We study the equation of state of pure SU(
Thermodynamic quantities (energy, pressure, thermal entropy, etc.) and transport coefficients (heat capacity, shear and bulk viscosities, etc.) play an important role in understanding features of quantum chromodynamics (QCD). In
As for the thermodynamic quantities, recently, Suzuki proposed a new method to calculate the energy–momentum tensor (EMT) using the gradient flow [
Other important quantities are the transport coefficients. For intermediate temperatures, experimental data show the small shear viscosity to thermal entropy ratio (
Based on these situations, we focus on the pure SU(
Several works have obtained the thermodynamic quantities for the SU(
It is reported that for intermediate temperatures (
The trace anomaly (
In this work, we precisely determine the scalesetting function using the
The structure of the paper is as follows: In Sect.
The Yang–Mills gradient flow equation, which is a key equation in this work, is defined by
Now, we introduce a length scale of the order of
Here,
The
Here,
The solution of the
For the SU(
For
Similar scales for the SU(
Perturbative analysis of Eq. (
In this work, we will obtain the thermodynamic quantities from the EMT. In general, measurements of the EMT using the lattice numerical simulation are essentially difficult, since the lattice regularization manifestly breaks the general covariance, while EMT is a generator of the corresponding invariance [
The key relation is given in Ref. [
Let us briefly review how to obtain the relation. There are two gaugeinvariant local products of dimension
As an advantage of using the gradient flow, it is not necessary to calculate the wave function renormalization of operators thanks to their UV finiteness in pure gauge theories [
The EMT (
Although the coefficients are derived perturbatively, we assume that the relation is available in the nonperturbative regime. Then, we can obtain the renormalized EMT from Eq. (
In the finitetemperature system, the following components of the EMT correspond to a combination of the energy density (
Here,
We use the standard Wilson–Plaquette gauge action defined on a fourdimensional Euclidean lattice,
Here,
In this work, we adopt periodic boundary conditions for all directions. Gauge configurations are generated by using the pseudoheatbath algorithm with overrelaxation. We use “a sweep” to refer to the combination of one pseudoheatbath update step followed by multiple overrelaxation steps. The mixed ratio of the combination is
In Sect.
Here,
The equation is an ordinary firstorder differential equation, so we numerically integrate it using the Runge–Kutta (RK) method. The thirdorder RK formula with the initial configuration
The error per step is of order
Before carrying out the scalesetting simulation, we compare the operator
The operator
It is worth summarizing several values for the SU(
The running coupling constant in the Schrödinger function (SF) scheme for the SU(
Here,
On the treelevel SF action with
Comparison
Now, we measure
Flowtime dependence of
The values of
# of conf.  

