseudocritical temperature discussed so far have been obtained using different improved versions of the staggered quarks. For these fermion species, the so called “rooting” problem may alter the continuum limit due to breaking of the

The continuum extrapolated renormalized chiral condensate (a) and the Polakov loop (b) are compared for Wilson and stout-smeared staggered fermions, from [

Since the effects of chiral symmetry persist in the crossover region, it is important to compare the existing results for

The renormalized chiral condensate for the overlap quarks is compared to the continuum extrapolated results using the stout smeared staggered quarks in (a), from [

Thermodynamic observables characterize the different phases across a phase transition. From the behaviour of these observables, one can infer about the degrees of freedom of the different phases and the nature of the interactions among the constituents. It was already known from an important lattice study that the pressure in high temperature phase of QCD showed a strong dependence on the number of quark flavours [

The entropy density, pressure, and the speed of sound for the stout-smeared fermions as a function of temperature, from [

The effects of charm quarks to the pressure in the QGP phase were estimated sometime ago, using next-to leading order perturbation theory [

In (a), the effects of quenched charm quark to the pressure, energy density, and trace anomaly are shown as a function of temperature, from [

The chiral phase transition for

The order parameter that characterizes the chiral phase transition is the chiral condensate. A suitable dimensionless definition of the chiral condensate used in the lattice study by the BNL-Bielefeld collaboration [

The interpolated data for

The QCD partition function breaks

From dilute instanton gas approximation,

Analyticity of

If

If

In fact, to understand the effect of anomaly, it is desirable to use fermions with exact chiral symmetry on the lattice. The overlap and the domain wall fermions are such candidates, for which the chiral anomaly can be defined. Indeed, the overlap fermions satisfy an exact index theorem on the lattice [

The susceptibilities for different meson quantum states constructed with the domain wall fermions are shown as a function of temperature in (a), from [

A recent theoretical study [

In (a), the quark mass dependence of eigenvalue distribution for the overlap quarks is compared at different temperatures, from [

With the chiral fermions, the fate of

The density of eigenvalues at

QCD with a finite number of baryons is relevant for the physics of neutron stars and supernovae. It is the theoretical setup for the heavy ion physics phenomena occurring at low center of mass energy,

To explain these experimental results from first principles, we need to extend the lattice QCD formulation to include the information of finite baryon density. One of the methods is to work in a grand canonical ensemble. In such an ensemble, the partition function is given by

reweighting of the

Taylor series expansion [

canonical ensemble method [

imaginary chemical potential approach [

complex Langevin algorithm [

worm algorithms [

The Taylor series method has been widely used in the lattice QCD studies in the recent years, which has led to interesting results relevant for the experiments. One such proposal is the determination of the line of chemical freezeout for the hadrons in the phase diagram at small baryon density, from first principles lattice study. It was first proposed that cumulants of baryon number fluctuations could be used for determining the freezeout parameters [

The contribution of the

The studies of fluctuations of the conserved charges are important to understand the nature of the degrees of freedom in a thermalized medium and the interactions among them [

To relate to the results of the heavy ion experiments at a lower collision energy,

The higher order susceptibilities

Other important quantities of relevance are the off-diagonal susceptibilities. These defined as

The HISQ data for

To relate the results from heavy ion experiments with the lattice data, it is crucial to map the center of mass energy of the colliding nuclei in the heavy ion collisions,

In (a), the leading term for

The estimates of the critical point from lattice studies are shown in (a), from [

It is known from models with the same symmetries as QCD that the chiral phase transition at

The other results for the critical point were obtained using the Taylor series method. In this method, the baryon number susceptibility at finite density is expanded in powers of

Though there is growing evidence in support for the existence of the critical end-point, the systematics for all these lattice studies are still not under control. It would be desirable to follow a different strategy to determine its existence. The alternate method suggested [

It is equally important to understand the possible experimental signatures of the critical point. The search of the critical end-point is one of the important aims for the extensive BES program at RHIC. In a heavy ion experiment, one measures the number of charged hadrons at the chemical freezeout and its cumulants. During the expansion of the fireball, the hot and dense QCD medium would pass through the critical region and cool down eventually forming hadrons. If the freezeout and the critical regions are far separated, the system would have no memory of the critical fluctuations and the baryon number susceptibility measured from the experiments could be consistent with the predictions from thermal HRG models which has no critical behaviour. If the freezeout region is within the critical region, the critical fluctuations would be larger than the thermal fluctuations. It is thus important to estimate the chiral critical line for QCD from first principles. The curvature of the chiral critical line has been estimated by the BNL-Bielefeld collaboration [

In (a), the width of the pseudocritical region for chiral condensate is shown as a blue curve and that for strange quark susceptibility is shown as a red curve, from [

Another complimentary study about the fate of the critical region at finite density was done by the Budapest-Wuppertal group [

It was noted that the higher order fluctuations are more strongly dependent on the correlation length of the system [

The EoS at finite density would be the important input for understanding the hydrodynamical evolution of the fireball formed at low values of the collisional energy, at the RHIC and the future experiments at FAIR and NICA. It is believed that there is no generation of entropy once the fireball thermalizes [

The EoS for different isentropes using asqtad quarks is shown in (a), from [

As emphasized in the introduction, I have tried to compile together some of the important instances to show that the lattice results have already entered into the precision regime with different fermion discretizations giving consistent continuum results for the pseudocritical temperature and fluctuations of different quantum numbers. The continuum result for the EoS would be available in very near future, with consistency already observed for different discretizations. The lattice community has opened the door for a very active collaboration between the theorists and experimentalists. With the EoS as an input, one can study the phenomenology of the hot and dense matter created at the heavy ion colliders. On the hand, there is a proposal of nonperturbative determination of the freezeout curve using lattice techniques, once the experimental data on cumulants of the charged hadrons are available.

A good understanding of the QCD phase diagram at zero baryon density has been achieved from the lattice studies. While the early universe transition from the QGP to the hadron phase is now known to be an analytic crossover and not a real phase transition, it is observed that the chiral dynamics will have observable effects in the crossover region. One of the remnant effects of the chiral symmetry would be the presence of a critical end-point. The search for the still elusive critical endpoint is one of the focus areas of lattice studies, and the important developments made so far in this area are reviewed.

While QCD at small baryon density is reasonably well understood with lattice techniques, the physics of baryon rich systems cannot be formulated satisfactorily on the lattice due to the infamous sign-problem. A lot of conceptual work, in understanding the severity and consequences of the sign problem as well algorithmic developments in circumventing this problem is ongoing which is one of the challenging problems in the field of lattice thermodynamics.

Sayantan Sharma would like to thank all the members of the Theoretical Physics Group at Bielefeld University, and in particular, Frithjof Karsch, Edwin Laermann, Olaf Kaczmarek, and Christian Schmidt for a lot of discussions that have enriched the author’s knowledge about QCD thermodynamics and lattice QCD. The author expresses gratitude to Edwin Laermann for a careful reading of the paper and his helpful suggestions and Toru Kojo and Amaresh Jaiswal for their constructive criticism that has led to a considerable improvement of this paper. The author also acknowledges Rajiv Gavai and Rajamani Narayanan for very enjoyable collaboration, in which the author learnt many aspects of the subject.

_{B}= 0 versus phase quenched QCD