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The recent determination of αs(mZ) =0.11852(84)by the ALPHA collaboration [1] distinguishes itself by the very good control of perturbative truncation and other systematic errors. A variety of tools and methods had to be deployed to enable this result. In this contribution I will give a short account of the step-scaling method and its application to QCD couplings in finite volume renormalization schemes. Tracing the running couplings non-perturbatively between scales μ0 and 32μ0(corresponding roughly to the range 4–128 GeV) leads to the intermediate result Λ(3)MS/μ0=0.0791(19)in 3-flavour QCD. By computing this ratio in variety of ways, using perturbation theory in different schemes and at different energy scales at intermediate stages, gives us confidence in the error estimate and also enables a number of useful tests of perturbation theory. The remaining steps required for αs(mZ)will be discussed by Mattia Dalla Brida in these proceedings [2].

International audience; A summary of the main points of discussion raised during the talks and their follow-up questions,as well as in the round table of the last day of the $α_s$ (2019) workshop, is presented. The discussions not only focused on particular issues affecting each one of the individual $α_s$ extractions, but also on the current PDG categorization of $α_s$ measurements and on the methods used to average them into a single $α_s$ ($m_Z$) value. Most of the listed points are open and sources of potential controversies, which we highlight here as one might expect that ongoing progress in the field will lead to their clarification and resolution.

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

We present a lattice determination of the $\Lambda$ 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 $0.2\,$GeV to $70\,$GeV, and using experimental input values for the masses and decay constants of the pion and the kaon, we obtain $\Lambda_{\overline{\rm MS}}^{(3)}=341(12)\,$MeV. The nonperturbative running up to very high energies guarantees that systematic effects associated with perturbation theory are well under control. Using the four-loop prediction for $\Lambda_{\overline{\rm MS}}^{(5)}/\Lambda_{\overline{\rm MS}}^{(3)}$ yields $\alpha^{(5)}_{\overline{\rm MS}}(m_{\rm Z}) = 0.11852(84)$.

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 L\"uscher and renders lattice perturbation theory more practical at finite flow time and with SF, open, SF-open or open-SF boundary conditions., Comment: 7 pages, 2 figures

Using a finite volume gradient flow renormalization scheme with Schr\"odinger Functional boundary conditions, we compute the nonperturbative running coupling in the range $2.2\ensuremath{\lesssim}{\overline{g}}_{\mathrm{GF}}^{2}(L)\ensuremath{\lesssim}13$. 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 $200\text{ }\text{ }\mathrm{MeV}\ensuremath{\lesssim}\ensuremath{\mu}=\phantom{\rule{0ex}{0ex}}1/L\ensuremath{\lesssim}4\text{ }\text{ }\mathrm{GeV}$. While there is no convincing contact to two-loop running, we match nonperturbatively to the Schr\"odinger functional coupling with background field. In this case, we know the $\ensuremath{\mu}$-dependence up to $\ensuremath{\sim}100\text{ }\text{ }\mathrm{GeV}$ and can thus connect to the $\mathrm{\ensuremath{\Lambda}}$-parameter.

34th annual International Symposium on Lattice Field Theory, Southampton, UK, 24 Jul 2016 - 30 Jul 2016 ; Proceedings of Science (2016)., We present results by the ALPHA collaboration for the $\Lambda$-parameter in 3-flavour QCD and the strong coupling constant at the electroweak scale, $\alpha_s(m_Z)$, in terms of hadronic quantities computed on the CLS gauge configurations. The first part of this proceedings contribution contains a review of published material \cite{Brida:2016flw,DallaBrida:2016kgh} and yields the $\Lambda$-parameter in units of a low energy scale, $1/L_{\rm had}$. We then discuss how to determine this scale in physical units from experimental data for the pion and kaon decay constants. We obtain $\Lambda_{\overline{\rm MS}}^{(3)} = 332(14)$ MeV which translates to $\alpha_s(M_Z)=0.1179(10)(2)$ using perturbation theory to match between 3-, 4- and 5-flavour QCD., Published by SISSA, Trieste

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 chirally rotated Schr\"odinger functional ($\chi$SF) with massless Wilson-type fermions provides an alternative lattice regularization of the Schr\"odinger 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 $\chi$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 $\chi$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 $\chi$SF framework and prepares the ground for non-perturbative applications., Comment: 51 pages, 16 figures; improved presentation and added references; results unchanged; matches published version

Sixth Workshop on Theory, Phenomenology and Experiments in Flavour Physics Interplay of Flavour Physics with Electroweak symmetry breaking, Anacapri, Italy, 11 Jun 2016 - 13 Jun 2016; Nuclear and particle physics proceedings 285-286, 132 - 138 (2017). doi:10.1016/j.nuclphysbps.2017.03.024, 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 $\alpha \gtrsim 0.2$ and at the same time has the physical scales small compared to the cutoff 1/a in all stages of the computation. The result $\Lambda \frac{(3)}{MS} 0 332(14)MeV$ is translated to $\alpha_\overline{MS}(m_{Z}) 0 0.1179(10)(2)$ 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., Published by Elsevier, Amsterdam [u.a.]

