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Register a new userResults and techniques for higher order calculations within the gradient-flow formalism
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We describe in detail the implementation of a systematic perturbative approach to observables in the QCD gradient-flow formalism. This includes a collection of all relevant Feynman rules of the five-dimensional field theory and the composite operators considered in this paper. Tools from standard perturbative calculations are used to obtain Greens functions at finite flow time $t$ at higher orders in perturbation theory. The three-loop results for the quark condensate at finite $t$ and the conversion factor for the ringed quark fields to the $overline{mbox{MS}}$ scheme are presented as applications. We also re-evaluate an earlier result for the three-loop gluon condensate, improving on its accuracy.
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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 $tto0$ 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.
To obtain the precise values of the bulk quantities and transport coefficients in quark-gluon-plasma phase, we propose that a direct calculation of the renormalized energy-momentum tensor (EMT) on the lattice using the gradient flow. From one-point function of EMT, authors in Ref.[1] obtained the interaction measure and thermal entropy. The results are consistent with the one obtained by the integral method. Based on the success, we try to measure the two-point function of EMT, which is related to the transport coefficients. Advantages of our method are (1) a clear signal because of the smearing effects of the gradient flow and (2) no need to calculate the wave function renormalization of EMT. In addition, we give a short remark on a comparison of the numerical cost between the positive- and adjoint-flow methods for fermions, needed to obtain the EMT in the (2+1) flavor QCD.
The gradient-flow operator product expansion for QCD current correlators including operators up to mass dimension four is calculated through NNLO. This paves an alternative way for efficient lattice evaluations of hadronic vacuum polarization functions. In addition, flow-time evolution equations for flowed composite operators are derived. Their explicit form for the non-trivial dimension-four operators of QCD is given through order $alpha_s^3$.
Non-zero topological charge is prohibited in the chiral limit of gauge-fermion systems because any instanton would create a zero mode of the Dirac operator. On the lattice, however, the geometric $Q_text{geom}=langle F{tilde F}rangle /32pi^2$ definition of the topological charge does not necessarily vanish even when the gauge fields are smoothed for example with gradient flow. Small vacuum fluctuations (dislocations) not seen by the fermions may be promoted to instanton-like objects by the gradient flow. We demonstrate that these artifacts of the flow cause the gradient flow renormalized gauge coupling to increase and run faster. In step-scaling studies such artifacts contribute a term which increases with volume. The usual $a/Lto 0$ continuum limit extrapolations can hence lead to incorrect results. In this paper we investigate these topological lattice artifacts in the SU(3) 10-flavor system with domain wall fermions and the 8-flavor system with staggered fermions. Both systems exhibit nonzero topological charge at the strong coupling, especially when using Symanzik gradient flow. We demonstrate how this artifact impacts the determination of the renormalized gauge coupling and the step-scaling $beta$ function.
The energy-momentum tensor plays an important role in QCD thermodynamics. Its expectation value contains information of the pressure and the energy density as its diagonal part. Further properties like viscosity and specific heat can be extracted from its correlation function. Recently a new method based on the gradient flow was introduced to calculate the energy-momentum tensor on the lattice, and has been successfully applied to quenched QCD. In this paper, we apply the gradient flow method to calculate the energy-momentum tensor in (2+1)-flavor QCD. As the first application of the method with dynamical quarks, we study at a single but fine lattice spacing a=0.07 fm with heavy u and d quarks ($m_pi/m_rho=0.63$) and approximately physical s quark. Performing simulations on lattices with Nt=16 to 4, the temperature range of T=174-697 MeV is covered. We find that the results of the pressure and the energy density by the gradient flow method are consistent with the previous results using the T-integration method at T<280 MeV, while the results show disagreement at T>350 MeV (Nt<8), presumably due to the small-Nt lattice artifact of $O((aT)^2)=O(1/N_t^2)$. We also apply the gradient flow method to evaluate the chiral condensate taking advantage of the gradient flow method that renormalized quantities can be directly computed avoiding the difficulty of explicit chiral violation with lattice quarks. We compute the renormalized chiral condensate in the MS-bar scheme at renormalization scale $mu=2$ GeV with a high precision to study the temperature dependence of the chiral condensate and its disconnected susceptibility. Even with the Wilson-type quark action, we obtain the chiral condensate and its disconnected susceptibility showing a clear signal of pseudocritical temperature at T~190 MeV related to the chiral restoration crossover.
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