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

Abstract

We study thermodynamic properties of N f ¼ 2 þ 1 QCD on the lattice adopting a nonperturbatively OðaÞ-improved Wilson quark action and the renormalization group-improved Iwasaki gauge action. To cope with the problems due to explicit violation of the Poincaré 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 thanks to the asymptotic freedom around the small flow-time limit. In a previous study using one-loop matching coefficients, we found that the SFtX method works well for the equation of state extracted from diagonal components of the energy-momentum tensor and for the chiral condensates and susceptibilities. In this paper, we study the effect of two-loop matching coefficients which have been calculated by Harlander et al. recently. We also test the influence of the renormalization scale in the SFtX method. We find that, by adopting the μ 0 renormalization scale of Harlander et al. instead of the conventional μ d ¼ 1= ffiffiffiffi 8t p scale, the linear behavior at large flow-times 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 contamination of OððaTÞ 2 Þ ¼ Oð1=N 2 t Þ discretization errors in the equation of motion at N t ≲ 10.

Four quark operators for kaon bag parameter with gradient flow

“…The method has been applied to calculate the energymomentum tensor, which is the generator of the continuous Poincaré transformation and thus is not straightforward to evaluate on discrete lattices. From test studies around the deconfinement transition temperature in quenched QCD [26][27][28] and in 2 þ 1 flavor QCD with improved Wilson quarks [29][30][31][32][33], it was shown that the results of the energy-momentum tensor by the SFtX method correctly reproduce previous results of the equation of state estimated by the conventional integral methods.…”

“…In practice, however, because the perturbative expansions are truncated, the quality of the results may be affected by the choice of μ. Recently, we found that an optimal choice of μ can improve the reliability and applicability of the SFtX method [33]. Such improvement may be important in evaluating complicated operators, such as O AE for the kaon bag parameter.…”

Latent heat and pressure gap at the first-order deconfining phase transition of SU(3) Yang-Mills theory using the small flow-time expansion method

“…Though the flowed observable is not the observable itself, we can get the latter by extrapolating the result of the flowed observable to the vanishing flow time limit 𝑡 → 0. The SF𝑡X method has been successfully applied to evaluate thermodynamic quantities in the Yang-Mills gauge theory [7][8][9][10] and in QCD with 2 + 1 flavors of dynamical quarks [11][12][13][14].…”

The effective electroweak Hamiltonian in the gradient-flow formalism

“…Refs. [5][6][7]. More recently, the matching matrix has also been calculated for the quark dipole operators at nextto-leading order (NLO) QCD [8], and for the hadronic vacuum polarization through NNLO QCD [9].…”