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 $\Lambda$-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 $\beta$-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 $\Lambda$-parameter requires non-perturbative data up to scales where $\alpha_s\approx 0.1$, whereas the apparent precision obtained from applying perturbation theory around $\alpha_s \approx 0.2$ can be misleading. This should be taken as a general warning to practitioners of QCD perturbation theory., Comment: 10 pages, 4 figures; contributed to XIIth Confinement and the Hadron Spectrum, Thessaloniki, 28 August - 4 September 2016, to appear in the proceedings

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 $L_{\rm max} F_K$.

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 setup 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.

The chirally rotated Schr\"odinger 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., Comment: Talk given at the 32nd International Symposium on Lattice Field Theory, 23-28 June, 2014, New York, US; LaTeX source, 7 pages, 6 figures

We review our new strategy and current status towards a high precision computation of the Lambda parameter from three-flavour simulations in QCD. To reach this goal we combine specific advantages of the Schr\"odinger functional and gradient flow couplings., Comment: 7 pages, 3 figures; Proceedings of the 32nd International Symposium on Lattice Field Theory; 23-28 June, 2014, Columbia University, New York

The Schrodinger functional (SF) is a powerful and widely used tool for the treatment of a variety of problems in renormalization and related areas. Albeit offering many conceptual advantages, one major downside of the SF scheme is the fact that perturbative calculations quickly become cumbersome with the inclusion of higher orders in the gauge coupling and hence the use of an automated perturbation theory framework is desirable. We present the implementation of the SF in numerical stochastic perturbation theory (NSPT) and compare first results for the running coupling at two loops in pure SU(3) Yang-Mills theory with the literature.

We determine the improvement coefficients b_m and b_a-bp in quenched lattice QCD for a range of beta-values, which is relevant for current large scale simulations. At fixed beta, the results are rather sensitive to the precise choices of parameters. We therefore impose improvement conditions at constant renormalized parameters, and the coefficients are then obtained as smooth functions of g_0^2. Other improvement conditions yield a different functional dependence, but the difference between the coefficients vanishes with a rate proportional to the lattice spacing. We verify this theoretical expectation in a few examples and are therefore confident that O(a) improvement is achieved for physical quantities. As a byproduct of our analysis we also obtain the finite renormalization constant which relates the subtracted bare quark mass to the bare PCAC mass., Comment: 25 pages, 8 figures, minor change at figures

We present results at one-loop order of perturbation theory for various improvement coefficients in on-shell O($a$) improved lattice QCD. In particular we determine the additive counterterm required for on-shell improvement of the isovector vector current. Employing a general mass-independent renormalization scheme we also obtain the coefficients of the O($a$) counterterms which are proportional to the quark mass in the improved isovector pseudo-scalar, axial vector and vector operators. In the latter case a comparison with a recent non-perturbative study is made., Comment: 24 pages plain TeX, 2 postscript figures, macros and format files included

A finite-size technique is employed to compute the normalization constant $Z_A$ of the isovector axial current in lattice QCD. The calculation is carried out in the quenched approximation for values of the bare gauge coupling $g_0$ ranging from 0 to 1. In the lattice action and the lattice expression for the axial current we include the counterterms required for O(a) improvement, with non-perturbatively determined coefficients. With little additional work the normalization constant $Z_V$ of the improved isospin current is also obtained.

We present the results of a precision computation of BK with Wilson fermions. Simulations are performed at different lattice spacings, enabling continuum limit extrapolations. Two different twisted mass QCD (tmQCD) regularisations are considered for the computation of bare matrix elements. In both cases the relevant four-fermion operator renormalises multiplicatively. In one regularisation it is possible to perform the computation directly at the physical kaon mass value, thus avoiding extrapolations in the mass. Nonperturbative renormalisation is carried out using available Schrodinger Functional results.

Renormalization constants for multiplicatively renormalizable parity-odd four–fermion operators are computed in various different Schrodinger Functional (SF) schemes and lattice regularizations with Wilson quarks at one–loop order in perturbation theory. Our results are used in the calculation of their NLO anomalous dimensions, through matching to continuum schemes. They also enable a comparison of the two–loop perturbative RG running to the previously obtained nonperturbative one in the region of small renormalized coupling.

