Search

We present the preliminary results of the non-perturbative running of the flavour non-singlet tensor operator in the high-energy range $2~\rm{GeV}\lesssim \mu\lesssim 128~\rm{GeV}$ in $N_f=3$ massless QCD, comparing four different definitions of the renormalisation constant. We use the configuration ensembles of arXiv:1802.05243 and arXiv:1607.06423, subject to Schr\"odinger functional (SF) boundary conditions, and valence quarks with chirally rotated Schr\"odinger functional ($\chi$SF) boundary conditions. Provided that boundary counterterms have been appropriately tuned, this results in O($a$) improvement of the tensor operator, without the need of a dimension-4 Symanzik counterterm (proportional to $c_T$)., Comment: 10 pages, 4 figures. Presented at the 39th International Symposium on Lattice Field Theory, 8th-13th August, 2022, Bonn, Germany

We study the Renormalisation Group (RG) running of the quark mass, for $N_f=3$ QCD with Wilson fermions in a mixed action setup, with standard Schr\"odinger Functional (SF) boundary conditions for sea quarks and chirally rotated Schr\"odinger Functional ($\chi$SF) boundary conditions for valence quarks. This necessitates the tuning of the boundary factor $z_f(g_0^2)$ of the $\chi$SF valence action, in order to ensure that QCD symmetries are fully recovered in the continuum. The properties of this novel setup are monitored through the ratios $Z_S/Z_P$ and $\Sigma_S/\Sigma_P$ of the renormalisation parameters and step scaling functions of the scalar and pseudoscalar densities. Where comparison is possible, our $Z_S/Z_P$ results are found to agree with previous determinations, based on a mass ratio method arXiv:1906.03445 and Ward identities arXiv:2005.01352, arXiv:2101.10969, with Schr\"odinger Functional boundary conditions. The behaviour of $\Sigma_S/\Sigma_P$ confirms the theoretical expectations of $\chi$SF QCD, related to the restoration of the theory's symmetries in the continuum limit. From the step scaling function of the pseudoscalar density we obtain the quark mass RG-running function from hadronic to perturbative energy scales. This is fully compatible with the earlier result obtained in a similar setup for Wilson quarks with Schr\"odinger Functional boundary conditions arXiv:1802.05243 and provides a strong universality test for the two lattice setups., Comment: 18 pages, 13 figures

We compute the nonperturbative quark mass RG-running in the range $\Lambda_{QCD}\lessapprox\mu\lessapprox M_W$ for $N_f=3$ massless QCD with a mixed action approach: sea quarks are regularised using nonperturbatively $O(a)$-improved Wilson fermions with Schr\"odinger functional (SF) boundary conditions, employing the configurations of 1802.05243, while valence quarks are regularised using nonperturbatively $O(a)$-improved Wilson fermions with chirally rotated Schr\"odinger functional boundary conditions ($\chi$SF). Our result is compatible with its SF counterpart of ref.1802.05243, confirming the universality of $\chi$SF and SF in the continuum limit. We also establish the optimal tuning strategy for the critical hopping parameter $\kappa_c$ and the $\chi$SF boundary counterterm coefficient $z_{\rm f}$. We work in two energy regimes with two different definitions of the coupling: SF-coupling for 2 GeV $\lessapprox\mu\lessapprox M_W$ and GF-coupling for $\Lambda_{QCD} \lessapprox\mu\lessapprox 2 GeV$., Comment: 8 pages, 5 figures, presented at the 38th International Symposium on Lattice Field Theory, LATTICE2021, 26th-30th July, 2021. Updated acknowledgements

We present preliminary results of the non-perturbative renormalization group (RG) running of the flavor non-singlet tensor operator. We employ the $\chi$SF scheme for $N_f=3$ QCD using ensembles generated by the ALPHA collaboration for the computation of the quark mass running. The $\chi$SF property of automatic $O(a)$ improvement prevents the $O(a)$ mixing of the correlation functions., Comment: 8 pages, 4 figures, Presented at the 38th International Symposium on Lattice Field Theory, LATTICE2021, 26th-30th July, 2021. v3: Updated Acknowledgements

