Search

Modern particle physics experiments, e.g. at the Large Hadron Collider (LHC) at CERN, crucially depend on the precise description of the scattering processes in terms of the known fundamental forces. This is limited by our current understanding of the strong nuclear force, as quantified by the strong coupling, $\alpha_s$, between quarks and gluons. Relating $\alpha_s$ to experiments poses a major challenge as the strong interactions lead to the confinement of quarks and gluons inside hadronic bound states. At high energies, however, the strong interactions become weaker ("asymptotic freedom") and thus amenable to an expansion in powers of the coupling. Attempts to relate both regimes usually rely on modeling of the bound state problem in one way or another. Using large scale numerical simulations of a first principles formulation of Quantum Chromodynamics on a space-time lattice, we have carried out a model-independent determination of $\alpha_s$ with unprecedented precision. The uncertainty, about half that of all other results combined, originates predominantly from the statistical Monte Carlo evaluation and has a clear probabilistic interpretation. The result for $\alpha_s$ describes a variety of physical phenomena over a wide range of energy scales. If used as input information, it will enable significantly improved analyses of many high energy experiments, by removing an important source of theoretical uncertainty. This will increase the likelihood to uncover small effects of yet unknown physics, and enable stringent precision tests of the Standard Model. In summary, this result boosts the discovery potential of the LHC and future colliders, and the methods developed in this work pave the way for even higher precision in the future., Comment: 49 pages, LaTeX

Close to the continuum limit, lattice QCD with mass-degenerate Wilson quarks can be described by Symanzik's effective continuum action, which contains the dimension 5 operator, $m\,{\rm tr}(F_{\mu\nu}F_{\mu\nu})$. Its effect can be eliminated by an O($am_{\rm q}$) rescaling of the bare lattice coupling constant. Until recently, the corresponding improvement coefficient, $b_{\rm g}$, was only known perturbatively to 1-loop order and an estimate of the remaining uncertainty is the dominant systematic error in the ALPHA collaboration's recent determination of $\alpha_s(m_Z)$ with the decoupling method. To remove this error we have determined $b_{\rm g}$ non-perturbatively for the corresponding parameter range. We here briefly review improvement conditions for $b_{\rm g}$, perform a perturbative test and report on our non-perturbative results for $b_{\rm g}$., Comment: 8 pages, 1 figure, contribution to "The 40th International Symposium on Lattice Field Theory (Lattice 2023)", CERN-TH-2024-004

With Wilson quarks, on-shell O($a$) improvement of the lattice QCD action is achieved by including the Sheikholeslami-Wohlert term and two further operators of mass dimension 5, which amount to a mass-dependent rescaling of the bare parameters. We here focus on the rescaled bare coupling, $\tilde{g}_0^2 = g_0^2(1 + b_{\rm g} am_{\rm q})$, and the determination of $b_{\rm g}(g_0^2)$, which is currently only known to 1-loop order of perturbation theory. We derive suitable improvement conditions in the chiral limit and in a finite space-time volume and evaluate these for different gluonic observables, both with and without the gradient flow. The choice of $\beta$-values and the line of constant physics are motivated by the ALPHA collaboration's decoupling strategy to determine $\alpha_s(m_Z)$. However, the improvement conditions and some insight into systematic effects may prove useful in other contexts, too., Comment: 26 pages, 5 figures. Included additional references and fixed some typos. Matches published version

In a somewhat forgotten paper [1] it was shown how to perform interpolations between relativistic and static computations in order to obtain results for heavy-light observables for masses from, say, $m_{\rm charm}$ to $m_{\rm bottom}$. All quantities are first continuum extrapolated and then interpolated in $1/m_h=1/m_{\rm heavy}$. Large volume computations are combined with finite volume ones where a relativistic bottom quark is accessible with small $am_{\rm bottom}$. We discuss how this strategy is extended to semi-leptonic form factors and other quantities of phenomenological interest. The essential point is to form quantities where the limit $m_h\to\infty$ is approached with power corrections O$(1/m_h)$ only. Perturbative corrections $\sim\alpha_s(m_h)^{\gamma+n}$ are cancelled in the construction of the observables. We also point out how such an approach can help to control systematics in semi-leptonic decays with just large volume data. First numerical results with $N_f = 2 + 1$ and lattice spacings down to 0.039 fm are presented in [2]., Comment: Lattice 2023 talk

