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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

It has been demonstrated that distillation profiles can be employed to build optimized quarkonium interpolators for spectroscopy calculations in lattice QCD. We test their usefulness for heavy-light systems on (3+1)-flavor ensembles with mass-degenerate light and a charm quark in the sea in preparation for a future $D\bar{D}$-scattering analysis. The additional cost of light inversions naturally leads to the question if knowledge of optimal profiles can be used to avoid superfluous computations. We show such optimal profiles for different lattice sizes and pion masses and discuss general trends. Furthermore, we discuss the handling of momenta in this framework.

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

We use the method of optimal distillation profiles to compute the low-lying charmonium spectrum in an $N_f = 3+1$ ensemble at the $SU(3)$ light flavor symmetric point ($m_{\pi} \approx 420$ MeV), physical charm quark mass and lattice spacing $a\approx 0.0429$ fm. The spectrum and mass splittings display good agreement with their values in nature and the statistical errors are comparable, if not smaller, than those of state-of-the-art lattice calculations. We also present first results on the mixing of charmonium with glueballs and light hadrons obtained in a similar $N_f = 3+1$ ensemble but at larger pion mass., Comment: 10 pages, 7 figures. Proceedings of the 40th International Symposium on Lattice Field Theory (Lattice 2023), Fermilab, 2023. Added a reference, corrected a typo and modified legend of figure 5

We determine the scales $r_0$, $r_1$, the ratio $r_0/r_1$ for $N_f$ = 2 + 1 flavor QCD ensembles generated by CLS. These scales are determined from an improved definition of the static force, which we measure using Wilson loops and furthermore use to study the shape of the potential. Our analysis involves various continuum and chiral extrapolations of data that covers pion masses between 134 MeV and 420 MeV and five lattice spacings down to 0.038 fm., Comment: 9 pages, 13 figures

Optimized meson operators in the distillation framework are used to study the charmonium spectrum in two ensembles with two heavy dynamical quarks at half the physical charm quark mass but different lattice spacings. The use of optimal meson distillation profiles is shown to increase the overlap with the ground state significantly, as well as grant access to excited states, for multiple quantum numbers including hybrid states with very little additional cost. These same operators are also employed for the calculation of meson-glueball mixing., Comment: 10 pages, 6 figures, Proceedings of the 39th International Symposium on Lattice Field Theory, LATTICE2022, August, 8-13, 2022, in Bonn, Germany

We investigate static quark anti-quark operators based on trial states formed from eigenvectors of the covariant three-dimensional lattice Laplace operator. We test the method by computing the static quark-anti-quark potential and comparing results to standard Wilson loop measurements. The new method is efficient not only for on-axis, but also for many off-axis quark-anti-quark separations when a fine spatial resolution is required. We further improve the ground-state overlap by using multiple eigenvector pairs, weighted with Gaussian profile functions of the eigenvalues, providing a variational basis. The method presented here can be applied to potential functions for all possible excitations of a gluonic string with fixed ends, hybrid or tetra-quark potentials, as well as static-light systems and allows visualization of the spatial distribution of the Laplace trial states.

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

Smearing the bare quantum fields in lattice calculations before applying composite hadron creation operators has a long record of substantially improving overlaps onto low-lying energy eigenstates. A technique called distillation which defines smearing for quark fields as a low-rank linear projection operator into a small vector space of smooth gauge-covariant fields has proven to be both effective and versatile in hadron spectroscopy calculations albeit with significant computational cost . In this paper, more general operators in this space of smooth fields are introduced and optimised, which enhances the performance of the method when tested on systems of heavy quark-anti-quark pairs close to the charmonium energy scale., Comment: 19 pages, 19 figures, 6 tables

Compared to standard tomographic reconstruction, iterative approaches offer the possibility to account for extraneous experimental influences, which allows for a suppression of related artifacts. However, the inclusion of corresponding parameters in the iterative forward model typically leads to longer computation times. Here, we demonstrate experimentally for phase sensitive X-ray imaging based on the edge illumination principle that inadequately sampled illumination curves result in ring artifacts in tomographic reconstructions. We take advantage of appropriately sampled illumination curves instead, which enables us to eliminate the corresponding parameter from the forward model and substantially increase computational speed. In addition, we demonstrate a 30\% improvement in spatial resolution of the iterative approach compared with the standard non-iterative single shot approach. Further, we report on several significant improvements in our numerical implementation of the iterative approach, which we make available online with this publication. Finally, we show that the combination of both experimental and algorithmic advancement lead to a total speed increase by one order of magnitude and an improved contrast to noise ratio in the reconstructions.

