The gradient flow exact renormalization group (GFERG) is an exact renormalization group motivated by the Yang--Mills gradient flow and its salient feature is a manifest gauge invariance. We generalize this GFERG, originally formulated for the pure Ya
ng--Mills theory, to vector-like gauge theories containing fermion fields, keeping the manifest gauge invariance. For the chiral symmetry we have two options: one possible formulation preserves the conventional form of the chiral symmetry and the other simpler formulation realizes the chiral symmetry in a modified form `a la Ginsparg--Wilson. We work out a gauge-invariant local Wilson action in quantum electrodynamics to the lowest nontrivial order of perturbation theory. This Wilson action reproduces the correct axial anomaly in~$D=2$.
We theoretically clarify the functional form to be used in $t to 0$ extrapolation in the small flow time expansion (SF$t$X) method for the energy-momentum tensor (EMT), which facilitates lattice simulation of the EMT based on the gradient flow. We ar
gue that in the $t to 0$ extrapolation analysis, lattice data should be fitted by a power function in $g(mu(t))$, the flow time dependent running coupling, where the power is determined by the perturbation order we consider. From actual lattice data, we confirm the validity of the extrapolation function. Using the new extrapolation function, we present updated lattice results for thermodynamics quantities in quenched QCD; our results are consistent with the previous study [arXiv:1812.06444] but we obtain smaller errors due to reduction of systematic errors.
The gradient flow bears a close resemblance to the coarse graining, the guiding principle of the renormalization group (RG). In the case of scalar field theory, a precise connection has been made between the gradient flow and the RG flow of the Wilso
n action in the exact renormalization group (ERG) formalism. By imitating the structure of this connection, we propose an ERG differential equation that preserves manifest gauge invariance in Yang--Mills theory. Our construction in continuum theory can be extended to lattice gauge theory.
We present additional observations to previous studies on the infrared (IR) renormalon in $SU(N)$ QCD(adj.), the $SU(N)$ gauge theory with $n_W$-flavor adjoint Weyl fermions on~$mathbb{R}^3times S^1$ with the $mathbb{Z}_N$ twisted boundary condition.
First, we show that, for arbitrary finite~$N$, a logarithmic factor in the vacuum polarization of the photon (the gauge boson associated with the Cartan generators of~$SU(N)$) disappears under the $S^1$~compactification. Since the IR renormalon is attributed to the presence of this logarithmic factor, it is concluded that there is no IR renormalon in this system with finite~$N$. This result generalizes the observation made by Anber and~Sulejmanpasic [J. High Energy Phys. textbf{1501}, 139 (2015)] for $N=2$ and~$3$ to arbitrary finite~$N$. Next, we point out that, although renormalon ambiguities do not appear through the Borel procedure in this system, an ambiguity appears in an alternative resummation procedure in which a resummed quantity is given by a momentum integration where the inverse of the vacuum polarization is included as the integrand. Such an ambiguity is caused by a simple zero at non-zero momentum of the vacuum polarization. Under the decompactification~$Rtoinfty$, where $R$ is the radius of the $S^1$, this ambiguity in the momentum integration smoothly reduces to the IR renormalon ambiguity in~$mathbb{R}^4$. We term this ambiguity in the momentum integration renormalon precursor. The emergence of the IR renormalon ambiguity in~$mathbb{R}^4$ under the decompactification can be naturally understood with this notion.
We study the infrared renormalon in the gluon condensate in the $SU(N)$ gauge theory with $n_W$-flavor adjoint Weyl fermions (QCD(adj.)) on~$mathbb{R}^3times S^1$ with the $mathbb{Z}_N$ twisted boundary conditions. We rely on the so-called large-$bet
a_0$ approximation as a conventional tool to analyze the renormalon, in which only Feynman diagrams that dominate in the large-$n_W$ limit are considered while the coefficient of the vacuum polarization is set by hand to the one-loop beta function~$beta_0=11/3-2n_W/3$. In the large~$N$ limit within the large-$beta_0$ approximation, the W-boson, which acquires the twisted Kaluza--Klein momentum, produces the renormalon ambiguity corresponding to the Borel singularity at~$u=2$. This provides an example that the system in the compactified space~$mathbb{R}^3times S^1$ possesses the renormalon ambiguity identical to that in the uncompactified space~$mathbb{R}^4$. We also discuss the subtle issue that the location of the Borel singularity can change depending on the order of two necessary operations.
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 stud
y 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.
In K.~Hieda, A.~Kasai, H.~Makino, and H.~Suzuki, Prog. Theor. Exp. Phys. textbf{2017}, 063B03 (2017), a properly normalized supercurrent in the four-dimensional (4D) $mathcal{N}=1$ super Yang--Mills theory (SYM) that works within on-mass-shell correl
ation functions of gauge-invariant operators is expressed in a regularization-independent manner by employing the gradient flow. In the present paper, this construction is extended to the supercurrent in the 4D $mathcal{N}=2$ SYM. The so-constructed supercurrent will be useful, for instance, for fine tuning of lattice parameters toward the supersymmetric continuum limit in future lattice simulations of the 4D $mathcal{N}=2$ SYM.
We propose a clear definition of the gluon condensate within the large-$beta_0$ approximation as an attempt toward a systematic argument on the gluon condensate. We define the gluon condensate such that it is free from a renormalon uncertainty, consi
stent with the renormalization scale independence of each term of the operator product expansion (OPE), and an identical object irrespective of observables. The renormalon uncertainty of $mathcal{O}(Lambda^4)$, which renders the gluon condensate ambiguous, is separated from a perturbative calculation by using a recently suggested analytic formulation. The renormalon uncertainty is absorbed into the gluon condensate in the OPE, which makes the gluon condensate free from the renormalon uncertainty. As a result, we can define the OPE in a renormalon-free way. Based on this renormalon-free OPE formula, we discuss numerical extraction of the gluon condensate using the lattice data of the energy density operator defined by the Yang--Mills gradient flow.
It is believed that the two-dimensional massless $mathcal{N}=2$ Wess--Zumino model becomes the $mathcal{N}=2$ superconformal field theory (SCFT) in the infrared (IR) limit. We examine this theoretical conjecture of the Landau--Ginzburg (LG) descripti
on of the $mathcal{N}=2$ SCFT by numerical simulations on the basis of a supersymmetric-invariant momentum-cutoff regularization. We study a single supermultiplet with cubic and quartic superpotentials. From two-point correlation functions in the IR region, we measure the scaling dimension and the central charge, which are consistent with the conjectured LG description of the $A_2$ and $A_3$ minimal models, respectively. Our result supports the theoretical conjecture and, at the same time, indicates a possible computational method of correlation functions in the $mathcal{N}=2$ SCFT from the LG description.
