Szabolcs Borsanyi
We discuss recent developments in lattice QCD for the bulk thermodynamics of the transition. We review the current status of the equation of state, the possible implications of a magnetic field and the fluctuations of conserved charges, like the net electric charge or baryon number. We also show predictions for higher cumulants, which will be experimentally available in near future.
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We study the thermodynamic properties of QCD at nonzero isospin chemical potential using improved staggered quarks at physical quark masses. In particular, we discuss the determination of the equation of state at zero and nonzero temperatures and show results. Using the results for the isospin density $n_I$, we also determine the phase diagram in the $(n_I,T)$-plane.
We review results from lattice QCD calculations on the thermodynamics of strong-interaction matter with emphasis on input these calculations can provide to the exploration of the phase diagram and properties of hot and dense matter created in heavy ion experiments. This review is organized as follows: 1) Introduction, 2) QCD thermodynamics on the lattice, 3) QCD phase diagram at high temperature, 4) Bulk thermodynamics, 5) Fluctuations of conserved charges, 6) Transport properties, 7) Open heavy flavors and heavy quarkonia, 8) QCD in external magnetic fields, 9) Summary.
What Thermodynamics tells about QCD Plasma near Phase Transition
由于熵密度$s(T)$在QCD相变的临界温度$T_c$上的快速变化，高于$T_c$的压力$P(T)$和能量密度$e(T)$通常偏离它们的Stefan-Boltzmann值。我们将证明这两个解析和数值的一般类的$s(T)$一致的物理约束，并作出定性比较的结果与格点QCD数据。讨论了与$ds(T)/dT$有关的比热、声速等问题。
Lattice Results on QCD Thermodynamics
本文综述了有限温度下QCD的最新研究结果。重点讨论了对实验研究夸克-胶子等离子体有直接意义的观测量，即相变温度、2味和3味QCD的状态方程、重夸克自由能和强子质量的热效应。
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 those which require both conceptual as well as 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 wou…
The weak-coupling expansion of the QCD free energy is known to order g_s^6log{g_s}, however, the resulting series is poorly convergent at phenomenologically relevant temperatures. In this proceedings, I discuss hard-thermal-loop perturbation theory (HTLpt) which is a gauge-invariant reorganization of the perturbative expansion for gauge theories. I review a recent NNLO HTLpt calculation of QCD thermodynamic functions. I show that the NNLO HTLpt results are consistent with lattice data down to temperatures T~2T_c.
We compare lattice QCD results for appropriate combinations of net strangeness fluctuations and their correlations with net baryon number fluctuations with predictions from two hadron resonance gas (HRG) models having different strange hadron content. The conventionally used HRG model based on experimentally established strange hadrons fails to describe the lattice QCD results in the hadronic phase close to the QCD crossover. Supplementing the conventional HRG with additional, experimentally uncharted strange hadrons predicted by quark model calculations and observed in lattice QCD spectrum ca…
QCD Thermodynamics on lattice
我讨论了格点量子色动力学热力学在有限温度和密度下的跃迁性质、状态方程、静电荷屏蔽和高温下的介子谱函数等方面的最新进展。
We extend our work on QCD thermodynamics with 2+1 quark flavors at nonzero chemical potential to finer lattices with $N_t=6$. We study the equation of state and other thermodynamic quantities, such as quark number densities and susceptibilities, and compare them with our previous results at $N_t=4$. We also calculate the effects of the addition of the charm and bottom quarks on the equation of state at zero and nonzero chemical potential. These effects are important for cosmological studies of the early Universe.
