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 rati
o $Lambda/mu_{rm dec}$ 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.
We present a new theoretical and practical strategy to renormalize non-perturbatively the energy-momentum tensor in lattice QCD based on the framework of shifted boundary conditions. As a preparatory step for the fully non-perturbative calculation, w
e apply the strategy at 1-loop order in perturbation theory determining the renormalization constants of both the gluonic and the fermionic components of the energy-momentum tensor. Using shifted boundary conditions, the entropy density of QCD is directly related to the expectation value of the space-time components of the renormalized energy-momentum tensor. We then discuss its practical implementation by numerical simulations of QCD with 3 flavours of Wilson quarks for temperatures between 2.5 GeV and 80 GeV.
We employ the chirally rotated Schrodinger functional ($chi$SF) to study two-point fermion bilinear correlation functions used in the determination of $Z_{A,V,S,P,T}$ on a series of well-tuned ensembles. The gauge configurations, which span renormali
sation scales from 4 to 70~GeV, are generated with $N_{rm f}=3$ massless flavors and Schrodinger Functional (SF) boundary conditions. Valence quarks are computed with $chi$SF boundary conditions. We show preliminary results on the tuning of the $chi$SF Symanzik coefficient $z_f$ and the scaling of the axial current normalization $Z_{rm A}$. Moreover we carry out a detailed comparison with the expectations from one-loop perturbation theory. Finally we outline how automatically $mathrm{O}(a)$-improved $B_{rm K}$ matrix elements, including BSM contributions, can be computed in a $chi$SF renormalization scheme.
Using finite size scaling techniques and a renormalization scheme based on the Gradient Flow, we determine non-perturbatively the $beta$-function of the $SU(3)$ Yang-Mills theory for a range of renormalized couplings $bar g^2sim 1-12$. We perform a d
etailed study of the matching with the asymptotic NNLO perturbative behavior at high-energy, with our non-perturbative data showing a significant deviation from the perturbative prediction down to $bar{g}^2sim1$. We conclude that schemes based on the Gradient Flow are not competitive to match with the asymptotic perturbative behavior, even when the NNLO expansion of the $beta$-function is known. On the other hand, we show that matching non-perturbatively the Gradient Flow to the Schrodinger Functional scheme allows us to make safe contact with perturbation theory with full control on truncation errors. This strategy allows us to obtain a precise determination of the $Lambda$-parameter of the $SU(3)$ Yang-Mills theory in units of a reference hadronic scale ($sqrt{8t_0},Lambda_{overline{rm MS}} = 0.6227(98)$), showing that a precision on the QCD coupling below 0.5% per-cent can be achieved using these techniques.
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 Schrodinger 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.
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 Schrodinger 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 $1/L_0approx 4$ 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 $alpha_s=bar{g}^2/(4pi)$. In the present work we reach this precision by studying scales that vary by a factor $2^5= 32$, reaching down to $alpha_sapprox 0.1$. We here provide the details of our analysis and an extended discussion.
We investigate the quark content of the scalar meson $a_0(980)$ using lattice QCD. To this end we consider correlation functions of six different two- and four-quark interpolating fields. We evaluate all diagrams, including diagrams, where quarks pro
pagate within a timeslice, e.g. with closed quark loops. We demonstrate that diagrams containing such closed quark loops have a drastic effect on the final results and, thus, may not be neglected. Our analysis shows that in addition to the expected spectrum of two-meson scattering states there is an additional energy level around the two-particle thresholds of $K + bar{K}$ and $eta + pi$. This additional state, which is a candidate for the $a_0(980)$ meson, couples to a quark-antiquark as well as to a diquark-antidiquark interpolating field, indicating that it is a superposition of an ordinary $bar{q} q$ and a tetraquark structure. The analysis is performed using AMIAS, a novel statistical method based on the sampling of all possible spectral decompositions of the considered correlation functions, as well as solving standard generalized eigenvalue problems.
Numerical stochastic perturbation theory is a powerful tool for estimating high-order perturbative expansions in lattice field theory. The standard algorithms based on the Langevin equation, however, suffer from several limitations which in practice
restrict the potential of this technique. In this work we investigate some alternative methods which could in principle improve on the standard approach. In particular, we present a study of the recently proposed Instantaneous Stochastic Perturbation Theory, as well as a formulation of numerical stochastic perturbation theory based on Generalized Hybrid Molecular Dynamics algorithms. The viability of these methods is investigated in $varphi^4$ theory.
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 $0.0002$.
The chirally rotated Schrodinger functional ($chi$SF) renders the mechanism of automatic $O(a)$ improvement compatible with Schrodinger functional (SF) renormalization schemes. Here we define a family of renormalization schemes based on the $chi$SF f
or a complete basis of $Delta F = 2$ parity-odd four-fermion operators. We compute the corresponding scale-dependent renormalization constants to one-loop order in perturbation theory and obtain their NLO anomalous dimensions by matching to the $overline{textrm{MS}}$ scheme. Due to automatic $O(a)$ improvement, once the $chi$SF is renormalized and improved at the boundaries, the step scaling functions (SSF) of these operators approach their continuum limit with $O(a^{2})$ corrections without the need of operator improvement.
