Table 1 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

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 strong-interaction effects can be quantified. Large-scale 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 low-energy 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 nonlattice-experts. This is generally not an easy question to answer; different collaborations use different lattice actions (discretizations of QCD) with a variety of lattice spacings and volumes, and with a range of masses for the u- and d-quarks. Not only are the systematic errors different, but also the methodology used to estimate these uncertainties varies between collaborations. In the present work we summarize the main features of each of the calculations and provide a framework for judging and combining the different results. Sometimes it is a single result that provides the “best” value; more often it is a combination of results from different collaborations. Indeed, the consistency of values obtained using different formulations adds significantly to our confidence in the results.

The first 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 light-quark 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 low-energy constants of \(SU(2)_L \times SU(2)_R\) and \(SU(3)_L \times SU(3)_R\) Chiral Perturbation Theory. The quantities related to D- and B-meson physics that were reviewed were the B- and D-meson-decay 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 unitarity-triangle 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 above-mentioned 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.

Our plan is to continue providing FLAG updates, in the form of a peer reviewed paper, roughly on a biennial basis. This effort is supplemented by our more frequently updated website [3], where figures as well as pdf-files for the individual sections can be downloaded. The papers reviewed in the present edition have appeared before the closing date 30 November 2015.

Table 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

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.

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 particle-physics 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.

The current list of FLAG members and their Working Group assignments is:

  • 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,Footnote 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

As some members of the WG on quark masses were faced with unexpected hindrances, S. Simula has kindly assisted in the completion of the relevant section during the final phases of its composition.

The most important FLAG guidelines and rules are the following:

  • the composition of the AB reflects the main geographical areas in which lattice collaborations are active, with members from America, Asia/Oceania and Europe;

  • the mandate of regular members is not limited in time, but we expect that a certain turnover will occur naturally;

  • whenever a replacement becomes necessary this has to keep, and possibly improve, the balance in FLAG, so that different collaborations, from different geographical areas are represented;

  • in all working groups the three members must belong to three different lattice collaborations;Footnote 2

  • a paper is in general not reviewed (nor colour-coded, 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.

Citation policy

We draw attention to this particularly important point. As stated above, our aim is to make lattice QCD results easily accessible to nonlattice-experts 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 bibtex-file available on our webpage [3] which contains the bibtex-entries 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.

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 low-energy 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 requirementsFootnote 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 b-quark 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 light-quark sector. An overview of the issues specific to heavy-quark 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 heavy-quark 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 sea-quark fluctuations. The quenched approximation (\(N_f=0\)), in which sea-quark contributions are omitted, has uncontrolled systematic errors and is no longer used in modern lattice simulations with relevance to phenomenology. Accordingly, we will review results obtained with \(N_f=2\), \(N_f=2+1\), and \(N_f = 2+1+1\), but omit earlier results with \(N_f=0\). The only exception concerns the QCD coupling constant \(\alpha _s\). Since this observable does not require valence light quarks, it is theoretically well defined also in the \(N_f=0\) theory, which is simply pure gluon-dynamics. 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 B-parameters, are for quantities defined in a given renormalization scheme and at a specific renormalization scale. The schemes employed (e.g. regularization-independent 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 so-called 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 finite-size scaling techniques).


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 (four-dimensional) volume. In order to obtain physical results, lattice data are obtained at a sequence of pion masses and a sequence of lattice spacings, and then extrapolated to the physical-pion mass and to the continuum limit. In principle, an extrapolation to infinite volume is also required. However, for most quantities discussed in this review, finite-volume 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 light-quark actions, the lattice-spacing 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 Heavy-Quark Effective Theory (HQET). The pion-mass 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 Heavy-Light Meson Chiral Perturbation Theory (HM\(\chi \)PT) to extrapolate quantities involving mesons composed of one heavy (b or c) and one light quark. One can combine Symanzik’s effective theory with \(\chi \)PT to simultaneously extrapolate to the physical-pion mass and the continuum; in this case, the form of the effective theory depends on the discretization. See Appendix A.4 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 finite-volume effects measured in units of the inverse pion mass, thus providing information on the systematic error due to finite-volume 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 large-scale 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 lattice-QCD 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 speed-up the computation, though it may produce systematic effects. Currently available tests indicate that errors from the use of inexact algorithms are under control.

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.

Systematic errors and colour code

The major sources of systematic error are common to most lattice calculations. These include, as discussed in detail below, the chiral, continuum and infinite-volume extrapolations. To each such source of error for which systematic improvement is possible we assign one of three coloured symbols: green star, unfilled green circle (which replaced in Ref. [2] the amber disk used in the original FLAG review [1]) or red square. These correspond to the following ratings:


the parameter values and ranges used to generate the datasets allow for a satisfactory control of the systematic uncertainties;


the parameter values and ranges used to generate the datasets allow for a reasonable attempt at estimating systematic uncertainties, which, however, could be improved;


the parameter values and ranges used to generate the datasets are unlikely to allow for a reasonable control of systematic uncertainties.

The appearance of a red tag, even in a single source of systematic error of a given lattice result, disqualifies it from inclusion in the global average.

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 well-controlled 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 colour-coding 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 field-theoretic point of view (namely nonperturbatively, or with two-loop or one-loop 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.

Systematic effects and rating criteria

The precise criteria used in determining the colour coding are unavoidably time-dependent; 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 light-quark 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 heavy-quark physics [2]. The criteria used for light- and heavy-flavour quantities were not always the same. For the continuum limit, the difference was more a matter of choice: the light-flavour Working Groups defined the ratings using conditions involving specific values of the lattice spacing, whereas the heavy-flavour Working Groups preferred more data-driven criteria. Also, for finite-volume effects, the heavy-flavour groups slightly relaxed the boundary between   and , compared to the light-quark case, to account for the fact that heavy-quark quantities are less sensitive to the finiteness of the volume.

In the present edition we have opted for simplicity and adopted unified criteria for both light- and heavy-flavoured quantities.Footnote 4 The colour code used in the tables is specified as follows:

  • Chiral extrapolation:


\(M_{\pi ,\mathrm {min}}< 200\) MeV


200 MeV \(\le M_{\pi ,{\mathrm {min}}} \le \) 400 MeV


400 MeV \( < M_{\pi ,\mathrm {min}}\)

It is assumed that the chiral extrapolation is performed with at least a 3-point analysis; otherwise this will be explicitly mentioned. This condition is unchanged from Ref. [2].

  • Continuum extrapolation:


at least three lattice spacings and at least 2 points below 0.1 fm and a range of lattice spacings satisfying \([a_{\mathrm {max}}/a_{\mathrm {min}}]^2 \ge 2\)


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



It is assumed that the lattice action is \(\mathcal {O}(a)\)-improved (i.e. the discretization errors vanish quadratically with the lattice spacing); otherwise this will be explicitly mentioned. For unimproved actions an additional lattice spacing is required. This condition has been tightened compared to that of Ref. [2] by the requirements concerning the range of lattice spacings.

  • Finite-volume effects:


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


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



  • It is assumed here that calculations are in the p-regimeFootnote 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. Finite-volume 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   and \(M_{\pi ,\text {min}} L>3\) for ). However, as pion masses decrease, one must also account for the weakening of the pion couplings. In particular, one-loop 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):




one-loop perturbation theory or higher with a reasonable estimate of truncation errors



  • In Ref. [1], we assigned a red square to all results which were renormalized at one-loop 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 one-loop, is often estimated conservatively and reliably.

  • Renormalization Group (RG) running (where applicable):

    For scale-dependent 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): Regularization-independent 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.

The pion mass plays an important role in the criteria relevant for chiral extrapolation and finite volume. For some of the regularizations used, however, it is not a trivial matter to identify this mass.

In the case of twisted-mass fermions, discretization effects give rise to a mass difference between charged and neutral pions even when the up- and down-quark masses are equal: the charged pion is found to be the heavier of the two for twisted-mass 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 chiral-extrapolation rating criterion. On the other hand, sea quarks (corresponding to both charged and neutral “sea pions“ in an effective-chiral-theory logic) as well as valence quarks are intertwined with finite-volume 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 finite-volume rating criterion.Footnote 6

In the case of staggered fermions, discretization effects give rise to several light states with the quantum numbers of the pion.Footnote 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 chiral-extrapolation and finite-volume criteria.Footnote 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.

Heavy-quark actions

In most cases, and in particular for the b quark, the discretization of the heavy-quark 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 b-quark 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 light-quark 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 light-quark quantities.

In order to address these complications, we add a new heavy-quark 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 heavy-quark 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 () rating for both the heavy-quark actions and the weak operators. Heavy-quark implementations without the level of improvement described below are rated not acceptable (). The matching is evaluated together with renormalization, using the renormalization criteria described in Sect. 2.1.1. We emphasize that the heavy-quark 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 heavy-quark actions:

at least tree-level \(\mathcal {O}(a)\) improved action and weak operators.

This is similar to the requirements for light-quark actions. All current implementations of relativistic heavy-quark actions satisfy this criterion.


tree-level 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 tree-level corrections of \(\mathcal {O}(1/m_h^2)\) in the action.


tree-level 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 static-limit action do not satisfy this criterion, since the static-limit 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.

Light-quark actions for heavy quarks

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 charm-quark quantities. It also applies to calculations that use these light-quark 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 heavy-quark masses used in the calculation.

Conventions for the figures

For a coherent assessment of the present situation, the quality of the data plays a key role, but the colour coding cannot be carried over to the figures. On the other hand, simply showing all data on equal footing would give the misleading impression that the overall consistency of the information available on the lattice is questionable. Therefore, in the figures we indicate the quality of the data in a rudimentary way, using the following symbols:


corresponds to results included in the average or estimate (i.e. results that contribute to the black square below);


corresponds to results that are not included in the average but pass all quality criteria;


corresponds to all other results;


corresponds to FLAG averages or estimates; they are also highlighted by a grey vertical band.

The reason for not including a given result in the average is not always the same: the result may fail one of the quality criteria; the paper may be unpublished; it may be superseded by newer results; or it may not offer a complete error budget.

Symbols other than squares are used to distinguish results with specific properties and are always explained in the caption.Footnote 9

Often nonlattice data are also shown in the figures for comparison. For these we use the following symbols:

  • corresponds to nonlattice results;

  • corresponds to Particle Data Group (PDG) results.

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 colour-coded 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.

There are two other important criteria that also play a role in this respect, but that cannot be colour coded, because a systematic improvement is not possible. These are: (i) the publication status, and (ii) the number of sea-quark flavours \(N_{ f}\). As far as the former criterion is concerned, we adopt the following policy: we average only results that have been published in peer-reviewed journals, i.e. they have been endorsed by referee(s). The only exception to this rule consists in straightforward updates of previously published results, typically presented in conference proceedings. Such updates, which supersede the corresponding results in the published papers, are included in the averages. Note that updates of earlier results rely, at least partially, on the same gauge-field-configuration ensembles. For this reason, we do not average updates with earlier results. Nevertheless, all results are listed in the tables,Footnote 10 and their publication status is identified by the following symbols:

  • Publication status:

    A   published or plain update of published results

    P   preprint

    C   conference contribution.

In the present edition, the publication status on the 30th of November 2015 is relevant. If the paper appeared in print after that date, this is accounted for in the bibliography, but does not affect the averages.

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 Zweig-rule violations related to strange-quark 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.

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 gauge-field configurations, produced with a given fermion discretization, are often used by different research teams with different valence-quark lattice actions, obtaining results that are not really independent. Our averaging procedure takes such correlations into account.