Recently, we proposed a novel method to define and calculate the energy-momentum tensor (EMT) in lattice gauge theory on the basis of the Yang-Mills gradient flow [1]. In this proceedings, we summarize the basic idea and technical steps to obtain the bulk thermodynamic quantities in lattice gauge theory using this method for the quenched and $(2+1)$-flavor QCD. The revised results of integration measure (trace anomaly) and entropy density of the quenched QCD with corrected coefficients are shown. Furthermore, we also show the flow time dependence of the parts of EMT including the dynamical fer…
Theory and Phenomenology of the Polyakov loop in QCD Thermodynamics
将退禁闭跃迁的Polyakov圈模型和胶子等离子体热力学的准粒子模型综合起来，得到一类胶子在非平凡Polyakov圈背景中运动的模型。这些模型成功地解释了有限温度下SU（3）规范理论的临界行为和状态方程。两个敏感的晶格探针，可以区分模型之间的波利亚科夫环P和e-3 p。
Flavor and Quark Mass Dependence of QCD Thermodynamics
我们在时间尺度Nt=4的格点上用改进的规范和交错费米子作用量计算了2味和3味QCD的转变温度。当nf=2和3时，Tc分别为173（8）MeV和154（8）MeV。在三味QCD的情况下，我们证明了手征临界点，即一阶手征相变线的二阶端点，属于3d伊辛模型的普适类.
We calculate the thermodynamic functions of a quark-gluon plasma for general N_c and N_f to three-loop order using hard-thermal-loop perturbation theory. At this order, all the ultraviolet divergences can be absorbed into renormalizations of the vacuum, the HTL mass parameters, and the strong coupling constant.We show that at three loops, the results for the pressure and trace anomaly are in very good agreement with recent lattice data down to temperatures T~2T_c.
Large Nc Continuum Reduction and the Thermodynamics of QCD
值得注意的是，如果大的Nc连续约减适用于一个可观察的，那么该可观察的是独立于温度的所有温度低于某个临界值。这一事实，加上介子和胶球弱相互作用在大Nc的事实是用来作为基础的推导大Nc连续约化的手征凝聚体。这种推导的结构是相当一般的，可以扩展到广泛的一类可观的。
I review recent developments in QCD thermodynamics and collective excitations from the hard-thermal-loop effective theory. I begin by motivating the discussion with open questions from heavy-ion collisions. I then discuss a finite-temperature and -density calculation of QCD thermodynamics at NNLO from the hard-thermal-loop perturbation theory. Finally I discuss a recent exploration of generalizing the hard-thermal-loop framework to the (chromo)magnetic scale $g^2T$, from which a novel massless mode is uncovered.
Taste symmetry violations in staggered fermion formulations correlate strongly with the cut-off (lattice spacing) dependence in thermodynamic quantities. Better taste symmetry on the lattice can be achieved either by decreasing the lattice spacing and going to larger temporal extent in finite-temperature calculations, or by further improving the action. The highly improved staggered quark (HISQ) action offers a further degree of improvement and substantially reduces taste violations. We report on our studies of the 2+1 flavor QCD thermodynamics with the HISQ/tree action. By systematically comp…
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.
We summarize recent results on the hadron resonance gas description of QCD. In particular, we apply this approach to describe the equation of state and the vacuum expectation value of the Polyakov loop in several representations. Ambiguities related to exactly which states should be included are discussed.
HTL-resummed thermodynamics of hot and dense QCD: An update
本文回顾了在夸克-胶子等离子体的热力学中，通过由Phi可导的双圈近似得到的熵和密度的非微扰表达式来恢复硬热圈物理的建议.与最近解决的热QCD的大N_f极限进行了比较，并作了一些更新，特别是在夸克数可调性方面。
Monte-Carlo simulations of QCD Thermodynamics in the PNJL model
本文将Monte-Carlo方法应用于双味Polyakov圈扩展Nambu和Jona-Lasinio（PNJL）模型。通过这种方式，我们可以超越平均场计算引入场的波动。我们研究了涨落对模型热力学的影响。我们计算了热力学巨正则配分函数相对于化学势的二阶导数，并与晶格数据进行了比较。
Thermodynamics of strong coupling 2-color QCD with chiral and diquark condensates
2-本文讨论了在有限温度T和有限化学势μ下，在强耦合极限下的交错费米子格点上的色QCD（quantum chromodynamics with Nc =2）.利用平均场近似和1/d展开（d为空间维数），阐明了T、μ和夸克质量m空间中的相结构。重点讨论了手征凝聚< \bar{q} q>、双夸克凝聚<qq>和夸克密度<q^{ \dagger }q>在T-μ-m空间中的相互作用。在不假定μ或m很小的情况下，也导出了简单的解析公式。我们的结果与最近的双色QCD蒙特卡罗模拟的结果作了定性的比较。
QCD Thermodynamics and Neutral Pion in a Uniform Magnetic Field: Finite Volume Effects
本文利用次领头阶手征微扰理论研究了背景磁场中格点QCD计算的有限体积效应，计算了手征凝聚等热力学量并推导了中性π介子有效作用量，发现部分平均量与横向格点尺寸无关，但有限体积修正对多项物理量仍不可忽略。
We use results from a 6-th order Taylor expansion of the QCD equation of state to construct expansions for cumulants of conserved charge fluctuations and their correlations. We show that these cumulants strongly constrain the range of applicability of hadron resonance gas model calculations. We point out that the latter is inappropriate to describe equilibrium properties of QCD at zero and non-zero values of the baryon chemical potential already at T~155 MeV.