We present preliminary results for B-physics from a combination of non-perturbative results in the static limit with relativistic computations satisfying $am_{\mathrm{heavy}}\ll 1$. Relativistic measurements are carried out at the physical b-quark mass using the Schr\"{o}dinger Functional in a $0.5 \ \mathrm{fm}$ box. They are connected to large volume observables through step scaling functions that trace the mass dependence between the physical charm region and the static limit, such that B-physics results can be obtained by interpolation; the procedure is designed to exactly cancel the troublesome $\alpha_s(m_{\mathrm{heavy}})^{n+\gamma}$ corrections to large mass scaling. Large volume computations for both static and relativistic quantities use CLS $N_f=2+1$ ensembles at $m_u=m_d=m_s$, and with five values of the lattice spacing down to $0.039$ fm. Our preliminary results for the b-quark mass and leptonic decay constants have competitive uncertainties, which are furthermore dominated by statistics, allowing for substantial future improvement. Here we focus on numerical results, while the underlying strategy is discussed in a companion contribution., Comment: Lattice 2023 talk

Multi-particle states with additional pions are expected to result in a difficult-to-control excited-state contamination in lattice simulations. We show that heavy meson chiral perturbation theory can be employed to estimate the contamination due to two-particle $B\pi$ states in various $B$-meson observables like the $B$-meson decay constant and the $BB^*\pi$ coupling. We work in the static limit and to next-to-leading order in the chiral expansion, i.e. including $\rm O(p)$. The $B\pi$ states are found to typically overestimate the observables at the few percent level. We determine two of the LECs from $B\to \pi$ form factor computations of the KEK group and discuss ways to determine the others. In particular two LECs which are associated with smeared interpolating fields seem to be easily accessible and thus open up a way to systematically study the effect of smearing on excited state effects., Comment: 37 pages, 14 figures

When designing lattice actions, gauge field smearing is frequently used to define the lattice Dirac operator. Since the smearing procedure removes effects of ultraviolet fluctuations, the fermions effectively see a larger lattice spacing than the gauge fields. Creutz ratios, formed from ratios of rectangular Wilson loops, based on smeared gauge fields are adequate observables to investigate the effect of smearing since they do not need renormalisation and provide a measure of the physical forces felt by the fermions. We study their behaviour at various smearing radii (fixed in lattice units) and in particular how the smearing influences the scaling towards the continuum limit. Since we employ the Wilson gradient flow as smearing, the same Creutz ratios have another, well defined continuum limit, when the flow time is fixed in physical units. That continuum limit is reached with smaller corrections at finite $a$., Comment: 10 pages, 5 figures, proceedings to talk presented at the 39th International Symposium on Lattice Field Theory, 8th-13th August, 2022, Rheinische Friedrich-Wilhelms-Universit\"at Bonn, Bonn, Germany

Integrated time-slice correlation functions $G(t)$ with weights $K(t)$ appear, e.g., in the moments method to determine $\alpha_s$ from heavy quark correlators, in the muon g-2 determination or in the determination of smoothed spectral functions. For the (leading-order-)normalised moment $R_4$ of the pseudo-scalar correlator we have non-perturbative results down to $a=10^{-2}$ fm and for masses, $m$, of the order of the charm mass in the quenched approximation. A significant bending of $R_4$ as a function of $a^2$ is observed at small lattice spacings. Starting from the Symanzik expansion of the integrand we derive the asymptotic convergence of the integral at small lattice spacing in the free theory and prove that the short distance part of the integral leads to $\log(a)$-enhanced discretisation errors when $G(t)K(t) \sim\, t $ for small $t$. In the interacting theory an unknown, function $K(a\Lambda)$ appears. For the $R_4$-case, we modify the observable to improve the short distance behavior and demonstrate that it results in a very smooth continuum limit. The strong coupling and the $\Lambda$-parameter can then be extracted. In general, and in particular for $g-2$, the short distance part of the integral should be determined by perturbation theory. The (dominating) rest can then be obtained by the controlled continuum limit of the lattice computation.

Using Heavy Meson Chiral Perturbation Theory (HMChPT), the $B^* \pi$ excited states contamination of the $B \to \pi$ vector form factors is computed to NLO in the chiral expansion and in the static limit. The results suggest that the excited states for $h_\parallel$ are of the order of a few percent whereas $h_\perp$ receives large negative contributions and thus might be significantly underestimated in lattice simulations., Comment: 8 pages, 5 figures, Proceedings of the 39th International Symposium on Lattice Field Theory, 8th-13th August 2022, Bonn, Germany

Multi-particle states with additional pions are expected to result in a non-negligible excited-state contamination in lattice simulations. We show that heavy meson chiral perturbation theory can be employed to calculate the contamination due to two-particle $B\pi$ states in various $B$-meson observables like the $B$-meson decay constant and the $BB^*\pi$ coupling. We work in the static limit and to next-to-leading order in the chiral expansion. The $B\pi$ states are found to typically overestimate the observables at the few percent level depending on the size of two currently unknown NLO low-energy coefficients. A strategy to independently measure one of them with the 3-point function of the light axial vector current will be discussed., Comment: 9 pages, 5 figures. Talk given at the 39th International Symposium on Lattice Field Theory (LATTICE2022)