Consider a given measurable quantity Q, measured by M distinct, not necessarily uncorrelated, numerical experiments (simulations). The result of each of these measurement is expressed as

$$\begin{aligned} Q_i = x_i \, \pm \, \sigma ^{(1)}_i \pm \, \sigma ^{(2)}_i \pm \cdots \pm \, \sigma ^{(E)}_i, \end{aligned}$$

where \(x_i\) is the value obtained by the ith experiment (\(i = 1, \ldots , M\)) and \(\sigma ^{(k)}_i\) (for \(k = 1, \ldots , E\)) are the various errors. Typically \(\sigma ^{(1)}_i\) stands for the statistical error and \(\sigma ^{(k)}_i\) (\(k \ge 2\)) are the different systematic errors from various sources. For each individual result, we estimate the total error \(\sigma _i \) by adding statistical and systematic errors in quadrature:

$$\begin{aligned} Q_i= & {} x_i \pm \sigma _i,\nonumber \\ \sigma _i\equiv & {} \sqrt{\sum _{k=1}^E [\sigma ^{(k)}_i]^2}. \end{aligned}$$

With the weight factor of each total error estimated in standard fashion:

$$\begin{aligned} \omega _i =\dfrac{\sigma _i^{-2}}{\sum _{i=1}^M \sigma _i^{-2}}, \end{aligned}$$

the central value of the average over all simulations is given by

$$\begin{aligned} x_\mathrm{av} =\sum _{i=1}^M x_i\, \omega _i. \end{aligned}$$

The above central value corresponds to a \(\chi _\mathrm{min}^2\) weighted average, evaluated by adding statistical and systematic errors in quadrature. If the fit is not of good quality (\(\chi _\mathrm{min}^2/\hbox {d.o.f.} > 1\)), the statistical and systematic error bars are stretched by a factor \(S = \sqrt{\chi ^2/\hbox {d.o.f.}}\)

Next we examine error budgets for individual calculations and look for potentially correlated uncertainties. Specific problems encountered in connection with correlations between different data sets are described in the text that accompanies the averaging. If there is reason to believe that a source of error is correlated between two calculations, a \(100\%\) correlation is assumed. The correlation matrix \(C_{ij}\) for the set of correlated lattice results is estimated by a prescription due to Schmelling [91]. This consists in defining

$$\begin{aligned} \sigma _{i;j} =\sqrt{\underset{(k)}{\displaystyle \sum \nolimits ^\prime }[ \sigma _i^{(k)}]^2}, \end{aligned}$$

with \(\sum _{(k)}^\prime \) running only over those errors of \(x_i\) that are correlated with the corresponding errors of measurement \(x_j\). This expresses the part of the uncertainty in \(x_i\) that is correlated with the uncertainty in \(x_j\). If no such correlations are known to exist, then we take \(\sigma _{i;j} =0\). The diagonal and off-diagonal elements of the correlation matrix are then taken to be

$$\begin{aligned} C_{ii}= & {} \sigma _i^2 \quad (i = 1, \ldots , M),\nonumber \\ C_{ij}= & {} \sigma _{i;j} \, \sigma _{j;i} \quad (i \ne j). \end{aligned}$$

Finally the error of the average is estimated by

$$\begin{aligned} \sigma ^2_\mathrm{av} =\sum _{i=1}^M \sum _{j=1}^M \omega _i \,\omega _j \,C_{ij}, \end{aligned}$$

and the FLAG average is

$$\begin{aligned} Q_\mathrm{av} =x_\mathrm{av} \pm \sigma _\mathrm{av}. \end{aligned}$$

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 heavy-quark expansions. The up-, down- and strange-quark 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 bottom-quark 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], quark-mass determinations came from two- and three-flavour QCD calculations. Moreover, these calculations were most often performed in the isospin limit, where the up- and down-quark masses (especially those in the sea) are set equal. In addition, some of the results retained in our light-quark 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 down-quark mass point. Among the calculations performed near physical \(m_{ud}\) by PACS-CS [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 quark-mass 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 up-down-quark 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 \((m_d-m_u)\), strong isospin-breaking effects. As we will see below these are particularly important for determining the individual up- and down-quark masses. But with the level of precision being reached in calculations, these effects are also becoming important for other quark masses.

Three-flavour QCD has four free parameters: the strong coupling, \(\alpha _s\) (alternatively \(\Lambda _\mathrm {QCD}\)) and the up-, down- and strange-quark masses, \(m_u\), \(m_d\) and \(m_s\). Four-flavour calculations have an additional parameter, the charm-quark mass \(m_c\). When the calculations are performed in the isospin limit, up- and down-quark masses are replaced by a single parameter: the isospin-averaged up- and down-quark mass, \(m_{ud}=\frac{1}{2}(m_u+m_d)\). A lattice determination of these parameters, and in particular of the quark masses, proceeds in two steps:

  1. 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 valence-quark observable containing this quark must be computed and the mass of that quark must be tuned to reproduce experiment.

  2. 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 short-distance calculation, which may be performed perturbatively. Experience shows that one-loop 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 higher-order terms. Nevertheless we will include in our averages one-loop results if they carry a solid estimate of the systematic uncertainty due to the truncation of the series.

In the absence of electromagnetic corrections, the renormalization factors for all quark masses are the same at a given lattice spacing. Thus, uncertainties due to renormalization are absent in ratios of quark masses if the tuning of the masses to their physical values can be done lattice spacing by lattice spacing and significantly reduced otherwise.

We mention that lattice QCD calculations of the b-quark mass have an additional complication which is not present in the case of the charm- and light-quarks. 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 b-quark 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 b-quark mass will be briefly described in Sect. 3.3.

Masses of the light quarks

Light-quark 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 light-quark masses: the masses of the resulting Nambu–Goldstone bosons (NGB), i.e. pions, kaons and etas. Indeed, the Gell-Mann–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 higher-order 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 lattice-QCD calculations. In addition, the experimental uncertainties on these meson masses are negligible. Thus, in lattice calculations, light-quark masses are typically obtained by renormalizing the input quark mass and tuning them to reproduce NGB masses, as described above.

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 isospin-breaking 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 symmetry-breaking 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.Footnote 11

Such a convention allows us to distinguish the physical mass \(M_P\), \(P\in \{\pi ^+,\) \(\pi ^0\), \(K^+\), \(K^0\}\), from the mass \(\hat{M}_P\) within QCD. The e.m. self-energy is the difference between the two, \(M_P^\gamma \equiv M_P-\hat{M}_P\). Because the self-energy of the Nambu–Goldstone bosons diverges in the chiral limit, it is convenient to replace it by the contribution of the e.m. interaction to the square of the mass,

$$\begin{aligned} \Delta _{P}^\gamma \equiv M_P^2-\hat{M}_P^2= 2\,M_P M_P^\gamma +\mathcal {O}(e^4). \end{aligned}$$

The main effect of the e.m. interaction is an increase in the mass of the charged particles, generated by the photon cloud that surrounds them. The self-energies of the neutral ones are comparatively small, particularly for the Nambu–Goldstone bosons, which do not have a magnetic moment. Dashen’s theorem [102] confirms this picture, as it states that, to leading order (LO) of the chiral expansion, the self-energies of the neutral NGBs vanish, while the charged ones obey \(\Delta _{K^+}^\gamma = \Delta _{\pi ^+}^\gamma \). It is convenient to express the self-energies of the neutral particles as well as the mass difference between the charged and neutral pions within QCD in units of the observed mass difference, \(\Delta _\pi \equiv M_{\pi ^+}^2-M_{\pi ^0}^2\):

$$\begin{aligned} \Delta _{\pi ^0}^\gamma \equiv \epsilon _{\pi ^0}\,\Delta _\pi ,\quad \Delta _{K^0}^\gamma \equiv \epsilon _{K^0}\,\Delta _\pi ,\quad \!\!\hat{M}_{\pi ^+}^2- \hat{M}_{\pi ^0}^2\equiv \epsilon _m\,\Delta _\pi .\!\!\!\!\!\nonumber \\ \end{aligned}$$

In this notation, the self-energies of the charged particles are given by

$$\begin{aligned}&\Delta _{\pi ^+}^\gamma =(1+\epsilon _{\pi ^0}-\epsilon _m)\,\Delta _\pi ,\nonumber \\&\Delta _{K^+}^\gamma =(1+\epsilon +\epsilon _{K^0}-\epsilon _m)\,\Delta _\pi ,\end{aligned}$$

where the dimensionless coefficient \(\epsilon \) parameterizes the violation of Dashen’s theorem,Footnote 12

$$\begin{aligned} \Delta _{K^+}^\gamma -\Delta _{K^0}^\gamma - \Delta _{\pi ^+}^\gamma +\Delta _{\pi ^0}^\gamma \equiv \epsilon \,\Delta _\pi .\end{aligned}$$

Any determination of the light-quark masses based on a calculation of the masses of \(\pi ^+,K^+\) and \(K^0\) within QCD requires an estimate for the coefficients \(\epsilon \), \(\epsilon _{\pi ^0}\), \(\epsilon _{K^0}\) and \(\epsilon _m\).

The first determination of the self-energies 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 self-energy difference of the kaons amount to 50% of the leading term. The result for the self-energy 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 self-energy 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 self-energy 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 self-energy 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 self-energies 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 finite-volume 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 2-flavour 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, finite-volume, 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 light-quark masses, they find \(\epsilon = 0.50 (6)\), where the error is statistical only.

The effective Lagrangian that governs the self-energies to next-to-leading 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 quark-mass 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 post-2010 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 self-energies and then specify the estimates used as an input in our analysis of the data. The Cottingham formula [124] represents the self-energy 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 self-energy 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 self-energy of the \(\pi ^+\). The low-energy theorem of Das, Guralnik, Mathur, Low and Young [128] ensures that, in the limit \(m_u,m_d\rightarrow 0\), the e.m. self-energy of the \(\pi ^0\) vanishes, while the one of the \(\pi ^+\) is given by an integral over the difference between the vector and axial-vector 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 self-energy 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 self-energy 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 self-energy 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 \(m_u-m_d\). There is a parameter-free 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 self-energy 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 pion-mass splitting in QCD, the result corresponds to \(\epsilon _m=0.04(5)\). The corrections of next-to-next-to-leading 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.

In summary, we use the following estimates for the e.m. corrections:

$$\begin{aligned}&\epsilon ={0.7(3)},\quad \epsilon _{\pi ^0}=0.07(7),\quad \epsilon _{K^0}=0.3(3),\nonumber \\&\quad \epsilon _m=0.04(2).\end{aligned}$$

While the range used for the coefficient \(\epsilon \) affects our analysis in a significant way, the numerical values of the other coefficients only serve to set the scale of these contributions. The range given for \(\epsilon _{\pi ^0}\) and \(\epsilon _{K^0}\) may be overly generous, but because of the exploratory nature of the lattice determinations, we consider it advisable to use a conservative estimate.

Treating the uncertainties in the four coefficients as statistically independent and adding errors in quadrature, the numbers in Eq. (13) yield the following estimates for the e.m. self-energies,

$$\begin{aligned}&M_{\pi ^+}^\gamma = 4.7(3)~\hbox {MeV},\quad M_{\pi ^0}^\gamma = 0.3(3)~\hbox {MeV},\nonumber \\&\quad M_{\pi ^+}^\gamma -M_{\pi ^0}^\gamma =4.4(1)~\hbox {MeV},\nonumber \\&M_{K^+}^\gamma = 2.5(5)~\hbox {MeV},\quad M_{K^0}^\gamma =0.4(4)\,\hbox {MeV},\nonumber \\&\quad M_{K^+}^\gamma -M_{K^0}^\gamma = 2.1(4)~\hbox {MeV}, \end{aligned}$$

and for the pion and kaon masses occurring in the QCD sector of the Standard Model,

$$\begin{aligned}&\hat{M}_{\pi ^+}= 134.8(3)~\hbox {MeV},\quad \hat{M}_{\pi ^0} = 134.6(3)~\hbox {MeV} ,\nonumber \\&\quad \hat{M}_{\pi ^+}-\hat{M}_{\pi ^0}=0.2(1)~\hbox {MeV},\nonumber \\&\hat{M}_{K^+}=491.2(5)~\hbox {MeV},\quad \hat{M}_{K^0} =497.2(4)~\hbox {MeV},\nonumber \\&\quad \hat{M}_{K^+}-\hat{M}_{K^0}=-6.1(4)~\hbox {MeV}.\end{aligned}$$

The self-energy difference between the charged and neutral pion involves the same coefficient \(\epsilon _m\) that describes the mass difference in QCD – this is why the estimate for \( M_{\pi ^+}^\gamma -M_{\pi ^0}^\gamma \) is so precise.