We determine non-perturbatively the normalisation factor $r_\mathrm{m}\equiv Z_{\rm S}/Z_{\rm S}^{0}$, where $Z_{\rm S}$ and $Z_{\rm S}^{0}$ are the renormalisation parameters of the flavour non-singlet and singlet scalar densities, respectively. This quantity is required in the computation of quark masses with Wilson fermions and for instance the renormalisation of nucleon matrix elements of scalar densities. Our calculation involves simulations of finite-volume lattice QCD with the tree-level Symanzik-improved gauge action, $N_\mathrm{f} = 3$ mass-degenerate $\mathrm{O}(a)$ improved Wilson fermions and Schr\"odinger functional boundary conditions. The slope of the current quark mass, as a function of the subtracted Wilson quark mass is extracted both in a unitary setup (where nearly chiral valence and sea quark masses are degenerate) and in a non-unitary setup (where all valence flavours are chiral and the sea quark masses are small). These slopes are then combined with $Z \equiv Z_{\rm P}/(Z_{\rm S}Z_{\rm A})$ in order to obtain $r_\mathrm{m}$. A novel chiral Ward identity is employed for the calculation of the normalisation factor $Z$. Our results cover the range of gauge couplings corresponding to lattice spacings below $0.1\,$fm, for which $N_\mathrm{f} = 2+1$ QCD simulations in large volumes with the same lattice action are typically performed., Comment: 28 pages, 9 figures

We derive chiral Ward identities for lattice QCD with Wilson quarks and $N_\mathrm{f} \geq 3$ flavours, on small lattices with Schr\"odinger functional boundary conditions and vanishingly small quark masses. These identities relate the axial variation of the non-singlet pseudoscalar density to the scalar one, thus enabling the non-perturbative determination of the scale-independent ratio $Z_\mathrm{S}/Z_\mathrm{P}$ of the renormalisation parameters of these operators. We obtain results for $N_\mathrm{f}=3$ QCD with tree-level Symanzik-improved gluons and Wilson-Clover quarks, for bare gauge couplings which cover the typical range of large-volume $N_\mathrm{f} = 2+1$ simulations with Wilson fermions at lattice spacings below $0.1\,$fm. The precision of our results varies from 0.3\% to 1\%, except for the coarsest lattice, where it is 2\%. We discuss how the $Z_\mathrm{S}/Z_\mathrm{P}$ ratio can be used in the non-perturbative calculations of $\mathrm{O}(a)$ improved renormalised quark masses., Comment: 24 pages, 8 figures, corrected a few misprints of the published version

We present a lattice QCD determination of light quark masses with three sea-quark flavours ($N_f = 2+1$). Bare quark masses are known from PCAC relations in the framework of CLS lattice computations with a non-perturbatively improved Wilson-Clover action and a tree-level Symanzik improved gauge action. They are fully non-perturbatively improved, including the recently computed Symanzik counter-term $b_{\rm A} - b_{\rm P}$. The mass renormalisation at hadronic scales and the renormalisation group running over a wide range of scales are known non-perturbatively in the Schr\"odinger functional scheme. In the present paper we perform detailed extrapolations to the physical point, obtaining (for the four-flavour theory) $m_{u/d}(2{\rm GeV}) = 3.54(12)(9)$ MeV and $m_s(2{\rm GeV}) = 95.7(2.5)(2.4)$ MeV in the $\bar{MS}$ scheme. For the mass ratio we have $m_s/m_{u/d} = 27.0(1.0)(0.4)$. The RGI values in the three-flavour theory are $M_{u/d} = 4.70(15)(12)$ MeV and $M_s = 127.0(3.1)(3.2)$ MeV., Comment: 27 pages, 5 figures. Published version of the paper. The present version contains additions and improvements proposed by the referee; the results remain unchanged

We employ the chirally rotated Schr\"odinger 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 renormalisation scales from 4 to 70~GeV, are generated with $N_{\rm f}=3$ massless flavors and Schr\"odinger 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., Comment: 7 pages, 4 figures. Presented as a poster at the 37th International Symposium on Lattice Field Theory, 16-22 June, 2019, Wuhan, China

