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FLAG Review 2019

Flavour Lattice Averaging Group (FLAG)

A preprint version of the article is available at arXiv.

Abstract

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

Introduction

Flavour physics provides an important opportunity for exploring the limits of the Standard Model of particle physics and for constraining possible extensions that go beyond it. As the LHC explores a new energy frontier and as experiments continue to extend the precision frontier, the importance of flavour physics will grow, both in terms of searches for signatures of new physics through precision measurements and in terms of attempts to construct the theoretical framework behind direct discoveries of new particles. Crucial to such searches for new physics is the ability to quantify strong-interaction effects. 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 that are important for flavour physics. Set up in November 2007, it comprises experts in Lattice Field Theory, Chiral Perturbation Theory and Standard Model phenomenology. Our aim is to provide an answer to the frequently posed question “What is currently the best lattice value for a particular quantity?” in a way that is readily accessible to those who are not expert in lattice methods. This is generally not an easy question to answer; different collaborations use different lattice actions (discretizations of QCD) with a variety of lattice spacings and volumes, and with a range of masses for the u- and d-quarks. Not only are the systematic errors different, but also the methodology used to estimate these uncertainties varies between collaborations. In the present work, we summarize the main features of each of the calculations and provide a framework for judging and combining the different results. Sometimes it is a single result that provides the “best” value; more often it is a combination of results from different collaborations. Indeed, the consistency of values obtained using different formulations adds significantly to our confidence in the results.

The first three editions of the FLAG review were made public in 2010 [1], 2013 [2], and 2016 [3] (and will be referred to as FLAG 10, FLAG 13 and FLAG 16, respectively). The third edition reviewed results related to both light (u-, d- and s-), and heavy (c- and b-) flavours. The quantities related to pion and kaon physics were 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 \(f_\pi \), the \(B_K\) parameter from neutral kaon mixing, and the kaon mixing matrix elements of new operators that arise in theories of physics beyond the Standard Model. Their implications for the CKM matrix elements \(V_{us}\) and \(V_{ud}\) were also discussed. Furthermore, results were reported for some of the 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 masses of the charm and bottom quarks together with the decay constants, form factors, and mixing parameters of B- and D-mesons. These are the 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.

Table 1 Summary of the main results of this review concerning quark masses, light-meson decay constants, LECs, and kaon mixing parameters. These are grouped in terms of \(N_{ f}\), the number of dynamical quark flavours in lattice simulations. Quark masses and the quark condensate are given in the \({\overline{\mathrm {MS}}}\) scheme at running scale \(\mu =2\,\mathrm {GeV}\) or as indicated. BSM bag parameters \(B_{2,3,4,5}\) are given in the \({\overline{\mathrm {MS}}}\) scheme at scale \(\mu =3\,\mathrm {GeV}\). Further specifications of the quantities are given in the quoted sections. Results for \(N_f=2\) quark masses are unchanged since FLAG 16 [3]. For each result we list the references that enter the FLAG average or estimate, and we stress again the importance of quoting these original works when referring to FLAG results. 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 consulting the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors

In the present paper we provide updated results for all the above-mentioned quantities, but also extend the scope of the review by adding a section on nucleon matrix elements. This presents results for matrix elements of flavor nonsinglet and singlet bilinear operators, including the nucleon axial charge \(g_A\) and the nucleon sigma terms. These results are relevant for constraining \(V_{ud}\), for searches for new physics in neutron decays and other processes, and for dark matter searches. In addition, the section on up and down quark masses has been largely rewritten, replacing previous estimates for \(m_u\), \(m_d\), and the mass ratios R and Q that were largely phenomenological with those from lattice QED+QCD calculations. We have also updated the discussion of the phenomenology of isospin-breaking effects in the light meson sector, and their relation to quark masses, with a lattice-centric discussion. A short review of QED in lattice-QCD simulations is also provided, including a discussion of ambiguities arising when attempting to define “physical” quantities in pure QCD.

Our main results are collected in Tables 1, 2 and 3. As is clear from the tables, for most quantities there are results from ensembles with different values for \(N_f\). In most cases, there is reasonable agreement among results with \(N_f=2\), \(2\,+\,1\), and \(2\,+\,1\,+\,1\). As precision increases, we may some day be able to distinguish among the different values of \(N_f\), in which case, presumably \(2\,+\,1\,+\,1\) would be the most realistic. (If isospin violation is critical, then \(1\,+\,1+1\) or \(1\,+\,1+1\,+\,1\) might be desired.) At present, for some quantities the errors in the \(N_f=2\,+\,1\) results are smaller than those with \(N_f=2\,+\,1\,+\,1\) (e.g., for \(m_c\)), while for others the relative size of the errors is reversed. Our suggestion to those using the averages is to take whichever of the \(N_f=2\,+\,1\) or \(N_f=2\,+\,1\,+\,1\) results has the smaller error. We do not recommend using the \(N_f=2\) results, except for studies of the \(N_f\)-dependence of condensates and \(\alpha _s\), as these have an uncontrolled systematic error coming from quenching the strange quark.

Our plan is to continue providing FLAG updates, in the form of a peer reviewed paper, roughly on a triennial basis. This effort is supplemented by our more frequently updated website http://flag.unibe.ch [4], 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 September 2018.Footnote 1

Table 2 Summary of the main results of this review concerning heavy–light mesons and the strong coupling constant. These are grouped in terms of \(N_{ f}\), the number of dynamical quark flavours in lattice simulations. The quantities listed are specified in the quoted sections. For each result we list the references that enter the FLAG average or estimate, and we stress again the importance of quoting these original works when referring to FLAG results. 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 consulting the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors
Table 3 Summary of the main results of this review concerning nuclear matrix elements, grouped in terms of \(N_{ f}\), the number of dynamical quark flavours in lattice simulations. The quantities listed are specified in the quoted sections. For each result we list the references that enter the FLAG average or estimate, and we stress again the importance of quoting these original works when referring to FLAG results. 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 consulting 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 set of closely connected physical quantities. Each of these sections is accompanied by an Appendix with explicatory notes.Footnote 2 Finally, in Appendix A we provide a glossary in which we introduce some standard lattice terminology (e.g., concerning the gauge, light-quark and heavy-quark actions), and in addition we summarize and describe the most commonly used lattice techniques and methodologies (e.g., related to renormalization, chiral extrapolations, scale setting).

FLAG composition, guidelines and rules

FLAG strives to be representative of the lattice community, both in terms of the geographical location of its members and the lattice collaborations to which they belong. We aspire to provide the nuclear- and particle-physics communities with a single source of reliable information on lattice results.

In order to work reliably and efficiently, we have adopted a formal structure and a set of rules by which all FLAG members abide. The collaboration presently consists of an Advisory Board (AB), an Editorial Board (EB), and eight Working Groups (WG). The rôle of the Advisory Board is to provide oversight of the content, procedures, schedule and membership of FLAG, to help resolve disputes, to serve as a source of advice to the EB and to FLAG as a whole, and to provide a critical assessment of drafts. They also give their approval of the final version of the preprint before it is rendered public. The Editorial Board coordinates the activities of FLAG, sets priorities and intermediate deadlines, organizes votes on FLAG procedures, writes the introductory sections, and takes care of the editorial work needed to amalgamate the sections written by the individual working groups into a uniform and coherent review. The working groups concentrate on writing the review of the physical quantities for which they are responsible, which is subsequently circulated to the whole collaboration for critical evaluation.

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

  • Advisory Board (AB):    S. Aoki, M. Golterman, R. Van De Water, and A. Vladikas

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

  • Working Groups (coordinator listed first):

    • Quark masses: T. Blum, A. Portelli, and A. Ramos;

    • \(V_{us},V_{ud}\): S. Simula, T. Kaneko, and J. N. Simone;

    • LEC: S. Dürr, H. Fukaya, and U.M. Heller;

    • \(B_K\): P. Dimopoulos, G. Herdoiza, and R. Mawhinney;

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

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

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

    • NME: R. Gupta, S. Collins, A. Nicholson, and H. Wittig;

The most important FLAG guidelines and rules are the following:

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

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

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

  • in all working groups the three members must belong to three different lattice collaborations;Footnote 3\(^{,}\)Footnote 4

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

  • lattice collaborations will be consulted on the colour coding of their calculation;

  • there are also internal rules regulating our work, such as voting procedures.

For this edition of the FLAG review, we sought the advice of external reviewers once a complete draft of the review was available. For each review section, we have asked one lattice expert (who could be a FLAG alumnus/alumna) and one nonlattice phenomenologist for a critical assessment. This is similar to the procedure followed by the Particle Data Group in the creation of the Review of Particle Physics. The reviewers provide comments and feedback on scientific and stylistic matters. They are not anonymous, and enter into a discussion with the authors of the WG. Our aim with this additional step is to make sure that a wider array of viewpoints enter into the discussions, so as to make this review more useful for its intended audience.

Citation policy

We draw attention to this particularly important point. As stated above, our aim is to make lattice-QCD results easily accessible to those without lattice expertise, and we are well aware that it is likely that some readers will only consult the present paper and not the original lattice literature. It is very important that this paper not be the only one cited when our results are quoted. We strongly suggest that readers also cite the original sources. In order to facilitate this, in Tables 1, 2, and 3, besides summarizing the main results of the present review, we also cite the original references from which they have been obtained. In addition, for each figure we make a bibtex file available on our webpage [4] 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 section, we focus on a few important points. Though the discussion has been duly updated, it is similar to that of Sect. 1.2 in the previous two reviews [2, 3], with the addition of comments on the contributions from excited states that are particularly relevant for the new section on NMEs.

The present review aims to achieve two distinct goals: first, to provide a description of the relevant work done on the lattice; and, second, to draw conclusions on the basis of that work, summarizing the results obtained for the various quantities of physical interest.

The core of the information about the work done on the lattice is presented in the form of tables, which not only list the various results, but also describe the quality of the data that underlie them. We consider it important that this part of the review represents a generally accepted description of the work done. For this reason, we explicitly specify the quality requirements used and provide sufficient details in appendices so that the reader can verify the information given in the tables.Footnote 5

On the other hand, the conclusions drawn on the basis of the available lattice results are the responsibility of FLAG alone. Preferring to err on the side of caution, in several cases we draw conclusions that are more conservative than those resulting from a plain weighted average of the available lattice results. This cautious approach is usually adopted when the average is dominated by a single lattice result, or when only one lattice result is available for a given quantity. In such cases, one does not have the same degree of confidence in results and errors as when there is agreement among several different calculations using different approaches. The reader should keep in mind that the degree of confidence cannot be quantified, and it is not reflected in the quoted errors.

Each discretization has its merits, but also its shortcomings. For most topics covered in this review we have an increasingly broad database, and for most quantities lattice calculations based on totally different discretizations are now available. This is illustrated by the dense population of the tables and figures in most parts of this review. Those calculations that do satisfy our quality criteria indeed lead, in almost all cases, to consistent results, confirming universality within the accuracy reached. In our opinion, the consistency between independent lattice results, obtained with different discretizations, methods, and simulation parameters, is an important test of lattice QCD, and observing such consistency also provides further evidence that systematic errors are fully under control.

In the sections dealing with heavy quarks and with \(\alpha _s\), the situation is not the same. Since the b-quark mass can barely be resolved with current lattice spacings, most 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 exists 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 all sea-quark contributions are omitted, has uncontrolled systematic errors and is no longer used in modern lattice simulations with relevance to phenomenology. Accordingly, we will review results obtained with \(N_f=2\), \(N_f=2\,+\,1\), and \(N_f = 2\,+\,1\,+\,1\), but omit earlier results with \(N_f=0\). The only exception concerns the QCD coupling constant \(\alpha _s\). Since this observable does not require valence light quarks, it is theoretically well defined also in the \(N_f=0\) theory, which is simply pure gluodynamics. The \(N_f\)-dependence of \(\alpha _s\), or more precisely of the related quantity \(r_0 \Lambda _{\overline{\mathrm {MS}}}\), is a theoretical issue of considerable interest; here \(r_0\) is a quantity with the dimension of length that sets the physical scale, as discussed in Appendix A.2. We stress, however, that only results with \(N_f \ge 3\) are used to determine the physical value of \(\alpha _s\) at a high scale.

Lattice actions, simulation parameters, and scale setting

The remarkable progress in the precision of lattice calculations is due to improved algorithms, better computing resources, and, last but not least, conceptual developments. Examples of the latter are improved actions that reduce lattice artifacts and actions that preserve chiral symmetry to very good approximation. A concise characterization of the various discretizations that underlie the results reported in the present review is given in Appendix A.1.

Physical quantities are computed in lattice simulations in units of the lattice spacing so that they are dimensionless. For example, the pion decay constant that is obtained from a simulation is \(f_\pi a\), where a is the spacing between two neighboring lattice sites. (All simulations with results quoted in this review use hypercubic lattices, i.e., with the same spacing in all four Euclidean directions.) To convert these results to physical units requires knowledge of the lattice spacing a at the fixed values of the bare QCD parameters (quark masses and gauge coupling) used in the simulation. This is achieved by requiring agreement between the lattice calculation and experimental measurement of a known quantity, which thus “sets the scale” of a given simulation. A few details on this procedure are provided in Appendix A.2.

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 \({\overline{\mathrm {MS}}}\)-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 \({\overline{\mathrm {MS}}}\), 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).

Extrapolations

Because of limited computing resources, lattice simulations are often performed at unphysically heavy pion masses, although results at the physical point have become increasingly common. Further, numerical simulations must be done at nonzero lattice spacing, and in a finite (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 fixed. 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 [93, 94]; 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.

Excited-state contamination

In all the hadronic matrix elements discussed in this review, the hadron in question is the lightest state with the chosen quantum numbers. This implies that it dominates the required correlation functions as their extent in Euclidean time is increased. Excited-state contributions are suppressed by \(e^{-\Delta E \Delta \tau }\), where \(\Delta E\) is the gap between the ground and excited states, and \(\Delta \tau \) the relevant separation in Euclidean time. The size of \(\Delta E\) depends on the hadron in question, and in general is a multiple of the pion mass. In practice, as discussed at length in Sect. 10, the contamination of signals due to excited-state contributions is a much more challenging problem for baryons than for the other particles discussed here. This is in part due to the fact that the signal-to-noise ratio drops exponentially for baryons, which reduces the values of \(\Delta \tau \) that can be used.

Critical slowing down

The lattice spacings reached in recent simulations go down to 0.05 fm or even smaller. In this regime, long autocorrelation times slow down the sampling of the configurations [95,96,97,98,99,100,101,102,103,104]. 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 [105], instead of the more common antiperiodic ones. More recently two other approaches have been proposed, one based on a multiscale thermalization algorithm [106, 107] and another based on defining QCD on a nonorientable manifold [108]. The problem is also touched upon in Sect. 9.2.1, where it is stressed that attention must be paid to this issue. While large scale simulations with open boundary conditions are already far advanced [109], only one result reviewed here has been obtained with any of the above methods (results for \(\alpha _s\) from Ref. [79] which use open boundary conditions). It is usually assumed that the continuum limit can be reached by extrapolation from the existing simulations, and that potential systematic errors due to the long autocorrelation times have been adequately controlled. Partially or completely frozen topology would produce a mixture of different \(\theta \) vacua, and the difference from the desired \(\theta =0\) result may be estimated in some cases using chiral perturbation theory, which gives predictions for the \(\theta \)-dependence of the physical quantity of interest [110, 111]. These ideas have been systematically and successfully tested in various models in [112, 113], and a numerical test on MILC ensembles indicates that the topology dependence for some of the physical quantities reviewed here is small, consistent with theoretical expectations [114].

Simulation algorithms and numerical errors

Most of the modern lattice-QCD simulations use exact algorithms such as those of Refs. [115, 116], 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 [117].

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 about 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 data sets allow for a satisfactory control of the systematic uncertainties;

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

  • the parameter values and ranges used to generate the data sets 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.

Note that in the first two editions [1, 2], FLAG used the three symbols in order to rate the reliability of the systematic errors attributed to a given result by the paper’s authors. Starting with the previous edition [3] the meaning of the symbols has changed slightly – they now 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.

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 2-loop or 1-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 FLAG reassesses criteria with each edition and as a result some of the quality criteria (the one on chiral extrapolation for instance) have been tightened up over time [1,2,3].

In the following, we present the rating criteria used in the current report. While these criteria apply to most quantities without modification there are cases where they need to be amended or additional criteria need to be defined. For instance, when discussing results obtained in the \(\epsilon \)-regime of chiral perturbation theory in Sect. 5 the finite volume criterion listed below for the p-regime is no longer appropriate.Footnote 6 Similarly, the discussion of the strong coupling constant in Sect. 9 requires tailored criteria for renormalization, perturbative behaviour, and continuum extrapolation. In such cases, the modified criteria are discussed in the respective sections. Apart from only a few exceptions the following colour code applies in the tables:

  • Chiral extrapolation:

    • \(M_{\pi ,\mathrm {min}}< 200\) MeV, with three or more pion masses used in the extrapolation

      or two values of \(M_\pi \) with one lying within 10 MeV of 135MeV (the physical neutral pion mass) and the other one below 200 MeV

    • 200 MeV \(\le M_{\pi ,{\mathrm {min}}} \le \) 400 MeV, with three or more pion masses used in the extrapolation

      or two values of \(M_\pi \) with \(M_{\pi ,{\mathrm {min}}}<\) 200 MeV

      or a single value of \(M_\pi \), lying within 10 MeV of 135 MeV (the physical neutral pion mass)

    • otherwise

    This criterion has changed with respect to the previous edition [3].

  • Continuum extrapolation:

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

    • otherwise

    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 is unchanged from Ref. [3].

  • Finite-volume effects:

    The finite-volume colour code used for a result is chosen to be the worse of the QCD and the QED codes, as described below. If only QCD is used the QED colour code is ignored.

    – For QCD:

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

    • otherwise

    where we have introduced \([L(M_{\pi ,\mathrm {min}})]_{\mathrm {max}}\), which is the maximum box size used in the simulations performed at the 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). It is assumed here that calculations are in the p-regime of chiral perturbation theory, and that all volumes used exceed 2 fm. This condition has been improved between the second [2] and the third [3] edition of the FLAG review but remains unchanged since. 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, 1-loop chiral perturbation theory [118] 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 \(L~M_{\pi ,\mathrm {min}} = 4\).

    – For QED (where applicable):

    • \(1/([M_{\pi ,\mathrm {min}}L(M_{\pi ,\mathrm {min}})]_{\mathrm {max}})^{n_{\mathrm {min}}}<0.02\), or at least four volumes

    • \(1/([M_{\pi ,\mathrm {min}}L(M_{\pi ,\mathrm {min}})]_{\mathrm {max}})^{n_{\mathrm {min}}}<0.04\), or at least three volumes

    • otherwise

    Because of the infrared-singular structure of QED, electromagnetic finite-volume effects decay only like a power of the inverse spatial extent. In several cases like mass splittings [119, 120] or leptonic decays [121], the leading corrections are known to be universal, i.e., independent of the structure of the involved hadrons. In such cases, the leading universal effects can be directly subtracted exactly from the lattice data. We denote \(n_{\mathrm {min}}\) the smallest power of \(\frac{1}{L}\) at which such a subtraction cannot be done. In the widely used finite-volume formulation \(\mathrm {QED}_L\), one always has \(n_{\mathrm {min}}\le 3\) due to the nonlocality of the theory [122]. While the QCD criteria have not changed with respect to Ref. [3] the QED criteria are new. They are used here only in Sect. 3.

  • Isospin breaking effects (where applicable):

    • all leading isospin breaking effects are included in the lattice calculation

    • isospin breaking effects are included using the electro-quenched approximation

    • otherwise

    This criterion is used for quantities which are breaking isospin symmetry or which can be determined at the sub-percent accuracy where isospin breaking effects, if not included, are expected to be the dominant source of uncertainty. In the current edition, this criterion is only used for the up and down quark masses, and related quantities (\(\epsilon \), \(Q^2\) and \(R^2\)). The criteria for isospin breaking effects feature for the first time in the FLAG review.