Pion and kaon masses in the isospin limit

As mentioned above, most of the lattice calculations concerning the properties of the light mesons are performed in the isospin limit of QCD (\(m_u-m_d\rightarrow 0\) at fixed \(m_u+m_d\)). We denote the pion and kaon masses in that limit by \(\overline{M}_{\pi }\) and \(\overline{M}_{K}\), respectively. Their numerical values can be estimated as follows. Since the operation \(u\leftrightarrow d\) interchanges \(\pi ^+\) with \(\pi ^-\) and \(K^+\) with \(K^0\), the expansion of the quantities \(\hat{M}_{\pi ^+}^2\) and \(\frac{1}{2}(\hat{M}_{K^+}^2+\hat{M}_{K^0}^2)\) in powers of \(m_u-m_d\) only contains even powers. As shown in Ref. [131], the effects generated by \(m_u-m_d\) in the mass of the charged pion are strongly suppressed: the difference \(\hat{M}_{\pi ^+}^2-\overline{M}_{\pi }^{\,2}\) represents a quantity of \(\mathcal {O}[(m_u-m_d)^2(m_u+m_d)]\) and is therefore small compared to the difference \(\hat{M}_{\pi ^+}^2-\hat{M}_{\pi ^0}^2\), for which an estimate was given above. In the case of \(\frac{1}{2}(\hat{M}_{K^+}^2+\hat{M}_{K^0}^2)-\overline{M}_{K}^{\,2}\), the expansion does contain a contribution at NLO, determined by the combination \(2L_8{-}L_5\) of low-energy constants, but the lattice results for that combination show that this contribution is very small, too. Numerically, the effects generated by \(m_u-m_d\) in \(\hat{M}_{\pi ^+}^2\) and in \(\frac{1}{2}(\hat{M}_{K^+}^2+\hat{M}_{K^0}^2)\) are negligible compared to the uncertainties in the electromagnetic self-energies. The estimates for these given in Eq. (15) thus imply

$$\begin{aligned}&\overline{M}_{\pi }= \hat{M}_{\pi ^+}=134.8(3)\,\mathrm{MeV},\nonumber \\&\overline{M}_{K}= \sqrt{\frac{1}{2}(\hat{M}_{K^+}^2+\hat{M}_{K^0}^2)}= 494.2(3)\,\mathrm{MeV}. \end{aligned}$$

This shows that, for the convention used above to specify the QCD sector of the Standard Model, and within the accuracy to which this convention can currently be implemented, the mass of the pion in the isospin limit agrees with the physical mass of the neutral pion: \(\overline{M}_{\pi }-M_{\pi ^0}=-0.2(3)\) MeV.

Lattice determination of \(m_s\) and \(m_{ud}\)

We now turn to a review of the lattice calculations of the light-quark masses and begin with \(m_s\), the isospin-averaged up- and down-quark mass, \(m_{ud}\), and their ratio. Most groups quote only \(m_{ud}\), not the individual up- and down-quark 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 mid-1990s. 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 physical-pion 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 CP-PACS 01 [134]. Moreover, the latter calculation is performed on very coarse lattices, with lattice spacings \(a\ge 0.11\,\mathrm{fm}\) and only one-loop 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 twisted-mass 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.

The ALPHA 12 [12] calculation of \(m_s\) is an update of ALPHA 05 [135], which pushes computations to finer lattices and much lighter pion masses. It also importantly includes a determination of the lattice spacing with the decay constant \(F_K\), whereas ALPHA 05 converted results to physical units using the scale parameter \(r_0\) [136], defined via the force between static quarks. In particular, the conversion relied on measurements of \(r_0/a\) by QCDSF/UKQCD 04 [137] which differ significantly from the new determination by ALPHA 12. As in ALPHA 05, in ALPHA 12 both nonperturbative running and nonperturbative renormalization are performed in a controlled fashion, using Schrödinger functional methods.

Table 3 \(N_{ f}=2\) 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
Table 4 \(N_{ f}=2+1\) lattice results for the masses \(m_{ud}\) and \(m_s\) (see Table 3 for notation)
Table 5 \(N_{ f}=2+1+1\) lattice results for the masses \(m_{ud}\) and \(m_s\) (see Table 3 for notation)

The conclusion of our analysis of \(N_{ f}=2\) calculations is that the results of ALPHA 12 [12] and ETM 10B [11] (which update and extend ALPHA 05 [135] and ETM 07 [133], respectively), are the only ones to date which satisfy our selection criteria. Thus we average those two results for \(m_s\), obtaining 101(3) MeV. Regarding \(m_{ud}\), for which only ETM 10B [11] gives a value, we do not offer an average but simply quote ETM’s number. Thus, we quote as our estimates:

$$\begin{aligned}&m_s&= 101(3)~\hbox {MeV}&\,\mathrm {Refs.}~ [11, 12],\nonumber \\&N_{ f}=2 :&\\&m_{ud}&= 3.6(2) ~\hbox {MeV}&\,\mathrm {Ref.}~[11].\nonumber \end{aligned}$$

The errors on these results are 3 and 6%, respectively. However, these errors do not account for the fact that sea strange-quark mass effects are absent from the calculation, a truncation of the theory whose systematic effects cannot be estimated a priori. Thus, these results carry an additional unknown systematic error. It is worth remarking that the difference between ALPHA 12’s [12] central value for \(m_s\) and that of ETM 10B [11] is 7(7) MeV.

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 charm-quark 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 isospin-breaking 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 sea-quarks affect the determination of the light-quark 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 domain-wall fermion simulations to three used previously. Two of the new simulations are performed at essentially physical-pion 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 Wilson-flow 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 light-quark 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 valence-pion 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 two-loop 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 PACS-CS 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 physical-mass point. This was achieved by reweighting the simulations performed in PACS-CS 08 [93] at \(M_\pi \simeq 160\,\mathrm{MeV}\). If adequately controlled, this procedure eliminates the need to extrapolate to the physical-mass point and, hence, the corresponding systematic error. The new calculation now applies similar reweighting techniques to include electromagnetic and \(m_u\ne m_d\) isospin-breaking effects directly at the physical-pion mass. Further, as in PACS-CS 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 finite-volume 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 down-quark 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 infinite-volume extrapolation are controlled. It does neglect, however, isospin-breaking 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 charm-quark mass, \(m_c\), with the result of a calculation of the quark-mass ratio, \(m_c/m_s\). As described in Ref. [152] and in Sect. 3.2, HPQCD determines \(m_c\) by fitting Euclidean-time moments of the \(\bar{c}c\) pseudoscalar density 2-point functions, obtained numerically in lattice QCD, to fourth-order, 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 quark-mass 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 charm-quark mass, \(m_c(m_c)=1.268(6)\),Footnote 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 \(\chi ^2/\hbox {d.o.f.}\! =\! 1.8.\)

For the light-quark mass \(m_{ud}\), the results satisfying our criteria are RBC/UKQCD 14B, BMW 10A, 10B, HPQCD 10, and MILC 10A. For the error, we include the same 100% correlation between statistical errors for the latter two as for the strange case, resulting in \(m_{ud}=3.373(43)\) at 2 GeV in the \(\overline{\mathrm{MS}}\) scheme \((\chi ^2/\hbox {d.o.f.}=1.5).\) Adding the 2% estimate for the missing charm contribution, our final estimates for the light-quark masses are

$$\begin{aligned}&m_{ud}&= 3.373 (80)\;\mathrm{MeV}&\,\mathrm {Refs.}~ [7{-}10, 13],\nonumber \\&N_{ f}=2+1 :&\nonumber \\&m_s&=92.0(2.1)\;\;\mathrm{MeV}&\,\mathrm {Refs.}~ [6{-}10]. \nonumber \\ \end{aligned}$$

\(N_{ f}=2+1+1\) lattice calculations One of the novelties since the last edition of this review [2] is the fact that \(N_f=2+1+1\) results for the light-quark masses have been published. These and the features of the corresponding calculations are summarized in Table 5. Note that the results of Ref. [5] are reported as \(m_s(2\,\mathrm{GeV};N_f=3)\) and those of Ref. [4] as \(m_{ud(s)}(2\,\mathrm{GeV};N_f=4)\). We convert the former to \(N_f=4\) and obtain \(m_s(2\,\mathrm{GeV};N_f=4)=93.7(8)~\mathrm{MeV}\). The average of ETM 14 and HPQCD 14A is 93.9(1.1) \(\mathrm{MeV}\) with \(\chi ^2/\hbox {d.o.f.}=1.8.\) For the light0quark average we use the sole available value from ETM 14A. Our averages are

$$\begin{aligned}&m_{ud}&= 3.70 (17)\;\mathrm{MeV}&\,\mathrm {Ref.}~[4],\nonumber \\&N_{ f}=2+1+1{:}&\nonumber \\&m_s&=93.9(1.1)\; \mathrm{MeV}&\,\mathrm {Refs.}~ [4,5].\nonumber \\ \end{aligned}$$

In Figs. 1 and 2 the lattice results listed in Tables 3, 4 and 5 and the FLAG averages obtained at each value of \(N_f\) are presented and compared with various phenomenological results.

Fig. 1
figure 1

\({\overline{\text {MS}}}\) mass of the strange quark (at 2 GeV scale) in MeV. The upper three panels show the lattice results listed in Tables 3, 4 and 5, while the bottom panel collects a few sum rule results and also indicates the current PDG estimate. Diamonds and squares represent results based on perturbative and nonperturbative renormalization, respectively. The black squares and the grey bands represent our estimates (17), (18) and (19). The significance of the colours is explained in Sect. 2

Fig. 2
figure 2

Mean mass of the two lightest quarks, \(m_{ud}=\frac{1}{2}(m_u+m_d)\) (for details see Fig. 1)

Lattice determinations of \(m_s/m_{ud}\)

Table 6 Lattice results for the ratio \(m_s/m_{ud}\)

The lattice results for \(m_s/m_{ud}\) are summarized in Table 6. In the ratio \(m_s/m_{ud}\), one of the sources of systematic error – the uncertainties in the renormalization factors – drops out. Also, we can compare the lattice results with the leading-order formula of \(\chi \)PT,

$$\begin{aligned} \frac{m_s}{m_{ud}}\mathop {=}\limits ^{{\mathrm{LO}}}\frac{\hat{M}_{K^+}^2+ \hat{M}_{K^0}^2-\hat{M}_{\pi ^+}^2}{\hat{M}_{\pi ^+}^2},\end{aligned}$$

which relates the quantity \(m_s/m_{ud}\) to a ratio of meson masses in QCD. Expressing these in terms of the physical masses and the four coefficients introduced in Eqs. (10)–(12), linearizing the result with respect to the corrections and inserting the observed mass values, we obtain

$$\begin{aligned} \frac{m_s}{m_{ud}} \mathop {=}\limits ^{{\mathrm{LO}}}25.9 - 0.1\, \epsilon + 1.9\, \epsilon _{\pi ^0} - 0.1\, \epsilon _{K^0} -1.8 \,\epsilon _m.\end{aligned}$$

If the coefficients \(\epsilon \), \(\epsilon _{\pi ^0}\), \(\epsilon _{K^0}\) and \(\epsilon _m\) are set equal to zero, the right hand side reduces to the value \(m_s/m_{ud}=25.9\), which follows from Weinberg’s leading-order formulae for \(m_u/m_d\) and \(m_s/m_d\) [161], in accordance with the fact that these do account for the e.m. interaction at leading chiral order, and neglect the mass difference between the charged and neutral pions in QCD. Inserting the estimates (13) gives the effect of chiral corrections to the e.m. self-energies and of the mass difference between the charged and neutral pions in QCD. With these, the LO prediction in QCD becomes

$$\begin{aligned} \frac{m_s}{m_{ud}}\mathop {=}\limits ^{{\mathrm{LO}}}25.9(1), \end{aligned}$$

leaving the central value unchanged at 25.9. The corrections parameterized by the coefficients of Eq. (13) are small for this quantity. Note that the quoted uncertainty does not include an estimate of higher-order chiral contributions to this LO QCD formula, but only accounts for the error bars in the coefficients. However, even this small uncertainty is no longer irrelevant given the high precision reached in lattice determinations of the ratio \(m_s/m_{ud}\).

