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Register a new userExploring Nf=2+1 QCD thermodynamics from the gradient flow
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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.
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We study thermodynamic properties of Nf=2+1 QCD on the lattice adopting O(a)-improved Wilson quark action and Iwasaki gauge action. To cope with the problems due to explicit violation of the Poincare and chiral symmetries, we apply the Small Flow-time eXpansion (SFtX) method based on the gradient flow, which is a general method to correctly calculate any renormalized observables on the lattice. In this method, the matching coefficients in front of operators in the small flow-time expansion are calculated by perturbation theory. In a previous study using one-loop matching coefficients, we found that the SFtX method works well for the equation of state, chiral condensates and susceptibilities. In this paper, we study the effect of two-loop matching coefficients by Harlander et al. We also test the influence of the renormalization scale in the SFtX method. We find that, by adopting the mu_0 renormalization scale of Harlander et al. instead of the conventional mu_d=1/sqrt{8t} scale, the linear behavior at large t is improved so that we can perform the t -> 0 extrapolation of the SFtX method more confidently. In the calculation of the two-loop matching coefficients by Harlander et al., the equation of motion for quark fields was used. For the entropy density in which the equation of motion has no effects, we find that the results using the two-loop coefficients agree well with those using one-loop coefficients. On the other hand, for the trace anomaly which is affected by the equation of motion, we find discrepancies between the one- and two-loop results at high temperatures. By comparing the results of one-loop coefficients with and without using the equation of motion, the main origin of the discrepancies is suggested to be attributed to O((aT)^2)=O(1/N_t^2) discretization errors in the equation of motion at N_t =< 10.
To obtain the precise values of the bulk quantities and transport coefficients in quark-gluon-plasma phase, we propose that a direct calculation of the renormalized energy-momentum tensor (EMT) on the lattice using the gradient flow. From one-point function of EMT, authors in Ref.[1] obtained the interaction measure and thermal entropy. The results are consistent with the one obtained by the integral method. Based on the success, we try to measure the two-point function of EMT, which is related to the transport coefficients. Advantages of our method are (1) a clear signal because of the smearing effects of the gradient flow and (2) no need to calculate the wave function renormalization of EMT. In addition, we give a short remark on a comparison of the numerical cost between the positive- and adjoint-flow methods for fermions, needed to obtain the EMT in the (2+1) flavor QCD.
We compute the topological charge and its susceptibility in finite temperature (2+1)-flavor QCD on the lattice applying a gradient flow method. With the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson quarks, we perform simulations on a fine lattice with~$asimeq0.07,mathrm{fm}$ at a heavy $u$, $d$ quark mass with $m_pi/m_rhosimeq0.63$ but approximately physical $s$ quark mass with $m_{eta_{ss}}/m_phisimeq0.74$. In a temperature range from~$Tsimeq174,mathrm{MeV}$ ($N_t=16$) to $697,mathrm{MeV}$ ($N_t=4$), we study two topics on the topological susceptibility. One is a comparison of gluonic and fermionic definitions of the topological susceptibility. Because the two definitions are related by chiral Ward-Takahashi identities, their equivalence is not trivial for lattice quarks which violate the chiral symmetry explicitly at finite lattice spacings. The gradient flow method enables us to compute them without being bothered by the chiral violation. We find a good agreement between the two definitions with Wilson quarks. The other is a comparison with a prediction of the dilute instanton gas approximation, which is relevant in a study of axions as a candidate of the dark matter in the evolution of the Universe. We find that the topological susceptibility shows a decrease in $T$ which is consistent with the predicted $chi_mathrm{t}(T) propto (T/T_{rm pc})^{-8}$ for three-flavor QCD even at low temperature $T_{rm pc} < Tle1.5 T_{rm pc}$.
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 (epsilon) and the pressure (P) of SU(3) gauge theory at fixed temperature are calculated directly on 32^3 x (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 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 ($N_tmathrel{rlap{raise 0.511ex hbox{$>$}}{lower 0.511ex hbox{$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.
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