We present some preliminary results for $B_K$ at $\beta=6.0$, using the twisted mass QCD formalism for the computation of bare matrix elements of the $\Delta S=2$ operator. The main advantage of the method is that mixing with other $d=6$ operators under renormalisation is avoided. Moreover the operator renormalisation is performed in the Schr\"odinger functional (SF) framework, using earlier results of our collaboration for the corresponding step scaling function., Comment: Lattice2003(matrix), 3 pages, 2 figures

Models of Dynamical Electroweak Symmetry Breaking are expected to display a quasi-conformal scaling behaviour in order to accommodate experimental constraints. The scaling properties of a theory can be studied using finite volume renormalisation schemes. Among these, the most practical ones are based on the Schrodinger Functional (SF). However, lattices accessible in numerical simulations suffer from potentially large cutoff effects and special care has to be taken to remove these effects. Here we will study the standard setup of the SF with Wilson quarks and a setup with chirally rotated boundary conditions. We study the step scaling function for SU(2) and SU(3) gauge groups in the fundamental, 2-index symmetric and adjoint representations. We perform the O(a)-improvement of both setups to 1-loop order in perturbation theory, and we describe a way of minimising higher order cutoff effects by a redefinition of the renormalised coupling.

In a previous publication, we have constructed the Schr\"odinger functional in Wilson's lattice QCD. It was found that the naive continuum limit leads to a well-defined classical continuum theory. Starting from the latter, a formal continuum definition of the Schr\"odinger functional is given and its saddle point expansion is carried out to one-loop order of perturbation theory. Dimensional regularization and heat kernel techniques are used to determine the one-loop divergences. These are partly canceled by the usual renormalizations of the quark mass and the coupling constant in QCD. An additional divergence can be absorbed in a multiplicative renormalization of the quark boundary fields. The corresponding boundary counterterm is a local polynomial in the fields, so that we confirm a general expectation of Symanzik., Comment: 31 pages, uuencoded postscript file, no figures

We show that the application of twisted mass QCD (tmQCD) with four (Wilson) quark flavours to the computation of lattice weak matrix elements relevant toΔ I =½ transitions has important advantages: the renormalisation of K → π matrix elements does not require the subtraction of other dimension six operators, the divergence arising from the subtraction of lower dimensional operators is softened by one power of the lattice spacing and quenched simulations do not suffer from exceptional configurations at small pion mass. This last feature is also retained in the tmQCD computation of K → ππ matrix elements, which, as far as renormalisation and power subtractions axe concerned, has properties analogous to the standard Wilson case.

We present our preliminary result for the charmed quark mass, which follows from taking the D_s and K meson masses from experiment and r0=0.5 fm (or, equivalently F_K=160 MeV) to set the scale. For the renormalization group invariant quark mass we obtain M_c = 1684(64) MeV, which translates to m_c(m_c)= 1314 (40)(20)(7) MeV for the running mass in the MSbar scheme. Renormalization is treated non-perturbatively, and the continuum limit has been taken, so that the only uncontrolled systematic error consists in the use of the quenched approximation., Lattice2001(spectrum), 3 pages

The set-up of the QCD Schrodinger functional (SF) on the lattice with staggered quarks requires an even number of points L/a in the spatial directions, while the Euclidean time extent of the lattice, T/a, must be odd. Identifying a unique renormalisation scale, L = T, is then only possible up to O(a) lattice artefacts. In this article we study such lattices in the pure SU(3) gauge theory, where we can also compare to the standard set-up. We consider the SF coupling as obtained from the variation of an SU(3) Abelian and spatially constant background field. The O(a) lattice artefacts can be cancelled by the existing O(a) boundary counterterm. However, its coefficient, ct, differs at the tree-level from its standard value, so that one first needs to re-determine the induced background gauge field. The perturbative one-loop correction to the coupling allows to determine ct to one-loop order. A few numerical simulations serve to demonstrate that residual cutoff effects in the step scaling function are small in both cases, T = L ± a and comparable to the standard case with T = L.

