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A non-perturbative and continuous definition of RG transformations as stochastic processes is proposed, inspired by the observation that the functional RG equations for effective Boltzmann factors may be interpreted as Fokker-Planck equations. The result implies a new approach to Monte Carlo RG that is amenable to lattice simulation. Long-distance correlations of the effective theory are shown to approach gradient-flowed correlations, which are simpler to measure. The Markov property of the stochastic RG transformation implies an RG scaling formula which allows for the measurement of anomalous dimensions when transcribed into gradient flow expectation values., 18 pages

We study the discrete beta function of SU(3) gauge theory with Nf=12 massless fermions in the fundamental representation. Using an nHYP-smeared staggered lattice action and an improved gradient flow running coupling $\tilde g_c^2(L)$ we determine the continuum-extrapolated discrete beta function up to $g_c^2 \approx 8.2$. We observe an IR fixed point at $g_{\star}^2 = 7.3\left(_{-2}^{+8}\right)$ in the $c = \sqrt{8t} / L = 0.25$ scheme, and $g_{\star}^2 = 7.3\left(_{-3}^{+6}\right)$ with c=0.3, combining statistical and systematic uncertainties in quadrature. The systematic effects we investigate include the stability of the $(a / L) \to 0$ extrapolations, the interpolation of $\tilde g_c^2(L)$ as a function of the bare coupling, the improvement of the gradient flow running coupling, and the discretization of the energy density. In an appendix we observe that the resulting systematic errors increase dramatically upon combining smaller $c \lesssim 0.2$ with smaller $L \leq 12$, leading to an IR fixed point at $g_{\star}^2 = 5.9(1.9)$ in the c=0.2 scheme, which resolves to $g_{\star}^2 = 6.9\left(_{-1}^{+6}\right)$ upon considering only $L \geq 16$. At the IR fixed point we measure the leading irrelevant critical exponent to be $\gamma_g^{\star} = 0.26(2)$, comparable to perturbative estimates., Comment: One of four consistency checks corrected, changing some systematic uncertainty estimates

We evaluate the $\Lambda$-parameter in the $\overline{\mathrm{MS}}$ scheme for the pure SU(3) gauge theory with the twisted gradient flow (TGF) method. A running coupling constant $g_{\mathrm{TGF}}^2(1/L)$ is defined in a finite volume box with size of $L^4$ with the twisted boundary condition. This defines the TGF scheme. Using the step scaling method for the TGF coupling with lattice simulations, we can evaluate the $\Lambda$-parameter non-perturbatively in the TGF scheme. In this paper we determine the dimensionless ratios, $\Lambda_{\mathrm{TGF}}/\sqrt{\sigma}$ and $r_{0}\Lambda_{\mathrm{TGF}}$ together with the $\Lambda$-parameter ratio $\Lambda_{\mathrm{SF}}/\Lambda_{\mathrm{TGF}}$ on the lattices numerically. Combined with the known ratio $\Lambda_{\overline{\mathrm{MS}}}/\Lambda_{\mathrm{SF}}$, we obtain $\Lambda_{\overline{\mathrm{MS}}}/\sqrt{\sigma} = 0.517(10)(^{+8}_{-7})$ and $r_{0}\Lambda_{\overline{\mathrm{MS}}}=0.593(12)(^{+12}_{-9})$, where the first error is statistical one and the second is our estimate of systematic uncertainty., Comment: 23 pages, 11 figures, LaTeX

We propose a new approach to determining quasi parton distribution functions (PDFs) from lattice quantum chromodynamics. By incorporating the gradient flow, this method guarantees that the lattice quasi PDFs are finite in the continuum limit and evades the thorny, and as yet unresolved, issue of the renormalization of quasi PDFs on the lattice. In the limit that the flow time is much smaller than the length scale set by the nucleon momentum, the moments of the smeared quasi PDF are proportional to those of the light-front PDF. We use this relation to derive evolution equations for the matching kernel that relates the smeared quasi PDF and the light-front PDF. As part of this discussion, we elucidate the relationship between the quasi and light-front PDFs., 15 pages, no figures. Version published by JHEP

