A status update on the determination of ${\Lambda}_{\overline{\rm MS}}^{N_{\rm f}=3}$ by the ALPHA collaboration
Abstract
The ALPHA collaboration aims to determine $\alpha_s(m_Z)$ with a total error below the percent level. A further step towards this goal can be taken by combining results from the recent simulations of 2+1flavour QCD by the CLS initiative with a number of tools developed over the years: renormalized couplings in finite volume schemes, recursive finite size techniques, twoloop renormalized perturbation theory and the (improved) gradient flow on the lattice. We sketch the strategy, which involves both the standard SF coupling in the high energy regime and a gradient flow coupling at low energies. This implies the need for matching both schemes at an intermediate switching scale, $L_{\rm swi}$, which we choose roughly in the range 24 GeV. In this contribution we present a preliminary result for this matching procedure, and we then focus on our almost final results for the scale evolution of the SF coupling from $L_{\rm swi}$ towards the perturbative regime, where we extract the $N_{\rm f} = 3$ ${\Lambda}$parameter, ${\Lambda}_{\overline{\rm MS}}^{N_{\rm f}=3}$, in units of $L_{\rm swi}$ . Connecting $L_{\rm swi}$ and thus the ${\Lambda}$parameter to a hadronic scale such as $F_K$ requires 2 further ingredients: first, the connection of $L_{\rm swi}$ to $L_{\rm max}$ using a few steps with the stepscaling function of the gradient flow coupling, and, second, the continuum extrapolation of $L_{\rm max} F_K$.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 arXiv:
 arXiv:1511.05831
 Bibcode:
 2015arXiv151105831D
 Keywords:

 High Energy Physics  Lattice
 EPrint:
 7 pages, 4 figures, Proceedings of the 33rd International Symposium on Lattice Field Theory (Lattice 2015), 1418 July 2015, Kobe, Japan