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 study the charmonium spectrum on an ensemble with two heavy dynamical quarks with a mass at half the physical charm quark mass. Operators for different quantum numbers are used in the framework of distillation with different smearing profiles to increase the overlap with ground and excited states. The use of exact distillation, large statistics and the absence of light quarks gives robust results for the charmonium spectrum. We also present preliminary results for the glueball spectrum in this theory., Comment: 10 pages, 8 figures, Proceedings of the 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We estimate the effects on the decay constants of charmonium and on heavy meson masses due to the charm quark in the sea. Our goal is to understand whether for these quantities $N_f=2+1$ lattice QCD simulations provide results that can be compared with experiments or whether $N_f=2+1+1$ QCD including the charm quark in the sea needs to be simulated. We consider two theories, $N_f=0$ QCD and QCD with $N_f=2$ charm quarks in the sea. The charm sea effects (due to two charm quarks) are estimated comparing the results obtained in these two theories, after matching them and taking the continuum limit. The absence of light quarks allows us to simulate the $N_f=2$ theory at lattice spacings down to $0.023$ fm that are crucial for reliable continuum extrapolations. We find that sea charm quark effects are below $1\%$ for the decay constants of charmonium. Our results show that decoupling of charm works well up to energies of about $500$ MeV. We also compute the derivatives of the decay constants and meson masses with respect to the charm mass. For these quantities we again do not see a significant dynamical charm quark effect, albeit with a lower precision. For mesons made of a charm quark and a heavy antiquark, whose mass is twice that of the charm quark, sea effects are only about $0.1\%$ in the ratio of vector to pseudoscalar masses., Comment: Matches version accepted for publication in Eur.Phys.J.C. 30 pages, 7 figures

We present the scale setting for a new set of gauge configurations generated with $N_f=3+1$ Wilson quarks with a non-perturbatively determined clover coefficient in a massive O($a$) improvement scheme. The three light quarks are degenerate, with the sum of their masses being equal to its value in nature and the charm quark has its physical mass. We use open boundary conditions in time direction to avoid the problem of topological freezing at small lattice spacings and twisted-mass reweighting for improved stability of the simulations. The decoupling of charm at low energy allows us to set the scale by measuring the value of the low-energy quantity $t_0^\star/a^2$, which is the flow scale $t_0$ at our mass point, and comparing it to an $N_f=2+1$ result in physical units. We present the details of the algorithmic setup and tuning procedure and give the bare parameters of ensembles with two lattice spacings a=0.054 fm and a=0.043 fm. We discuss finite volume effects and lattice artifacts and present physical results for the charmonium spectrum. In particular the hyperfine splitting between the $\eta_c$ and $J/\psi$ mesons agrees very well with its physical value., Comment: 25 pages, 9 figures. Version accepted for publication in the European Physical Journal C

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 present first results of the scale setting for QCD with Nf=3+1 dynamical quarks on the lattice. We use a recently proposed massive renormalization scheme with a non-perturbatively determined clover coefficient. To relate the bare coupling of the simulations to a lattice spacing in fm, we measure the flow scale t0 in lattice units at a mass point with degenerate light quark masses and a physical charm quark mass, where its value in physical units is known. We discuss the setup, tuning procedure, simulation parameters and measurement results for ensembles with three different volumes and present a charmonium spectrum., Comment: 7 pages, 4 figures, v2: version submitted to the journal, Fig. 3 changed