  • Renormalization (where applicable):

    • nonperturbative

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

    • otherwise

    In Ref. [1], we assigned a red square to all results which were renormalized at 1-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 1-loop, is often estimated conservatively and reliably. We did not change these criteria since.

  • 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 about 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. [123]). In early works, typically referring to \(N_f=2\) simulations (e.g., Refs. [123] and [48]), chiral extrapolations are based on chiral perturbation theory formulae which do not take these regularization effects into account. After the importance of accounting for isospin breaking when doing chiral fits was shown in Ref. [124], later works, typically referring to \(N_f=2\,+\,1\,+\,1\) simulations, have taken these effects into account [9]. We use \(M_{\pi ^\pm }\) for \(M_{\pi ,\mathrm {min}}\) in the chiral-extrapolation rating criterion. On the other hand, 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 7

In the case of staggered fermions, discretization effects give rise to several light states with the quantum numbers of the pion.Footnote 8 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 artifacts 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 9

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. 9.2.1.

In the new section on nuclear matrix elements, Sect. 10, an additional criterion has been introduced. This concerns the level of control over contamination from excited states, which is a more challenging issue for nucleons than for mesons. In addition, the chiral-extrapolation criterion in this section is somewhat stricter than that given above.

Heavy-quark actions

For the b quark, the discretization of the heavy-quark action follows a very different approach from that used for light flavours. There are several different methods for treating heavy quarks on the lattice, each with its own issues and considerations. Most 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 earlier reviews, the calculations involving charm quarks often treated it using one of the approaches adopted for the b quark. Since the last report [3], however, we found more recent calculations to simulate the charm quark using light-quark actions. 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 focused 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.

NRQCD

   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.

HQET

  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 include 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, domain wall fermions 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 might 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 gray 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 10

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 analyzed 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 errors 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 11 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 September 2018 is relevant. If the paper appeared in print after that date, this is accounted for in the bibliography, but does not affect the averages.Footnote 12

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 \( r_0 \Lambda _{\overline{\mathrm {MS}}}\) 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 \(N_{ f}\) are not provided.

To date, no significant differences between results with different values of \(N_f\) have been observed in the quantities listed in Tables 1, 2, and 3. 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 Refs. [130, 131]. 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.

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. Here we provide details on how averages are obtained.

Averaging: generic case

We follow the procedure of the previous two editions [2, 3], which we describe here in full detail.

One of the problems arising when forming averages is that not all of the data sets 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

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

where \(x_i\) is the value obtained by the \(i{\mathrm{th}}\) 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 \(\sigma ^{(\alpha )}_i\) (\(\alpha \ge 2\)) 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:

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

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

$$\begin{aligned} \omega _i = \dfrac{\sigma _i^{-2}}{\sum _{i=1}^M \sigma _i^{-2}} , \end{aligned}$$
(3)

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

$$\begin{aligned} x_{\mathrm{av}}= & {} \sum _{i=1}^M x_i\, \omega _i . \end{aligned}$$
(4)

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 (\(\chi _\mathrm{min}^2/\mathrm{dof} > 1\)), the statistical and systematic error bars are stretched by a factor \(S = \sqrt{\chi ^2/\mathrm{dof}}\).

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 [132]. This consists in defining

$$\begin{aligned} \sigma _{i;j} = \sqrt{{\sum _{\alpha }}^\prime \Big [ \sigma _i^{(\alpha )} \Big ]^2 } , \end{aligned}$$
(5)

with \(\sum _{\alpha }^\prime \) running only over those errors of \(x_i\) that are correlated with the corresponding errors of the 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

$$\begin{aligned} C_{ii}= & {} \sigma _i^2 \quad (i = 1, \ldots , M) , \nonumber \\ C_{ij}= & {} \sigma _{i;j} \, \sigma _{j;i} \quad (i \ne j) . \end{aligned}$$
(6)

Finally, the error of the average is estimated by

$$\begin{aligned} \sigma ^2_{\mathrm{av}} = \sum _{i=1}^M \sum _{j=1}^M \omega _i \,\omega _j \,C_{ij}\,\,, \end{aligned}$$
(7)

and the FLAG average is

$$\begin{aligned} Q_{\mathrm{av}} = x_{\mathrm{av}} \, \pm \, \sigma _{\mathrm{av}} . \end{aligned}$$
(8)

Nested averaging

We have encountered one case where the correlations between results are more involved, and a nested averaging scheme is required. This concerns the B-meson bag parameters discussed in Sect. 8.2. In the following, we describe the details of the nested averaging scheme. This is an updated version of the section added in the web update of the FLAG 16 report.

The issue arises for a quantity Q that is given by a ratio, \(Q=Y/Z\). In most simulations, both Y and Z are calculated, and the error in Q can be obtained in each simulation in the standard way. However, in other simulations only Y is calculated, with Z taken from a global average of some type. The issue to be addressed is that this average value \({\overline{Z}}\) has errors that are correlated with those in Q.

In the example that arises in Sect. 8.2, \(Q=B_B\), \(Y=B_B f_B^2\) and \(Z=f_B^2\). In one of the simulations that contribute to the average, Z is replaced by \({\overline{Z}}\), the PDG average for \(f_B^2\) [133] (obtained with an averaging procedure similar to that used by FLAG). This simulation is labeled with \(i=1\), so that

$$\begin{aligned} Q_1 = \frac{Y_1}{{\overline{Z}}}. \end{aligned}$$
(9)

The other simulations have results labeled \(Q_j\), with \(j\ge 2\). In this set up, the issue is that \({\overline{Z}}\) is correlated with the \(Q_j\), \(j\ge 2\).Footnote 13

We begin by decomposing the error in \(Q_1\) in the same schematic form as above,

$$\begin{aligned} Q_1 = x_1 \pm \frac{\sigma _{Y_1}^{(1)}}{{\overline{Z}}} \pm \frac{\sigma _{Y_1}^{(2)}}{{\overline{Z}}} \pm \cdots \pm \frac{\sigma _{Y_1}^{(E)}}{{\overline{Z}}} \pm \frac{Y_1 \sigma _{{\overline{Z}}}}{{\overline{Z}}^2}. \end{aligned}$$
(10)

Here the last term represents the error propagating from that in \({\overline{Z}}\), while the others arise from errors in \(Y_1\). For the remaining \(Q_j\) (\(j\ge 2\)) the decomposition is as in Eq. (1). The total error of \(Q_1\) then reads

$$\begin{aligned} \sigma _1^2= & {} \left( \frac{\sigma _{Y_1}^{(1)}}{{\overline{Z}}}\right) ^2 + \left( \frac{\sigma _{Y_1}^{(2)}}{{\overline{Z}}}\right) ^2 +\cdots + \left( \frac{\sigma _{Y_1}^{(E)}}{{\overline{Z}}}\right) ^2\nonumber \\&+ \left( \frac{Y_1}{{\overline{Z}}^2}\right) ^2 \sigma _{{\overline{Z}}}^2, \end{aligned}$$
(11)

while that for the \(Q_j\) (\(j\ge 2\)) is

$$\begin{aligned} \sigma _j^2 = \left( \sigma _j^{(1)}\right) ^2 + \left( \sigma _j^{(2)}\right) ^2 +\cdots + \left( \sigma _j^{(E)}\right) ^2. \end{aligned}$$
(12)

Correlations between \(Q_j\) and \(Q_k\) (\(j,k\ge 2\)) are taken care of by Schmelling’s prescription, as explained above. What is new here is how the correlations between \(Q_1\) and \(Q_j\) (\(j\ge 2\)) are taken into account.

To proceed, we recall from Eq. (7) that \(\sigma _{{\overline{Z}}}\) is given by

$$\begin{aligned} \sigma _{{\overline{Z}}}^2 = \sum _{{i'},{j'}=1}^{M'} \omega [Z]_{i'} \omega [Z]_{j'} C[Z]_{i'j'}. \end{aligned}$$
(13)

Here the indices \(i'\) and \(j'\) run over the \(M'\) simulations that contribute to \({\overline{Z}}\), which, in general, are different from those contributing to the results for Q. The weights \(\omega [Z]\) and correlation matrix C[Z] are given an explicit argument Z to emphasize that they refer to the calculation of this quantity and not to that of Q. C[Z] is calculated using the Schmelling prescription [Eqs. (5)–(7)] in terms of the errors, \(\sigma [Z]_{i'}^{(\alpha )}\), taking into account the correlations between the different calculations of Z.

We now generalize Schmelling’s prescription for \(\sigma _{i;j}\), Eq. (5), to that for \(\sigma _{1;k}\) (\(k\ge 2\)), i.e., the part of the error in \(Q_1\) that is correlated with \(Q_k\). We take

$$\begin{aligned} \sigma _{1;k} = \sqrt{ \frac{1}{{\overline{Z}}^2} \sum ^\prime _{(\alpha )\leftrightarrow k} \Big [\sigma _{Y_1}^{(\alpha )} \Big ]^2 + \frac{Y_1^2}{{\overline{Z}}^4} \sum _{i',j'}^{M'} \omega [Z]_{i'} \omega [Z]_{j'} C[Z]_{i'j'\leftrightarrow k} }. \nonumber \\ \end{aligned}$$
(14)

The first term under the square root sums those sources of error in \(Y_1\) that are correlated with \(Q_k\). Here we are using a more explicit notation from that in Eq. (5), with \((\alpha ) \leftrightarrow k\) indicating that the sum is restricted to the values of \(\alpha \) for which the error \(\sigma _{Y_1}^{(\alpha )}\) is correlated with \(Q_k\). The second term accounts for the correlations within \({\overline{Z}}\) with \(Q_k\), and is the nested part of the present scheme. The new matrix \(C[Z]_{i'j'\leftrightarrow k}\) is a restriction of the full correlation matrix C[Z], and is defined as follows. Its diagonal elements are given by

$$\begin{aligned} C[Z]_{i'i'\leftrightarrow k}= & {} (\sigma [Z]_{i'\leftrightarrow k})^2 \quad (i' = 1, \ldots , M') , \end{aligned}$$
(15)
$$\begin{aligned} (\sigma [Z]_{i'\leftrightarrow k})^2= & {} \sum ^\prime _{(\alpha )\leftrightarrow k} (\sigma [Z]_{i'}^{(\alpha )})^2, \end{aligned}$$
(16)

where the summation \(\sum ^\prime _{(\alpha )\leftrightarrow k}\) over \((\alpha )\) is restricted to those \(\sigma [Z]_{i'}^{(\alpha )}\) that are correlated with \(Q_k\). The off-diagonal elements are

$$\begin{aligned} C[Z]_{i'j'\leftrightarrow k}= & {} \sigma [Z]_{i';j'\leftrightarrow k} \, \sigma [Z]_{j';i'\leftrightarrow k} \quad (i' \ne j') , \end{aligned}$$
(17)
$$\begin{aligned} \sigma [Z]_{i';j'\leftrightarrow k}= & {} \sqrt{ \sum ^\prime _{(\alpha )\leftrightarrow j'k} \left( \sigma [Z]_{i'}^{(\alpha )}\right) ^2}, \end{aligned}$$
(18)

where the summation \(\sum ^\prime _{(\alpha )\leftrightarrow j'k}\) over \((\alpha )\) is restricted to \(\sigma [Z]_{i'}^{(\alpha )}\) that are correlated with both \(Z_{j'}\) and \(Q_k\).

The last quantity that we need to define is \(\sigma _{k;1}\).

$$\begin{aligned} \sigma _{k;1} = \sqrt{\sum ^\prime _{(\alpha )\leftrightarrow 1} \Big [ \sigma _k^{(\alpha )} \Big ]^2 } , \end{aligned}$$
(19)

where the summation \(\sum ^\prime _{(\alpha )\leftrightarrow 1}\) is restricted to those \(\sigma _k^{(\alpha )}\) that are correlated with one of the terms in Eq. (11).

In summary, we construct the correlation matrix \(C_{ij}\) using Eq. (6), as in the generic case, except the expressions for \(\sigma _{1;k}\) and \(\sigma _{k;1}\) are now given by Eqs. (14) and (19), respectively. All other \(\sigma _{i;j}\) are given by the original Schmelling prescription, Eq. (5). In this way we extend the philosophy of Schmelling’s approach while accounting for the more involved correlations.

Quark masses

Authors: T. Blum, A. Portelli, A. Ramos

Quark masses are fundamental parameters of the Standard Model. An accurate determination of these parameters is important for both phenomenological and theoretical applications. The bottom- and charm-quark masses, for instance, are important sources of parametric uncertainties in several Higgs decay modes. 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 about 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 [134], 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 below 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. From a theoretical point of view, in QCD with \(N_f\) flavours, a precise definition of quark masses requires one to choose a particular renormalization scheme. This renormalization procedure introduces a renormalization scale \(\mu \), and quark masses depend on this renormalization scale according to the Renormalization Group (RG) equations. In mass-independent renormalization schemes the RG equations reads

$$\begin{aligned} \mu \frac{\mathrm{d} {{\bar{m}}}_i(\mu )}{\mathrm{d}{\mu }} = {{\bar{m}}}_i(\mu ) \tau ({{\bar{g}}})\,, \end{aligned}$$
(20)

where the function \(\tau ({{\bar{g}}})\) is the anomalous dimension, which depends only on the value of the strong coupling \(\alpha _s=\bar{g}^2/(4\pi )\). Note that in QCD \(\tau ({{\bar{g}}})\) is the same for all quark flavours. The anomalous dimension is scheme dependent, but its perturbative expansion

(21)

has a leading coefficient \(d_0 = 8/(4\pi )^2\), which is scheme independent.Footnote 14 Equation (20), being a first order differential equation, can be solved exactly by using Eq. (21) as boundary condition. The formal solution of the RG equation reads

$$\begin{aligned} M_i= & {} {{\bar{m}}}_i(\mu )[2b_0{{\bar{g}}}^2(\mu )]^{-d_0/(2b_0)} \nonumber \\&\times \exp \left\{ - \int _0^{{{\bar{g}}}(\mu )}\mathrm{d} x\, \left[ \frac{\tau (x)}{\beta (x)} - \frac{d_0}{b_0x} \right] \right\} \,, \end{aligned}$$
(22)

where \(b_0 = (11-2N_f/3) / (4\pi )^2\) is the universal leading perturbative coefficient in the expansion of the \(\beta \)-function \(\beta ({{\bar{g}}})\). The renormalization group invariant (RGI) quark masses \(M_i\) are formally integration constants of the RG Eq. (20). They are scale independent, and due to the universality of the coefficient \(d_0\), they are also scheme independent. Moreover, they are nonperturbatively defined by Eq. (22). They only depend on the number of flavours \(N_f\), making them a natural candidate to quote quark masses and compare determinations from different lattice collaborations. Nevertheless, it is customary in the phenomenology community to use the \(\overline{\mathrm{MS}}\) scheme at a scale \(\mu = 2\) GeV to compare different results for light-quark masses, and use a scale equal to its own mass for the charm and bottom quarks. In this review, we will quote the final averages of both quantities.

Results for quark masses are always quoted in the four-flavour theory. \(N_{\mathrm{f}}=2\,+\,1\) results have to be converted to the four flavour theory. Fortunately, the charm quark is heavy \((\Lambda _\mathrm{QCD}/m_c)^2<1\), and this conversion can be performed in perturbation theory with negligible (\(\sim 0.2\%\)) perturbative uncertainties. Nonperturbative corrections in this matching are more difficult to estimate. Since these effects are suppressed by a factor of \(1/N_{\mathrm{c}}\), and a factor of the strong coupling at the scale of the charm mass, naive power counting arguments would suggest that the effects are \(\sim 1\%\). In practice, numerical nonperturbative studies [130, 131] have found this power counting argument to be an overestimate by one order of magnitude in the determination of simple hadronic quantities or the \(\Lambda \)-parameter. Moreover, lattice determinations do not show any significant deviation between the \(N_{\mathrm{f}}=2\,+\,1\) and \(N_\mathrm{f}=2\,+\,1\,+\,1\) simulations. For example, the difference in the final averages for the mass of the strange quark \(m_s\) between \(N_f=2\,+\,1\) and \(N_f=2\,+\,1\,+\,1\) determinations is about a 0.8%, and negligible from a statistical point of view.

We quote all final averages at 2 GeV in the \(\overline{\mathrm{MS}}\) scheme and also the RGI values (in the four flavour theory). We use the exact RG Eq. (22). Note that to use this equation we need the value of the strong coupling in the \(\overline{\mathrm{MS}}\) scheme at a scale \(\mu = 2\) GeV. All our results are obtained from the RG equation in the \(\overline{\mathrm{MS}}\) scheme and the 5-loop beta function together with the value of the \(\Lambda \)-parameter in the four-flavour theory \(\Lambda ^{(4)}_{\overline{\mathrm{MS}}} = 294(12)\, \mathrm{MeV}\) obtained in this review (see Sect. 9). In the uncertainties of the RGI massses we separate the contributions from the determination of the quark masses and the propagation of the uncertainty of \(\Lambda ^{(4)}_{\overline{\mathrm{MS}}}\). These are identified with the subscripts m and \(\Lambda \), respectively.

Conceptually, all lattice determinations of quark masses contain three basic ingredients:

  1. 1.

    Tuning the lattice bare-quark masses to match the experimental values of some quantities. Pseudo-scalar meson masses provide the most common choice, since they have a strong dependence on the values of quark masses. In pure QCD with \(N_f\) quark flavours these values are not known, since the electromagnetic interactions affect the experimental values of meson masses. Therefore, pure QCD determinations use model/lattice information to determine the location of the physical point. This is discussed at length in Sect. 3.1.1.

  2. 2.

    Renormalization of the bare-quark masses. Bare-quark masses determined with the above-mentioned criteria have to be renormalized. Many of the latest determinations use some nonperturbatively defined scheme. One can also use perturbation theory to connect directly the values of the bare-quark masses to the values in the \(\overline{\mathrm{MS}}\) scheme at 2 GeV. Experience shows that 1-loop calculations are unreliable for the renormalization of quark masses: usually at least two loops are required to have trustworthy results.

  3. 3.

    If quark masses have been nonperturbatively renormalized, for example, to some MOM/SF scheme, the values in this scheme must be converted to the phenomenologically useful values in the \(\overline{\mathrm{MS}}\) scheme (or to the scheme/scale independent RGI masses). Either option requires the use of perturbation theory. The larger the energy scale of this matching with perturbation theory, the better, and many recent computations in MOM schemes do a nonperturbative running up to \(3{-}4\) GeV. Computations in the SF scheme allow us to perform this running nonperturbatively over large energy scales and match with perturbation theory directly at the electro-weak scale \(\sim 100\) GeV.

Note that quark masses are different from other quantities determined on the lattice since perturbation theory is unavoidable when matching to schemes in the continuum.

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 is very challenging. Only one determination of the b-quark mass uses this approach, reaching the physical b-quark mass region at two lattice spacings with \(am\sim 0.9\) and 0.64, respectively (see Sect. 3.3). 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 Sect. 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 eta. Indeed, the Gell-Mann-Oakes-Renner relation [136] 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.

The physical point and isospin symmetry

As mentioned in Sect. 2.1, the present review relies on the hypothesis that, at low energies, the Lagrangian \(\mathcal{L}_{ \text{ QCD }}+{{{\mathcal {L}}}}_{ \text{ QED }}\) 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 discussed. How the matching is done is generally ambiguous because it relies on the unphysical separation of QCD and QED contributions. In this section, and in the following, we discuss this issue in detail. Of course, once QED is included in lattice calculations, the subtraction of electromagnetic contributions is no longer necessary.