We determine non-perturbatively the normalisation constant Z_m*Z_P/Z_A as well as the Symanzik coefficients b_m and b_A-b_P, required in O(a) improved quark mass renormalisation with Wilson fermions. The strategy underlying their computation involves simulations in N_f=3 QCD with O(a) improved massless sea and non-degenerate valence quarks in the finite-volume Schroedinger functional scheme. Our results, which cover the typical gauge coupling range of large-volume N_f=2+1 QCD simulations with Wilson fermions at lattice spacings below 0.1 fm, are of particular use for the non-perturbative calculation of O(a) improved renormalised quark masses., Comment: 36 pages including figures and tables, latex2e; version published in Eur. Phys. J. C; small additions to the text and reference added; results unchanged

We present a nearly final analysis of the $u/d$ and $s$ quark masses, extracted using the PCAC quark masses reported in [PRD 95 (2017) 074504]. The data is based on the CLS $N_f = 2 + 1$ simulations with Wilson/Clover quarks and L\"uscher-Weisz gauge action, at four $\beta$ values (i.e. lattice spacings) and a range of quark masses. We use the ALPHA results of [EPJC 78 (2018) 387] for non-perturbative quark mass renormalisation and RG-running from hadronic to electroweak scales in the Schr\"odinger Functional scheme. Quark masses are quoted both in the $\overline{\rm MS}$ scheme and as RGI quantities., Comment: Talk presented at the 36th Annual International Symposium on Lattice Field Theory (Lattice 2018), 22-28 July 2018, Michigan, USA

We review lattice results related to pion, kaon, $D$-meson, $B$-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $f_+(0)$ arising in the semileptonic $K \to \pi$ transition at zero momentum transfer, as well as the decay constant ratio $f_K/f_\pi$ and its consequences for the CKM matrix elements $V_{us}$ and $V_{ud}$. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $SU(2)_L\times SU(2)_R$ and $SU(3)_L\times SU(3)_R$ Chiral Perturbation Theory. We review the determination of the $B_K$ parameter of neutral kaon mixing as well as the additional four $B$ parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $m_c$ and $m_b$ as well as those for $D$- and $B$-meson decay constants, form factors, and mixing parameters. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant $\alpha_s$. Finally, in this review we have added a new section reviewing results for nucleon matrix elements of the axial, scalar and tensor bilinears, both isovector and flavor diagonal., Comment: 458 pages, 46 figures, 209 tables, 1146 references. Minor changes, version as published in EPJC. arXiv admin note: substantial text overlap with arXiv:1607.00299, arXiv:1310.8555

We report on advances in the non-perturbative determination of the ratio $Z_S/Z_P$ of the pseudoscalar to the scalar renormalization constants in three-flavour lattice QCD with Wilson-clover quarks and tree-level Symanzik improved gluons. The computations are based on the Ward identity approach, using Schr\"odinger functional boundary conditions. Our results for $Z_S/Z_P$ cover a range of couplings along a line of constant physics with lattice spacings of about 0.09 fm and below, relevant for phenomenological applications such as the calculation of renormalized quark masses., Comment: 7 pages including figures and tables, latex2e; Proceedings of the 36th International Symposium on Lattice Field Theory (Lattice 2018), 22-28 July 2018, Michigan State University, East Lansing, Michigan, USA

We determine from first principles the quark mass anomalous dimension in Nf=3 QCD between the electroweak and hadronic scales. This allows for a fully non-perturbative connection of the perturbative and non-perturbative regimes of the Standard Model in the hadronic sector. The computation is carried out to high accuracy, employing massless O(a)-improved Wilson quarks and finite-size scaling techniques. We also provide the matching factors required in the renormalisation of light quark masses from lattice computations with O(a)-improved Wilson fermions and a tree-level Symanzik improved gauge action. The total uncertainty due to renormalisation and running in the determination of light quark masses in the SM is thus reduced to about 1%., Comment: 41 pages, 10 tables, 7 figures, published version (minimal text improvements)