The properties of charmonium states are or will be intensively studied by the B-factories Belle II and BESIII, the LHCb and PANDA experiments and at a future Super-c-tau Factory. Precise lattice calculations provide valuable input and several results have been obtained by simulating up, down and strange quarks in the sea. We investigate the impact of a charm quark in the sea on the charmonium spectrum, the renormalization group invariant charm-quark mass and the scalar charm-quark content of charmonium. The latter is obtained by the direct computation of the mass-derivatives of the charmonium masses. We do this investigation in a model, QCD with two degenerate charm quarks. The absence of light quarks allows us to reach very small lattice spacings down to 0.023 fm. By comparing to pure gauge theory we find that charm quarks in the sea affect the hyperfine splitting at a level below 2%. The most significant effects are 5% in the quark mass and 3% in the value of the charm quark content of the eta_c meson. Given that we simulate two charm quarks these estimates are upper bounds for the contribution of a single charm quark. We show that lattice spacings <0.06 fm are needed for safe continuum extrapolations of the charmonium spectrum with O(a) improved Wilson quarks. A useful relation for the projection to the desired parity of operators in two-point functions computed with twisted mass fermions is proven., Comment: 25 pages, 8 figures

We investigate the influence of dynamical charm quarks on observables that depend explicitly on the charm quark fields, like the pseudo-scalar and vector masses $m_{\eta_c}$ and $m_{J/\psi}$, the hyperfine splitting $(m_{J/\psi}-m_{\eta_c})/m_{\eta_c}$, the charm quark mass and the meson decay constants. For this purpose, instead of working in full QCD we study a simplified setup. We simulate two theories: $N_f= 0$ QCD and QCD with $N_f= 2$ dynamical quarks at the charm mass. The absence of light quarks allows us to reach extremely fine lattice spacings (0.02 fm < a < 0.05 fm) which are crucial for reliable continuum extrapolations. Our main result is a comparison of various quantities in the continuum limit. For the hyperfine splitting we find that the effects of a dynamical charm quark are below our statistical precision of $2\%$., Comment: 7 pages, 3 figures, The 36th International Symposium on Lattice Field Theory (Lattice 2018)

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

We determine the non-perturbatively renormalized axial current for O($a$) improved lattice QCD with Wilson quarks. Our strategy is based on the chirally rotated Schr\"odinger functional and can be generalized to other finite (ratios of) renormalization constants which are traditionally obtained by imposing continuum chiral Ward identities as normalization conditions. Compared to the latter we achieve an error reduction up to one order of magnitude. Our results have already enabled the setting of the scale for the $N_{\rm f}=2+1$ CLS ensembles [1] and are thus an essential ingredient for the recent $\alpha_s$ determination by the ALPHA collaboration [2]. In this paper we shortly review the strategy and present our results for both $N_{\rm f}=2$ and $N_{\rm f}=3$ lattice QCD, where we match the $\beta$-values of the CLS gauge configurations. In addition to the axial current renormalization, we also present precise results for the renormalized local vector current., Comment: 43 pages, 11 figures. Added Appendix C containing fits to the final results for Z_{A,V}. Matches published version

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

A high precision determination of the strong coupling constant in the MS-bar scheme at the Z-mass scale, using low energy quantities, namely pion/kaon decay constants and masses, as experimental input is presented. The computation employs two different massless finite volume renormalization schemes to non-perturbatively trace the scale dependence of the respective running couplings from a scale of about 200 MeV to 100 GeV. At the largest energies perturbation theory is reliable. At high energies the Schroedinger-Functional scheme is used, while the running at low and intermediate energies is computed in a novel renormalization scheme based on an improved gradient flow. Large volume Nf=2+1 QCD simulations by CLS are used to set the overall scale. The result is compared to world averages by FLAG and the PDG., Comment: 18 pages, 8 figures. Talk presented at the 35th International Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spain

We study the effective theory of decoupling of a charm quark at low energies. We do this by simulating a model, QCD with two mass-degenerate charm quarks. At leading order the effective theory is a pure gauge theory. By computing ratios of hadronic scales we have direct access to the power corrections in the effective theory. We show that these corrections follow the expected leading behavior, which is quadratic in the inverse charm quark mass., Comment: 8 pages, 4 figures. Proceedings from the 35th International Symposium on Lattice Field Theory (Lattice2017), Granada, Spain

