Review of lattice results concerning lowenergy particle physics
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Abstract
We review lattice results related to pion, kaon, D and Bmeson physics with the aim of making them easily accessible to the particlephysics community. More specifically, we report on the determination of the lightquark masses, the form factor \(f_+(0)\), arising in the semileptonic \(K \rightarrow \pi \) 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. The latter quantities are an addition compared to the previous review. For the heavyquark sector, we provide results for \(m_c\) and \(m_b\) (also new compared to the previous review), as well as those for D and Bmesondecay 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. Finally, we review the status of lattice determinations of the strong coupling constant \(\alpha _s\).
1 Introduction
Summary of the main results of this review, 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 \({\overline{\text {MS}}}\) scheme at running scale \(\mu =2~\mathrm{GeV}\) or as indicated; the other quantities listed are specified in the quoted sections. For each result we list the references that entered the FLAG average or estimate. From the entries in this column one can also read off the number of results that enter our averages for each quantity. We emphasize that these numbers only give a very rough indication of how thoroughly the quantity in question has been explored on the lattice and recommend to consult the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors
Quantity  Sects.  \(N_f=2+1+1\)  Refs.  \(N_f=2+1\)  Refs.  \(N_f=2\)  Refs. 

\( m_s\) [MeV]  93.9(1.1)  92.0(2.1)  101(3)  
\( m_{ud}\) [MeV]  3.70(17)  [4]  3.373(80)  3.6(2)  [11]  
\( m_s / m_{ud} \)  27.30(34)  27.43(31)  27.3(9)  [11]  
\( m_u \) [MeV]  2.36(24)  [4]  2.16(9)(7)  \({}^{\mathrm{a}}\)  2.40(23)  [16]  
\( m_d \) [MeV]  5.03(26)  [4]  4.68(14)(7)  \({}^{\mathrm{a}}\)  4.80(23)  [16]  
\( {m_u}/{m_d} \)  0.470(56)  [4]  0.46(2)(2)  \({}^{\mathrm{a}}\)  0.50(4)  [16]  
\(\overline{m}_c(3~\hbox {GeV})\) [GeV]  0.996(25)  0.987(6)  1.03(4)  [11]  
\( m_c / m_s \)  11.70(6)  11.82(16)  11.74(35)  
\(\overline{m}_b(\overline{m}_b)\) [GeV]  4.190(21)  4.164(23)  [9]  4.256(81)  
\( f_+(0) \)  0.9704(24)(22)  [22]  0.9677(27)  0.9560(57)(62)  [25]  
\( f_{K^\pm } / f_{\pi ^\pm } \)  1.193(3)  1.192(5)  1.205(6)(17)  [32]  
\( f_{\pi ^\pm }\) [MeV]  130.2(1.4)  
\( f_{K^\pm } \) [MeV]  155.6(4)  155.9(9)  157.5(2.4)  [32]  
\( \Sigma ^{1/3}\) [MeV]  280(8)(15)  [33]  274(3)  266(10)  
\( {F_\pi }/{F}\)  1.076(2)(2)  [39]  1.064(7)  1.073(15)  
\( \bar{\ell }_3\)  3.70(7)(26)  [39]  2.81(64)  3.41(82)  
\( \bar{\ell }_4\)  4.67(3)(10)  [39]  4.10(45)  4.51(26)  
\( \bar{\ell }_6\)  15.1(1.2)  
\(\hat{B}_\mathrm{{K}} \)  0.717(18)(16)  [42]  0.7625(97)  0.727(22)(12)  [46] 
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. A major theoretical limitation consists in the precision with which stronginteraction effects can be quantified. 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 in lowenergy 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 nonlatticeexperts. 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 dquarks. 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 two editions of the FLAG review were published in 2011 [1] and 2014 [2]. The second 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 \(f_\pi \), and the \(B_\mathrm{K}\) parameter from neutral kaon mixing. 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 Bmeson physics that were reviewed were the B and Dmesondecay constants, form factors, and mixing parameters. These are the heavy–light 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 in two ways. First, we now present results for the charm and bottom quark masses, in addition to those of the three lightest quarks. Second, we review results obtained for the kaon mixing matrix elements of new operators that arise in theories of physics beyond the Standard Model. Our main results are collected in Tables 1 and 2.
Summary of the main results of this review, 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 list the references that entered the FLAG average or estimate. From the entries in this column one can also read off the number of results that enter our averages for each quantity. We emphasize that these numbers only give a very rough indication of how thoroughly the quantity in question has been explored on the lattice and recommend to consult the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors
Quantity  Sects.  \(N_f=2+1+1\)  Refs.  \(N_f=2+1\)  Refs.  \(N_f=2\)  Refs. 

\( f_D\) [MeV]  212.15(1.45)  209.2(3.3)  208(7)  [20]  
\( f_{D_s}\) [MeV]  248.83(1.27)  249.8(2.3)  250(7)  [20]  
\( {{f_{D_s}}/{f_D}}\)  1.1716(32)  1.187(12)  1.20(2)  [20]  
\( f_+^{D\pi }(0)\)  0.666(29)  [50]  
\( f_+^{DK}(0) \)  0.747(19)  [51]  
\( f_B\) [MeV]  186(4)  [52]  192.0(4.3)  188(7)  
\( f_{B_s}\) [MeV]  224(5)  [52]  228.4(3.7)  227(7)  
\( {{f_{B_s}}/{f_B}}\)  1.205(7)  [52]  1.201(16)  1.206(23)  
\( f_{B_d}\sqrt{\hat{B}_{B_d}} \) [MeV]  219(14)  216(10)  [20]  
\( f_{B_s}\sqrt{\hat{B}_{B_s}} \) [MeV]  270(16)  262(10)  [20]  
\( \hat{B}_{B_d} \)  1.26(9)  1.30(6)  [20]  
\( \hat{B}_{B_s} \)  1.32(6)  1.32(5)  [20]  
\( \xi \)  1.239(46)  1.225(31)  [20]  
\( B_{B_s}/B_{B_d} \)  1.039(63)  1.007(21)  [20] 
Quantity  Sects.  \(N_f=2+1\) and \(N_f=2+1+1\)  Refs.  

\( \alpha _{\overline{\mathrm{MS}}}^{(5)}(M_Z) \)  0.1182(12)  
\( \Lambda _{\overline{\mathrm{MS}}}^{(5)} \) [MeV]  211(14) 
This review is organized as follows. In the remainder of Sect. 1 we summarize the composition and rules of FLAG and discuss general issues that arise in modern lattice calculations. In Sect. 2 we explain our general methodology for evaluating the robustness of lattice results. We also describe the procedures followed for combining results from different collaborations in a single average or estimate (see Sect. 2.2 for our definition of these terms). The rest of the paper consists of sections, each dedicated to a single (or groups of closely connected) physical quantity(ies). Each of these sections is accompanied by an Appendix with explicatory notes.
1.1 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 particlephysics community 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 seven Working Groups (WG). The rôle of the Advisory Board is that of general supervision and consultation. Its members may interfere at any point in the process of drafting the paper, expressing their opinion and offering advice. 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, 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 up the review of the physical quantities for which they are responsible, which is subsequently circulated to the whole collaboration for critical evaluation.

Advisory Board (AB): S. Aoki, C. Bernard, M. Golterman, H. Leutwyler, and C. Sachrajda

Editorial Board (EB): G. Colangelo, A. Jüttner, S. Hashimoto, S. Sharpe, A. Vladikas, and U. Wenger
 Working Groups (coordinator listed first):

Quark masses L. Lellouch, T. Blum, and V. Lubicz

\(V_{us},V_{ud}\) S. Simula, P. Boyle,^{1} and T. Kaneko

LEC S. Dürr, H. Fukaya, and U.M. Heller

\(B_K\) H. Wittig, P. Dimopoulos, and R. Mawhinney

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

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

\(\alpha _s\) R. Sommer, R. Horsley, and T. Onogi


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;^{2}

a paper is in general not reviewed (nor colourcoded, as described in the next section) by any of its authors;

lattice collaborations not represented in FLAG will be consulted on the colour coding of their calculation;

there are also internal rules regulating our work, such as voting procedures.
1.2 Citation policy
We draw attention to this particularly important point. As stated above, our aim is to make lattice QCD results easily accessible to nonlatticeexperts 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 be not 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 and 2, besides summarizing the main results of the present review, we also cite the original references from which they have been obtained. In addition, for each figure we make a bibtexfile available on our webpage [3] which contains the bibtexentries of all the calculations contributing to the FLAG average or estimate. 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.
1.3 General issues
Several general issues concerning the present review are thoroughly discussed in Sect. 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 essentially that of Sect. 1.2 of the 2013 review [2].
The present review aims to achieve two distinct goals: first, to provide a description of the work done on the lattice concerning lowenergy particle physics; 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 as regards 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^{3} 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 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 bquark mass cannot 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 Sect. 8. For B and D meson leptonic decay constants, there already exist 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 Sect. 9.
Number of sea quarks in lattice simulations:
Lattice QCD 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 seaquark 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 gluondynamics. The \(N_f\)dependence of \(\alpha _s\), or more precisely of the related quantity \(r_0 \Lambda _{\overline{\text {MS}}}\), is a theoretical issue of considerable interest; here \(r_0\) is a quantity with the dimension of length, which sets the physical scale, as discussed in Appendix A.2. 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 artefacts 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.
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 neighbouring lattice sites. 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 of this procedure are provided in Appendix A.2.
Renormalization and scheme dependence:
Several of the results covered by this review, such as quark masses, the gauge coupling, and Bparameters, 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. The conversion of the results, obtained in these socalled intermediate schemes, to more familiar regularization schemes, such as the \({\overline{\text {MS}}}\)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 \({\overline{\text {MS}}}\), 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 physicalpion 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 equal. 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 [64, 65]; 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 Nambu–Goldstone 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 physicalpion mass and the continuum; in this case, the form of the effective theory depends on the discretization. See Appendix A.4 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.
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 [66, 67, 68, 69, 70, 71, 72, 73, 74, 75]. 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 [76]. More recently two other approaches have been proposed, one based on a multiscale thermalization algorithm [77] and another based on defining QCD on a nonorientable manifold [78]. The problem is also touched upon in Sect. 9.2, where it is stressed that attention must be paid to this issue. While largescale simulations with open boundary conditions are already far advanced [79], unfortunately so far no results reviewed here have 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.
Simulation algorithms and numerical errors:
Most of the modern latticeQCD simulations use exact algorithms such as those of Refs. [80, 81], 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 speedup the computation, though it may produce systematic effects. Currently available tests indicate that errors from the use of inexact algorithms are under control.
2 Quality criteria, averaging and error estimation
The essential characteristics of our approach to the problem of rating and averaging lattice quantities have been outlined in our first publication [1]. Our aim is to help the reader assess the reliability of a particular lattice result without necessarily studying the original article in depth. This is a delicate issue, since the ratings may make things appear simpler than they are. Nevertheless, it safeguards against the common practice of using lattice results, and drawing physics conclusions from them, without a critical assessment of the quality of the various calculations. We believe that, despite the risks, it is important to provide some compact information as regards the quality of a calculation. We stress, however, the importance of the accompanying detailed discussion of the results presented in the various sections of the present review.
2.1 Systematic errors and colour code
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the parameter values and ranges used to generate the datasets allow for a satisfactory control of the systematic uncertainties;
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the parameter values and ranges used to generate the datasets allow for a reasonable attempt at estimating systematic uncertainties, which, however, could be improved;
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the parameter values and ranges used to generate the datasets are unlikely to allow for a reasonable control of systematic uncertainties.
The attentive reader will notice that these criteria differ from those used in Refs. [1, 2]. In the previous FLAG editions we used the three symbols in order to rate the reliability of the systematic errors attributed to a given result by the paper’s authors. This sometimes proved to be a daunting task, as the methods used by some collaborations for estimating their systematics are not always explained in full detail. Moreover, it is sometimes difficult to disentangle and rate different uncertainties, since they are interwoven in the error analysis. Thus, in the present edition we have opted for a different approach: the three symbols rate the quality of a particular simulation, based on the values and range of the chosen parameters, and its aptness to obtain wellcontrolled systematic uncertainties. They do not rate the quality of the analysis performed by the authors of the publication. The latter question is deferred to the relevant sections of the present review, which contain detailed discussions of the results contributing (or not) to each FLAG average or estimate. As a result of this different approach to the rating criteria, as well as changes of the criteria themselves, the colour coding of some papers in the current FLAG version differs from that of Ref. [2].
For most quantities the colourcoding system refers to the following sources of systematic errors: (i) chiral extrapolation; (ii) continuum extrapolation; (iii) finite volume. As we will see below, renormalization is another source of systematic uncertainties in several quantities. This we also classify using the three coloured symbols listed above, but now with a different rationale: they express how reliably these quantities are renormalized, from a fieldtheoretic point of view (namely nonperturbatively, or with twoloop or oneloop perturbation theory).
Given the sophisticated status that the field has attained, several aspects, besides those rated by the coloured symbols, need to be evaluated before one can conclude whether a particular analysis leads to results that should be included in an average or estimate. Some of these aspects are not so easily expressible in terms of an adjustable parameter such as the lattice spacing, the pion mass or the volume. As a result of such considerations, it sometimes occurs, albeit rarely, that a given result does not contribute to the FLAG average or estimate, despite not carrying any red tags. This happens, for instance, whenever aspects of the analysis appear to be incomplete (e.g. an incomplete error budget), so that the presence of inadequately controlled systematic effects cannot be excluded. This mostly refers to results with a statistical error only, or results in which the quoted error budget obviously fails to account for an important contribution.
Of course any colour coding has to be treated with caution; we emphasize that the criteria are subjective and evolving. Sometimes a single source of systematic error dominates the systematic uncertainty and it is more important to reduce this uncertainty than to aim for green stars for other sources of error. In spite of these caveats we hope that our attempt to introduce quality measures for lattice simulations will prove to be a useful guide. In addition we would like to stress that the agreement of lattice results obtained using different actions and procedures provides further validation.
2.1.1 Systematic effects and rating criteria
The precise criteria used in determining the colour coding are unavoidably timedependent; as lattice calculations become more accurate, the standards against which they are measured become tighter. For this reason, some of the quality criteria related to the lightquark sector have been tightened up between the first [1] and second [2] editions of FLAG.
In the second edition we have also reviewed quantities related to heavyquark physics [2]. The criteria used for light and heavyflavour quantities were not always the same. For the continuum limit, the difference was more a matter of choice: the lightflavour Working Groups defined the ratings using conditions involving specific values of the lattice spacing, whereas the heavyflavour Working Groups preferred more datadriven criteria. Also, for finitevolume effects, the heavyflavour groups slightly relaxed the boundary between Open image in new window and Open image in new window , compared to the lightquark case, to account for the fact that heavyquark quantities are less sensitive to the finiteness of the volume.

Chiral extrapolation:
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\(M_{\pi ,\mathrm {min}}< 200\) MeV
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200 MeV \(\le M_{\pi ,{\mathrm {min}}} \le \) 400 MeV
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400 MeV \( < M_{\pi ,\mathrm {min}}\)

Continuum extrapolation:
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at least three lattice spacings and at least 2 points below 0.1 fm and a range of lattice spacings satisfying \([a_{\mathrm {max}}/a_{\mathrm {min}}]^2 \ge 2\)
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at least two lattice spacings and at least 1 point below 0.1 fm and a range of lattice spacings satisfying \([a_{\mathrm {max}}/a_{\mathrm {min}}]^2 \ge 1.4\)
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otherwise

Finitevolume effects:
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\([M_{\pi ,\mathrm {min}} / M_{\pi ,\mathrm {fid}}]^2 \exp \{4M_{\pi ,\mathrm {min}}[L(M_{\pi ,\mathrm {min}})]_{\mathrm {max}}\} < 1\), or at least 3 volumes
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\([M_{\pi ,\mathrm {min}} / M_{\pi ,\mathrm {fid}}]^2 \exp \{3M_{\pi ,\mathrm {min}}[L(M_{\pi ,\mathrm {min}})]_{\mathrm {max}}\} < 1\), or at least 2 volumes
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otherwise

It is assumed here that calculations are in the pregime^{5} of chiral perturbation theory, and that all volumes used exceed 2 fm. Here we are using a more sophisticated condition than that of Ref. [2]. The new condition involves the quantity \([L(M_{\pi ,\mathrm {min}})]_{\mathrm {max}}\), which is the maximum box size used in the simulations performed at smallest pion mass \(M_{\pi ,\mathrm {min}}\), as well as a fiducial pion mass \(M_{\pi ,\mathrm {fid}}\), which we set to 200 MeV (the cutoff value for a green star in the chiral extrapolation).
The rationale for this condition is as follows. Finitevolume effects contain the universal factor \(\exp \{ L~M_\pi \}\), and if this were the only contribution a criterion based on the values of \(M_{\pi ,\text {min}} L\) would be appropriate. This is what we used in Ref. [2] (with \(M_{\pi ,\text {min}} L>4\) for Open image in new window and \(M_{\pi ,\text {min}} L>3\) for Open image in new window ). However, as pion masses decrease, one must also account for the weakening of the pion couplings. In particular, oneloop chiral perturbation theory [82] reveals a behaviour proportional to \(M_\pi ^2 \exp \{ L~M_\pi \}\). Our new condition includes this weakening of the coupling and ensures, for example, that simulations with \(M_{\pi ,\mathrm {min}} = 135~\mathrm{MeV}\) and \(L~M_{\pi ,\mathrm {min}} = 3.2\) are rated equivalently to those with \(M_{\pi ,\mathrm {min}} = 200~\mathrm{MeV}\) and \(L~M_{\pi ,\mathrm {min}} = 4\).

Renormalization (where applicable):
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nonperturbative
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oneloop perturbation theory or higher with a reasonable estimate of truncation errors
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otherwise

In Ref. [1], we assigned a red square to all results which were renormalized at oneloop in perturbation theory. In Ref. [2] we decided that this was too restrictive, since the error arising from renormalization constants, calculated in perturbation theory at oneloop, is often estimated conservatively and reliably.

Renormalization Group (RG) running (where applicable):
For scaledependent quantities, such as quark masses or \(B_K\), it is essential that contact with continuum perturbation theory can be established. Various different methods are used for this purpose (cf. Appendix A.3): Regularizationindependent Momentum Subtraction (RI/MOM), the Schrödinger functional, and direct comparison with (resummed) perturbation theory. Irrespective of the particular method used, the uncertainty associated with the choice of intermediate renormalization scales in the construction of physical observables must be brought under control. This is best achieved by performing comparisons between nonperturbative and perturbative running over a reasonably broad range of scales. These comparisons were initially only made in the Schrödinger functional approach, but are now also being performed in RI/MOM schemes. We mark the data for which information as regards nonperturbative running checks is available and give some details, but do not attempt to translate this into a colour code.
In the case of twistedmass fermions, discretization effects give rise to a mass difference between charged and neutral pions even when the up and downquark masses are equal: the charged pion is found to be the heavier of the two for twistedmass Wilson fermions (cf. Ref. [83]). In early work, typically referring to \(N_f=2\) simulations (e.g. Refs. [83] and [36]), chiral extrapolations are based on chiral perturbation theory formulae which do not take these regularization effects into account. After the importance of keeping the isospin breaking when doing chiral fits was shown in Ref. [84], later work, typically referring to \(N_f=2+1+1\) simulations, has taken these effects into account [4]. We use \(M_{\pi ^\pm }\) for \(M_{\pi ,\mathrm {min}}\) in the chiralextrapolation rating criterion. On the other hand, sea quarks (corresponding to both charged and neutral “sea pions“ in an effectivechiraltheory logic) as well as valence quarks are intertwined with finitevolume effects. Therefore, we identify \(M_{\pi ,\mathrm {min}}\) with the root mean square (RMS) of \(M_{\pi ^+}\), \(M_{\pi ^}\) and \(M_{\pi ^0}\) in the finitevolume rating criterion.^{6}
In the case of staggered fermions, discretization effects give rise to several light states with the quantum numbers of the pion.^{7} The mass splitting among these “taste” partners represents a discretization effect of \(\mathcal {O}(a^2)\), which can be significant at large lattice spacings but shrinks as the spacing is reduced. In the discussion of the results obtained with staggered quarks given in the following sections, we assume that these artefacts are under control. We conservatively identify \(M_{\pi ,\mathrm {min}}\) with the root mean square (RMS) average of the masses of all the taste partners, both for chiralextrapolation and finitevolume criteria.^{8}
The strong coupling \(\alpha _s\) is computed in lattice QCD with methods differing substantially from those used in the calculations of the other quantities discussed in this review. Therefore we have established separate criteria for \(\alpha _s\) results, which will be discussed in Sect. 9.2.
2.1.2 Heavyquark actions
In most cases, and in particular for the b quark, the discretization of the heavyquark action follows a very different approach to that used for light flavours. There are several different methods for treating heavy quarks on the lattice, each with their own issues and considerations. All of these methods use Effective Field Theory (EFT) at some point in the computation, either via direct simulation of the EFT, or by using EFT as a tool to estimate the size of cutoff errors, or by using EFT to extrapolate from the simulated lattice quark masses up to the physical bquark mass. Because of the use of an EFT, truncation errors must be considered together with discretization errors.
The charm quark lies at an intermediate point between the heavy and light quarks. In our previous review, the bulk of the calculations involving charm quarks treated it using one of the approaches adopted for the b quark. Many recent calculations, however, simulate the charm quark using lightquark actions, in particular the \(N_f=2+1+1\) calculations. This has become possible thanks to the increasing availability of dynamical gauge field ensembles with fine lattice spacings. But clearly, when charm quarks are treated relativistically, discretization errors are more severe than those of the corresponding lightquark quantities.
In order to address these complications, we add a new heavyquark treatment category to the rating system. The purpose of this criterion is to provide a guideline for the level of action and operator improvement needed in each approach to make reliable calculations possible, in principle.
A description of the different approaches to treating heavy quarks on the lattice is given in Appendix A.1.3, including a discussion of the associated discretization, truncation, and matching errors. For truncation errors we use HQET power counting throughout, since this review is focussed on heavyquark quantities involving B and D mesons rather than bottomonium or charmonium quantities. Here we describe the criteria for how each approach must be implemented in order to receive an acceptable ( Open image in new window ) rating for both the heavyquark actions and the weak operators. Heavyquark implementations without the level of improvement described below are rated not acceptable ( Open image in new window ). The matching is evaluated together with renormalization, using the renormalization criteria described in Sect. 2.1.1. We emphasize that the heavyquark implementations rated as acceptable and described below have been validated in a variety of ways, such as via phenomenological agreement with experimental measurements, consistency between independent lattice calculations, and numerical studies of truncation errors. These tests are summarized in Sect. 8.
Relativistic heavyquark actions:
Open image in new window at least treelevel \(\mathcal {O}(a)\) improved action and weak operators.
This is similar to the requirements for lightquark actions. All current implementations of relativistic heavyquark actions satisfy this criterion.
NRQCD
Open image in new window treelevel matched through \(\mathcal {O}(1/m_h)\) and improved through \(\mathcal {O}(a^2)\).
The current implementations of NRQCD satisfy this criterion, and also include treelevel corrections of \(\mathcal {O}(1/m_h^2)\) in the action.
HQET
Open image in new window treelevel matched through \(\mathcal {O}(1/m_h)\) with discretization errors starting at \(\mathcal {O}(a^2)\).
The current implementation of HQET by the ALPHA Collaboration satisfies this criterion, since both action and weak operators are matched nonperturbatively through \(\mathcal {O}(1/m_h)\). Calculations that exclusively use a staticlimit action do not satisfy this criterion, since the staticlimit action, by definition, does not include \(1/m_h\) terms. We therefore consider static computations in our final estimates only if truncation errors (in \(1/m_h\)) are discussed and included in the systematic uncertainties.
Lightquark actions for heavy quarks
Open image in new window discretization errors starting at \(\mathcal {O}(a^2)\) or higher.
This applies to calculations that use the tmWilson action, a nonperturbatively improved Wilson action, or the HISQ action for charmquark quantities. It also applies to calculations that use these lightquark actions in the charm region and above together with either the static limit or with an HQET inspired extrapolation to obtain results at the physical b quark mass. In these cases, the continuum extrapolation criteria described earlier must be applied to the entire range of heavyquark masses used in the calculation.
2.1.3 Conventions for the figures
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corresponds to results included in the average or estimate (i.e. results that contribute to the black square below);
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corresponds to results that are not included in the average but pass all quality criteria;
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corresponds to all other results;
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corresponds to FLAG averages or estimates; they are also highlighted by a grey vertical band.
Symbols other than squares are used to distinguish results with specific properties and are always explained in the caption.^{9}