We present the details and first results of a new strategy for the determination of $\alpha_s(m_Z)$. By simultaneously decoupling 3 fictitious heavy quarks we establish a relation between the $\Lambda$-parameters of three-flavor QCD and pure gauge theory. Very precise recent results in the pure gauge theory can thus be leveraged to obtain the three-flavour $\Lambda$-parameter in units of a common decoupling scale. Connecting this scale to hadronic physics in 3-flavour QCD leads to our result in physical units, $\Lambda^{(3)}_{\bar{\rm MS}} = 336(12)\, {\rm MeV}$, which translates to $\alpha_s(m_Z) = 0.11823(84)$. This is compatible with both the FLAG average and the previous ALPHA result, with a comparable, yet still statistics dominated, error. This constitutes a highly non-trivial check, as the decoupling strategy is conceptually very different from the 3-flavour QCD step-scaling method, and so are their systematic errors. These include the uncertainties of the combined decoupling and continuum limits, which we discuss in some detail. We also quantify the correlation between both results, due to some common elements, such as the scale determination in physical units and the definition of the energy scale where we apply decoupling., Comment: 62 pages, 10 figures, LaTeX

Observables in particle physics and specifically in lattice QCD calculations are often extracted from fits. Standard $\chi^2$ tests require a reliable determination of the covariance matrix and its inverse from correlated and auto-correlated data, a challenging task often leading to close-to-singular estimates. These motivate modifications of the definition of $\chi^2$ such as uncorrelated fits. We show how the goodness-of-fit measured by their p-value can still be estimated robustly for a broad class of such fits., Comment: 15 pages, 4 figures, matching published version

Lattice quantum chromodynamics has proven to be an indispensable method to determine nonperturbative strong contributions to weak decay processes. In this white paper for the Snowmass community planning process we highlight achievements and future avenues of research for lattice calculations of weak $b$ and $c$ quark decays, and point out how these calculations will help to address the anomalies currently in the spotlight of the particle physics community. With future increases in computational resources and algorithmic improvements, percent level (and below) lattice determinations will play a central role in constraining the standard model or identifying new physics., Comment: contribution to Snowmass 2021; 19 pages; v2 corrected typo and added references

The QCD-coupling is a necessary input in the computation of many observables, and the parametric error on input parameters can be a dominant source of uncertainty. The coupling can be extracted by comparing high order perturbative computations and lattice evaluated moments of mesonic two-point functions with heavy quarks, which provide a high energy scale for perturbation theory. The truncation of the perturbative series is an important systematic uncertainty. We report on our attempt to study this issue by measuring pseudo-scalar two-point functions in volumes of $L=2\, \text{fm}$ with twisted-mass Wilson fermions in the quenched approximation. We use full twist, the non-perturbative clover term and lattice spacings down to $a=0.015\,\text{fm}$ to tame the sizable discretization effects. Our preliminary results indicate that either higher order perturbative corrections or the continuum limit are not under sufficient control despite our small lattice spacings and quark masses extending beyond $2\,m_{\text{charm}}$.

We present analysis details and new results for the strong coupling $\alpha_s(m_Z)$, determined by the decoupling strategy. We measure a massive gradient flow (GF) coupling defined in finite volume with Schr\"odinger functional (SF) boundary conditions in a theory with $N_\text{f}=3$ degenerate heavy quarks of mass $M$. The massive couplings are matched to effective couplings in pure gauge. Using the running in the pure gauge theory and the perturbative relation of the Lambda parameters, the Lambda parameter of the three flavor theory is obtained by an extrapolation to infinite M. Our final result is compatible both with the FLAG average and with the previous ALPHA result, albeit with a slightly smaller, yet still statistics dominated, error. This constitutes a non-trivial check, as the decoupling strategy is conceptually very different from the 3-flavor QCD step-scaling method, and so are most of its systematic errors. These include the uncertainties of the decoupling and continuum limits, which we discuss in some detail. Furthermore, by relying on decoupling once again, we could estimate the small $O(a)$ and $O(1/M)$ contaminations to the massive GF coupling stemming from the SF boundaries by means of pure gauge simulations., Comment: Lattice 2021 contribution 492

