The third moments of conserved charges, the baryon and electric charge numbers, and energy, as well as their mixed moments, carry more information on the state around the QCD phase boundary than previously proposed fluctuation observables and higher order moments. In particular, their signs give plenty of information on the location of the state created in relativistic heavy ion collisions in the temperature and baryon chemical potential plane. We demonstrate this with an effective model.

Baryon number cumulants are invaluable tools to diagnose the primordial stage of heavy ion collisions if they can be measured. In experiments, however, proton number cumulants have been measured as substitutes. In fact, proton number fluctuations are further modified in the hadron phase and are different from those of the baryon number. We show that the isospin distribution of nucleons at kinetic freeze-out is binomial and factorized. This leads to formulas that express the baryon number cumulants solely in terms of proton number fluctuations, which are experimentally observable.

We explore the relation between proton and nucleon number fluctuations in the final state in relativistic heavy ion collisions. It is shown that the correlations between the isospins of nucleons in the final state are almost negligible over a wide range of collision energy. This leads to a factorization of the distribution function of the proton, the neutron, and their antiparticles in the final state with binomial distribution functions. Using the factorization, we derive formulas to determine nucleon number cumulants, which are not direct experimental observables, from proton number fluctuations, which are experimentally observable in event-by-event analyses. With a simple treatment for strange baryons, the nucleon number cumulants are further promoted to the baryon number ones. Experimental determination of the baryon number cumulants makes it possible to compare various theoretical studies directly with experiments. The effects of nonzero isospin density on this formula are addressed quantitatively. It is shown that the effects are well suppressed over a wide energy range.

The energy density and the pressure of SU(3) gauge theory at finite temperature are studied by direct lattice measurements of the renormalized energy-momentum tensor obtained by the gradient flow. Numerical analyses are carried out with $\beta =6.287–7.500$ corresponding to the lattice spacing $a=0.013–0.061\mathrm{fm}$. The spatial (temporal) sizes are chosen to be $N_s=64$, 96, 128 ($N_{\tau }=12$, 16, 20, 22, 24) with the aspect ratio, $5.33\le N_s/N_{\tau }\le 8$. Double extrapolation, $a\to 0$ (the continuum limit) followed by $t\to 0$ (the zero flow-time limit), is taken using the numerical data. Above the critical temperature, the thermodynamic quantities are obtained with a few percent precision including statistical and systematic errors. The results are in good agreement with previous high-precision data obtained by using the integral method.

A novel method to study the bulk thermodynamics in lattice gauge theory is proposed on the basis of the Yang-Mills gradient flow with a fictitious time $t$. The energy density $ϵ$ and the pressure $P$ of $SU(3)$ gauge theory at fixed temperature are calculated directly on ${32}^{3}\ifmmode\times\else\texttimes\fi{}(6,8,10)$ lattices from the thermal average of the well-defined energy-momentum tensor ${T}_{$\mu${}$\nu${}}^{R}(x)$ obtained by the gradient flow. It is demonstrated that the continuum limit can be taken in a controlled manner from the $t$ dependence of the flowed data.

The energy-momentum tensor plays an important role in QCD thermodynamics. Its expectation value contains information of the pressure and the energy density as its diagonal part. Further properties like viscosity and specific heat can be extracted from its correlation function. A nonperturbative evaluation of it on the lattice is called. Recently, a new method based on the gradient flow was introduced to calculate the energy-momentum tensor on the lattice and has been successfully applied to quenched QCD. In this paper, we apply the gradient flow method to calculate the energy-momentum tensor in ($2+1$)-flavor QCD adopting a nonperturbatively $O(a)$-improved Wilson quark action and the renormalization group-improved Iwasaki gauge action. As the first application of the method with dynamical quarks, we study at a single but fine lattice spacing $a\simeq 0.07\mathrm{fm}$ with heavy $u$ and $d$ quarks ($m_{\pi }/m_{\rho }\simeq 0.63$) and approximately physical $s$ quark ($m_{{\eta }_{ss}}/m_{\varphi }\simeq 0.74$). With the fixed-scale approach, temperature is varied by the temporal lattice size $N_t$ at a fixed lattice spacing. Performing simulations on lattices with $N_t=16$ to 4, the temperature range of $T\simeq 174–697\mathrm{MeV}$ is covered. We find that the results of the pressure and the energy density by the gradient flow method are consistent with the previous results using the $T$-integration method at $T\lesssim 280\mathrm{MeV}$ ($N_t\gtrsim 10$), while the results show disagreement at $T\gtrsim 350\mathrm{MeV}$ ($N_t\lesssim 8$), presumably due to the small-$N_t$ lattice artifact of $O((aT{)}^2)=O(1/N_t^2)$. We also apply the gradient flow method to evaluate the chiral condensate taking advantage of the gradient flow method that renormalized quantities can be directly computed avoiding the difficulty of explicit chiral violation with lattice quarks. We compute the renormalized chiral condensate in the $\overline{\mathrm{MS}}$ scheme at renormalization scale $\mu =2\mathrm{GeV}$ with a high precision to study the temperature dependence of the chiral condensate and its disconnected susceptibility. Even with the Wilson-type quark action which violates the chiral symmetry explicitly, we obtain the chiral condensate and its disconnected susceptibility showing a clear signal of pseudocritical temperature at $T\sim 190\mathrm{MeV}$ related to the chiral restoration crossover.