arXiv subjects

Stefan Sint

Publications and source records attributed to Stefan Sint.

At least 19 recordsLinked to original sources

arXiv2025

The strength of the interaction between quarks and gluons

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, $α_s$, between quarks and gluons. Relating $α_s$ 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 $α_s$ 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 $α_s$ 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.

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arXiv2024

Heavy Wilson Quarks and O($a$) Improvement: Nonperturbative Results for $b_{\rm g}$

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 $β$-values and the line of constant physics are motivated by the ALPHA collaboration's decoupling strategy to determine $α_s(m_Z)$. However, the improvement conditions and some insight into systematic effects may prove useful in other contexts, too.

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arXiv2024

A non-perturbative determination of $b_{\rm g}$

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, $m\,{\rm tr}(F_{μν}F_{μν})$. 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 $α_s(m_Z)$ 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 $b_{\rm g}$.

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arXiv2022

Determination of $α_s(m_Z)$ by the non-perturbative decoupling method

We present the details and first results of a new strategy for the determination of $α_s(m_Z)$. By simultaneously decoupling 3 fictitious heavy quarks we establish a relation between the $Λ$-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 $Λ$-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, $Λ^{(3)}_{\bar{\rm MS}} = 336(12)\, {\rm MeV}$, which translates to $α_s(m_Z) = 0.11823(84)$. 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.

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arXiv2021

Results for $α_s$ from the decoupling strategy

We present analysis details and new results for the strong coupling $α_s(m_Z)$, determined by the decoupling strategy. We measure a massive gradient flow (GF) coupling defined in finite volume with Schrödinger 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.

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arXiv2019

High precision renormalization of the flavour non-singlet Noether currents in lattice QCD with Wilson quarks

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 Schrödinger 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 $α_s$ 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 $β$-values of the CLS gauge configurations. In addition to the axial current renormalization, we also present precise results for the renormalized local vector current.

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arXiv2018

A non-perturbative exploration of the high energy regime in $N_\text{f}=3$ QCD

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 Schrödinger 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 $L_0 Λ^{N_{\rm f}=3}_{\overline{\rm MS}} =0.0791(21)$. 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 $L_0 Λ^{N_{\rm f}=3}_{\overline{\rm MS}}$, a reliable estimate of the truncation error requires non-perturbative data for a sufficiently large range of values of $α_s=\bar{g}^2/(4π)$. In the present work we reach this precision by studying scales that vary by a factor $2^5= 32$, reaching down to $α_s\approx 0.1$. We here provide the details of our analysis and an extended discussion.

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arXiv2016

Perturbative O($a^2$) effects in gradient flow couplings with SF and SF-open boundary conditions

The gradient flow provides a new class of renormalized observables which can be measured with high precision in lattice simulations. In principle this allows for many interesting applications to renormalization and improvement problems. In practice, however, such applications are made difficult by the rather large cutoff effects found in many gradient flow observables. At lowest order of perturbation theory we here study the leading cutoff effects in a finite volume gradient flow coupling with SF and SF-open boundary conditions. We confirm that O($a^2$) Symanzik improvement is achieved at tree-level, provided the action, observable and the flow are O($a^2$) improved. O($a^2$) effects from the time boundaries are found to be absent at this order, both with SF and SF-open boundary conditions. For the calculation we have used a convenient representation of the free gauge field propagator at finite flow times which follows from a recently proposed set-up by Lüscher and renders lattice perturbation theory more practical at finite flow time and with SF, open, SF-open or open-SF boundary conditions.

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arXiv2016

Probing QCD perturbation theory at high energies with continuum extrapolated lattice data

Precision tests of QCD perturbation theory are not readily available from experimental data. The main reasons are systematic uncertainties due to the confinement of quarks and gluons, as well as kinematical constraints which limit the accessible energy scales. We here show how continuum extrapolated lattice data may overcome such problems and provide excellent probes of renormalized perturbation theory. This work corresponds to an essential step in the ALPHA collaboration's project to determine the $Λ$-parameter in 3-flavour QCD. I explain the basic techniques used in the high energy regime, namely the use of mass-independent renormalization schemes for the QCD coupling constant in a finite Euclidean space time volume. When combined with finite size techniques this allows one to iteratively step up the energy scale by factors of 2, thereby quickly covering two orders of magnitude in scale. We may then compare perturbation theory (with $β$-functions available up to 3-loop order) to our non-perturbative data for a 1-parameter family of running couplings. We conclude that a target precision of 3 percent for the $Λ$-parameter requires non-perturbative data up to scales where $α_s\approx 0.1$, whereas the apparent precision obtained from applying perturbation theory around $α_s \approx 0.2$ can be misleading. This should be taken as a general warning to practitioners of QCD perturbation theory.

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arXiv2016

The determination of $α_s$ by the ALPHA collaboration

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 $α> 0.2$ and at the same time has the physical scales small compared to the cutoff $1/a$ in all stages of the computation. The result $Λ_\overline{MS}^{(3)}=332(14)$~MeV is translated to $α_\overline{MS}(m_Z)=0.1179(10)(2)$ by use of (high order) perturbative relations between the effective theory couplings at the charm and beauty quark "thresholds". The error of this perturbative step is discussed and estimated as $0.0002$.

