^{3}

Recently, Harlander et al. [Eur. Phys. J. C

The energy–momentum tensor (EMT) ^{1}

One important application of the lattice EMT is the thermodynamics of gauge theory at finite temperature; see Refs. [^{2}

In the gradient-flow approach, moreover, the computation of isotropic/anisotropic Karsch coefficients (i.e., the lattice

In this paper, we investigate the thermodynamics in quenched QCD (quantum chromodynamics), i.e., the pure Yang–Mills theory, in the gradient-flow approach. The EMT in the gradient-flow representation is obtained as follows. Assuming dimensional regularization, the EMT in the pure Yang–Mills theory is given by
^{3}

One can express any composite operator in gauge theory such as the EMT (^{4}

The parameter

Here,

In Eq. (

This paper is organized as follows. In

The

To compute expectation values of the EMT by employing the representation (

Although the above argument shows that in principle the coefficients

A conventional choice of

All the numerical experiments on the basis of the representation (^{5}

Let us now list the known coefficients in Eq. (

In the one-loop level, we have [

The number

The two-loop order coefficients in Ref. [

In the pure Yang–Mills theory, if one has the small flow-time expansion of the renormalized operator

We recall the trace anomaly [

Plugging this into Eq. (

Comparing this with the trace of Eq. (

One can also confirm that Eqs. (

In what follows, we use the lattice data obtained in Ref. [

Let us start with the entropy density,

In ^{6}

Equation (

We then take the continuum limit ^{7}

We carry out the ^{8}

In ^{9}

Linear coefficients in the

NLO | ||

0.93 | ||

1.02 | ||

1.12 | ||

1.40 | ||

1.68 | ||

2.10 | ||

2.31 | ||

2.69 | ||

0.93 | ||

1.02 | ||

1.12 | ||

1.40 | ||

1.68 |

In

Summary of the entropy density and the trace anomaly obtained by using coefficients in different orders of perturbation theory. The statistical errors are shown in the first parentheses. The numbers in the other parentheses show systematic errors: the error associated with the fit range, the

NLO | FlowQCD 2016 | ||

0.93 | 0.082(33)( |
||

1.02 | 2.104(63)( |
||

1.12 | 3.603(46)( |
||

1.40 | 4.706(35)( |
||

1.68 | 5.285(35)( |
||

2.10 | 5.617(34)( |
||

2.31 | 5.657(55)( |
||

2.69 | 5.914(32)( |
||

FlowQCD 2016 | |||

0.93 | 0.066(32)( |
||

1.02 | 1.945(57)( |
||

1.12 | 2.560(33)( |
||

1.40 | 1.777(24)( |
||

1.68 | 1.201(19)( |

In

Summary of the entropy density

Reference [

We now turn to the trace anomaly,

Equation (

Summary of the trace anomaly

We investigated the thermodynamics in quenched QCD using the gradient-flow representation of the EMT. In particular, we studied the effect of the N

We would like to thank Robert V. Harlander, Kazuyuki Kanaya, Yusuke Taniguchi, and Ryosuke Yanagihara for helpful discussions. This work was supported by Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research, Grant Numbers JP17K05442 (M.K.) and JP16H03982 (H.S.). Numerical simulation was carried out on the IBM System Blue Gene Solution at KEK under its Large-Scale Simulation Program (No. 16/17-07).

Open Access funding: SCOAP

In this appendix, we show the plots of Eqs. (

Same as

Same as

Same as

Same as

Same as

Same as

Same as

Same as

Same as

Same as

Same as

^{1}See Ref. [

^{2}For simplicity of expression, here and in what follows we omit the subtraction of the vacuum expectation value of an expression; this is always assumed.

^{3}

^{4}Note that our convention for

^{5}The reduction of the renormalization-scale dependence from the one-loop order to the two-loop order is studied in detail in Ref. [

^{6}In Ref. [

^{7}An explanation for this flatter behavior will be provided in another paper (H. Suzuki and H. Takaura, manuscript in preparation).

^{8}The finite lattice spacing and volume effects are controlled by

^{9}Recall that the renormalization scale