# $\Lambda$ parameter of the SU(3) Yang-Mills theory from the continuous $\beta$ function

## TL;DR

This paper nonperturbatively determines the SU(3) Yang-Mills theory's $eta$ function using the gradient flow scheme, enabling a precise calculation of the $
mar{MS}$ Lambda parameter across various coupling regimes.

## Contribution

It provides the first comprehensive nonperturbative calculation of the $eta$ function for SU(3) Yang-Mills theory in the gradient flow scheme, covering a wide coupling range.

## Key findings

- Calculated $eta$ function in the gradient flow scheme for SU(3) Yang-Mills.
- Determined $
mar{MS}$ Lambda parameter as 0.622(10).
- Results agree with recent direct determinations.

## Abstract

Nonperturbative determinations of the renormalization group $\beta$ function are essential to connect lattice results to perturbative predictions of strongly coupled gauge theories and to determine the $\Lambda$ parameter or the strong coupling constant. The continuous $\beta$ function is very well suited for this task because it is applicable both in the weakly coupled deconfined regime as well as the strongly coupled confined regime. Here we report on our results for the $\beta$ function of the pure gauge SU(3) Yang-Mills theory in the gradient flow scheme. Our calculations cover the renormalized coupling range $g^2_{\textrm{GF}} \sim 1.2 - 27$, allowing for a direct determination of $\sqrt{8t_0} \Lambda_{\overline{\textrm{MS}}}$ in this system. Our prediction, $\sqrt{8 t_0} \Lambda_{\overline{\textrm{MS}}}=0.622(10)$, is in good agreement with recent direct determinations of this quantity.