FLAG Review 2019

The version of the FLAG 19 review as of December 2020 is accessible here. It contains updated sections as follows:
$V_{ud}$ and $V_{us}$: updated December 2020
$B$meson decay constants, mixing parameters, and form factors: updated December 2020
The original 2019 edition of the FLAG review can be downloaded here. The version dated March 4, 2020 is the one accepted for publication in EPJC. The EPJC version can be obtained from here.
The separate sections can be downloaded as separate pdffiles following the links in the table of contents below, or via the menu in the sidebar. Clicking on the FLAG logo in the upper left corner links back to this main page.
The latest figures can be downloaded in eps, pdf and png format, together with a bibfile containing the bibtexentries for the calculations which contribute to the FLAG averages and estimates. The downloads are available via the menu in the sidebar.
In the notes we compile detailed information on the simulations used to calculate the quantities discussed in the review. Here we provide the complete tables, in contrast to the paper version of the review which contains this information only for results that have appeared since FLAG 16.
The original complete 2015/2016 review is still accessible from EPJC. The 2016/2017 updates are available from here. The 2013/2014 review is accessible here or from EPJC.
Contents
 Introduction
 Quality criteria
 Quark masses
 $\small{V_{ud}}$ and $\small{V_{us}}$
 Lowenergy constants
 Kaon mixing
 $\small{D}$meson decay constants and form factors
 $\small{B}$meson decay constants, mixing parameters, and form factors
 The strong coupling $\alpha_s$
 Nucleon matrix elements
 Glossary
 Notes
Introduction
The introduction with the updated summary tables can be downloaded here.
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 lightquark 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 lowenergy 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 heavyquark 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 heavyquark quantities most relevant for the determination of CKM matrix elements and the global CKM unitaritytriangle 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.
Flavour physics provides an important opportunity for exploring the limits of the Standard Model of particle physics and for constraining possible extensions that go beyond it. As the LHC explores a new energy frontier and as experiments continue to extend the precision frontier, the importance of flavour physics will grow, both in terms of searches for signatures of new physics through precision measurements and in terms of attempts to construct the theoretical framework behind direct discoveries of new particles. Crucial to such searches for new physics is the ability to quantify stronginteraction effects. Largescale numerical simulations of lattice QCD allow for the computation of these effects from first principles. The scope of the Flavour Lattice Averaging Group (FLAG) is to review the current status of lattice results for a variety of physical quantities that are important for flavour physics. Set up in November 2007, it comprises experts in Lattice Field Theory, Chiral Perturbation Theory and Standard Model phenomenology. Our aim is to provide an answer to the frequently posed question What is currently the best lattice value for a particular quantity?" in a way that is readily accessible to those who are not expert in lattice methods. This is generally not an easy question to answer; different collaborations use different lattice actions (discretizations of QCD) with a variety of lattice spacings and volumes, and with a range of masses for the $u$ and $d$quarks. Not only are the systematic errors different, but also the methodology used to estimate these uncertainties varies between collaborations. In the present work, we summarize the main features of each of the calculations and provide a framework for judging and combining the different results. Sometimes it is a single result that provides the best" value; more often it is a combination of results from different collaborations. Indeed, the consistency of values obtained using different formulations adds significantly to our confidence in the results.