We perform a non-perturbative study of the scale-dependent renormalisation factors of a complete set of dimension-six four-fermion operators. The renormalisation-group (RG) running is determined in the continuum limit for a specific Schr\"dinger Functional (SF) renormalisation scheme in the framework of lattice QCD with two dynamical flavours ( $N_f = 2$ ). The theory is regularised on a lattice with a plaquette Wilson action and $\mathcal{O}(a)$-improved Wilson fermions. For one of these operators, the computation had been performed in ref. [1]; the present work completes the study for the rest of the operator basis, on the same simulations (configuration ensembles). The related weak matrix elements arise in several operator product expansions; in $\Delta F = 2$ transitions they contain the QCD long-distance effects, including contributions from beyond-Standard Model (BSM) processes. Some of these operators mix under renormalisation and their RG-running is governed by anomalous dimension matrices. In ref. [2] the RG formalism for the operator basis has been worked out in full generality and the anomalous dimension matrix has been calculated in NLO perturbation theory. Here the discussion is extended to the matrix step-scaling functions (matrix-SSFs), which are used in finite-size recursive techniques. We rely on these matrix-SSFs to obtain non-perturbative estimates of the operator anomalous dimensions for scales ranging from $\mathcal{O}(\Lambda_{\rm QCD})$ to $\mathcal{O}(M_W)$., Comment: 45 pages, 17 figures

We report on non-perturbative computations of the improvement coefficient c_V and the renormalization factor Z_V of the vector current in three-flavour O(a) improved lattice QCD with Wilson quarks and tree-level Symanzik improved gauge action. To reduce finite quark mass effects, our improvement and normalization conditions exploit massive chiral Ward identities formulated in the Schroedinger functional setup, which also allow deriving a new method to extract the ratio Z_S/Z_P of scalar to pseudoscalar renormalization constants. We present preliminary results of a numerical evaluation of Z_V and c_V along a line of constant physics with gauge couplings corresponding to lattice spacings of about 0.09 fm and below, relevant for phenomenological applications., Comment: 8 pages including figures and tables, latex2e; Proceedings of the 35th International Symposium on Lattice Field Theory (Lattice 2017), 18-24 June 2017, Granada, Spain; typo in parametrization of c_V corrected, overall results unchanged

We present our preliminary results of the non-perturbative determination of the valence mass dependent coefficients $b_\mathrm{A}-b_\mathrm{P}$ and $b_\mathrm{m}$ as well as the ratio $Z_\mathrm{P} Z_\mathrm{m}/ Z_\mathrm{A}$ entering the flavour non-singlet PCAC relation in lattice QCD with $N_f=3$ dynamical flavours. We apply the method proposed in the past for quenched approximation and $N_f=2$ cases, employing a set of finite-volume ALPHA configurations with Schr\"odinger functional boundary conditions, generated with $O(a)$ improved Wilson fermions and the tree-level Symanzik-improved gauge action for a range of couplings relevant for simulations at lattice spacings of about $0.09 \,$fm and below., Comment: 8 pages, 6 figures. Talk given at Lattice2017, the 35th International Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spain

We present our preliminary results for the computation of the non-perturbative running of renormalized quark masses in $N_f = 3$ QCD, between the electroweak and hadronic scales, using standard finite-size scaling techniques. The computation is carried out to very high precision, using massless $\mathcal{O}(a)$-improved Wilson quarks. Following the strategy adopted by the ALPHA Collaboration for the running coupling, different schemes are used above and below a scale $\mu_0 \sim m_b$, which differ by using either the Schr\"odinger Functional or Gradient Flow renormalized coupling. We discuss our results for the running in both regions, and the procedure to match the two schemes., Comment: 7 pages, 3 figures, 34th annual International Symposium on Lattice Field Theory

The determination of quark masses from lattice QCD simulations requires a non-perturbative renormalization procedure and subsequent scale evolution to high energies, where a conversion to the commonly used MS-bar scheme can be safely established. We present our results for the non-perturbative running of renormalized quark masses in Nf=3 QCD between the electroweak and a hadronic energy scale, where lattice simulations are at our disposal. Recent theoretical advances in combination with well-established techniques allows to follow the scale evolution to very high statistical accuracy, and full control of systematic effects., Comment: 10 pages, 4 figures, Proc. (CONF12) XIIth Quark Confinement and the Hadron Spectrum, published version (typo corrected)

