# The twisted gradient flow coupling at one loop

## TL;DR

This paper calculates the one-loop running of the SU(N) 't Hooft coupling in a finite volume using twisted boundary conditions, providing a scheme for regularization and scheme matching with potential implications for non-commutative gauge theories.

## Contribution

It introduces a method to compute the one-loop running of the coupling in a twisted gradient flow scheme with explicit scheme matching and N dependence analysis.

## Key findings

- Derived the one-loop running of the coupling in a twisted scheme.
- Provided a numerical evaluation of the scheme matching coefficient.
- Analyzed the N dependence and implications for non-commutative theories.

## 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 $\overline{\rm MS}$ 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.