Open image in new window corresponds to nonlattice results;

Open image in new window corresponds to Particle Data Group (PDG) results.
2.2 Averages and estimates
FLAG results of a given quantity are denoted either as averages or as estimates. Here we clarify this distinction. To start with, both averages and estimates are based on results without any red tags in their colour coding. For many observables there are enough independent lattice calculations of good quality, with all sources of error (not merely those related to the colourcoded criteria), as analysed in the original papers, appearing to be under control. In such cases it makes sense to average these results and propose such an average as the best current lattice number. The averaging procedure applied to this data and the way the error is obtained is explained in detail in Sect. 2.3. In those cases where only a sole result passes our rating criteria (colour coding), we refer to it as our FLAG average, provided it also displays adequate control of all other sources of systematic uncertainty.
On the other hand, there are some cases in which this procedure leads to a result that, in our opinion, does not cover all uncertainties. Systematic error estimates are by their nature often subjective and difficult to estimate, and may thus end up being underestimated in one or more results that receive green symbols for all explicitly tabulated criteria. Adopting a conservative policy, in these cases we opt for an estimate (or a range), which we consider as a fair assessment of the knowledge acquired on the lattice at present. This estimate is not obtained with a prescribed mathematical procedure, but reflects what we consider the best possible analysis of the available information. The hope is that this will encourage more detailed investigations by the lattice community.

Publication status:
A published or plain update of published results
P preprint
C conference contribution.
As noted above, in this review we present results from simulations with \(N_f=2\), \(N_f=2+1\) and \(N_f=2+1+1\) (except for \( r_0 \Lambda _{\overline{\text {MS}}}\) where we also give the \(N_f=0\) result). We are not aware of an a priori way to quantitatively estimate the difference between results produced in simulations with a different number of dynamical quarks. We therefore average results at fixed \(N_{ f}\) separately; averages of calculations with different \(N_{ f}\) will not be provided.
To date, no significant differences between results with different values of \(N_f\) have been observed in the quantities listed in Tables 1 and 2. In the future, as the accuracy and the control over systematic effects in lattice calculations increases, it will hopefully be possible to see a difference between results from simulations with \(N_{ f}= 2\) and \(N_{ f}= 2 + 1\), and thus determine the size of the Zweigrule violations related to strangequark loops. This is a very interesting issue per se, and one which can be quantitatively addressed only with lattice calculations.
The question of differences between results with \(N_{ f}=2+1\) and \(N_{ f}=2+1+1\) is more subtle. The dominant effect of including the charm sea quark is to shift the lattice scale, an effect that is accounted for by fixing this scale nonperturbatively using physical quantities. For most of the quantities discussed in this review, it is expected that residual effects are small in the continuum limit, suppressed by \(\alpha _s(m_c)\) and powers of \(\Lambda ^2/m_c^2\). Here \(\Lambda \) is a hadronic scale that can only be roughly estimated and depends on the process under consideration. Note that the \(\Lambda ^2/m_c^2\) effects have been addressed in Ref. [90]. Assuming that such effects are small, it might be reasonable to average the results from \(N_{ f}=2+1\) and \(N_{ f}=2+1+1\) simulations. This is not yet a pressing issue in this review, since there are relatively few results with \(N_{ f}=2+1+1\), but it will become a more important question in the future.
2.3 Averaging procedure and error analysis
In the present report we repeatedly average results obtained by different collaborations and estimate the error on the resulting averages. We follow the procedure of the previous edition [2], which we describe here in full detail.
One of the problems arising when forming averages is that not all of the datasets are independent. In particular, the same gaugefield configurations, produced with a given fermion discretization, are often used by different research teams with different valencequark lattice actions, obtaining results that are not really independent. Our averaging procedure takes such correlations into account.
3 Quark masses
Quark masses are fundamental parameters of the Standard Model. An accurate determination of these parameters is important for both phenomenological and theoretical applications. The charm and bottom masses, for instance, enter the theoretical expressions of several cross sections and decay rates in heavyquark expansions. The up, down and strangequark masses govern the amount of explicit chiral symmetry breaking in QCD. From a theoretical point of view, the values of quark masses provide information as regards the flavour structure of physics beyond the Standard Model. The Review of Particle Physics of the Particle Data Group contains a review of quark masses [92], which covers light as well as heavy flavours. Here we also consider light and heavy quark masses, but focus on lattice results and discuss them in more detail. We do not discuss the top quark, however, because it decays weakly before it can hadronize, and the nonperturbative QCD dynamics described by present day lattice simulations is not relevant. The lattice determination of light (up, down, strange), charm and bottomquark masses is considered in Sects. 3.1, 3.2, and 3.3, respectively.
Quark masses cannot be measured directly in experiment because quarks cannot be isolated, as they are confined inside hadrons. On the other hand, quark masses are free parameters of the theory and, as such, cannot be obtained on the basis of purely theoretical considerations. Their values can only be determined by comparing the theoretical prediction for an observable, which depends on the quark mass of interest, with the corresponding experimental value.
In the last edition of this review [2], quarkmass determinations came from two and threeflavour QCD calculations. Moreover, these calculations were most often performed in the isospin limit, where the up and downquark masses (especially those in the sea) are set equal. In addition, some of the results retained in our lightquark mass averages were based on simulations performed at values of \(m_{ud}\) which were still substantially larger than its physical value imposing a significant extrapolation to reach the physical up and downquark mass point. Among the calculations performed near physical \(m_{ud}\) by PACSCS [93, 94, 95], BMW [7, 8] and RBC/UKQCD [31], only the ones in Refs. [7, 8] did so while controlling all other sources of systematic error.
Today, however, the effects of the charm quark in the sea are more and more systematically considered and most of the new quarkmass results discussed below have been obtained in \(N_f=2+1+1\) simulations by ETM [4], HPQCD [14] and FNAL/MILC [5]. In addition, RBC/UKQCD [10], HPQCD [14] and FNAL/MILC [5] are extending their calculations down to updownquark masses at or very close to their physical values while still controlling other sources of systematic error. Another aspect that is being increasingly addressed are electromagnetic and \((m_dm_u)\), strong isospinbreaking effects. As we will see below these are particularly important for determining the individual up and downquark masses. But with the level of precision being reached in calculations, these effects are also becoming important for other quark masses.
 1.
One computes as many experimentally measurable quantities as there are quark masses. These observables should obviously be sensitive to the masses of interest, preferably straightforward to compute and obtainable with high precision. They are usually computed for a variety of input values of the quark masses which are then adjusted to reproduce experiment. Another observable, such as the pion decay constant or the mass of a member of the baryon octet, must be used to fix the overall scale. Note that the mass of a quark, such as the b, which is not accounted for in the generation of gauge configurations, can still be determined. For that an additional valencequark observable containing this quark must be computed and the mass of that quark must be tuned to reproduce experiment.
 2.
The input quark masses are bare parameters which depend on the lattice spacing and particulars of the lattice regularization used in the calculation. To compare their values at different lattice spacings and to allow a continuum extrapolation they must be renormalized. This renormalization is a shortdistance calculation, which may be performed perturbatively. Experience shows that oneloop calculations are unreliable for the renormalization of quark masses: usually at least two loops are required to have trustworthy results. Therefore, it is best to perform the renormalizations nonperturbatively to avoid potentially large perturbative uncertainties due to neglected higherorder terms. Nevertheless we will include in our averages oneloop results if they carry a solid estimate of the systematic uncertainty due to the truncation of the series.
We mention that lattice QCD calculations of the bquark mass have an additional complication which is not present in the case of the charm and lightquarks. At the lattice spacings currently used in numerical simulations the direct treatment of the b quark with the fermionic actions commonly used for light quarks will result in large cutoff effects, because the bquark mass is of order one in lattice units. There are a few widely used approaches to treat the b quark on the lattice, which have been already discussed in the FLAG 13 review (see Section 8 of Ref. [2]). Those relevant for the determination of the bquark mass will be briefly described in Sect. 3.3.
3.1 Masses of the light quarks
Lightquark masses are particularly difficult to determine because they are very small (for the up and down quarks) or small (for the strange quark) compared to typical hadronic scales. Thus, their impact on typical hadronic observables is minute, and it is difficult to isolate their contribution accurately.
Fortunately, the spontaneous breaking of \(SU(3)_L\times SU(3)_R\) chiral symmetry provides observables which are particularly sensitive to the lightquark masses: the masses of the resulting Nambu–Goldstone bosons (NGB), i.e. pions, kaons and etas. Indeed, the GellMann–Oakes–Renner relation [96] predicts that the squared mass of a NGB is directly proportional to the sum of the masses of the quark and antiquark which compose it, up to higherorder mass corrections. Moreover, because these NGBs are light and are composed of only two valence particles, their masses have a particularly clean statistical signal in latticeQCD calculations. In addition, the experimental uncertainties on these meson masses are negligible. Thus, in lattice calculations, lightquark masses are typically obtained by renormalizing the input quark mass and tuning them to reproduce NGB masses, as described above.
3.1.1 Contributions from the electromagnetic interaction
As mentioned in Sect. 2.1, the present review relies on the hypothesis that, at low energies, the Lagrangian \(\mathcal{L}_{\mathrm{QCD}}+\mathcal{L}_{\mathrm{QED}}\) describes nature to a high degree of precision. However, most of the results presented below are obtained in pure QCD calculations, which do not include QED. Quite generally, when comparing QCD calculations with experiment, radiative corrections need to be applied. In pure QCD simulations, where the parameters are fixed in terms of the masses of some of the hadrons, the electromagnetic contributions to these masses must be accounted for. Of course, once QED is included in lattice calculations, the subtraction of e.m. contributions is no longer necessary.
The electromagnetic interaction plays a particularly important role in determinations of the ratio \(m_u/m_d\), because the isospinbreaking effects generated by this interaction are comparable to those from \(m_u\ne m_d\) (see Sect. 3.1.5). In determinations of the ratio \(m_s/m_{ud}\), the electromagnetic interaction is less important, but at the accuracy reached, it cannot be neglected. The reason is that, in the determination of this ratio, the pion mass enters as an input parameter. Because \(M_\pi \) represents a small symmetrybreaking effect, it is rather sensitive to the perturbations generated by QED.
The decomposition of the sum \(\mathcal{L}_{\mathrm{QCD}}+\mathcal{L}_{\mathrm{QED}}\) into two parts is not unique and specifying the QCD part requires a convention. In order to give results for the quark masses in the Standard Model at scale \(\mu =2\,\hbox {GeV}\), on the basis of a calculation done within QCD, it is convenient to match the parameters of the two theories at that scale. We use this convention throughout the present review.^{11}
The first determination of the selfenergies on the lattice was carried out by Duncan, Eichten and Thacker [104]. Using the quenched approximation, they arrived at \(M_{K^+}^\gamma M_{K^0}^\gamma = 1.9\,\hbox {MeV}\). Actually, the parameterization of the masses given in that paper yields an estimate for all but one of the coefficients introduced above (since the mass splitting between the charged and neutral pions in QCD is neglected, the parameterization amounts to setting \(\epsilon _m=0\) ab initio). Evaluating the differences between the masses obtained at the physical value of the electromagnetic coupling constant and at \(e=0\), we obtain \(\epsilon = 0.50(8)\), \(\epsilon _{\pi ^0} = 0.034(5)\) and \(\epsilon _{K^0} = 0.23(3)\). The errors quoted are statistical only: an estimate of lattice systematic errors is not possible from the limited results of Ref. [104]. The result for \(\epsilon \) indicates that the violation of Dashen’s theorem is sizeable: according to this calculation, the nonleading contributions to the selfenergy difference of the kaons amount to 50% of the leading term. The result for the selfenergy of the neutral pion cannot be taken at face value, because it is small, comparable to the neglected mass difference \(\hat{M}_{\pi ^+}\hat{M}_{\pi ^0}\). To illustrate this, we note that the numbers quoted above are obtained by matching the parameterization with the physical masses for \(\pi ^0\), \(K^+\) and \(K^0\). This gives a mass for the charged pion that is too high by 0.32 MeV. Tuning the parameters instead such that \(M_{\pi ^+}\) comes out correctly, the result for the selfenergy of the neutral pion becomes larger: \(\epsilon _{\pi ^0}=0.10(7)\) where, again, the error is statistical only.
In an update of this calculation by the RBC Collaboration [105] (RBC 07), the electromagnetic interaction is still treated in the quenched approximation, but the strong interaction is simulated with \(N_{ f}=2\) dynamical quark flavours. The quark masses are fixed with the physical masses of \(\pi ^0\), \(K^+\) and \(K^0\). The outcome for the difference in the electromagnetic selfenergy of the kaons reads \(M_{K^+}^\gamma M_{K^0}^\gamma = 1.443(55)\,\hbox {MeV}\). This corresponds to a remarkably small violation of Dashen’s theorem. Indeed, a recent extension of this work to \(N_{ f}=2+1\) dynamical flavours [103] leads to a significantly larger selfenergy difference: \(M_{K^+}^\gamma M_{K^0}^\gamma = 1.87(10)\,\hbox {MeV}\), in good agreement with the estimate of Eichten et al. Expressed in terms of the coefficient \(\epsilon \) that measures the size of the violation of Dashen’s theorem, it corresponds to \(\epsilon =0.5(1)\).
The input for the electromagnetic corrections used by MILC is specified in Ref. [106]. In their analysis of the lattice data, \(\epsilon _{\pi ^0}\), \(\epsilon _{K^0}\) and \(\epsilon _m\) are set equal to zero. For the remaining coefficient, which plays a crucial role in determinations of the ratio \(m_u/m_d\), the very conservative range \(\epsilon =1(1)\) was used in MILC 04 [107], while in MILC 09 [89] and MILC 09A [6] this input has been replaced by \(\epsilon =1.2(5)\), as suggested by phenomenological estimates for the corrections to Dashen’s theorem [108, 109]. Results of an evaluation of the electromagnetic selfenergies based on \(N_{ f}=2+1\) dynamical quarks in the QCD sector and on the quenched approximation in the QED sector have also been reported by MILC in Refs. [110, 111, 112] and updated recently in Refs. [113, 114]. Their latest (preliminary) result is \(\bar{\epsilon }= 0.84(5)(19)\), where the first error is statistical and the second systematic, coming from discretization and finitevolume uncertainties added in quadrature. With the estimate for \(\epsilon _m\) given in Eq. (13), this result corresponds to \(\epsilon = 0.81(5)(18)\).
Preliminary results have also been reported by the BMW Collaboration in conference proceedings [115, 116, 117], with the updated result being \(\epsilon = 0.57(6)(6)\), where the first error is statistical and the second systematic.
The RM123 Collaboration employs a new technique to compute e.m. shifts in hadron masses in 2flavour QCD: the effects are included at leading order in the electromagnetic coupling \(\alpha \) through simple insertions of the fundamental electromagnetic interaction in quark lines of relevant Feynman graphs [16]. They find \(\epsilon =0.79(18)(18)\), where the first error is statistical and the second is the total systematic error resulting from chiral, finitevolume, discretization, quenching and fitting errors all added in quadrature.
Recently [118] the QCDSF/UKQCD Collaboration has presented results for several pseudoscalar meson masses obtained from \(N_f = 2+1\) dynamical simulations of QCD + QED (at a single lattice spacing \( a \simeq 0.07\) fm). Using the experimental values of the \(\pi ^0\), \(K^0\) and \(K^+\) mesons masses to fix the three lightquark masses, they find \(\epsilon = 0.50 (6)\), where the error is statistical only.
The effective Lagrangian that governs the selfenergies to nexttoleading order (NLO) of the chiral expansion was set up in Ref. [119]. The estimates made in Refs. [108, 109] are obtained by replacing QCD with a model, matching this model with the effective theory and assuming that the effective coupling constants obtained in this way represent a decent approximation to those of QCD. For alternative model estimates and a detailed discussion of the problems encountered in models based on saturation by resonances, see Refs. [120, 121, 122]. In the present review of the information obtained on the lattice, we avoid the use of models altogether.
There is an indirect phenomenological determination of \(\epsilon \), which is based on the decay \(\eta \rightarrow 3\pi \) and does not rely on models. The result for the quarkmass ratio Q, defined in Eq. (32) and obtained from a dispersive analysis of this decay, implies \(\epsilon = 0.70(28)\) (see Sect. 3.1.5). While the values found in older lattice calculations [103, 104, 105] are a little less than one standard deviation lower, the most recent determinations [16, 110, 111, 112, 113, 114, 115, 116, 123], though still preliminary, are in excellent agreement with this result and have significantly smaller error bars. However, even in the more recent calculations, e.m. effects are treated in the quenched approximation. Thus, we choose to quote \(\epsilon = 0.7(3)\), which is essentially the \(\eta \rightarrow 3\pi \) result and covers the range of post2010 lattice results. Note that this value has an uncertainty which is reduced by about 40% compared to the result quoted in the first edition of the FLAG review [1].
We add a few comments concerning the physics of the selfenergies and then specify the estimates used as an input in our analysis of the data. The Cottingham formula [124] represents the selfenergy of a particle as an integral over electron scattering cross sections; elastic as well as inelastic reactions contribute. For the charged pion, the term due to elastic scattering, which involves the square of the e.m. form factor, makes a substantial contribution. In the case of the \(\pi ^0\), this term is absent, because the form factor vanishes on account of charge conjugation invariance. Indeed, the contribution from the form factor to the selfenergy of the \(\pi ^+\) roughly reproduces the observed mass difference between the two particles. Furthermore, the numbers given in Refs. [125, 126, 127] indicate that the inelastic contributions are significantly smaller than the elastic contributions to the selfenergy of the \(\pi ^+\). The lowenergy theorem of Das, Guralnik, Mathur, Low and Young [128] ensures that, in the limit \(m_u,m_d\rightarrow 0\), the e.m. selfenergy of the \(\pi ^0\) vanishes, while the one of the \(\pi ^+\) is given by an integral over the difference between the vector and axialvector spectral functions. The estimates for \(\epsilon _{\pi ^0}\) obtained in Ref. [104] and more recently in Ref. [118] are consistent with the suppression of the selfenergy of the \(\pi ^0\) implied by chiral \(SU(2)\times SU(2)\). In our opinion, as already done in the FLAG 13 review [2], the value \(\epsilon _{\pi ^0}=0.07(7)\) still represents a quite conservative estimate for this coefficient. The selfenergy of the \(K^0\) is suppressed less strongly, because it remains different from zero if \(m_u\) and \(m_d\) are taken massless and only disappears if \(m_s\) is turned off as well. Note also that, since the e.m. form factor of the \(K^0\) is different from zero, the selfenergy of the \(K^0\) does pick up an elastic contribution. The recent lattice result \(\epsilon _{K^0} = 0.2(1)\) obtained in Ref. [118] indicates that the violation of Dashen’s theorem is smaller than in the case of \(\epsilon \). Following the FLAG 13 review [2] we confirm the choice of the conservative value \(\epsilon _{K^0} = 0.3(3)\).
Finally, we consider the mass splitting between the charged and neutral pions in QCD. This effect is known to be very small, because it is of second order in \(m_um_d\). There is a parameterfree prediction, which expresses the difference \(\hat{M}_{\pi ^+}^2\hat{M}_{\pi ^0}^2\) in terms of the physical masses of the pseudoscalar octet and is valid to NLO of the chiral perturbation series. Numerically, the relation yields \(\epsilon _m=0.04\) [129], indicating that this contribution does not play a significant role at the present level of accuracy. We attach a conservative error also to this coefficient: \(\epsilon _m=0.04(2)\). The lattice result for the selfenergy difference of the pions, reported in Ref. [103], \(M_{\pi ^+}^\gamma M_{\pi ^0}^\gamma = 4.50(23)\,\hbox {MeV}\), agrees with this estimate: expressed in terms of the coefficient \(\epsilon _m\) that measures the pionmass splitting in QCD, the result corresponds to \(\epsilon _m=0.04(5)\). The corrections of nexttonexttoleading order (NNLO) have been worked out in Ref. [130], but the numerical evaluation of the formulae again meets with the problem that the relevant effective coupling constants are not reliably known.
3.1.2 Pion and kaon masses in the isospin limit
3.1.3 Lattice determination of \(m_s\) and \(m_{ud}\)
We now turn to a review of the lattice calculations of the lightquark masses and begin with \(m_s\), the isospinaveraged up and downquark mass, \(m_{ud}\), and their ratio. Most groups quote only \(m_{ud}\), not the individual up and downquark masses. We then discuss the ratio \(m_u/m_d\) and the individual determination of \(m_u\) and \(m_d\).
Quark masses have been calculated on the lattice since the mid1990s. However, early calculations were performed in the quenched approximation, leading to unquantifiable systematics. Thus in the following, we only review modern, unquenched calculations, which include the effects of light sea quarks.
Tables 3, 4 and 5 list the results of \(N_{ f}=2\), \(N_{ f}=2+1\) and \(N_{ f}=2+1+1\) lattice calculations of \(m_s\) and \(m_{ud}\). These results are given in the \({\overline{\text {MS}}}\) scheme at \(2\,\mathrm{GeV}\), which is standard nowadays, though some groups are starting to quote results at higher scales (e.g. Ref. [31]). The tables also show the colour coding of the calculations leading to these results. As indicated earlier in this review, we treat calculations with different numbers, \(N_f\), of dynamical quarks separately.
\(N_{ f}=2\) lattice calculations For \(N_{ f}=2\), no new calculations have been performed since the previous edition of the FLAG review [2]. A quick inspection of Table 3 indicates that only the more recent calculations, ALPHA 12 [12] and ETM 10B [11], control all systematic effects – the special case of Dürr 11 [132] is discussed below. Only ALPHA 12 [12], ETM 10B [11] and ETM 07 [133] really enter the chiral regime, with pion masses down to about 270 MeV for ALPHA and ETM. Because this pion mass is still quite far from the physicalpion mass, ALPHA 12 refrain from determining \(m_{ud}\) and give only \(m_s\). All the other calculations have significantly more massive pions, the lightest being about 430 MeV, in the calculation by CPPACS 01 [134]. Moreover, the latter calculation is performed on very coarse lattices, with lattice spacings \(a\ge 0.11\,\mathrm{fm}\) and only oneloop perturbation theory is used to renormalize the results.
ETM 10B’s [11] calculation of \(m_{ud}\) and \(m_s\) is an update of the earlier twistedmass determination of ETM 07 [133]. In particular, they have added ensembles with a larger volume and three new lattice spacings, \(a = 0.054, 0.067\) and \(0.098\,\mathrm{fm}\), allowing for a continuum extrapolation. In addition, it features analyses performed in SU(2) and SU(3) \(\chi \)PT.
\(N_{ f}=2\) lattice results for the masses \(m_{ud}\) and \(m_s\) (MeV, running masses in the \({\overline{\text {MS}}}\) scheme at scale 2 GeV). The significance of the colours is explained in Sect. 2. If information as regards nonperturbative running is available, this is indicated in the column “running”, with details given at the bottom of the table
Collaboration  Refs.  Publication status  Chiral extrapolation  Continuum extrapolation  Finite volume  Renormalization  Running  \(m_{ud} \)  \(m_s \) 