We present an update of the scale setting for $N_f=2+1$ flavor QCD using gradient flow scales and pseudo-scalar decay constants. We analyze the latest ensembles with $2+1$ flavors of non-perturbatively improved Wilson fermions generated by CLS for improved precision. Special care is taken to correct for mistuning by measuring directly the mass derivatives of the various observables. We determine $t_0$ with input taken from a combination of leptonic decay rates of the Pion and the Kaon., Comment: 10 pages, 6 figures, The 38th International Symposium on Lattice Field Theory, LATTICE2021

We consider three-flavor QCD and perform a determination of the low-energy coupling $\hat{g}_\chi$ of SU(2) Heavy Meson Chiral Perturbation Theory. It is the $B^*B\pi$ coupling in the limit of static heavy and chiral light quarks and has not been determined with precision thus far. The calculation is performed on a large set of the $2+1$ flavor CLS ensembles with pion masses from 420 MeV down to 130 MeV. This allows us to significantly reduce the systematic uncertainty from the chiral extrapolation compared to previous works. Only a weak dependence on the lattice spacing is visible in our results., Comment: 9 pages, 4 figures, The 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We analyse the leading logarithmic corrections to the $a^2$ scaling of lattice artefacts in QCD, following the seminal work of Balog, Niedermayer and Weisz in the O(n) non-linear sigma model. Limiting the discussion to contributions from the action, the leading logarithmic corrections can be determined by the anomalous dimensions of mass-dimension 6 operators. These operators form a minimal on-shell basis of the Symanzik Effective Theory. We present results for non-perturbatively O($a$) improved Wilson and Ginsparg-Wilson quarks., Comment: 7 pages, 3 figures, Proceedings of The 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021 Zoom/Gather@Massachusetts Institute of Technology. Multiplied a proper power of $(2b_0)^{n_j^\mathrm{I}}$ where needed

We consider spectral quantities in lattice QCD and determine the asymptotic behavior of their discretization errors. Wilson fermion with O$(a)$-improvement, (M\"obius) Domain wall fermion (DWF), and overlap Dirac operators are considered in combination with the commonly used gauge actions. Wilson fermions and DWF with domain wall height $M_5=1+{\rm O}(g_0^2)$ have the same, approximate, form of the asymptotic cutoff effects: $ K\,a^2\left[\bar g^2(a^{-1})\right]^{0.760}$. A domain wall height $M_5=1.8$, as often used, introduces large mass-dependent $K'(m)\,a^2\left[\bar g^2(a^{-1})\right]^{0.518}$ effects. Massless twisted mass fermions have the same form as Wilson fermions when the Sheikholeslami-Wohlert term [1] is included. For their mass-dependent cutoff effects we have information on the exponents $\hat\Gamma_i$ of $\bar g^2(a^{-1})$ but not for the pre-factors. For staggered fermions there is only partial information on the exponents. We propose that tree-level ${\rm O}(a^2)$ improvement, which is easy to do [2], should be used in the future -- both for the fermion and the gauge action. It improves the asymptotic behavior in all cases., Comment: corrected the normalisation of some of the operators in the massive basis (sect. 6.2 and Table 3)

Discretization effects of lattice QCD are described by Symanzik's effective theory when the lattice spacing, $a$, is small. Asymptotic freedom predicts that the leading asymptotic behavior is $\sim a^n [\bar g^2(a^{-1})]^{\hat\gamma_1} \sim a^n \left[\frac{1}{-\log(a\Lambda)}\right]^{\hat\gamma_1}$. For spectral quantities, $n=d$ is given in terms of the (lowest) canonical dimension, $d+4$, of the operators in the local effective Lagrangian and $\hat\gamma_1$ is proportional to the leading eigenvalue of their one-loop anomalous dimension matrix $\gamma^{(0)}$. We determine $\gamma^{(0)}$ for Yang-Mills theory ($n=2$) and discuss consequences in general and for perturbatively improved short distance observables. With the help of results from the literature, we also discuss the $n=1$ case of Wilson fermions with perturbative O$(a)$ improvement and the discretization effects specific to the flavor currents. In all cases known so far, the discretization effects are found to disappear faster than the naive $\sim a^n$ and the log-corrections are a rather weak modification -- in contrast to the two-dimensional O(3) sigma model.