We compute the fermionic contribution to the strong coupling $\alpha_{qq}$ extracted from the static force in Lattice QCD up to order $g^4$ in perturbation theory. This allows us to subtract the leading fermionic lattice artifacts from recent determinations of $\alpha_{qq}$ produced in simulations of two dynamical charm quarks. Moreover, by using a suitable parametrization of the $\beta_{qq}$-function, we can evaluate the charm loop effects on $\alpha_{qq}$ in the continuum limit., Comment: 8 pages, 2 figures; Proceedings of the 35th International Symposium on Lattice Field Theory, Granada, Spain

Decoupling of heavy quarks at low energies can be described by means of an effective theory as shown by S. Weinberg in Ref. [1]. We study the decoupling of the charm quark by lattice simulations. We simulate a model, QCD with two degenerate charm quarks. In this case the leading order term in the effective theory is a pure gauge theory. The higher order terms are proportional to inverse powers of the charm quark mass $M$ starting at $M^{-2}$. Ratios of hadronic scales are equal to their value in the pure gauge theory up to power corrections. We show, by precise measurements of ratios of scales defined from the Wilson flow, that these corrections are very small and that they can be described by a term proportional to $M^{-2}$ down to masses in the region of the charm quark mass., Comment: 20 pages, 3 figures, 3 tables, v2: Introduction and Conclusions expanded, new fits added to Figure 2 and 3, References updated

We compute and compare the continuum limits of several quantities in QCD with and without a dynamical charm quark. We consider both low energy quantities, like the hadronic scales $r_0$ and $t_0$, and high energy quantities, like the charmonium masses., Comment: 7 pages, 7 figures, 34th International Symposium on Lattice Field Theory (lattice 2016), 24-30 July 2016, Southampton, UK

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

We present measurements of a combination of the decay constants of the light pseudoscalar mesons and the gradient flow scale $t_0$, which allow to set the scale of the lattices generated by CLS with $2 + 1$ flavors of non-perturbatively improved Wilson fermions. Mistunings of the quark masses are corrected for by measuring the derivatives of observables with respect to the bare quark masses., Comment: 21 pages, 4 figures, Updated to match the published version

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

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 define a finite size renormalization scheme for $\phi^4$ theory which in the thermodynamic limit reduces to the standard scheme used in the broken phase. We use it to re-investigate the question of triviality for the four dimensional infinite bare coupling (Ising) limit. The relevant observables all rely on two-point functions and are very suitable for a precise estimation with the worm algorithm. This contribution updates an earlier publication by analysing a much larger dataset., Comment: 6 pages, 2 figures, 1 table, presented at the 32nd International Symposium on Lattice Field Theory (Lattice 2014), June 23-28 2014, New York, USA

We discuss methods to extract decay constants, meson masses and gluonic observables in the presence of open boundary conditions. The ensembles have been generated by the CLS effort and have 2+1 flavors of O(a)-improved Wilson fermions with a small twisted-mass term as proposed by L\"uscher and Palombi. We analyse the effect of the associated reweighting factors on the computation of different observables., Comment: 7 pages, talk presented at the 32nd International Symposium on Lattice Field Theory - Lattice 2014, Columbia University New York

We describe a new set of gauge configurations generated within the CLS effort. These ensembles have N_f=2+1 flavors of non-perturbatively improved Wilson fermions in the sea with the Luescher-Weisz action used for the gluons. Open boundary conditions in time are used to address the problem of topological freezing at small lattice spacings and twisted-mass reweighting for improved stability of the simulations. We give the bare parameters at which the ensembles have been generated and how these parameters have been chosen. Details of the algorithmic setup and its performance are presented as well as measurements of the pion and kaon masses alongside the scale parameter t_0., Comment: 29 pages, 7 figures, version accepted for publication by JHEP

We present a Monte-Carlo algorithm for the simulation of the all-order strong coupling expansion of the Z2 gauge theory. This random surface ensemble is equivalent to the standard formulation, but allows to measure some quantities, like Polyakov loop correlators or excess free energies, with an accuracy that could not have been easily achieved with traditional simulation methods. One interesting application of the algorithm is the investigation of effective string theories., Comment: presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany

All-order strong coupling simulations have been used to derive precise energies of string states in the confined phase of three dimensional Z(2) lattice gauge theory. The behavior of the ground state energy is here compared with predictions of effective string theory. Our new data reported here are consistent with known universal terms of the long string length ($L_0$) expansion known from effective string models in the continuum limit. Our precision is however still not sufficient to disentangle non-univeral terms of order $L_0^{-7}, Comment: 7 pages, 1 figure, 2 tables. PoS (Lattice 2013) 2013