Let us start from the unambiguous case of QCD+QED. As explained in the introduction of this section, the physical quark masses are the parameters of the Lagrangian such that a given set of experimentally measured, dimensionful hadronic quantities are reproduced by the theory. Many choices are possible for these quantities, but in practice many lattice groups use pseudoscalar meson masses, as they are easily and precisely obtained both by experiment, and through lattice simulations. For example, in the four-flavour case, one can solve the system

$$\begin{aligned} M_{\pi ^+}(m_u,m_d,m_s,m_c,\alpha )= & {} M_{\pi ^+}^{\mathrm {exp.}}\,, \end{aligned}$$
(23)
$$\begin{aligned} M_{K^+}(m_u,m_d,m_s,m_c,\alpha )= & {} M_{K^+}^{\mathrm {exp.}}\,, \end{aligned}$$
(24)
$$\begin{aligned} M_{K^0}(m_u,m_d,m_s,m_c,\alpha )= & {} M_{K^0}^{\mathrm {exp.}}\,, \end{aligned}$$
(25)
$$\begin{aligned} M_{D^0}(m_u,m_d,m_s,m_c,\alpha )= & {} M_{D^0}^{\mathrm {exp.}}\,, \end{aligned}$$
(26)

where we assumed that

  • all the equations are in the continuum and infinite-volume limits;

  • the overall scale has been set to its physical value, generally through some lattice-scale setting procedure involving a fifth dimensionful input;

  • the quark masses \(m_q\) are assumed to be renormalized from the bare, lattice ones in some given continuum renormalization scheme;

  • \(\alpha =\frac{e^2}{4\pi }\) is the fine-structure constant expressed as function of the positron charge e, generally set to the Thomson limit \(\alpha =0.007297352\dots \) [137];

  • the mass \(M_{h}(m_u,m_d,m_s,m_c,\alpha )\) of the meson h is a function of the quark masses and \(\alpha \). The functional dependence is generally obtained by choosing an appropriate parameterization and performing a global fit to the lattice data;

  • the superscript exp. indicates that the mass is an experimental input, lattice groups use in general the values in the Particle Data Group review [137].

However, ambiguities arise with simulations of QCD only. In that case, there is no experimentally measurable quantity that emerges from the strong interaction only. The missing QED contribution is tightly related to isospin-symmetry breaking effects. Isospin symmetry is explicitly broken by the differences between the up- and down-quark masses \(\delta m=m_u-m_d\), and electric charges \(\delta Q=Q_u-Q_d\). Both these effects are, respectively, of order \({\mathcal {O}}(\delta m/\Lambda _{\mathrm {QCD}})\) and \({\mathcal {O}}(\alpha )\), and are expected to be \({\mathcal {O}}(1\%)\) of a typical isospin-symmetric hadronic quantity. Strong and electromagnetic isospin-breaking effects are of the same order and therefore cannot, in principle, be evaluated separately without introducing strong ambiguities. Because these effects are small, they can be treated as a perturbation:

$$\begin{aligned}&X(m_u,m_d,m_s,m_c,\alpha )\nonumber \\&\quad ={\bar{X}}(m_{ud}, m_s, m_c) +\delta mA_X(m_{ud}, m_s, m_c) \nonumber \\&\qquad +\,\alpha B_X(m_{ud}, m_s, m_c)\,, \end{aligned}$$
(27)

for a given hadronic quantity X, where \(m_{ud}=\frac{1}{2}(m_u+m_d)\) is the average light-quark mass. There are several things to notice here. Firstly, the neglected higher-order \({\mathcal {O}}(\delta m^2,\alpha \delta m,\alpha ^2)\) corrections are expected to be \({\mathcal {O}}(10^{-4})\) relatively to X, which at the moment is way beyond the relative statistical accuracy that can be delivered by a lattice calculation. Secondly, this is not strictly speaking an expansion around the isospin-symmetric point, the electromagnetic interaction has also symmetric contributions. From this last expression the previous statements about ambiguities become clearer. Indeed, the only unambiguous prediction one can perform is to solve Eqs. (23)–(26) and use the resulting parameters to obtain a prediction for X, which is represented by the left-hand side of Eq. (27). This prediction will be the sum of the QCD isospin-symmetric part \({\bar{X}}\), the strong isospin-breaking effects \( X^{SU(2)}=\delta mA_X\), and the electromagnetic effects \(X^{\gamma }=\alpha B_X\). Obtaining any of these terms individually requires extra, unphysical conditions to perform the separation. To be consistent with previous editions of FLAG, we also define \({\hat{X}}={\bar{X}}+X^{SU(2)}\) to be the \(\alpha \rightarrow 0\) limit of X.

With pure QCD simulations, one typically solves Eqs. (23)–(26) by equating the QCD, isospin-symmetric part of a hadron mass \({\bar{M}}_h\), result of the simulations, with its experimental value \(M_h^{\mathrm {exp.}}\). This will result in an \({\mathcal {O}}(\delta m,\alpha )\) mis-tuning of the theory parameters which will propagate as an error on predicted quantities. Because of this, in principle, one cannot predict hadronic quantities with a relative accuracy higher than \({\mathcal {O}}(1\%)\) from pure QCD simulations, independently on how the target X is sensitive to isospin breaking effects. If one performs a complete lattice prediction of the physical value of X, it can be of phenomenological interest to define in some way \({\bar{X}}\), \(X^{SU(2)}\), and \(X^{\gamma }\). If we keep \(m_{ud}\), \(m_s\) and \(m_c\) at their physical values in physical units, for a given renormalization scheme and scale, then these three quantities can be extracted by setting successively and simultaneously \(\alpha \) and \(\delta m\) to 0. This is where the ambiguity lies: in general the \(\delta m=0\) point will depend on the renormalization scheme used for the quark masses. In the next section, we give more details on that particular aspect and discuss the order of scheme ambiguities.

Ambiguities in the separation of isospin-breaking contributions

In this section, we discuss the ambiguities that arise in the individual determination of the QED contribution \(X^{\gamma }\) and the strong-isospin correction \(X^{SU(2)}\) defined in the previous section. Throughout this section, we assume that the isospin-symmetric quark masses \(m_{ud}\), \(m_s\) and \(m_c\) are always kept fixed in physical units to the values they take at the QCD+QED physical point in some given renormalization scheme. Let us assume that both up and down masses have been renormalized in an identical mass-independent scheme which depends on some energy scale \(\mu \). We also assume that the renormalization procedure respects chiral symmetry so that quark masses renormalize multiplicatively. The renormalization constants of the quark masses are identical for \(\alpha =0\) and therefore the renormalized mass of a quark has the general form

$$\begin{aligned} m_q(\mu )= & {} Z_m(\mu )\left[ 1+\alpha Q_{\mathrm {tot.}}^2\delta _{Z}^{(0)}(\mu ) +\alpha Q_{\mathrm {tot.}}Q_q\delta _{Z}^{(1)}(\mu )\right. \nonumber \\&\left. +\,\alpha Q_q^2\delta _{Z}^{(2)}(\mu ) \right] m_{q,0} \,, \end{aligned}$$
(28)

up to \({\mathcal {O}}(\alpha ^2)\) corrections, where \(m_{q,0}\) is the bare quark mass, and where \(Q_{\mathrm {tot.}}\) and \(Q_{\mathrm {tot.}}^2\) are the sum of all quark charges and squared charges, respectively. Throughout this section, a subscript ud generally denotes the average between up and down quantities and \(\delta \) the difference between the up and the down quantities. The source of the ambiguities described in the previous section is the mixing of the isospin-symmetric mass \(m_{ud}\) and the difference \(\delta m\) through renormalization. Using Eq. (28) one can make this mixing explicit at leading order in \(\alpha \):

$$\begin{aligned} \begin{pmatrix}m_{ud}(\mu )\\ \delta m(\mu )\end{pmatrix}= & {} Z_m(\mu )\Bigg [1+\alpha Q_{\mathrm {tot.}}^2\delta _{Z}^{(0)}(\mu )+\alpha M^{(1)}(\mu )\nonumber \\&+\,\alpha M^{(2)}(\mu )\Bigg ] \begin{pmatrix}m_{ud,0}\\ \delta m_0\end{pmatrix} \end{aligned}$$
(29)

with the mixing matrices

$$\begin{aligned} M^{(1)}(\mu )= & {} \delta _Z^{(1)}(\mu )Q_{\mathrm {tot.}}\begin{pmatrix} Q_{ud} &{}\quad \frac{1}{4}\delta Q\\ \delta Q &{}\quad Q_{ud} \end{pmatrix}\qquad \text {and}\qquad \nonumber \\ M^{(2)}(\mu )= & {} \delta _Z^{(2)}(\mu )\begin{pmatrix} Q_{ud}^2 &{}\quad \frac{1}{4}\delta Q^2\\ \delta Q^2 &{}\quad Q_{ud}^2 \end{pmatrix}\,. \end{aligned}$$
(30)

Now let us assume that for the purpose of determining the different components in Eq. (27), one starts by tuning the bare masses to obtain equal up and down masses, for some small coupling \(\alpha _0\) at some scale \(\mu _0\), i.e., \(\delta m(\mu _0)=0\). At this specific point, one can extract the pure QCD, and the QED corrections to a given quantity X by studying the slope of \(\alpha \) in Eq. (27). From these quantities the strong isospin contribution can then readily be extracted using a nonzero value of \(\delta m(\mu _0)\). However, if now the procedure is repeated at another coupling \(\alpha \) and scale \(\mu \) with the same bare masses, it appears from Eq. (29) that \(\delta m(\mu )\ne 0\). More explicitly,

$$\begin{aligned} \delta m(\mu )=m_{ud}(\mu _0)\frac{Z_m(\mu )}{Z_m(\mu _0)} [\alpha \Delta _Z(\mu ) -\alpha _0\Delta _Z(\mu _0)]\,, \end{aligned}$$
(31)

with

$$\begin{aligned} \Delta _Z(\mu )=Q_{\mathrm {tot.}}\delta Q\delta _Z^{(1)}(\mu )+\delta Q^2\delta _Z^{(2)}(\mu )\,,\end{aligned}$$
(32)

up to higher-order corrections in \(\alpha \) and \(\alpha _0\). In other words, the definitions of \({\bar{X}}\), \(X^{SU(2)}\), and \(X^{\gamma }\) depend on the renormalization scale at which the separation was made. This dependence, of course, has to cancel in the physical sum X. One can notice that at no point did we mention the renormalization of \(\alpha \) itself, which, in principle, introduces similar ambiguities. However, the corrections coming from the running of \(\alpha \) are \({\mathcal {O}}(\alpha ^2)\) relatively to X, which, as justified above, can be safely neglected. Finally, important information is provided by Eq. (31): the scale ambiguities are \({\mathcal {O}}(\alpha m_{ud})\). For physical quark masses, one generally has \(m_{ud}\simeq \delta m\). So by using this approximation in the first-order expansion Eq. (27), it is actually possible to define unambiguously the components of X up to second-order isospin-breaking corrections. Therefore, in the rest of this review, we will not keep track of the ambiguities in determining pure QCD or QED quantities. However, in the context of lattice simulations, it is crucial to notice that \(m_{ud}\simeq \delta m\) is only accurate at the physical point. In simulations at larger-than-physical pion masses, scheme ambiguities in the separation of QCD and QED contributions are generally large. Once more, the argument made here assumes that the isospin-symmetric quark masses \(m_{ud}\), \(m_s\), and \(m_c\) are kept fixed to their physical value in a given scheme while varying \(\alpha \). Outside of this assumption there is an additional isospin-symmetric \(O(\alpha m_q)\) ambiguity between \({\bar{X}}\) and \(X^{\gamma }\).

Such separation on lattice-QCD+QED simulation results appeared for the first time in RBC 07 [138] and Blum 10 [139], where the scheme was implicitly defined around the \(\chi \)PT expansion. In that setup, the \(\delta m(\mu _0)=0\) point is defined in pure QCD, i.e., \(\alpha _0=0\) in the previous discussion. The QCD part of the kaon-mass splitting from the first FLAG review [1] is used as an input in RM123 11 [140], which focuses on QCD isospin corrections only. It therefore inherits from the convention that was chosen there, which is also to set \(\delta m(\mu _0)=0\) at zero QED coupling. The same convention was used in the follow-up works RM123 13 [141] and RM123 17 [19]. The BMW collaboration was the first to introduce a purely hadronic scheme in its electro-quenched study of the baryon octet mass splittings [142]. In this work, the quark mass difference \(\delta m(\mu )\) is swapped with the mass splitting \(\Delta M^2\) between the connected \({\bar{u}}u\) and \({\bar{d}}d\) pseudoscalar masses. Although unphysical, this quantity is proportional [143] to \(\delta m(\mu )\) up to \({\mathcal {O}}(\alpha m_{ud})\) chiral corrections. In this scheme, the quark masses are assumed to be equal at \(\Delta M^2=0\), and the \({\mathcal {O}}(\alpha m_{ud})\) corrections to this statement are analogous to the scale ambiguities mentioned previously. The same scheme was used with the same data set for the determination of light-quark masses BMW 16 [20]. The BMW collaboration used a different hadronic scheme for its determination of the nucleon-mass splitting BMW 14 [119] using full QCD+QED simulations. In this work, the \(\delta m=0\) point was fixed by imposing the baryon splitting \(M_{\Sigma ^+}-M_{\Sigma ^-}\) to cancel. This scheme is quite different from the other ones presented here, in the sense that its intrinsic ambiguity is not \({\mathcal {O}}(\alpha m_{ud})\). What motivates this choice here is that \(M_{\Sigma ^+}-M_{\Sigma ^-}=0\) in the limit where these baryons are point particles, so the scheme ambiguity is suppressed by the compositeness of the \(\Sigma \) baryons. This may sounds like a more difficult ambiguity to quantify, but this scheme has the advantage of being defined purely by measurable quantities. Moreover, it has been demonstrated numerically in BMW 14 [119] that, within the uncertainties of this study, the \(M_{\Sigma ^+}-M_{\Sigma ^-}=0\) scheme is equivalent to the \(\Delta M^2=0\) one, explicitly \(M_{\Sigma ^+}-M_{\Sigma ^-}=-0.18(12)(6)\,\mathrm {MeV}\) at \(\Delta M^2=0\). The calculation QCDSF/UKQCD 15 [144] uses a “Dashen scheme,” where quark masses are tuned such that flavour-diagonal mesons have equal masses in QCD and QCD+QED. Although not explicitly mentioned by the authors of the paper, this scheme is simply a reformulation of the \(\Delta M^2=0\) scheme mentioned previously. Finally, the recent preprint MILC 18 [145] also used the \(\Delta M^2=0\) scheme and noticed its connection to the “Dashen scheme” from QCDSF/UKQCD 15.

In the previous edition of this review, the contributions \({\bar{X}}\), \(X^{SU(2)}\), and \(X^{\gamma }\) were given for pion and kaon masses based on phenomenological information. Considerable progress has been achieved by the lattice community to include isospin-breaking effects in calculations, and it is now possible to determine these quantities precisely directly from a lattice calculation. However, these quantities generally appear as intermediate products of a lattice analysis, and are rarely directly communicated in publications. These quantities, although unphysical, have a phenomenological interest, and we encourage the authors of future calculations to quote them explicitly.

Inclusion of electromagnetic effects in lattice-QCD simulations

Electromagnetism on a lattice can be formulated using a naive discretization of the Maxwell action \(S[A_{\mu }]=\frac{1}{4}\int d^4 x\,\sum _{\mu ,\nu }[\partial _{\mu }A_{\nu }(x)-\partial _{\nu }A_{\mu }(x)]^2\). Even in its noncompact form, the action remains gauge-invariant. This is not the case for non-Abelian theories for which one uses the traditional compact Wilson gauge action (or an improved version of it). Compact actions for QED feature spurious photon-photon interactions which vanish only in the continuum limit. This is one of the main reason why the noncompact action is the most popular so far. It was used in all the calculations presented in this review. Gauge-fixing is necessary for noncompact actions. It was shown [146, 147] that gauge fixing is not necessary with compact actions, including in the construction of interpolating operators for charged states.

Although discretization is straightforward, simulating QED in a finite volume is more challenging. Indeed, the long range nature of the interaction suggests that important finite-size effects have to be expected. In the case of periodic boundary conditions, the situation is even more critical: a naive implementation of the theory features an isolated zero-mode singularity in the photon propagator. It was first proposed in [148] to fix the global zero-mode of the photon field \(A_{\mu }(x)\) in order to remove it from the dynamics. This modified theory is generally named \(\mathrm {QED}_{\mathrm {TL}}\). Although this procedure regularizes the theory and has the right classical infinite-volume limit, it is nonlocal because of the zero-mode fixing. As first discussed in [119], the nonlocality in time of \(\mathrm {QED}_{\mathrm {TL}}\) prevents the existence of a transfer matrix, and therefore a quantum-mechanical interpretation of the theory. Another prescription named \(\mathrm {QED}_{\mathrm {L}}\), proposed in [149], is to remove the zero-mode of \(A_{\mu }(x)\) independently for each time slice. This theory, although still nonlocal in space, is local in time and has a well-defined transfer matrix. Wether these nonlocalities constitute an issue to extract infinite-volume physics from lattice-QCD+\(\mathrm {QED}_{\mathrm {L}}\) simulations is, at the time of this review, still an open question. However, it is known through analytical calculations of electromagnetic finite-size effects at \(O(\alpha )\) in hadron masses [119, 120, 122, 141, 149,150,151], meson leptonic decays [151], and the hadronic vacuum polarization [152] that \(\mathrm {QED}_{\mathrm {L}}\) does not suffer from a problematic (e.g., UV divergent) coupling of short and long-distance physics due to its nonlocality. Another strategy, first prosposed in [153] and used by the QCDSF collaboration, is to bound the zero-mode fluctuations to a finite range. Although more minimal, it is still a nonlocal modification of the theory and so far finite-size effects for this scheme have not been investigated. More recently, two proposals for local formulations of finite-volume QED emerged. The first one described in [154] proposes to use massive photons to regulate zero-mode singularities, at the price of (softly) breaking gauge invariance. The second one presented in [147] avoids the zero-mode issue by using anti-periodic boundary conditions for \(A_{\mu }(x)\). In this approach, gauge invariance requires the fermion field to undergo a charge conjugation transformation over a period, breaking electric charge conservation. These local approaches have the potential to constitute cleaner approaches to finite-volume QED. All the calculations presented in this review used \(\mathrm {QED}_{\mathrm {L}}\) or \(\mathrm {QED}_{\mathrm {TL}}\), with the exception of QCDSF.

Once a finite-volume theory for QED is specified, there are various ways to compute QED effects themselves on a given hadronic quantity. The most direct approach, first used in [148], is to include QED directly in the lattice simulations and assemble correlation functions from charged quark propagators. Another approach proposed in [141], is to exploit the perturbative nature of QED, and compute the leading-order corrections directly in pure QCD as matrix elements of the electromagnetic current. Both approaches have their advantages and disadvantages and as shown in [19], are not mutually exclusive. A critical comparative study can be found in [155].

Finally, most of the calculations presented here made the choice of computing electromagnetic corrections in the electro-quenched approximation. In this limit, one assumes that only valence quarks are charged, which is equivalent to neglecting QED corrections to the fermionic determinant. This approximation reduces dramatically the cost of lattice-QCD + QED calculations since it allows the reuse of previously generated QCD configurations. It also avoids computing disconnected contributions coming from the electromagnetic current in the vacuum, which are generally challenging to determine precisely. The electromagnetic contributions from sea quarks are known to be flavour-SU(3) and large-\(N_c\) suppressed, thus electro-quenched simulations are expected to have an \(O(10\%)\) accuracy for the leading electromagnetic effects. This suppression is in principle rather weak and results obtained from electro-quenched simulations might feature uncontrolled systematic errors. For this reason, the use of the electro-quenched approximation constitutes the difference between  and  in the FLAG criterion for the inclusion of isospin breaking effects.

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 determinations of \(m_u\) and \(m_d\).