We study the hadron resonance gas model and describe the equation of state of QCD and the vacuum expectation value of the Polyakov loop in the confined phase, in terms of hadronic states with light quarks in the first case, and with exactly one heavy quark in the second case. Comparison with lattice simulations is made.
A precise lattice determination of the equation of state in SU(3) Yang-Mills theory is carried out by means of a simulation algorithm, based on Jarzynski's theorem, that allows one to compute physical quantities in thermodynamic equilibrium, by driving the field configurations of the system out of equilibrium. The physical results and the computational efficiency of the algorithm are compared with other state-of-the-art lattice calculations, and the extension to full QCD with dynamical fermions and to other observables is discussed.
Lectures given at the Summer School on "Modern perspectives in lattice QCD", Les Houches, August 3-28, 2009
QCD thermodynamics at zero and non-zero density
本文介绍了近物理轻夸克质量和物理奇夸克质量的QCD热力学的最新结果。这些计算是在改进的staggerd作用量下进行的，该作用量是专门为有限温度格点量子色动力学设计的。详细地说，我们提出了一个计算的转变温度，使用相结合的手性和连续外推。此外，我们提出的相互作用措施和能量密度在几乎现实的夸克质量的初步结果。最后讨论了压力对有限夸克化学势的响应。在泰勒展开形式下，我们计算了夸克数的可调性和对有限化学势的首阶修正。这对于绘制QCD相图中的临界区域特别有用。
The Polyakov-quark-meson (PQM) model, which combines chiral as well as deconfinement aspects of strongly interacting matter is introduced for three light quark flavors. An analysis of the chiral and deconfinement phase transition of the model and its thermodynamics at finite temperatures is given. Three different forms of the effective Polyakov loop potential are considered. The findings of the (2+1)-flavor model investigations are confronted to corresponding recent QCD lattice simulations of the RBC-Bielefeld, HotQCD and Wuppertal-Budapest collaborations. The influence of the heavier quark ma…
We compare recent lattice QCD calculations of higher order cumulants of net-strangeness fluctuations with hadron resonance gas (HRG) model calculations. Up to the QCD transition temperature Tc=( 154 +/- 9) MeV we find good agreement between QCD and HRG model calculations of second and fourth order cumulants, even when subtle aspects of net-baryon number, strangeness and electric charge fluctuations are probed. In particular, the fourth order cumulants indicate that also in the strangeness sector of QCD the failure of HRG model calculations sets in quite abruptly in the vicinity of the QCD tran…
Fluctuations of conserved charges allow to study the chemical composition of hadronic matter. A comparison between lattice simulations and the Hadron Resonance Gas (HRG) model suggested the existence of missing strange resonances. To clarify this issue we calculate the partial pressures of mesons and baryons with different strangeness quantum numbers using lattice simulations in the confined phase of QCD. In order to make this calculation feasible, we perform simulations at imaginary strangeness chemical potentials. We systematically study the effect of different hadronic spectra on thermodyna…
Dynamical lattice QCD thermodynamics and the U(1)_A symmetry with domain wall fermions