We exactly rewrite the Z(2) lattice gauge theory with standard plaquette action as a random surface model equivalent to the untruncated set of its strong coupling graphs. By extending the worm approach applied to spin models we simulate such surfaces including Polyakov line defects that randomly walk over the lattice. Our Monte Carlo algorithms for the graph ensemble are reasonably efficient but not free of critical slowing down. Polyakov line correlators can be measured in this approach with small relative errors that are independent of the separation. As a first application our results are confronted with effective string theory predictions. In addition, the excess free energy due to twisted boundary conditions becomes an easily accessible observable. Our numerical experiments are in three dimensions, but the method is expected to work in any dimension., Comment: 25 pages, 7 figures, V2: small changes in the text, refs. added, new footnote p.18, to appear in Nucl. Phys. B [FS]

Worm algorithms have been very successful with the simulation of sigma models with fixed length spins which result from scalar field theories in the limit of infinite quartic coupling lambda. Here we investigate closer their algorithmic efficiency at finite and even vanishing lambda for the one component model in dimensions D = 2, 3, 4., Comment: 10 pages, 2 Figs

We present an algorithm in which the all-order strong coupling expansion of the Abelian U(1) gauge theory with Wilson plaquette action is sampled. In addition to the vacuum closed surface graphs of the partition function we propose to also allow for a class of defects (boundaries) related to Wilson loops in the ensemble. The efficiency of our scheme in estimating various observables is compared to a standard Metropolis algorithm., Comment: 7 pages, 2 figures. Presented at the XXVIII International Symposium on Lattice Field Theory (Lattice 2010), June 14-19 2010, Villasimius, Italy

We present first results on the axial and pseudoscalar $\Delta$ form factors. The analysis is carried out in the quenched approximation where statistical errors are small and the lattice set-up can be investigated relatively quickly. We also present an analysis with a hybrid action using staggered sea quarks and domain-wall valence fermions., Comment: 7 pages, 10 figures, presented at the XXVIII International Symposium on Lattice Field Theory (Lattice 2010), Villasimius, Sardinia, Italy, June 14-19, 2010

We extend the formalism relating electromagnetic form factors to transverse quark charge densities in the light-front frame to the case of a spin-3/2 baryon and calculate these transverse densities for the $\Delta(1232)$ isobar using lattice QCD. The transverse charge densities for a transversely polarized spin-3/2 particle are characterized by monopole, dipole, quadrupole, and octupole patterns representing the structure beyond that of a pure point-like spin-3/2 particle. We present lattice QCD results for the $\Delta$-isobar electromagnetic form factors for pion masses down to approximatively 350 MeV for three cases: quenched QCD, two-degenerate flavors of dynamical Wilson quarks, and three flavors of quarks using a mixed action that combines domain wall valence quarks and dynamical staggered sea quarks. We extract transverse quark charge densities from these lattice results and find that the $\Delta$ is prolately deformed, as indicated by the fact that the quadrupole moment $G_{E2}(0$) is larger than the value -3 characterizing a point particle and the fact that the transverse charge density in a $\Delta^+$ of maximal transverse spin projection is elongated along the axis of the spin., Comment: 35 pages, 10 figures

The electromagnetic and axial form factors of the nucleon are evaluated in twisted mass QCD with two degenerate flavors of light, dynamical quarks. The axial charge g_A, magnetic moment and the Dirac and Pauli radii are determined for pion masses in the range 300 MeV to 500 MeV.

We develop a methodology that enables us to extract accurately the electromagnetic Delta form factors and their momentum dependence. We test our approach in the quenched approximation as a preparation for a study using dynamical fermions. Our calculation of the four form factors covers pion masses between about 410 MeV and 560 MeV on lattices with a size of 2.9 fm and a lattice spacing 0.09 fm. From the form factors we are able to obtain estimates of the magnetic moment and the charge radius of the Delta, which we compare to existing experimental and theoretical results.Our non-zero result for the electric quadrupole form factor signals a deformation of the Delta, pointing to an oblate charge distribution.