Quark masses have been calculated on the lattice since the mid-nineties. 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 4 and 5 list the results of \(N_{ f}=2\,+\,1\) and \(N_{ f}=2\,+\,1\,+\,1\) lattice calculations of \(m_s\) and \(m_{ud}\). These results are given in the \({\overline{\mathrm {MS}}}\) scheme at \(2\,\mathrm {GeV}\), which is standard nowadays, though some groups are starting to quote results at higher scales (e.g., Ref. [156]). 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.

\(N_{ f}=2\,+\,1\) 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. Isospin-breaking and electromagnetic 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. We have already commented that the effect of the omission of the charm quark in the sea is expected to be small, below our current precision. In contrast with previous editions of the FLAG report, we do not add any additional uncertainty due to these effects in the final averages.

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

The only new calculation since FLAG 16 is the \(m_s\) determination of Maezawa 16 [157]. This new result agrees well with other determinations; however because it is computed with a single pion mass of about 160 MeV, it does not meet our criteria for entering the average. RBC/UKQCD 14 [10] significantly improves on their RBC/UKQCD 12 [156] 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 (\(m_{\mathrm{sea}}=m_{\mathrm{val}}\)) \(M_\pi \)’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 [166, 167]. They use rooted staggered fermions. By 2009 their simulations covered an impressive range of parameter space, with lattice spacings going down to 0.045 fm, and valence-pion masses down to approximately 180 MeV [17]. The most recent MILC \(N_{ f}=2\,+\,1\) results, i.e., MILC 10A [14] and MILC 09A [17], feature large statistics and 2-loop renormalization. Since these data sets 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 [158] calculation represents an important extension of the collaboration’s earlier 2010 computation [159], 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 [162] 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 [159], renormalization of quark masses is implemented nonperturbatively, through the Schrödinger functional method [172]. 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 [11, 12] 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 [13]. It updates the staggered-fermions calculation of HPQCD 09A [24]. 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. [171] and in Sect. 3.2, HPQCD determines \(m_c\) by fitting Euclidean-time moments of the \({{\bar{c}}}c\) pseudoscalar density two-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 [13] use the MILC 09 determination of the quark-mass ratio \(m_s/m_{ud}\) [129].

HPQCD 09A [24] obtains \(m_c/m_s=11.85(16)\) [24] 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 15 is even more precise, achieving an accuracy better than 0.5%.

This discussion leaves us with five results for our final average for \(m_s\): MILC 09A [17], BMW 10A, 10B [11, 12], HPQCD 10 [13] 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 \(m_s=92.03(88)\,\mathrm {MeV}\) with a \(\chi ^2/\)dof = 1.2.

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 \(m_{ud}=3.364(41)\) at 2 GeV in the \(\overline{\mathrm{MS}}\) scheme (\(\chi ^2/\)d.of.=1.1). Our final estimates for the light-quark masses are

(33)

And the RGI values

(34)

\(N_{ f}=2\,+\,1\,+\,1\) lattice calculations

Since the previous FLAG review, two new results for the strange-quark mass have appeared, HPQCD 18 [15] and FNAL/MILC/TUMQCD 18 [8]. In the former quark masses are renormalized nonperturbatively in the RI-SMOM scheme. The mass of the (fictitious) \({{\bar{s}}} s\) meson is used to tune the bare strange mass. The “physical” \({{\bar{s}}} s\) mass is given in QCD from the pion and kaon masses. In addition, they use the same HISQ ensembles and valence quarks as those in HPQCD 14A, where the quark masses were computed from time moments of vector-vector correlation functions. The new results are consistent with the old, with roughly the same size error, but of course with different systematics. In particular the new results avoid the use of high-order perturbation theory in the matching between lattice and continuum schemes. It is reassuring that the two methods, applied to the same ensembles, agree well.

The \(N_{ f}=2\,+\,1\,+\,1\) results are summarized in Table 5. Note that the results of Ref. [16] are reported as \(m_s(2\,\mathrm {GeV};N_f=3)\) and those of Ref. [9] as \(m_{ud(s)}(2\,\mathrm {GeV};N_f=4)\). We convert the former to \(N_f=4\) and obtain \(m_s(2\,\mathrm {GeV}; N_f=4)=93.12(69)\mathrm {MeV}\). The average of FNAL/MILC/TUMQCD 18, HPQCD 18, ETM 14 and HPQCD 14A is 93.44(68)\(\mathrm {MeV}\) with \(\chi ^2/\text{ dof }=1.7\). For the light-quark average we use ETM 14A and FNAL/MILC/TUMQCD 18 with an average 3.410(43) and a \(\chi ^2/\text{ dof }=3\). We note these \(\chi ^2\) values are large. For the case of the light-quark masses this is mostly due to ETM 14(A) masses lying significantly above the rest, but in the case of \(m_s\) there is also some tension between the recent and very precise results of HPQCD 18 and FNAL/MILC/TUMQCD 18. Also note that the 2 + 1-flavour values are consistent with the four-flavour ones, so in all cases we have decided to simply quote averages according to FLAG rules, including stretching factors for the errors based on \(\chi ^2\) values of our fits.

(35)

and the RGI values

(36)

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

Fig. 1
figure1

\({\overline{\mathrm {MS}}}\) mass of the strange quark (at 2 GeV scale) in MeV. The upper two panels show the lattice results listed in Tables 4 and 5, while the bottom panel collects sum rule results [173,174,175,176,177]. Diamonds and squares represent results based on perturbative and nonperturbative renormalization, respectively. The black squares and the grey bands represent our estimates (33) and (35). The significance of the colours is explained in Sect. 2

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

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.

\(N_{ f}=2\,+\,1\) lattice calculations

For \(N_f = 2\,+\,1\) our average has not changed since the last version of the review and is based on the result RBC/UKQCD 14B, which replaces RBC/UKQCD 12 (see Sect. 3.1.4), 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 \(m_s / m_{ud} = 27.42(12)\) with \(\chi ^2/\text{ dof } \simeq 0.2\). 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%.

Fig. 2
figure2

Mean mass of the two lightest quarks, \(m_{ud}=\frac{1}{2}(m_u+m_d)\). The bottom panel shows results based on sum rules [173, 176, 178] (for more details see Fig. 1)

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

At this level of precision, the uncertainties in the electromagnetic and strong isospin-breaking corrections might not be completely negligible. Nevertheless, we decide not to add any uncertainty associated with this effect. The main reason is that most recent determinations try to estimate this uncertainty themselves and found an effect smaller than naive power counting estimates (see \(N_{ f}=2\,+\,1\,+\,1\) section).

$$\begin{aligned} N_f = 2+1: \quad {m_s}/{m_{ud}} = 27.42 ~ (12) \quad \,\mathrm {Refs.}~{[10{-}12,17]}.\nonumber \\ \end{aligned}$$
(37)

\(N_{ f}=2\,+\,1\,+\,1\) lattice calculations

For \(N_f = 2\,+\,1\,+\,1\) there are three results, MILC 17 [5], ETM 14 [9] and FNAL/MILC 14A [18], all of which satisfy our selection criteria.

MILC 17 uses 24 HISQ staggered-fermion ensembles at six values of the lattice spacing in the range \(0.15\, \mathrm{fm}\)\(0.03\, \mathrm{fm}\).

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 four 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 electromagnetic effects with \(\epsilon = 0.84(20)\) [179]. All of our selection criteria are satisfied with the tag . Thus our average is given by \(m_s / m_{ud} = 27.23 ~ (10)\), where the error includes a large stretching factor equal to \(\sqrt{\chi ^2/\text{ dof }} \simeq 1.6\), coming from our rules for the averages discussed in Sect. 2.2. As mentioned already this is mainly due to ETM 14(A) values lying significantly above the averages for the individual masses.

$$\begin{aligned} {N_f = 2\,+\,1\,+\,1:}\quad m_s / m_{ud} = 27.23 ~ (10)\quad \,\mathrm {Refs.}~\text{[5,9,18] },\!\!\nonumber \\ \end{aligned}$$
(38)

which corresponds to an overall uncertainty equal to 0.4%. It is worth noting that [5] estimates the EM effects in this quantity to be \(\sim 0.18\%\).

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 and sum rules.

Fig. 3
figure3

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 [176, 180,181,182,183]

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

In addition to reviewing computations of individual \(m_u\) and \(m_d\) quark masses, we will also determine FLAG averages for the parameter \(\epsilon \) related to the violations of Dashen’s theorem

$$\begin{aligned} \epsilon =\frac{\left( \Delta M_{K}^{2}-\Delta M_{\pi }^{2}\right) ^{\gamma }}{\Delta M_{\pi }^{2}}\,, \end{aligned}$$
(39)

where \(\Delta M_{\pi }^{2}=M_{\pi ^+}^{2}-M_{\pi ^0}^{2}\) and \(\Delta M_{K}^{2}=M_{K^+}^{2}-M_{K^0}^{2}\) are the pion and kaon squared mass splittings, respectively. The superscript \(\gamma \), here and in the following, denotes corrections that arise from electromagnetic effects only. This parameter is often a crucial intermediate quantity in the extraction of the individual light-quark masses. Indeed, it can be shown using the G-parity symmetry of the pion triplet that \(\Delta M_{\pi }^{2}\) does not receive \(O(\delta m)\) isospin-breaking corrections. In other words

$$\begin{aligned} \Delta M_{\pi }^{2}=\left( \Delta M_{\pi }^{2}\right) ^{\gamma } \quad \text {and} \quad \epsilon =\frac{\left( \Delta M_{K}^{2}\right) ^{\gamma }}{\Delta M_{\pi }^{2}}-1\,, \end{aligned}$$
(40)

at leading-order in the isospin-breaking expansion. The difference \((\Delta M_{\pi }^{2})^{SU(2)}\) was estimated in previous editions of FLAG through the \(\epsilon _m\) parameter. However, consistent with our leading-order truncation of the isospin-breaking expansion, it is simpler to ignore this term. Once known, \(\epsilon \) allows one to consistently subtract the electromagnetic part of the kaon splitting to obtain the QCD splitting \((\Delta M_{K}^{2})^{SU(2)}\). In contrast with the pion, the kaon QCD splitting is sensitive to \(\delta m\), and, in particular, proportional to it at leading order in \(\chi \)PT. Therefore, the knowledge of \(\epsilon \) allows for the determination of \(\delta m\) from a chiral fit to lattice-QCD data. Originally introduced in another form in [184], \(\epsilon \) vanishes in the SU(3) chiral limit, a result known as Dashen’s theorem. However, in the 1990’s numerous phenomenological papers pointed out that \(\epsilon \) might be an O(1) number, indicating a significant failure of SU(3) \(\chi \)PT in the description of electromagnetic effects on light meson masses. However, the phenomenological determinations of \(\epsilon \) feature some level of controversy, leading to the rather imprecise estimate \(\epsilon =0.7(5)\) given in the first edition of FLAG. In this edition of the review, we quote below more precise averages for \(\epsilon \), directly obtained from lattice-QCD+QED simulations. We refer the reader to the previous editions of FLAG, and to the review [185] for discusions of the phenomenological determinations of \(\epsilon \).

Regarding finite-volume effects for calculations including QED, this edition of FLAG uses a new quality criterion presented in Sect. 2.1.1. Indeed, due to the long-distance nature of the electromagnetic interaction, these effects are dominated by a power law in the lattice spatial size. The coefficients of this expansion depend on the chosen finite-volume formulation of QED. For \(\mathrm {QED}_{\mathrm {L}}\), these effects on the squared mass \(M^2\) of a charged meson are given by [119, 120, 122]

$$\begin{aligned} \Delta _{\mathrm {FV}}M^2= \alpha M^2\left\{ \frac{c_{1}}{ML}+\frac{2c_1}{(ML)^2}+ {\mathcal {O}}\left[ \frac{1}{(ML)^3}\right] \right\} \,,\end{aligned}$$
(41)

with \(c_1\simeq -2.83730\). It has been shown in [119] that the two first orders in this expansion are exactly known for hadrons, and are equal to the pointlike case. However, the \({\mathcal {O}}[1/(ML)^{3}]\) term and higher orders depend on the structure of the hadron. The universal corrections for \(\mathrm {QED}_{\mathrm {TL}}\) can also be found in [119]. In all this part, for all computations using such universal formulae, the QED finite-volume quality criterion has been applied with \(n_{\mathrm {min}}=3\), otherwise \(n_{\mathrm {min}}=1\) was used.

Since FLAG 16, six new results have been reported for nondegenerate light-quark masses. In the \(N_f=2\,+\,1\,+\,1\) sector, MILC 18 [145] computed \(\epsilon \) using \(N_f=2\,+\,1\) asqtad electro-quenched QCD+\(\mathrm {QED}_{\mathrm {TL}}\) simulations and extracted the ratio \(m_u/m_d\) from a new set of \(N_f=2\,+\,1\,+\,1\) HISQ QCD simulations. Although \(\epsilon \) comes from \(N_f=2\,+\,1\) simulations, \((\Delta M_{K}^{2})^{SU(2)}\), which is about three times larger than \((\Delta M_{K}^{2})^{\gamma }\), has been determined in the \(N_f=2\,+\,1\,+\,1\) theory. We therefore chose to classify this result as a four-flavour one. This result is explicitly described by the authors as an update of MILC 17 [5]. In MILC 17 [5], \(m_u/m_d\) is determined as a side-product of a global analysis of heavy-meson decay constants, using a preliminary version of \(\epsilon \) from MILC 18 [145]. In FNAL/MILC/TUMQCD 18 [8] the ratio \(m_u/m_d\) from MILC 17 [5] is used to determine the individual masses \(m_u\) and \(m_d\) from a new calculation of \(m_{ud}\). The work RM123 17 [19] is the continuation the \(N_f=2\) result named RM123 13 [141] in the previous edition of FLAG. This group now uses \(N_f=2\,+\,1\,+\,1\) ensembles from ETM 10 [186], however still with a rather large minimum pion mass of \(270~\mathrm {MeV}\), leading to the  rating for chiral extrapolations. In the \(N_f=2\,+\,1\) sector, BMW 16 [20] reuses the data set produced from their determination of the light baryon octet mass splittings [142] using electro-quenched QCD+\(\mathrm {QED}_{\mathrm {TL}}\) smeared clover fermion simulations. Finally, MILC 16 [187], which is a preliminary result for the value of \(\epsilon \) published in MILC 18 [145], also provides a \(N_f=2\,+\,1\) computation of the ratio \(m_u/m_d\).

MILC 09A [17] uses the mass difference between \(K^0\) and \(K^+\), from which they subtract electromagnetic effects using Dashen’s theorem with corrections, as discussed in the introduction of this section. 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 [11, 12] 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 \(Q=22.3(8)\), from \(\eta \rightarrow 3\pi \) decays [188] (the decay \(\eta \rightarrow 3\pi \) is very sensitive to QCD isospin breaking but fairly insensitive to QED isospin breaking). Instead of subtracting electromagnetic effects using phenomenology, RBC 07 [138] and Blum 10 [139] 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 [158] 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. However, they do not correct for the large finite-volume effects coming from electromagnetism in their \(M_{\pi }L\sim 2\) simulations, but provide rough estimates for their size, based on Ref. [149]. QCDSF/UKQCD 15 [189] 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. The smallest partially quenched (\(m_{\mathrm{sea}}\ne m_{\mathrm{val}}\)) pion mass is greater than 200 MeV, so our chiral-extrapolation criteria require a rating. Concerning finite-volume effects, this work uses three spatial extents L of \(1.6~\mathrm {fm}\), \(2.2~\mathrm {fm}\), and \(3.3~\mathrm {fm}\). QCDSF/UKQCD 15 claims that the volume dependence is not visible on the two largest volumes, leading them to assume that finite-size effects are under control. As a consequence of that, the final result for quark masses does not feature a finite-volume extrapolation or an estimation of the finite-volume uncertainty. However, in their work on the QED corrections to the hadron spectrum [189] based on the same ensembles, a volume study shows some level of compatibility with the \(\mathrm {QED}_{\mathrm {L}}\) finite-volume effects derived in [120]. We see two issues here. Firstly, the analytical result quoted from [120] predicts large, \(O(10\%)\) finite-size effects from QED on the meson masses at the values of \(M_{\pi }L\) considered in QCDSF/UKQCD 15, which is inconsistent with the statement made in the paper. Secondly, it is not known that the zero-mode regularization scheme used here has the same volume scaling as \(\mathrm {QED}_{\mathrm {L}}\). We therefore chose to assign the  rating for finite volume to QCDSF/UKQCD 15. Finally, for \(N_f=2\,+\,1\,+\,1\), ETM 14 [9] uses simulations in pure QCD, but determines \(m_u-m_d\) from the slope \(\partial M_K^2/\partial m_{ud}\) and the physical value for the QCD kaon-mass splitting taken from the phenomenological estimate in FLAG 13 (Fig. 4).

Fig. 4
figure4

Lattice results and FLAG averages at \(N_f = 2\,+\,1\) and \(2\,+\,1\,+\,1\) for the up–down quark masses ratio \(m_u/m_d\), together with the current PDG estimate

Table 7 Lattice results for \(m_u\), \(m_d\) (MeV) and for the ratio \(m_u/m_d\). The values refer to the \({\overline{\mathrm {MS}}}\) scheme at scale 2 GeV. The top part of the table lists the result obtained with \(N_{ f}=2\,+\,1\,+\,1\), while the lower part presents calculations with \(N_f = 2\,+\,1\)

Lattice results for \(m_u\), \(m_d\) and \(m_u/m_d\) are summarized in Table 7. It is important to notice two major changes in the grading of these results: the introduction of an “isospin breaking” criterion and the modification of the “finite volume” criterion in the presence of QED. The colour coding is specified in detail in Sect. 2.1. Considering the important progress in the last years on including isospin-breaking effects in lattice simulations, we are now in a position where averages for \(m_u\) and \(m_d\) can be made without the need of phenomenological inputs. Therefore, lattice calculations of the individual quark masses using phenomenological inputs for isospin-breaking effects will be coded .