We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle physics community. More specifically, we report on the determination of the light-quark masses, the form factor f+(0), arising in the semileptonic K -> pi transition at zero momentum transfer, as well as the decay constant ratio fK/fpi and its consequences for the CKM matrix elements Vus and Vud. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)LxSU(2)R and SU(3)LxSU(3)R Chiral Perturbation Theory. We review the determination of the BK parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. The latter quantities are an addition compared to the previous review. For the heavy-quark sector, we provide results for mc and mb (also new compared to the previous review), as well as those for D- and B-meson decay constants, form factors, and mixing parameters. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. Finally, we review the status of lattice determinations of the strong coupling constant alpha_s., Comment: 383 pages, 32 figures, 160 tables, 795 references

After a short presentation of the FLAG collaboration, we review lattice results related to pion, $K$-, $D$- and $B$-meson physics with the aim of making them easily accessible to the particle-physics community. Only a selection of FLAG averages or estimates is presented. For light flavours, we present results on the form factor $f_+(0)$, arising in semileptonic $K \rightarrow \pi$ transition at zero momentum transfer, as well as the decay-constants $f_K,f_\pi$ and their ratio. The consequences of these results for the CKM matrix elements $|V_{us}|$ and $|V_{ud}|$ are discussed. For heavy flavours we focus on $D$- and $B$-meson decay constants and form factors, as well as the CKM matrix elements $|V_{cs}|$, $|V_{cd}|$ and $|V_{ub}|$. In addition we briefly cover the recent advances stemming from the calculation the $B_K$-parameters and touch upon related current results relevant to the Physics beyond the Standard Model, which will be the subject of the next FLAG edition., Comment: To appear in the proceedings of "Flavour Physics and CP Violation 2015", May 25-29 2015, Nagoya, Japan; 16 pages, 7 figures

We present the current status of a revised strategy to compute the running of renormalized quark masses in QCD with three flavours of massless O(a) improved Wilson quarks. The strategy employed uses the standard finite-size scaling method in the Schr\"odinger functional and accommodates for the non-perturbative scheme-switch which becomes necessary at intermediate renormalized couplings as discussed in [arXiv:1411.7648]., Comment: 7 pages, 3 figures, 1 table; Proceedings of the 33rd International Symposium on Lattice Field Theory, 14-18 July 2015, Kobe, Japan

We determine a basis of dimension-7 operators which arise at O($a$) in the Symanzik expansion of the $\Delta B=2$ operators with static heavy quarks. We consider both Wilson-like and Ginsparg-Wilson light quarks. Exact chiral symmetry reduces the number of these O($a$) counterterms by a factor of two. Only a subset of these operators has previously appeared in the literature. We then extend the analysis to the O($1/m$) operators contributing beyond the static approximation., Comment: 7 pages, presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany

We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle physics community. More specifically, we report on the determination of the light-quark masses, the form factor f+(0), arising in semileptonic K -> pi transition at zero momentum transfer, as well as the decay constant ratio fK/fpi of decay constants and its consequences for the CKM matrix elements Vus and Vud. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)LxSU(2)R and SU(3)LxSU(3)R Chiral Perturbation Theory and review the determination of the BK parameter of neutral kaon mixing. The inclusion of heavy-quark quantities significantly expands the FLAG scope with respect to the previous review. Therefore, for this review, we focus on D- and B-meson decay constants, form factors, and mixing parameters, since these are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. In addition we review the status of lattice determinations of the strong coupling constant alpha_s., Comment: Several errors of transcription in the entries of Table 1 have been corrected. The entries of Table 1 now agree with the final results quoted in the main text. 324 pages, 26 figures, 118 tables, 697 references