The first three editions of the FLAG review were made public in 2010 ^{[1]}, 2013 ^{[2]}, and 2016 ^{[3]} (and will be referred to as FLAG 10, FLAG 13 and FLAG 16, respectively). The third edition reviewed results related to both light ($u$, $d$ and $s$), and heavy ($c$ and $b$) flavours. The quantities related to pion and kaon physics were lightquark masses, the form factor $f_+(0)$ arising in semileptonic $K \rightarrow \pi$ transitions (evaluated at zero momentum transfer), the decay constants $f_K$ and $f_\pi$, the $B_K$ parameter from neutral kaon mixing, and the kaon mixing matrix elements of new operators that arise in theories of physics beyond the Standard Model. Their implications for the CKM matrix elements $V_{us}$ and $V_{ud}$ were also discussed. Furthermore, results were reported for some of the lowenergy constants of $SU(2)_L \times SU(2)_R$ and $SU(3)_L \times SU(3)_R$ Chiral Perturbation Theory. The quantities related to $D$ and $B$meson physics that were reviewed were the masses of the charm and bottom quarks together with the decay constants, form factors, and mixing parameters of $B$ and $D$mesons. These are the heavylight quantities most relevant to the determination of CKM matrix elements and the global CKM unitaritytriangle fit. Last but not least, the current status of lattice results on the QCD coupling $\alpha_s$ was reviewed.
In the present paper we provide updated results for all the abovementioned quantities, but also extend the scope of the review by adding a section on nucleon matrix elements. This presents results for matrix elements of flavor nonsinglet and singlet bilinear operators, including the nucleon axial charge $g_A$ and the nucleon sigma terms. These results are relevant for constraining $V_{ud}$, for searches for new physics in neutron decays and other processes, and for dark matter searches. In addition, the section on up and down quark masses has been largely rewritten, replacing previous estimates for $m_u$, $m_d$, and the mass ratios $R$ and $Q$ that were largely phenomenological with those from lattice QED+QCD calculations. We have also updated the discussion of the phenomenology of isospinbreaking effects in the light meson sector, and their relation to quark masses, with a latticecentric discussion. A short review of QED in latticeQCD simulations is also provided, including a discussion of ambiguities arising when attempting to define physical quantities in pure QCD.
Our main results are collected in Tables 1, 2 and 3. As is clear from the tables, for most quantities there are results from ensembles with different values for $N_f$. In most cases, there is reasonable agreement among results with $N_f=2$, $2+1$, and $2+1+1$. As precision increases, we may some day be able to distinguish among the different values of $N_f$, in which case, presumably $2+1+1$ would be the most realistic. (If isospin violation is critical, then $1+1+1$ or $1+1+1+1$ might be desired.) At present, for some quantities the errors in the $N_f=2+1$ results are smaller than those with $N_f=2+1+1$ (e.g., for $m_c$), while for others the relative size of the errors is reversed. Our suggestion to those using the averages is to take whichever of the $N_f=2+1$ or $N_f=2+1+1$ results has the smaller error. We do not recommend using the $N_f=2$ results, except for studies of the $N_f$dependence of condensates and $\alpha_s$, as these have an uncontrolled systematic error coming from quenching the strange quark.
Table 1 