We start by recalling the \(N_f=2\) FLAG estimate for the light-quark masses, entirely coming from RM123 13 [141],

$$\begin{aligned}&\quad m_u =2.40(23) \,\mathrm {MeV}\quad \,\mathrm {Ref.}~\text{[141] },\nonumber \\ N_{ f}= 2:&\quad m_d = 4.80(23) \,\mathrm {MeV}\quad \,\mathrm {Ref.}~\text{[141] },\nonumber \\&\quad {m_u}/{m_d} = 0.50(4) \quad \,\mathrm {Ref.}~\text{[141] }, \end{aligned}$$
(42)

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 \(N_{ f}=2\,+\,1\), the only result, which qualifies for entering the FLAG average for quark masses, is BMW 16 [20],

$$\begin{aligned}&\quad m_u =2.27(9)\,\mathrm {MeV}\quad \,\mathrm {Ref.}~ \text{[20] }\,, \nonumber \\ N_{ f}= 2\,+\,1:&\quad m_d = 4.67(9) \,\mathrm {MeV}\quad \,\mathrm {Ref.}~ \text{[20] }\,,\nonumber \\&\quad {m_u}/{m_d} = 0.485(19)\quad \,\mathrm {Ref.}~ \text{[20] }\,, \end{aligned}$$
(43)

with errors of roughly 4%, 2% and 4%, respectively. This estimate is slightly more precise than in the previous edition of FLAG. More importantly, it now comes entirely from a lattice-QCD+QED calculation, whereas phenomenological input was used in previous editions. These numbers result in the following RGI averages

$$\begin{aligned}&\quad M_u^{\mathrm{RGI}} =3.16(13)_m(4)_\Lambda \,\mathrm {MeV}= 3.16(13)\,\mathrm {MeV}\quad \,\mathrm {Ref.}~ \text{[20] }\,, \nonumber \\ N_{ f}= 2\,+\,1:&\nonumber \\&\quad M_d^{\mathrm{RGI}} = 6.50(13)_m(8)_\Lambda \,\mathrm {MeV}= 6.50(15)\,\mathrm {MeV}\quad \,\mathrm {Ref.}~ \text{[20] }\,. \end{aligned}$$
(44)

Finally, for \(N_{ f}=2\,+\,1\,+\,1\), only RM123 17 [19] enters the average, giving

$$\begin{aligned}&\quad m_u =2.50(17)\,\mathrm {MeV}\quad \,\mathrm {Ref.}~ \text{[19] }\,,\nonumber \\ N_{ f}= 2\,+\,1\,+\,1:&\quad m_d = 4.88(20)\,\mathrm {MeV}\quad \,\mathrm {Ref.}~ \text{[19] }\,,\nonumber \\&\quad {m_u}/{m_d} = 0.513(31)\quad \,\mathrm {Ref.}~ \text{[19] }\,. \end{aligned}$$
(45)

with errors of roughly 7%, 4% and 6%, respectively. In the previous edition of FLAG, ETM 14 [9] was used for the average. The RM123 17 result used here is slightly more precise and is free of phenomenological input. The value of \(m_u/m_d\) in MILC 17 [5] depends critically on the value of \(\epsilon \) given in MILC 18 [145], which was unpublished at the time of the review deadline. As a consequence we did not include the result MILC 17 [5] in the average. The value will appear in the average of the online version of the review. It is, however important to point out that both MILC 17 and MILC 18 results show a marginal discrepancy with RM123 17 [19] of 1.7 standard deviations. The RGI averages are

$$\begin{aligned}&\quad M_u^{\mathrm{RGI}} =3.48(24)_m(4)_\Lambda \,\mathrm {MeV}= 3.48(24) \,\mathrm {MeV}\quad \,\mathrm {Ref.}~ \text{[19] }\,,\nonumber \\ N_{ f}= 2\,+\,1\,+\,1:&\nonumber \\&\quad M_d^{\mathrm{RGI}} = 6.80(28)_m(8)_\Lambda \,\mathrm {MeV}= 6.80(29) \,\mathrm {MeV}\quad \,\mathrm {Ref.}~ \text{[19] }\,. \end{aligned}$$
(46)

Every result for \(m_u\) and \(m_d\) used here to produce the FLAG averages relies on electro-quenched calculations, so there is some interest to comment on the size of quenching effects. Considering phenomenology and the lattice results presented here, it is reasonable for a rough estimate to use the value \((\Delta M_{K}^{2})^{\gamma }\sim 2000~\mathrm {MeV}^2\) for the QED part of the kaon splitting. Using the arguments presented in Sect. 3.1.3, one can assume that the QED sea contribution represents \(O(10\%)\) of \((\Delta M_{K}^{2})^{\gamma }\). Using SU(3) PQ\(\chi \)PT+QED [143, 191] gives a \(\sim 5\%\) effect. Keeping the more conservative \(10\%\) estimate and using the experimental value of the kaon splitting, one finds that the QCD kaon splitting \((\Delta M_{K}^{2})^{SU(2)}\) suffers from a reduced \(3\%\) quenching uncertainty. Considering that this splitting is proportional to \(m_u-m_d\) at leading order in SU(3) \(\chi \)PT, we can estimate that a similar error will propagate to the quark masses. So the individual up and down masses look mildly affected by QED quenching. However, one notices that \(\sim 3\%\) is the level of error in the new FLAG averages, and increasing significantly this accuracy will require using fully unquenched calculations.

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. (43) differs from zero by 25 standard deviations. We conclude that nature solves the strong CP-problem differently.

Finally, we conclude this section by giving the FLAG averages for \(\epsilon \) defined in Eq. (39). For \(N_{ f}=2\,+\,1\,+\,1\), we average the RM123 17 [19] result with the value of \((\Delta M_{K}^{2})^{\gamma }\) from BMW 14 [119] combined with Eq. (40), giving

$$\begin{aligned} \epsilon =0.79(7)\,. \end{aligned}$$
(47)

Although BMW 14 [119] focuses on hadron masses and did not extract the light-quark masses, they are the only fully unquenched QCD+QED calculation to date that qualifies to enter a FLAG average. With the exception of renormalization which is not discussed in the paper, this work has a  rating for every FLAG criterion considered for the \(m_u\) and \(m_d\) quark masses. For \(N_{ f}=2\,+\,1\) we use the results from BMW 16 [20]

$$\begin{aligned} \epsilon =0.73(17)\,. \end{aligned}$$
(48)

These results are entirely determined from lattice-QCD+QED and represent an improvement of the error by a factor of two to three on the FLAG 16 phenomenological estimate.

It is important to notice that the \(\epsilon \) uncertainties from BMW 16 and RM123 17 are dominated by estimates of the QED quenching effects. Indeed, in contrast with the quark masses, \(\epsilon \) is expected to be rather sensitive to the sea quark-QED constributions. Using the arguments presented in Sect. 3.1.3, if one conservatively assumes that the QED sea contributions represent \(O(10\%)\) of \((\Delta M_{K}^{2})^{\gamma }\), then Eq. (40) implies that \(\epsilon \) will have a quenching error of \(\sim 0.15\) for \((\Delta M_{K}^{2})^{\gamma }\sim 2000~\mathrm {MeV}^2\), representing a large \(\sim 20\%\) relative error. It is interesting to observe that such a discrepancy does not appear between BMW 15 and RM123 17, although the \(\sim 10\%\) accuracy of both results might not be sufficient to resolve these effects. To conclude, although the controversy around the value of \(\epsilon \) has been significantly reduced by lattice-QCD+QED determinations, computing this quantity precisely requires fully unquenched simulations.

Estimates for R and Q

The quark-mass ratios

$$\begin{aligned} R\equiv \frac{m_s-m_{ud}}{m_d-m_u}\quad \text{ and }\quad Q^2\equiv \frac{m_s^2-m_{ud}^2}{m_d^2-m_u^2} \end{aligned}$$
(49)

compare SU(3) breaking with isospin breaking. Both numbers only depend on the ratios \(m_s/m_{ud}\) and \(m_u/m_d\),

$$\begin{aligned} R=\frac{1}{2}\left( \frac{m_s}{m_{ud}}-1\right) \frac{1+\frac{m_u}{m_d}}{1-\frac{m_u}{m_d}} \quad \text {and}\quad Q^2=\frac{1}{2}\left( \frac{m_s}{m_{ud}}+1\right) R.\nonumber \\ \end{aligned}$$
(50)

The quantity Q is of particular interest because of a low-energy theorem [192], which relates it to a ratio of meson masses,

$$\begin{aligned} Q^2_M\equiv & {} \frac{{\hat{M}}_K^2}{{\hat{M}}_\pi ^2}\frac{{\hat{M}}_K^2-{\hat{M}}_\pi ^2}{{\hat{M}}_{K^0}^2- {\hat{M}}_{K^+}^2}\,,\quad {\hat{M}}^2_\pi \equiv {\frac{1}{2}}( {\hat{M}}^2_{\pi ^+}+ {\hat{M}}^2_{\pi ^0}) \,,\nonumber \\ {\hat{M}}^2_K\equiv & {} {\frac{1}{2}}( {\hat{M}}^2_{K^+}+ {\hat{M}}^2_{K^0})\,.\end{aligned}$$
(51)
Table 8 Our estimates for the strange-quark and the average up-down-quark masses in the \({\overline{\mathrm {MS}}}\) scheme at running scale \(\mu =2\,\mathrm {GeV}\). Mass 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. (33) and (37)
Table 9 Our estimates for the masses of the two lightest quarks and related, strong isospin-breaking ratios. Again, the masses refer to the \({\overline{\mathrm {MS}}}\) scheme at running scale \(\mu =2\,\mathrm {GeV}\). Mass values are given in MeV

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:

(52)

We recall here the \(N_f=2\) estimates for Q and R from FLAG 16,

$$\begin{aligned} R=40.7(3.7)(2.2)\,,\quad Q=24.3(1.4)(0.6)\ , \end{aligned}$$
(53)

where the second error comes from the phenomenological inputs that were used. For \(N_{ f}=2\,+\,1\), we use Eqs. (37) and (43) and obtain

$$\begin{aligned} R=38.1(1.5)\,,\quad Q=23.3(0.5)\ , \end{aligned}$$
(54)

where now only lattice results have been used. For \(N_{ f}=2\,+\,1\,+\,1\) we obtain

$$\begin{aligned} R=40.7(2.7)\,,\quad Q=24.0(0.8)\ , \end{aligned}$$
(55)

which are quite compatible with two- and three-flavour results. It is interesting to notice that the most recent phenomenological determination of R and Q from \(\eta \rightarrow 3\pi \) decay [193] gives the values \(R=34.4(2.1)\) and \(Q=22.1(7)\), which are marginally discrepant with the averages presented here. For \(N_{ f}=2\,+\,1\), the discrepancy is 1.4 standard deviations for both R and Q. For \(N_{ f}=2\,+\,1\,+\,1\) it is 1.8 standard deviations. The authors of [193] point out that this discrepancy is due to surprisingly large corrections to the approximation (52) used in the phenomenological analysis.

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 following, we collect and discuss 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 two-point heavy–light- or heavy–heavy-meson correlation functions, using as input the experimental values of the D, \(D_s\), and charmonium mesons. Other groups use the moments method. 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 various lattice collaborations have been discussed in previous FLAG reviews [2, 3], 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 two-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 [170]), we have collected in Table 10 the lattice results for both \({\overline{m}}_c({\overline{m}}_c)\) and \({\overline{m}}_c(\text{3 } \text{ GeV })\), obtained for \(N_f =2\,+\,1\) and \(2\,+\,1\,+\,1\). This year’s review does not contain results for \(N_f=2\), and interested readers are referred to previous reviews [2, 3].

When not directly available in the published work, 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 \({\overline{\mathrm {MS}}}\)-charm-quark mass \({\overline{m}}_c({\overline{m}}_c)\) and \({\overline{m}}_c(\text{3 } \text{ GeV })\) in GeV, together with the colour coding of the calculations used to obtain these. When not directly available in a publication, we employ a conversion factor equal to 0.900 between the scales \(\mu = 2\) GeV and \(\mu = 3\) GeV (or, 0.766 between \(\mu = {\overline{m}}_c\) and \(\mu = 3\) GeV)

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

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

The HPQCD 10 [13] result is computed from moments, using a subset of \(N_f = 2\,+\,1\) Asqtad-staggered-fermion ensembles from MILC [129] and HISQ valence fermions. The charm mass is fixed from the \(\eta _c\) meson, \(M_{\eta _c} = 2.9852 (34) ~ \mathrm {GeV}\), corrected for \({{\bar{c}}}c\) annihilation and electromagnetic effects. HPQCD 10 supersedes the HPQCD 08B [171] result using valence-Asqtad-staggered fermions.

\(\chi \)QCD 14 [22] uses a mixed-action approach based on overlap fermions for the valence quarks and domain-wall fermions for the sea quarks. They adopt six of the gauge ensembles generated by the RBC/UKQCD collaboration [160] 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, \(M_\pi L=4.1\), which satisfies the tag for 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 [194] determines the charm mass by using the moments method and Möbius domain-wall fermions at three values of the lattice spacing, ranging from 0.044 to 0.083 fm. They employ 15 ensembles in all, including several different pion masses and volumes. The lightest pion mass is \(\simeq 230\) MeV with \(M_\pi L\) is \(\simeq 4.4\). The linear size of their lattices is in the range 2.6–3.8 fm.

Since FLAG 16 there have been two new results, JLQCD 16 [23] and Maezawa 16 [157]. The former supersedes JLQCD 15B as it is a published update of their previous preliminary result. The latter employs the moments method using pseudoscalar correlation functions computed with HISQ fermions on a set of 11 ensembles with lattices spacing in the range 0.04 to 0.14 fm. Only a single pion mass of 160 MeV is studied. The linear size of the lattices take on values between 2.5 and 5.2 fm.

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 results HPQCD 10, \(\chi \)QCD 14, and JLQCD 16,

$$\begin{aligned}&\quad {\overline{m}}_c({\overline{m}}_c) = 1.275 ~ (5) ~ \mathrm {GeV}\quad \,\mathrm {Refs.}~ \text{[13,22,23] }\,, \nonumber \\ {N_f = 2\,+\,1:}&\end{aligned}$$
(56)
$$\begin{aligned}&\quad {\overline{m}}_c(\text{3 } \text{ GeV }) = 0.992 ~ (6)~ \mathrm {GeV}\quad \,\mathrm {Refs.}~ \text{[13,22,23] }, \end{aligned}$$
(57)

where the error on \( {\overline{m}}_c(\text{3 } \text{ GeV })\) includes a stretching factor \(\sqrt{\chi ^2/\text{ dof }} \simeq 1.18\) as discussed in Sect. 2.2. This result corresponds to the following RGI average

$$\begin{aligned} M_c^{\mathrm{RGI}}&= 1.529(9)_m(14)_\Lambda ~ \mathrm {GeV}= 1.529(17) ~ \mathrm {GeV}\nonumber \\&\quad \ \ \,\mathrm {Refs.}~ \text{[13,22,23] }. \end{aligned}$$
(58)

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

In FLAG 16 three results employing four dynamical quarks in the sea were discussed. ETM 14 [9] uses 15 twisted-mass gauge ensembles at three lattice spacings ranging from 0.062 to 0.089 fm, 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 [21] 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 [16] employs the moments method with HISQ fermions. Their results are based on 9 out of the 21 ensembles produced by the MILC collaboration [18]. Lattice spacings range from 0.057 to 0.153 fm, with box sizes up to 5.8 fm and taste-Goldstone-pion masses down to 130 MeV. The RMS-pion masses go down to 173 MeV. The strange- and charm-quark masses are fixed using \(M_{{{\bar{s}}}s} = 688.5 (2.2)~\mathrm {MeV}\), 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 electromagnetic effects. All of the selection criteria of Sect. 2.1.1 are satisfied with the tag .Footnote 16

Since FLAG 16 two groups, FNAL/MILC/TUMQCD and HPQCD have produced new values for the charm-quark mass [8, 15]. The latter use nonperturbative renormalization in the RI-SMOM scheme as described in the strange quark section and the same HISQ ensembles and valence quarks as those described in HPQCD 14A [16].

The FNAL/MILC/TUMQCD groups use a new minimal-renormalon-subtraction scheme (MRS) [195] and a sophisticated, but complex, fit strategy incorporating three effective field theories: heavy quark effective theory (HQET), heavy-meson rooted all-staggered chiral perturbation theory (HMrAS\(\chi \)PT), and Symanzik effective theory for cutoff effects. heavy–light meson masses are computed from fits to lattice-QCD correlation functions. They employ HISQ quarks on 20 MILC \(2\,+\,1\,+\,1\) flavour ensembles with six lattice spacings between 0.03 and 0.15 fm (the largest is used only in the estimation of the systematic error in the continuum-limit extrapolation). The pion mass is physical on several ensembles except the finest, and \(M_\pi L=3.7\)–3.9 on the physical mass ensembles. The light-quark masses are fixed from meson masses in pure QCD, which have been shifted from their physical values using \(O(\alpha )\) electromagnetic effects recently computed by the MILC collaboration [145], see Sect. 3.1.6 for details. The heavy–light mesons are shifted using a phenomenological formula. Using chiral perturbation theory at NLO and NNLO, the results are corrected for exponentially small finite-volume effects. They find that nonexponential finite-volume effects due to nonequilibration of topological charge are negligible compared to other quoted errors. These allow for a combined continuum, chiral, and infinite-volume limit from a global fit including 77 free parameters to 324 data points which satisfies all of the FLAG criteria.

All four results enter the FLAG average for \(N_f = 2\,+\,1\,+\,1\) quark flavours. We note however that while the determinations of \({\overline{m}}_c\) by ETM 14 and 14A agree well with each other, they are incompatible with HPQCD 14A, HPQCD 18, and FNAL/MILC/TUMQCD 18 by several standard deviations. While the latter use the same configurations, the analyses are quite different and independent. As mentioned earlier, \(m_{ud}\) and \(m_s\) are also systematically high compared to their respective averages. In addition, the other four-flavour values are consistent with the three-flavour average. Combining all four results yields

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
(59)
(60)

where the errors include large stretching factors \(\sqrt{\chi ^2/\text{ dof }}\approx 2.0\) and 1.7, respectively. We have assumed 100% correlation for statistical errors between ETM results. For HPQCD 14A, HPQCD 18, and FNAL/MILC/TUMQCD 18 we use the correlations given in Ref. [15]. Our fits have \(\chi ^2/\text{ dof }=3.9\) and 2.8, respectively. The RGI average reads as follows

$$\begin{aligned} M_c^{\mathrm{RGI}}&= 1.523(11)_m(14)_\Lambda ~ \mathrm {GeV}= 1.523(18) ~ \mathrm {GeV}\nonumber \\&\quad \,\mathrm {Refs.}~ \text{[8,9,15,16,21] }. \end{aligned}$$
(61)

Figure 5 presents the results given in Table 10 along with the FLAG averages obtained for \(2\,+\,1\) and \(2\,+\,1\,+\,1\) flavours.

Fig. 5
figure5

The charm quark mass for \(2\,+\,1\) and \(2\,+\,1\,+\,1\) flavours. For the latter a large stretching factor is used for the FLAG average due to poor \(\chi ^2\) from our fit

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 review these lattice calculations, which are listed in Table 11.

The \(N_f = 2\,+\,1\) results from \(\chi \)QCD 14 and HPQCD 09A [24] are the same as described for the charm-quark mass, and in addition the latter fixes the strange mass using \(M_{{{\bar{s}}}s} = 685.8(4.0)\,\mathrm {MeV}\). Since FLAG 16 another result has appeared, Maezawa 16 which does not pass our chiral-limit test (as described in the previous section), though we note that it is quite consistent with the other values. Combining \(\chi \)QCD 14 and HPQCD 09A, we obtain the same result reported in FLAG 16,

$$\begin{aligned} N_f = 2\,+\,1: \quad m_c / m_s = 11.82 ~ (16)\quad \,\mathrm {Refs.}~ \text{[22,24] },\nonumber \\ \end{aligned}$$
(62)

with a \(\chi ^2/\text{ dof } \simeq 0.85\).

Fig. 6
figure6

Lattice results for the ratio \(m_c / m_s\) listed in Table 11 and the FLAG averages corresponding to \(2\,+\,1\) and \(2\,+\,1\,+\,1\) quark flavours. The latter average includes a large stretching factor on the error due a poor \(\chi ^2\) from our fit

Table 12 Lattice results for the \({\overline{\mathrm {MS}}}\)-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

Turning to \(N_f = 2\,+\,1\,+\,1\), in addition to the HPQCD 14A and ETM 14 calculations, already described in Sect. 3.2.2, we consider the recent FNAL/MILC/TUMQCD 18 value [8] (which updates and replaces [18]), where HISQ fermions are employed as described in the previous section. As for the HPQCD 14A result, all of our selection criteria are satisfied with the tag . However, some tension exists between the HPQCD and FNAL/MILC/TUMQCD results. Combining all three yields

$$\begin{aligned} {N_f = 2\,+\,1\,+\,1:} \quad m_c / m_s = 11.768~ (33)\quad \,\mathrm {Refs.}~ \text{[8,9,16] },\nonumber \\ \end{aligned}$$
(63)

where the error includes the stretching factor \(\sqrt{\chi ^2/\text{ dof }} \simeq 1.5\), and \(\chi ^2/dof=2.28\). We have assumed a 100% correlation of statistical errors for FNAL/MILC/TUMQCD 18 and HPQCD 14A.

Results for \(m_c/m_s\) are shown in Fig. 6 together with the FLAG averages for \(2\,+\,1\) and \(2\,+\,1\,+\,1\) flavours.

