Perturbative analysis of the gradient flow in non-abelian gauge theories

Abstract

Abstract:The gradient flow in non-abelian gauge theories on R 4 is defined by a local diffusion equation that evolves the gauge field as a function of the flow time in a gaugecovariant manner. Similarly to the case of the Langevin equation, the correlation functions of the time-dependent field can be expanded in perturbation theory, the Feynman rules being those of a renormalizable field theory on R 4 ×[0, ∞). For any matter multiplet and to all loop orders, we show that the correlation functions are finite, i.e. do not require additional renormalization, once the theory in four dimensions is renormalized in the usual way. The flow thus maps the gauge field to a one-parameter family of smooth renormalized fields.

The running coupling of 8 flavors and 3 colors

“…If the β-function in one scheme has a non-trivial zero indicating conformal behavior in the infrared, its existence is universal in every other well-defined scheme. In the current work the recently proposed finite volume gradient flow scheme [9,10] is used, which is based on Luscher's Wilson flow [11][12][13][14] related to earlier constructions by Morningstar and Peardon [15] as well as Lohmayer and Neuberger [16]. In this scheme a 1-parameter family of couplings is defined in finite 4-volume by 1) where N corresponds to the gauge group SU(N ), t is the flow parameter, c = √ 8t/L is a constant, E(t) is the field strength squared at t and…”

“…The numerical factors are chosen such that at leading order g 2 c = g 2 MS for all c. The gauge field is chosen to be periodic and the massless fermions are anti-periodic in all 4 directions. The coupling g c (µ) runs via the scale µ = 1/L; for more details on the gradient flow in general see [11][12][13][14] while more details on the finite volume gradient flow scheme can be found in [9,10].…”

High-precision scale setting in lattice QCD

“…It was studied earlier by Narayanan and Neuberger [11] in a different context, too. Its important renormalization properties were clarified in [12,13]. Its application to scale setting, which we build upon, was suggested recently in [12].…”

“…In [12,13], where the Wilson action is used, Z(V t ) is the derivative of the plaquette action and the corresponding flow is called the Wilson flow. As it can be seen from eq.…”

The SU(∞) twisted gradient flow running coupling

“…At positive gradient flow time, the action density of SU(N ) gauge theory is a renormalized quantity with a perturbative expansion in the thermodynamic limit given by [10,15],…”