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 $\beta$-function, for an SU(3) gauge theory with twelve flavors. The gauge coupling of this system runs very slowly, which is reflected in a small step scaling function, making numerical simulations particularly challenging. We present a detailed analysis including the study of systematic effects of our extensive data set generated with twelve 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 $\beta$ function for a scale change $s=2$ on up to five different lattice volume pairs. Our preferred analysis is fully $O(a^2)$ Symanzik improved and uses Zeuthen flow combined with the Symanzik operator. We find an infrared fixed point within the range $5.2\le g_c^2\le 6.4$ in the $c=0.250$ finite volume gradient flow scheme. We account for systematic effects by calculating the step-scaling function based on alternative flows (Wilson or Symanzik) as well as operators (Wilson plaquette, clover) and also explore the effects of the perturbative tree-level improvement.

22 pages, 14 figures, 5 tables; v2 version published in Phys.Rev.D

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• Spectral Functions of Gauge Theories with Banks-Zaks Fixed Points
• Gradient flow step-scaling function for SU(3) with $N_f=8$ fundamental flavors
• Spectrum of SU(2) gauge theory at large number of flavors
• Dislocations under gradient flow and their effect on the renormalized coupling
• On the phase structure of extra-dimensional gauge theories with fermions
• Gradient flow step-scaling function for SU(3) with $N_f$ = 6 or 4 fundamental flavors
• Holography of Strongly Coupled Gauge Theories
• Taming lattice artifacts with Pauli--Villars fields