We propose a new strategy for the determination of the QCD coupling. It relies on a coupling computed in QCD with $N_{\rm f} \geq 3$ degenerate heavy quarks at a low energy scale $\mu_{\rm dec}$, together with a non-perturbative determination of the ratio $\Lambda/\mu_{\rm dec}$ in the pure gauge theory. We explore this idea using a finite volume renormalization scheme for the case of $N_{\rm f} = 3$ QCD, demonstrating that a precise value of the strong coupling $\alpha_s$ can be obtained. The idea is quite general and can be applied to solve other renormalization problems, using finite or infinite volume intermediate renormalization schemes., Comment: LaTeX, 6 pages, 2 figures

We analyse the leading logarithmic corrections to the $a^2$ scaling of lattice artefacts in QCD, following the seminal work of Balog, Niedermayer and Weisz in the O(n) non-linear sigma model. Restricting our attention to contributions from the action, the leading logarithmic corrections can be determined by the anomalous dimensions of a minimal on-shell basis of mass-dimension 6 operators. We present results for the SU(N) pure gauge theory. In this theory the logarithmic corrections reduce the cutoff effects. These computations are the first step towards a study of full lattice QCD at O($a^2$), which is in progress., Comment: 7 pages. Contribution to the 37th International Symposium on Lattice Field Theory (LATTICE2019), 16-22 June 2019, Wuhan, China

We discuss the extraction of the ground state $\langle \mathrm{K} ({\bf p})|V_\mu(0)|\mathrm{B} ({\bf 0})\rangle$ matrix elements from Euclidean lattice correlation functions. The emphasis is on the elimination of excited state contributions. Two typical gauge-field ensembles with lattice spacings $0.075, \; 0.05$ fm and pion masses $330,\;270$ MeV are used from the O($a$)- improved CLS 2-flavour simulations and the final state momentum is $|{\bf p}|=0.5\,{\rm GeV}$. The b-quark is treated in HQET including the $1/m_\mathrm{b}$ corrections. Fits to two-point and three-point correlation functions and suitable ratios including summed ratios are used, yielding consistent results with precision of around 2% which is $not$ limited by the $1/m_\mathrm{b}$ corrections but by the dominating static form factors. Excited state contributions are under reasonable control but are the bottleneck towards precision. We do not yet include a specific investigation of multi-hadron contaminations, a gap in the literature which ought to be filled soon., Comment: 38 pages, 23 figures

The `XIIIth Quark Confinement and the Hadron Spectrum' conference (Confinement 2018) contained a `Round Table Discussion' on the status of the determinations of the strong coupling $\alpha_s(m_Z)$ as well as prospects for future improvements. In this contribution, we summarize the different aspects of the discussion. In particular, we cover $\alpha_s$ determinations from inclusive observables, such as the hadronic decays of the $Z$ boson and the $\tau$ lepton; from global fits of parton distribution functions (PDFs); from high-energy collider observables, and from event shapes; as well as from various observables computed by lattice QCD, specifically from the QCD static energy. There is overall good agreement between these various determinations, but there are also outliers, differing from the world average by up to $-5\%$. Nevertheless, the general agreement constitutes a beautiful and significant test of the detailed nature of the strong interactions, and provides a crucial input for high-precision calculations in QCD and beyond., Comment: Round Table Discussion at the XIIIth Quark Confinement and the Hadron Spectrum conference. 32 pages, 11 figures. Added references

Effects of heavy sea quarks on the low energy physics are described by an effective theory where the expansion parameter is the inverse quark mass, 1/$M$. At leading order in 1/$M$ (and neglecting light quark masses) the dependence of any low energy quantity on $M$ is given in terms of the ratio of $\Lambda$ parameters of the effective and the fundamental theory. We define a function describing the scaling with the mass $M$. We find that its perturbative expansion is very reliable for the bottom quark and also seems to work very well at the charm quark mass. The same is then true for the ratios of $\Lambda^{(4)}/\Lambda^{(5)}$ and $\Lambda^{(3)}/\Lambda^{(4)}$, which play a major r\^ole in connecting lattice determinations of $\alpha^{(3)}_{MSbar}$ from the three-flavor theory with $\alpha^{(5)}_{MSbar}(M_Z)$. Also the charm quark content of the nucleon, relevant for dark matter searches, can be computed accurately from perturbation theory. We investigate a very closely related model, namely QCD with $N_f=2$ heavy quarks. Our non-perturbative information is derived from simulations on the lattice, with masses up to the charm quark mass and lattice spacings down to about 0.023 fm followed by a continuum extrapolation. The non-perturbative mass dependence agrees within rather small errors with the perturbative prediction at masses around the charm quark mass. Surprisingly, from studying solely the massive theory we can make a prediction for the ratio $Q^{1/\sqrt{t_0}}_{0,2}=[\Lambda \sqrt{t_0(0)}]_{N_f=2}/[\Lambda \sqrt{t_0}]_{N_f=0}$, which refers to the chiral limit in $N_f=2$. Here $t_0$ is the Gradient Flow scale of [1]. The uncertainty for $Q$ is estimated to be 2.5%. For the phenomenologically interesting $\Lambda^{(3)}/\Lambda^{(4)}$, we conclude that perturbation theory introduces errors which are at most at the 1.5% level, far smaller than other current uncertainties., Comment: 38 pages, 14 figures; version accepted for publication in Nuclear Physics B; abstract slightly abridged