Summary of the main results of this review concerning quark masses, lightmeson decay constants, LECs, and kaon mixing parameters. These are grouped in terms of $N_f$, the number of dynamical quark flavours in lattice simulations. Quark masses and the quark condensate are given in the $\overline{\rm MS}$ scheme at running scale $\mu=2\,$GeV, unless otherwise indicated. BSM bag parameters $B_{2,3,4,5}$ are given in the $\overline{\rm MS}$ scheme at scale $\mu=3\,$GeV. Further specifications of the quantities are given in the quoted sections. Results for $N_f = 2$ quark masses are unchanged since FLAG 16^{[3]}. For each result we provide the list of references that entered the FLAG average or estimate in the bibtex file for download. We recommend consulting the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors. 

Quantity 
Sec. 
$N_f=2+1+1$ 
$N_f=2+1$ 
$N_f=2$ 
Refs. 
$m_{ud}$ [MeV] 
3.410(43) 
3.364(41) 


$m_s$ [MeV] 
93.44(68) 
92.03(88) 


$m_s/m_{ud}$ 
27.23(10) 
27.42(12) 


$m_u$ [MeV] 
2.50(17) 
2.27(9) 


$m_d$ [MeV] 
4.88(20) 
4.67(9) 


$m_u/m_d$ 
0.513(31) 
0.485(19) 


$\overline{m}_c(3\;{\rm GeV})$ [GeV] 
0.988(7) 
0.992(6) 


$m_c/m_s$ 
11.768(33) 
11.82(16) 


$\overline{m}_b(\overline{m}_b)$ [GeV] 
4.198 (12) 
4.164 (23) 


$f_+(0)$ 
0.9698(17) 
0.9677(27) 
0.9560(57)(62) 

$f_{K^\pm}/f_{\pi^\pm}$ 
1.1932(21) 
1.1917(37) 
1.205(18) 

$f_{\pi^\pm}$ [MeV] 

130.2(8) 


$f_{K^\pm}$ [MeV] 
155.7(3) 
155.7(7) 
157.5(2.4) 

$\Sigma^{1/3}$ [MeV] 
286(23) 
272(5) 
266(10) 

$F_\pi/F$ [MeV] 
1.077(3) 
1.062(7) 
1.073(15) 

$\bar{\ell}_3$ [MeV] 
3.53(26) 
3.07(64) 
3.41(82) 

$\bar{\ell}_4$ [MeV] 
4.73(10) 
4.02(45) 
4.40(28) 

$\bar{\ell}_6$ [MeV] 


15.1(1.2) 

$\hat B_K$ 
0.717(18)(16) 
0.7625(97) 
0.727(22)(12) 

$B_2$ 
0.46(1)(3) 
0.502(14) 
0.47(2)(1) 

$B_3$ 
0.79(2)(4) 
0.766(32) 
0.78(4)(2) 

$B_4$ 
0.78(2)(4) 
0.926(19) 
0.76(2)(2) 

$B_5$ 
0.49(3)(3) 
0.720(38) 
0.58(2)(2) 
Table 2 

Summary of the main results of this review concerning heavylight mesons and the strong coupling constant. These are grouped in terms of $N_f$, the number of dynamical quark flavours in lattice simulations. The quantities listed are specified in the quoted sections. For each result we provide the list of references that entered the FLAG average or estimate in the bibtex file for download. We recommend consulting the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors. 

Quantity 
Sec. 
$N_f=2+1+1$ 
$N_f=2+1$ 
$N_f=2$ 
Refs. 
$f_D$ [MeV] 
212.0(7) 
209.0(2.4) 
208(7) 

$f_{D_s}$ [MeV] 
249.9(5) 
248.0(1.6) 
242.5(5.8) 

$f_{D_s}/f_D$ 
1.1783(16) 
1.174(7) 
1.20(2) 

$f_+^{D\pi}(0)$ [MeV] 
0.612(35) 
0.666(29) 


$f_+^{DK}(0)$ [MeV] 
0.765(31) 
0.747(19) 


$f_B$ [MeV] 
190.0(1.3) 
192.0(4.3) 
188(7) 

$f_{B_s}$ [MeV] 
230.3(1.3) 
228.4(3.7) 
227(7) 

$f_{B_s}/f_B$ 
1.209(5) 
1.201(16) 
1.206(23) 

$f_{B_d}\sqrt{\hat{B}_{B_d}}$ [MeV] 

225(9) 
216(10) 

$f_{B_s}\sqrt{\hat{B}_{B_s}}$ [MeV] 

274(8) 
262(10) 

$\hat{B}_{B_d}$ 

1.30(10) 
1.30(6) 

$\hat{B}_{B_s}$ 

1.35(6) 
1.32(5) 

$\xi$ 

1.206(17) 
1.225(31) 

$B_{B_s}/B_{B_d}$ 

1.032(38) 
1.007(21) 

$\alpha^{(5)}_{\overline{\text{MS} } }(M_Z)$ 
0.1182(8) 


$\Lambda^{(5)}_{\overline{\text{MS} } }$ [MeV] 
211(10) 


$\Lambda^{(4)}_{\overline{\text{MS} } }$ [MeV] 
294(12) 


$\Lambda^{(3)}_{\overline{\text{MS} } }$ [MeV] 
343(12) 

Table 3 

Summary of the main results of this review concerning nuclear matrix elements, grouped in terms of $N_f$, the number of dynamical quark flavours in lattice simulations. The quantities listed are specified in the quoted sections. For each result we provide the list of references that entered the FLAG average or estimate in the bibtex file for download. We recommend consulting the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors. 