The gradient flow in QCD is treated perturbatively through next-to-next-to-leading order in the strong coupling constant. The evaluation of the relevant momentum and flow-time integrals is described, including various means of validation. For the vacuum expectation value of the action density, which turns out to be a useful quantity in lattice calculations, we find a very well-behaved perturbative series through NNLO. Quark mass effects are taken into account through NLO. The theoretical uncertainty due to renormalization scale variation is significantly reduced with respect to LO and NLO, as long as the flow time is smaller than about 0.1 fm., Comment: 25 pages, 6 figures, 3 tables. v2: typos fixed; matches published version

Journal of high energy physics 1603(03), 116(2016). doi:10.1007/JHEP03(2016)116, We describe and test a nonperturbatively improved single-plaquette lattice action for 4-d SU(2) and SU(3) pure gauge theory, which suppresses large fluctuations of the plaquette, without requiring the naive continuum limit for smooth fields. We tune the action parameters based on torelon masses in moderate cubic physical volumes, and investigate the size of cut-off effects in other physical quantities, including torelon masses in asymmetric spatial volumes, the static quark potential, and gradient flow observables. In 2-d O(N) models similarly constructed nearest-neighbor actions have led to a drastic reduction of cut-off effects, down to the permille level, in a wide variety of physical quantities. In the gauge theories, we find significant reduction of lattice artifacts, and for some observables, the coarsest lattice result is very close to the continuum value. We estimate an improvement factor of 40 compared to using the Wilson gauge action to achieve the same statistical accuracy and suppression of cut-off effects., Published by Springer, Berlin

The a-function is a proposed quantity defined in even dimensions which has a monotonic behaviour along RG flows, related to the beta-functions via a gradient flow equation. We study the a-function for a general scalar theory in six dimensions, using the beta-functions up to three-loop order for both the MSbar and MOM schemes (the latter presented here for the first time at three loops)., 27 pages, seven figures, uses axodraw. Minor improvements in wording

A gradient flow equation for λϕ 4 theory in D = 4 is formulated. In this scheme the gradient flow equation is written in terms of the renormalized probe variable Φ(t, x) and renormalized parameters m 2 and λ in a manner analogous to the higher derivative regularization. No extra divergence is induced in the interaction of the probe variable Φ(t, x) and the 4-dimensional dynamical variable ϕ(x) which is defined in renormalized perturbation theory. The finiteness to all orders in perturbation theory is established by power counting argument in the context of D + 1 dimensional field theory. This illustrates that one can formulate the gradient flow for the simple but important λϕ 4 theory in addition to the well-known Yang-Mills flow, and it shows the generality of the gradient flow for a wider class of field theory.

We show that compatibility of supersymmetry with exact semi-classics demands that in calculating multi-instanton amplitudes, the "separation" quasi-zeromode must be complexified and the integration cycles must be found by using complex gradient flow (or Picard-Lefschetz equations.) As a non-trivial application, we study $\mathcal N=2$ extended supersymmetric quantum mechanics. Even though in this case supersymmetry is unbroken, the instanton-anti-instanton amplitude (naively calculated) seems to contribute to the ground state energy. We show, however, that the instanton-anti-instanton event consists of two parts: a fermion-correlated and a scalar-correlated event. Although both of these contributions are naively of the same sign and the latter is superficially higher order in the perturbative coupling, we show that the two contributions exactly cancel when they are evaluated on Lefschetz thimbles due to their relative Hidden Topological Angles (HTAs). This gives strong evidence that the semi-classical expansion using Lefschetz thimbles is not only a meaningful prescription for higher order semi-classics, but a necessary one. This deduction seems to be universal and applicable to both supersymmetric and non-supersymmetric theories. In conclusion we speculate that similar conspiracies are responsible for the non-formation of certain molecular contributions in theories where instantons have more than two fermionic zeromodes and do not contribute to the superpotential., 18 pages, 2 figures. Some extensions and minor corrections were implemented. This version was accepted for publication at JHEP