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Register a new userThe determination of $alpha_s$ by the ALPHA collaboration
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We review the ALPHA collaboration strategy for obtaining the QCD coupling at high scale. In the three-flavor effective theory it avoids the use of perturbation theory at $alpha > 0.2$ and at the same time has the physical scales small compared to the cutoff $1/a$ in all stages of the computation. The result $Lambda_overline{MS}^{(3)}=332(14)$~MeV is translated to $alpha_overline{MS}(m_Z)=0.1179(10)(2)$ by use of (high order) perturbative relations between the effective theory couplings at the charm and beauty quark thresholds. The error of this perturbative step is discussed and estimated as $0.0002$.
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We present results by the ALPHA collaboration for the $Lambda$-parameter in 3-flavour QCD and the strong coupling constant at the electroweak scale, $alpha_s(m_Z)$, in terms of hadronic quantities computed on the CLS gauge configurations. The first part of this proceedings contribution contains a review of published material cite{Brida:2016flw,DallaBrida:2016kgh} and yields the $Lambda$-parameter in units of a low energy scale, $1/L_{rm had}$. We then discuss how to determine this scale in physical units from experimental data for the pion and kaon decay constants. We obtain $Lambda_{overline{rm MS}}^{(3)} = 332(14)$ MeV which translates to $alpha_s(M_Z)=0.1179(10)(2)$ using perturbation theory to match between 3-, 4- and 5-flavour QCD.
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+1-flavour QCD by the CLS initiative with a number of tools developed over the years: renormalized couplings in finite volume schemes, recursive finite size techniques, two-loop 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 2-4 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 step-scaling function of the gradient flow coupling, and, second, the continuum extrapolation of $L_{rm max} F_K$.
We revisit the earlier determination of alpha_s(M_Z) via perturbative analyses of short-distance-sensitive lattice observables, incorporating new lattice data and performing a modified version of the original analysis. We focus on two high-intrinsic-scale observables, log(W_11) and log(W_12), and one lower-intrinsic scale observable, log(W_{12}/u_0^6), finding improved consistency among the values extracted using the different observables and a final result, alpha_s(M_Z)=0.1192(11), 2 sigma higher than the earlier result, in excellent agreement with recent non-lattice determinations and, in addition, in good agreement with the results of a similar, but not identical, re-analysis by the HPQCD Collaboration. A discussion of the relation between the two re-analyses is given, focussing on the complementary aspects of the two approaches.
We determine f_K for lattice QCD in the two flavor approximation with non-perturbatively improved Wilson fermions. The result is used to set the scale for dimensionful quantities in CLS/ALPHA simulations. To control its dependence on the light quark mass, two different strategies for the chiral extrapolation are applied. Combining f_K and the bare strange quark mass with non-perturbative renormalization factors and step scaling functions computed in the Schroedinger Functional, we determine the RGI strange quark mass and the Lambda parameter in units of f_K.
We describe the first lattice determination of the strong coupling constant with 3 flavors of dynamical quarks. The method follows previous analyses in using a perturbative expansion for the plaquette and Upsilon spectroscopy to set the scale. Using dynamical configurations from the MILC collaboration with 2+1 flavors of dynamical quarks we are able to avoid previous problems of having to extrapolate to 3 light flavors from 0 and 2. Our results agree with our previous work: alpha_s_MSbar(M_Z) = 0.121(3).
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