Bottom quark mass

Now we review the lattice results for the \(\overline{\mathrm{MS}}\)-bottom-quark mass \({\overline{m}}_b\). Related heavy-quark actions and observables have been discussed in the FLAG 13 and 17 reviews [2, 3], and descriptions can be found in Sect. A.1.3. In Table 12 we collect results for \({\overline{m}}_b({\overline{m}}_b)\) obtained with \(N_f =2\,+\,1\) and \(2\,+\,1\,+\,1\) quark flavours in the sea. Available results for the quark-mass ratio \(m_b / m_c\) are also reported. After discussing the various results we evaluate the corresponding FLAG averages.

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

HPQCD 13B [197] 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. This result is not used in our average.

The value of \({\overline{m}}_b({\overline{m}}_b)\) reported in HPQCD 10 [13] is computed in a very similar fashion to the one in HPQCD 14A described in the following section on \(2\,+\,1\,+\,1\) flavour results, except that MILC \(2\,+\,1\)-flavour-Asqtad ensembles are used under HISQ 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.

Maezawa 16 reports a new result for the b-quark mass since the last FLAG review. However as discussed in the charm-quark section, this calculation does not satisfy the criteria to be used in the FLAG average. As in the previous review, we take the HPQCD 10 result as our average,

$$\begin{aligned}&N_f= 2\,+\,1 : \quad \overline{m}_b(\overline{m}_b) = 4.164 (23) ~ \mathrm {GeV}\nonumber \\&\quad \,\mathrm {Ref.}~ \text{[13] }\,, \end{aligned}$$
(64)

Since HPQCD quotes \({\overline{m}}_b({\overline{m}}_b)\) using \(N_f = 5\) running, we used that value in the average. The corresponding 4-flavour RGI average is

$$\begin{aligned}&N_f= 2\,+\,1 : M_b^\mathrm{RGI} = 6.874(38)_m(54)_\Lambda \nonumber \\&\quad \mathrm {GeV}= 6.874(66) ~ \mathrm {GeV}\quad \ \ \,\mathrm {Ref.}~ \text{[13] }. \end{aligned}$$
(65)

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

Results have been published by HPQCD using NRQCD and HISQ-quark actions (HPQCD 14B [25] and HPQCD 14A [16], respectively). In both works the b-quark mass is computed with the moments method, that is, from Euclidean-time moments of two-point, heavy–heavy-meson correlation functions (see also 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,

$$\begin{aligned} \left[ \frac{R_n r_{n-2}}{R_{n-2}r_n}\right] ^{1/2} \frac{{\bar{M}}_\mathrm{kin}}{2 m_b} = \frac{{\bar{M}}_{\Upsilon ,\eta _b}}{2 {{\bar{m}}}_b(\mu )} ~ , \end{aligned}$$
(66)

where \(r_n\) are the perturbative moments calculated at \(\hbox {N}^3\)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 average 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. (66), 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 HISQ (MILC) ensembles with \(a \approx \) 0.15, 0.12 and 0.09 fm and two pion masses, one of which is the physical one. 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). Since the physical b-quark mass in units of the lattice spacing is always greater than one in these calculations, fits to correlation functions are restricted to \(am_h \le 0.8\), and a high-degree polynomial in \(a m_{\eta _{h}}\), the corresponding pseudoscalar mass, is used in the fits to remove the lattice-spacing errors. Finally, to obtain the physical b-quark mass, the moments are extrapolated to \(m_{\eta _b}\). 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 [26] is from the ETM collaboration and updates their preliminary result appearing in a conference proceedings [196]. The calculation is performed on a set of configurations generated with twisted-Wilson fermions with three lattice spacings in the range 0.06–0.09 fm and with pion masses in the range 210–440 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 \(\hbox {N}^3\)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 \({\overline{m}}_b({\overline{m}}_b) = 4.26(3)(10)\) GeV, where the first error is statistical and the second systematic. The dominant errors come from setting the lattice scale and fit systematics.

The next new result since FLAG 16 is from Gambino, et al. [27]. The authors use twisted-mass-fermion ensembles from the ETM collaboration and the ETM ratio method as in ETM 16. Three values of the lattice spacing are used, ranging from 0.062 to 0.089 fm. Several volumes are also used. The light-quark masses produce pions with masses from 210 to 450 MeV. The main difference with ETM 16 is that the authors use the kinetic mass defined in the heavy-quark expansion (HQE) to extract the b-quark mass instead of the pole mass.

The final b-quark mass result is FNAL/MILC/TUM 18 [8]. The mass is extracted from the same fit and analysis that is described in the charm quark mass section. Note that relativistic HISQ quarks are used (almost) all the way up to the b-quark mass (0.9 \(am_b\)) on the finest two lattices, \(a=0.03\) and 0.042 fm. The authors investigated the effect of leaving out the heaviest points from the fit, and the result did not noticeably change.

All of the above results enter our average. We note that here the updated ETM result is consistent with the average and a stretching factor on the error is not used. The average and error is dominated by the very precise FNAL/MILC/TUM 18 value.

$$\begin{aligned}&N_f = 2\,+\,1\,+\,1:\quad {\overline{m}}_b({\overline{m}}_b) = 4.198 (12) \quad \mathrm {GeV}\nonumber \\&\quad \,\mathrm {Refs.}~{ [8,16,25{-}27]}. \end{aligned}$$
(67)

Since HPQCD quotes \({\overline{m}}_b({\overline{m}}_b)\) using \(N_f= 5\) running, we used that value in the average. We have included a 100% correlation on the statistical errors of ETM 16 and Gambino 17 since the same ensembles are used in both. This translates to the following RGI average

$$\begin{aligned}&N_f= 2\,+\,1\,+\,1:\quad M_b^{\mathrm{RGI}} = 6.936(20)_m(54)_\Lambda ~ \nonumber \\&\quad \mathrm {GeV}= 6.936(57) ~ \mathrm {GeV}\quad \,\mathrm {Refs.}{ [8,16,25{-}27]}. \end{aligned}$$
(68)

All the results for \({\overline{m}}_b({\overline{m}}_b)\) discussed above are shown in Fig. 7 together with the FLAG averages corresponding to \(N_f=2\,+\,1\) and \(2\,+\,1\,+\,1\) quark flavours.

Fig. 7
figure7

The b-quark mass, \(N_f =2\,+\,1\) and \(2\,+\,1\,+\,1\). The updated PDG value from Ref. [137] is reported for comparison

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

Authors: T. Kaneko, J. N. Simone, S. Simula

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 [3] the data in this section has been updated. As in Ref. [3], 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 (\(\chi \)PT), 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 \(|V_{us}|f_+(0)\) [200] and the ratio \(|V_{us}/V_{ud}|f_{K^\pm }/f_{\pi ^\pm }\) [200, 201]:

$$\begin{aligned} |V_{us}| f_+(0) = 0.2165(4)\,,\quad \left| \frac{V_{us}}{V_{ud}}\right| \frac{ f_{K^\pm }}{ f_{\pi ^\pm }} \; =0.2760(4)\,.\nonumber \\ \end{aligned}$$
(69)

Here and in the following, \(f_{K^\pm }\) and \(f_{\pi ^\pm }\) are the isospin-broken decay constants, respectively, in QCD. 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 \(f_{\pi ^\pm }\)). The parameters \(|V_{ud}|\) and \(|V_{us}|\) are elements of the Cabibbo-Kobayashi-Maskawa matrix and \(f_+(q^2)\) represents one of the form factors relevant for the semileptonic decay \(K^0\rightarrow \pi ^-\ell \,\nu \), which depends on the momentum transfer q between the two mesons. What matters here is the value at \(q^2 = 0\): . The pion and kaon decay constants are defined byFootnote 17

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

In Eq. (69), the electromagnetic effects have already been subtracted in the experimental analysis using \(\chi \)PT. Recently, a new method [206] has been proposed for calculating the leptonic decay rates of hadrons including both QCD and QED on the lattice, and successfully applied to the case of the ratio of the leptonic decay rates of kaons and pions [207]. The correction to the tree-level \(K_{\mu 2} / \pi _{\mu 2}\) decay rate, including both electromagnetic and strong isospin-breaking effects, is found to be equal to \(-1.22 (16) \%\) to be compared to the estimate \(-1.12 (21) \%\) based on \(\chi \)PT [133, 208]. Using the experimental values of the \(K_{\mu 2} \) and \(\pi _{\mu 2}\) decay rates the result of Ref. [207] implies

$$\begin{aligned} \left| \frac{V_{us}}{V_{ud}}\right| \frac{f_K}{f_\pi } = 0.27673 \, (29)_{\mathrm{exp}} \, (23)_{\mathrm{th}} \, [37] ~ , \end{aligned}$$
(70)

where the last error in brackets is the sum in quadrature of the experimental and theoretical uncertainties, and the ratio of the decay constants is the one corresponding to isosymmetric QCD. The single calculation of Ref. [207] is clearly not ready for averaging, but it demonstrates that the determination of \(V_{us} / V_{ud}\) using only lattice-QCD+QED and the ratio of the experimental values of the \(K_{\mu 2} \) and \(\pi _{\mu 2}\) decay rates is feasible with good accuracy.

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 [209], which is based on 20 different superallowed transitions, readsFootnote 18

$$\begin{aligned} |V_{ud}| = 0.97420(21)\,.\end{aligned}$$
(71)

The matrix element \(|V_{us}|\) can be determined from semi-inclusive \(\tau \) decays [217,218,219,220]. By separating the inclusive decay \(\tau \rightarrow \text{ hadrons }+\nu \) into nonstrange and strange final states, e.g., HFLAV 16 [221] obtains \(|V_{us}|=0.2186(21)\) and both Maltman et al. [219, 222, 223] and Gamiz et al. [224, 225] arrive at very similar values. Inclusive hadronic \(\tau \) decay offers an interesting way to measure \(|V_{us}|\), but the above value of \(|V_{us}|\) differs from the result one obtains from assuming three-flavour SM-unitarity by more than three standard deviations [221]. This apparent tension has been recently solved in Ref. [226] thanks to the use of a different experimental input and to a new treatment of higher orders in the operator product expansion and of violations of quark-hadron duality. A much larger value of \(|V_{us}|\) is obtained, namely,

$$\begin{aligned} |V_{us}| = 0.2231 (27)_{\mathrm{exp}} (4)_{\mathrm{th}} ~ , \end{aligned}$$
(72)

which is in much better agreement with CKM unitarity. Recently, in Ref. [227], a new method, which includes also the lattice calculation of the hadronic vacuum polarization function, has been proposed for the determination of \(|V_{us}|\) from inclusive strange \(\tau \) decays.

Table 13 Colour code for the data on \(f_+(0)\). With respect to the previous edition [3] old results with two red tags have been dropped

The experimental results in Eq. (69) 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 [228]. This is quite a convenient procedure as long as lattice-QCD simulations do not include strong or QED isospin-breaking effects. Several lattice results for \(f_K/f_\pi \) are quoted for QCD with (squared) pion and kaon masses of \(M_\pi ^2=M_{\pi ^0}^2\) and \(M_K^2=\frac{1}{2} \left( M_{K^\pm }^2\,+\,M_{K^0}^2-M_{\pi ^\pm }^2\,+\,M_{\pi ^0}^2\right) \) for which the leading strong and electromagnetic isospin violations cancel. While the modern trend is to include strong and electromagnetic isospin breaking in the lattice simulations (e.g., Refs. [140, 141, 162, 185, 206, 207, 229,230,231]), in this section contact with 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. With respect to the previous edition [3] old results with two red tags have been dropped

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 [241]. 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.

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 data sets are characterized by means of the colour code specified in Sect. 2.1. Note that with respect to the previous edition [3] we have dropped old results with two red tags. More detailed information on individual computations are compiled in Appendix B.2, which in this edition is limited to new results and to those entering the FLAG averages. For other calculations the reader should refer to the Appendix B.2 of Ref. [3].

The quantity \(f_+(0)\) represents a matrix element of a strangeness-changing null-plane charge, \(f_+(0)\,{=}\,\langle K|Q^{{\bar{u}}s}|\pi \rangle \) (see Ref. [242]). The vector charges obey the commutation relations of the Lie algebra of SU(3), in particular \([Q^{{\bar{u}}s},Q^{{\bar{s}}u}]=Q^{{\bar{u}}u-{\bar{s}}s}\). This relation implies the sum rule \(\sum _n |\langle K|Q^{{\bar{u}}s}|n \rangle |^2-\sum _n |\langle K|Q^{{\bar{s}}u}|n \rangle |^2=1\). 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 [243]:

$$\begin{aligned} f_+(0)^2=1-\sum _{n\ne \pi } |\langle K|Q^{{\bar{u}}s}|n \rangle |^2\,+\,\sum _n |\langle K |Q^{{\bar{s}}u}|n \rangle |^2\,.\nonumber \\ \end{aligned}$$
(73)

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 \(f_+(0)=1+f_2\,+\,f_4+\ldots \,\) [244]. Since all of the low-energy constants occurring in \(f_2\) can be expressed in terms of \(M_\pi \), \(M_K\), \(M_\eta \) and \(f_\pi \) [242], the NLO correction is known. In the language of the sum rule (73), \(f_2\) 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 [30, 245]. At the same order in the SU(2) expansion [246], \(f_+(0)\) 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. (Lattice results concerning the value of the ratio \(f_\pi /f_0\) are reviewed in Sect. 5.3.)

Fig. 8
figure8

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 lattice results shown in the left panel of Fig. 8 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. [251], 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 [250, 252]. The corresponding formula for \(f_4\) accounts for the chiral logarithms occurring at NNLO and is not subject to the ambiguity mentioned above.Footnote 19 The numerical result, however, depends on the model used to estimate the low-energy constants occurring in \(f_4\) [247,248,249,250]. 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 [30] and Ref. [253] 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 \(f_{K^\pm }/f_{\pi ^\pm }\)

Many lattice results for the form factor \(f_+(0)\) and 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 a few simulations of isospin-breaking effects in lattice QCD, which is ultimately the cleanest way for predicting these effects [139,140,141, 148, 185, 206, 207, 231, 254, 255]. In the meantime, one relies either on chiral perturbation theory [166, 244] 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 results that qualify for inclusion into the FLAG average include the strong SU(2) isospin-breaking correction, we confirm the choice made in the previous edition of the FLAG review [3] and we provide in Fig. 8 the overview of the world data of \(f_{K^\pm }/f_{\pi ^\pm }\). For all the results of Table 14 provided only in the isospin-symmetric limit we apply individually an isospin correction that will be described later on (see Eqs. (78)–(79)).

The plots in Fig. 8 illustrate our compilation of data for \(f_+(0)\) and \(f_{K^\pm }/f_{\pi ^\pm }\). The lattice data for the latter quantity is 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.

Results for \(f_+(0)\)

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 \(S = {\bar{s}} u\). Peculiarities of the staggered fermion discretization used by FNAL/MILC (see Ref. [30]) makes this the favoured choice. The other collaborations are instead computing the vector current matrix element \(\langle \pi | {\bar{s}} \gamma _\mu u |K\rangle \). Apart from FNAL/MILC 13C, FNAL/MILC 13E and RBC/UKQCD 15A 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 \(q^2=0\) [266, 267], 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 [29, 240], while keeping at the same time the advantage of the high-precision determination of the scalar form factor at the kinematical end-point \(q_{max}^2 = (M_K - M_\pi )^2\) [32, 268] 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 results FNAL/MILC 13E and ETM 16 with \(N_{ f}=2\,+\,1\,+\,1\) dynamical flavours of fermions, respectively, can enter the FLAG averages.

At \(N_{ f}=2\,+\,1\,+\,1\) the 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 tailored to reduce staggered taste-breaking effects, and includes simulations with three lattice spacings and physical light-quark masses. These features allow to keep the uncertainties due to the chiral extrapolation and to the discretization artifacts well below the statistical error. The remaining largest systematic uncertainty comes from finite-size effects, which have been investigated in Ref. [269] using 1-loop \(\chi \)PT (with and without taste-violating effects). Recently [232] the FNAL/MILC collaboration presented a more precise determination of \(f_+(0)\), \(f_+(0) = 0.9696 (15) (11)\) (see the entry FNAL/MILC 18 in Table 13), in which the improvement of the precision with respect to FNAL/MILC 13E is obtained mainly by using an estimate of finite-size effects based on ChPT only. We do not consider FNAL/MILC 18 as a plain update of FNAL/MILC 13E.

The new result from the ETM collaboration, \(f_+(0) = 0.9709 (45) (9)\) (ETM 16), makes use of the twisted-mass discretization adopting three values of the lattice spacing in the range \(0.06{-}0.09\) fm and pion masses simulated in the range \(210{-}450\) MeV. The chiral and continuum extrapolations are performed in a combined fit together with the momentum dependence, using both a SU(2)-\(\chi \)PT inspired ansatz (following Ref. [240]) and a modified z-expansion fit. The uncertainties coming from the chiral extrapolation, the continuum extrapolation and the finite-volume effects turn out to be well below the dominant statistical error, which includes also the error due to the fitting procedure. A set of synthetic data points, representing both the vector and the scalar semileptonic form factors at the physical point for several selected values of \(q^2\), is provided together with the corresponding correlation matrix.

At \(N_{ f}=2\,+\,1\) there is a new result from the JLQCD collaboration [234], which however does not satisfy all FLAG criteria for entering the average. The two results eligible to enter the FLAG average at \(N_{ f}=2\,+\,1\) are the one from RBC/UKQCD 15A, \(f_+(0) = 0.9685 (34) (14)\) [31], and the one from FNAL/MILC 12I, \(f_+(0)=0.9667(23)(33)\) [30]. These 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 analyzed 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 cut-off 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 [250] 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 [270].

The ETM collaboration uses the twisted-mass discretization and provides at \(N_{ f}=2\) a comprehensive study of the systematics [32, 240], 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) [242] and SU(2) [246] 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 [40] with the one in the quenched approximation, obtained earlier by the SPQcdR collaboration [268].

We now compute the \(N_f = 2\,+\,1\,+\,1\) FLAG-average for \(f_+(0)\) using the FNAL/MILC 13E and ETM 16 (uncorrelated) results, the \(N_f =2\,+\,1\) FLAG-average based on FNAL/MILC 12I and RBC/UKQCD 15A, which we consider uncorrelated, while for \(N_f = 2\) we consider directly the ETM 09A result, respectively:

$$\begin{aligned}&\text{ direct },\,N_{ f}=2\,+\,1\,+\,1:\quad f_+(0) = 0.9706(27)\quad \,\mathrm {Refs.}~ \text{[28,29] }, \end{aligned}$$
(74)
$$\begin{aligned}&\text{ direct },\,N_{ f}=2\,+\,1: \quad f_+(0) = 0.9677(27) \quad \,\mathrm {Refs.}~\text{[30,31] }, \nonumber \\\end{aligned}$$
(75)
$$\begin{aligned}&\text{ direct },\,N_{ f}=2: \quad f_+(0) = 0.9560(57)(62)\quad \,\mathrm {Ref.}~\text{[32] }, \end{aligned}$$
(76)

where the brackets in the third line indicate the statistical and systematic errors, respectively. We stress that the results (74) and (75), corresponding to \(N_f = 2\,+\,1\,+\,1\) and \(N_f = 2\,+\,1\), respectively, include already simulations with physical light-quark masses.