Quantity 
Sec. 
$N_f=2+1+1$ 
$N_f=2+1$ 
$N_f=2$ 
Refs. 
$g_A^{ud}$ 
1.251(33) 
1.254(16)(30) 
1.278(86) 

$g_S^{ud}$ 
1.022(80)(60) 



$g_T^{ud}$ 
0.989(32)(10) 



$g_A^{u}$ 
0.777(25)(30) 
0.847(18)(32) 


$g_A^{d}$ 
−0.438(18)(30) 
−0.407(16)(18) 


$g_A^{s}$ 
−0.053(8) 
−0.035(6)(7)(32) 


$\sigma_{\pi N}$ [MeV] 
64.9(1.5)(13.2) 
39.7(3.6) 
37(8)(6) 

$\sigma_{s}$ [MeV] 
41.0(8.8) 
52.9(7.0) 


$g_T^{u}$ 
0.784(28)(10) 



$g_T^{d}$ 
−0.204(11)(10) 



$g_T^{s}$ 
−0.027(16) 


Our plan is to continue providing FLAG updates, in the form of a peer reviewed paper, roughly on a triennial basis. This effort is supplemented by our more frequently updated website (this page), where figures as well as pdffiles for the individual sections can be downloaded. The papers reviewed in the present edition have appeared before the closing date 30 September 2018. ^{[4]}
FLAG composition, guidelines and rules
FLAG strives to be representative of the lattice community, both in terms of the geographical location of its members and the lattice collaborations to which they belong. We aspire to provide the nuclear and particlephysics communities with a single source of reliable information on lattice results.
In order to work reliably and efficiently, we have adopted a formal structure and a set of rules by which all FLAG members abide. The collaboration presently consists of an Advisory Board (AB), an Editorial Board (EB), and eight Working Groups (WG). The role of the Advisory Board is to provide oversight of the content, procedures, schedule and membership of FLAG, to help resolve disputes, to serve as a source of advice to the EB and to FLAG as a whole, and to provide a critical assessment of drafts. They also give their approval of the final version of the preprint before it is rendered public. The Editorial Board coordinates the activities of FLAG, sets priorities and intermediate deadlines, organizes votes on FLAG procedures, writes the introductory sections, and takes care of the editorial work needed to amalgamate the sections written by the individual working groups into a uniform and coherent review. The working groups concentrate on writing the review of the physical quantities for which they are responsible, which is subsequently circulated to the whole collaboration for critical evaluation.
The current list of FLAG members and their Working Group assignments is:
Advisory Board (AB) 
S. Aoki, M. Golterman, R. Van de Water, T. Vladikas 

Editorial Board (EB) 
G. Colangelo, A. Jüttner, S. Hashimoto, S. Sharpe, U. Wenger 

Working Groups (WG) 

Quark masses 
T. Blum, A. Portelli, A. Ramos 

$V_{us}, V_{ud}$ 
S. Simula, T. Kaneko, J.N. Simone 

LEC 
S. Dürr, H. Fukaya, U.M. Heller 

$B_K$ 
P. Dimopoulos, G. Herdoiza, R. Mawhinney 

$f_{B_{(s)}}$, $f_{D_{(s)}}$, $B_B$ 
D. Lin, Y. Aoki, M. Della Morte 

$B_{(s)}$, $D$ semileptonic and radiative decays 
E. Lunghi, D. Becirevic, S. Gottlieb, C. Pena 