ALPHA 12  [12]  A  \(\,a,b\)  102(3)(1)  
Dürr 11\(^{\mathrm{a}}\)  [132]  A  −  −  3.52(10)(9)  97.0(2.6)(2.5)  
ETM 10B  [11]  A  \(\,c\)  3.6(1)(2)  95(2)(6)  
JLQCD/TWQCD 08A  [138]  A  −  4.452(81)(38)\(\left( {\begin{array}{c}+0\\ 227\end{array}}\right) \)  –  
RBC 07\(^\mathrm{b}\)  [105]  A  −  4.25(23)(26)  119.5(5.6)(7.4)  
ETM 07  [133]  A  −  3.85(12)(40)  105(3)(9)  
QCDSF/UKQCD 06  [139]  A  −  4.08(23)(19)(23)  111(6)(4)(6)  
SPQcdR 05  [140]  A  −  \(4.3(4)(^{+1.1}_{0.0})\)  \(101(8)(^{+25}_{0})\)  
ALPHA 05  [135]  A  \(\,a\)  97(4)(18)\(^{\mathrm{c}}\)  
QCDSF/UKQCD 04  [137]  A  −  4.7(2)(3)  119(5)(8)  
JLQCD 02  [141]  A  −  \(3.223(^{+46}_{69})\)  \(84.5(^{+12.0}_{1.7})\)  
CPPACS 01  [134]  A  −  \(3.45(10)(^{+11}_{18})\)  \(89(2)(^{+2}_{6})^{\mathrm{d}}\) 
\(N_{ f}=2+1\) lattice results for the masses \(m_{ud}\) and \(m_s\) (see Table 3 for notation)
Collaboration  Refs.  Publication status  Chiral extrapolation  Continuum extrapolation  Finite volume  Renormalization  Running  \(m_{ud} \)  \(m_s \) 

RBC/UKQCD 14B\(^{\mathrm{a}}\)  [10]  P  d  3.31(4)(4)  90.3(0.9)(1.0)  
RBC/UKQCD 12\(^{\mathrm{a}}\)  [31]  A  d  3.37(9)(7)(1)(2)  92.3(1.9)(0.9)(0.4)(0.8)  
PACSCS 12\(^{\mathrm{b}}\)  [143]  A  \(\,b\)  3.12(24)(8)  83.60(0.58)(2.23)  
Laiho 11  [44]  C  −  3.31(7)(20)(17)  94.2(1.4)(3.2)(4.7)  
BMW 10A, 10B\(^{\mathrm{c}}\)  A  \(\,c\)  3.469(47)(48)  95.5(1.1)(1.5)  
PACSCS 10  [95]  A  \(\,b\)  2.78(27)  86.7(2.3)  
MILC 10A  [13]  C  −  3.19(4)(5)(16)  –  
HPQCD 10\(^{\mathrm{d}}\)  [9]  A  −  −  3.39(6)  92.2(1.3)  
RBC/UKQCD 10A  [144]  A  \(\,a\)  3.59(13)(14)(8)  96.2(1.6)(0.2)(2.1)  
Blum 10\(^{\mathrm{e}}\)  [103]  A  −  3.44(12)(22)  97.6(2.9)(5.5)  
PACSCS 09  [94]  A  \(\,b\)  2.97(28)(3)  92.75(58)(95)  
HPQCD 09A\(^{\mathrm{f}}\)  [18]  A  −  −  3.40(7)  92.4(1.5)  
MILC 09A  [6]  C  −  3.25 (1)(7)(16)(0)  89.0(0.2)(1.6)(4.5)(0.1)  
MILC 09  [89]  A  −  3.2(0)(1)(2)(0)  88(0)(3)(4)(0)  
PACSCS 08  [93]  A  −  2.527(47)  72.72(78)  
RBC/UKQCD 08  [145]  A  −  3.72(16)(33)(18)  107.3(4.4)(9.7)(4.9)  
CPPACS/JLQCD 07  [146]  A  −  \(3.55(19)(^{+56}_{20})\)  \(90.1(4.3)(^{+16.7}_{4.3})\)  
HPQCD 05  [147]  A  −  \(3.2(0)(2)(2)(0)^{\mathrm{g}}\)  \(87(0)(4)(4)(0)^{\mathrm{g}}\)  
MILC 04, HPQCD/MILC/UKQCD 04  A  −  2.8(0)(1)(3)(0)  76(0)(3)(7)(0) 
\(N_{ f}=2+1+1\) lattice results for the masses \(m_{ud}\) and \(m_s\) (see Table 3 for notation)
We have not included the results of Dürr 11 [132] in the averages of Eq. (17), despite the fact that they satisfy our selection criteria. The reason for this is that the observable which they actually compute on the lattice is \(m_c/m_s=11.27(30)(26)\), reviewed in Sect. 3.2.4. They obtain \(m_s\) by combining that value of \(m_c/m_s\) with already existing phenomenological calculations of \(m_c\). Subsequently they obtain \(m_{ud}\) by combining this result for \(m_s\) with the \(N_f=2+1\) calculation of \(m_s/m_{ud}\) of BMW 10A, 10B [7, 8] discussed below. Thus, their results for \(m_s\) and \(m_{ud}\) are not per se lattice results, nor do they correspond to \(N_f=2\). The value of the charmquark mass which they use is an average of phenomenological determinations, which they estimate to be \(m_c(2\,\mathrm{GeV})=1.093(13)\,\mathrm{GeV}\), with a 1.2% total uncertainty. This value for \(m_c\) leads to the results for \(m_s\) and \(m_{ud}\) in Table 3 which are compatible with the averages given in Eq. (17) and have similar uncertainties. Note, however, that their determination of \(m_c/m_s\) is about 1.5 combined standard deviations below the only other \(N_{ f}=2\) result which satisfies our selection criteria, ETM 10B’s [11] result, as discussed in Sect. 3.2.4.
\(N_{ f}=2+1\) lattice calculations We turn now to \(N_{ f}=2+1\) calculations. These and the corresponding results for \(m_{ud}\) and \(m_s\) are summarized in Table 4. Given the very high precision of a number of the results, with total errors on the order of 1%, it is important to consider the effects neglected in these calculations. Since isospinbreaking and e.m. effects are small on \(m_{ud}\) and \(m_s\), and have been approximately accounted for in the calculations that will be retained for our averages, the largest potential source of uncontrolled systematic error is that due to the omission of the charm quark in the sea. Beyond the small perturbative corrections that come from matching the \(N_{ f}=3\) to the \(N_{ f}=4\) \({\overline{\text {MS}}}\) scheme at \(m_c\) (\({\sim } 0.2\%\)), the charm seaquarks affect the determination of the lightquark masses through contributions of order \(1/m_c^2\). As these are further suppressed by the Okubo–Zweig–Iizuka rule, they are also expected to be small, but are difficult to quantify a priori. Fortunately, as we will see below, \(m_s\) has been directly computed with \(N_f=2+1+1\) simulations. In particular, HPQCD 14 [5] has computed \(m_s\) in QCD\(_4\) with very much the same approach as it had used to obtain the QCD\(_3\) result of HPQCD 10 [9]. Their results for \(m_s(N_f=3, 2~\mathrm{GeV})\) are \(93.8(8)\,\mathrm{MeV}\) [5] and \(92.2(1.3)\,\mathrm{MeV}\) [9], where the \(N_f=4\) result has been converted perturbatively to \(N_f=3\) in Ref. [5]. This leads to a relative difference of \(1.7(1.6)\%\). While the two results are compatible within one combined standard deviation, a \({\sim } 2\%\) effect cannot be excluded. Thus, we will retain this 2% uncertainty and add it to the averages for \(m_s\) and \(m_{ud}\) given below.
The only new calculation since the last FLAG report [2] is that of RBC/UKQCD 14 [10]. It significantly improves on their RBC/UKQCD 12 [31] work by adding three new domainwall fermion simulations to three used previously. Two of the new simulations are performed at essentially physicalpion masses (\(M_\pi \simeq 139\,\mathrm{MeV}\)) on lattices of about \(5.4\,\mathrm{fm}\) in size and with lattice spacings of \(0.114\,\mathrm{fm}\) and \(0.084\,\mathrm{fm}\). It is complemented by a third simulation with \(M_\pi \simeq 371\,\mathrm{MeV}\), \(a\simeq 0.063\) and a rather small \(L\simeq 2.0\,\mathrm{fm}\). Altogether, this gives them six simulations with six unitary \(M_\pi \)’s in the range of 139 to \(371\,\mathrm{MeV}\) and effectively three lattice spacings from 0.063 to \(0.114\,\mathrm{fm}\). They perform a combined global continuum and chiral fit to all of their results for the \(\pi \) and K masses and decay constants, the \(\Omega \) baryon mass and two Wilsonflow parameters. Quark masses in these fits are renormalized and run nonperturbatively in the RI/SMOM scheme. This is done by computing the relevant renormalization constant for a reference ensemble and determining those for other simulations relative to it by adding appropriate parameters in the global fit. This new calculation passes all of our selection criteria. Its results will replace the older RBC/UKQCD 12 results in our averages.
\(N_{ f}=2+1\) MILC results for lightquark masses go back to 2004 [107, 148]. They use rooted staggered fermions. By 2009 their simulations covered an impressive range of parameter space, with lattice spacings which go down to 0.045 fm and valencepion masses down to approximately 180 MeV [6]. The most recent MILC \(N_{ f}=2+1\) results, i.e. MILC 10A [13] and MILC 09A [6], feature large statistics and twoloop renormalization. Since these datasets subsume those of their previous calculations, these latest results are the only ones that must be kept in any world average.
The PACSCS 12 [143] calculation represents an important extension of the collaboration’s earlier 2010 computation [95], which already probed pion masses down to \(M_\pi \simeq 135\,\mathrm{MeV}\), i.e. down to the physicalmass point. This was achieved by reweighting the simulations performed in PACSCS 08 [93] at \(M_\pi \simeq 160\,\mathrm{MeV}\). If adequately controlled, this procedure eliminates the need to extrapolate to the physicalmass point and, hence, the corresponding systematic error. The new calculation now applies similar reweighting techniques to include electromagnetic and \(m_u\ne m_d\) isospinbreaking effects directly at the physicalpion mass. Further, as in PACSCS 10 [95], renormalization of quark masses is implemented nonperturbatively, through the Schrödinger functional method [153]. As it stands, the main drawback of the calculation, which makes the inclusion of its results in a world average of lattice results inappropriate at this stage, is that for the lightest quark mass the volume is very small, corresponding to \(LM_\pi \simeq 2.0\), a value for which finitevolume effects will be difficult to control. Another problem is that the calculation was performed at a single lattice spacing, forbidding a continuum extrapolation. Further, it is unclear at this point what might be the systematic errors associated with the reweighting procedure.
The BMW 10A, 10B [7, 8] calculation still satisfies our stricter selection criteria. They reach the physical up and downquark mass by interpolation instead of by extrapolation. Moreover, their calculation was performed at five lattice spacings ranging from 0.054 to 0.116 fm, with full nonperturbative renormalization and running and in volumes of up to (6 fm)\(^3\) guaranteeing that the continuum limit, renormalization and infinitevolume extrapolation are controlled. It does neglect, however, isospinbreaking effects, which are small on the scale of their error bars.
Finally we come to another calculation which satisfies our selection criteria, HPQCD 10 [9]. It updates the staggered fermions calculation of HPQCD 09A [18]. In these papers the renormalized mass of the strange quark is obtained by combining the result of a precise calculation of the renormalized charmquark mass, \(m_c\), with the result of a calculation of the quarkmass ratio, \(m_c/m_s\). As described in Ref. [152] and in Sect. 3.2, HPQCD determines \(m_c\) by fitting Euclideantime moments of the \(\bar{c}c\) pseudoscalar density 2point functions, obtained numerically in lattice QCD, to fourthorder, continuum perturbative expressions. These moments are normalized and chosen so as to require no renormalization with staggered fermions. Since \(m_c/m_s\) requires no renormalization either, HPQCD’s approach displaces the problem of lattice renormalization in the computation of \(m_s\) to one of computing continuum perturbative expressions for the moments. To calculate \(m_{ud}\) HPQCD 10 [9] use the MILC 09 determination of the quarkmass ratio \(m_s/m_{ud}\) [89].
HPQCD 09A [18] obtains \(m_c/m_s=11.85(16)\) [18] fully nonperturbatively, with a precision slightly larger than 1%. HPQCD 10’s determination of the charmquark mass, \(m_c(m_c)=1.268(6)\),^{13} is even more precise, achieving an accuracy better than 0.5%. While these errors are, perhaps, surprisingly small, we take them at face value as we do those of RBC/UKQCD 14, since we will add a 2% error due to the quenching of the charm on the final result.
This discussion leaves us with four results for our final average for \(m_s\): MILC 09A [6], BMW 10A, 10B [7, 8], HPQCD 10 [9] and RBC/UKQCD 14 [10]. Assuming that the result from HPQCD 10 is 100% correlated with that of MILC 09A, as it is based on a subset of the MILC 09A configurations, we find \(m_s\!=\!92.0(1.1)\,\mathrm{MeV}\) with a \(\chi ^2/\hbox {d.o.f.}\! =\! 1.8.\)
3.1.4 Lattice determinations of \(m_s/m_{ud}\)
Lattice results for the ratio \(m_s/m_{ud}\)
Collaboration  Refs.  \(N_{ f}\)  Publication status  Chiral extrapolation  Continuum extrapolation  Finite volume  \(m_s/m_{ud}\) 

FNAL/MILC 14A  [14]  \(2+1+1\)  A  \(27.35(5)^{+10}_{7}\)  
ETM 14  [4]  \(2+1+1\)  A  26.66(32)(2)  
RBC/UKQCD 14B  [10]  \(2+1\)  P  27.34(21)  
RBC/UKQCD 12\(^{\mathrm{a}}\)  [31]  \(2+1\)  A  27.36(39)(31)(22)  
PACSCS 12\(^{\mathrm{b}}\)  [143]  \(2+1\)  A  26.8(2.0)  
Laiho 11  [44]  \(2+1\)  C  28.4(0.5)(1.3)  
BMW 10A, 10B\(^{\mathrm{c}}\)  \(2+1\)  A  27.53(20)(8)  
RBC/UKQCD 10A  [144]  \(2+1\)  A  26.8(0.8)(1.1)  
Blum 10\(^{\mathrm{d}}\)  [103]  \(2+1\)  A  28.31(0.29)(1.77)  
PACSCS 09  [94]  \(2+1\)  A  31.2(2.7)  
MILC 09A  [6]  \(2+1\)  C  27.41(5)(22)(0)(4)  
MILC 09  [89]  \(2+1\)  A  27.2(1)(3)(0)(0)  
PACSCS 08  [93]  \(2+1\)  A  28.8(4)  
RBC/UKQCD 08  [145]  \(2+1\)  A  28.8(0.4)(1.6)  
MILC 04, HPQCD/MILC/UKQCD 04  \(2+1\)  A  27.4(1)(4)(0)(1)  
ETM 14D  [160]  2  C  27.63(13)  
ETM 10B  [11]  2  A  27.3(5)(7)  
RBC 07\(^{\mathrm{d}}\)  [105]  2  A  28.10(38)  
ETM 07  [133]  2  A  27.3(0.3)(1.2)  
QCDSF/UKQCD 06  [139]  2  A  27.2(3.2) 
The lattice results in Table 6, which satisfy our selection criteria, indicate that the corrections generated by the nonleading terms of the chiral perturbation series are remarkably small, in the range 3–10%. Despite the fact that the SU(3)flavoursymmetrybreaking effects in the Nambu–Goldstone boson masses are very large (\(M_K^2\simeq 13\, M_\pi ^2\)), the mass spectrum of the pseudoscalar octet obeys the \(SU(3)\times SU(3)\) Eq. (20) very well.
\(N_{ f}=2\) lattice calculations With respect to the FLAG 13 review [2] there is only one new result, ETM 14D [160], based on recent ETM gauge ensembles generated close to the physical point with the addition of a clover term to the tmQCD action. The new simulations are performed at a single lattice spacing of \({\simeq } 0.09\) fm and at a single box size \(L \simeq 4\) fm and therefore their calculations do not pass our criteria for the continuum extrapolation and finitevolume effects.
\(N_{ f}=2+1\) lattice calculations For \(N_f = 2+1\) our average of \(m_s/m_{ud}\) is based on the new result RBC/UKQCD 14B, which replaces RBC/UKQCD 12 (see Sect. 3.1.3), and on the results MILC 09A and BMW 10A, 10B. The value quoted by HPQCD 10 does not represent independent information as it relies on the result for \(m_s/m_{ud}\) obtained by the MILC Collaboration. Averaging these results according to the prescriptions of Sect. 2.3 gives \(m_s / m_{ud} = 27.43(13)\) with \(\chi ^2/\hbox {d.o.f.} \simeq 0.2\). Since the errors associated with renormalization drop out in the ratio, the uncertainties are even smaller than in the case of the quark masses themselves: the above number for \(m_s/m_{ud}\) amounts to an accuracy of 0.5%.
\(N_{ f}=2+1+1\) lattice calculations For \(N_f = 2+1+1\) there are two results, ETM 14 [4] and FNAL/MILC 14A [14], both of which satisfy our selection criteria.
ETM 14 uses 15 twistedmass gauge ensembles at three lattice spacings ranging from 0.062 to 0.089 fm (using \(f_\pi \) as input), in boxes of size ranging from 2.0 to 3.0 fm and pion masses from 210 to 440 MeV (explaining the tag Open image in new window in the chiral extrapolation and the tag Open image in new window for the continuum extrapolation). The value of \(M_\pi L\) at their smallest pion mass is 3.2 with more than two volumes (explaining the tag Open image in new window in the finitevolume effects). They fix the strange mass with the kaon mass.
Note that our averages (23), (24) and (25), obtained for \(N_f = 2\), \(2+1\) and \(2+1+1\), respectively, agree very well within the quoted errors. They also show that the LO prediction of \(\chi \)PT in Eq. (22) receives only small corrections from higher orders of the chiral expansion: according to Eqs. (24) and (25), these generate shifts of 5.9(1.1) and \(5.4(1.2) \%\) relative to Eq. (22), respectively.
The ratio \(m_s/m_{ud}\) can also be extracted from the masses of the neutral Nambu–Goldstone bosons: neglecting effects of order \((m_um_d)^2\) also here, the leadingorder formula reads \(m_s / m_{ud} \mathop {=}\limits ^{{\mathrm{LO}}}\frac{3}{2} \hat{M}_\eta ^2 / \hat{M}_\pi ^2  \frac{1}{2}\). Numerically, this gives \(m_s / m_{ud} \mathop {=}\limits ^{{\mathrm{LO}}}24.2\). The relation has the advantage that the e.m. corrections are expected to be much smaller here, but it is more difficult to calculate the \(\eta \)mass on the lattice. The comparison with Eqs. (24) and (25) shows that, in this case, the NLO contributions are somewhat larger: 11.9(9) and \(11.4( 1.1) \%\).
3.1.5 Lattice determination of \(m_u\) and \(m_d\)
Since FLAG 13, two new results have been reported for nondegenerate lightquark masses, ETM 14 [4], and QCDSF/UKQCD 15 [166], for \(N_f=2+1+1\), and 3 flavours respectively. The former uses simulations in pure QCD, but determines \(m_um_d\) from the slope of the square of the kaon mass and the neutralcharged masssquares difference, evaluated at the isospinsymmetric point. The latter uses QCD+QED dynamical simulations performed at the SU(3)flavoursymmetric point, but at a single lattice spacing, so they do not enter our average. While QCDSF/UKQCD 15 use three volumes, the smallest has linear size roughly 1.7 fm, and the smallest partially quenched pion mass is greater than 200 MeV, so our finitevolume and chiralextrapolation criteria require Open image in new window ratings. In Ref. [166] results for \(\epsilon \) and \(m_{u}/m_{d}\) are computed in the socalled Dashen scheme. A subsequent paper [118] gives formulae to convert the \(\epsilon \) parameters to the \(\overline{\mathrm{MS}}\) scheme.
As the above implies, the determination of \(m_u\) and \(m_d\) separately requires additional input. MILC 09A [6] uses the mass difference between \(K^0\) and \(K^+\), from which they subtract electromagnetic effects using Dashen’s theorem with corrections, as discussed in Sect. 3.1.1. The up and down sea quarks remain degenerate in their calculation, fixed to the value of \(m_{ud}\) obtained from \(M_{\pi ^0}\).
To determine \(m_u/m_d\), BMW 10A, 10B [7, 8] follow a slightly different strategy. They obtain this ratio from their result for \(m_s/m_{ud}\) combined with a phenomenological determination of the isospinbreaking quarkmass ratio \(Q=22.3(8)\), defined below in Eq. (32), from \(\eta \rightarrow 3\pi \) decays [101] (the decay \(\eta \rightarrow 3\pi \) is very sensitive to QCD isospin breaking but fairly insensitive to QED isospin breaking). As discussed in Sect. 3.1.6, the central value of the e.m. parameter \(\epsilon \) in Eq. (13) is taken from the same source.
Lattice results for \(m_u\), \(m_d\) (MeV) and for the ratio \(m_u/m_d\). The values refer to the \({\overline{\text {MS}}}\) scheme at scale 2 GeV. The top part of the table lists the result obtained with \(N_{ f}=2+1+1\), while the middle and lower part presents calculations with \(N_f = 2+1 \) and \(N_f = 2\), respectively
Collaboration  Refs.  Publication status  Chiral extrapolation  Continuum extrapolation  Finite volume  Renormalization  Running  \(m_u\)  \(m_d\)  \(m_u/m_d\) 