The large $N$ limit of SU($N$) gauge theories is well understood in perturbation theory. Also non-perturbative lattice studies have yielded important positive evidence that 't Hooft's predictions are valid. We go far beyond the statistical and systematic precision of previous studies by making use of the Yang-Mills gradient flow and detailed Monte Carlo simulations of SU($N$) pure gauge theories in 4 dimensions. With results for $N=3,4,5,6,8$ we study the limit and the approach to it. We pay particular attention to observables which test the expected factorization in the large $N$ limit. The investigations are carried out both in the continuum limit and at finite lattice spacing. Large $N$ scaling is verified non-perturbatively and with high precision; in particular, factorization is confirmed. For quantities which only probe distances below the typical confinement length scale, the coefficients of the $1/N$ expansion are of $\mathrm{O}(1)$, but we found that large (smoothed) Wilson loops have rather large $\mathrm{O}(1/N^2)$ corrections. The exact size of such corrections does, of course, also depend on what is kept fixed when the limit is taken., Comment: 25 pages, 11 figures

We discuss the problem of lattice artefacts in QCD simulations enhanced by the introduction of dynamical charmed quarks. In particular, we advocate the use of a massive renormalization scheme with a close to realistic charm mass. To maintain O(a) improvement for Wilson type fermions in this case we define a finite size scheme and carry out a nonperturbative estimation of the clover coefficient $c_\mathrm{sw}$. It is summarized in a fit formula $c_\mathrm{sw}(g_0^2)$ that defines an improved action suitable for future dynamical charm simulations., Comment: 17 pages, 3 tables, 2 figures, published version (extended introduction)

Using continuum extrapolated lattice data we trace a family of running couplings in three-flavour QCD over a large range of scales from about 4 to 128 GeV. The scale is set by the finite space time volume so that recursive finite size techniques can be applied, and Schr\"odinger functional (SF) boundary conditions enable direct simulations in the chiral limit. Compared to earlier studies we have improved on both statistical and systematic errors. Using the SF coupling to implicitly define a reference scale $1/L_0\approx 4$ GeV through $\bar{g}^2(L_0) =2.012$, we quote $L_0 \Lambda^{N_{\rm f}=3}_{\overline{\rm MS}} =0.0791(21)$. This error is dominated by statistics; in particular, the remnant perturbative uncertainty is negligible and very well controlled, by connecting to infinite renormalization scale from different scales $2^n/L_0$ for $n=0,1,\ldots,5$. An intermediate step in this connection may involve any member of a one-parameter family of SF couplings. This provides an excellent opportunity for tests of perturbation theory some of which have been published in a letter [1]. The results indicate that for our target precision of 3 per cent in $L_0 \Lambda^{N_{\rm f}=3}_{\overline{\rm MS}}$, a reliable estimate of the truncation error requires non-perturbative data for a sufficiently large range of values of $\alpha_s=\bar{g}^2/(4\pi)$. In the present work we reach this precision by studying scales that vary by a factor $2^5= 32$, reaching down to $\alpha_s\approx 0.1$. We here provide the details of our analysis and an extended discussion., Comment: 40 pages, 11 figures

We investigate the SU(3) Yang Mills theory at small gradient flow time and at short distances. Lattice spacings down to $a=0.015$ fm are simulated with open boundary conditions to allow topology to flow in and out. We study the behaviour of the action density $E(t)$ close to the boundaries, the feasibility of the small flow-time expansion and the extraction of the $\Lambda$-parameter from the static force at small distances. For the latter, significant deviations from the 4-loop perturbative $\beta$-function are visible at $\alpha\approx 0.2\,$. We still can extrapolate to extract $r_0\Lambda$., Comment: 8 pages, 5 figures, Proceedings of the 35th International Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spain

We compute semi-leptonic $B_s$ decay form factors using Heavy Quark Effective Theory on the lattice. To obtain good control of the $1/m_b$ expansion, one has to take into account not only the leading static order but also the terms arising at $O(1/m_b)$: kinetic, spin and current insertions. We show results for these terms calculated through the ratio method, using our prior results for the static order. After combining them with non-perturbative HQET parameters they can be continuum-extrapolated to give the QCD form factor correct up to $O(1/m_b^2)$ corrections and without $O(\alpha_s(m_b)^n)$ corrections., Comment: 7 pages, 4 figures. Proceedings of the 35th International Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spain; v2: small corrections, published version