本文给出了在N_t=4时，考虑畴壁费米子的全、双味QCD热力学的数值模拟结果。数值模拟首次显示了完整的QCD相变的低温相与自发的手征对称性破缺，但完整的味对称性和高温相的完整的$SU(2) \times SU(2)$手征味对称性。发现在高温相转变附近，U（1）_A轴对称性仅发生少量破坏。由于U（1）_A对称性破缺与跃迁的期望阶数之间的联系，这个结果是特别有意义的。
Lattice QCD thermodynamics with Wilson quarks
本文综述了用动力学Wilson夸克进行的格点量子色动力学模拟对QCD热力学的研究。在解释了有限温度下Wilson夸克QCD的基本性质，包括相结构和手征相变附近的标度性质之后，我们讨论了临界温度、状态方程和重夸克自由能。
Equilibrium Thermodynamics of Lattice QCD
格点量子色动力学允许我们在非零温度和/或密度下模拟量子色动力学。这种平衡热力学计算与相对论性重离子碰撞的物理学有关。我简要回顾一下这一领域，重点是我们的工作。
In this talk we extend the Polyakov-quark-meson model to N_f=2+1 quark flavors and study its bulk thermodynamics at finite temperatures in mean-field approximation. Three different Polyakov-loop potentials are considered. Our findings are confronted to recent QCD lattice simulations of the RBC-Bielefeld and HotQCD collaborations. Furthermore, the finite chemical potential expansion of the quark-number susceptibility in a Taylor series around vanishing chemical potential is analyzed. By means of a novel algorithmic differentiation technique, we have calculated Taylor coefficients up to 24th ord…
Thermodynamics of 1-flavor QCD
我们提出的第一个结果，重到中等夸克质量，1味QCD的热力学性质的研究，使用多玻色子算法。
Thermodynamics of (2+1)-flavor QCD
我们报告我们的QCD热力学项目的状态。它在布鲁克海文国家实验室的QCDOC机器和比勒费尔德大学的APEnext机器上进行。在几乎真实的夸克质量下，我们用2+1味QCD公式计算了几个量子力学量。在这一过程中，我们展示了手征凝聚体和Polyakov圈，静态夸克势和空间弦张力。
QCD thermodynamics is considered using Wilson fermions in the fixed scale approach. The temperature dependence of the renormalized chiral condensate, quark number susceptibility and Polyakov loop is measured at four lattice spacings allowing for a controlled continuum limit. The light quark masses are fixed to heavier than physical values in this first study. Finite volume effects are ensured to be negligible by using approriately large box sizes. The final continuum results are compared with staggered fermion simulations performed in the fixed N_t approach. The same continuum renormalization …
Over the past few years new physics methods and algorithms as well as the latest supercomputers have enabled the study of the QCD thermodynamic phase transition using lattice gauge theory numerical simulations with unprecedented control over systematic errors. This is largely a consequence of the ability to perform continuum extrapolations with physical quark masses. Here we review recent progress in lattice QCD thermodynamics, focussing mainly on results that benefit from the use of physical quark masses: the crossover temperature, the equation of state, and fluctuations of the quark number s…
I give a brief introduction to the goals, challenges, and technical difficulties of lattice QCD thermodynamics and present some recent results from the HotQCD collaboration for the crossover temperature, equation of state, and other observables.
We discuss an effective Polyakov loop model for QCD thermodynamics with a chemical potential. Using high temperature expansion techniques the partition sum is mapped exactly onto the partition sum of a flux model. In the flux representation the complex action problem is resolved and a simulation with worm-type algorithms becomes possible also at finite chemical potential.
