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We study the latent heat and the pressure gap between the hot and cold phases at the first-order transition temperature $T=T_c$ of SU(3) Yang-Mills theory, using the small flow-time expansion (SF$t$X) method based on the gradient flow. We first examine alternative procedures in the SFtX method -- the order of the continuum and vanishing flow-time extrapolations. We confirm that the final results adopting the two orders, as well as other alternatives in which the perturbative order of the matching coefficients and the renormalization scale of the flow scheme are varied, are all consistent with each other. We also confirm $\Delta p$ is consistent with zero, as expected from the dynamical balance of two phases at $T_c$. For the latent heat in the continuum limit, we find $\Delta \epsilon /T^4 = 1.117(40)$ for the spatial volume $L^3$ corresponding to the aspect ratio $N_s/N_t=T_cL=8$ and $1.349(38)$ for $N_s/N_t=6$. From hysteresis curves, we show that the entropy density in the hot phase is sensitive to the spatial volume, while that in the confined phase is insensitive., Comment: 9 pages, 6 figures, poster presented at the 38th International Symposium on Lattice Field Theory (Lattice 2021), July 26-30 2021, Zoom/Gather@MIT, USA

We study latent heat and the pressure gap between the hot and cold phases at the first-order deconfining phase transition temperature of the SU(3) Yang-Mills theory. Performing simulations on lattices with various spatial volumes and lattice spacings, we calculate the gaps of the energy density and pressure using the small flow-time expansion (SFtX) method. We find that the latent heat $\Delta \epsilon$ in the continuum limit is $\Delta \epsilon /T^4 = 1.117 \pm 0.040$ for the aspect ratio $N_s/N_t=8$ and $1.349 \pm 0.038$ for $N_s/N_t=6$ at the transition temperature $T=T_c$. We also confirm that the pressure gap is consistent with zero, as expected from the dynamical balance of two phases at $T_c$. From hysteresis curves of the energy density near $T_c$, we show that the energy density in the (metastable) deconfined phase is sensitive to the spatial volume, while that in the confined phase is insensitive. Furthermore, we examine the effect of alternative procedures in the SFtX method - the order of the continuum and the vanishing flow-time extrapolations, and also the renormalization scale and higher-order corrections in the matching coefficients. We confirm that the final results are all very consistent with each other for these alternatives., Comment: 29 pages, 22 figures

We study the endpoint of the first order deconfinement phase transition of 2 and 2+1 flavor QCD in the heavy quark region. We perform simulations of quenched QCD and apply the reweighting method to study the heavy quark region. The quark determinant for the reweighting is evaluated by a hopping parameter expansion. To reduce the overlap problem, we introduce an external source term of the Polyakov loop in the simulation. We study the location of critical point at which the first order phase transition changes to crossover by investigating the histogram of the Polyakov loop and applying the finite-size scaling analysis. We estimate the truncation error of the hopping parameter expansion, and discuss the lattice spacing dependence and the spatial volume dependence in the result of the critical point., Comment: 7 pages, 7 figures, 1 table, talk presented at the 37th International Symposium on Lattice Field Theory - Lattice 2019, 16-22 June 2019, Wuhan, China

We study the end point of the first-order deconfinement phase transition in two and 2+1 flavor QCD in the heavy quark region of the quark mass parameter space. We determine the location of critical point at which the first-order deconfinement phase transition changes to crossover, and calculate the pseudo-scalar meson mass at the critical point. Performing quenched QCD simulations on lattices with the temporal extents Nt=6 and 8, the effects of heavy quarks are determined using the reweighting method. We adopt the hopping parameter expansion to evaluate the quark determinants in the reweighting factor. We estimate the truncation error of the hopping parameter expansion by comparing the results of leading and next-to-leading order calculations, and study the lattice spacing dependence as well as the spatial volume dependence of the result for the critical point. The overlap problem of the reweighting method is also examined. Our results for Nt=4 and 6 suggest that the critical quark mass decreases as the lattice spacing decreases and increases as the spatial volume increases., Comment: 25 pages, 31 figures, 8 tables

We study energy gap (latent heat) between the hot and cold phases at the first order phase transition point of the SU(3) gauge theory. Performing simulations on lattices with various spatial volumes and lattice spacings, we calculate the energy gap by a method using the Yang-Mills gradient flow and compare it with that by the conventional derivative method., Comment: 7 pages, 10 figures, talk presented at the 36th Annual International Symposium on Lattice Field Theory (Lattice 2018)