Results for \(f_{K^\pm }/f_{\pi ^\pm }\)

In the case of the ratio of decay constants the data sets that meet the criteria formulated in the introduction are HPQCD 13A [33], ETM 14E [34] and FNAL/MILC 17 [5] (which updates FNAL/MILC 14A [18]) with \(N_f=2\,+\,1\,+\,1\), HPQCD/UKQCD 07 [35], MILC 10 [36], BMW 10 [37], RBC/UKQCD 14B [10], Dürr 16 [38, 260] and QCDSF/UKQCD 16 [39] with \(N_{ f}=2\,+\,1\) and ETM 09 [40] 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) [246] 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 17 [5] 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 24 ensembles for six values of the lattice spacing (\(0.03 - 0.15\) 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 FNAL/MILC 14A they have increased the statistics and added three ensembles at very fine lattice spacings, \(a \simeq 0.03\) and 0.042 fm, including for the latter case also a simulation at the physical value of the light-quark mass. The final result of their analysis is \({f_{K^\pm }}/{f_{\pi ^\pm }}=1.1950(14)_{\mathrm{stat}}(_{-17}^{+0})_{\mathrm{a}^2} (2)_{FV} (3)_{f_\pi , PDG} (3)_{EM} (2)_{Q^2}\), where the errors are statistical, due to the continuum extrapolation, finite-volume, pion decay constant from PDG, electromagnetic effects and sampling of the topological charge distribution.

HPQCD 13A has analyzed 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 ones at the finest lattice spacings (namely, only \(a = 0.09 - 0.15\) fm, scale set with \(f_{\pi ^+}\) and relative scale set with the Wilson flow [271, 272]) 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 two previous editions of the FLAG review [2, 3] the error budget of HPQCD 13A was compared with the ones of MILC 13A and FNAL/MILC 14A and discussed in detail. It was pointed out that, despite the overlap in primary lattice data, both collaborations arrive at surprisingly different error budgets, particularly in the cases of the cutoff dependence and of the finite volume effects. The error budget of the latest update FNAL/MILC 17, which has a richer lattice setup with respect to HPQCD 13A, is consistent with the one of HPQCD 13A.

Adding in quadrature all the uncertainties one gets: \(f_{K^\pm }/f_{\pi ^\pm } = 1.1916(22)\) (HPQCD 13A) and \({f_{K^\pm }}/{f_{\pi ^\pm }}=1.1944(18)\)Footnote 20 (FNAL/MILC 17). It can be seen that the total errors are very similar and the central values are consistent within approximately one standard deviation. Thus, the HPQCD 13A and FNAL/MILC 17 are averaged, assuming a \(100 \%\) statistical and systematic correlations between them, together with the (uncorrelated) ETM 14E result, obtaining

$$\begin{aligned}&\text{ direct },\,N_{ f}=2\,+\,1\,+\,1: \quad {f_{K^\pm }}/{f_{\pi ^\pm }}=1.1932(19)\nonumber \\&\quad \,\mathrm {Refs.}~\text{[5,33,34] }. \end{aligned}$$
(77)

For \(N_f=2\,+\,1\) the result Dürr 16 [38, 260] is now eligible to enter the FLAG average as well as the new result [39] from the QCDSF collaboration. Dürr 16 [38, 260] has analyzed the decay constants evaluated for 47 gauge ensembles generated using tree-level clover-improved fermions with two HEX-smearings and the tree-level Symanzik-improved gauge action. The ensembles correspond to five values of the lattice spacing (\(0.05{-}0.12\) fm, scale set by \(\Omega \) mass), to pion masses in the range \(130{-}680\) MeV and to values of the lattice size from 1.7 to 5.6 fm, obtaining a good control over the interpolation to the physical mass point and the extrapolation to the continuum and infinite volume limits.

QCDSF/UKQCD 16 [39] has used the nonperturbatively \(\mathcal{{O}}(a)\)-improved clover action for the fermions (mildly stout-smeared) and the tree-level Symanzik action for the gluons. Four values of the lattice spacing (\(0.06{-}0.08\) fm) have been simulated with pion masses down to \(\sim 220\) MeV and values of the lattice size in the range \(2.0{-}2.8\) fm. The decay constants are evaluated using an expansion around the symmetric SU(3) point \(m_u = m_d = m_s = (m_u + m_d + m_s)^{phys}/3\).

Note that for \(N_f=2\,+\,1\) MILC 10 and HPQCD/UKQCD 07 are based on staggered fermions, BMW 10, Dürr 16 and QCDSF/UKQCD 16 have used improved Wilson fermions and RBC/UKQCD 14B’s result is based on the domain-wall formulation. In contrast to RBC/UKQCD 14B and Dürr 16 the other simulations are for unphysical values of the light-quark masses (corresponding to smallest pion masses in the range \(220 - 260\) MeV in the case of MILC 10, HPQCD/UKQCD 07 and QCDSF/UKQCD 16) 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, RBC/UKQCD 14B and QCDSF/UKQCD 16 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 both statistical and systematic uncertainties to be correlated.

For \(N_f=2\) no new result enters the corresponding FLAG average with respect to the previous edition of the FLAG review [3], which therefore remains the ETM 09 result, which has simulated twisted-mass fermions down to (charged) pion masses equal to 260 MeV.

We note that the overall uncertainties quoted by ETM 14E at \(N_{ f}=2\,+\,1\,+\,1\) and by Dürr 16 and QCDSF/UKQCD 16 at \(N_{ f}=2\,+\,1\) are much larger than the overall uncertainties obtained with staggered (HPQCD 13A, FNAL/MILC 17 at \(N_{ f}=2\,+\,1\,+\,1\) and MILC 10, HPQCD/UKQCD 07 at \(N_{ f}=2\,+\,1\)) and domain-wall fermions (RBC/UKQCD 14B at \(N_{ f}=2\,+\,1\)).

Before determining the average for \(f_{K^\pm }/f_{\pi ^\pm }\), which should be used for applications to Standard Model phenomenology, we apply the strong isospin correction individually to all those results that have been published only in the isospin-symmetric limit, i.e., BMW 10, HPQCD/UKQCD 07 and RBC/UKQCD 14B at \(N_f = 2\,+\,1\) and ETM 09 at \(N_f = 2\). To this end, as in the previous edition of the FLAG reviews [2, 3], we make use of NLO SU(3) chiral perturbation theory [208, 244], which predicts

$$\begin{aligned} \frac{f_{K^\pm }}{f_{\pi ^\pm }}= \frac{f_K}{f_\pi } ~ \sqrt{1 + \delta _{SU(2)}} ~ , \end{aligned}$$
(78)

where [208]

(79)

We use as input \(\epsilon _{SU(2)} = \sqrt{3} / (4 R)\) with the FLAG result for R of Eq. (54), \(F_0 = f_0 / \sqrt{2} = 80\,(20)\) 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 last 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 \(f_{K^\pm }/f_{\pi ^\pm }\)
Fig. 9
figure9

The plot compares the information for \(|V_{ud}|\), \(|V_{us}|\) obtained on the lattice for \(N_f = 2\,+\,1\) and \(N_f = 2\,+\,1\,+\,1\) 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. For the \(N_f = 2\) results see the previous FLAG edition [3]

For \(N_f=2\) and \(N_f=2\,+\,1\,+\,1\) dedicated studies of the strong-isospin correction in lattice QCD do exist. The updated \(N_f=2\) result of the RM123 collaboration [141] amounts to \(\delta _{SU(2)}=-0.0080(4)\) and we use this result for the isospin correction of the ETM 09 result. Note that the above RM123 value for the strong-isospin correction is 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 [33], FNAL/MILC [5] and ETM [273] estimate a value for \(\delta _{SU(2)}\) equal to \(-0.0054(14)\), \(-0.0052(9)\) and \(-0.0073(6)\), respectively. Note that the RM123 and ETM results are obtained using the insertion of the isovector scalar current according to the expansion method of Ref. [140], while the HPQCD and FNAL/MILC results correspond to the difference between the values of the decay constant ratio extrapolated to the physical u-quark mass \(m_u\) and to the average \((m_u + m_d) / 2\) light-quark mass.

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

$$\begin{aligned}&\text{ direct },\,N_{ f}=2\,+\,1\,+\,1:\quad f_{K^\pm } / f_{\pi ^\pm } = 1.1932(19)\nonumber \\&\quad \,\mathrm {Refs.}~\text{[5,33,34] }, \end{aligned}$$
(80)
$$\begin{aligned}&\text{ direct },\,N_{ f}=2\,+\,1: \quad f_{K^\pm } / f_{\pi ^\pm } = 1.1917(37)\nonumber \\&\quad \,\mathrm {Refs.}~{ [10,35{-}39]}, \end{aligned}$$
(81)
$$\begin{aligned}&\text{ direct },\,N_{ f}=2: \quad f_{K^\pm } / f_{\pi ^\pm } = 1.205(18)\nonumber \\&\quad \,\mathrm {Ref.}~\text{[40] }, \end{aligned}$$
(82)

for QCD with broken isospin.

The averages obtained for \(f_+(0)\) and \({f_{K^\pm }}/{f_{\pi ^\pm }}\) at \(N_{ f}=2\,+\,1\) and \(N_{ f}=2\,+\,1\,+\,1\) [see Eqs. (74-75) and (80-81)] exhibit a precision better than \(\sim 0.3 \%\). At such a level of precision QED effects cannot be ignored and a consistent lattice treatment of both QED and QCD effects in leptonic and semileptonic decays becomes mandatory.

Extraction of \(|V_{ud}|\) and \(|V_{us}|\)

It is instructive to convert the averages 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 (69). Consider first the results for \(N_{ f}=2\,+\,1\,+\,1\). The range for \(f_+(0)\) in Eq. (74) is mapped into the interval \(|V_{us}|=0.2231(7)\), depicted as a horizontal red band in Fig. 9, while the one for \({f_{K^\pm }}/{f_{\pi ^\pm }}\) in Eq. (80) is converted into \(|V_{us}|/|V_{ud}|= 0.2313(5)\), shown as a tilted red band. The red ellipse is the intersection of these two bands and represents the 68% likelihood contour,Footnote 21 obtained by treating the above two results as independent measurements. Repeating the exercise for \(N_{ f}=2\,+\,1\) leads to the green ellipse. The plot indicates a slight tension of both the \(N_f=2\,+\,1\,+\,1\) and \(N_f=2\,+\,1\) results with the one from nuclear \(\beta \) decay.

Tests of the Standard Model

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

$$\begin{aligned} |V_u|^2\equiv |V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 1\,.\end{aligned}$$
(83)

The tiny contribution from \(|V_{ub}|\) is known much better than needed in the present context: \(|V_{ub}|= 3.94 (36) \cdot 10^{-3}\) [201]. In the following, we first discuss the evidence for the validity of the relation (83) and only then use it to analyse the lattice data within the Standard Model.

Fig. 10
figure10

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 is analysed within the standard model

In Fig. 9, 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 \(|V_u|^2 = 0.9797(74)\), which deviates from unity at the level of \(\simeq 2.7\) 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. (74) and (80) with the \(\beta \) decay value of \(|V_{ud}|\) quoted in Eq. (71), the test sharpens considerably: the lattice result for \(f_+(0)\) leads to \(|V_u|^2 = 0.99884(53)\), which highlights again a \(\simeq 2.2\sigma \)-tension with unitarity, while the one for \({f_{K^\pm }}/{f_{\pi ^\pm }}\) implies \(|V_u|^2 = 0.99986(46)\), confirming the first-row CKM unitarity below the permille level.Footnote 22 Note that the largest contribution to the uncertainty on \(|V_u|^2\) comes from the error on \(|V_{ud}|\) given in Eq. (71).

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

The situation is similar for \(N_{ f}=2\,+\,1\): with the lattice data alone one has \(|V_u|^2 = 0.9832(89)\), which deviates from unity at the level of \(\simeq 1.9\) standard deviations. Combining the lattice results for \(f_+(0)\) and \({f_{K^\pm }}/{f_{\pi ^\pm }}\) in Eqs. (75) and (81) with the \(\beta \) decay value of \(|V_{ud}|\), the test sharpens again considerably: the lattice result for \(f_+(0)\) leads to \(|V_u|^2 = 0.99914(53)\), implying only a \(\simeq 1.6\sigma \)-tension with unitarity, while the one for \({f_{K^\pm }}/{f_{\pi ^\pm }}\) implies \(|V_u|^2 = 0.99999(54)\), thus confirming again CKM unitarity below the permille level.

For the analysis corresponding to \(N_f = 2\) the reader should refer to the previous FLAG edition [3].

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 the lifetime of the muon. In certain modifications of the Standard Model, this is not the case and 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 is 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 (69) and the unitarity condition (83) 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.

As Fig. 10 shows, 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 \({f_{K^\pm }}/{f_{\pi ^\pm }}\) that are obtained by analysing the lattice data within the Standard Model (see text). 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

For \(N_{ f}=2\,+\,1\,+\,1\) we consider the data both for \(f_+(0)\) and \({f_{K^\pm }}/{f_{\pi ^\pm }}\), treating ETM 16 and ETM 14E on the one hand and FNAL/MILC 13E, FNAL/MILC 17 and HPQCD 13A on the other hand, as statistically correlated according to the prescription of Sect. 2.3. We obtain \(|V_{us}|=0.2249(7)\), where the error includes the inflation factor due to the value of \(\chi ^2/\mathrm{dof} \simeq 2.5\). This result is indicated on the left hand side of Fig. 10 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 14B 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 \(|V_{us}| = 0.2249(5)\) with \(\chi ^2/\mathrm{dof} \simeq 0.8\). For \(N_{ f}=2\) we consider ETM 09A and ETM 09 as statistically correlated, obtaining \(|V_{us}|=0.2256(19)\) with \(\chi ^2/\mathrm{dof} \simeq 0.7\). The figure shows that the results 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 \(|V_{ud}|=0.97437(16)\), which follows from the lattice data with \(N_{ f}=2\,+\,1\,+\,1\), is perfectly consistent with the values \(|V_{ud}|=0.97438(12)\) and \(|V_{ud}|=0.97423(44)\) 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. 9: 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 \(f_+(0) = 0.9627(35)\) for \(N_{ f}=2\,+\,1\,+\,1\), \(f_+(0) = 0.9627(28)\) for \(N_{ f}=2\,+\,1\), \(f_+(0) = 0.9597(83)\) for \(N_{ f}=2\) and \({f_{K^\pm }}/{f_{\pi ^\pm }}= 1.196(3)\) for \(N_{ f}=2\,+\,1\,+\,1\), \({f_{K^\pm }}/{f_{\pi ^\pm }}= 1.196(3)\) 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 is also remarkably precise. On the other hand, the values of \(|V_{ud}|\), \(f_+(0)\) and \({f_{K^\pm }}/{f_{\pi ^\pm }}\) that follow from the \(\tau \)-decay data if the Standard Model is assumed to be valid were initially not all in 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 [223]: the discrepancy then amounts to little more than one standard deviation. The disagreement disappears when recent implementations of the relevant sum rules and a different experimental input are considered [226].

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. With respect to the previous edition [3] old results with two red tags have been dropped

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 the PDG value [201] of this quantity, based on the use of the value of \(|V_{ud}|\) obtained from superallowed nuclear \(\beta \) decays [209], 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 and it is unchanged from the corresponding one in the previous FLAG review [3].

In comparison 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. For several lattice formulations the use of the nonsinglet axial-vector Ward identity allows 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 data sets can form the average of \(f_{K^\pm }\) only: ETM 14E [34], FNAL/MILC 14A [18] and HPQCD 13A [33]. 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 data sets can form the average of \(f_{\pi ^\pm }\) and \(f_{K^\pm }\) : RBC/UKQCD 14B [10] (update of RBC/UKQCD 12), HPQCD/UKQCD 07 [35] and MILC 10 [36], 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

$$\begin{aligned}&N_f = 2 + 1: \quad f_{\pi ^\pm }= 130.2 ~ (0.8)~ \text{ MeV } \quad \,\mathrm {Refs.}~ \text{[10,35,36] }, \end{aligned}$$
(84)
$$\begin{aligned}&N_f = 2 + 1 + 1: \quad f_{K^\pm } = 155.7 ~ (0.3)~ \text{ MeV } \quad \,\mathrm {Refs.}~ \text{[18,33,34] } ,\nonumber \\&N_f = 2 + 1: \quad f_{K^\pm } = 155.7 ~ (0.7)~ \text{ MeV } \quad \,\mathrm {Refs.}~\text{[10,35,36] }, \nonumber \\&N_f = 2: \quad f_{K^\pm } = 157.5 ~ (2.4)~ \text{ MeV } \quad \,\mathrm {Ref.}~ \text{[40] }. \end{aligned}$$
(85)

The lattice results of Table 18 and our estimates (84)–(85) are reported in Fig. 11. Note that the FLAG estimates of \(f_{K^\pm }\) for \(N_f = 2\) and \(N_f = 2 + 1 + 1\) are based on calculations in which \(f_{\pi ^\pm }\) is used to set the lattice scale, while the \(N_f = 2 + 1\) estimate does not rely on that.

Low-energy constants

Authors: S. Dürr, H. Fukaya, U. M. Heller

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 first 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,

(86)

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 \(p^2\),

(87)

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 23 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 \(\mu =M_{\pi ,\mathrm {phys}}\) (as indicated by the bar). For the precise definition of these constants and their scale dependence we refer the reader to Refs. [244, 277].

Fig. 11
figure11

Values of \(f_\pi \) and \(f_K\). The black squares and grey bands indicate our estimates (84) and (85)

Patterns of chiral symmetry breaking

If the box size 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 with periodic boundary conditions in the spatial directions. 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. 5.1.6 and 5.1.7, 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 an important by-product for two reasons:

  1. 1.

    All LECs up to order \(p^4\) (with the exception of B and \(B_0\), since only the product of these times the quark masses can be estimated from phenomenology) have either been determined by comparison to experiment or estimated theoretically, e.g., in large-\(N_c\) QCD. A lattice determination of the better known LECs thus provides a test of the \(\chi \)PT approach.

  2. 2.

    The less well-known LECs are those which describe the quark-mass dependence of observables – these cannot be determined from experiment, and therefore the lattice, where quark masses can be varied, provides unique quantitative information. This information is essential for improving phenomenological \(\chi \)PT predictions in which these LECs play a role.

We stress that this program is based on the nonobvious assumption that \(\chi \)PT is valid in the region of masses and momenta used in the lattice simulations under consideration, something that can and should be checked. With the ability to create data at multiple values of the light-quark masses, lattice QCD offers the possibility to check the convergence of \(\chi \)PT. Lattice data may be used to verify that higher order contributions, for small enough quark masses, become increasingly unimportant. In the end one wants to compare lattice and phenomenological determinations of LECs, much in the spirit of Ref. [278]. An overview of many of the conceptual issues involved in matching lattice data to an effective field theory framework like \(\chi \)PT is given in Refs. [279,280,281].

The fact that, at large volume, the finite-size effects, which occur if a system undergoes spontaneous symmetry breakdown, are controlled by the Nambu-Goldstone modes, was first noted in solid state physics, in connection with magnetic systems [282, 283]. As pointed out in Ref. [284] in the context of QCD, the thermal properties of such systems can be studied in a systematic and model-independent manner by means of the corresponding effective field theory, provided the temperature is low enough. While finite volumes are not of physical interest in particle physics, lattice simulations are necessarily carried out in a finite box. As shown in Refs. [285,286,287], the ensuing finite-size effects can be studied on the basis of the effective theory – \(\chi \)PT in the case of QCD – provided the simulation is close enough to the continuum limit, the volume is sufficiently large and the explicit breaking of chiral symmetry generated by the quark masses is sufficiently small. Indeed, \(\chi \)PT represents a useful tool for the analysis of the finite-size effects in lattice simulations.

In the remainder of this section we collect the relevant \(\chi \)PT formulae that will be used in the two following sections to extract SU(2) and SU(3) LECs from lattice data.