MILC 14  [113]  C  −  −  \(0.4482(48)({}^{+\phantom {0}21}_{115})(1)(165)\)  
ETM 14  [4]  A  \(\,b\)  2.36(24)  5.03(26)  0.470(56)  
QCDSF/UKQCD 15\(^{\mathrm{a}}\)  [166]  P  −  −  0.52(5)  
PACSCS 12\(^{\mathrm{b}}\)  [143]  A  \(\,a\)  2.57(26)(7)  3.68(29)(10)  0.698(51)  
Laiho 11  [44]  C  −  1.90(8)(21)(10)  4.73(9)(27)(24)  0.401(13)(45)  
HPQCD 10\(^{\mathrm{c}}\)  [9]  A  −  2.01(14)  4.77(15)  
BMW 10A, 10B\(^{\mathrm{d}}\)  A  \(\,b\)  2.15(03)(10)  4.79(07)(12)  0.448(06)(29)  
Blum 10\(^{\mathrm{g}}\)  [103]  A  −  2.24(10)(34)  4.65(15)(32)  0.4818(96)(860)  
MILC 09A  [6]  C  −  1.96(0)(6)(10)(12)  4.53(1)(8)(23)(12)  0.432(1)(9)(0)(39)  
MILC 09  [89]  A  −  1.9(0)(1)(1)(1)  4.6(0)(2)(2)(1)  0.42(0)(1)(0)(4)  
MILC 04, HPQCD/ MILC/UKQCD 04  A  −  1.7(0)(1)(2)(2)  3.9(0)(1)(4)(2)  0.43(0)(1)(0)(8)  
RM123 13  [16]  A  \(\,c\)  2.40(15)(17)  4.80 (15)(17)  0.50(2)(3)  
RM123 11\(^{\mathrm{f}}\)  [167]  A  \(\,c\)  2.43(11)(23)  4.78(11)(23)  0.51(2)(4)  
Dürr 11\(^{\mathrm{e}}\)  [132]  A  −  −  2.18(6)(11)  4.87(14)(16)  
RBC 07\(^{\mathrm{g}}\)  [105]  A  −  3.02(27)(19)  5.49(20)(34)  0.550(31) 
Instead of subtracting electromagnetic effects using phenomenology, RBC 07 [105] and Blum 10 [103] actually include a quenched electromagnetic field in their calculation. This means that their results include corrections to Dashen’s theorem, albeit only in the presence of quenched electromagnetism. Since the up and down quarks in the sea are treated as degenerate, very small isospin corrections are neglected, as in MILC’s calculation.
PACSCS 12 [143] takes the inclusion of isospinbreaking effects one step further. Using reweighting techniques, it also includes electromagnetic and \(m_um_d\) effects in the sea.
It is interesting to compare the assumptions made or results obtained by the different collaborations for the violation of Dashen’s theorem. The input used in MILC 09A is \(\epsilon =1.2(5)\) [6], while the \(N_f=2\) computation of RM123 13 finds \(\epsilon =0.79(18)(18)\) [16]. As discussed in Sect. 3.1.6, the value of Q used by BMW 10A, 10B [7, 8] gives \(\epsilon =0.70(28)\) at NLO (see Eq. (40)). On the other hand, RBC 07 [105] and Blum 10 [103] obtain the results \(\epsilon =0.13(4)\) and \(\epsilon =0.5(1)\). The new results from QCDSF/UKQCD 15 give \(\epsilon =0.50(6)\) [118]. Note that PACSCS 12 [143] do not provide results which allow us to determine \(\epsilon \) directly. However, using their result for \(m_u/m_d\), together with Eq. (27), and neglecting NLO terms, one finds \(\epsilon =1.6(6)\), which is difficult to reconcile with what is known from phenomenology (see Sects. 3.1.1 and 3.1.6). Since the values assumed or obtained for \(\epsilon \) differ, it does not come as a surprise that the determinations of \(m_u/m_d\) are different.
These values of \(\epsilon \) are also interesting because they allow us to estimate the chiral corrections to the LO prediction (28) for \(m_u/m_d\). Indeed, evaluating the relation (27) for the values of \(\epsilon \) given above, and neglecting all other corrections in this equation, yields the LO values \((m_u/m_d)^\mathrm {LO}=0.46(4)\), 0.547(3), 0.52(1), 0.50(2), 0.49(2) and 0.51(1) for MILC 09A, RBC 07, Blum 10, BMW 10A, 10B, RM123 13, and QCDSF/UKQCD 15, respectively. However, in comparing these numbers to the nonperturbative results of Table 7 one must be careful not to double count the uncertainty arising from \(\epsilon \). One way to obtain a sharp comparison is to consider the ratio of the results of Table 7 to the LO values \((m_u/m_d)^\mathrm{LO}\), in which the uncertainty from \(\epsilon \) cancels to good accuracy. Here we will assume for simplicity that they cancel completely and will drop all uncertainties related to \(\epsilon \). For \(N_f = 2\) we consider RM123 13 [16], which updates RM123 11 and has no red dots. Since the uncertainties common to \(\epsilon \) and \(m_u/m_d\) are not explicitly given in Ref. [16], we have to estimate them. For that we use the leadingorder result for \(m_u/m_d\), computed with RM123 13’s value for \(\epsilon \). Its error bar is the contribution of the uncertainty on \(\epsilon \) to \((m_u/m_d)^\mathrm{LO}\). To good approximation this contribution will be the same for the value of \(m_u/m_d\) computed in Ref. [16]. Thus, we subtract it in quadrature from RM123 13’s result in Table 7 and compute \((m_u/m_d)/(m_u/m_d)^\mathrm{LO}\), dropping uncertainties related to \(\epsilon \). We find \((m_u/m_d)/(m_u/m_d)^\mathrm{LO} = 1.02(6)\). This result suggests that chiral corrections in the case of \(N_{ f}=2\) are negligible. For the two most accurate \(N_{ f}=2+1\) calculations, those of MILC 09A and BMW 10A, 10B, this ratio of ratios is 0.94(2) and 0.90(1), respectively. Though these two numbers are not fully consistent within our rough estimate of the errors, they indicate that higherorder corrections to Eq. (28) are negative and about 8% when \(N_{ f}=2+1\). In the following, we will take them to be 8(4)%. The fact that these corrections are seemingly larger and of opposite sign than in the \(N_{ f}=2\) case is not understood at this point. It could be an effect associated with the quenching of the strange quark. It could also be due to the fact that the RM123 13 calculation does not probe deeply enough into the chiral regime – it has \(M_\pi \,{\mathop {\sim }\limits ^{{>}}}\,270\,\mathrm{MeV}\) – to pick up on important chiral corrections. Of course, being less than a twostandarddeviation effect, it may be that there is no problem at all and that differences from the LO result are actually small.
Given the exploratory nature of the RBC 07 calculation, its results do not allow us to draw solid conclusions about the e.m. contributions to \(m_u/m_d\) for \(N_{ f}=2\). As discussed in Sect. 3.1.3 and here, the \(N_{ f}=2+1\) results of Blum 10, PACSCS 12, and QCDSF/UKQCD 15 do not pass our selection criteria either. We therefore resort to the phenomenological estimates of the electromagnetic selfenergies discussed in Sect. 3.1.1, which are validated by recent, preliminary lattice results.
It is interesting to note that in the results above, the errors are no longer dominated by the uncertainties in the input used for the electromagnetic corrections, though these are still significant at the level of precision reached in the \(N_f=2+1\) results. This is due to the reduction in the error on \(\epsilon \) discussed in Sect. 3.1.1. Nevertheless, the comparison of Eqs. (28) and (30) indicates that more than half of the difference between the prediction \(m_u/m_d=0.558\) obtained from Weinberg’s mass formulae [161] and the result for \(m_u/m_d\) obtained on the lattice stems from electromagnetism, the higher orders in the chiral perturbation generating a comparable correction.
In view of the fact that a massless upquark would solve the strong CPproblem, many authors have considered this an attractive possibility, but the results presented above exclude this possibility: the value of \(m_u\) in Eq. (30) differs from zero by 20 standard deviations. We conclude that nature solves the strong CPproblem differently. This conclusion relies on lattice calculations of kaon masses and on the phenomenological estimates of the e.m. selfenergies discussed in Sect. 3.1.1. The uncertainties therein currently represent the limiting factor in determinations of \(m_u\) and \(m_d\). As demonstrated in Refs. [16, 103, 104, 105, 110, 111, 112, 113, 114, 115, 116, 123], lattice methods can be used to calculate the e.m. selfenergies. Further progress on the determination of the lightquark masses hinges on an improved understanding of the e.m. effects.
3.1.6 Estimates for R and Q
Comparison of \(R_M\) and \(Q_M\) with the full results quoted above gives a handle on higherorder terms in the chiral expansion. Indeed, the ratios \(R_M/R\) and \(Q_M/Q\) give NLO and NNLO (and higher)corrections to the relations \(R \mathop {=}\limits ^{{\mathrm{LO}}}R_M\) and \(Q\mathop {=}\limits ^{{\mathrm{NLO}}}Q_M\), respectively. The uncertainties due to the use of the e.m. corrections of Eq. (13) are highly correlated in the numerators and denominators of these ratios, and we make the simplifying assumption that they cancel in the ratio. Thus, for \(N_f=2\) we evaluate Eqs. (38) and (39) using \(\epsilon =0.79(18)(18)\) from RM123 13 [16] and the other corrections from Eq. (13), dropping all uncertainties. We divide them by the results for R and Q in Eq. (35), omitting the uncertainties due to e.m. We obtain \(R_M/R\simeq 0.88(8)\) and \(Q_M/Q\simeq 0.91(5)\). We proceed analogously for \(N_f=2+1\) and 2+1+1, using \(\epsilon =0.70(3)\) from Eq. (13) and R and Q from Eqs. (36) and (37), and find \(R_M/R\simeq 1.02(5)\) and 1.03(17), and \(Q_M/Q\simeq 0.99(3)\) and 1.00(8). The chiral corrections appear to be small for three and four flavours, especially those in the relation of Q to \(Q_M\). This is less true for \(N_f=2\), where the NNLO and higher corrections to \(Q=Q_M\) could be significant. However, as for other quantities which depend on \(m_u/m_d\), this difference is not significant.
Our estimates for the strangequark and the average updownquark masses in the \({\overline{\text {MS}}}\) scheme at running scale \(\mu =2~\mathrm{GeV}\). Numerical values are given in MeV. In the results presented here, the error is the one which we obtain by applying the averaging procedure of Sect. 2.3 to the relevant lattice results. We have added an uncertainty to the \(N_f=2+1\) results, associated with the neglect of the charm seaquark and isospinbreaking effects, as discussed around Eqs. (18) and (24). This uncertainty is not included in the \(N_f=2\) results, as it should be smaller than the uncontrolled systematic associated with the neglect of strange seaquark effects
\(N_{ f}\)  \(m_{ud}\)  \( m_s \)  \(m_s/m_{ud}\) 

\(2+1+1\)  3.70(17)  93.9(1.1)  27.30(34) 
\(2+1\)  3.373(80)  92.0(2.1)  27.43(31) 
2  3.6(2)  101(3)  27.3(9) 
Our estimates for the masses of the two lightest quarks and related, strong isospinbreaking ratios. Again, the masses refer to the \({\overline{\text {MS}}}\) scheme at running scale \(\mu =2\,\mathrm{GeV}\). Numerical values are given in MeV. In the results presented here, the first error is the one that comes from lattice computations, while the second for \(N_f=2+1\) is associated with the phenomenological estimate of e.m. contributions, as discussed after Eq. (30). The second error on the \(N_f=2\) results for R and Q is also an estimate of the e.m. uncertainty, this time associated with the lattice computation of Ref. [16], as explained after Eq. (35). We present these results in a separate table, because they are less firmly established than those in Table 8. For \(N_f=2+1\) and \(2+1+1\) they still include information coming from phenomenology, in particular on e.m. corrections, and for \(N_f=2\) the e.m. contributions are computed neglecting the feedback of sea quarks on the photon field
\(N_{ f}\)  \(m_u \)  \(m_d \)  \(m_u/m_d\)  R  Q 

\(2+1+1\)  2.36(24)  5.03(26)  0.470(56)  35.6(5.1)  22.2 (1.6) 
\(2+1\)  2.16(9)(7)  4.68(14)(7)  0.46(2)(2)  35.0(1.9)(1.8)  22.5(6)(6) 
2  2.40(23)  4.80(23)  0.50(4)  40.7(3.7)(2.2)  24.3(1.4)(0.6) 
Our final results for the masses \(m_u\), \(m_d\), \(m_{ud}\), \(m_s\) and the mass ratios \(m_u/m_d\), \(m_s/m_{ud}\), R, Q are collected in Tables 8 and 9. We separate \(m_u\), \(m_d\), \(m_u/m_d\), R and Q from \(m_{ud}\), \(m_s\) and \(m_s/m_{ud}\), because the latter are completely dominated by lattice results while the former still include some phenomenological input.
3.2 Charmquark mass
In the present review we collect and discuss for the first time the lattice determinations of the \(\overline{\mathrm{MS}}\) charmquark mass \(\overline{m}_c\). Most of the results have been obtained by analyzing the latticeQCD simulations of 2point heavy–light or heavy–heavymeson correlation functions, using as input the experimental values of the D, \(D_s\) and charmonium mesons. The exceptions are represented by the HPQCD 14A [5] result at \(N_f = 2+1+1\), the HPQCD 08B [152], HPQCD 10 [9] and JLQCD 15B [174] results at \(N_f = 2 +1\), and the ETM 11F [175] result at \(N_f = 2\), where the moments method has been employed. The latter is based on the lattice calculation of the Euclidean time moments of pseudoscalarpseudoscalar correlators for heavyquark currents followed by an OPE expansion dominated by perturbative QCD effects, which provides the determination of both the heavyquark mass and the strong coupling constant \(\alpha _s\).
Lattice results for the \({\overline{\text {MS}}}\)charmquark mass \(\overline{m}_c(\overline{m}_c)\) and \(\overline{m}_c(3~\hbox {GeV})\) in GeV, together with the colour coding of the calculations used to obtain these. When not directly available in the publications, a conversion factor equal to 0.900 between the scales \(\mu = 2\) GeV and \(\mu = 3\) GeV (or equal to 0.766 between the scales \(\mu = \overline{m}_c\) and \(\mu = 3\) GeV) has been considered
Collaboration  Refs.  \(N_f\)  Publication status  Chiral extrapolation  Continuum extrapolation  Finite volume  Renormalization  \(\overline{m}_c(\overline{m}_c)\)  \(\overline{m}_c(3~\hbox {GeV})\) 

HPQCD 14A  [5]  \(2+1+1\)  A  −  1.2715(95)  0.9851(63)  
ETM 14A  [176]  \(2+1+1\)  A  1.3478(27)(195)  1.0557(22)(153)  
ETM 14  [4]  \(2+1+1\)  A  1.348(46)  1.058(35)  
JLQCD 15B  [174]  \(2+1\)  C  −  1.2769(21)(89)  0.9948(16)(69)  
\(\chi \)QCD 14  [17]  \(2+1\)  A  1.304(5)(20)  1.006(5)(22)  
HPQCD 10  [9]  \(2+1\)  A  −  1.273(6)  0.986(6)  
HPQCD 08B  [152]  \(2+1\)  A  −  1.268(9)  0.986(10)  
ALPHA 13B  [177]  2  C  1.274(36)  0.976(28)  
ETM 11F  [175]  2  C  −  1.279(12)/1.296(18)\(^{\mathrm{a}}\)  0.979(09)/0.998(14)\(^{\mathrm{a}}\)  
ETM 10B  [11]  2  A  1.28(4)  1.03(4)  
PDG  [151]  1.275(25) 
In the next subsections we review separately the results of \(\overline{m}_c(\overline{m}_c)\) for the various values of \(N_f\).
3.2.1 \(N_f = 2+1+1\) results
There are three recent results employing four dynamical quarks in the sea. ETM 14 [4] uses 15 twistedmass gauge ensembles at three lattice spacings ranging from 0.062 to 0.089 fm (using \(f_\pi \) as input), in boxes of size ranging from 2.0 to 3.0 fm and pion masses from 210 to 440 MeV (explaining the tag Open image in new window in the chiral extrapolation and the tag Open image in new window for the continuum extrapolation). The value of \(M_\pi L\) at their smallest pion mass is 3.2 with more than two volumes (explaining the tag Open image in new window in the finitevolume effects). They fix the strange mass with the kaon mass and the charm one with that of the \(D_s\) and D mesons.
ETM 14A [176] uses 10 out of the 15 gauge ensembles adopted in ETM 14 spanning the same range of values for the pion mass and the lattice spacing, but the latter is fixed using the nucleon mass. Two lattice volumes with size larger than 2.0 fm are employed. The physical strange and the charm mass are obtained using the masses of the \(\Omega ^\) and \(\Lambda _c^+\) baryons, respectively.
HPQCD 14A [5] works with the moments method adopting HISQ staggered fermions. Their results are based on 9 out of the 21 ensembles carried out by the MILC Collaboration [14] at 4 values of the coupling \(\beta \) corresponding to lattice spacings in the range from 0.057 to 0.153 fm, in boxes of sizes up to 5.8 fm and with tasteGoldstonepion masses down to 130 MeV and RMSpion masses down to 173 MeV. The strange and charmquark masses are fixed using as input the lattice result \(M_{\bar{s}s} = 688.5 (2.2)~\mathrm{MeV}\), calculated without including \(\bar{s}s\) annihilation effects, and \(M_{\eta _c} = 2.9863(27)~\mathrm{GeV}\), obtained from the experimental \(\eta _c\) mass after correcting for \(\bar{c}c\) annihilation and e.m. effects. All of the selection criteria of Sect. 2.1.1 are satisfied with the tag Open image in new window .^{14}
3.2.2 \(N_f = 2+1\) results
The HPQCD 10 [9] result is based on the moments method adopting a subset of \(N_f = 2+1\) Asqtadstaggeredfermion ensembles from MILC [89], on which HISQ valence fermions are studied. The charm mass is fixed from that of the \(\eta _c\) meson, \(M_{\eta _c} = 2.9852 (34) ~ \mathrm{GeV}\) corrected for \(\bar{c}c\) annihilation and e.m. effects. HPQCD 10 replaces the result HPQCD 08B [152], in which Asqtad staggered fermions have been used also for the valence quarks.
\(\chi \)QCD 14 [17] uses a mixedaction approach based on overlap fermions for the valence quarks and on domainwall fermions for the sea quarks. They adopt six of the gauge ensembles generated by the RBC/UKQCD Collaboration [144] at two values of the lattice spacing (0.087 and 0.11 fm) with unitary pion masses in the range from 290 to 420 MeV. For the valence quarks no lightquark masses are simulated. At the lightest pion mass \(M_\pi \simeq \) 290 MeV, the value of \(M_\pi L\) is 4.1, which satisfies the tag Open image in new window for the finitevolume effects. The strange and charmquark masses are fixed together with the lattice scale by using the experimental values of the \(D_s\), \(D_s^*\) and \(J/\psi \) meson masses.
JLQCD 15B [174] determines the charm mass through the moments method using Möbius domainwall fermions at three values of the lattice spacing, ranging from 0.044 to 0.083 fm. The lightest pion mass is \({\simeq } 230\) MeV and the corresponding value of \(M_\pi L\) is \({\simeq } 4.4\).
3.2.3 \(N_f = 2\) results
We turn now to the three results at \(N_f = 2\).
ETM 10B [11] is based on tmQCD simulations at four values of the lattice spacing in the range from 0.05 fm to 0.1 fm, with pion masses as low as 270 MeV at two lattice volumes. They fix the strangequark mass with either \(M_K\) or \(M_{\bar{s}s}\) and the charm mass using alternatively the D, \(D_s\) and \(\eta _c\) masses.
ETM 11F [175] is based on the same gauge ensemble as ETM 10B, but the moments method is adopted.
ALPHA 13B uses a subset of the CLS gauge ensembles with \(\mathcal {O}(a)\)improved Wilson fermions generated at two values of the lattice spacing (0.048 fm and 0.065 fm), using the kaon decay constant to fix the scale. The pion masses are as low as 190 MeV with the value of \(M_\pi L\) equal to \(\simeq 4\) at the lightest pion mass (explaining the tag Open image in new window for finitevolume effects).
3.2.4 Lattice determinations of the ratio \(m_c/m_s\)
Lattice results for the quarkmass ratio \(m_c/m_s\), together with the colour coding of the calculations used to obtain these
Collaboration  Refs.  \(N_{ f}\)  Publication status  Chiral extrapolation  Continuum extrapolation  Finite volume  \(m_c/m_s\) 