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)-flavour-symmetry-breaking 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 finite-volume effects.

Therefore the FLAG average at \(N_f = 2\) is still obtained by considering only the ETM 10B result (described already in the previous section), namely

$$\begin{aligned} N_f = 2:\quad m_s / m_{ud} = 27.3 ~ (9)\quad \,\mathrm {Ref.}~[11], \end{aligned}$$

with an overall uncertainty equal to 3.3%.

\(N_{ f}=2+1\) 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%.

At this level of precision, the uncertainties in the electromagnetic and strong isospin-breaking corrections are not completely negligible. The error estimate in the LO result (22) indicates the expected order of magnitude. In view of this, we ascribe conservatively a 1.0% uncertainty to this source of error. Thus, our final conservative estimate is

$$\begin{aligned}&N_f = 2+1 : \quad \nonumber \\&{m_s}/{m_{ud}} = 27.43 ~ (13) ~ (27) = 27.43 ~ (31) \,\mathrm {Refs.}~[6{-}8, 10],\quad \end{aligned}$$

with a total 1.1% uncertainty. It is also fully consistent with the ratio computed from our individual quark masses in Eq. (18), \(m_s / m_{ud} = 27.6(6)\), which has a larger 2.2% uncertainty. In Eq. (24) the first error comes from the averaging of the lattice results, and the second is the one that we add to account for the neglect of isospin-breaking effects.

\(N_{ f}=2+1+1\) 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 twisted-mass 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 in the chiral extrapolation and the tag 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 in the finite-volume effects). They fix the strange mass with the kaon mass.

FNAL/MILC 14A employs HISQ staggered fermions. Their result is based on 21 ensembles 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 taste-Goldstone pion masses down to 130 MeV and RMS pion masses down to 143 MeV. They fix the strange mass with \(M_{\bar{s}s}\), corrected for e.m. effects with \(\bar{\epsilon }= 0.84(20)\) [113]. All of our selection criteria are satisfied with the tag . Thus our average is given by \(m_s / m_{ud} = 27.30 ~ (20)\), where the error includes a large stretching factor equal to \(\sqrt{\chi ^2/\hbox {d.o.f.}} \simeq 2.1\), coming from our rules for the averages discussed in Sect. 2.2. Nevertheless the above number amounts still to an accuracy of 0.7%. As in the case of our average for \(N_f = 2+1\), we add a 1.0% uncertainty related to the neglect of isospin-breaking effects, leading to

$$\begin{aligned} N_f= & {} 2+1+1 :\quad m_s / m_{ud} = 27.30 ~ (20) ~ (27) \nonumber \\= & {} 27.30 ~ (34)\,\mathrm {Refs.}~[4, 14], \end{aligned}$$

which corresponds to an overall uncertainty equal to 1.3%.

All the lattice results listed in Table 6 as well as the FLAG averages for each value of \(N_f\) are reported in Fig. 3 and compared with \(\chi \)PT, sum rules and the updated PDG estimate \(m_s / m_{ud} = 27.5(3)\) [151].

Fig. 3
figure 3

Results for the ratio \(m_s/m_{ud}\). The upper part indicates the lattice results listed in Table 6 together with the FLAG averages for each value of \(N_f\). The lower part shows results obtained from \(\chi \)PT and sum rules, together with the current PDG estimate

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 \((m_u-m_d)^2\) also here, the leading-order 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 \(11.4( 1.1) \%\).

Lattice determination of \(m_u\) and \(m_d\)

Since FLAG 13, two new results have been reported for nondegenerate light-quark 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 \(m_u-m_d\) from the slope of the square of the kaon mass and the neutral-charged mass-squares difference, evaluated at the isospin-symmetric point. The latter uses QCD+QED dynamical simulations performed at the SU(3)-flavour-symmetric 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 finite-volume and chiral-extrapolation criteria require ratings. In Ref. [166] results for \(\epsilon \) and \(m_{u}/m_{d}\) are computed in the so-called 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 isospin-breaking quark-mass 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.

RM123 11 [167] actually uses the e.m. parameter \(\epsilon =0.7(5)\) from the first edition of the FLAG review [1]. However, they estimate the effects of strong isospin breaking at first nontrivial order, by inserting the operator \(\frac{1}{2}(m_u-m_d)\int (\bar{u}u-\bar{d}d)\) into correlation functions, while performing the gauge averages in the isospin limit. Applying these techniques, they obtain \((\hat{M}_{K^0}^2-\hat{M}_{K^+}^2)/(m_d-m_u)=2.57(8)\,\mathrm{MeV}\). Combining this result with the phenomenological \((\hat{M}_{K^0}^2-\hat{M}_{K^+}^2)=6.05(63)\times 10^3\) determined with the above value of \(\epsilon \), they get \((m_d-m_u)=2.35(8)(24)\,\mathrm{MeV}\), where the first error corresponds to the lattice statistical and systematic uncertainties combined in quadrature, while the second arises from the uncertainty on \(\epsilon \). Note that below we quote results from RM123 11 for \(m_u\), \(m_d\) and \(m_u/m_d\). As described in Table 7, we obtain them by combining RM123 11’s result for \((m_d-m_u)\) with ETM 10B’s result for \(m_{ud}\).

Table 7 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

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.

PACS-CS 12 [143] takes the inclusion of isospin-breaking effects one step further. Using reweighting techniques, it also includes electromagnetic and \(m_u-m_d\) effects in the sea.

Lattice results for \(m_u\), \(m_d\) and \(m_u/m_d\) are summarized in Table 7. In order to discuss them, we consider the LO formula

$$\begin{aligned} \frac{m_u}{m_d}\mathop {=}\limits ^{{\mathrm{LO}}}\frac{\hat{M}_{K^+}^2-\hat{M}_{K^0}^2+\hat{M}_{\pi ^+}^2}{\hat{M}_{K^0}^2-\hat{M}_{K^+}^2+\hat{M}_{\pi ^+}^2} .\end{aligned}$$

Using Eqs. (10)–(12) to express the meson masses in QCD in terms of the physical ones and linearizing in the corrections, this relation takes the form

$$\begin{aligned} \frac{m_u}{m_d}\mathop {=}\limits ^{{\mathrm{LO}}}0.558 - 0.084\, \epsilon - 0.02\, \epsilon _{\pi ^0} + 0.11\, \epsilon _m .\end{aligned}$$

Inserting the estimates (13) and adding errors in quadrature, the LO prediction becomes

$$\begin{aligned} \frac{m_u}{m_d}\mathop {=}\limits ^{{\mathrm{LO}}}0.50(3).\end{aligned}$$

Again, the quoted error exclusively accounts for the errors attached to the estimates (13) for the epsilons – contributions of nonleading order are ignored. The uncertainty in the leading-order prediction is dominated by the one in the coefficient \(\epsilon \), which specifies the difference between the meson squared-mass splittings generated by the e.m. interaction in the kaon and pion multiplets. The reduction in the error on this coefficient since the previous review [1] results in a reduction of a factor of a little less than 2 in the uncertainty on the LO value of \(m_u/m_d\) given in Eq. (28).

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 PACS-CS 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 leading-order 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 higher-order 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 two-standard-deviation 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, PACS-CS 12, and QCDSF/UKQCD 15 do not pass our selection criteria either. We therefore resort to the phenomenological estimates of the electromagnetic self-energies discussed in Sect. 3.1.1, which are validated by recent, preliminary lattice results.

Since RM123 13 [16] includes a lattice estimate of e.m. corrections, for the \(N_{ f}=2\) final results we simply quote the values of \(m_u\), \(m_d\), and \(m_{u}/m_{d}\) from RM123 13 given in Table 7:


with errors of roughly 10, 5 and 8%, respectively. In these results, the errors are obtained by combining the lattice statistical and systematic errors in quadrature.

For \(N_{ f}=2+1\) there is to date no final, published computation of e.m. corrections. Thus, we take the LO estimate for \(m_u/m_d\) of Eq. (28) and use the −8(4)% obtained above as an estimate of the size of the corrections from higher orders in the chiral expansion. This gives \(m_u/m_d=0.46(3)\). The two individual masses can then be worked out from the estimate (18) for their mean. Therefore, for \(N_{ f}=2+1\) we obtain


In these results, the first error represents the lattice statistical and systematic errors, combined in quadrature, while the second arises from the uncertainties associated with e.m. corrections of Eq. (13). The estimates in Eq. (30) have uncertainties of order 5, 3 and 7%, respectively.

Finally, for four flavours we simply adopt the results of ETM 14A which meet all of our criteria.


Naively propagating errors to the end, we obtain \((m_u/m_d)_{N_f=2}/(m_u/m_d)_{N_f=2+1}=1.09(10)\). If instead of Eq. (29) we use the results from RM123 11, modified by the e.m. corrections in Eq. (13), as was done in our previous review, we obtain \((m_u/m_d)_{N_f=2}/(m_u/m_d)_{N_f=2+1}=1.11(7)(1)\), confirming again the strong cancellation of e.m. uncertainties in the ratio. The \(N_f=2\) and \(2+1\) results are compatible at the 1 to 1.5 \(\sigma \) level. Clearly the difference between three and four flavours is even smaller, and completely covered by the quoted uncertainties.

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 up-quark would solve the strong CP-problem, 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 CP-problem differently. This conclusion relies on lattice calculations of kaon masses and on the phenomenological estimates of the e.m. self-energies 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. self-energies. Further progress on the determination of the light-quark masses hinges on an improved understanding of the e.m. effects.

Estimates for R and Q

The quark-mass ratios

$$\begin{aligned} R\equiv \frac{m_s-m_{ud}}{m_d-m_u}\quad \hbox {and}\quad Q^2\equiv \frac{m_s^2-m_{ud}^2}{m_d^2-m_u^2} \end{aligned}$$

compare SU(3) breaking with isospin breaking. The quantity Q is of particular interest because of a low-energy theorem [168], which relates it to a ratio of meson masses,

$$\begin{aligned}&Q^2_M\equiv \frac{\hat{M}_K^2}{\hat{M}_\pi ^2}\cdot \frac{\hat{M}_K^2-\hat{M}_\pi ^2}{\hat{M}_{K^0}^2- \hat{M}_{K^+}^2},\quad \hat{M}^2_\pi \equiv \frac{1}{2}( \hat{M}^2_{\pi ^+}+ \hat{M}^2_{\pi ^0}) ,\nonumber \\&\quad \hat{M}^2_K\equiv \frac{1}{2}(\hat{M}^2_{K^+}+ \hat{M}^2_{K^0}).\end{aligned}$$

Chiral symmetry implies that the expansion of \(Q_M^2\) in powers of the quark masses (i) starts with \(Q^2\) and (ii) does not receive any contributions at NLO:

$$\begin{aligned} Q_M\mathop {=}\limits ^{{\mathrm{NLO}}}Q .\end{aligned}$$

Inserting the estimates for the mass ratios \(m_s/m_{ud}\), and \(m_u/m_d\) given for \(N_{ f}=2\) in Eqs. (17) and (29) respectively, we obtain

$$\begin{aligned} R=40.7(3.7)(2.2),\quad Q=24.3(1.4)(0.6), \end{aligned}$$

where the errors have been propagated naively and the e.m. uncertainty has been separated out, as discussed in the third paragraph after Eq. (28). Thus, the meaning of the errors is the same as in Eq. (30). These numbers agree within errors with those reported in Ref. [16] where values for \(m_s\) and \(m_{ud}\) are taken from ETM 10B [11].