QCD strings and the thermodynamics of the metastable phase of QCD at large $N_c$
在大$N_C$下QCD亚稳强子相的热变性与有效QCD弦的性质有关。特别是，它表明，在大的$N_c$极限和附近的最大强子温度，$T_H$，能量密度和压力的亚稳相规模为${\cal E} \sim (T_H-T)^{-(D_\perp-6)/2}$（$D_\perp <6$）和$P \sim (T_H-T)^{-(D_\perp-4)/2}$（$D_\perp <4$），其中$D_\perp$是弦理论的横向尺寸的有效数量。然而，它示出，对于感兴趣的热力学量的限制$T \to T_H$和$N_c \to \infty$不交换。讨论了在中等大的$N_c$上通过亚稳强子相的晶格模拟提取$D_\perp$的前景。
In nonzero magnetic field, the magnetic properties and thermodynamics of the quantum-chromodynamic (QCD) matter is studied in the hadron resonance gas and the Polyakov linear-sigma models and compared with recent lattice calculations. Both models are fairly suited to describe the degrees of freedom in the hadronic phase. The partonic ones are only accessible by the second model. It is found that the QCD matter has paramagnetic properties, which monotonically depend on the temperature and are not affected by the hadron-quark phase-transition. Furthermore, raising the magnetic field strength inc…
Dynamical lattice QCD thermodynamics with domain wall fermions
本文给出了在N_t=4时，用畴壁费米子模拟两味QCD热力学的结果.与其他晶格费米子公式相反，畴壁费米子在有限的晶格间距（直到在一个额外的参数中指数地小）下保持连续体的完全手征对称性。在相变的正上方，我们发现轴向U（1）对称性仅被少量破缺。我们讨论了一个正在进行的计算，以确定使用畴壁费米子的相变的顺序和属性，因为全球对称性的理论，预计在这里是重要的。
Thermodynamics of Lattice QCD with massless quarks and chiral 4-fermion interactions
我们模拟了具有无关的4-费米子相互作用和2个零质量夸克的格点量子色动力学。观察到手性相变是二级的，我们讨论了临界指数的提取。
Thermodynamics of 2 and 3 flavour QCD
本文讨论了在夸克动力学自由度存在下QCD热力学的最新结果。特别是，我们集中在味和夸克质量依赖的QCD相图，状态方程和转变温度的分析。此外，我们提出了最近的结果，重夸克自由能。
Thermodynamics of fermionic excitations in heavy-quark QCD
本文在格点重夸克QCD中用hopping参数展开解析计算巨势与夸克数磁化率，揭示退禁闭相与禁闭相中四阶与二阶磁化率比值分别为1和9，并给出夸克及重子激发能的解析公式。
Dynamical QCD thermodynamics with domain wall fermions
本文给出了N_t=4时具有畴壁费米子的完全双味QCD热力学的数值模拟结果。对QCD全相变的数值模拟首次显示了低温相和高温相，低温相具有自发的手征对称性破缺，但味对称性完整，高温相具有SU（2）x SU（2）手征味对称性。
Recent Progress in Lattice QCD Thermodynamics
这篇评论对QCD热力学的晶格模拟的现状以及它告诉我们的关于热强子物质的内容进行了批判性的评估。它简要概述了在非零温度和零重子数密度下研究QCD的格点方法，特别强调评估和减少截止效应。它讨论了确定转变温度的方法的各种困难。它使用最近在文献和本次会议上报告的结果进行说明，特别是那些来自HotQCD合作进行的一项重大研究的结果。
In a previous paper (JHEP {\bf 05} (2014) 27), we calculated the three-loop thermodynamic potential of QCD at finite temperature $T$ and quark chemical potentials $μ_q$ using the hard-thermal-loop perturbation theory (HTLpt) reorganization of finite temperature and density QCD. The result allows us to study the thermodynamics of QCD at finite temperature and isospin chemical potential $μ_I$. We calculate the pressure, energy density, and entropy density, the trace anomaly, and the speed of sound at zero and nonzero $μ_I$. The second, fourth, and sixth-order isospin susceptibilities are calcula…
The hard-thermal-loop perturbation theory (HTLpt) framework is used to calculate the thermodynamic functions of a quark-gluon plasma to three-loop order. This is the highest order accessible by finite temperature perturbation theory applied to a non-Abelian gauge theory before the high-temperature infrared catastrophe. All ultraviolet divergences are eliminated by renormalization of the vacuum, the HTL mass parameters, and the strong coupling constant. After choosing a prescription for the mass parameters, the three-loop results for the pressure and trace anomaly are found to be in very good a…
