The gradient flow running coupling with twisted boundary conditions

Abstract

We study the gradient flow for Yang-Mills theories with twisted boundary conditions. The perturbative behavior of the energy density E(t) is used to define a running coupling at a scale given by the linear size of the finite volume box. We compute the non-perturbative running of the pure gauge SU(2) coupling constant and conclude that the technique is well suited for further applications due to the relatively mild cutoff effects of the step scaling function and the high numerical precision that can be achieved in lattice simulations. We also comment on the inclusion of matter fields.

The SU(∞) twisted gradient flow running coupling

“…Before proceeding any further, let us mention that twisted boundary conditions have already been used in combination with the Yang-Mills gradient flow [9,10] to define a running coupling for SU(N ) gauge theories [11,12]. Here we define an analogous coupling that runs in terms of the effective scalel.…”

“…3 Twisted Gradient Flow (TGF) running coupling: λ TGF To determine the running coupling we will make use of the recently proposed Twisted Gradient Flow (TGF) scheme [12]. The gradient flow [9,10] smoothes gauge fields along a flow-time trajectory defined by the equation:…”

“…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]. Among them, the absence of zero momentum modes and the manifest invariance of the theory under space-time translations.…”

“…A significant improvement is obtained if one adjusts the normalization constant N (c) entering the definition of the coupling to preserve the equality between renormalized and bare coupling at leading order on the lattice [12,13,17,22]. For our purposes we will only need N (c) on the TEK single-site lattices when the Wilson plaquette action is used for the simulation and the flow.…”

“…To define the coupling we have proposed a modification of the Twisted Gradient Flow running coupling scheme introduced by A. Ramos in ref. [12]. TGF is a finite volume renormalization scheme that uses the gradient flow [10] combined with twisted boundary conditions to define the SU(N ) running coupling at a scale set by the size of the box.…”