2.400  0.9549(5)  300 
2.420  1.083(2)  100 
2.500  1.839(3)  300 
2.600  3.522(10)  300 
2.700  6.628(36)  300 
2.800  11.96(12)  300 
2.850  16.95(17)  600 
The data of
Ratio of the lattice spacing (
We also perform a similar calculation for the other reference scales. The results are summarized in Appendix
The relationship between the following typical scales in the theory is useful in understanding the dynamics. The left (right) panel of Fig.
Left (right) panel shows the continuum extrapolation for the ratio between
The filled squares (blue) and open circles (red) in each panel are obtained as functions of
The values in the continuum limit with the clover definition of the operator
As a consistency check, we consider
Continuum extrapolation of
This shows
Finally, using
Our
The simulation parameters (
Lattice parameters (
0.95  (2.41)  2.50  2.57  2.62 
0.98  2.42  2.51  2.58  2.63 
1.01  2.43  2.52  2.59  2.64 
1.04  2.44  2.53  2.60  2.65 
1.08  2.45  2.54  2.61  2.66 
1.12  2.46  2.55  2.62  2.67 
1.28  2.50  2.59  2.66  2.72 
1.50  2.55  2.64  2.71  2.77 
1.76  2.60  2.69  2.76  2.82 
2.07  2.65  2.74  2.81  (2.87) 
The thermodynamic quantities have been obtained using
The procedure to calculate the EMT on the lattice is summarized as the following four steps in Ref. [
We have to carefully estimate the propagation of errors, in particular, taking the double limits in Step 4, since each flowtime data item after taking the continuum extrapolation is correlated with each other. In this work, we use the jackknife method. Thus, first we generate the jackknife samples for the lattice raw data of
The left two panels of Fig.
Flowtime dependence for the expectation values of the trace anomaly (top two panels) and entropy density (bottom two panels) in each lattice parameter. The left two panels show the
In Step 2, we carry out both constant and linear extrapolations for
Continuum extrapolation for the fixed flowtime
Plots of the extrapolations in the
We also estimate the other systematic error coming from the uncertainty of the lattice determination of
We plot the trace anomaly (red circles) and entropy density (black squares) as a function of
(Left) Results of the trace anomaly (red circles) and the entropy density (black squares) as a function of temperature. The triangles (cyan) and diamonds (brown) denote the trace anomaly and the entropy density at
The right panel of Fig.
Now, let us compare our results with the analytic prediction, namely the results of the HTL model. The left panel of Fig.
(Left) Results of the energy density normalized by those in the SB limit. (Right) Rescaled trace anomaly,
Finally, to see the scaling law of the trace anomaly more precisely, we also compare the results between the lattice data and the HTL analyses. The trace anomaly has a leading correction term of
In this work, we numerically investigate the thermodynamics of the pure SU(
For the precise scale setting of lattice parameters, we propose a reference scale
We also obtain the thermodynamic quantities, which are directly calculated from the EMT by the smallflowtime expansion of the gradient flow method. This work is the first application of the gradient flow method to the thermodynamic quantities for the SU(
For future works, we address the following points.
The finitetemperature phase transition in the pure SU(
In our analysis, we utilize the oneloop calculation for the coefficients
One of the motivations for this work is to prepare the determination of
Once the temperature dependences of the energy density (or pressure) and the expectation value of the Polyakov loop are determined by the pure gauge lattice simulations, one can construct the effective Polyakovloop potential used in an effective model such as the PNJL model [
We would like to thank M. Yahiro and M. Yamazaki deeply for valuable comments and discussions. E.I. would like to thank K. Iida and M. Panero for useful comments. H.K. would like to thank H. Yoneyama for valuable discussions. Numerical simulations were performed on the xc40 at YITP, Kyoto University and on the SXACE at the Research Center for Nuclear Physics (RCNP), Osaka University. The work of H. K. is supported in part by a GrantinAid for Scientific Research No. 17K05446.
Open Access funding: SCOAP
To investigate the autocorrelation of the generated configuration, we measure the topological charge using the gradient flow. The topological charge is related to the vacuum structure, and it has a long autocorrelation among the observables in Yang–Mills theory. The gluonic definition of the topological charge in Euclidean spacetime is given by
History of topological charge at
In our analysis in Sect. 4.2, the scalesetting function is valid for
Results of



2.400  0.7264(9)  0.8400(12)  0.9549(5)  1.171(23) 
2.420  0.8202(11)  0.9529(14)  1.083(2)  1.336(2) 
2.500  1.372(2)  1.609(2)  1.839(3)  2.279(4) 
2.600  2.612(7)  3.075(8)  3.522(10)  4.370(14) 
2.700  4.881(25)  5.770(33)  6.628(36)  8.247(54) 
2.800  8.780(74)  10.40(10)  11.96(12)  14.92(17) 
2.850  12.25(10)  14.63(14)  16.95(17)  21.43(25) 
Table
To see the scale violation coming from the choice of the reference values, we show the ratio of the lattice spacing (
Ratios of the lattice spacings
2.400  1.896(3)  1.913(3)  1.921(3)  1,932(4) 
2.420  1.784(3)  1.797(3)  1.803(3)  1.809(3) 
2.500  1.380(2)  1.383(2)  1.384(2)  1.385(3) 
2.600  1.000(2)  1.000(2)  1.000(2)  1.000(2) 
2.700  0.7315(21)  0.7301(23)  0.7290(22)  0.7279(27) 
2.800  0.5454(24)  0.5438(26)  0.5427(28)  0.5412(31) 
2.850  0.4617(20)  0.4586(22)  0.4558(24)  0.4516(18) 
Finally, we obtain the scalesetting function with the following interpolating function:
The coefficients of the scalesetting function for
Fit range  

0.08  0.9590(22)  6.343(16)  
0.09  1.122(2)  6.382(18)  
0.10  1.258(2)  6.409(14)  
0.12  1.474(2)  6.439(15) 
Data of thermodynamic quantities: trace anomaly (
0.95  0.246(35)  0.232(27)  0.235(24)  −0.00348(1041) 
0.98  0.365(25)  0.307(29)  0.322(24)  −0.0145(87) 
1.01  0.551(22)  0.585(22)  0.576(18)  0.00848(739) 
1.04  0.822(22)  0.844(29)  0.838(21)  0.00543(1019) 
1.08  0.844(28)  0.931(28)  0.909(23)  0.0217(93) 
1.12  0.871(13)  1.18(2)  1.10(1)  0.772(69) 
1.28  0.819(20)  1.57(2)  1.38(1)  0.188(7) 
1.50  0.615(13)  1.81(2)  1.51(2)  0.298(8) 
1.76  0.463(14)  1.92(3)  1.56(2)  0.365(9) 
2.07  0.322(14)  2.12(4)  1.67(3)  0.451(10) 