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arXiv2016

Determination of the QCD $Λ$-parameter and the accuracy of perturbation theory at high energies

We discuss the determination of the strong coupling $α_\mathrm{\overline{MS}}^{}(m_\mathrm{Z})$ or equivalently the QCD $Λ$-parameter. Its determination requires the use of perturbation theory in $α_s(μ)$ in some scheme, $s$, and at some energy scale $μ$. The higher the scale $μ$ the more accurate perturbation theory becomes, owing to asymptotic freedom. As one step in our computation of the $Λ$-parameter in three-flavor QCD, we perform lattice computations in a scheme which allows us to non-perturbatively reach very high energies, corresponding to $α_s = 0.1$ and below. We find that (continuum) perturbation theory is very accurate there, yielding a three percent error in the $Λ$-parameter, while data around $α_s \approx 0.2$ 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.

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arXiv2016

The chirally rotated Schrödinger functional: theoretical expectations and perturbative tests

The chirally rotated Schrödinger functional ($χ$SF) with massless Wilson-type fermions provides an alternative lattice regularization of the Schrödinger functional (SF), with different lattice symmetries and a common continuum limit expected from universality. The explicit breaking of flavour and parity symmetries needs to be repaired by tuning the bare fermion mass and the coefficient of a dimension 3 boundary counterterm. Once this is achieved one expects the mechanism of automatic O($a$) improvement to be operational in the $χ$SF, in contrast to the standard formulation of the SF. This is expected to significantly improve the attainable precision for step-scaling functions of some composite operators. Furthermore, the $χ$SF offers new strategies to determine finite renormalization constants which are traditionally obtained from chiral Ward identities. In this paper we consider a complete set of fermion bilinear operators, define corresponding correlation functions and explain the relation to their standard SF counterparts. We discuss renormalization and O($a$) improvement and then use this set-up to formulate the theoretical expectations which follow from universality. Expanding the correlation functions to one-loop order of perturbation theory we then perform a number of non-trivial checks. In the process we obtain the action counterterm coefficients to one-loop order and reproduce some known perturbative results for renormalization constants of fermion bilinears. By confirming the theoretical expectations, this perturbative study lends further support to the soundness of the $χ$SF framework and prepares the ground for non-perturbative applications.

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arXiv2016

Slow running of the Gradient Flow coupling from 200 MeV to 4 GeV in $N_{\rm f}=3$ QCD

Using a finite volume Gradient Flow (GF) renormalization scheme with Schrödinger 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 $200\,{\rm MeV} \lesssim μ=1/L \lesssim 4\,{\rm GeV}$. 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 $μ$ dependence up to $\sim 100\,{\rm GeV}$ and can thus connect to the $Λ$-parameter.

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arXiv2015

Renormalizability of the Schrödinger Functional

Symanzik showed that quantum field theory can be formulated on a space with boundaries by including suitable surface interactions in the action to implement boundary conditions. We show that to all orders in perturbation theory all the divergences induced by these surface interactions can be absorbed by a renormalization of their coefficients.

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arXiv2015

A status update on the determination of $Λ_{\overline{\rm MS}}^{N_{\rm f}=3}$ by the ALPHA collaboration

The ALPHA collaboration aims to determine $α_s(m_Z)$ 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 $N_{\rm f} = 3$ $Λ$-parameter, $Λ_{\overline{\rm MS}}^{N_{\rm f}=3}$, in units of $L_{\rm swi}$ . Connecting $L_{\rm swi}$ and thus the $Λ$-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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arXiv2015

On O($a^2$) effects in gradient flow observables

In lattice gauge theories, the gradient flow has been used extensively both, for scale setting and for defining finite volume renormalization schemes for the gauge coupling. Unfortunately, rather large cutoff effects have been observed in some cases. We here investigate these effects to leading order in perturbation theory, considering various definitions of the lattice observable, the lattice flow equation and the Yang Mills lattice action. These considerations suggest an improved set- up for which we perform a scaling test in the pure SU(3) gauge theory, demonstrating strongly reduced cutoff effects. We then attempt to obtain a more complete understanding of the structure of O($a^2$) effects by applying Symanzik's effective theory approach to the 4+1 dimensional local field theory with flow time as the fifth dimension. From these considerations we are led to a fully O($a^2$) improved set-up the study of which is left to future work.

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arXiv2014

A dynamical study of the chirally rotated Schrödinger functional in QCD

The chirally rotated Schrödinger functional for Wilson-fermions allows for finite-volume, mass-independent renormalization schemes compatible with automatic O($a$) improvement. So far, in QCD, the set-up has only been studied in the quenched approximation. Here we present first results for $N_{\rm f} = 2$ dynamical quark-flavours for several renormalization factors of quark-bilinears. We discuss how these renormalization factors can be easily obtained from simple ratios of two-point functions, and show how automatic O($a$) improvement is at work. As a by-product of this investigation the renormalization of the non-singlet axial current, $Z_A$, is determined very precisely.

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arXiv2014

A perturbative study of the chirally rotated Schrödinger Functional in QCD

The chirally rotated Schrödinger functional ($χ$SF) renders the mechanism of automatic $O(a)$ improvement compatible with the Schrödinger functional (SF) formulation. Here we report on the determination to 1-loop order in perturbation theory of the renormalization coefficients necessary to achieve automatic $O(a)$ improvement and the boundary improvement coefficients needed to eliminate the extra boundary $O(a)$ effects present in any SF formulation. After this is done, we perform a set of tests of automatic $O(a)$ improvement and of the universality between standard and chirally rotated SF formulations.

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