The twisted gradient flow coupling at one loop

Abstract

We compute the one-loop running of the $SU(N)$ 't Hooft coupling in a finite volume gradient flow scheme using twisted boundary conditions. The coupling is defined in terms of the energy density of the gradient flow fields at a scale $\tilde{l}$ given by an adequate combination of the torus size and the rank of the gauge group, and is computed in the continuum using dimensional regularization. We present the strategy to regulate the divergences for a generic twist tensor, and determine the matching to the $\overline{\mathrm{MS}}$ scheme at one-loop order. For the particular case in which the twist tensor is non-trivial in a single plane, we evaluate the matching coefficient numerically and determine the ratio of $\Lambda$ parameters between the two schemes. We analyze the $N$ dependence of the results and the possible implications for non-commutative gauge theories and volume independence.

52 pages, 12 figures

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• FLAG Review 2021
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• Scale setting for large-$N$ SUSY Yang-Mills on the lattice
• Center-vortex semiclassics with non-minimal 't Hooft fluxes on ${\mathbb{R}}^2\times T^2$ and center stabilization at large $N$
• Scale setting of SU($N$) Yang--Mills theory, topology and large-$N$ volume independence