For \(N_{ f}=2+1\), we use Eqs. (24) and (30) and obtain

$$\begin{aligned} R=35.7(1.9)(1.8),\quad Q=22.5(6)(6), \end{aligned}$$

where the meaning of the errors is the same as above. The \(N_{ f}=2\) and \(N_{ f}=2+1\) results are compatible within 2\(\sigma \), even taking the correlations between e.m. effects into account.

Again, for \(N_{ f}=2+1+1\), we simply take values from ETM 14A,

$$\begin{aligned} R=35.6(5.1),\quad Q=22.2(1.6), \end{aligned}$$

which are quite compatible with two and three flavour results.

It is interesting to use these results to study the size of chiral corrections in the relations of R and Q to their expressions in terms of meson masses. To investigate this issue, we use \(\chi \)PT to express the quark-mass ratios in terms of the pion and kaon masses in QCD and then again use Eqs. (10)–(12) to relate the QCD masses to the physical ones. Linearizing in the corrections, this leads to

$$\begin{aligned} R&\mathop {=}\limits ^{{\mathrm{LO}}}&R_M = 43.9 - 10.8\, \epsilon + 0.2\, \epsilon _{\pi ^0} - 0.2\, \epsilon _{K^0}- 10.7\, \epsilon _m,\nonumber \\ \end{aligned}$$
$$\begin{aligned} Q&\mathop {=}\limits ^{{\mathrm{NLO}}}&Q_M = 24.3 - 3.0\, \epsilon + 0.9\, \epsilon _{\pi ^0} - 0.1\, \epsilon _{K^0} + 2.6 \,\epsilon _m .\nonumber \\ \end{aligned}$$

While the first relation only holds to LO of the chiral perturbation series, the second remains valid at NLO, on account of the low-energy theorem mentioned above. The first terms on the right hand side represent the values of R and Q obtained with the Weinberg leading-order formulae for the quark-mass ratios [161]. Inserting the estimates (13), we find that the e.m. corrections lower the Weinberg values to \(R_M= 36.7(3.3)\) and \(Q_M= 22.3(9)\), respectively.

Comparison of \(R_M\) and \(Q_M\) with the full results quoted above gives a handle on higher-order 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.

As mentioned in Sect. 3.1.1, there is a phenomenological determination of Q based on the decay \(\eta \rightarrow 3\pi \) [169, 170]. The key point is that the transition \(\eta \rightarrow 3\pi \) violates isospin conservation. The dominating contribution to the transition amplitude stems from the mass difference \(m_u-m_d\). At NLO of \(\chi \)PT, the QCD part of the amplitude can be expressed in a parameter-free manner in terms of Q. It is well known that the electromagnetic contributions to the transition amplitude are suppressed (a thorough recent analysis is given in Ref. [171]). This implies that the result for Q is less sensitive to the electromagnetic uncertainties than the value obtained from the masses of the Nambu–Goldstone bosons. For a recent update of this determination and for further references to the literature, we refer to Ref. [172]. Using dispersion theory to pin down the momentum dependence of the amplitude, the observed decay rate implies \(Q=22.3(8)\) (since the uncertainty quoted in Ref. [172] does not include an estimate for all sources of error, we have retained the error estimate given in Ref. [165], which is twice as large). The formulae for the corrections of NNLO are available also in this case [173] – the poor knowledge of the effective coupling constants, particularly of those that are relevant for the dependence on the quark masses, is currently the limiting factor encountered in the application of these formulae.

Table 8 Our estimates for the strange-quark and the average up-down-quark 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 sea-quark and isospin-breaking 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 sea-quark effects
Table 9 Our estimates for the masses of the two lightest quarks and related, strong isospin-breaking 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

As was to be expected, the central value of Q obtained from \(\eta \)-decay agrees exactly with the central value obtained from the low-energy theorem: we have used that theorem to estimate the coefficient \(\epsilon \), which dominates the e.m. corrections. Using the numbers for \(\epsilon _m\), \(\epsilon _{\pi ^0}\) and \(\epsilon _{K^0}\) in Eq. (13) and adding the corresponding uncertainties in quadrature to those in the phenomenological result for Q, we obtain

$$\begin{aligned} \epsilon \mathop {=}\limits ^{{\mathrm{NLO}}}0.70(28).\end{aligned}$$

The estimate (13) for the size of the coefficient \(\epsilon \) is taken from this, as is confirmed by the most recent, preliminary lattice determinations [16, 110,111,112, 115, 116].

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.

Charm-quark mass

In the present review we collect and discuss for the first time the lattice determinations of the \(\overline{\mathrm{MS}}\) charm-quark mass \(\overline{m}_c\). Most of the results have been obtained by analyzing the lattice-QCD simulations of 2-point heavy–light- or heavy–heavy-meson 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 pseudoscalar-pseudoscalar correlators for heavy-quark currents followed by an OPE expansion dominated by perturbative QCD effects, which provides the determination of both the heavy-quark mass and the strong coupling constant \(\alpha _s\).

The heavy-quark actions adopted by the various lattice collaborations have been reviewed already in the FLAG 13 review [2], and their descriptions can be found in Sect. A.1.3. While the charm mass determined with the moments method does not need any lattice evaluation of the mass renormalization constant \(Z_m\), the extraction of \(\overline{m}_c\) from 2-point heavy-meson correlators does require the nonperturbative calculation of \(Z_m\). The lattice scale at which \(Z_m\) is obtained, is usually at least of the order \(2{-}3\) GeV, and therefore it is natural in this review to provide the values of \(\overline{m}_c(\mu )\) at the renormalization scale \(\mu = 3~\mathrm{GeV}\). Since the choice of a renormalization scale equal to \(\overline{m}_c\) is still commonly adopted (as by PDG [151]), we have collected in Table 10 the lattice results for both \(\overline{m}_c(\overline{m}_c)\) and \(\overline{m}_c(\hbox {3 GeV})\), obtained at \(N_f = 2\), \(2+1\) and \(2+1+1\). When not directly available in the publications, we apply a conversion factor equal either to 0.900 between the scales \(\mu = 2\) GeV and \(\mu = 3\) GeV or to 0.766 between the scales \(\mu = \overline{m}_c\) and \(\mu = 3\) GeV, obtained using perturbative QCD evolution at four loops assuming \(\Lambda _{QCD} = 300\) MeV for \(N_f = 4\).

Table 10 Lattice results for the \({\overline{\text {MS}}}\)-charm-quark 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

In the next subsections we review separately the results of \(\overline{m}_c(\overline{m}_c)\) for the various values of \(N_f\).

\(N_f = 2+1+1\) results

There are three recent results employing four dynamical quarks in the sea. ETM 14 [4] uses 15 twisted-mass 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 in the chiral extrapolation and the tag 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 in the finite-volume 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 taste-Goldstone-pion masses down to 130 MeV and RMS-pion masses down to 173 MeV. The strange- and charm-quark 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 .Footnote 14

According to our rules on the publication status all the three results can enter the FLAG average at \(N_f = 2+1+1\). The determinations of \(\overline{m}_c\) obtained by ETM 14 and 14A agree quite well with each other, but they are not compatible with the HPQCD 14A result. Therefore we first combine the two ETM results with a 100\(\%\) correlation in the statistical error, yielding \(\overline{m}_c(\overline{m}_c) = 1.348 (20)~ \mathrm{GeV}\). Then we perform the average with the HPQCD 14A result, obtaining the final FLAG averages,

$$\begin{aligned}& \overline{m}_c(\overline{m}_c) = 1.286 ~ (30) ~ \mathrm{GeV}&\,\mathrm {Refs.}~[4,5], \end{aligned}$$
$$\begin{aligned}&N_f = 2+1+1:&\nonumber \\& \overline{m}_c(\hbox {3 GeV}) = 0.996 ~ (25)~ \mathrm{GeV}&\,\mathrm {Refs.}~[4,5], \end{aligned}$$

where the errors include a quite large value (3.5 and 4.4, respectively) for the stretching factor \(\sqrt{\chi ^2/\hbox {d.o.f.}}\) coming from our rules for the averages discussed in Sect. 2.2.

\(N_f = 2+1\) results

The HPQCD 10 [9] result is based on the moments method adopting a subset of \(N_f = 2+1\) Asqtad-staggered-fermion 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 mixed-action approach based on overlap fermions for the valence quarks and on domain-wall 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 light-quark 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 for the finite-volume effects. The strange- and charm-quark 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 domain-wall 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 \({\simeq } 4.4\).

Thus, according to our rules on the publication status, the FLAG average for the charm-quark mass at \(N_f = 2+1\) is obtained by combining the two results HPQCD 10 and \(\chi \)QCD 14, leading to

$$\begin{aligned}&\overline{m}_c(\overline{m}_c) = 1.275 ~ (8) ~ \mathrm{GeV}&\,\mathrm {Refs.}~[9,17], \end{aligned}$$
$$\begin{aligned}&N_f = 2+1:&\nonumber \\&\overline{m}_c(\hbox {3 GeV}) = 0.987 ~ (6)~ \mathrm{GeV}&\,\mathrm {Refs.}~[9,17], \end{aligned}$$

where the error on \( \overline{m}_c(\overline{m}_c)\) includes a stretching factor \(\sqrt{\chi ^2/\hbox {d.o.f.}} \simeq 1.4\) as discussed in Sect. 2.2.

\(N_f = 2\) 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 strange-quark 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 for finite-volume effects).

According to our rules on the publication status ETM 10B becomes the FLAG average at \(N_f = 2\), namely

$$\begin{aligned}&\overline{m}_c(\overline{m}_c) = 1.28~(4) ~ \mathrm{GeV}&\,\mathrm {Ref.}~[11], \end{aligned}$$
$$\begin{aligned}&N_f = 2:&\nonumber \\&\overline{m}_c(\hbox {3 GeV}) = 1.03~(4)~ \mathrm{GeV}&\,\mathrm {Ref.}~[11]. \end{aligned}$$

In Fig. 4 the lattice results of Table 10 and the FLAG averages obtained at \(N_f = 2\), \(2+1\) and \(2+1+1\) are presented.

Fig. 4
figure 4

Lattice results and FLAG averages at \(N_f = 2\), \(2+1\), and \(2+1+1\) for the charm-quark mass \(\overline{m}_c(3~\mathrm{GeV})\)

Lattice determinations of the ratio \(m_c/m_s\)

Because some of the results for the light-quark masses given in this review are obtained via the quark-mass ratio \(m_c/m_s\), we now review also these lattice calculations, which are listed in Table 11.