$\alpha_s$ 
R. Sommer, R. Horsley, T. Onogi 

Nucleon Matrix Elements (NME) 
R. Gupta, S. Collins, A. Nicholson, H. Wittig 
The most important FLAG guidelines and rules are the following:
 the composition of the AB reflects the main geographical areas in which lattice collaborations are active, with members from America, Asia/Oceania, and Europe;
 the mandate of regular members is not limited in time, but we expect that a certain turnover will occur naturally;
 whenever a replacement becomes necessary this has to keep, and possibly improve, the balance in FLAG, so that different collaborations, from different geographical areas are represented;
in all working groups the three members must belong to three different lattice collaborations;^{[5]} ^{[6]}
 a paper is in general not reviewed (nor colourcoded, as described in the next section) by any of its authors;
 lattice collaborations will be consulted on the colour coding of their calculation;
 there are also internal rules regulating our work, such as voting procedures.
For this edition of the FLAG review, we sought the advice of external reviewers once a complete draft of the review was available. For each review section, we have asked one lattice expert (who could be a FLAG alumnus/alumna) and one nonlattice phenomenologist for a critical assessment. This is similar to the procedure followed by the Particle Data Group in the creation of the Review of Particle Physics. The reviewers provide comments and feedback on scientific and stylistic matters. They are not anonymous, and enter into a discussion with the authors of the WG. Our aim with this additional step is to make sure that a wider array of viewpoints enter into the discussions, so as to make this review more useful for its intended audience.
Citation policy
We draw attention to this particularly important point. As stated above, our aim is to make latticeQCD results easily accessible to those without lattice expertise, and we are well aware that it is likely that some readers will only consult the present paper and not the original lattice literature. It is very important that this paper not be the only one cited when our results are quoted. We strongly suggest that readers also cite the original sources. In order to facilitate this, in Tables 1, 2 and 3, besides summarizing the main results of the present review, we also provide here a bibtex file containing the original references from which the results have been obtained. In addition, for each figure, we make a corresponding bibtex file available. The bibliography at the end of this paper should also make it easy to cite additional papers. Indeed, we hope that the bibliography will be one of the most widely used elements of the whole paper.
General issues
Several general issues concerning the present review are thoroughly discussed in Sec.~1.1 of our initial 2010 paper ^{[1]}, and we encourage the reader to consult the relevant pages. In the remainder of the present subsection, we focus on a few important points. Though the discussion has been duly updated, it is similar to that of Sec.~1.2 in the previous two reviews ^{[2]}^{[3]}, with the addition of comments on the contributions from excited states that are particularly relevant for the new section on NMEs.
The present review aims to achieve two distinct goals: first, to provide a description of the relevant work done on the lattice; and, second, to draw conclusions on the basis of that work, summarizing the results obtained for the various quantities of physical interest.
The core of the information about the work done on the lattice is presented in the form of tables, which not only list the various results, but also describe the quality of the data that underlie them. We consider it important that this part of the review represents a generally accepted description of the work done. For this reason, we explicitly specify the quality requirements^{[7]} used and provide sufficient details in appendices so that the reader can verify the information given in the tables.
On the other hand, the conclusions drawn on the basis of the available lattice results are the responsibility of FLAG alone. Preferring to err on the side of caution, in several cases we draw conclusions that are more conservative than those resulting from a plain weighted average of the available lattice results. This cautious approach is usually adopted when the average is dominated by a single lattice result, or when only one lattice result is available for a given quantity. In such cases, one does not have the same degree of confidence in results and errors as when there is agreement among several different calculations using different approaches. The reader should keep in mind that the degree of confidence cannot be quantified, and it is not reflected in the quoted errors.