We present the first unquenched, continuum limit, lattice QCD results for the matrix elements of the operators describing neutral kaon oscillations in extensions of the Standard Model. Owing to the accuracy of our calculation on \Delta S=2 weak Hamiltonian matrix elements, we are able to provide a refined Unitarity Triangle analysis improving the bounds coming from model independent constraints on New Physics. In our non-perturbative computation we use a combination of Nf=2 maximally twisted sea quarks and Osterwalder-Seiler valence quarks in order to achieve both O(a)-improvement and continuum-like renormalization properties for the relevant four-fermion operators. The calculation of the renormalization constants has been performed non-perturbatively in the RI-MOM scheme. Based on simulations at four values of the lattice spacing and a number of quark masses we have extrapolated/interpolated our results to the continuum limit and physical light/strange quark masses., Comment: small corrections to RC matrix Z_Q for i,j=2,3; marginal corrections to bag parameters B_4, B_5, and ratios R_4, R_5; results on NP scales from UTfit analysis unchanged; 66 pages, 12 figures

The mass m_K* and vector coupling f_K* of the K*-meson, as well as the ratio of the tensor to vector couplings fT/fV|_K*, are computed in lattice QCD. Our simulations are performed in a partially quenched setup, with two dynamical (sea) Wilson quark flavours, having a maximally twisted mass term. Valence quarks are either of the standard or the Osterwalder-Seiler maximally twisted variety. Results obtained at three values of the lattice spacing are extrapolated to the continuum, giving m_K* = 981(33) MeV, f_K* = 240(18) MeV and fT(2 GeV)/fV|_K* = 0.704(41)., Comment: 1+11 pages

This is an expanded version of lecture notes, delivered at the XCIII Les Houches Summer School (August 2009). Our aim is to present three very specific topics: (i) The consequences of the loss of chiral symmetry in the Wilson lattice regularization of the fermionic action and its recovery in the continuum limit. The treatment of these arguments involves lattice Ward identities. (ii) The definition and properties of mass independent renormalization schemes, which are suitable for a non-perturbative computation of various operator renormalization constants. (iii) The modification of the Wilson fermion action, by the introduction of a chirally twisted mass term (known as twisted mass QCD - tmQCD), which results to improved (re)normalization and scaling properties for physical quantities of interest., Comment: 67 pages, Lecture Notes delivered at the XCIII Les Houches Summer School "Modern perspectives in lattice QCD" (August 2009); comment and reference concerning RG-running in RI/MOM added; minor typos corrected

We present preliminary results on the of neutral kaon oscillations in extensions of the Standard Model. Using Nf=2 maximally twisted sea quarks and Osterwalder-Seiler valence quarks, we achieve both O(a)-improvement and continuum-like renormalization pattern for the relevant four-fermion operators. We perform simulations at three values of the lattice spacing and extrapolate/interpolate our results to the continuum limit and physical light/strange quark mass. The calculation of the renormalization constants of the complete operator basis is performed non- perturbatively in the RI-MOM scheme., Comment: 7 pages, 2 figures; presented at the XXVIII International Symposium on Lattice Field Theory Villasimius, Sardinia, Italy

We review lattice results relevant for pion and kaon physics with the aim of making them easily accessible to the particle physics community. Specifically, we review the determination of the light-quark masses, the form factor f_+(0), relevant for the semileptonic K -> pi transition at zero momentum transfer as well as the ratio f_K/f_pi of decay constants and discuss the consequences for the elements V_{us} and V_{ud} of the CKM matrix. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)_LxSU(2)_R and SU(3)_LxSU(3)_R Chiral Perturbation Theory and review the determination of the B_K parameter of neutral kaon mixing. We introduce quality criteria and use these when forming averages. Although subjective and imperfect, these criteria may help the reader to judge different aspects of current lattice computations. Our main results are summarized in section 1.2, but we stress the importance of the detailed discussion that underlies these results and constitutes the bulk of the present review., Comment: 126 pages, 11 figures. Plain Latex. Review provided by the FLAG (Flavianet Lattice Averaging Group) working group. Updated to February 28, 2011. To appear in EPJ