HPQCD 14A  [5]  \(2+1+1\)  A  11.652(35)(55)  
FNAL/MILC 14A  [14]  \(2+1+1\)  A  11.747(19)(\(^{+59}_{43}\))  
ETM 14  [4]  \(2+1+1\)  A  11.62(16)  
\(\chi \)QCD 14  [17]  \(2+1\)  A  11.1(8)  
HPQCD 09A  [18]  \(2+1\)  A  11.85(16)  
ETM 14D  [160]  2  C  12.29(10)  
Dürr 11  [132]  2  A  11.27(30)(26)  
ETM 10B  [11]  2  A  12.0(3) 
We begin with the \(N_f = 2\) results. Besides the result ETM 10B, already discussed in Sect. 3.2.3, there are two more results, Dürr 11 [132] and ETM 14D [160]. Dürr 11 [132] is based on QCDSF \(N_f = 2\) \(\mathcal {O}(a)\)improved Wilsonfermion simulations [139, 178] on which valence, Brillouinimproved Wilson quarks [179] are considered. It features only 2 ensembles with \(M_\pi < 400~\mathrm{MeV}\). The bare axialWardidentity (AWI) masses for \(m_s\) and \(m_c\) are tuned to simultaneously reproduce the physical values of \(M_{\bar{s}s}^2/(M_{D_s^*}^2M_{D_s}^2)\) and \((2M_{D_s^*}^2M_{\bar{s}s}^2)/(M_{D_s^*}^2M_{D_s}^2)\), where \(M_{\bar{s}s}^2 = 685.8 (8)\) MeV is the quarkconnected\(\bar{s}s\) pseudoscalar mass.
3.3 Bottomquark mass
Lattice results for the \({\overline{\text {MS}}}\)bottomquark mass \(\overline{m}_b(\overline{m}_b)\) in GeV, together with the systematic error ratings for each. Available results for the quark mass ratio \(m_b / m_c\) are also reported
Collaboration  Refs.  \(N_f\)  Publication status  Chiral extrapolation  Continuum extrapolation  Finite volume  Renormalization  Heavyquark treatment  \(\overline{m}_b(\overline{m}_b)\)  \(m_b / m_c\) 

HPQCD 14B  [19]  \(2+1+1\)  A  4.196(23)\(^{\mathrm{a}}\)  
ETM 14B  [180]  \(2+1+1\)  C  4.26(7)(14)  4.40(6)(5)  
HPQCD 14A  [5]  \(2+1+1\)  A  −  4.162(48)  4.528(14)(52)  
HPQCD 13B  [181]  \(2+1\)  A  −  −  4.166(43)  
HPQCD 10  [9]  \(2+1\)  A  −  4.164(23)\(^{\mathrm{b}}\)  4.51(4)  
ETM 13B  [20]  2  A  4.31(9)(8)  
ALPHA 13C  [21]  2  A  4.21(11)  
ETM 11A  [182]  2  A  4.29(14)  
PDG  [151]  4.18(3) 
3.3.1 \(N_f=2+1+1\)
Results have been published by HPQCD using NRQCD and HISQquark actions (HPQCD 14B [19] and HPQCD 14A [5], respectively). In both works the bquark mass is computed with the moments method, that is, from Euclideantime moments of 2point, heavy–heavy meson correlation functions (see Sect. 9.7 for a description of the method).
In HPQCD 14A the quark mass is computed using a similar strategy as above but with HISQ heavy quarks instead of NRQCD. The gaugefield ensembles are the same as in HPQCD 14B above plus the one with \(a = 0.06\) fm (four lattice spacings in all). Bare heavyquark masses are tuned to their physical values using the \(\eta _h\) mesons, and ratios of ratios yield \(m_h/m_c\). The \(\overline{\mathrm{MS}}\)charmquark mass determined as described in Sect. 3.2 then gives \(m_b\). The moment ratios are expanded using the OPE, and the quark masses and \(\alpha _S\) are determined from fits of the lattice ratios to this expansion. The fits are complicated: HPQCD uses cubic splines for valence and seamass dependence, with several knots, and many priors for 21 ratios to fit 29 data points. Taking this fit at face value results in a Open image in new window rating for the continuum limit since they use four lattice spacings down to 0.06 fm. See, however, the detailed discussion of the continuum limit given in Sect. 9.7 on \(\alpha _S\).
The third fourflavour result is from the ETM Collaboration and appears in a conference proceedings, so it is not included in our final average. The calculation is performed on a set of configurations generated with twisted Wilson fermions with three lattice spacings in the range 0.06 to 0.09 fm and with pion masses in the range 210 to 440 MeV. The bquark mass is determined from a ratio of heavy–light pseudoscalar meson masses designed to yield the quark pole mass in the static limit. The pole mass is related to the \(\overline{\mathrm{MS}}\) mass through perturbation theory at N\(^3\)LO. The key idea is that by taking ratios of ratios, the bquark mass is accessible through fits to heavy–light(strange)meson correlation functions computed on the lattice in the range \({\sim }1{}2\times m_c\) and the static limit, the latter being exactly 1. By simulating below \(\overline{m}_b\), taking the continuum limit is easier. They find \(\overline{m}_b(\overline{m}_b) = 4.26(7)(14)\) GeV, where the first error is statistical and the second systematic. The dominant errors come from setting the lattice scale and fit systematics.
3.3.2 \(N_f=2+1\)
HPQCD 13B [181] extracts \(\overline{m}_b\) from a lattice determination of the \(\Upsilon \) energy in NRQCD and the experimental value of the meson mass. The latter quantities yield the pole mass which is related to the \(\overline{\mathrm{MS}}\) mass in 3loop perturbation theory. The MILC coarse (0.12 fm) and fine (0.09 fm) Asqtad\(2+1\)flavour ensembles are employed in the calculation. The bare light(sea)quark masses correspond to a single, relatively heavy, pion mass of about 300 MeV. No estimate of the finitevolume error is given.
The value of \(\overline{m}_b(\overline{m}_b)\) reported in HPQCD 10 [9] is computed in a very similar fashion to the one in HPQCD 14A described in the last section, except that MILC \(2+1\)flavourAsqtad ensembles are used under HISQheavyvalence quarks. The lattice spacings of the ensembles range from 0.18 to 0.045 fm and pion masses down to about 165 MeV. In all, 22 ensembles were fit simultaneously. An estimate of the finitevolume error based on leadingorder perturbation theory for the moment ratio is also provided. Details of perturbation theory and renormalization systematics are given in Sect. 9.7.
3.3.3 \(N_f=2\)
The ETM Collaboration computes \(\overline{m}_b(\overline{m}_b)\) using the ratio method described above on twoflavour twistedmass gauge ensembles with four values of the lattice spacing in the range 0.10 to 0.05 fm and pion masses between 280 and 500 MeV (ETM 13B updates ETM 11). The heavyquark masses cover a range from charm to a little more than three GeV, plus the exact staticlimit point. They find \(\overline{m}_b(\overline{m}_b) = 4.31(9)(8)\) GeV for twoflavour running, while \(\overline{m}_b(\overline{m}_b) = 4.27(9)(8)\) using fourflavour running, from the 3 GeV scale used in the N\(^3\)LO perturbative matching calculation from the pole mass to the \(\overline{\mathrm{MS}}\) mass. The latter are computed nonperturbatively in the RIMOM scheme at 3 GeV and matched to \(\overline{\mathrm{MS}}\). The dominant errors are combined statistical \(+\) fit(continuum \(+\) chiral limits) and the uncertainty in setting the lattice scale. ETM quotes the average of two and fiveflavour results, \(\overline{m}_b(\overline{m}_b) = 4.29(9)(8)(2)\) where the last error is onehalf the difference between the two. In our average (see below), we use the twoflavour result.
The Alpha Collaboration uses HQET for heavy–light mesons to obtain \(m_b\) [21] (ALPHA 13C). They employ CLS, nonperturbatively improved, Wilson gauge field ensembles with three lattice spacings (0.075–0.048 fm), pion masses from 190 to 440 MeV, and three or four volumes at each lattice spacing, with \(m_\pi L > 4.0\). The barequark mass is related to the RGIscheme mass using the Schrödinger Functional technique with conversion to \(\overline{\mathrm{MS}}\) through fourloop anomalous dimensions for the mass. The final result, extrapolated to the continuum and chiral limits, is \(\overline{m}_b(\overline{m}_b) = 4.21(11)\) with twoflavour running, where the error combines statistical and systematic uncertainties. The value includes all corrections in HQET through \(\Lambda ^2/m_b\), but repeating the calculation in the static limit yields the identical result, indicating the HQET expansion is under very good control.
3.3.4 Averages for \(\overline{m}_b(\overline{m}_b)\)
4 Leptonic and semileptonic kaon and pion decay and \(V_{ud}\) and \(V_{us}\)
This section summarizes stateoftheart lattice calculations of the leptonic kaon and pion decay constants and the kaon semileptonicdecay form factor and provides an analysis in view of the Standard Model. With respect to the previous edition of the FLAG review [2] the data in this section has been updated. As in Ref. [2], when combining lattice data with experimental results, we take into account the strong SU(2) isospin correction, either obtained in lattice calculations or estimated by using chiral perturbation theory, both for the kaon leptonic decay constant \(f_{K^\pm }\) and for the ratio \(f_{K^\pm } / f_{\pi ^\pm }\).
4.1 Experimental information concerning \(V_{ud}\), \(V_{us}\), \(f_+(0)\) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}\)
Colour code for the data on \(f_+(0)\)
Collaboration  Refs.  \(N_{ f}\)  Publication status  Chiral extrapolation  Continuum extrapolation  Finitevolume errors  \(f_+(0)\) 

ETM 15C  [208]  \(2+1+1\)  C  0.9709(45)(9)  
FNAL/MILC 13E  [22]  \(2+1+1\)  A  0.9704(24)(22)  
FNAL/MILC 13C  [209]  \(2+1+1\)  C  0.9704(24)(32)  
RBC/UKQCD 15A  [24]  \(2+1\)  A  0.9685(34)(14)  
RBC/UKQCD 13  [210]  \(2+1\)  A  0.9670(20)(\(^{+18}_{46}\))  
FNAL/MILC 12I  [23]  \(2+1\)  A  0.9667(23)(33)  
JLQCD 12  [211]  \(2+1\)  C  0.959(6)(5)  
JLQCD 11  [212]  \(2+1\)  C  0.964(6)  
RBC/UKQCD 10  [213]  \(2+1\)  A  0.9599(34)(\(^{+31}_{47}\))(14)  
RBC/UKQCD 07  [214]  \(2+1\)  A  0.9644(33)(34)(14)  
ETM 10D  [215]  2  C  0.9544(68)\(_{\mathrm{stat}}\)  
ETM 09A  [25]  2  A  0.9560(57)(62)  
QCDSF 07  [216]  2  C  0.9647(15)\(_{\mathrm{stat}}\)  
RBC 06  [217]  2  A  0.968(9)(6)  
JLQCD 05  [218]  2  C  0.967(6), 0.952(6) 
Inclusive hadronic \(\tau \) decay offers an interesting way to measure \(V_{us}\), but a number of open issues yet remain to be clarified. In particular, the value of \(V_{us}\) as determined from \(\tau \) decays differs from the result one obtains from assuming threeflavour SMunitarity by more than three standard deviations [197]. It is important to understand this apparent tension better. A possibility is that at the current level of precision the treatment of higher orders in the operator product expansion and violations of quarkhadron duality may play a role. Very recently [202] a new implementation of the relevant sum rules has been elaborated suggesting a much larger value of \(V_{us}\) with respect to the result (56), namely \(V_{us} = 0.2228 (23)\), which is in much better agreement with CKM unitarity. Another possibility is that \(\tau \) decay involves new physics, but more work both on the theoretical and experimental side is required.
Colour code for the data on the ratio of decay constants: \(f_K/f_\pi \) is the pure QCD SU(2)symmetric ratio, while \(f_{K^\pm }/f_{\pi ^\pm }\) is in pure QCD including the SU(2) isospinbreaking correction
Collaboration  Refs.  \(N_{ f}\)  Publication status  Chiral extrapolation  Continuum extrapolation  Finitevolume errors  \(f_K/f_\pi \)  \(f_{K^\pm }/f_{\pi ^\pm }\) 