Table 11 Lattice results for the quark-mass ratio \(m_c/m_s\), together with the colour coding of the calculations used to obtain these

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 Wilson-fermion simulations [139, 178] on which valence, Brillouin-improved Wilson quarks [179] are considered. It features only 2 ensembles with \(M_\pi < 400~\mathrm{MeV}\). The bare axial-Ward-identity (AWI) masses for \(m_s\) and \(m_c\) are tuned to simultaneously reproduce the physical values of \(M_{\bar{s}s}^2/(M_{D_s^*}^2-M_{D_s}^2)\) and \((2M_{D_s^*}^2-M_{\bar{s}s}^2)/(M_{D_s^*}^2-M_{D_s}^2)\), where \(M_{\bar{s}s}^2 = 685.8 (8)\) MeV is the quark-connected-\(\bar{s}s\) pseudoscalar mass.

The ETM 14D result [160] is 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 finite-volume effects. The FLAG average at \(N_f = 2\) can be therefore obtained by averaging ETM 10B and Dürr 11, obtaining

$$\begin{aligned} N_f = 2: \quad m_c / m_s = 11.74 ~ (35)\quad \,\mathrm {Refs.}~[11,132], \end{aligned}$$

where the error includes the stretching factor \(\sqrt{\chi ^2/\hbox {d.o.f.}} \simeq 1.5\).

The situation is similar also for the \(N_f = 2+1\) results, as besides \(\chi \)QCD 14 there is only the result HPQCD 09A [18]. The latter is based on a subset of \(N_f = 2+1\) Asqtad-staggered-fermion simulations from MILC, on which HISQ-valence fermions are studied. The strange mass is fixed with \(M_{\bar{s}s} = 685.8(4.0),\mathrm{MeV}\) and the charm’s from that of the \(\eta _c\), \(M_{\eta _c} = 2.9852(34)~\mathrm{GeV}\) corrected for \(\bar{c}c\) annihilation and e.m. effects. By combing the results \(\chi \)QCD 14 and HPQCD 09A we obtain

$$\begin{aligned} N_f = 2+1: \quad m_c / m_s = 11.82 ~ (16)\quad \,\mathrm {Refs.}~[17,18], \end{aligned}$$

with a \(\chi ^2/\hbox {d.o.f.} \simeq 0.85\).

Turning now to the \(N_f = 2+1+1\) results, in addition to the HPQCD 14A and ETM 14 calculations, already described in Sect. 3.2.1, we consider the recent FNAL/MILC 14 result [14], where HISQ staggered fermions are employed. Their result is based on the use of 21 gauge ensembles 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 taste-Goldstone-pion masses down to 130 MeV and RMS-pion masses down to 143 MeV. They fix the strange mass with \(M_{\bar{s}s}\), corrected for e.m. effects with \(\bar{\epsilon }= 0.84(20)\) [113]. The charm mass is fixed with the mass of the \(D_s\) meson. As for the HPQCD 14A result, all of our selection criteria are satisfied with the tag . However, a slight tension exists between the two results. Indeed by combining HPQCD 14A and FNAL/MILC 14 results, assuming a 100 \(\%\) correlation between the statistical errors (since the two results share the same gauge configurations), we obtain \(m_c / m_s = 11.71 (6)\), where the error includes the stretching factor \(\sqrt{\chi ^2/\hbox {d.o.f.}} \simeq 1.35\). A further average with the ETM 14A result leads to our final average

$$\begin{aligned} N_f = 2+1+1: \quad m_c / m_s = 11.70 ~ (6)\quad \,\mathrm {Refs.}~[4,5,14],\nonumber \\ \end{aligned}$$

which has a remarkable overall precision of 0.5\(\%\).

All of the results for \(m_c/m_s\) discussed above are shown in Fig. 5 together with the FLAG averages corresponding to \(N_f =2\), \(2+1\) and \(2+1+1\).

Fig. 5
figure 5

Lattice results for the ratio \(m_c / m_s\) listed in Table 11 and the FLAG averages corresponding to \(N_f =2\), \(2+1\) and \(2+1+1\)

Bottom-quark mass

We now give the lattice results for the \(\overline{\mathrm{MS}}\)-bottom-quark mass \(\overline{m}_b\) for the first time as part of this review. Related heavy-quark actions and observables have been discussed in the FLAG 13 review [2], and descriptions can be found in Sect. A.1.3. In Table 12 we have collected the lattice results for \(\overline{m}_b(\overline{m}_b)\) obtained at \(N_f = 2\), \(2+1\) and \(2+1+1\), which in the following we review separately. Available results for the quark-mass ratio \(m_b / m_c\) are also reported. Afterwards we evaluate the corresponding FLAG averages.

Table 12 Lattice results for the \({\overline{\text {MS}}}\)-bottom-quark 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


Results have been published by HPQCD using NRQCD and HISQ-quark actions (HPQCD 14B  [19] and HPQCD 14A [5], respectively). In both works the b-quark mass is computed with the moments method, that is, from Euclidean-time moments of 2-point, heavy–heavy meson correlation functions (see Sect. 9.7 for a description of the method).

In HPQCD 14B the b-quark mass is computed from ratios of the moments \(R_n\) of heavy current–current correlation functions, namely

$$\begin{aligned} \left[ \frac{R_n r_{n-2}}{R_{n-2}r_n}\right] ^{1/2} \frac{\bar{M}_\mathrm{kin}}{2 m_b} = \frac{\bar{M}_{\Upsilon ,\eta _b}}{2 \bar{m}_b(\mu )}, \end{aligned}$$

where \(r_n\) are the perturbative moments calculated at N\(^3\)LO, \(\bar{M}_\mathrm{kin}\) is the spin-averaged kinetic mass of the heavy-heavy vector and pseudoscalar mesons and \(\bar{M}_{\Upsilon ,\eta _b}\) is the experimental spin average of the \(\Upsilon \) and \(\eta _b\) masses. The kinetic mass \(\bar{M}_\mathrm{kin}\) is chosen since in the lattice calculation the splitting of the \(\Upsilon \) and \(\eta _b\) states is inverted. In Eq. (50) the bare mass \(m_b\) appearing on the left hand side is tuned so that the spin-averaged mass agrees with experiment, while the mass \(\overline{m}_b\) at the fixed scale \(\mu = 4.18\) GeV is extrapolated to the continuum limit using three HISC (MILC) ensembles with \(a \approx 0.15, 0.12\) and 0.09 fm and two pion masses, one of which is the physical one. Therefore according to our rules on the chiral extrapolation a warning must be given. Their final result is \(\overline{m}_b(\mu = 4.18~\mathrm{GeV}) = 4.207(26)\) GeV, where the error is from adding systematic uncertainties in quadrature only (statistical errors are smaller than \(0.1 \%\) and ignored). The errors arise from renormalization, perturbation theory, lattice spacing, and NRQCD systematics. The finite-volume uncertainty is not estimated, but at the lowest pion mass they have \( m_\pi L \simeq 4\), which leads to the tag .

In HPQCD 14A the quark mass is computed using a similar strategy as above but with HISQ heavy quarks instead of NRQCD. The gauge-field ensembles are the same as in HPQCD 14B above plus the one with \(a = 0.06\) fm (four lattice spacings in all). Bare heavy-quark masses are tuned to their physical values using the \(\eta _h\) mesons, and ratios of ratios yield \(m_h/m_c\). The \(\overline{\mathrm{MS}}\)-charm-quark 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 sea-mass dependence, with several knots, and many priors for 21 ratios to fit 29 data points. Taking this fit at face value results in a 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 four-flavour 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 b-quark 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 b-quark 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.


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 3-loop 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 finite-volume 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 \(2+1\)-flavour-Asqtad ensembles are used under HISQ-heavy-valence 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 finite-volume error based on leading-order perturbation theory for the moment ratio is also provided. Details of perturbation theory and renormalization systematics are given in Sect. 9.7.


The ETM Collaboration computes \(\overline{m}_b(\overline{m}_b)\) using the ratio method described above on two-flavour twisted-mass 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 heavy-quark masses cover a range from charm to a little more than three GeV, plus the exact static-limit point. They find \(\overline{m}_b(\overline{m}_b) = 4.31(9)(8)\) GeV for two-flavour running, while \(\overline{m}_b(\overline{m}_b) = 4.27(9)(8)\) using four-flavour 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 RI-MOM 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 five-flavour results, \(\overline{m}_b(\overline{m}_b) = 4.29(9)(8)(2)\) where the last error is one-half the difference between the two. In our average (see below), we use the two-flavour 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 bare-quark mass is related to the RGI-scheme mass using the Schrödinger Functional technique with conversion to \(\overline{\mathrm{MS}}\) through four-loop 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 two-flavour 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.

Averages for \(\overline{m}_b(\overline{m}_b)\)

Taking the results that meet our rating criteria, , or better, we compute the averages from HPQCD 14A and 14B for \(N_f = 2+1+1\), ETM 13B and ALPHA 13C for \(N_f = 2\), and we take HPQCD 10 as estimate for \(N_f = 2+1\), obtaining

$$\begin{aligned}&N_f= 2+1+1 :&\overline{m}_b(\overline{m}_b)&= 4.190 (21)&\,\mathrm {Refs.}~[5,19],\end{aligned}$$
$$\begin{aligned}&N_f= 2+1 :&\overline{m}_b(\overline{m}_b)&= 4.164 (23)&\,\mathrm {Ref.}~[9], \end{aligned}$$
$$\begin{aligned}&N_f= 2 :&\overline{m}_b(\overline{m}_b)&= 4.256 (81)&\,\mathrm {Refs.}~[20,21]. \end{aligned}$$

Since HPQCD quotes \(\overline{m}_b(\overline{m}_b)\) values using \(N_f = 5\) running, we used those values directly in these \(N_f=2+1+1\) and \(2+1\) averages. The results ETM 13B and ALPHA 13C, entering the average at \(N_f = 2\), correspond to the \(N_f =2 \) running.

All the results for \(\overline{m}_b(\overline{m}_b)\) discussed above are shown in Fig. 6 together with the FLAG averages corresponding to \(N_f = 2\), \(2+1\) and \(2+1+1\).

Fig. 6
figure 6

Lattice results and FLAG averages at \(N_f = 2\), \(2+1\), and \(2+1+1\) for the b-quark mass \(\overline{m}_b(\overline{m}_b)\). The updated PDG value from Ref. [151] is reported for comparison

Leptonic and semileptonic kaon and pion decay and \(|V_{ud}|\) and \(|V_{us}|\)

This section summarizes state-of-the-art lattice calculations of the leptonic kaon and pion decay constants and the kaon semileptonic-decay 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 \(f_{K^\pm } / f_{\pi ^\pm }\).

Experimental information concerning \(|V_{ud}|\), \(|V_{us}|\), \(f_+(0)\) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}\)

The following review relies on the fact that precision experimental data on kaon decays very accurately determine the product \(|V_{us}|f_+(0)\) [183] and the ratio \(|V_{us}/V_{ud}|f_{K^\pm }/f_{\pi ^\pm }\) [183, 184]:

$$\begin{aligned} |V_{us}| f_+(0) = 0.2165(4),\quad \left| \frac{V_{us}}{V_{ud}}\right| \frac{ f_{K^\pm }}{ f_{\pi ^\pm }} \; =0.2760(4).\end{aligned}$$

Here and in the following \(f_{K^\pm }\) and \(f_{\pi ^\pm }\) are the isospin-broken decay constants, respectively, in QCD (the electromagnetic effects have already been subtracted in the experimental analysis using chiral perturbation theory). We will refer to the decay constants in the SU(2) isospin-symmetric limit as \(f_K\) and \(f_\pi \) (the latter at leading order in the mass difference (\(m_u - m_d\)) coincides with \(f_{\pi ^\pm }\)). \(|V_{ud}|\) and \(|V_{us}|\) are elements of the Cabibbo–Kobayashi–Maskawa matrix and \(f_+(t)\) represents one of the form factors relevant for the semileptonic decay \(K^0\rightarrow \pi ^-\ell \,\nu \), which depends on the momentum transfer t between the two mesons. What matters here is the value at \(t=0\): \(f_+(0)\equiv f_+^{K^0\pi ^-}(t)\,\Big |_{\;t\rightarrow 0}\). The pion and kaon decay constants are defined byFootnote 15

$$\begin{aligned}&\langle 0|\,\,\overline{d}\gamma _\mu \gamma _5 u|\pi ^+(p)\rangle =i p_\mu f_{\pi ^+},\\&\quad \langle 0|\,\,\overline{s}\gamma _\mu \gamma _5 u|K^+(p)\rangle =i p_\mu f_{K^+}.\end{aligned}$$

In this normalization, \(f_{\pi ^\pm } \simeq 130\) MeV, \(f_{K^\pm }\simeq 155\) MeV.