We study the energy-momentum tensor and the equation of state as well as the chiral condensate in (2+1)-flavor QCD at the physical point applying the method of Makino and Suzuki based on the gradient flow. We adopt a nonperturbatively O(a)-improved Wilson quark action and the renormalization group-improved Iwasaki gauge action. At Lattice 2016, we have presented our preliminary results of our study in (2+1)-flavor QCD at a heavy u, d quark mass point. We now extend the study to the physical point and perform finite-temperature simulations in the range T \simeq 155--544 MeV (Nt = 4--14 including odd Nt's) at a \simeq 0.09 fm. We show our final results of the heavy QCD study and present some preliminary results obtained at the physical point so far., Comment: 8 pages, 15 figures, talk presented at the 35th International Symposium on Lattice Field Theory (LATTICE 2017), 18-24 June 2017, Granada, Spain

We calculate the energy gap (latent heat) and pressure gap between the hot and cold phases of the SU(3) gauge theory at the first order deconfining phase transition point. We perform simulations around the phase transition point with the lattice size in the temporal direction $N_t=6,$ 8 and 12 and extrapolate the results to the continuum limit. The energy density and pressure are evaluated by the derivative method with nonperturabative anisotropy coefficients. We find that the pressure gap vanishes at all values of $N_t$. The spatial volume dependence in the latent heat is found to be small on large lattices. Performing extrapolation to the continuum limit, we obtain $\Delta \epsilon/T^4 = 0.75 \pm 0.17$ and $\Delta (\epsilon -3 p)/T^4 = 0.623 \pm 0.056.$ We also tested a method using the Yang-Mills gradient flow. The preliminary results are consistent with those by the derivative method within the error., Comment: 7 pages, 6 figures, talk presented at the 34th International Symposium on Lattice Field Theory (LATTICE 2016), 24-30 July, 2016, University of Southampton, UK

We study temperature dependence of the topological susceptibility with the $N_{f}=2+1$ flavors Wilson fermion. We have two major interests in this paper. One is a comparison of gluonic and fermionic definitions of the topological susceptibility. Two definitions are related by the chiral Ward-Takahashi identity but their coincidence is highly non-trivial for the Wilson fermion. By applying the gradient flow both for the gauge and quark fields we find a good agreement of these two measurements. The other is a verification of a prediction of the dilute instanton gas approximation at low temperature region $T_{pc}< T<1.5T_{pc}$, for which we confirm the prediction that the topological susceptibility decays with power $\chi_{t}\propto(T/T_{pc})^{-8}$ for three flavors QCD., Comment: 6 pages, 8 figures. Talk presented at the 34th International Symposium on Lattice Field Theory, 24-30 July 2016, University of Southampton, UK

The energy-momentum tensor and equation of state are studied in finite-temperature (2+1)-flavor QCD with improved Wilson quarks using the method proposed by Makino and Suzuki based on the gradient flow. We find that the results of the gradient flow are consistent with the previous results using the $T$-integration method at $T \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 280$ MeV ($N_t\mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}}10$), while a disagreement is found at $T \mathrel{\rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}} 350$ MeV ($N_t \mathrel{\rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}} 8$) presumably due to the small-$N_t$ lattice artifact. We also report on the results on the renormalized chiral condensate and its disconnected susceptibility using the method of Hieda and Suzuki. The results show a clear signal of the expected chiral restoration crossover even with Wilson-type quarks which violate the chiral symmetry explicitly., Comment: 7 pages, 10 figures, talk presented at the 34th International Symposium on Lattice Field Theory (LATTICE 2016), 24-30 July 2016, University of Southampton, UK

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. 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. As the first application of the method with dynamical quarks, we study at a single but fine lattice spacing a=0.07 fm with heavy u and d quarks ($m_\pi/m_\rho=0.63$) and approximately physical s quark. Performing simulations on lattices with Nt=16 to 4, the temperature range of T=174-697 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<280 MeV, while the results show disagreement at T>350 MeV (Nt<8), presumably due to the small-Nt 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 MS-bar scheme at renormalization scale $\mu=2$ 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, we obtain the chiral condensate and its disconnected susceptibility showing a clear signal of pseudocritical temperature at T~190 MeV related to the chiral restoration crossover., Comment: 49 pages, 20 figures, published version, a typo in Eq. (C7) is corrected; all the simulation results are not changed