Each discretization has its merits, but also its shortcomings. For most topics covered in this review we have an increasingly broad database, and for most quantities lattice calculations based on totally different discretizations are now available. This is illustrated by the dense population of the tables and figures in most parts of this review. Those calculations that do satisfy our quality criteria indeed lead, in almost all cases, to consistent results, confirming universality within the accuracy reached. In our opinion, the consistency between independent lattice results, obtained with different discretizations, methods, and simulation parameters, is an important test of lattice QCD, and observing such consistency also provides further evidence that systematic errors are fully under control.
In the sections dealing with heavy quarks and with $\alpha_s$, the situation is not the same. Since the $b$quark mass can barely be resolved with current lattice spacings, all lattice methods for treating $b$ quarks use effective field theory at some level. This introduces additional complications not present in the lightquark sector. An overview of the issues specific to heavyquark quantities is given in the introduction of Sec. 8. For $B$ and $D$ meson leptonic decay constants, there already exists a good number of different independent calculations that use different heavyquark methods, but there are only one or two independent calculations of semileptonic $B$ and $D$ meson form factors and $B$ meson mixing parameters. For $\alpha_s$, most lattice methods involve a range of scales that need to be resolved and controlling the systematic error over a large range of scales is more demanding. The issues specific to determinations of the strong coupling are summarized in Sec. 9.
Number of sea quarks in lattice simulations:
LatticeQCD simulations currently involve two, three or four flavours of dynamical quarks. Most simulations set the masses of the two lightest quarks to be equal, while the strange and charm quarks, if present, are heavier (and tuned to lie close to their respective physical values). Our notation for these simulations indicates which quarks are nondegenerate, e.g., $N_f=2+1$ if $m_u=m_d < m_s$ and $N_f =2+1+1$ if $m_u=m_d < m_s < m_c$. Calculations with $N_f =2$, i.e., two degenerate dynamical flavours, often include strange valence quarks interacting with gluons, so that bound states with the quantum numbers of the kaons can be studied, albeit neglecting strange seaquark fluctuations. The quenched approximation ($N_f=0$), in which all sea quark contributions are omitted, has uncontrolled systematic errors and is no longer used in modern lattice simulations with relevance to phenomenology. Accordingly, we will review results obtained with $N_f=2$, $N_f=2+1$, and $N_f = 2+1+1$, but omit earlier results with $N_f=0$. The only exception concerns the QCD coupling constant $\alpha_s$. Since this observable does not require valence light quarks, it is theoretically well defined also in the $N_f=0$ theory, which is simply pure gluodynamics. The $N_f$dependence of $\alpha_s$, or more precisely of the related quantity $r_0 \Lambda_{\overline{\rm MS}}$, is a theoretical issue of considerable interest; here $r_0$ is a quantity with the dimension of length that sets the physical scale, as discussed in Appendix A.2 in the Glossary. We stress, however, that only results with $N_f \ge 3$ are used to determine the physical value of $\alpha_s$ at a high scale.
Lattice actions, simulation parameters and scale setting:
The remarkable progress in the precision of lattice calculations is due to improved algorithms, better computing resources, and, last but not least, conceptual developments. Examples of the latter are improved actions that reduce lattice artifacts and actions that preserve chiral symmetry to very good approximation. A concise characterization of the various discretizations that underlie the results reported in the present review is given in Appendix A.1 in the Glossary.
Physical quantities are computed in lattice simulations in units of the lattice spacing so that they are dimensionless. For example, the pion decay constant that is obtained from a simulation is $f_\pi a$, where $a$ is the spacing between two neighboring lattice sites. (All simulations with results quoted in this review use hypercubic lattices, i.e., with the same spacing in all four Euclidean directions.) To convert these results to physical units requires knowledge of the lattice spacing $a$ at the fixed values of the bare QCD parameters (quark masses and gauge coupling) used in the simulation. This is achieved by requiring agreement between the lattice calculation and experimental measurement of a known quantity, which thus sets the scale" of a given simulation. A few details on this procedure are provided in Appendix A.2 in the Glossary.