The chirally rotated Schroedinger functional provides a test bed for universality and automatic O(a) improvement. We here report on extensive quenched simulations of lattice QCD with Wilson quarks in the massless limit. We demonstrate that, after proper tuning of a dimension 3 boundary counterterm, the expected chirally rotated boundary conditions are indeed obtained. This implies automatic O(a) improvement which we then verify in a few examples. Universality of properly renormalized correlation functions is confirmed by comparing to the standard set-up of the Schroedinger functional. As a by-product of this study the non-singlet current renormalisation constants Z_A and Z_V are obtained from ratios of 2-point functions., Comment: 7 pages, 10 figures, contribution to Lattice 2010, to appear in the proceedings

We present preliminary results for the running coupling in the Schrödinger functional scheme in QCD with four flavours. A single‐component staggered quark field is used on lattices of size (L/a)3×(L/a±1). This provides us with 2 different regularisations of the same renormalized coupling, and thus some control over the size of lattice artefacts. These are found to be comparatively large, calling for a more refined analysis, which still remains to be done., AIP Conf. Proc. 1343

A modified formulation of the Schr\"odinger functional (SF) is proposed. In the continuum it is related to the standard SF by a non-singlet chiral field rotation and therefore referred to as the chirally rotated SF ($\chi$SF). On the lattice with Wilson fermions the relation is not exact, suggesting some interesting tests of universality. The main advantage of the $\chi$SF consists in its compatibility with the mechanism of automatic O($a$) improvement. In this paper the basic set-up is introduced and discussed. Chirally rotated SF boundary conditions are implemented on the lattice using an orbifold construction. The lattice symmetries imply a list of counterterms, which determine how the action and the basic fermionic two-point functions are renormalised and O($a$) improved. As with the standard SF, a logarithmically divergent boundary counterterm leads to a multiplicative renormalisation of the fermionic boundary fields. In addition, a finite dimension 3 boundary counterterm must be tuned in order to preserve the chirally rotated boundary conditions in the interacting theory. Once this is achieved, O($a$) effects originating from the bulk action or from insertions of composite operators in the bulk can be avoided by the mechanism of automatic O($a$) improvement. The remaining O($a$) effects arise from the boundaries and can be cancelled by tuning a couple of O($a$) boundary counterterms. The general results are illustrated in the free theory where the Sheikholeslami-Wohlert term is shown to affect correlation functions only at O($a^2$), irrespective of its coefficient., Comment: 51 pages, 2 figures, revised version: improved and extended discussion of Ward identities in section 3 and of the inclusion of counterterms in section 5; eliminated some typos, introduced new ones, results unchanged

7 pages, 2 figures.-- En: Proceedings of science. The XXVI International Symposium on Lattice Field Theory 14-19 July 2008.

Due to the absence of an exact chiral symmetry in lattice QCD with Wilson fermions, the iso-singlet scalar density has to be renormalized both additively and multiplicatively. We propose to use chiral Ward identities between correlation functions derived from the Schrodinger functional to determine the relevant renormalization constants directly in the chiral limit. Although the method does not rely on perturbation theory, we here use it to determine the renormalization constants and to obtain an idea of the typical size of cutoff effects. Finally we comment on the prospects for a direct determination of the chiral condensate as expectation value of a renormalized scalar density.

We introduce a new algorithm which we call the {Rational Hybrid Monte Carlo} Algorithm (RHMC). This method uses a rational approximation to the fermionic kernel together with a noisy Kennedy-Kuti acceptance step to give an efficient algorithm with no molecular dynamics integration step-size errors., LATTICE98(algorithms)

In order to study the running coupling in four-flavour QCD, we review the set-up of the Schr\"odinger functional (SF) with staggered quarks. Staggered quarks require lattices which, in the usual counting, have even spatial lattice extent $L/a$ while the time extent $T/a$ must be odd. Setting $T=L$ is therefore only possible up to ${\rm O}(a)$, which introduces different cutoff effects already in the pure gauge theory. We re-define the SF such as to cope with this situation and determine the corresponding classical background field. A perturbative calculation yields the coefficient of the pure gauge ${\rm O}(a)$ boundary counterterm to one-loop order., Comment: 7 pages,1 figure, talk presented at Lattice2007 (Renormalization). Compilation problems solved

We simulate two variants of quenched twisted mass QCD (tmQCD), with degenerate Wilson quarks of masses equal to or heavier than half the strange quark mass. We use Ward identities in order to measure the twist angles of the theory and thus check the quality of the tuning of mass parameters to a physics condition which stays constant as the lattice spacing is varied. Flavour symmetry breaking in tmQCD is studied in a framework of two fully twisted and two standard Wilson quark flavours, tuned to be degenerate in the continuum. Comparing pseudoscalar masses, obtained from connected quark diagrams made of tmQCD and/or standard Wilson quark propagators, we confirm that flavour symmetry breaking effects, which are at most 5%, decrease as we approach the continuum limit. We also compute the pseudoscalar decay constant in the continuum limit, with reduced systematics. As a consequence of improved tuning of the mass parameters at $\beta = 6.1$, we reanalyse our previous $B_K$ results. Our main phenomenological findings are $r_0 f_K = 0.421(7)$ and $\hat B_K = 0.735(71)$., Comment: 41 pages, figures included, one reference added. Final version as accepted for publication on Nucl.Phys.B

