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.
Highlights
The energy–momentum tensor (EMT) Tμν(x) is a fundamental physical observable in quantum field theory
For the trace part of the EMT, we examine the use of the threeloop order coefficient, k2(3), which is presented in this paper; this higher-order coefficient can be obtained for quenched QCD by combining a two-loop result in Ref. [34] and the trace anomaly [37,38,39], as we will explain below
We investigated the thermodynamics in quenched QCD using the gradient-flow representation of the EMT
Summary
The energy–momentum tensor (EMT) Tμν(x) is a fundamental physical observable in quantum field theory. For the trace part of the EMT, we examine the use of the threeloop order coefficient, k2(3), which is presented in this paper; this higher-order coefficient can be obtained for quenched QCD by combining a two-loop result in Ref. This improvement brought about by the two-loop order coefficients persists in wider applications of the gradient-flow representation of the EMT, such as the thermodynamics of full QCD. In the pure Yang–Mills theory, if one has the small flow-time expansion of the renormalized operator {FμaνFμaν}R(x) in the th-loop order, it is possible to further obtain the coefficient of c2(t) one loop higher, k2( +1), by using information on the trace anomaly [1]. We will examine the use of this N3LO coefficient for the trace anomaly in the numerical analyses below
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