We calculate the energy gap (latent heat) and pressure gap between the hot and cold phases of the SU(3) gauge theory at the first order deconfining phase transition point. We perform simulations around the phase transition point with the lattice size in the temporal direction Nt=6, 8 and 12 and extrapolate the results to the continuum limit. We also investigate the spatial volume dependence. The energy density and pressure are evaluated by the derivative method with non-perturabative anisotropy coefficients. We adopt a multi-point reweighting method to determine the anisotropy coefficients. We confirm that the anisotropy coefficients approach the perturbative values as Nt increases. We find that the pressure gap vanishes at all values of Nt when the non-perturbative anisotropy coefficients are used. The spatial volume dependence in the latent heat is found to be small on large lattices. Performing extrapolation to the continuum limit, we obtain $ \Delta \epsilon/T^4 = 0.75 \pm 0.17 $ and $ \Delta (\epsilon -3 p)/T^4 = 0.623 \pm 0.056.$, Comment: 20 pages, 20 figures

We perform lattice numerical simulations to study the phase transition of QCD at finite temperature to clarify the nature of the transition of massless two flavor QCD. We investigate QCD with two light and Nf heavy quarks instead of two-flavor QCD, and focus on the light quark mass dependence of the critical heavy mass, below which the transition is of first order. The heavy quarks are incorporated into two flavor configurations in the form of the hopping parameter expansion through the reweighting technique. The nature of the transition is identified by the shape of the constraint effective potential at the critical temperature. Our result indicates that the critical heavy mass remains finite in the chiral limit of the two flavors, suggesting the phase transition of massless two-flavor QCD is of second order., Comment: 7 pages, 4 figures, Proceedings of the 33rd International Symposium on Lattice Field Theory (LATTICE2015), Kobe, Japan, 14-18 July 2015

Aiming to understand the phase structure of lattice QCD at nonzero temperature and density, we study the phase transitions of QCD in an extended parameter space, where the number of flavor and quark masses are considered as parameters. Performing simulations of 2 flavor QCD and using the reweighting method, we investigate (2+Nf) flavor QCD at finite density, where two light flavors and Nf massive flavors exist. Calculating probability distribution functions, we determine the critical surface terminating first order phase transitions in the parameter space of the light quark mass, the heavy quark mass and the chemical potential. Through the study of the many flavor system, we discuss the phase structure of QCD at finite density., Comment: 4 pages, 5 figures, Talk presented at Quark Matter 2015. arXiv admin note: text overlap with arXiv:1512.08317

We discuss the QCD critical point at finite density through the study of many flavor QCD, in which two light flavors and Nf massive flavors exist. Performing simulations of QCD with two flavors of improved Wilson fermions, we calculate probability distribution functions of many flavor QCD at finite temperature and density. The dynamical effects of massive flavors and the chemical potential are added using the reweighting technique. From the shape of the distribution functions, we determine the critical surface separating the first order transition and crossover regions in the parameter space of the light and massive quark masses and the chemical potentials. It is found that the critical massive quark mass becomes larger as the chemical potential increases in (2+Nf) flavor QCD. The indication to the (2+1) flavor QCD is then discussed., Comment: 7 pages, 10 figures, Proceedings of the 33rd International Symposium on Lattice Field Theory (Lattice 2015), 14-18 July 2015, Kobe, Japan

Chiral phase transition of two flavor QCD at finite quark masses is known to be crossover except near the chiral limit, but it can turn to a first order transition when adding many extra flavors. This property is used to explore the nature of the phase transition of massless two flavor QCD using lattice numerical simulations. The extra heavy flavors being incorporated in the form of the hopping parameter expansion through the reweighting, the number of the extra flavors and their masses appear only in a single parameter, defined by $h$. We determine the critical value of the parameter, at which the first order and the crossover regions are separated, and examine its dependence on the two flavor mass. The lattice calculations are carried out at $N_t$=4, and show that the critical value does not depend on the two flavor mass in the range we have studied ($0.46 \le m_\pi/m_\rho \le 0.66$) and appears to remain finite and positive in the chiral limit, suggesting that the phase transition of massless two flavor QCD is of second order., Comment: 30 pages, 13 figures. v2:published version

We study a reweighting method aiming at numerical studies of QCD at finite density, in which the conventional Monte-Carlo method cannot be applied directly. One of the most important problems in the reweighting method is the overlap problem. To solve it, we propose to perform simulations at several simulation points and combine their results in the data analyses. In this report, we introduce this multipoint reweighting method and test if the method works well by measuring histograms of physical quantities. Using this method, we calculate the meson masses as continuous functions of the gauge coupling beta and the hopping parameters kappa in QCD at zero density. We then determine lines of constant physics in the (beta, kappa) space and evaluate the derivatives of the lattice spacing with respect to beta and kappa along the lines of constant physics (inverse of the beta functions), which are needed in a calculation of the equation of state., Comment: 7 pages, 9 figures, talk presented at the 32nd International Symposium on Lattice Field Theory - Lattice 2014, June 23-28, 2014, Columbia University New York, NY