In these lectures I explain how chiral symmetry of continuum QCD naturally leads to a class of lattice regularisations known as twisted mass QCD (tmQCD). As compared to standard Wilson quarks, its advantages are the absence of unphysical zero modes, the possibility to circumvent lattice renormalisation problems and automatic O(a) improvement. On the other hand, the physical parity and flavour symmetries are explicitly broken. I discuss these aspects and then turn to the theory in a finite space-time volume with Schroedinger functional boundary conditions. Again, chiral transformations of the continuum theory may be used as a guide to formulate an alternative lattice regularisation of the Schr\"odinger functional, with interesting applications to renormalization problems in QCD., Comment: 40 pages, 7 figures; lectures presented at the School "Perspectives in Lattice Gauge Theories", Nara, Japan, October 31 - November 14, 2005, to appear in the proceedings

Using the Schr\"odinger functional we have computed a variety of renormalized on-shell correlation functions to one-loop order of perturbation theory. By studying their approach to the continuum limit we have determined the O($a$) counterterms needed to improve the quark mass and a number of isovector quark bilinear operators., Comment: 3 pages Latex using espcrc2.sty, to appear in the conference proceedings of Lattice '97, Edinburgh

The $B_K$ parameter is computed in quenched lattice QCD with Wilson twisted mass fermions. Two variants of tmQCD are used; in both of them the relevant $\Delta S = 2$ four-fermion operator is renormalised multiplicatively. The renormalisation adopted is non-perturbative, with a Schroedinger functional renormalisation condition. Renormalisation group running is also non-perturbative, up to very high energy scales. In one of the two tmQCD frameworks the computations have been performed at the physical $K$-meson mass, thus eliminating the need of mass extrapolations. Simulations have been performed at several lattice spacings and the continuum limit was reached by combining results from both tmQCD regularisations. Finite volume effects have been partially checked and turned out to be small. Exploratory studies have also been performed with non-degenerate valence flavours. The final result for the RGI bag parameter, with all sources of uncertainty (except quenching) under control, is $\hat B_K =0.789 \pm 0.046$., Comment: 54 pages, 11 figures

I report on a recent determination by the ALPHA collaboration of the kaon B-parameter using lattice QCD with Wilson type quarks. An effort is made to control all systematic errors except for the quenched approximation. The preliminary result for the renormalization group invariant parameter is $\hat{B}_K = 0.834(37)$, which translates to $B_K^MSbar(2 GeV)=0.604(27)$ in the MSbar scheme with anticommuting $\gamma_5$., Comment: 4 pages, parallel talk at International Conference on QCD and Hadronic Physics, Beijing, June 2005

Starting from the QCD Schroedinger functional (SF), we define a family of renormalization schemes for two four-quark operators, which are, in the chiral limit, protected against mixing with other operators. With the appropriate flavour assignments these operators can be interpreted as part of either the $\Delta F=1$ or $\Delta F=2$ effective weak Hamiltonians. In view of lattice QCD with Wilson-type quarks, we focus on the parity odd components of the operators, since these are multiplicatively renormalized both on the lattice and in continuum schemes. We consider 9 different SF schemes and relate them to commonly used continuum schemes at one-loop order of perturbation theory. In this way the two-loop anomalous dimensions in the SF schemes can be inferred. As a by-product of our calculation we also obtain the one-loop cutoff effects in the step-scaling functions of the respective renormalization constants, for both O(a) improved and unimproved Wilson quarks. Our results will be needed in a separate study of the non-perturbative scale evolution of these operators., Comment: 22 pages, 4 figures

The Schr\"odinger functional in Wilson's lattice QCD leads to a sensible classical continuum theory which can be taken as starting point for a perturbative analysis. In dimensional regularization, the saddle point expansion of the Schr\"odinger functional is performed to one-loop order of perturbation theory. The divergences are partly cancelled by the usual coupling constant and quark mass renormalization. An additional divergence can be absorbed in a multiplicative renormalization of the quark boundary fields. The corresponding boundary counterterm being a local polynomial in the fields we confirm the general expectation expressed by Symanzik~\cite{Symanzik}., Comment: 3 pages in postscript (no figures), talk presented at Lattice '94 in Bielefeld 9/27--10/1/94