The measurement of \(|V_{ud}|\) based on superallowed nuclear \(\beta \) transitions has now become remarkably precise. The result of the update of Hardy and Towner [186], which is based on 20 different superallowed transitions, readsFootnote 16

$$\begin{aligned} |V_{ud}| = 0.97417(21).\end{aligned}$$
Table 13 Colour code for the data on \(f_+(0)\)

The matrix element \(|V_{us}|\) can be determined from semiinclusive \(\tau \) decays [193,194,195,196]. Separating the inclusive decay \(\tau \rightarrow \hbox {hadrons}+\nu \) into nonstrange and strange final states, e.g. HFAG 14 [197] obtain

$$\begin{aligned} |V_{us}|=0.2176(21) .\end{aligned}$$

Maltman et al. [195, 198, 199] and Gamiz et al. [200, 201] arrive at very similar values.

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 three-flavour SM-unitarity 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 quark-hadron 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.

The experimental results in Eq. (54) are for the semileptonic decay of a neutral kaon into a negatively charged pion and the charged pion and kaon leptonic decays, respectively, in QCD. In the case of the semileptonic decays the corrections for strong and electromagnetic isospin breaking in chiral perturbation theory at NLO have allowed for averaging the different experimentally measured isospin channels [203]. This is quite a convenient procedure as long as lattice QCD does not include strong or QED isospin-breaking effects. Lattice results for \(f_K/f_\pi \) are typically quoted for QCD with (squared) pion and kaon masses of \(M_\pi ^2=M_{\pi ^0}^2\) and \(M_K^2=\frac{1}{2} (M_{K^\pm }^2+M_{K^0}^2-M_{\pi ^\pm }^2+M_{\pi ^0}^2)\) for which the leading strong and electromagnetic isospin violations cancel. While progress is being made for including strong and electromagnetic isospin breaking in the simulations (e.g. Refs. [16, 93, 167, 204,205,206,207]), for now contact to experimental results is made by correcting leading SU(2) isospin breaking guided either by chiral perturbation theory or by lattice calculations.

Table 14 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) isospin-breaking correction

Lattice results for \(f_+(0)\) and \(f_{K^\pm }/f_{\pi ^\pm }\)

The traditional way of determining \(|V_{us}|\) relies on using estimates for the value of \(f_+(0)\), invoking the Ademollo–Gatto theorem [219]. Since this theorem only holds to leading order of the expansion in powers of \(m_u\), \(m_d\) and \(m_s\), theoretical models are used to estimate the corrections. Lattice methods have now reached the stage where quantities like \(f_+(0)\) or \(f_K/f_\pi \) can be determined to good accuracy. As a consequence, the uncertainties inherent in the theoretical estimates for the higher-order effects in the value of \(f_+(0)\) do not represent a limiting factor any more and we shall therefore not invoke those estimates. Also, we will use the experimental results based on nuclear \(\beta \) decay and \(\tau \) decay exclusively for comparison – the main aim of the present review is to assess the information gathered with lattice methods and to use it for testing the consistency of the SM and its potential to provide constraints for its extensions.

Fig. 7
figure 7

Comparison of lattice results (squares) for \(f_+(0)\) and \(f_{K^\pm }/ f_{\pi ^\pm }\) with various model estimates based on \(\chi \)PT (blue circles). The ratio \(f_{K^\pm }/f_{\pi ^\pm }\) is obtained in pure QCD including the SU(2) isospin-breaking correction (see Sect. 4.3). The black squares and grey bands indicate our estimates. The significance of the colours is explained in Sect. 2

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, finite-size 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 quantity \(f_+(0)\) represents a matrix element of a strangeness-changing null-plane charge, \(f_+(0)=\langle K|Q^{us}|\pi \rangle \). The vector charges obey the commutation relations of the Lie algebra of SU(3), in particular \([Q^{us},Q^{su}]=Q^{uu-ss}\). This relation implies the sum rule \(\sum _n |\langle K|Q^{us}|n \rangle |^2-\sum _n |\langle K|Q^{su}|n \rangle |^2=1\). Since the contribution from the one-pion intermediate state to the first sum is given by \(f_+(0)^2\), the relation amounts to an exact representation for this quantity [220]:

$$\begin{aligned} f_+(0)^2=1-\sum _{n\ne \pi } |\langle K|Q^{us}|n \rangle |^2+\sum _n |\langle K |Q^{su}|n \rangle |^2.\end{aligned}$$

While the first sum on the right extends over nonstrange intermediate states, the second runs over exotic states with strangeness \(\pm 2\) and is expected to be small compared to the first.

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 low-energy 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 higher-order contributions \(\Delta f\equiv f_+(0)-1-f_2\) 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 symmetry-breaking 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.Footnote 17 The numerical result, however, depends on the model used to estimate the low-energy 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 low-energy constants appearing in \(f_4\) from lattice data.

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 isospin-symmetric QCD. The reason for this unphysical parameter choice is that there are only few simulations of SU(2) isospin-breaking 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 \((m_u-m_d)\) 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 light-quark 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) isospin-breaking 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 isospin-symmetric 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 \(\langle \pi |S|K\rangle \) of the flavour-changing 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 vector-current 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 physical-pion 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 form-factor 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 high-precision determination of the scalar form factor at the kinematical end-point \(q_{\mathrm{max}}^2 = (M_K - M_\pi )^2\) [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 taste-breaking effects, and includes simulations with three lattice spacings and physical light-quark 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 finite-size 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 domain-wall 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 light-quark 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 finite-volume effects, for which an estimate based on effective-theory 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 finite-lattice spacing inherent in their approach. The \(p^4\) low-energy constants entering the NNLO expression have been fixed in terms of external input [130].

The ETM Collaboration uses the twisted-mass 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].

We now compute the \(N_f =2+1\) FLAG average for \(f_+(0)\) based on FNAL/MILC 12I and RBC/UKQCD 15A, which we consider uncorrelated, while for \(N_f = 2+1+1\) and \(N_f = 2\) we consider directly the FNAL/MILC 13E and ETM 09A results, respectively:

$$\begin{aligned}&\hbox {direct},\,N_{ f}=2+1+1:&f_+(0)&= 0.9704(24)(22)\quad \,\mathrm {Ref.}~[22],\end{aligned}$$
$$\begin{aligned}&\hbox {direct},\,N_{ f}=2+1:&f_+(0)&= 0.9677(27) \quad \,\mathrm {Refs.}~[23,24], \end{aligned}$$
$$\begin{aligned}&\hbox {direct},\,N_{ f}=2:&f_+(0)&= 0.9560(57)(62)\quad \,\mathrm {Ref.}~[25], \end{aligned}$$

where the brackets in the first and third lines indicate the statistical and systematic errors, respectively. We stress that the results (58) and (59), corresponding to \(N_f = 2+1+1\) and \(N_f = 2+1\) respectively, include already simulations with physical light-quark masses.

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 twisted-mass 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 mass-renormalization constant, the finite-volume and (strong) isospin-breaking 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 sea-quark 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 light-quark 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, finite-volume 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 finite-volume 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 higher-order terms including discretization and finite-volume 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 finite-volume effects and the last error contains the combined uncertainties from the chiral extrapolation, the scale-setting 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 taste-breaking 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 finite-volume effects, NLO staggered chiral perturbation theory (MILC) or continuum chiral perturbation theory (HPQCD) was used for correcting the lattice data towards the infinite-volume limit. MILC then compared the finite-volume correction to the one obtained by the NNLO expression and took the difference as their estimate for the residual finite-volume error. In addition they checked the compatibility of the effective-theory 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.

Adding in quadrature all the uncertainties one gets \(f_{K^\pm }/f_{\pi ^\pm } = 1.1916(22)\) (HPQCD 13A) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}=1.1960(24)\) Footnote 18 (FNAL/MILC 14A). It can be seen that the total errors turn out to be very similar, but the central values seem to show a slight tension of about two standard deviations. While FLAG is looking forward to independent confirmations of the result for \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) at the same level of precision, we evaluate the FLAG average using a two-step procedure. First, the HPQCD 13A and FNAL/MILC 14A are averaged assuming a \(100 \%\) statistical correlation, obtaining \( {f_{K^\pm }}/{f_{\pi ^\pm }}=1.1936(29)\), where, following the prescription of Sect. 2.3, the error has been inflated by the factor \(\sqrt{(\chi ^2/\mathrm{d.o.f.})} \simeq \sqrt{2.5}\) as a result of the tension between the two central values. Then, the above finding is averaged with the (uncorrelated) ETM 14E result, obtaining

$$\begin{aligned}&\hbox {direct},\,N_{ f}=2+1+1: \quad {f_{K^\pm }}/{f_{\pi ^\pm }}=1.1933(29)\nonumber \\&\quad \,\mathrm {Refs.}~ [14,26,27]. \end{aligned}$$

For both \(N_f=2+1\) and \(N_f=2\) no new result enters the corresponding FLAG averages with respect to the previous edition of the FLAG review [2] and before the closing date specified in Sect. 1. Here we limit ourselves to note that for \(N_f=2+1\) MILC 10 and HPQCD/UKQCD 07 are based on staggered fermions, BMW 10 has used improved Wilson fermions and RBC/UKQCD 12’s result is based on the domain-wall formulation. Concerning simulations with \(N_f=2\) the FLAG average remains the ETM 09 result, which has simulated twisted-mass fermions. In contrast to FNAL/MILC 14A all these simulations are for unphysical values of the light-quark masses (corresponding to smallest pion masses in the range \(240{-}260\) MeV in the case of MILC 10, HPQCD/UKQCD 07 and ETM 09 and around 170 MeV for RBC/UKQCD 12) and therefore slightly more sophisticated extrapolations needed to be controlled. Various ansätze for the mass and cutoff dependence comprising SU(2) and SU(3) chiral perturbation theory or simply polynomials were used and compared in order to estimate the model dependence. While BMW 10 and RBC/UKQCD 12 are entirely independent computations, subsets of the MILC gauge ensembles used by MILC 10 and HPQCD/UKQCD 07 are the same. MILC 10 is certainly based on a larger and more advanced set of gauge configurations than HPQCD/UKQCD 07. This allows them for a more reliable estimation of systematic effects. In this situation we consider only their statistical but not their systematic uncertainties to be correlated.