Renormalization and scheme dependence:
Several of the results covered by this review, such as quark masses, the gauge coupling, and $B$parameters, are for quantities defined in a given renormalization scheme and at a specific renormalization scale. The schemes employed (e.g., regularizationindependent MOM schemes) are often chosen because of their specific merits when combined with the lattice regularization. For a brief discussion of their properties, see Appendix A.3 in the Glossary. The conversion of the results obtained in these socalled intermediate schemes to more familiar regularization schemes, such as the $\overline{\rm MS}$scheme, is done with the aid of perturbation theory. It must be stressed that the renormalization scales accessible in simulations are limited, because of the presence of an ultraviolet (UV) cutoff of $\sim \pi/a$. To safely match to $\overline{\rm MS}$, a scheme defined in perturbation theory, Renormalization Group (RG) running to higher scales is performed, either perturbatively or nonperturbatively (the latter using finitesize scaling techniques).
Extrapolations:
Because of limited computing resources, lattice simulations are often performed at unphysically heavy pion masses, although results at the physical point have become increasingly common. Further, numerical simulations must be done at nonzero lattice spacing, and in a finite (fourdimensional) volume. In order to obtain physical results, lattice data are obtained at a sequence of pion masses and a sequence of lattice spacings, and then extrapolated to the physical pion mass and to the continuum limit. In principle, an extrapolation to infinite volume is also required. However, for most quantities discussed in this review, finitevolume effects are exponentially small in the linear extent of the lattice in units of the pion mass, and, in practice, one often verifies volume independence by comparing results obtained on a few different physical volumes, holding other parameters fixed. To control the associated systematic uncertainties, these extrapolations are guided by effective theories. For lightquark actions, the latticespacing dependence is described by Symanzik's effective theory; for heavy quarks, this can be extended and/or supplemented by other effective theories such as HeavyQuark Effective Theory (HQET). The pionmass dependence can be parameterized with Chiral Perturbation Theory ($\chi$PT), which takes into account the NambuGoldstone nature of the lowest excitations that occur in the presence of light quarks. Similarly, one can use HeavyLight Meson Chiral Perturbation Theory (HM$\chi$PT) to extrapolate quantities involving mesons composed of one heavy ($b$ or $c$) and one light quark. One can combine Symanzik's effective theory with $\chi$PT to simultaneously extrapolate to the physical pion mass and the continuum; in this case, the form of the effective theory depends on the discretization. See Appendix A.4 in the Glossary for a brief description of the different variants in use and some useful references. Finally, $\chi$PT can also be used to estimate the size of finitevolume effects measured in units of the inverse pion mass, thus providing information on the systematic error due to finitevolume effects in addition to that obtained by comparing simulations at different volumes.
Excitedstate Contamination:
In all the hadronic matrix elements discussed in this review, the hadron in question is the lightest state with the chosen quantum numbers. This implies that it dominates the required correlation functions as their extent in Euclidean time is increased. Excitedstate contributions are suppressed by $e^{\Delta E \Delta \tau}$, where $\Delta E$ is the gap between the ground and excited states, and $\Delta \tau$ the relevant separation in Euclidean time. The size of $\Delta E$ depends on the hadron in question, and in general is a multiple of the pion mass. In practice, as discussed at length in Sec. 10, the contamination of signals due to excitedstate contributions is a much more challenging problem for baryons than for the other particles discussed here. This is in part due to the fact that the signaltonoise ratio drops exponentially for baryons, which reduces the values of $\Delta \tau$ that can be used.
Critical slowing down:
