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We study thermodynamic properties of Nf=2+1 QCD on the lattice adoptingO(a)-improved Wilson quark action and Iwasaki gauge action. To cope with theproblems due to explicit violation of the Poincare and chiral symmetries, weapply the Small Flow-time eXpansion (SFtX) method based on the gradient flow,which is a general method to correctly calculate any renormalized observableson the lattice. In this method, the matching coefficients in front of operatorsin the small flow-time expansion are calculated by perturbation theory. In aprevious study using one-loop matching coefficients, we found that the SFtXmethod works well for the equation of state, chiral condensates andsusceptibilities. In this paper, we study the effect of two-loop matchingcoefficients by Harlander et al. We also test the influence of therenormalization scale in the SFtX method. We find that, by adopting the mu_0renormalization scale of Harlander et al. instead of the conventionalmu_d=1/sqrt{8t} scale, the linear behavior at large t is improved so that wecan perform the t -> 0 extrapolation of the SFtX method more confidently. Inthe calculation of the two-loop matching coefficients by Harlander et al., theequation of motion for quark fields was used. For the entropy density in whichthe equation of motion has no effects, we find that the results using thetwo-loop coefficients agree well with those using one-loop coefficients. On theother hand, for the trace anomaly which is affected by the equation of motion,we find discrepancies between the one- and two-loop results at hightemperatures. By comparing the results of one-loop coefficients with andwithout using the equation of motion, the main origin of the discrepancies issuggested to be attributed to O((aT)^2)=O(1/N_t^2) discretization errors in theequation of motion at N_t =< 10.

Nf=2+1 QCD thermodynamics with gradient flow using two-loop matching coefficients