We present results for Wilson loops smoothed with the Yang-Mills gradient flow and matched through the scale $t_0$. They provide renormalized and precise operators allowing to test the $1/N^2$ scaling both at finite lattice spacing and in the continuum limit. Our results show an excellent scaling up to $1/N = 1/3$. Additionally, we obtain a very precise non-perturbative confirmation of factorization in the large $N$ limit., Comment: 8 pages, 7 figures. Presented at the 35th International Symposium on Lattice Field Theory, Granada, Spain, 18-24 June 2017 (Lattice 2017)

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 $0.0002$., Comment: 7 pages, proceedings of FPCapri2016 conference

Using a finite volume Gradient Flow (GF) renormalization scheme with Schr\"odinger Functional (SF) boundary conditions, we compute the non-perturbative running coupling in the range $2.2 \lesssim {\bar g}_\mathrm{GF}^2(L) \lesssim 13$. Careful continuum extrapolations turn out to be crucial to reach our high accuracy. The running of the coupling is always between one-loop and two-loop and very close to one-loop in the region of $200\,{\rm MeV} \lesssim \mu=1/L \lesssim 4\,{\rm GeV}$. While there is no convincing contact to two-loop running, we match non-perturbatively to the SF coupling with background field. In this case we know the $\mu$ dependence up to $\sim 100\,{\rm GeV}$ and can thus connect to the $\Lambda$-parameter., Comment: 34 pages, LaTeX

We discuss the determination of the strong coupling $\alpha_\mathrm{\overline{MS}}^{}(m_\mathrm{Z})$ or equivalently the QCD $\Lambda$-parameter. Its determination requires the use of perturbation theory in $\alpha_s(\mu)$ in some scheme, $s$, and at some energy scale $\mu$. The higher the scale $\mu$ the more accurate perturbation theory becomes, owing to asymptotic freedom. As one step in our computation of the $\Lambda$-parameter in three-flavor QCD, we perform lattice computations in a scheme which allows us to non-perturbatively reach very high energies, corresponding to $\alpha_s = 0.1$ and below. We find that (continuum) perturbation theory is very accurate there, yielding a three percent error in the $\Lambda$-parameter, while data around $\alpha_s \approx 0.2$ is clearly insufficient to quote such a precision. It is important to realize that these findings are expected to be generic, as our scheme has advantageous properties regarding the applicability of perturbation theory., Comment: 6.5 pages including figures and references; version accepted for publication in PRL

We discuss the computation of form factors for semi-leptonic decays of $\rm B$-, $\rm B_s$- mesons in lattice QCD. Considering in particular the example of the static $\rm B_s$ form factors we demonstrate that after non-perturbative renormalization the continuum limit can be taken with confidence. The resulting precision is of interest for extractions of $V_{\rm ub}$. The size of the corrections of order $1/m_{\rm b}$ is just estimated at present but it is expected that their inclusion does not pose significant difficulties., Comment: 16 pages, 3 figures; minor corrections and typo fixes, published version

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 $L_{\rm max} F_K$., Comment: 7 pages, 4 figures, Proceedings of the 33rd International Symposium on Lattice Field Theory (Lattice 2015), 14-18 July 2015, Kobe, Japan

We simulate a theory with $N_f=2$ heavy quarks of mass $M$. At energies much smaller than $M$ the heavy quarks decouple and the theory can be described by an effective theory which is a pure gauge theory to leading order in $1/M$. We present results for the mass dependence of ratios such as $t_0(M)/t_0(0)$. We compute these ratios from simulations and compare them to the perturbative prediction. The latter relies on a factorisation formula for the ratios which is valid to leading order in $1/M$., Comment: 7 pages, 3 figures, Proceedings of the 33rd International Symposium on Lattice Field Theory, 14-18 July 2015, Kobe, Japan

We develop a strategy for the non-perturbative determination of the O(a)-improvement coefficient c_sw for Wilson fermions with massive sea quarks. The improvement condition is defined via the PCAC relation in the Schr\"odinger functional. It is imposed along a line of constant physics designed to be close to the correct mass of the charm quark. The numerical work uses the tree-level improved L\"uscher-Weisz gauge action in N_f=3+1 Lattice QCD.

We review a lattice strategy how to non-perturbatively determine the coefficients in the HQET expansion of all components of the heavy-light axial and vector currents, including 1/m_h-corrections. We also discuss recent preliminary results on the form factors parameterizing semi-leptonic B-decays at the leading order in 1/m_h., Comment: 10 pages, 7 figures, Contribution to the Proceedings of the Final Meeting of DFG SFB-TR-9 (Durbach, Germany, Sept. 2014), to appear in Nucl. Phys. (Proc. Suppl.)