The lattice spacings reached in recent simulations go down to 0.05 fm or even smaller. In this regime, long autocorrelation times slow down the sampling of the configurations. Many groups check for autocorrelations in a number of observables, including the topological charge, for which a rapid growth of the autocorrelation time is observed with decreasing lattice spacing. This is often referred to as topological freezing. A solution to the problem consists in using open boundary conditions in time, instead of the more common antiperiodic ones. More recently two other approaches have been proposed, one based on a multiscale thermalization algorithm and another based on defining QCD on a nonorientable manifold. The problem is also touched upon in Sec. 9.2, where it is stressed that attention must be paid to this issue. While large scale simulations with open boundary conditions are already far advanced, only one result reviewed here has been obtained with any of the above methods. It is usually assumed that the continuum limit can be reached by extrapolation from the existing simulations, and that potential systematic errors due to the long autocorrelation times have been adequately controlled. Partially or completely frozen topology would produce a mixture of different $\theta$ vacua, and the difference from the desired $\theta=0$ result may be estimated in some cases using chiral perturbation theory, which gives predictions for the $\theta$dependence of the physical quantity of interest. A numerical test on MILC ensembles indicates that the topology dependence for some of the physical quantities reviewed here is small, consistent with theoretical expectations.
Simulation algorithms and numerical errors:
Most of the modern latticeQCD simulations use exact algorithms, which do not produce any systematic errors when exact arithmetic is available. In reality, one uses numerical calculations at double (or in some cases even single) precision, and some errors are unavoidable. More importantly, the inversion of the Dirac operator is carried out iteratively and it is truncated once some accuracy is reached, which is another source of potential systematic error. In most cases, these errors have been confirmed to be much less than the statistical errors. In the following we assume that this source of error is negligible. Some of the most recent simulations use an inexact algorithm in order to speed up the computation, though it may produce systematic effects. Currently available tests indicate that errors from the use of inexact algorithms are under control.
References
G. Colangelo et al., Review of lattice results concerning low energy particle physics, Eur. Phys. J. C71 (2011) 1695, arXiv:1011.4408 (1 2)
S. Aoki et al., Review of lattice results concerning lowenergy particle physics, Eur. Phys. J. C74 (2014) 2890, arXiv:1310.8555. (1 2)
S. Aoki et al., Review of lattice results concerning lowenergy particle physics, Eur. Phys. J. C77 (2017) 112, arXiv:1607.00299. (1 2 3)
Working groups were given the option of including papers submitted to arXiv.org before the closing date but published after this date. This flexibility allows this review to be uptodate at the time of submission. Three papers of this type were included: Ref. [5] in Secs. 7 and 8, and Refs. [6] and [7] in Sec. 10. (1)
The WG on semileptonic D and B decays has currently four members, but only three of them belong to lattice collaborations. (1)
The NME WG, new in this addition of the FLAG review, has been formed with four members (all members of lattice collaborations) rather than three. This reflects the large amount of work needed to create a section for which some of the systematic errors are substantially different from those described in other sections, and to provide a better representation of relevant collaborations. (1)
We also use terms like “quality criteria”, “rating”, “colour coding” etc. when referring to the classification of results, as described in Sec. 2. (1)
Quality criteria
The section on the quality criteria can be downloaded here.
Quark masses
The section on the quark masses can be downloaded here.
$\small{V_{ud}}$ and $\small{V_{us}}$
The section on $V_{ud}$ and $V_{us}$ can be downloaded here.
Lowenergy constants
The section on the lowenergy constants can be downloaded here.
Kaon mixing
The section on the kaon mixing can be downloaded here.
$\small{D}$meson decay constants and form factors
The section on the $D$meson decay constants and form factors can be downloaded here.
$\small{B}$meson decay constants, mixing parameters, and form factors
The section on the $B$meson decay constants, mixing parameters, and form factors can be downloaded here.
The strong coupling $\alpha_s$
The section on the strong coupling $\alpha_s$ can be downloaded here.
Nucleon matrix elements
The section on nucleon matrix elements can be downloaded here.
Glossary
The glossary can be downloaded here.
Notes
Notes to the various sections can be downloaded here. A reduced version containing only the notes on calculations which are new w.r.t. FLAG 13 is available here.
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