Before determining the average for \(f_{K^\pm }/f_{\pi ^\pm }\), which should be used for applications to Standard-Model phenomenology, we apply the isospin correction individually to all those results which have been published in the isospin-symmetric limit, i.e. BMW 10, HPQCD/UKQCD 07 and RBC/UKQCD 12 at \(N_f = 2+1\) and ETM 09 at \(N_f = 2\). To this end, as in the previous edition of the FLAG review [2], we make use of NLO SU(3) chiral perturbation theory [129, 247], which predicts

$$\begin{aligned} \frac{f_{K^\pm }}{f_{\pi ^\pm }}= \frac{f_K}{f_\pi } ~ \sqrt{1 + \delta _{SU(2)}}, \end{aligned}$$

where [247]

$$\begin{aligned} \delta _{SU(2)}\approx & {} \sqrt{3}\,\epsilon _{SU(2)} \left[ -\frac{4}{3} \left( f_K/f_\pi -1\right) +\frac{2}{3 (4\pi )^2 f_0^2} \right. \nonumber \\&\times \left. \left( M_K^2-M_\pi ^2-M_\pi ^2\ln \frac{M_K^2}{M_\pi ^2}\right) \right] . \end{aligned}$$

We use as input \(\epsilon _{SU(2)}=\sqrt{3}/4/R\) with the FLAG result for R of Eq. (36), \(F_0=f_0/\sqrt{2}=80(20)\) MeV, \(M_\pi =135\) MeV and \(M_K=495\) MeV (we decided to choose a conservative uncertainty on \(f_0\) in order to reflect the magnitude of potential higher-order corrections). The results are reported in Table 15, where in the last column the first error is statistical and the second error is due to the isospin correction (the remaining errors are quoted in the same order as in the original data).

Table 15 Values of the SU(2) isospin-breaking 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 \(f_{K^\pm }/f_{\pi ^\pm }\)

For \(N_f=2\) a dedicated study of the strong-isospin 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 strong-isospin 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 sea-quark contributions to be responsible for such a large effect. Whether higher-order effects in chiral perturbation theory or other sources are responsible still needs to be understood. More lattice QCD simulations of SU(2) isospin-breaking 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.

Using the results of Table 15 for \(N_f = 2+1\) we obtain

$$\begin{aligned}&\hbox {direct},\,N_{ f}=2+1+1:f_{K^\pm } / f_{\pi ^\pm } = 1.193(3)\nonumber \\&\quad \,\mathrm {Refs.}~[14,26,27], \end{aligned}$$
$$\begin{aligned}&\hbox {direct},\,N_{ f}=2+1: f_{K^\pm } / f_{\pi ^\pm } = 1.192(5)\nonumber \\&\quad \,\mathrm {Refs.}~[28{-}31], \end{aligned}$$
$$\begin{aligned}&\hbox {direct},\,N_{ f}=2: f_{K^\pm } / f_{\pi ^\pm } = 1.205(6)(17)\nonumber \\&\quad \,\mathrm {Ref.}~[32], \end{aligned}$$

for QCD with broken isospin.

It is instructive to convert the above results for \(f_+(0)\) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) into a corresponding range for the CKM matrix elements \(|V_{ud}|\) and \(|V_{us}|\), using the relations (54). Consider first the results for \(N_{ f}=2+1+1\). The range for \(f_+(0)\) in Eq. (58) is mapped into the interval \(|V_{us}|=0.2231(9)\), depicted as a horizontal red band in Fig. 8, while the one for \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) in Eq. (64) is converted into \(|V_{us}|/|V_{ud}|= 0.2313(7)\), shown as a tilted red band. The red ellipse is the intersection of these two bands and represents the 68% likelihood contour,Footnote 19 obtained by treating the above two results as independent measurements. Repeating the exercise for \(N_{ f}=2+1\) and \(N_{ f}=2\) leads to the green and blue ellipses, respectively. The plot indicates a slight tension between the \(N_f=2+1+1\) and the nuclear \(\beta \) decay results.

Fig. 8
figure 8

The plot compares the information for \(|V_{ud}|\), \(|V_{us}|\) obtained on the lattice with the experimental result extracted from nuclear \(\beta \) transitions. The dotted line indicates the correlation between \(|V_{ud}|\) and \(|V_{us}|\) that follows if the CKM-matrix is unitary

Tests of the Standard Model

In the Standard Model, the CKM matrix is unitary. In particular, the elements of the first row obey

$$\begin{aligned} |V_u|^2\equiv |V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 1.\end{aligned}$$

The tiny contribution from \(|V_{ub}|\) is known much better than needed in the present context: \(|V_{ub}|= 4.13 (49) \times 10^{-3}\) [151]. In the following, we first discuss the evidence for the validity of the relation (67) and only then use it to analyse the lattice data within the Standard Model.

In Fig. 8, the correlation between \(|V_{ud}|\) and \(|V_{us}|\) imposed by the unitarity of the CKM matrix is indicated by a dotted line (more precisely, in view of the uncertainty in \(|V_{ub}|\), the correlation corresponds to a band of finite width, but the effect is too small to be seen here). The plot shows that there is a slight tension with unitarity in the data for \(N_f = 2 + 1 + 1\): Numerically, the outcome for the sum of the squares of the first row of the CKM matrix reads \(|V_u|^2 = 0.980(9)\), which deviates from unity at the level of two standard deviations. Still, it is fair to say that at this level the Standard Model passes a nontrivial test that exclusively involves lattice data and well-established kaon decay branching ratios. Combining the lattice results for \(f_+(0)\) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) in Eqs. (58) and (64) with the \(\beta \) decay value of \(|V_{ud}|\) quoted in Eq. (55), the test sharpens considerably: the lattice result for \(f_+(0)\) leads to \(|V_u|^2 = 0.9988(6)\), which highlights again a \(2\sigma \)-tension with unitarity, while the one for \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) implies \(|V_u|^2 = 0.9998(5)\), confirming the first-row CKM unitarity below the permille level.

Fig. 9
figure 9

Results for \(|V_{us}|\) and \(|V_{ud}|\) that follow from the lattice data for \(f_+(0)\) (triangles) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) (squares), on the basis of the assumption that the CKM matrix is unitary. The black squares and the grey bands represent our estimates, obtained by combining these two different ways of measuring \(|V_{us}|\) and \(|V_{ud}|\) on a lattice. For comparison, the figure also indicates the results obtained if the data on nuclear \(\beta \) decay and \(\tau \) decay are analysed within 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.

Repeating the analysis for \(N_f = 2\), we find \(|V_u|^2 = 1.029(34)\) with the lattice data alone. This number is fully compatible with unity and perfectly consistent with the value of \(|V_{ud}|\) found in nuclear \(\beta \) decay: combining this value with the result (60) for \(f_+(0)\) yields \(|V_u|^2=1.0003(10)\), combining it with the data (66) on \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) gives \(|V_u|^2= 0.9988(15)\).

Table 16 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

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.

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.

Figure 9 shows that the results obtained for \(|V_{us}|\) and \(|V_{ud}|\) from the data on \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) (squares) are quite consistent with the determinations via \(f_+(0)\) (triangles). In order to calculate the corresponding average values, we restrict ourselves to those determinations that we have considered best in Sect. 4.3. The corresponding results for \(|V_{us}|\) are listed in Table 16 (the error in the experimental numbers used to convert the values of \(f_+(0)\) and \( {f_{K^\pm }}/{f_{\pi ^\pm }}\) into values for \(|V_{us}|\) is included in the statistical error).

Table 17 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
Table 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 isospin-breaking correction has been applied to the quoted result (see Sect. 4.3). The numerical values are listed in MeV units

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.

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 light-quark 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 lattice-QCD 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 scale-setting 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 axial-vector 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 \(f_{K^\pm }\).

Thus, our estimates read

$$\begin{aligned}&N_f = 2 + 1: f_{\pi ^\pm }= 130.2 ~ (1.4)~ \hbox {MeV}\nonumber \\&\quad \,\mathrm {Refs.}~[28,29,31],\end{aligned}$$
$$\begin{aligned}&N_f = 2 + 1 + 1: f_{K^\pm } = 155.6 ~ (0.4)~ \hbox {MeV}\nonumber \\&\quad \,\mathrm {Refs.}~[14,26,27] ,\nonumber \\&N_f = 2 + 1: f_{K^\pm } = 155.9 ~ (0.9)~ \hbox {MeV}\nonumber \\&\quad \,\mathrm {Refs.}~[28,29,31],\\&N_f = 2: f_{K^\pm } = 157.5 ~ (2.4)~ \hbox {MeV}\nonumber \\&\quad \,\mathrm {Ref.}~[32].\nonumber \end{aligned}$$

The lattice results of Table 18 and our estimates (68)–(69) are reported in Fig. 10. The latter ones agree within the errors with the latest experimental determinations of \(f_\pi \) and \(f_K\) from the PDG [151]:

$$\begin{aligned}&f_{\pi ^\pm }^{(PDG)} = 130.41 ~ (0.20)~\hbox {MeV},\nonumber \\&\quad f_{K^\pm }^{(PDG)} = 156.2 ~ (0.7)~\hbox {MeV}. \end{aligned}$$

Moreover, the values of \(f_{\pi ^\pm }\) and \(f_{K^\pm }\) quoted by the PDG are obtained assuming Eq. (55) for the value of \(|V_{ud}|\) and adopting the average of FNAL/MILC 12I and RBC-UKQCD 10 results for \(f_+(0)\).

Fig. 10
figure 10

Values of \(f_\pi \) and \(f_K\). The black squares and grey bands indicate our estimates (68) and (69). The black triangles represent the experimental values quoted by the PDG; see Eq. (70)

Low-energy constants

In the study of the quark-mass 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 quark-mass 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. process-independent) low-energy 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.

Chiral perturbation theory

\(\chi \)PT is an effective field theory approach to the low-energy 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 quark-mass 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 one expands around the SU(2) chiral limit, there appear two LECs at order \(p^2\) in the chiral effective Lagrangian,

$$\begin{aligned}&F\equiv F_\pi \,\Big |_{\;m_u,m_d\rightarrow 0} \quad \hbox {and} \quad B\equiv \frac{\Sigma }{F^2},\nonumber \\&\quad \hbox {where}\ \Sigma \equiv -\langle \overline{u}u\rangle \,\Big |_{\;m_u,m_d\rightarrow 0}, \end{aligned}$$

and seven at order \(p^4\), indicated by \(\bar{\ell }_i\) with \(i=1,\ldots ,7\). In the analysis of the SU(3) chiral limit there are also just two LECs at order \(p^2\),

$$\begin{aligned}&F_0\equiv F_\pi \,\Big |_{\;m_u,m_d,m_s\rightarrow 0} \quad \hbox {and} \quad B_0\equiv \frac{\Sigma _0}{F_0^2},\nonumber \\&\quad \hbox {where}\ \Sigma _0\equiv -\langle \overline{u}u\rangle \,\Big |_{\;m_u,m_d,m_s\rightarrow 0}, \end{aligned}$$

but ten at order \(p^4\), indicated by the capital letter \(L_i(\mu )\) with \(i=1,\ldots ,10\). These constants are independent of the quark masses,Footnote 20 but they become scale dependent after renormalization (sometimes a superscript r is added). The SU(2) constants \(\bar{\ell }_i\) are scale independent, since they are defined at scale \(\mu =M_\pi \) (as indicated by the bar). For the precise definition of these constants and their scale dependence we refer the reader to Refs. [129, 131].

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 so-called p-regime of \(\chi \)PT. It coincides with the setting that is used for standard phenomenologically oriented lattice-QCD computations, and we shall consider the p-regime the default in the following. However, if the pion mass is so small that the box-length 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 finite-volume versions of \(\chi \)PT with correspondingly adjusted power-counting schemes, referred to as \(\epsilon \)- and \(\delta \)-regime, are described in Sects. 5.1.4 and 5.1.5, respectively.

Lattice calculations can be used to test if chiral symmetry is indeed spontaneously broken along the path \(SU(N_{ f})_L \times SU(N_{ f})_R \rightarrow SU(N_{ f})_{L+R}\) by measuring nonzero chiral condensates and by verifying the validity of the GMOR relation \(M_\pi ^2\propto m_q\) close to the chiral limit. If the chiral extrapolation of quantities calculated on the lattice is made with the help of fits to their \(\chi \)PT forms, apart from determining the observable at the physical value of the quark masses, one also obtains the relevant LECs. This is