ETM 14E  [27]  \(2+1+1\)  A  1.188(11)(11)  1.184(12)(11)  
FNAL/MILC 14A  [14]  \(2+1+1\)  A  1.1956(10)(\(_{18}^{+26}\))  
ETM 13F  [230]  \(2+1+1\)  C  1.193(13)(10)  1.183(14)(10)  
HPQCD 13A  [26]  \(2+1+1\)  A  1.1948(15)(18)  1.1916(15)(16)  
MILC 13A  [231]  \(2+1+1\)  A  1.1947(26)(37)  
MILC 11  [232]  \(2+1+1\)  C  1.1872(42)\(_\mathrm{stat.}{}^{\mathrm{a}}\)  
ETM 10E  [233]  \(2+1+1\)  C  1.224(13)\(_\mathrm{stat}\)  
RBC/UKQCD 14B  [10]  \(2+1\)  A  1.1945(45)  
RBC/UKQCD 12  [31]  \(2+1\)  A  1.199(12)(14)  
Laiho 11  [44]  \(2+1\)  C  \(1.202(11)(9)(2)(5)^{\mathrm{b}}\)  
MILC 10  [29]  \(2+1\)  C  1.197(2)(\(^{+3}_{7}\))  
JLQCD/TWQCD 10  [234]  \(2+1\)  C  1.230(19)  
RBC/UKQCD 10A  [144]  \(2+1\)  A  1.204(7)(25)  
PACSCS 09  [94]  \(2+1\)  A  1.333(72)  
BMW 10  [30]  \(2+1\)  A  1.192(7)(6)  
JLQCD/TWQCD 09A  [235]  \(2+1\)  C  \(1.210(12)_\mathrm{stat}\)  
MILC 09A  [6]  \(2+1\)  C  1.198(2)(\(^{+6}_{8}\))  
MILC 09  [89]  \(2+1\)  A  1.197(3)(\(^{\;+6}_{13}\))  
Aubin 08  [236]  \(2+1\)  C  1.191(16)(17)  
PACSCS 08, 08A  \(2+1\)  A  1.189(20)  
RBC/UKQCD 08  [145]  \(2+1\)  A  1.205(18)(62)  
HPQCD/UKQCD 07  [28]  \(2+1\)  A  1.189(2)(7)  
NPLQCD 06  [238]  \(2+1\)  A  1.218(2)(\(^{+11}_{24}\))  
MILC 04  [107]  \(2+1\)  A  1.210(4)(13)  
ETM 14D  [160]  2  C  1.203(5)\(_\mathrm{stat}\)  
ALPHA 13A  [239]  2  C  1.1874(57)(30)  
BGR 11  [240]  2  A  1.215(41)  
ETM 10D  [215]  2  C  1.190(8)\(_\mathrm{stat}\)  
ETM 09  [32]  2  A  1.210(6)(15)(9)  
QCDSF/UKQCD 07  [241]  2  C  1.21(3) 
4.2 Lattice results for \(f_+(0)\) and \(f_{K^\pm }/f_{\pi ^\pm }\)
The database underlying the present review of the semileptonic form factor and the ratio of decay constants is listed in Tables 13 and 14. The properties of the lattice data play a crucial role for the conclusions to be drawn from these results: range of \(M_\pi \), size of \(L M_\pi \), continuum extrapolation, extrapolation in the quark masses, finitesize effects, etc. The key features of the various datasets are characterized by means of the colour code specified in Sect. 2.1. More detailed information on individual computations are compiled in Appendix B.2.
The expansion of \(f_+(0)\) in SU(3) chiral perturbation theory in powers of \(m_u\), \(m_d\) and \(m_s\) starts with \(f_+(0)=1+f_2+f_4+\cdots \) [129]. Since all of the lowenergy constants occurring in \(f_2\) can be expressed in terms of \(M_\pi \), \(M_K\), \(M_\eta \) and \(f_\pi \) [221], the NLO correction is known. In the language of the sum rule (57), \(f_2\) stems from nonstrange intermediate states with three mesons. Like all other nonexotic intermediate states, it lowers the value of \(f_+(0)\): \(f_2=0.023\) when using the experimental value of \(f_\pi \) as input. The corresponding expressions have also been derived in quenched or partially quenched (staggered) chiral perturbation theory [23, 222]. At the same order in the SU(2) expansion [223], \(f_+(0)\) is parameterized in terms of \(M_\pi \) and two a priori unknown parameters. The latter can be determined from the dependence of the lattice results on the masses of the quarks. Note that any calculation that relies on the \(\chi \)PT formula for \(f_2\) is subject to the uncertainties inherent in NLO results: instead of using the physical value of the pion decay constant \(f_\pi \), one may, for instance, work with the constant \(f_0\) that occurs in the effective Lagrangian and represents the value of \(f_\pi \) in the chiral limit. Although trading \(f_\pi \) for \(f_0\) in the expression for the NLO term affects the result only at NNLO, it may make a significant numerical difference in calculations where the latter are not explicitly accounted for (the lattice results concerning the value of the ratio \(f_\pi /f_0\) are reviewed in Sect. 5.3).
The lattice results shown in the left panel of Fig. 7 indicate that the higherorder contributions \(\Delta f\equiv f_+(0)1f_2\) are negative and thus amplify the effect generated by \(f_2\). This confirms the expectation that the exotic contributions are small. The entries in the lower part of the left panel represent various model estimates for \(f_4\). In Ref. [228] the symmetrybreaking effects are estimated in the framework of the quark model. The more recent calculations are more sophisticated, as they make use of the known explicit expression for the \(K_{\ell 3}\) form factors to NNLO in \(\chi \)PT [227, 229]. The corresponding formula for \(f_4\) accounts for the chiral logarithms occurring at NNLO and is not subject to the ambiguity mentioned above.^{17} The numerical result, however, depends on the model used to estimate the lowenergy constants occurring in \(f_4\) [224, 225, 226, 227]. The figure indicates that the most recent numbers obtained in this way correspond to a positive or an almost vanishing rather than a negative value for \(\Delta f\). We note that FNAL/MILC 12I [23] have made an attempt at determining a combination of some of the lowenergy constants appearing in \(f_4\) from lattice data.
4.3 Direct determination of \(f_+(0)\) and \(f_{K^\pm }/f_{\pi ^\pm }\)
All lattice results for the form factor \(f_+(0)\) and many available results for the ratio of decay constants, which we summarize here in Tables 13 and 14, respectively, have been computed in isospinsymmetric QCD. The reason for this unphysical parameter choice is that there are only few simulations of SU(2) isospinbreaking effects in lattice QCD, which is ultimately the cleanest way for predicting these effects [16, 103, 104, 110, 115, 167, 206, 207]. In the meantime one relies either on chiral perturbation theory [107, 129] to estimate the correction to the isospin limit or one calculates the breaking at leading order in \((m_um_d)\) in the valence quark sector by extrapolating the lattice data for the charged kaons to the physical value of the up(down)quark mass (the result for the pion decay constant is always extrapolated to the value of the average lightquark mass \(\hat{m}\)). This defines the prediction for \(f_{K^\pm }/f_{\pi ^\pm }\).
Since the majority of the collaborations present their newest results including the strong SU(2) isospinbreaking correction (as we will see this comprises the majority of results which qualify for inclusion into the FLAG average), we prefer to provide in Fig. 7 the overview of the world data of \(f_{K^\pm }/f_{\pi ^\pm }\), at variance with the choice made in the previous edition of the FLAG review [2]. For all the results of Table 14 provided only in the isospinsymmetric limit we apply individually an isospin correction which will be described later on (see equations Eqs. (62)–(63)).
The plots in Fig. 7 illustrate our compilation of data for \(f_+(0)\) and \(f_{K^\pm }/f_{\pi ^\pm }\). The lattice data for the latter quantity are largely consistent even when comparing simulations with different \(N_f\), while in the case of \(f_+(0)\) a slight tendency to get higher values for increasing \(N_f\) seems to be visible, even if it does not exceed one standard deviation. We now proceed to form the corresponding averages, separately for the data with \(N_{ f}=2+1+1\), \(N_{ f}=2+1\) and \(N_{ f}=2\) dynamical flavours and in the following we will refer to these averages as the “direct” determinations.
For \(f_+(0)\) there are currently two computational strategies: FNAL/MILC uses the Ward identity to relate the \(K\rightarrow \pi \) form factor at zero momentum transfer to the matrix element \(\langle \pi SK\rangle \) of the flavourchanging scalar current. Peculiarities of the staggered fermion discretization used by FNAL/MILC (see Ref. [23]) makes this the favoured choice. The other collaborations are instead computing the vectorcurrent matrix element \(\langle \pi V_\mu K\rangle \). Apart from FNAL/MILC 13C and the recent FNAL/MILC 13E all simulations in Table 13 involve unphysically heavy quarks and therefore the lattice data needs to be extrapolated to the physicalpion and kaon masses corresponding to the \(K^0\rightarrow \pi ^\) channel. We note also that the recent computations of \(f_+(0)\) obtained by the FNAL/MILC and RBC/UKQCD Collaborations make use of the partially twisted boundary conditions to determine the formfactor results directly at the relevant kinematical point \(q^2=0\) [242, 243], avoiding in this way any uncertainty due to the momentum dependence of the vector and/or scalar form factors. The ETM Collaboration uses partially twisted boundary conditions to compare the momentum dependence of the scalar and vector form factors with the one of the experimental data [215], while keeping at the same time the advantage of the highprecision determination of the scalar form factor at the kinematical endpoint \(q_{\mathrm{max}}^2 = (M_K  M_\pi )^2\) [25, 244] for the interpolation at \(q^2 = 0\).
According to the colour codes reported in Table 13 and to the FLAG rules of Sect. 2.2, only the result ETM 09A with \(N_{ f}=2\), the results FNAL/MILC 12I and RBC/UKQCD 15A with \(N_{ f}=2+1\) and the result FNAL/MILC 13E with \(N_{ f}=2+1+1\) dynamical flavours of fermions, respectively, can enter the FLAG averages.
At \(N_{ f}=2+1+1\) the new result from the FNAL/MILC Collaboration, \(f_+(0) = 0.9704 (24) (22)\) (FNAL/MILC 13E), is based on the use of the Highly Improved Staggered Quark (HISQ) action (for both valence and sea quarks), which has been taylored to reduce staggered tastebreaking effects, and includes simulations with three lattice spacings and physical lightquark masses. These features allow one to keep the uncertainties due to the chiral extrapolation and to the discretization artefacts well below the statistical error. The remaining largest systematic uncertainty comes from finitesize effects.
At \(N_{ f}=2+1\) there is a new result from the RBC/UKQCD Collaboration, \(f_+(0) = 0.9685 (34) (14)\) [24] (RBC/UKQCD 15A), which satisfies all FLAG criteria for entering the average. RBC/UKQCD 15A superseeds RBC/UKQCD 13 thanks to two new simulations at the physical point. The other result eligible to enter the FLAG average at \(N_{ f}=2+1\) is the one from FNAL/MILC 12I, \(f_+(0)=0.9667(23)(33)\). The two results, based on different fermion discretizations (staggered fermions in the case of FNAL/MILC and domainwall fermions in the case of RBC/UKQCD) are in nice agreement. Moreover, in the case of FNAL/MILC the form factor has been determined from the scalar current matrix element, while in the case of RBC/UKQCD it has been determined including also the matrix element of the vector current. To a certain extent both simulations are expected to be affected by different systematic effects.
RBC/UKQCD 15A has analysed results on ensembles with pion masses down to 140 MeV, mapping out the complete range from the SU(3)symmetric limit to the physical point. No significant cutoff effects (results for two lattice spacings) were observed in the simulation results. Ensembles with unphysical lightquark masses are weighted to work as a guide for small corrections toward the physical point, reducing in this way the model dependence in the fitting ansatz. The systematic uncertainty turns out to be dominated by finitevolume effects, for which an estimate based on effectivetheory arguments is provided.
The result FNAL/MILC 12I is from simulations reaching down to a lightest RMS pion mass of about 380 MeV (the lightest valence pion mass for one of their ensembles is about 260 MeV). Their combined chiral and continuum extrapolation (results for two lattice spacings) is based on NLO staggered chiral perturbation theory supplemented by the continuum NNLO expression [227] and a phenomenological parameterization of the breaking of the Ademollo–Gatto theorem at finitelattice spacing inherent in their approach. The \(p^4\) lowenergy constants entering the NNLO expression have been fixed in terms of external input [130].
The ETM Collaboration uses the twistedmass discretization and provides at \(N_{ f}=2\) a comprehensive study of the systematics [25, 215], by presenting results for four lattice spacings and by simulating at light pion masses (down to \(M_\pi = 260\) MeV). This makes it possible to constrain the chiral extrapolation, using both SU(3) [221] and SU(2) [223] chiral perturbation theory. Moreover, a rough estimate for the size of the effects due to quenching the strange quark is given, based on the comparison of the result for \(N_{ f}=2\) dynamical quark flavours [32] with the one in the quenched approximation, obtained earlier by the SPQcdR Collaboration [244].
In the case of the ratio of decay constants the datasets that meet the criteria formulated in the introduction are HPQCD 13A [26], FNAL/MILC 14A [14] (which updates MILC 13A [231]) and ETM 14E [27] with \(N_f=2+1+1\), MILC 10 [29], BMW 10 [30], HPQCD/UKQCD 07 [28] and RBC/UKQCD 12 [31] (which is an update of RBC/UKQCD 10A [144]) with \(N_{ f}=2+1\) and ETM 09 [32] with \(N_{ f}=2\) dynamical flavours.
ETM 14E uses the twistedmass discretization and provides a comprehensive study of the systematics by presenting results for three lattice spacings in the range 0.06–0.09 fm and for pion masses in the range 210–450 MeV. This makes it possible to constrain the chiral extrapolation, using both SU(2) [223] chiral perturbation theory and polynomial fits. The ETM Collaboration always includes the spread in the central values obtained from different ansätze into the systematic errors. The final result of their analysis is \( {f_{K^\pm }}/{f_{\pi ^\pm }}= 1.184(12)_\mathrm{stat+fit}(3)_\mathrm{Chiral}(9)_\mathrm{a^2}(1)_{Z_P}(3)_{FV}(3)_{IB}\) where the errors are (statistical + the error due to the fitting procedure), due to the chiral extrapolation, the continuum extrapolation, the massrenormalization constant, the finitevolume and (strong) isospinbreaking effects.
FNAL/MILC 14A has determined the ratio of the decay constants from a comprehensive set of HISQ ensembles with \(N_f = 2+1+1\) dynamical flavours. They have generated ensembles for four values of the lattice spacing (\(0.06{}0.15\) fm, scale set with \(f_{\pi ^+}\)) and with both physical and unphysical values of the light seaquark masses, controlling in this way the systematic uncertainties due to chiral and continuum extrapolations. With respect to MILC 13A they have increased the statistics and added an important ensemble at the finest lattice spacing and for physical values of the lightquark mass. The final result of their analysis is \( {f_{K^\pm }}/{f_{\pi ^\pm }}=1.1956(10)_\mathrm{stat}(_{14}^{+23})_\mathrm{a^2} (10)_{FV} (5)_{EM}\) where the errors are statistical, due to the continuum extrapolation, finitevolume and electromagnetic effects. With respect to MILC 13A a factor of \({\simeq } 2.6,~ 1.8\) and \(\simeq 1.7\) has been gained for the statistical, the discretization and the finitevolume errors.
HPQCD 13A analyses ensembles generated by MILC and therefore its study of \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) is based on the same set of ensembles bar the one for the finest lattice spacing (\(a = 0.09{}0.15\) fm, scale set with \(f_{\pi ^+}\) and relative scale set with the Wilson flow [245, 246]) supplemented by some simulation points with heavier quark masses. HPQCD employs a global fit based on continuum NLO SU(3) chiral perturbation theory for the decay constants supplemented by a model for higherorder terms including discretization and finitevolume effects (61 parameters for 39 data points supplemented by Bayesian priors). Their final result is \(f_{K^\pm }/f_{\pi ^\pm }=1.1916(15)_\mathrm{stat}(12)_\mathrm{a^2}(1)_{FV}(10)\), where the errors are statistical, due to the continuum extrapolation, due to finitevolume effects and the last error contains the combined uncertainties from the chiral extrapolation, the scalesetting uncertainty, the experimental input in terms of \(f_{\pi ^+}\) and from the uncertainty in \(m_u/m_d\).
In the previous edition of the FLAG review [2] the error budget of HPQCD 13A was compared with the one of MILC 13A and discussed in detail. It was pointed out that, despite the large overlap in primary lattice data, both collaborations arrive at surprisingly different error budgets. The same still holds when the comparison is made between HPQCD 13A and FNAL/MILC 14A.
Concerning the cutoff dependence, the finest lattice included into MILC’s analysis is \(a = 0.06\) fm while the finest lattice in HPQCD’s case is \(a = 0.09\) fm and both collaborations allow for tastebreaking terms in their analyses. MILC estimates the residual systematic after extrapolating to the continuum limit by taking the split between the result of an extrapolation with up to quartic and only up to quadratic terms in a as their systematic. HPQCD on the other hand models cutoff effects within their global fit ansatz up to including terms in \(a^8\), using priors for the unknown coefficients and without including the spread in the central values obtained from different ansätze into the systematic errors. In this way HPQCD arrives at a systematic error due to the continuum limit which is smaller than MILC’s estimate by about a factor \({\simeq } 1.8\).
Turning to finitevolume effects, NLO staggered chiral perturbation theory (MILC) or continuum chiral perturbation theory (HPQCD) was used for correcting the lattice data towards the infinitevolume limit. MILC then compared the finitevolume correction to the one obtained by the NNLO expression and took the difference as their estimate for the residual finitevolume error. In addition they checked the compatibility of the effectivetheory predictions (NLO continuum, staggered and NNLO continuum chiral perturbation theory) against lattice data of different spacial extent. The final verdict is that the related residual systematic uncertainty on \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) made by MILC is larger by an order of magnitude than the one made by HPQCD.
Values of the SU(2) isospinbreaking correction \(\delta _{SU(2)}\) applied to the lattice data for \(f_K/f_\pi \), entering the FLAG average at \(N_f=2+1\), for obtaining the corrected charged ratio \(f_{K^\pm }/f_{\pi ^\pm }\)
\(f_K/f_\pi \)  \(\delta _{SU(2)}\)  \(f_{K^\pm }/f_{\pi ^\pm }\)  

HPQCD/UKQCD 07  1.189(2)(7)  \(\)0.0040(7)  1.187(2)(2)(7) 
BMW 10  1.192(7)(6)  \(\)0.0041(7)  1.190(7)(2)(6) 
RBC/UKQCD 12  1.199(12)(14)  \(\)0.0043(9)  1.196(12)(2)(14) 
For \(N_f=2\) a dedicated study of the strongisospin correction in lattice QCD does exist. The (updated) result of the RM123 Collaboration [16] amounts to \(\delta _{SU(2)}=0.0080(4)\) and we use this result for the isospin correction of the ETM 09 result at \(N_f=2\).
Note that the RM123 value for the strongisospin correction is almost incompatible with the results based on SU(3) chiral perturbation theory, \(\delta _{SU(2)}=0.004(1)\) (see Table 15). Moreover, for \(N_f=2+1+1\) HPQCD 13A [26] and ETM 14E [27] estimate a value for \(\delta _{SU(2)}\) equal to \(0.0054(14)\) and \(0.0080(38)\), respectively. One would not expect the strange and heavier seaquark contributions to be responsible for such a large effect. Whether higherorder effects in chiral perturbation theory or other sources are responsible still needs to be understood. More lattice QCD simulations of SU(2) isospinbreaking effects are therefore required. To remain on the conservative side we add a \(100 \%\) error to the correction based on SU(3) chiral perturbation theory. For further analyses we add (in quadrature) such an uncertainty to the systematic error.
4.4 Tests of the Standard Model
The situation is similar for \(N_{ f}=2+1\): \(V_u^2 = 0.984(11)\) with the lattice data alone. Combining the lattice results for \(f_+(0)\) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) in Eqs. (59) and (65) with the \(\beta \) decay value of \(V_{ud}\), the test sharpens again considerably: the lattice result for \(f_+(0)\) leads to \(V_u^2 = 0.9991(6)\), while the one for \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) implies \(V_u^2 = 0.9999(6)\), thus confirming again CKM unitarity below the permille level.
Values of \(V_{us}\) and \(V_{ud}\) obtained from the lattice determinations of either \(f_+(0)\) or \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) assuming CKM unitarity. The first (second) number in brackets represents the statistical (systematic) error
Collaboration  Refs.  \(N_{ f}\)  From  \(V_{us}\)  \(V_{ud}\) 

FNAL/MILC 13E  [22]  \(2+1+1\)  \(f_+(0)\)  0.2231(7)(5)  0.97479(16)(12) 
ETM 14E  [27]  \(2+1+1\)  \( {f_{K^\pm }}/{f_{\pi ^\pm }}\)  0.2270(22)(20)  0.97388(51)(47) 
FNAL/MILC 14A  [14]  \(2+1+1\)  \( {f_{K^\pm }}/{f_{\pi ^\pm }}\)  0.2249(4)(4)  0.97438(8)(9) 
HPQCD 13A  [26]  \(2+1+1\)  \( {f_{K^\pm }}/{f_{\pi ^\pm }}\)  0.2256(4)(3)  0.97420(10)(7) 
RBC/UKQCD 15A  [24]  \(2+1\)  \(f_+(0)\)  0.2235(9)(3)  0.97469(20)(7) 
FNAL/MILC 12I  [23]  \(2+1\)  \(f_+(0)\)  0.2240(7)(8)  0.97459(16)(18) 
MILC 10  [29]  \(2+1\)  \( {f_{K^\pm }}/{f_{\pi ^\pm }}\)  0.2250(5)(9)  0.97434(11)(21) 
RBC/UKQCD 12  [144]  \(2+1\)  \( {f_{K^\pm }}/{f_{\pi ^\pm }}\)  0.2249(22)(25)  0.97438(50)(58) 
BMW 10  [30]  \(2+1\)  \( {f_{K^\pm }}/{f_{\pi ^\pm }}\)  0.2259(13)(11)  0.97413(30)(25) 
HPQCD/UKQCD 07  [28]  \(2+1\)  \( {f_{K^\pm }}/{f_{\pi ^\pm }}\)  0.2265(6)(13)  0.97401(14)(29) 
ETM 09A  [25]  2  \(f_+(0)\)  0.2265(14) (15)  0.97401(33)(34) 
ETM 09  [32]  2  \( {f_{K^\pm }}/{f_{\pi ^\pm }}\)  0.2233(11) (30)  0.97475(25)(69) 
Note that the above tests also offer a check of the basic hypothesis that underlies our analysis: we are assuming that the weak interaction between the quarks and the leptons is governed by the same Fermi constant as the one that determines the strength of the weak interaction among the leptons and determines the lifetime of the muon. In certain modifications of the Standard Model, this is not the case. In those models it need not be true that the rates of the decays \(\pi \rightarrow \ell \nu \), \(K\rightarrow \ell \nu \) and \(K\rightarrow \pi \ell \nu \) can be used to determine the matrix elements \(V_{ud}f_\pi \), \(V_{us}f_K\) and \(V_{us}f_+(0)\), respectively and that \(V_{ud}\) can be measured in nuclear \(\beta \) decay. The fact that the lattice data are consistent with unitarity and with the value of \(V_{ud}\) found in nuclear \(\beta \) decay indirectly also checks the equality of the Fermi constants.
4.5 Analysis within the Standard Model
The Standard Model implies that the CKM matrix is unitary. The precise experimental constraints quoted in (54) and the unitarity condition (67) then reduce the four quantities \(V_{ud},V_{us},f_+(0), {f_{K^\pm }}/{f_{\pi ^\pm }}\) to a single unknown: any one of these determines the other three within narrow uncertainties.
The upper half of the table shows our final results for \(V_{us}\), \(V_{ud}\), \(f_+(0)\) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}\), which are obtained by analysing the lattice data within the Standard Model. For comparison, the lower half lists the values that follow if the lattice results are replaced by the experimental results on nuclear \(\beta \) decay and \(\tau \) decay, respectively
Refs.  \(V_{us}\)  \(V_{ud}\)  \(f_+(0)\)  \( {f_{K^\pm }}/{f_{\pi ^\pm }}\)  

\(N_{ f}= 2+1+1\)  0.2250(11)  0.97440(19)  0.9622(50)  1.195(5)  
\(N_{ f}= 2+1\)  0.2243(10)  0.97451(23)  0.9652(47)  1.199(5)  
\(N_{ f}=2\)  0.2256(21)  0.97423(47)  0.9597(91)  1.192(9)  
\(\beta \) Decay  [186]  0.2258(9)  0.97417(21)  0.9588(42)  1.191(4) 
\(\tau \) Decay  [200]  0.2165(26)  0.9763(6)  1.0000(122)  1.245(12) 
\(\tau \) Decay  [199]  0.2208(39)  0.9753(9)  0.9805(174)  1.219(18) 
Colour code for the lattice data on \(f_{\pi ^\pm }\) and \(f_{K^\pm }\) together with information on the way the lattice spacing was converted to physical units and on whether or not an isospinbreaking correction has been applied to the quoted result (see Sect. 4.3). The numerical values are listed in MeV units
Collaboration  Refs.  \(N_{ f}\)  Publication status  Chiral extrapolation  Continuum extrapolation  Finitevolume errors  Renormalization  Physical scale  SU(2) breaking  \(f_{\pi ^\pm }\)  \(f_{K^\pm }\) 

