We present a lattice determination of the Λ parameter in three-flavor QCD and the strong coupling at the Z pole mass. Computing the nonperturbative running of the coupling in the range from 0.2 to 70 GeV, and using experimental input values for the masses and decay constants of the pion and the kaon, we obtain Λ

Using finite size scaling techniques and a renormalization scheme based on the Gradient Flow, we determine non-perturbatively the β-function of the SU (3) Yang-Mills theory for a range of renormalized couplingsḡ 2 ∼ 1 − 12. We perform a detailed study of the matching with the asymptotic NNLO perturbative behavior at high-energy, with our non-perturbative data showing a significant deviation from the perturbative prediction down toḡ 2 ∼ 1. We conclude that schemes based on the Gradient Flow are not competitive to match with the asymptotic perturbative behavior, even when the NNLO expansion of the β-function is known. On the other hand, we show that matching non-perturbatively the Gradient Flow to the Schrödinger Functional scheme allows us to make safe contact with perturbation theory with full control on truncation errors. This strategy allows us to obtain a precise determination of the Λ-parameter of the SU (3) Yang-Mills theory in units of a reference hadronic scale ( √ 8t 0 Λ MS = 0.6227 (98)), showing that a precision on the QCD coupling below 0.5% per-cent can be achieved using these techniques. References 512 / 55 1 We note while passing that, at present, the perturbative β-function is most accurately known in the MS scheme of dimensional regularization, where the b k -coefficients have been computed up to k = 4 [23,24,25,26,27]. Other specific cases will be presented in detail below.2 In the following we shall loosely refer to (RG invariant) low-energy scales of the SU (3) Yang-Mills theory as "hadronic" scales/quantities, although strictly speaking there are no hadrons in this theory. Popular examples of lowenergy scales are, for instance, the energies composing the spectrum of the theory, the distance r0 obtained from the potential between two static quarks [28], and the gradient flow time t0 [11]. We will come back to some of these in later sections.

The chirally rotated Schrödinger functional (χSF) 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 χSF, 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 χSF 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 χSF framework and prepares the ground for non-perturbative applications.

We discuss the determination of the strong coupling α MS (m Z ) or equivalently the QCD Λparameter. Its determination requires the use of perturbation theory in αs(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 αs = 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 αs ≈ 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.

Using a finite volume gradient flow renormalization scheme with Schrödinger Functional boundary conditions, we compute the nonperturbative running coupling in the range 2.2 ≲ḡ 2 GF ðLÞ ≲ 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 MeV ≲ μ ¼ 1=L ≲ 4 GeV. While there is no convincing contact to two-loop running, we match nonperturbatively to the Schrödinger functional coupling with background field. In this case, we know the μ-dependence up to ∼100 GeV and can thus connect to the Λ-parameter.

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 (3) MS = 332(14) MeV is translated to α 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.

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 by up to one order of magnitude. Our results have already enabled the setting of the scale for the N 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 f = 2 and N 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.

We present a strategy to define non-perturbatively the energy-momentum tensor in Quantum Chromodynamics (QCD) which satisfies the appropriate Ward identities and has the right trace anomaly. The tensor is defined by regularizing the theory on a lattice, and by fixing its renormalization constants non-perturbatively by suitable Ward identities associated to the Poincaré invariance of the continuum theory. The latter are derived in thermal QCD with a non-zero imaginary chemical potential formulated in a moving reference frame. A renormalization group analysis leads to simple renormalizationgroup-invariant definitions of the gluonic and fermionic contributions to either the singlet or the non-singlet components of the tensor, and therefore of their form factors among physical states. The lattice discussion focuses on the Wilson discretization of quark fields but the strategy is general. Specific to that case, we also carry out the analysis for the on-shell O(a)-improvement of the energy-momentum tensor. The renormalization and improvement programs profit from the fact that, as shown here, the thermal theory enjoys de-facto automatic O(a)-improvement at finite temperature. The validity of the proposal is scrutinized analytically by a study to 1-loop order in lattice perturbation theory with shifted and twisted (for quarks only) boundary conditions. The latter provides also additional useful insight for a precise non-perturbative calculation of the renormalization constants. The strategy proposed here is accessible to Monte Carlo computations, and in this sense it provides a practical way to define non-perturbatively the energy-momentum tensor in QCD.