We present an unquenched $N_f=2$ lattice computation of the $B_{K}$ parameter which controls $K^0-\bar K^0$ oscillations. A partially quenched setup is employed with two maximally twisted dynamical (sea) light Wilson quarks, and valence quarks of both the maximally twisted and the Osterwalder--Seiler variety. Suitable combinations of these two kinds of valence quarks lead to a lattice definition of the $B_{K}$ parameter which is both multiplicatively renormalizable and O($a$) improved. Employing the non-perturbative RI-MOM scheme, in the continuum limit and at the physical value of the pion mass we get $B^{\rm RGI}_K=0.729\pm 0.030$, a number well in line with the existing quenched and unquenched determinations., Comment: Accepted for publication in Phys. Rev. D; systematic error discussion slightly modified; total uncertainty estimate on the BK result unchanged; 42 pages, 25 figures

We present results for the renormalization constants of bilinear quark operators obtained by using the tree-level Symanzik improved gauge action and the Nf=2 twisted mass fermion action at maximal twist, which guarantees automatic O(a)-improvement. Our results are also relevant for the corresponding standard (un-twisted) Wilson fermionic action since the two actions only differ, in the massless limit, by a chiral rotation of the quark fields. The scale-independent renormalization constants ZV, ZA and the ratio ZP/ZS have been computed using the RI-MOM approach, as well as other alternative methods. For ZA and ZP/ZS, the latter are based on both standard twisted mass and Osterwalder-Seiler fermions, while for ZV a Ward Identity has been used. The quark field renormalization constant Zq and the scale dependent renormalization constants ZS, ZP and ZT are determined in the RI-MOM scheme. Leading discretization effects of O(g^2 a^2), evaluated in one-loop perturbation theory, are explicitly subtracted from the RI-MOM estimates., Comment: 31 pages, 11 figures

We compute non-perturbatively the B-parameters of the complete basis of four-fermion operators needed to study the Kaon oscillations in the SM and in its supersymmetric extension. We perform numerical simulations with two dynamical maximally twisted sea quarks at three values of the lattice spacing on configurations generated by the ETMC. Unwanted operator mixings and O(a) discretization effects are removed by discretizing the valence quarks with a suitable Osterwalder-Seiler variant of the Twisted Mass action. Operators are renormalized non-perturbatively in the RI/MOM scheme. Our preliminary result for BK(RGI) is 0.73(3)(3)., Comment: 7 pages, 3 figures, 1 table, proceedings of the XXVII Int'l Symposyum on Lattice Field Theory (LAT2009), July 26-31 2009, Peking University, Beijing (China)

We apply an Osterwalder-Seiler version of twisted mass QCD to a study of the $B_K$ parameter, in which three of the four quark fields making up the relevant $\Delta S =2$ operator are maximally twisted with the same twist angle, while the fourth one has a twist angle of opposite sign. It is known that this setup ensures automatic improvement of the bare $K^0$-$\overline K^0$ operator matrix element and multiplicative renormalization of the $\Delta S =2$ operator, at the price of breaking the $K^0$-$\overline K^0$ mass degeneracy by discretization effects. As a result, two dominant systematic errors of the $B_ K$ determination with Wilson fermions are kept under control. With the Clover term included in the fermion action, we perform a feasibility study and find, in the quenched approximation, a significant improvement of the scaling behaviour of $B_K$, compared to earlier standard tmQCD determinations. Moreover, we study in detail the $K^0$-$\overline K^0$ mass splitting that characterizes this approach and confirm that, in the presence of the Clover term, it is greatly reduced in a maximally twisted theory., Comment: 18 pages, 4 figures

We present work in progress on the computation of the K-meson vector and tensor decay constants, as well as the B-parameter in Kaon oscillations. Our simulations are performed in a partially quenched setup, with two dynamical (sea) Wilson quark flavours, having a maximally twisted mass term. Valence quarks are either of the standard or the Osterwalder-Seiler maximally twisted variety. These two regularizations can be suitably combined in order to obtain a BK parameter which is both multiplicatively renormalizable and O(a) improved., Comment: 7 pages, 4 figures, talk presented at the XXVI International Symposium on Lattice Field Theory, July 14 - 19, 2008, Williamsburg, Virginia, USA