ETM 14E  [27]  \(2+1+1\)  A  na  \(f_\pi \)  –  154.4(1.5)(1.3)  
FNAL/MILC 14A  [14]  \(2+1+1\)  A  na  \(f_\pi \)  –  155.92(13)(\(_{23}^{+34}\))  
HPQCD 13A  [26]  \(2+1+1\)  A  na  \(f_\pi \)  –  155.37(20)(27)  
MILC 13A  [231]  \(2+1+1\)  A  na  \(f_\pi \)  –  155.80(34)(54)  
ETM 10E  [233]  \(2+1+1\)  C  na  \(f_\pi \)  \(\checkmark \)  –  159.6(2.0)  
RBC/UKQCD 14B  [10]  \(2+1\)  A  NPR  \(m_\Omega \)  \(\checkmark \)  130.19(89)  155.18(89)  
RBC/UKQCD 12  [31]  \(2+1\)  A  NPR  \(m_\Omega \)  \(\checkmark \)  127.1(2.7)(2.7)  152.1(3.0)(1.7)  
Laiho 11  [44]  \(2+1\)  C  na  \({}^{\mathrm{a}}\)  130.53(87)(210)  156.8(1.0)(1.7)  
MILC 10  [29]  \(2+1\)  C  na  \({}^{\mathrm{a}}\)  129.2(4)(14)  –  
MILC 10  [29]  \(2+1\)  C  na  \(f_\pi \)  –  156.1(4)(\(_{9}^{+6}\))  
JLQCD/TWQCD 10  [234]  \(2+1\)  C  na  \(m_\Omega \)  \(\checkmark \)  118.5(3.6)\(_\mathrm{stat}\)  145.7(2.7)\(_\mathrm{stat}\)  
RBC/UKQCD 10A  [144]  \(2+1\)  A  NPR  \(m_\Omega \)  \(\checkmark \)  124(2)(5)  148.8(2.0)(3.0)  
PACSCS 09  [94]  \(2+1\)  A  NPR  \(m_\Omega \)  \(\checkmark \)  124.1(8.5)(0.8)  165.0(3.4)(1.1)  
JLQCD/TWQCD 09A  [235]  \(2+1\)  C  na  \(f_\pi \)  \(\checkmark \)  –  156.9(5.5)\(_\mathrm{stat}\)  
MILC 09A  [6]  \(2+1\)  C  na  \(\Delta M_\Upsilon \)  128.0(0.3)(2.9)  153.8(0.3)(3.9)  
MILC 09A  [6]  \(2+1\)  C  na  \(f_\pi \)  –  156.2(0.3)(1.1)  
MILC 09  [89]  \(2+1\)  A  na  \(\Delta M_\Upsilon \)  128.3(0.5)(\(^{+2.4}_{3.5}\))  154.3(0.4)(\(^{+2.1}_{3.4}\))  
MILC 09  [89]  \(2+1\)  A  na  \(f_\pi \)  156.5(0.4)(\(^{+1.0}_{2.7}\))  
Aubin 08  [236]  \(2+1\)  C  na  \(\Delta M_\Upsilon \)  129.1(1.9)(4.0)  153.9(1.7)(4.4)  
PACSCS 08, 08A  \(2+1\)  A  1lp  \(m_\Omega \)  \(\checkmark \)  134.0(4.2)\(_\mathrm{stat}\)  159.0(3.1)\(_\mathrm{stat}\)  
RBC/UKQCD 08  [145]  \(2+1\)  A  NPR  \(m_\Omega \)  \(\checkmark \)  124.1(3.6)(6.9)  149.4(3.6)(6.3)  
HPQCD/UKQCD 07  [28]  \(2+1\)  A  na  \(\Delta M_\Upsilon \)  \(\checkmark \)  132(2)  156.7(0.7)(1.9)  
MILC 04  [107]  \(2+1\)  A  na  \(\Delta M_\Upsilon \)  129.5(0.9)(3.5)  156.6(1.0)(3.6)  
ETM 14D  [160]  2  C  na  \(f_\pi \)  \(\checkmark \)  –  153.3(7.5)\(_\mathrm{stat}\)  
TWQCD 11  [249]  2  P  na  \({r_0}^{\mathrm{c}}\)  127.3(1.7)(2.0)\(^{\mathrm{d}}\)  –  
ETM 09  [32]  2  A  na  \(f_\pi \)  \(\checkmark \)  –  157.5(0.8)(2.0)(1.1)\(^{\mathrm{b}}\)  
JLQCD/TWQCD 08A  [138]  2  A  na  \(r_0\)  119.6(3.0)(\(^{+6.5}_{1.0}\))\(^{\mathrm{d}}\)  – 
For \(N_{ f}=2+1+1\) we consider the data both for \(f_+(0)\) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}\), treating FNAL/MILC 13E, FNAL/MILC 14A and HPQCD 13A as statistically correlated (according to the prescription of Sect. 2.3). We obtain \(V_{us}=0.2250(11)\), where the error includes the inflation factor due the value of \(\chi ^2/\mathrm{d.o.f.} \simeq 2.3\). This result is indicated on the left hand side of Fig. 9 by the narrow vertical band. In the case \(N_f = 2+1\) we consider MILC 10, FNAL/MILC 12I and HPQCD/UKQCD 07 on the one hand and RBC/UKQCD 12 and RBC/UKQCD 15A on the other hand, as mutually statistically correlated, since the analysis in the two cases starts from partly the same set of gauge ensembles. In this way we arrive at \(V_{us} = 0.2243(10)\) with \(\chi ^2/\mathrm{d.o.f.} \simeq 1.0\). For \(N_{ f}=2\) we consider ETM 09A and ETM 09 as statistically correlated, obtaining \(V_{us}=0.2256(21)\) with \(\chi ^2/\mathrm{d.o.f.} \simeq 0.7\). The figure shows that the result obtained for the data with \(N_{ f}=2\), \(N_{ f}=2+1\) and \(N_{ f}=2+1+1\) are consistent with each other.
Alternatively, we can solve the relations for \(V_{ud}\) instead of \(V_{us}\). Again, the result \(V_{ud}=0.97440(19)\), which follows from the lattice data with \(N_{ f}=2+1+1\), is perfectly consistent with the values \(V_{ud}=0.97451(23)\) and \(V_{ud}=0.97423(47)\) obtained from the data with \(N_{ f}=2+1\) and \(N_{ f}=2\), respectively. The reduction of the uncertainties in the result for \(V_{ud}\) due to CKM unitarity is to be expected from Fig. 8: the unitarity condition reduces the region allowed by the lattice results to a nearly vertical interval.
Next, we determine the values of \(f_+(0)\) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) that follow from our determinations of \(V_{us}\) and \(V_{ud}\) obtained from the lattice data within the Standard Model. We find \(f_+(0) = 0.9622(50)\) for \(N_{ f}=2+1+1\), \(f_+(0) = 0.9652(47)\) for \(N_{ f}=2+1\), \(f_+(0) = 0.9597(91)\) for \(N_{ f}=2\) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}= 1.195(5)\) for \(N_{ f}=2+1+1\), \( {f_{K^\pm }}/{f_{\pi ^\pm }}= 1.199(5)\) for \(N_{ f}=2+1\), \( {f_{K^\pm }}/{f_{\pi ^\pm }}= 1.192(9) \) for \(N_{ f}=2\), respectively. These results are collected in the upper half of Table 17. In the lower half of the table, we list the analogous results found by working out the consequences of the CKM unitarity using the values of \(V_{ud}\) and \(V_{us}\) obtained from nuclear \(\beta \) decay and \(\tau \) decay, respectively. The comparison shows that the lattice result for \(V_{ud}\) not only agrees very well with the totally independent determination based on nuclear \(\beta \) transitions, but it is also remarkably precise. On the other hand, the values of \(V_{ud}\), \(f_+(0)\) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) which follow from the \(\tau \)decay data if the Standard Model is assumed to be valid, are not in good agreement with the lattice results for these quantities. The disagreement is reduced considerably if the analysis of the \(\tau \) data is supplemented with experimental results on electroproduction [199]: the discrepancy then amounts to little more than one standard deviation.
4.6 Direct determination of \(f_{K^\pm }\) and \(f_{\pi ^\pm }\)
It is useful for flavour physics studies to provide not only the lattice average of \(f_{K^\pm } / f_{\pi ^\pm }\), but also the average of the decay constant \(f_{K^\pm }\). The case of the decay constant \(f_{\pi ^\pm }\) is different, since the experimental value of this quantity is often used for setting the scale in lattice QCD (see Appendix A.2). However, the physical scale can be set in different ways, namely by using as input the mass of the \(\Omega \)baryon (\(m_\Omega \)) or the \(\Upsilon \)meson spectrum (\(\Delta M_\Upsilon \)), which are less sensitive to the uncertainties of the chiral extrapolation in the lightquark mass with respect to \(f_{\pi ^\pm }\). In such cases the value of the decay constant \(f_{\pi ^\pm }\) becomes a direct prediction of the latticeQCD simulations. It is therefore interesting to provide also the average of the decay constant \(f_{\pi ^\pm }\), obtained when the physical scale is set through another hadron observable, in order to check the consistency of different scalesetting procedures.
Our compilation of the values of \(f_{\pi ^\pm }\) and \(f_{K^\pm }\) with the corresponding colour code is presented in Table 18. With respect to the case of \(f_{K^\pm } / f_{\pi ^\pm }\) we have added two columns indicating which quantity is used to set the physical scale and the possible use of a renormalization constant for the axial current. Indeed, for several lattice formulations the use of the nonsinglet axialvector Ward identity allows one to avoid the use of any renormalization constant.
One can see that the determinations of \(f_{\pi ^\pm }\) and \(f_{K^\pm }\) suffer from larger uncertainties with respect to the ones of the ratio \(f_{K^\pm } / f_{\pi ^\pm }\), which is less sensitive to various systematic effects (including the uncertainty of a possible renormalization constant) and, moreover, is not exposed to the uncertainties of the procedure used to set the physical scale.
According to the FLAG rules, for \(N_f = 2 + 1 + 1\) three datasets can form the average of \(f_{K^\pm }\) only: ETM 14E [27], FNAL/MILC 14A [14] and HPQCD 13A [26]. Following the same procedure already adopted in Sect. 4.3 in the case of the ratio of the decay constant we treat FNAL/MILC 14A and HPQCD 13A as statistically correlated. For \(N_f = 2 + 1\) three datasets can form the average of \(f_{\pi ^\pm }\) and \(f_{K^\pm }\) : RBC/UKQCD 12 [31] (update of RBC/UKQCD 10A), HPQCD/UKQCD 07 [28] and MILC 10 [29], which is the latest update of the MILC program. We consider HPQCD/UKQCD 07 and MILC 10 as statistically correlated and use the prescription of Sect. 2.3 to form an average. For \(N_f = 2\) the average cannot be formed for \(f_{\pi ^\pm }\), and only one data set (ETM 09) satisfies the FLAG rules in the case of \(f_{K^\pm }\).
5 Lowenergy constants
In the study of the quarkmass dependence of QCD observables calculated on the lattice, it is common practice to invoke chiral perturbation theory (\(\chi \)PT). For a given quantity this framework predicts the nonanalytic quarkmass dependence and it provides symmetry relations among different observables. These relations are best expressed with the help of a set of linearly independent and universal (i.e. processindependent) lowenergy constants (LECs), which appear as coefficients of the polynomial terms (in \(m_q\) or \(M_\pi ^2\)) in different observables. When numerical simulations are done at heavier than physical (light) quark masses, \(\chi \)PT is usually invoked in the extrapolation to physical quark masses.
5.1 Chiral perturbation theory
\(\chi \)PT is an effective field theory approach to the lowenergy properties of QCD based on the spontaneous breaking of chiral symmetry, \(SU(N_{ f})_L \times SU(N_{ f})_R \rightarrow SU(N_{ f})_{L+R}\), and its soft explicit breaking by quarkmass terms. In its original implementation, in infinite volume, it is an expansion in \(m_q\) and \(p^2\) with power counting \(M_\pi ^2 \sim m_q \sim p^2\).
If the box volume is finite but large compared to the Compton wavelength of the pion, \(L \gg 1/M_\pi \), the power counting generalizes to \(m_q \sim p^2 \sim 1/L^2\), as one would assume based on the fact that \(p_\mathrm {min}=2\pi /L\) is the minimum momentum in a finite box. This is the socalled pregime of \(\chi \)PT. It coincides with the setting that is used for standard phenomenologically oriented latticeQCD computations, and we shall consider the pregime the default in the following. However, if the pion mass is so small that the boxlength L is no longer large compared to the Compton wavelength that the pion would have, at the given \(m_q\), in infinite volume, then the chiral series must be reordered. Such finitevolume versions of \(\chi \)PT with correspondingly adjusted powercounting schemes, referred to as \(\epsilon \) and \(\delta \)regime, are described in Sects. 5.1.4 and 5.1.5, respectively.
 1.
All LECs up to order \(p^4\) (with the exception of B and \(B_0\), since only the product of these times the quark masses can be estimated from phenomenology) have either been determined by comparison to experiment or estimated theoretically, e.g. in large\(N_c\) QCD. A lattice determination of the better known LECs thus provides a test of the \(\chi \)PT approach.
 2.
The less wellknown LECs are those which describe the quarkmass dependence of observables – these cannot be determined from experiment, and therefore the lattice, where quark masses can be varied, provides unique quantitative information. This information is essential for improving phenomenological \(\chi \)PT predictions in which these LECs play a role.
The fact that, at large volume, the finitesize effects, which occur if a system undergoes spontaneous symmetry breakdown, are controlled by the Nambu–Goldstone modes, was first noted in solid state physics, in connection with magnetic systems [255, 256]. As pointed out in Ref. [257] in the context of QCD, the thermal properties of such systems can be studied in a systematic and modelindependent manner by means of the corresponding effective field theory, provided the temperature is low enough. While finite volumes are not of physical interest in particle physics, lattice simulations are necessarily carried out in a finite box. As shown in Refs. [258, 259, 260], the ensuing finitesize effects can be studied on the basis of the effective theory – \(\chi \)PT in the case of QCD – provided the simulation is close enough to the continuum limit, the volume is sufficiently large and the explicit breaking of chiral symmetry generated by the quark masses is sufficiently small. Indeed, \(\chi \)PT represents a useful tool for the analysis of the finitesize effects in lattice simulations.
In the remainder of this subsection we collect the relevant \(\chi \)PT formulae that will be used in the two following subsections to extract SU(2) and SU(3) LECs from lattice data.
5.1.1 Quarkmass dependence of pseudoscalar masses and decay constants
A. SU(2) formulae
5.1.2 Pion form factors and charge radii
The difference between the quarkline connected and the full (i.e. containing the connected and the disconnected pieces) scalar pion form factor has been investigated by means of \(\chi \)PT in Ref. [266]. It is expected that the technique used can be applied to a large class of observables relevant in QCD phenomenology.
As a point of practical interest let us remark that there are no finitevolume correction formulae for the meansquare radii \(\langle r^2\rangle _{V,S}\) and the curvatures \(c_{V,S}\). The lattice data for \(F_{V,S}(t)\) need to be corrected, point by point in t, for finitevolume effects. In fact, if a given t is realized through several inequivalent \(p_1\!\!p_2\) combinations, the level of agreement after the correction has been applied is indicative of how well higherorder effects are under control.
5.1.3 Partially quenched and mixed action formulations
The term “partially quenched QCD” is used in two ways. For heavy quarks (c, b and sometimes s) it usually means that these flavours are included in the valence sector, but not into the functional determinant, i.e. the sea sector. For the light quarks (u, d and sometimes s) it means that they are present in both the valence and the sea sector of the theory, but with different masses (e.g. a series of valence quark masses is evaluated on an ensemble with fixed seaquark masses).
The program of extending the standard (unitary) SU(3) theory to the (second version of) “partially quenched QCD” has been completed at the twoloop (NNLO) level for masses and decay constants [267]. These formulae tend to be complicated, with the consequence that a stateoftheart analysis with \(\mathcal {O}(2000)\) bootstrap samples on \(\mathcal {O}(20)\) ensembles with \(\mathcal {O}(5)\) masses each [and hence \(\mathcal {O}(200\,000)\) different fits] will require significant computational resources. For a summary of recent developments in \(\chi \)PT relevant to lattice QCD we refer to Ref. [268]. The SU(2) partially quenched formulae can be obtained from the SU(3) ones by “integrating out the strange quark.” At NLO, they can be found in Ref. [269] by setting the latticeartefact terms from the staggered \(\chi \)PT form to zero.
The theoretical underpinning of how “partial quenching” is to be understood in the (properly extended) chiral framework is given in Ref. [270]. Specifically, for partially quenched QCD with staggered quarks it is shown that a transfer matrix can be constructed which is not Hermitian but bounded, and can thus be used to construct correlation functions in the usual way. The program of calculating all observables in the pregime in finite volume to two loops, first completed in the unitary theory [271, 272], has been carried out for the partially quenched case, too [273].
A further extension of the \(\chi \)PT framework concerns the lattice effects that arise in partially quenched simulations where sea and valence quarks are implemented with different lattice fermion actions [222, 274, 275, 276, 277, 278, 279, 280].
5.1.4 Correlation functions in the \(\epsilon \)regime
The finitesize effects encountered in lattice calculations can be used to determine some of the LECs of QCD. In order to illustrate this point, we focus on the two lightest quarks, take the isospin limit \(m_u=m_d=m\) and consider a box of size \(L_s\) in the three space directions and size \(L_t\) in the time direction. If m is sent to zero at fixed box size, chiral symmetry is restored, and the zeromomentum mode of the pion field becomes nonperturbative. An intuitive way to understand the regime with \(ML<1\) (\(L=L_s\,\lesssim \,L_t\)) starts from considering the pion propagator \(G(p)=1/(p^2+M^2)\) in finite volume. For \(ML\,\gtrsim \,1\) and \(p\sim 1/L\), \(G(p)\sim L^2\) for small momenta, including \(p=0\). But when M becomes of order \(1/L^2\), \(G(0)\propto L^4\gg G(p\ne 0)\sim L^2\). The \(p=0\) mode of the pion field becomes nonperturbative, and the integration over this mode restores chiral symmetry in the limit \(m\rightarrow 0\).
The pion effective action for the zeromomentum field depends only on the combination \(\mu =m\Sigma V\), the symmetryrestoration parameter, where \(V=L_s^3 L_t\). In the \(\epsilon \)regime, in which \(m\sim 1/V\), all other terms in the effective action are subdominant in powers of \(\epsilon \sim 1/L\), leading to a reordering of the usual chiral expansion, which assumes that \(m\sim \epsilon ^2\) instead of \(m\sim \epsilon ^4\). In the pregime, with \(m\sim \epsilon ^2\) or equivalently \(ML\,\gtrsim \, 1\), finitevolume corrections are of order \(\int d^4p\,e^{ipx}\,G(p)_{x\sim L}\sim e^{ML}\), while in the \(\epsilon \)regime, the chiral expansion is an expansion in powers of \(1/(\Lambda _\mathrm {QCD}L)\sim 1/(FL)\).
The fact that the representation of the correlators to NLO is not “contaminated” by higherorder unknown LECs, makes the \(\epsilon \)regime potentially convenient for a clean extraction of the LO couplings. The determination of these LECs is then affected by different systematic uncertainties with respect to the standard case; simulations in this regime yield complementary information which can serve as a valuable crosscheck to get a comprehensive picture of the lowenergy properties of QCD.
The effective theory can also be used to study the distribution of the topological charge in QCD [284] and the various quantities of interest may be defined for a fixed value of this charge. The expectation values and correlation functions then not only depend on the symmetryrestoration parameter \(\mu \), but also on the topological charge \(\nu \). The dependence on these two variables can explicitly be calculated. It turns out that the 2point correlation functions considered above retain the form (90), but the coefficients \(a_A\), \(b_A\), \(a_P\), \(b_P\) now depend on the topological charge as well as on the symmetryrestoration parameter (see Refs. [285, 286, 287] for explicit expressions).
A specific issue with \(\epsilon \)regime calculations is the scale setting. Ideally one would perform a pregime study with the same bare parameters to measure a hadronic scale (e.g. the proton mass). In the literature, sometimes a gluonic scale (e.g. \(r_0\)) is used to avoid such expenses. Obviously the issues inherent in scale setting are aggravated if the \(\epsilon \)regime simulation is restricted to a fixed sector of topological charge.
It is important to stress that in the \(\epsilon \)expansion higherorder finitevolume corrections might be significant, and the physical box size (in fm) should still be large in order to keep these distortions under control. The criteria for the chiral extrapolation and finitevolume effects are obviously different from the pregime. For these reasons we have to adjust the colour coding defined in Sect. 2.1 (see Sect. 5.2 for more details).
Recently, the effective theory has been extended to the “mixed regime” where some quarks are in the pregime and some in the \(\epsilon \)regime [288, 289]. In Ref. [290] a technique is proposed to smoothly connect the p and \(\epsilon \)regimes. In Ref. [291] the issue is reconsidered with a counting rule which is essentially the same as in the pregime. In this new scheme, one can treat the IR fluctuations of the zeromode nonperturbatively, while keeping the logarithmic quark mass dependence of the pregime.
Also first steps towards calculating higher npoint functions in the \(\epsilon \)regime have been taken. For instance the electromagnetic pion form factor in QCD has been calculated to NLO in the \(\epsilon \)expansion, and a way to get rid of the pion zeromomentum part has been proposed [292].
5.1.5 Energy levels of the QCD Hamiltonian in a box and \(\delta \)regime
At low temperature, the properties of the partition function are governed by the lowest eigenvalues of the Hamiltonian. In the case of QCD, the lowest levels are due to the Nambu–Goldstone bosons and can be worked out with \(\chi \)PT [293]. In the chiral limit the level pattern follows the one of a quantummechanical rotator, i.e. \(E_\ell =\ell (\ell +1)/(2\,\Theta )\) with \(\ell = 0, 1,2,\ldots \) For a cubic spatial box and to leading order in the expansion in inverse powers of the box size \(L_s\), the moment of inertia is fixed by the value of the pion decay constant in the chiral limit, i.e. \(\Theta =F^2L_s^3\).
In order to analyse the dependence of the levels on the quark masses and on the parameters that specify the size of the box, a reordering of the chiral series is required, the socalled \(\delta \)expansion; the region where the properties of the system are controlled by this expansion is referred to as the \(\delta \)regime. Evaluating the chiral series in this regime, one finds that the expansion of the partition function goes in even inverse powers of \(FL_s\), that the rotator formula for the energy levels holds up to NNLO and the expression for the moment of inertia is now also known up to and including terms of order \((FL_s)^{4}\) [294, 295, 296]. Since the level spectrum is governed by the value of the pion decay constant in the chiral limit, an evaluation of this spectrum on the lattice can be used to measure F. More generally, the evaluation of various observables in the \(\delta \)regime offers an alternative method for a determination of some of the lowenergy constants occurring in the effective Lagrangian. At present, however, the numerical results obtained in this way [178, 297] are not yet competitive with those found in the p or \(\epsilon \)regime.
5.1.6 Other methods for the extraction of the lowenergy constants
The role that the topological susceptibility plays in attempts to determine whether there is a large paramagnetic suppression when going from the \(N_{ f}=2\) to the \(N_{ f}=2+1\) theory has been highlighted in Ref. [300]. And the potential usefulness of higher moments of the topological charge distribution to determine LECs has been investigated in Ref. [301].
Another method for computing the quark condensate has been proposed in Ref. [302], where it is shown that starting from the Banks–Casher relation [303] one may extract the condensate from suitable (renormalizable) spectral observables, for instance the number of Dirac operator modes in a given interval. For those spectral observables higherorder corrections can be systematically computed in terms of the chiral effective theory. For recent implementations of this strategy, see Refs. [33, 38, 304]. As an aside let us remark that corrections to the Banks–Casher relation that come from a finite quark mass, a finite fourdimensional volume and (with Wilsontype fermions) a finitelattice spacing can be parameterized in a properly extended version of the chiral framework [305, 306].
An alternative strategy is based on the fact that at LO in the \(\epsilon \)expansion the partition function in a given topological sector \(\nu \) is equivalent to the one of a chiral Random Matrix Theory (RMT) [307, 308, 309, 310]. In RMT it is possible to extract the probability distributions of individual eigenvalues [311, 312, 313] in terms of two dimensionless variables \(\zeta =\lambda \Sigma V\) and \(\mu =m\Sigma V\), where \(\lambda \) represents the eigenvalue of the massless Dirac operator and m is the seaquark mass. More recently this approach has been extended to the Hermitian (Wilson) Dirac operator [314] which is easier to study in numerical simulations. Hence, if it is possible to match the QCD lowlying spectrum of the Dirac operator to the RMT predictions, then one may extract^{23} the chiral condensate \(\Sigma \). One issue with this method is that for the distributions of individual eigenvalues higherorder corrections are still not known in the effective theory, and this may introduce systematic effects which are hard^{24} to control. Another open question is that, while it is clear how the spectral density is renormalized [318], this is not the case for the individual eigenvalues, and one relies on assumptions. There have been many lattice studies [319, 320, 321, 322, 323] which investigate the matching of the lowlying Dirac spectrum with RMT. In this review the results of the LECs obtained in this way^{25} are not included.
5.2 Extraction of SU(2) lowenergy constants
Cubic root of the SU(2) quark condensate \(\Sigma \equiv \langle \overline{u}u\rangle _{m_u,m_d\rightarrow 0}\) in \(\,\mathrm {MeV}\) units, in the \(\overline{\mathrm{MS}}\)scheme, at the renormalization scale \(\mu =2\) GeV. All ETM values which were available only in \(r_0\) units were converted on the basis of \(r_0=0.48(2)~\mathrm{fm}\) [333, 350, 351], with this error being added in quadrature to any existing systematic error
Collaboration  Refs.  \(N_{ f}\)  Publication status  Chiral extrapolation  Continuum extrapolation  Finite volume  Renormalization  \(\Sigma ^{1/3}\) 

