Lattice QCD has reached a mature status. State of the art lattice computations include $u,d,s$ (and even the $c$) sea quark effects, together with an estimate of electromagnetic and isospin breaking corrections for hadronic observables. This precise
and first principles description of the standard model at low energies allows the determination of multiple quantities that are essential inputs for phenomenology and not accessible to perturbation theory. One of the fundamental parameters that are determined from simulations of lattice QCD is the strong coupling constant, which plays a central role in the quest for precision at the LHC. Lattice calculations currently provide its best determinations, and will play a central role in future phenomenological studies. For this reason we believe that it is timely to provide a pedagogical introduction to the lattice determinations of the strong coupling. Rather than analysing individual studies, the emphasis will be on the methodologies and the systematic errors that arise in these determinations. We hope that these notes will help lattice practitioners, and QCD phenomenologists at large, by providing a self-contained introduction to the methodology and the possible sources of systematic error. The limiting factors in the determination of the strong coupling turn out to be different from the ones that limit other lattice precision observables. We hope to collect enough information here to allow the reader to appreciate the challenges that arise in order to improve further our knowledge of a quantity that is crucial for LHC phenomenology.
We present ADerrors.jl, a software for linear error propagation and analysis of Monte Carlo data. Although the focus is in data analysis in Lattice QCD, where estimates of the observables have to be computed from Monte Carlo samples, the software als
o deals with variables with uncertainties, either correlated or uncorrelated. Thanks to automatic differentiation techniques linear error propagation is performed exactly, even in iterative algorithms (i.e. errors in parameters of non-linear fits). In this contribution we present an overview of the capabilities of the software, including access to uncertainties in fit parameters and dealing with correlated data. The software, written in julia, is available for download and use in https://gitlab.ift.uam-csic.es/alberto/aderrors.jl
We propose a new strategy for the determination of the step scaling function $sigma(u)$ in finite size scaling studies using the Gradient Flow. In this approach the determination of $sigma(u)$ is broken in two pieces: a change of the flow time at fix
ed physical size, and a change of the size of the system at fixed flow time. Using both perturbative arguments and a set of simulations in the pure gauge theory we show that this approach leads to a better control over the continuum extrapolations. Following this new proposal we determine the running coupling at high energies in the pure gauge theory and re-examine the determination of the $Lambda$-parameter, with special care on the perturbative truncation uncertainties.
We examine coherent phonons in a strongly driven sample of optimally-doped high temperature superconductor YBa$_2$Cu$_3$O$_{7-delta}$. We observe a non-linear lattice response of the 4.5,THz copper-oxygen vibrational mode at high excitation densities
, evidenced by the observation of the phonon third harmonic and indicating the mode is strongly anharmonic. In addition, we observe how high-amplitude phonon vibrations modify the position of the electronic charge transfer resonance. Both of these results have important implications for possible phonon-driven non-equilibrium superconductivity.
Using finite size scaling techniques and a renormalization scheme based on the Gradient Flow, we determine non-perturbatively the $beta$-function of the $SU(3)$ Yang-Mills theory for a range of renormalized couplings $bar g^2sim 1-12$. We perform a d
etailed study of the matching with the asymptotic NNLO perturbative behavior at high-energy, with our non-perturbative data showing a significant deviation from the perturbative prediction down to $bar{g}^2sim1$. We conclude that schemes based on the Gradient Flow are not competitive to match with the asymptotic perturbative behavior, even when the NNLO expansion of the $beta$-function is known. On the other hand, we show that matching non-perturbatively the Gradient Flow to the Schrodinger Functional scheme allows us to make safe contact with perturbation theory with full control on truncation errors. This strategy allows us to obtain a precise determination of the $Lambda$-parameter of the $SU(3)$ Yang-Mills theory in units of a reference hadronic scale ($sqrt{8t_0},Lambda_{overline{rm MS}} = 0.6227(98)$), showing that a precision on the QCD coupling below 0.5% per-cent can be achieved using these techniques.
Automatic Differentiation (AD) allows to determine exactly the Taylor series of any function truncated at any order. Here we propose to use AD techniques for Monte Carlo data analysis. We discuss how to estimate errors of a general function of measur
ed observables in different Monte Carlo simulations. Our proposal combines the $Gamma$-method with Automatic differentiation, allowing exact error propagation in arbitrary observables, even those defined via iterative algorithms. The case of special interest where we estimate the error in fit parameters is discussed in detail. We also present a freely available fortran reference implementation of the ideas discussed in this work.