Lattice QCD thermodynamic results with improved staggered fermions
我们给出了用两种不同的改进的动力学交错费米子作用量和几乎物理的夸克质量得到的QCD状态方程的结果。详细讨论了三种不同晶格间距下的晶格截止效应，即时间范围为$N_τ=4,6$和8的晶格上的结果。讨论了非零重子化学势的泰勒展开方法，给出了等熵状态方程。
Thermodynamics using p4-improved staggered fermion action on QCDOC
利用QCDOC超级计算机对改进的交错费米子$N_f=3$ QCD的热力学性质进行了探索性研究。我们使用p4操作和MILC风格的涂抹链接（Fat 7）。讨论了QCDOC上p4操作的实施细节，并给出了性能基准。我们给出了几种格点体积的赝临界温度$T_c$与夸克质量关系的初步结果.我们还比较了p4fat7和旧的p4作用。
QCD thermodynamics with $N_f=2+1$ near the continuum limit at realistic quark masses
本文报道了我们对具有2+1味动力学夸克的QCD热力学的研究。在这篇文章中，我们提出了几个热力学量和我们最近计算的临界温度。为了研究QCD在连续极限附近的热力学性质，我们在$N_t=4$和6格点上采用改进的交错（p4）夸克与树能级Symanzik改进胶耦合.模拟的奇夸克质量和轻夸克质量的物理值都在$m_q/m_s=0.05-0.4$的范围内。最轻的夸克质量相当于约150 MeV的π介子质量。
We study thermodynamic properties of 2+1 flavor QCD with improved Wilson quarks coupled with the RG improved Iwasaki glue, using the fixed scale approach. We present the results for the equation of state, renormalized Polyakov loop, and chiral condensate.
Heavy quark thermodynamics in full QCD
我们分析了QCD中静态夸克-反夸克对关联的大距离行为。计算了单重态自由能，并确定了熵对它的贡献，从而使我们能够计算出多余的内能。自由能在临界区域急剧下降，导致过剩熵和内能都出现尖峰。
We study QCD thermodynamics using two flavors of dynamical overlap fermions with quark masses corresponding to a pion mass of 350 MeV. We determine several observables on N_t=6 and 8 lattices. All our runs are performed with fixed global topology. Our results are compared with staggered ones and a nice agreement is found.
Polyakov loops $L_a(T), a=3,8,...$ are shown to give the most important nonperturbative contribution to the thermodynamic potentials. Derived from the gluonic field correlators they enter as factors into free energy. It is shown in the $SU(3)$ case that $L_a (T)$ define to a large extent the behavior of the free energy and the trace anomaly $I(T)$, most sensitive to nonperturbative effects.
We discuss recent progress in lattice QCD studies on various aspects involving strange quarks. Appropriate combinations of conserved net strange and net charm fluctuations and their correlations with other conserved charges provide evidence that in the hadronic phase so far unobserved hadrons contribute to the thermodynamics and need to be included in hadron resonance gas models. In the strange sector this leads to significant reductions of the chemical freeze-out temperature of strange hadrons. In this context, a discussion of data from heavy-ion collisions at SPS, RHIC and LHC on the chemica…
QCD Thermodynamics with 2 and 3 Quark Flavors
本文讨论了在时间范围Nt=4的格点上，用改进的规范和交错费米子作用量计算的2，2+1和3味QCD中压力和临界温度的味依赖性。当T > 2Tc时，我们发现含2个轻夸克和1个重奇夸克的QCD的体热力学性质可以用3味QCD很好地描述，而转变温度更接近于2味QCD。进一步证明了三味QCD的手征临界点，即一阶手征相变线的二阶端点，属于3d伊辛模型的普适类.
Two-flavor QCD Thermodynamics using Anisotropic Lattices
在各向异性晶格上进行全量子色动力学的数值模拟，可以在固定物理尺度和减小晶格间距误差的条件下，方便地研究量子色动力学热力学。我们报告的计算结果与两种口味的动态交错费米子，其中所有裸参数和重整化各向异性保持不变，温度变化的小步骤，只改变时间片的数量。包括零温标设置模拟的结果，它确定的Karsch系数，允许在有限温度下的状态方程的计算。
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