Using Schroedinger Functional methods, we compute the non-perturbative renormalisation and renormalisation group running of several four-fermion operators, in the framework of lattice simulations with two dynamical Wilson quarks. Two classes of operators have been targeted: (i) those with left-left current structure and four propagating quark fields; (ii) all operators containing two static quarks. In both cases, only the parity-odd contributions have been considered, being the ones that renormalise multiplicatively. Our results, once combined with future simulations of the corresponding lattice hadronic matrix elements, may be used for the computation of phenomenological quantities of interest, such as B_K and B_B (the latter also in the static limit)., Comment: 26 pages, 8 figures

We present results for the non-perturbative renormalisation of four-fermion operators with two flavours of dynamical quarks. We consider both fully relativistic left current-left current operators, and a full basis for $\Delta B=2$ operators with static heavy quarks. The renormalisation group running of the operators to high energy scales is computed in the continuum limit for a family of Schroedinger Functional renormalisation schemes, via standard finite size scaling techniques. The total renormalisation factors relating renormalisation group invariant to bare operators are computed for a choice of lattice regularisations., Comment: 7 pages, 2 figures, Proceedings from the XXV International Symposium on Lattice Field Theory, Regensburg, Germany Typos corrected + typographic adjustments

We present results for the renormalisation constants of bilinear quark operators, using the Nf=2 twisted mass Wilson action at maximal twist (which guarantees automatic O(a) improvement) and the tree-level Symanzik improved gauge action. The scale-independent renormalisation constants are computed with a new method, which makes use of both standard twisted mass and Osterwalder-Seiler fermions. Moreover, the results from an RI-MOM calculation are presented for both scale independent and scale dependent renormalisation constants., Comment: 7 pages, talk at The XXV International Symposium on Lattice Field Theory, July 30 - August 4 2007, Regensburg

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

We apply finite-size recursion techniques based on the Schrodinger functional formalism to determine the renormalization group running of four-fermion operators which appear in the Delta S=2 effective weak Hamiltonian of the Standard Model. Our calculations are done using O(a) improved Wilson fermions with N_f=2 dynamical flavours. Preliminary results are presented for the four-fermion operator which determines the B_K parameter in tmQCD., Comment: 7 pages, 2 figures, talk presented at Lattice2006 (Renormalization)

We outline a general strategy for the non-perturbative renormalisation of composite operators in discretisations based on Neuberger fermions, via a matching to results obtained with Wilson-type fermions. As an application, we consider the renormalisation of the four-quark operators entering the Delta S=1 and Delta S=2 effective Hamiltonians. Our results are an essential ingredient for the determination of the low-energy constants governing non-leptonic kaon decays., Comment: 14 pages, 3 figures

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

We define a family of Schroedinger Functional renormalization schemes for the four-quark multiplicatively renormalizable operators of the $\Delta F = 1$ and $\Delta F = 2$ effective weak Hamiltonians. Using the lattice regularization with quenched Wilson quarks, we compute non-perturbatively the renormalization group running of these operators in the continuum limit in a large range of renormalization scales. Continuum limit extrapolations are well controlled thanks to the implementation of two fermionic actions (Wilson and Clover). The ratio of the renormalization group invariant operator to its renormalized counterpart at a low energy scale, as well as the renormalization constant at this scale, is obtained for all schemes., Comment: 52 pages, 6 figures (40 eps files)

We present the results of a precision computation of B_K 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 Schroedinger Functional results., Comment: 3 pages, 3 figures, talk presented at Lattice2004(weak), Fermilab, June 21-26, 2004

Twisted mass lattice QCD (tmQCD), generalised to four Wilson quark flavours, can be used for the computation of some weak matrix elements related to $\Delta I=1/2$ transitions. Besides eliminating unphysical zero modes, tmQCD may alleviate long-standing renormalisation problems of the four-quark operators which contribute to CP-conserving $K\to\pi$ transitions. With an active charm quark, the renormalisation of the $K\to\pi$ matrix elements requires at most the subtraction of a linearly divergent counterterm. Furthermore, in the (partially) quenched approximation the twist angles can be chosen so that only a finite counterterm needs to be subtracted., Comment: 35 pages;final version with references added