We give an introduction to Heavy Quark Effective Theory (HQET). Our emphasis is on its formulation non-perturbative in the strong coupling, including the non-perturbative determination of the parameters in the HQET Lagrangian. In a second part we review the present status of HQET on the lattice, largely based on work of the ALPHA collaboration in the last few years. We finally discuss opportunities and challenges., Comment: 31 pages, 15 figures, Contribution to the Proceedings of the Final Meeting of DFG SFB-TR-9 (Durbach, Germany, Sept. 2014), to appear in Nucl. Phys. (Proc. Suppl.)

We review the long term project of the ALPHA collaboration to compute in QCD the running coupling constant and quark masses at high energy scales in terms of low energy hadronic quantities. The adapted techniques required to numerically carry out the required multiscale non-perturbative calculation with our special emphasis on the control of systematic errors are summarized. The complete results in the two dynamical flavor approximation are reviewed and an outlook is given on the ongoing three flavor extension of the programme with improved target precision., Comment: 30 pages, 20 figures, Contribution to the Proceedings of the Final Meeting of DFG SFB-TR-9 (Durbach, Germany, Sept. 2014), to appear in Nucl. Phys. (Proc. Suppl.)

We compute the spectral density of the (Hermitean) Dirac operator in Quantum Chromodynamics with two light degenerate quarks near the origin. We use CLS/ALPHA lattices generated with two flavours of O(a)-improved Wilson fermions corresponding to pseudoscalar meson masses down to 190 MeV, and with spacings in the range 0.05-0.08 fm. Thanks to the coverage of parameter space, we can extrapolate our data to the chiral and continuum limits with confidence. The results show that the spectral density at the origin is non-zero because the low modes of the Dirac operator do condense as expected in the Banks-Casher mechanism. Within errors, the spectral density turns out to be a constant function up to eigenvalues of approximately 80 MeV. Its value agrees with the one extracted from the Gell-Mann-Oakes-Renner relation.

We present the current status of the computation of the form factor $f_+ (q^2)$ for the semi-leptonic decay $\mathrm B_\mathrm s \to \mathrm K \ell \nu$ by the ALPHA collaboration. We use gauge configurations which were generated as part of the Coordinated Lattice Simulations (CLS) effort. They have $N_\mathrm f=2$ non-perturbatively $O(a)$ improved Wilson fermions, and pion masses down to $\approx 250 \,\mathrm {MeV}$ with $m_\pi L \geq 4$. The heavy quark is treated in non-perturbative Heavy Quark Effective Theory (HQET). We discuss how to extract the form factors from the correlation functions and present first results for the form factor at $q^2 = 21.23\,\mathrm{GeV}^2$ extrapolated to the continuum. Next-to-leading order terms in HQET and the chiral extrapolation still need to be included in the analysis., Comment: 6 pages and 3 figures. Proceedings of the 8th International Workshop on the CKM Unitarity Triangle (CKM 2014), Vienna, Austria, September 8-12, 2014

We simulate a theory with two dynamical O($a$) improved Wilson quarks whose mass $M$ ranges from a factor eight up to a factor two below the charm quark mass and at three values of the lattice spacing ranging from 0.066 to 0.034 fm. This theory is a prototype to study the decoupling of heavy quarks. We measure the mass and cut-off dependence of ratios of gluonic observables defined from the Wilson flow or the static potential. The size of the 1/$M$ corrections can be determined and disentangled from the lattice artifacts. The difference with the pure gauge theory is at the percent level when two quarks with a mass of the charm quark are present., Comment: 7 pages, 6 figures, PoS(LATTICE2014)288

We present a factorisation formula for the dependence of light hadron masses and low energy hadronic scales on the mass $M$ of a heavy quark: apart from an overall factor $Q$, ratios such as $r_0(M)/r_0(0)$ are computable in perturbation theory at large $M$. The mass-independent factor $Q$ is obtained from the theory in the limit $M\to0$ and the decoupled theory with the heavy quark removed. The perturbation theory part is stable concerning different loop orders and our non-perturbative results match on quantitatively to the perturbative prediction. Upon taking ratios of different hadronic scales at the same mass, the perturbative function drops out and the ratios are given by the decoupled theory up to $M^{-2}$ corrections. Our present numerical results are obtained in a model calculation where there are no light quarks and a heavy doublet of quarks is decoupled. They are limited to masses a factor two below the charm. This is not large enough to see the $M^{-2}$ scaling predicted by the theory, but it is sufficient to verify - in the continuum limit - that the sea quark effects of quarks with masses around the charm mass are very small.