The Yang-Mills gradient flow in finite volume

Abstract

The Yang-Mills gradient flow is considered on the four dimensional torus T 4 for SU (N ) gauge theory coupled to N f flavors of massless fermions in arbitrary representations. The small volume dynamics is dominated by the constant gauge fields. The expectation value of the field strength tensor squared TrF µν F µν (t) is calculated for positive flow time t by treating the non-zero gauge modes perturbatively and the zero modes exactly. The finite volume correction to the infinite volume result is found to contain both algebraic and exponential terms. The leading order result is then used to define a one parameter family of running coupling schemes in which the coupling runs with the linear size of the box. The new scheme is tested numerically in SU (3) gauge theory coupled to N f = 4 flavors of massless fundamental fermions. The calculations are performed at several lattice spacings with a controlled continuum extrapolation. The continuum result agrees with the perturbative prediction for small renormalized coupling as expected.

The running coupling of 8 flavors and 3 colors

“…In any case conformal behavior was not ruled out. Finally [8] studied the running coupling using our finite volume running coupling scheme [9,10] as in our present work and in section 5 we will comment on the relationship between our results and the analysis in [8].…”

“…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].…”

“…In section 2 we briefly summarize the finite volume gradient flow running coupling scheme that we use; for more details see [9,10]. In section 3 the details of our numerical simulations are given and the collected data is presented, while in section 4 the continuum extrapolation of the discrete β-function is performed.…”

The SU(∞) twisted gradient flow running coupling

“…This quantity can be used to define a renormalized coupling at a renormalization scale µ = 1/ √ 8t. The identification of this scale with the linear size of the box gives rise to the finite volume gradient flow schemes [16,17]. In this context, the use of twisted boundary conditions, leading to the TGF scheme, has many advantages [12].…”