The gradient flow coupling from numerical stochastic perturbation theory

Perturbative calculations of gradient flow observables are technically challenging. Current results are limited to a few quantities and, in general, to low perturbative orders. Numerical stochastic perturbation theory is a potentially powerful tool that may be applied in this context. Precise results using these techniques, however, require control over both statistical and systematic uncertainties. In this contribution, we discuss some recent algorithmic developments that lead to a substantial reduction of the cost of the computations. The matching of the ${\overline{\rm MS}}$ coupling with the gradient flow coupling in a finite box with Schr\"odinger functional boundary conditions is considered for illustration.Comment: Talk given at the 34th annual International Symposium on Lattice Field Theory, 24-30 July 2016, University of Southampton, UK; LaTeX source, 7 pages, 2 figure

SMD-based numerical stochastic perturbation theory

The viability of a variant of numerical stochastic perturbation theory, where the Langevin equation is replaced by the SMD algorithm, is examined. In particular, the convergence of the process to a unique stationary state is rigorously established and the use of higher-order symplectic integration schemes is shown to be highly profitable in this context. For illustration, the gradient-flow coupling in finite volume with Schr\"odinger functional boundary conditions is computed to two-loop (i.e. NNL) order in the SU(3) gauge theory. The scaling behaviour of the algorithm turns out to be rather favourable in this case, which allows the computations to be driven close to the continuum limit.Comment: 35 pages, 4 figures; v2: corrected typos, coincides with published versio

Precision Determination of αs\alpha_s from Lattice QCD

We present an overview of the recent lattice determination of the QCD coupling $\alpha_s$ by the ALPHA Collaboration. The computation is based on the non-perturbative determination of the $\Lambda$-parameter of $N_{\rm f}=3$ QCD, and the perturbative matching of the $N_{\rm f}=3$ and $N_{\rm f}=5$ theories. The final result: $\alpha_s(m_Z)=0.11852(84)$, reaches sub-percent accuracy.Comment: 14 pages, 4 figures. Contribution from the "Selected Papers from the 7th International Conference on New Frontiers in Physics (ICNFP 2018)

A dynamical study of the chirally rotated Schr\"odinger functional in QCD

The chirally rotated Schr\"odinger functional for Wilson-fermions allows for finite-volume, mass-independent renormalization schemes compatible with automatic O($a$) improvement. So far, in QCD, the set-up has only been studied in the quenched approximation. Here we present first results for $N_{\rm f} = 2$ dynamical quark-flavours for several renormalization factors of quark-bilinears. We discuss how these renormalization factors can be easily obtained from simple ratios of two-point functions, and show how automatic O($a$) improvement is at work. As a by-product of this investigation the renormalization of the non-singlet axial current, $Z_A$, is determined very precisely.Comment: Talk given at the 32nd International Symposium on Lattice Field Theory, 23-28 June, 2014, New York, US; LaTeX source, 7 pages, 6 figure

Perturbative renormalization of DS = 2 four-fermion operators with the chirally rotated Schroedinger functional

The chirally rotated Schrödinger functional (χSF) renders the mechanism of automatic O(a) improvement compatible with Schrödinger functional (SF) renormalization schemes. Here we define a family of renormalization schemes based on the χSF for a complete basis of ΔF=2 parity-odd four-fermion operators. We compute the corresponding scale-dependent renormalization constants to one-loop order in perturbation theory and obtain their NLO anomalous dimensions by matching to the MSbar scheme. Due to automatic O(a) improvement, once the χSF is renormalized and improved at the boundaries, the step scaling functions (SSF) of these operators approach their continuum limit with O(a2) corrections without the need of operator improvement