ETM 13  [33]  \(2+1+1\)  A  280(8)(15)  
RBC/UKQCD 15E  [335]  \(2+1\)  P  274.2(2.8)(4.0)  
RBC/UKQCD 14B  [10]  \(2+1\)  A  275.9(1.9)(1.0)  
BMW 13  [35]  \(2+1\)  A  271(4)(1)  
Borsanyi 12  [34]  \(2+1\)  A  272.3(1.2)(1.4)  
MILC 10A  [13]  \(2+1\)  C  281.5(3.4)\(\left( {\begin{array}{c}+2.0\\ 5.9\end{array}}\right) \)(4.0)  
JLQCD/TWQCD 10A  [338]  \(2+1\)  A  234(4)(17)  
RBC/UKQCD 10A  [144]  \(2+1\)  A  256(5)(2)(2)  
JLQCD 09  [337]  \(2+1\)  A  242(4)\(\left( {\begin{array}{c}+19\\ 18\end{array}}\right) \)  
MILC 09A, SU(3)fit  [6]  \(2+1\)  C  279(1)(2)(4)  
MILC 09A, SU(2)fit  [6]  \(2+1\)  C  280(2)\(\left( {\begin{array}{c}+4\\ 8\end{array}}\right) \)(4)  
MILC 09  [89]  \(2+1\)  A  278(1)\(\left( {\begin{array}{c}+2\\ 3\end{array}}\right) \)(5)  
TWQCD 08  [340]  \(2+1\)  A  259(6)(9)  
JLQCD/TWQCD 08B  [341]  \(2+1\)  C  249(4)(2)  
PACSCS 08, SU(3)fit  [93]  \(2+1\)  A  312(10)  
PACSCS 08, SU(2)fit  [93]  \(2+1\)  A  309(7)  
RBC/UKQCD 08  [145]  \(2+1\)  A  255(8)(8)(13)  
Engel 14  [38]  2  A  263(3)(4)  
Brandt 13  [37]  2  A  261(13)(1)  
ETM 13  [33]  2  A  283(7)(17)  
ETM 12  [342]  2  A  299(26)(29)  
Bernardoni 11  [343]  2  C  306(11)  
TWQCD 11  [249]  2  A  230(4)(6)  
TWQCD 11A  [344]  2  A  259(6)(7)  
JLQCD/TWQCD 10A  [338]  2  A  242(5)(20)  
Bernardoni 10  [345]  2  A  262\(\left( {\begin{array}{c}+33\\ 34\end{array}}\right) \left( {\begin{array}{c}+4\\ 5\end{array}}\right) \)  
ETM 09C  [36]  2  A  270(5)\(\left( {\begin{array}{c}+3\\ 4\end{array}}\right) \)  
ETM 09B  [346]  2  C  245(5)  
ETM 08  [41]  2  A  264(3)(5)  
CERN 08  [302]  2  A  276(3)(4)(5)  
Hasenfratz 08  [347]  2  A  248(6)  
JLQCD/TWQCD 08A  [138]  2  A  235.7(5.0)(2.0)\(\left( {\begin{array}{c}+12.7\\ 0.0\end{array}}\right) \)  
JLQCD/TWQCD 07  [348]  2  A  239.8(4.0)  
JLQCD/TWQCD 07A  [349]  2  A  252(5)(10) 
Results for the SU(2) lowenergy constant F (in MeV) and for the ratio \(F_\pi /F\). All ETM values which were available only in \(r_0\) units were converted on the basis of \(r_0=0.48(2)~\mathrm{fm}\) [333, 350, 351], with this error being added in quadrature to any existing systematic error. Numbers in slanted fonts have been calculated by us, based on \(\sqrt{2}F_\pi ^\mathrm {phys}=130.41(20)~\,\mathrm {MeV}\) [151], with this error being added in quadrature to any existing systematic error
Collaboration  Refs.  \(N_{ f}\)  Publication status  Chiral extrapolation  Continuum extrapolation  Finite volume  F  \(F_\pi /F\) 

ETM 11  [352]  \(2+1+1\)  C  85.60(4)  1.077(1)  
ETM 10  [39]  \(2+1+1\)  A  85.66(6)(13)  1.076(2)(2)  
RBC/UKQCD 15E  [335]  \(2+1\)  P  85.8(1.1)(1.5)  1.0641(21)(49)  
RBC/UKQCD 14B  [10]  \(2+1\)  A  86.63(12)(13)  1.0645(15)(0)  
BMW 13  [35]  \(2+1\)  A  88.0(1.3)(0.3)  1.055(7)(2)  
Borsanyi 12  [34]  \(2+1\)  A  86.78(05)(25)  1.0627(06)(27)  
NPLQCD 11  [40]  \(2+1\)  A  86.8(2.1)\(\left( {\begin{array}{c}+3.3\\ 3.4\end{array}}\right) \)  1.062(26)\(\left( {\begin{array}{c}+42\\ 40\end{array}}\right) \)  
MILC 10  [29]  \(2+1\)  C  87.0(4)(5)  1.060(5)(6)  
MILC 10A  [13]  \(2+1\)  C  87.5(1.0)\(\left( {\begin{array}{c}+0.7\\ 2.6\end{array}}\right) \)  1.054(12)\(\left( {\begin{array}{c}+31\\ 09\end{array}}\right) \)  
MILC 09A, SU(3)fit  [6]  \(2+1\)  C  86.8(2)(4)  1.062(1)(3)  
MILC 09A, SU(2)fit  [6]  \(2+1\)  C  87.4(0.6)\(\left( {\begin{array}{c}+0.9\\ 1.0\end{array}}\right) \)  1.054(7)\(\left( {\begin{array}{c}+12\\ 11\end{array}}\right) \)  
MILC 09  [89]  \(2+1\)  A  87.66(17)\(\left( {\begin{array}{c}+28\\ 52\end{array}}\right) \)  1.052(2)\(\left( {\begin{array}{c}+6\\ 3\end{array}}\right) \)  
PACSCS 08, SU(3)fit  [93]  \(2+1\)  A  90.3(3.6)  1.062(8)  
PACSCS 08, SU(2)fit  [93]  \(2+1\)  A  89.4(3.3)  1.060(7)  
RBC/UKQCD 08  [145]  \(2+1\)  A  81.2(2.9)(5.7)  1.080(8)  
ETM 15A  [333]  2  P  86.3(2.8)  1.069(35)  
Engel 14  [38]  2  A  85.8(0.7)(2.0)  1.075(09)(25)  
Brandt 13  [37]  2  A  84(8)(2)  1.080(16)(6)  
QCDSF 13  [353]  2  A  86(1)  1.07(1)  
TWQCD 11  [249]  2  A  83.39(35)(38)  1.106(5)(5)  
ETM 09C  [36]  2  A  85.91(07)\(\left( {\begin{array}{c}+78\\ 07\end{array}}\right) \)  1.0755(6)\(\left( {\begin{array}{c}+08\\ 94\end{array}}\right) \)  
ETM 09B  [346]  2  C  92.1(4.9)  1.00(5)  
ETM 08  [41]  2  A  86.6(7)(7)  1.067(9)(9)  
Hasenfratz 08  [347]  2  A  90(4)  1.02(5)  
JLQCD/TWQCD 08A  [138]  2  A  79.0(2.5)(0.7)\(\left( {\begin{array}{c}+4.2\\ 0.0\end{array}}\right) \)  1.167(37)(10)\(\left( {\begin{array}{c}+02\\ 62\end{array}}\right) \)  
JLQCD/TWQCD 07  [348]  2  A  87.3(5.6)  1.06(7)  
Colangelo 03  [354]  86.2(5)  1.0719(52) 
Results for the SU(2) NLO lowenergy constants \(\bar{\ell }_3\) and \(\bar{\ell }_4\). For comparison, the last two lines show results from phenomenological analyses
Collaboration  Refs.  \(N_{ f}\)  Publication status  Chiral extrapolation  Continuum extrapolation  Finite volume  \(\bar{\ell }_3\)  \(\bar{\ell }_4\) 

ETM 11  [352]  \(2+1+1\)  C  3.53(5)  4.73(2)  
ETM 10  [39]  \(2+1+1\)  A  3.70(7)(26)  4.67(3)(10)  
RBC/UKQCD 15E  [335]  \(2+1\)  P  2.81(19)(45)  4.02(8)(24)  
RBC/UKQCD 14B  [10]  \(2+1\)  A  2.73(13)(0)  4.113(59)(0)  
BMW 13  [35]  \(2+1\)  A  2.5(5)(4)  3.8(4)(2)  
RBC/UKQCD 12  [31]  \(2+1\)  A  2.91(23)(07)  3.99(16)(09)  
Borsanyi 12  [34]  \(2+1\)  A  3.16(10)(29)  4.03(03)(16)  
NPLQCD 11  [40]  \(2+1\)  A  4.04(40)\(\left( {\begin{array}{c}+73\\ 55\end{array}}\right) \)  4.30(51)\(\left( {\begin{array}{c}+84\\ 60\end{array}}\right) \)  
MILC 10  [29]  \(2+1\)  C  3.18(50)(89)  4.29(21)(82)  
MILC 10A  [13]  \(2+1\)  C  2.85(81)\(\left( {\begin{array}{c}+37\\ 92\end{array}}\right) \)  3.98(32)\(\left( {\begin{array}{c}+51\\ 28\end{array}}\right) \)  
RBC/UKQCD 10A  [144]  \(2+1\)  A  2.57(18)  3.83(9)  
MILC 09A, SU(3)fit  [6]  \(2+1\)  C  3.32(64)(45)  4.03(16)(17)  
MILC 09A, SU(2)fit  [6]  \(2+1\)  C  3.0(6)\(\left( {\begin{array}{c}+9\\ 6\end{array}}\right) \)  3.9(2)(3)  
PACSCS 08, SU(3)fit  [93]  \(2+1\)  A  3.47(11)  4.21(11)  
PACSCS 08, SU(2)fit  [93]  \(2+1\)  A  3.14(23)  4.04(19)  
RBC/UKQCD 08  [145]  \(2+1\)  A  3.13(33)(24)  4.43(14)(77)  
ETM 15A  [333]  2  P  3.3(4)  
Gülpers 15  [355]  2  P  4.54(30)(0)  
Gülpers 13  [356]  2  A  4.76(13)  
Brandt 13  [37]  2  A  3.0(7)(5)  4.7(4)(1)  
QCDSF 13  [353]  2  A  4.2(1)  
Bernardoni 11  [343]  2  C  4.46(30)(14)  4.56(10)(4)  
TWQCD 11  [249]  2  A  4.149(35)(14)  4.582(17)(20)  
ETM 09C  [36]  2  A  3.50(9)\(\left( {\begin{array}{c}+09\\ 30\end{array}}\right) \)  4.66(4)\(\left( {\begin{array}{c}+04\\ 33\end{array}}\right) \)  
JLQCD/TWQCD 09  [357]  2  A  4.09(50)(52)  
ETM 08  [41]  2  A  3.2(8)(2)  4.4(2)(1)  
JLQCD/TWQCD 08A  [138]  2  A  3.38(40)(24)\(\left( {\begin{array}{c}+31\\ 00\end{array}}\right) \)  4.12(35)(30)\(\left( {\begin{array}{c}+31\\ 00\end{array}}\right) \)  
CERNTOV 06  [358]  2  A  3.0(5)(1)  
Colangelo 01  [261]  4.4(2)  
Gasser 84  [131]  2.9(2.4)  4.3(9) 
Top (vector form factor of the pion): Lattice results for the charge radius \(\langle r^2\rangle _V^\pi \) (in \(\hbox {fm}^2\)), the curvature \(c_V\) (in \(\hbox {GeV}^{4}\)) and the effective coupling constant \(\bar{\ell }_6\) are compared with the experimental value, as obtained by NA7, and some phenomenological estimates. Bottom (scalar form factor of the pion): Lattice results for the scalar radius \(\langle r^2 \rangle _S^\pi \) (in \(\hbox {fm}^2\)) and the combination \(\bar{\ell }_1\bar{\ell }_2\) are compared with a dispersive calculation of these quantities
Collaboration  Refs.  \(N_{ f}\)  Publication status  Chiral extrapolation  Continuum extrapolation  Finite volume  \(\langle r^2\rangle _V^\pi \)  \(c_V\)  \(\bar{\ell }_6\) 

HPQCD 15B  [336]  \(2+1+1\)  P  0.403(18)(6)  
JLQCD 15A , SU(2)fit  [359]  \(2+1\)  P  0.395(26)(32)  13.49(89)(82)  
JLQCD 14  [360]  \(2+1\)  A  0.49(4)(4)  7.5(1.3)(1.5)  
PACSCS 11A  [361]  \(2+1\)  A  0.441(46)  
RBC/UKQCD 08A  [339]  \(2+1\)  A  0.418(31)  12.2(9)  
LHP 04  [362]  \(2+1\)  A  0.310(46)  
Brandt 13  [37]  2  A  0.481(33)(13)  15.5(1.7)(1.3)  
JLQCD/TWQCD 09  [357]  2  A  0.409(23)(37)  3.22(17)(36)  11.9(0.7)(1.0)  
ETM 08  [41]  2  A  0.456(30)(24)  3.37(31)(27)  14.9(1.2)(0.7)  
QCDSF/UKQCD 06A  [363]  2  A  0.441(19)(63)  
Bijnens 98  [264]  0.437(16)  3.85(60)  16.0(0.5)(0.7)  
NA7 86  [364]  0.439(8)  
Gasser 84  [131]  16.5(1.1) 
Collaboration  Refs.  \(N_{ f}\)  Publication status  Chiral extrapolation  Continuum extrapolation  Finite volume  \(\langle r^2\rangle _S^\pi \)  \(\bar{\ell }_1\bar{\ell }_2\)  

HPQCD 15B  [336]  \(2+1+1\)  P  0.481(37)(50)  
RBC/UKQCD 15E  [335]  \(2+1\)  P  \(\)9.2(4.9)(6.5)  
Gülpers 15  [355]  2  P  0.600(52)(0)  
Gülpers 13  [356]  2  A  0.637(23)  
JLQCD/TWQCD 09  [357]  2  A  0.617(79)(66)  \(\)2.9(0.9)(1.3)  
Colangelo 01  [261]  0.61(4)  \(\)4.7(6) 
The lattice results for the SU(2) LECs are summarized in Tables 19, 20, 21 and 22 and Figs. 11, 12 and 13. The tables present our usual colour coding which summarizes the main aspects related to the treatment of the systematic errors of the various calculations.
 1.
Using both the x and the \(\xi \) expansion is a good way to test how the ambiguity of the chiral expansion (at a given order) affects the numerical values of the LECs that are determined from a particular set of data [35, 138]. For instance, to determine \(\bar{\ell }_4\) (or \(\Lambda _4\)) from lattice data for \(F_\pi \) as a function of the quark mass, one may compare the fits based on the parameterization \(F_\pi =F\{1+x\ln (\Lambda _4^2/M^2)\}\) [see Eq. (73)] with those obtained from \(F_\pi =F/\{1\xi \ln (\Lambda _4^2/M_\pi ^2)\}\) [see Eq. (78)]. The difference between the two results provides an estimate of the uncertainty due to the truncation of the chiral series. Which central value one chooses is in principle arbitrary, but we find it advisable to use the one obtained with the \(\xi \) expansion,^{27} in particular because it makes the comparison with phenomenological determinations (where it is standard practice to use the \(\xi \) expansion) more meaningful.
 2.
Alternatively one could try to estimate the influence of higher chiral orders by reshuffling irrelevant higherorder terms. For instance, in the example mentioned above one might use \(F_\pi =F/\{1x\ln (\Lambda _4^2/M^2)\}\) as a different functional form at NLO. Another way to establish such an estimate is through introducing by hand “analytical” higherorder terms (e.g. “analytical NNLO” as done, in the past, by MILC [89]). In principle it would be preferable to include all NNLO terms or none, such that the structure of the chiral expansion is preserved at any order (this is what ETM [36] and JLQCD/TWQCD [138] have done for SU(2) \(\chi \)PT and MILC for both SU(2) and SU(3) \(\chi \)PT [6, 13, 29]). There are different opinions in the field as to whether it is advisable to include terms to which the data are not sensitive. In the case one is willing to include external (typically: nonlattice) information, the use of priors is a theoretically wellfounded option (e.g. priors for NNLO LECs if one is interested exclusively in LECs at LO/NLO).
 3.
Another issue concerns the squark mass dependence of the LECs \(\bar{\ell }_i\) or \(\Lambda _i\) of the SU(2) framework. As far as variations of \(m_s\) around \(m_s^\mathrm {phys}\) are concerned (say for \(0<m_s<1.5m_s^\mathrm {phys}\) at best) the issue can be studied in SU(3) \(\chi \)PT, and this has been done in a series of papers [129, 324, 325]. However, the effect of sending \(m_s\) to infinity, as is the case in \(N_{ f}=2\) lattice studies of SU(2) LECs, cannot be addressed in this way. A way to analyse this difference is to compare the numerical values of LECs determined in \(N_{ f}=2\) lattice simulations to those determined in \(N_{ f}=2+1\) lattice simulations (see e.g. Ref. [326] for a discussion).
 4.
Last but not least let us recall that the determination of the LECs is affected by discretization effects, and it is important that these are removed by means of a continuum extrapolation. In this step invoking an extended version of the chiral Lagrangian [275, 327, 328, 329, 330, 331] may be useful^{28} in the case one aims for a global fit of lattice data involving several \(M_\pi \) and a values and several chiral observables.
In the tables and figures we summarize the results of various lattice collaborations for the SU(2) LECs at LO (F or \(F/F_\pi \), B or \(\Sigma \)) and at NLO (\(\bar{\ell }_1\bar{\ell }_2\), \(\bar{\ell }_3\), \(\bar{\ell }_4\), \(\bar{\ell }_6\)). Throughout we group the results into those which stem from \(N_{ f}=2+1+1\) calculations, those which come from \(N_{ f}=2+1\) calculations and those which stem from \(N_{ f}=2\) calculations (since, as mentioned above, the LECs are logically distinct even if the current precision of the data is not sufficient to resolve the differences). Furthermore, we make a distinction whether the results are obtained from simulations in the pregime or whether alternative methods (\(\epsilon \)regime, spectral densities, topological susceptibility, etc.) have been used (this should not affect the result). For comparison we add, in each case, a few representative phenomenological determinations.
A generic comment applies to the issue of the scale setting. In the past none of the lattice studies with \(N_{ f}\ge 2\) involved simulations in the pregime at the physical value of \(m_{ud}\). Accordingly, the setting of the scale \(a^{1}\) via an experimentally measurable quantity did necessarily involve a chiral extrapolation, and as a result of this dimensionful quantities used to be particularly sensitive to this extrapolation uncertainty, while in dimensionless ratios such as \(F_\pi /F\), \(F/F_0\), \(B/B_0\), \(\Sigma /\Sigma _0\) this particular problem is much reduced (and often finite latticetocontinuum renormalization factors drop out). Now, there is a new generation of lattice studies with \(N_{ f}=2\) [333], \(N_{ f}=2+1\) [7, 8, 10, 23, 31, 34, 35, 94, 334, 335], and \(N_{ f}=2+1+1\) [26, 336], which does involve simulations at physicalpion masses. In such studies the uncertainty that the scale setting has on dimensionful quantities is much mitigated.
It is worth repeating here that the standard colourcoding scheme of our tables is necessarily schematic and cannot do justice to every calculation. In particular there is some difficulty in coming up with a fair adjustment of the rating criteria to finitevolume regimes of QCD. For instance, in the \(\epsilon \)regime^{29} we reexpress the “chiralextrapolation” criterion in terms of \(\sqrt{2m_\mathrm {min}\Sigma }/F\), with the same threshold values (in MeV) between the three categories as in the pregime. Also the “infinitevolume” assessment is adapted to the \(\epsilon \)regime, since the \(M_\pi L\) criterion does not make sense here; we assign a green star if at least 2 volumes with \(L>2.5\,\mathrm{fm}\) are included, an open symbol if at least 1 volume with \(L>2\,\mathrm{fm}\) is invoked and a red square if all boxes are smaller than \(2\,\mathrm{fm}\). Similarly, in the calculation of form factors and charge radii the tables do not reflect whether an interpolation to the desired \(q^2\) has been performed or whether the relevant \(q^2\) has been engineered by means of “twisted boundary conditions” [339]. In spite of these limitations we feel that these tables give an adequate overview of the qualities of the various calculations.
5.2.1 Results for the LO SU(2) LECs
We begin with a discussion of the lattice results for the SU(2) LEC \(\Sigma \). We present the results in Table 19 and Fig. 11. We add that results which include only a statistical error are listed in the table but omitted from the plot. Regarding the \(N_{ f}=2\) computations there are six entries without a red tag. We form the average based on ETM 09C, ETM 13 (here we deviate from our “superseded” rule, since the two works use different methods), Brandt 13, and Engel 14. We add that the last one (with numbers identical to those given in Ref. [304]) is new compared to FLAG 13. Here and in the following we take into account that ETM 09C, ETM 13 share configurations, and the same statement holds true for Brandt 13 and Engel 14. Regarding the \(N_{ f}=2+1\) computations there are four published or updated papers (MILC 10A, Borsanyi 12, BMW 13, and RBC/UKQCD 14B) which qualify for the \(N_{ f}=2+1\) average. The last one is new compared to FLAG 13, and the last but one was not included in the FLAG 13 average, since at the time it was only a preprint.
As a rationale for using estimates (as opposed to averages) for \(N_{ f}=2\) and \(N_{ f}=2+1\), we add that for \(\Sigma ^{1/3}_{N_{ f}=2}\) and \(\Sigma ^{1/3}_{N_{ f}=2+1}\) the standard averaging method would yield central values as quoted in Eq. (94), but with (overall) uncertainties of \(4\,\mathrm {MeV}\) and \(1\,\mathrm {MeV}\), respectively. It is not entirely clear to us that the scale is sufficiently well known in all contributing works to warrant a precision of up to 0.36% on our \(\Sigma ^{1/3}\), and a similar statement can be made about the level of control over the convergence of the chiral expansion. The aforementioned uncertainties would suggest an \(N_{ f}\)dependence of the SU(2) chiral condensate which (especially in view of similar issues with other LECs; see below) seems premature to us. Therefore we choose to form the central value of our estimate with the standard averaging procedure, but its uncertainty is taken as the median of the uncertainties of the participating results. We hope that future highquality determinations with both \(N_f=2\), \(N_f=2+1\), and in particular with \(N_f=2+1+1\), will help determine whether there is a noticeable \(N_f\)dependence of the SU(2) chiral condensate or not.
The next quantity considered is F, i.e. the pion decay constant in the SU(2) chiral limit (\(m_{ud}\rightarrow 0\), at fixed physical \(m_s\) for \(N_f > 2\) simulations). As argued on previous occasions we tend to give preference to \(F_\pi /F\) (here the numerator is meant to refer to the physicalpionmass point) wherever it is available, since often some of the systematic uncertainties are mitigated. We collect the results in Table 20 and Fig. 12. In those cases where the collaboration provides only F, the ratio is computed on the basis of the phenomenological value of \(F_\pi \), and the respective entries in Table 20 are in slante