# Thermodynamics in quenched QCD: energy--momentum tensor with two-loop order coefficients in the gradient flow formalism

## TL;DR

This paper investigates how two-loop order corrections in the gradient flow formalism improve the lattice computation of thermodynamic quantities in quenched QCD by reducing systematic uncertainties and stabilizing the extrapolation to zero flow time.

## Contribution

It demonstrates that incorporating two-loop order coefficients in the gradient flow formalism enhances the precision of thermodynamic measurements in quenched QCD by minimizing flow-time dependence.

## Key findings

- Reduced flow-time dependence of EMT expectation values.
- Decreased systematic errors in the $t o 0$ extrapolation.
- Improved stability against fit function and scale choices.

## Abstract

Recently, Harlander et al.\ [Eur.\ Phys.\ J.\ C {\bf 78}, 944 (2018)] have computed the two-loop order (i.e., NNLO) coefficients in the gradient-flow representation of the energy--momentum tensor (EMT) in vector-like gauge theories. In this paper, we study the effect of the two-loop order corrections (and the three-loop order correction for the trace part of the EMT, which is available through the trace anomaly) on the lattice computation of thermodynamic quantities in quenched QCD. The use of the two-loop order coefficients generally reduces the $t$~dependence of the expectation values of the EMT in the gradient-flow representation, where $t$~is the flow time. With the use of the two-loop order coefficients, therefore, the $t\to0$ extrapolation becomes less sensitive to the fit function, the fit range, and the choice of the renormalization scale; the systematic error associated with these factors is considerably reduced.