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Register a new userStrange quark mass and Lambda parameter by the ALPHA collaboration
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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.
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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.
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 $0.0002$.
QCD lattice simulations with 2+1 flavours typically start at rather large up-down and strange quark masses and extrapolate first the strange quark mass to its physical value and then the up-down quark mass. An alternative method of tuning the quark masses is discussed here in which the singlet quark mass is kept fixed, which ensures that the kaon always has mass less than the physical kaon mass. It can also take into account the different renormalisations (for singlet and non-singlet quark masses) occurring for non-chirally invariant lattice fermions and so allows a smooth extrapolation to the physical quark masses. This procedure enables a wide range of quark masses to be probed, including the case with a heavy up-down quark mass and light strange quark mass. Results show the correct order for the baryon octet and decuplet spectrum and an extrapolation to the physical pion mass gives mass values to within a few percent of their experimental values.
QCD lattice simulations with 2+1 flavours typically start at rather large up-down and strange quark masses and extrapolate first the strange quark mass to its physical value and then the up-down quark mass. An alternative method of tuning the quark masses is discussed here in which the singlet quark mass is kept fixed, which ensures that the kaon always has mass less than the physical kaon mass. Using group theory the possible quark mass polynomials for a Taylor expansion about the flavour symmetric line are found, which enables highly constrained fits to be used in the extrapolation of hadrons to the physical pion mass. Numerical results confirm the usefulness of this expansion and an extrapolation to the physical pion mass gives hadron mass values to within a few percent of their experimental values.
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 $L_{rm max} F_K$.
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