Gradient flow step-scaling function for SU(3) with twelve flavors

Abstract

We calculate the step scaling function, the lattice analog of the renormalization group β-function, for an SU(3) gauge theory with ten fundamental flavors. We present a detailed analysis including the study of systematic effects of our extensive data set generated with ten dynamical flavors using the Symanzik gauge action and three times stout smeared Möbius domain wall fermions. Using up to 32 4 volumes, we calculate renormalized couplings for different gradient flow schemes and determine the step-scaling β function for a scale change s = 2 on up to five different lattice volume pairs. In an accompanying paper we discuss that gradient flow can promote lattice dislocations to instantonlike objects, introducing nonperturbative lattice artifacts to the step scaling function. Motivated by the observation that Wilson flow sufficiently suppresses these artifacts, we choose Wilson flow with the Symanzik operator as our preferred analysis. We study systematic effects by calculating the step-scaling function based on alternative flows (Zeuthen or Symanzik), alternative operators (Wilson plaquette, clover), and also explore the effects of the perturbative tree-level improvement. Further we investigate the effects due to the finite value of Ls. c 11. At that strong coupling we also discovered previously unaccounted lattice artifacts. In an accompanying paper we discuss that gradient flow on coarse configurations can promote dislocations to instanton-like objects. This introduces a nonperturbative lattice artifact to the step scaling beta function which leads to incorrect continuum limit extrapolations [25]. We find that the perturbatively arXiv:2004.00754v1 [hep-lat] 2 Apr 2020 3 The good properties of this action has been first demonstrated for simulations in QCD [42,43] but is now also used in large scale simulations of a composite Higgs model [44][45][46]

Continuous $\beta$ function for the SU(3) gauge systems with two and twelve fundamental flavors

“…The value of the FP is scheme dependent. The continuous β function corresponds to c = 0 renormalization scheme, the value of the FP is however similar to our determination in the c = 0.250 gradient flow step-scaling scheme [15]. An advantage of the continuous RG transfromation is the possibility to resolve the scaling dimension of the irrelevant operators around a non-perturbative fixed point.…”

“…The N f = 12 analysis uses configurations generated for the step-scaling study published in Refs. [14,15]. We have implemented three different flows, Wilson (W), Symanzik (S), and Zeuthen (Z), in Qlua [16,17] and three different operators Wilson plaquette (W), clover (C), and Symanzik (S) to estimate the energy density [18,19].…”

“…We use the ensembles generated for the step scaling study published in Refs. [14,15]. These (L/a) 4 .00, 5.20, 5.50, 6.00, 6.50, and 7.00 have antiperiodic fermion boundary conditions in all four directions.…”

Looking through the QCD conformal window with perturbation theory

“…The controversy on the n f = 12 case is still open, see e.g. [18][19][20] for some of the latest works that appeared after [17]. 3 The aim of this paper is to study the conformal window of gauge theories using perturbation theory, starting from the upper edge and going down as much as we can towards the strongly coupled regime.…”

Holography of Strongly Coupled Gauge Theories

“…Despite this our knowledge of the spectrum of the set of asymptotically free theories is rather poor. Lattice studies only exist for a smattering of such theories and there remains debate over issues such as where the edge of the conformal window lies for SU (3) theories with fundamental quarks [5,6]. Even for theories that are known to break chiral symmetries the spectrum has frequently not yet been computed.…”