Quark-mass dependence of pseudoscalar masses and decay constants

A. SU(2) formulae

The expansionsFootnote 24 of \(M_{\pi }^2\) and \(F_\pi \) in powers of the quark mass are known to next-to-next-to-leading order (NNLO) in the SU(2) chiral effective theory. In the isospin limit, \(m_u=m_d=m\), the explicit expressions may be written in the form [288]

$$\begin{aligned} M_{\pi }^2= & {} M^2\left\{ 1-\frac{1}{2}x\ln \frac{\Lambda _3^2}{M^2} +\frac{17}{8}x^2 \left( \ln \frac{\Lambda _M^2}{M^2} \right) ^2 \right. \nonumber \\&\quad \quad \quad \left. +x^2 k_M +{\mathcal {O}}(x^3) \right\} , \nonumber \\ F_\pi= & {} F\left\{ 1+x\ln \frac{\Lambda _4^2}{M^2} -\frac{5}{4}x^2 \left( \ln \frac{\Lambda _F^2}{M^2} \right) ^2 \right. \nonumber \\&\quad \quad \quad \left. +x^2 k_F +{\mathcal {O}}(x^3) \right\} . \end{aligned}$$
(88)

Here the expansion parameter is given by

$$\begin{aligned} x=\frac{M^2}{(4\pi F)^2},\quad M^2=2Bm=\frac{2\Sigma m}{F^2}, \end{aligned}$$
(89)

but there is another option as discussed below. The scales \(\Lambda _3,\Lambda _4\) are related to the effective coupling constants \({\bar{\ell }}_3,{\bar{\ell }}_4\) of the chiral Lagrangian at scale \(\mu =M_{\pi ,\mathrm {phys}}\) by

$$\begin{aligned} {\bar{\ell }}_n=\ln \frac{\Lambda _n^2}{M_{\pi ,\mathrm {phys}}^2},\quad n=1,...,7. \end{aligned}$$
(90)

Note that in Eq. (88) the logarithms are evaluated at \(M^2\), not at \(M_{\pi }^2\). The coupling constants \(k_M,k_F\) in Eq. (88) are mass-independent. The scales of the squared logarithms can be expressed in terms of the \({\mathcal {O}}(p^4)\) coupling constants as

$$\begin{aligned} \ln \frac{\Lambda _M^2}{M^2}= & {} \frac{1}{51}\left( 28\ln \frac{\Lambda _1^2}{M^2} +32\ln \frac{\Lambda _2^2}{M^2} -9\ln \frac{\Lambda _3^2}{M^2} +49 \right) , \nonumber \\ \ln \frac{\Lambda _F^2}{M^2}= & {} \frac{1}{30}\left( 14\ln \frac{\Lambda _1^2}{M^2} +16\ln \frac{\Lambda _2^2}{M^2}\nonumber \right. \\&\quad \quad \quad \left. +6\ln \frac{\Lambda _3^2}{M^2} - 6 \ln \frac{\Lambda _4^2}{M^2} +23 \right) . \end{aligned}$$
(91)

Hence by analysing the quark-mass dependence of \(M_{\pi }^2\) and \(F_\pi \) with Eq. (88), possibly truncated at NLO, one can determineFootnote 25 the \({\mathcal {O}}(p^2)\) LECs B and F, as well as the \({\mathcal {O}}(p^4)\) LECs \({{\bar{\ell }}}_3\) and \({{\bar{\ell }}}_4\). The quark condensate in the chiral limit is given by \(\Sigma =F^2B\). With precise enough data at several small enough pion masses, one could in principle also determine \(\Lambda _M\), \(\Lambda _F\) and \(k_M\), \(k_F\). To date this is not yet possible. The results for the LO and NLO constants will be presented in Sect. 5.2.

Alternatively, one can invert Eq. (88) and express \(M^2\) and F as an expansion in

$$\begin{aligned} \xi \equiv \frac{M_{\pi }^2}{16 \pi ^2 F_\pi ^2} \; \; , \end{aligned}$$
(92)

and the corresponding expressions then take the form

$$\begin{aligned} M^2= & {} M_{\pi }^2\,\left\{ 1+\frac{1}{2}\,\xi \,\ln \frac{\Lambda _3^2}{M_\pi ^2}- \frac{5}{8}\,\xi ^2 \left( \!\ln \frac{\Omega _M^2}{M_\pi ^2}\!\right) ^2\right. \nonumber \\&\quad \quad \quad \left. + \xi ^2 c_{\scriptscriptstyle M}+{\mathcal {O}}(\xi ^3)\right\} \,,\nonumber \\ F= & {} F_\pi \,\left\{ 1-\xi \,\ln \frac{\Lambda _4^2}{M_\pi ^2}-\frac{1}{4}\,\xi ^2 \left( \!\ln \frac{\Omega _F^2}{M_\pi ^2}\!\right) ^2 \right. \nonumber \\&\quad \quad \quad \left. +\xi ^2 c_{\scriptscriptstyle F}+{\mathcal {O}}(\xi ^3)\right\} \,.\end{aligned}$$
(93)

The scales of the quadratic logarithms are determined by \(\Lambda _1,\ldots ,\Lambda _4\) through

$$\begin{aligned} \ln \frac{\Omega _M^2}{M_\pi ^2}= & {} \frac{1}{15}\left( 28\,\ln \frac{\Lambda _1^2}{M_\pi ^2}+32\,\ln \frac{\Lambda _2^2}{M_\pi ^2}- 33\,\ln \frac{\Lambda _3^2}{M_\pi ^2}\nonumber \right. \\&\quad \quad \quad \left. -12\,\ln \frac{\Lambda _4^2}{M_\pi ^2}+52\right) \,,\nonumber \\ \ln \frac{\Omega _F^2}{M_\pi ^2}= & {} \frac{1}{3}\,\left( -7\,\ln \frac{\Lambda _1^2}{M_\pi ^2}-8\,\ln \frac{\Lambda _2^2}{M_\pi ^2}+ 18\,\ln \frac{\Lambda _4^2}{M_\pi ^2}- \frac{29}{2}\right) \,.\nonumber \\ \end{aligned}$$
(94)

In practice, many results are expressed in terms of the LO constants F and \(\Sigma \) and the NLO constants \({{\bar{\ell }}}_i\). The LO constants relate to the LO constants used above through \(B=\Sigma /F^2\). At the NLO the relation is a bit more involved, since the \({{\bar{\ell }}}_i\) bear the notion of the physical pion mass, see (90). For instance, Eqs. (93) may be rewritten as

$$\begin{aligned} M^2= & {} M_{\pi }^2\,\left\{ 1+\frac{1}{2}\,\xi \,{{\bar{\ell }}}_3+\frac{1}{2}\,\xi \ln \frac{M_{\pi ,\mathrm {phys}}^2}{M_{\pi }^2}\nonumber \right. \\&\quad \quad \quad \left. - \frac{5}{8}\,\xi ^2 \left( \!\ln \frac{\Omega _M^2}{M_\pi ^2}\!\right) ^2 + \xi ^2 c_{\scriptscriptstyle M}+{\mathcal {O}}(\xi ^3)\right\} \,,\nonumber \\ F= & {} F_\pi \,\left\{ 1-\xi \,{{\bar{\ell }}}_4-\xi \,\ln \frac{M_{\pi ,\mathrm {phys}}^2}{M_{\pi }^2}\nonumber \right. \\&\quad \quad \quad \left. -\frac{1}{4}\,\xi ^2 \left( \!\ln \frac{\Omega _F^2}{M_\pi ^2}\!\right) ^2 +\xi ^2 c_{\scriptscriptstyle F}+{\mathcal {O}}(\xi ^3)\right\} \,,\end{aligned}$$
(95)

and this implies that fitting some lattice data (say at a single lattice spacing a) with Eq. (95) requires some a-priori knowledge of the lattice spacing. On the other hand, doing the same job with Eq. (93) yields the scales \(a\Lambda _3, a\Lambda _4\) in lattice units (which may be converted to \({{\bar{\ell }}}_3,{{\bar{\ell }}}_4\) at a later stage of the analysis when the scale is known more precisely).

B. SU(3) formulae

While the formulae for the pseudoscalar masses and decay constants are known to NNLO for SU(3) as well [289], they are rather complicated and we restrict ourselves here to next-to-leading order (NLO). In the isospin limit, the relevant SU(3) formulae take the form [244]

(96)

where \(m_{ud}\) is the joint up/down quark mass in the simulation [which may be taken different from the average light-quark mass \(\frac{1}{2}(m_u^\mathrm {phys}+m_d^\mathrm {phys})\) in the real world]. And \(B_0=\Sigma _0/F_0^2\), \(F_0\) denote the condensate parameter and the pseudoscalar decay constant in the SU(3) chiral limit, respectively. In addition, we use the notation

$$\begin{aligned} \mu _P=\frac{M_P^2}{32\pi ^2F_0^2} \ln \!\left( \frac{M_P^2}{\mu ^2}\right) \;. \end{aligned}$$
(97)

At the order of the chiral expansion used in these formulae, the quantities \(\mu _\pi \), \(\mu _K\), \(\mu _\eta \) can equally well be evaluated with the leading-order expressions for the masses,

(98)

Throughout, \(L_i\) denotes the renormalized low-energy constant/coupling (LEC) at scale \(\mu \), and we adopt the convention that is standard in phenomenology, \(\mu =M_\rho =770\,\mathrm {MeV}\). The normalization used for the decay constants is specified in footnote 24.

Pion form factors and charge radii

The scalar and vector form factors of the pion are defined by the matrix elements

$$\begin{aligned} \begin{aligned}&\langle \pi ^i(p_2) |\, {\bar{q}}\, q \, | \pi ^k(p_1) \rangle = \delta ^{ik} F_S^\pi (t) \,,\\&\langle \pi ^i(p_2) | \,{\bar{q}}\, {\frac{1}{2}}\tau ^j \gamma ^\mu q\,| \pi ^k(p_1) \rangle = \mathrm {i} \,\epsilon ^{ijk} (p_1^\mu + p_2^\mu ) F_V^\pi (t) \,,\end{aligned} \end{aligned}$$
(99)

where the operators contain only the lightest two quark flavours, i.e., \(\tau ^1\), \(\tau ^2\), \(\tau ^3\) are the Pauli matrices, and \(t\equiv (p_1-p_2)^2\) denotes the momentum transfer.

The vector form factor has been measured by several experiments for time-like as well as for space-like values of t. The scalar form factor is not directly measurable, but it can be evaluated theoretically from data on the \(\pi \pi \) and \(\pi K\) phase shifts [290] by means of analyticity and unitarity, i.e., in a model-independent way. Lattice calculations can be compared with data or model-independent theoretical evaluations at any given value of t. At present, however, most lattice studies concentrate on the region close to \(t=0\) and on the evaluation of the slope and curvature, which are defined as

$$\begin{aligned} \begin{aligned} F^\pi _V(t)&= 1+{\frac{1}{6}}\langle r^2 \rangle ^\pi _V t + c_V t^2\,+\,\cdots , \\ F^\pi _S(t)&= F^\pi _S(0) \left[ 1+{\frac{1}{6}}\langle r^2 \rangle ^\pi _S t + c_S\, t^2\,+\, \cdots \right] . \end{aligned} \end{aligned}$$
(100)

The slopes are related to the mean-square vector and scalar radii, which are the quantities on which most experiments and lattice calculations concentrate.

In \(\chi \)PT, the form factors are known at NNLO for SU(2) [291]. The corresponding formulae are available in fully analytical form and are compact enough that they can be used for the chiral extrapolation of the data (as done, for example, in Refs. [53, 292]). The expressions for the scalar and vector radii and for the \(c_{S,V}\) coefficients at 2-loop level in SU(2) terminology read

$$\begin{aligned} \langle r^2 \rangle ^\pi _S= & {} \frac{1}{(4\pi F_\pi )^2} \left\{ 6 \ln \frac{\Lambda _4^2}{M_\pi ^2}-\frac{13}{2} -\frac{29}{3}\,\xi \left( \!\ln \frac{\Omega _{r_S}^2}{M_{\pi }^2} \!\right) ^2\right. \nonumber \\&\left. \quad \quad \quad \quad \quad + 6 \xi \, k_{r_S}+{\mathcal {O}}(\xi ^2)\right\} \,,\nonumber \\ \langle r^2 \rangle ^\pi _V= & {} \frac{1}{(4\pi F_\pi )^2} \left\{ \ln \frac{\Lambda _6^2}{M_\pi ^2}-1 +2\,\xi \left( \!\ln \frac{\Omega _{r_V}^2}{M_{\pi }^2} \!\right) ^2\right. \nonumber \\&\left. \quad \quad \quad \quad \quad +6 \xi \,k_{r_V}+{\mathcal {O}}(\xi ^2)\right\} \,,\nonumber \\ c_S= & {} \frac{1}{(4\pi F_\pi M_{\pi })^2} \left\{ \frac{19}{120} + \xi \left[ \frac{43}{36} \left( \! \ln \frac{\Omega _{c_S}^2}{M_{\pi }^2} \!\right) ^2 + k_{c_S} \right] \right\} \,,\nonumber \\ c_V= & {} \frac{1}{(4\pi F_\pi M_{\pi })^2} \left\{ \frac{1}{60}+\xi \left[ \frac{1}{72} \left( \! \ln \frac{\Omega _{c_V}^2}{M_{\pi }^2} \!\right) ^2 + k_{c_V} \right] \right\} \,,\nonumber \\ \end{aligned}$$
(101)

where

$$\begin{aligned} \ln \frac{\Omega _{r_S}^2}{M_{\pi }^2}= & {} \frac{1}{29}\,\left( 31\,\ln \frac{\Lambda _1^2}{M_\pi ^2}+34\,\ln \frac{\Lambda _2^2}{M_\pi ^2}-36\,\ln \frac{\Lambda _4^2}{M_\pi ^2}+\frac{145}{24}\right) \,,\nonumber \\ \ln \frac{\Omega _{r_V}^2}{M_{\pi }^2}= & {} \frac{1}{2}\,\left( \ln \frac{\Lambda _1^2}{M_\pi ^2}-\ln \frac{\Lambda _2^2}{M_\pi ^2}+\ln \frac{\Lambda _4^2}{M_\pi ^2}+\ln \frac{\Lambda _6^2}{M_\pi ^2}-\frac{31}{12}\right) \,,\nonumber \\ \ln \frac{\Omega _{c_S}^2}{M_{\pi }^2}= & {} \frac{43}{63}\,\left( 11\,\ln \frac{\Lambda _1^2}{M_\pi ^2}+14\,\ln \frac{\Lambda _2^2}{M_\pi ^2}+18\,\ln \frac{\Lambda _4^2}{M_\pi ^2}-\frac{6041}{120}\right) \,,\nonumber \\ \ln \frac{\Omega _{c_V}^2}{M_{\pi }^2}= & {} \frac{1}{72}\,\left( 2\ln \frac{\Lambda _1^2}{M_\pi ^2}-2\ln \frac{\Lambda _2^2}{M_\pi ^2}-\ln \frac{\Lambda _6^2}{M_\pi ^2}-\frac{26}{30}\right) \,,\nonumber \\ \end{aligned}$$
(102)

and \(k_{r_S},k_{r_V}\) and \(k_{c_S},k_{c_V}\) are independent of the quark masses. Their expression in terms of the \(\ell _i\) and of the \({\mathcal {O}}(p^6)\) constants \(c_M,c_F\) is known but will not be reproduced here.

The SU(3) formula for the slope of the pion vector form factor reads, to NLO [242],

(103)

while the expression \(\langle r^2\rangle _S^\mathrm {oct}\) for the octet part of the scalar radius does not contain any NLO low-energy constant at 1-loop order [242] – contrary to the situation in SU(2), see Eq. (101).

The difference between the quark-line connected and the full (i.e., containing the connected and the disconnected pieces) scalar pion form factor has been investigated by means of \(\chi \)PT in Ref. [293]. It is expected that the technique used can be applied to a large class of observables relevant in QCD phenomenology.

As a point of practical interest let us remark that there are no finite-volume correction formulae for the mean-square radii \(\langle r^2\rangle _{V,S}\) and the curvatures \(c_{V,S}\). The lattice data for \(F_{V,S}(t)\) need to be corrected, point by point in t, for finite-volume effects. In fact, if a given \(\sqrt{t}\) is realized through several inequivalent \(p_1\!-\!p_2\) combinations, the level of agreement after the correction has been applied is indicative of how well higher-order and finite-volume effects are under control.

Goldstone boson scattering in a finite volume

The scattering of pseudoscalar octet mesons off each other (mostly \(\pi \)\(\pi \) and \(\pi \)K scattering) is a useful approach to determine \(\chi \)PT low-energy constants [288, 294,295,296,297]. This statement holds true both in experiment and on the lattice. We would like to point out that the main difference between these approaches is not so much the discretization of space-time, but rather the Minkowskian versus Euclidean setup.

In infinite-volume Minkowski space-time, 4-point Green’s functions can be evaluated (e.g., in experiment) for a continuous range of (on-shell) momenta, as captured, for instance, by the Mandelstam variable s. For a given isospin channel \(I=0\) or \(I=2\) the \(\pi \)\(\pi \) scattering phase shift \(\delta ^{I}(s)\) can be determined for a variety of s values, and by matching to \(\chi \)PT some low-energy constants can be determined (see below). In infinite-volume Euclidean space-time, such 4-point Green’s functions can only be evaluated at kinematic thresholds; this is the content of the so-called Maiani-Testa theorem [298]. However, in the Euclidean case, the finite volume comes to our rescue, as first pointed out by Lüscher [299,300,301,302]. By comparing the energy of the (interacting) two-pion system in a box with finite spatial extent L to twice the energy of a pion (with identical bare parameters) in infinite volume information on the scattering length can be obtained. In particular in the (somewhat idealized) situation where one can “scan” through a narrowly spaced set of box-sizes L such information can be reconstructed in an efficient way.

We begin with a brief summary of the relevant formulae from \(\chi \)PT in SU(2) terminology. In the x-expansion the formulae for \(a_\ell ^I\) with \(\ell =0\) and \(I=0,2\) are found in Ref. [277]

$$\begin{aligned} a_0^0M_{\pi }= & {} +\frac{7M^2}{32\pi F^2} \bigg \{ 1+\frac{5M^2}{84\pi ^2 F^2}\nonumber \\&\times \left[ {{\bar{\ell }}}_1+2{{\bar{\ell }}}_2-\frac{9}{10}{{\bar{\ell }}}_3 +\frac{21}{8}\right] +{\mathcal {O}}(x^2) \bigg \} \;, \end{aligned}$$
(104)
$$\begin{aligned} a_0^2M_{\pi }= & {} -\frac{ M^2}{16\pi F^2} \bigg \{ 1-\frac{ M^2}{12\pi ^2 F^2}\left[ {{\bar{\ell }}}_1+2{{\bar{\ell }}}_2\,+\,\frac{ 3}{8}\right] \nonumber \\&\quad \quad \quad \quad \quad +{\mathcal {O}}(x^2) \bigg \} \;, \end{aligned}$$
(105)

where we deviate from the \(\chi \)PT habit of absorbing a factor \(-M_{\pi }\) into the scattering length (relative to the convention used in quantum mechanics), since we include just a minus sign but not the factor \(M_{\pi }\). Hence, our \(a_\ell ^I\) have the dimension of a length so that all quark- or pion-mass dependence is explicit (as is most convenient for the lattice community). But the sign convention is the one of the chiral community (where \(a_\ell ^IM_{\pi }>0\) means attraction and \(a_\ell ^IM_{\pi }<0\) means repulsion).

An important difference between the two scattering lengths is evident already at tree-level. The isospin-0 S-wave scattering length (104) is large and positive, while the isospin-2 counterpart (105) is by a factor \(\sim 3.5\) smaller (in absolute magnitude) and negative. Hence, in the channel with \(I=0\) the interaction is attractive, while in the channel with \(I=2\) the interaction is repulsive and significantly weaker. In this convention experimental results, evaluated with the unitarity constraint genuine to any local quantum field theory, read \(a_0^0M_{\pi }=0.2198(46)_\mathrm {stat}(16)_\mathrm {syst}(64)_\mathrm {theo}\) and \(a_0^2M_{\pi }=-0.0445(11)_\mathrm {stat}(4)_\mathrm {syst}(8)_\mathrm {theo}\) [288, 303,304,305]. The ratio between the two (absolute) central values is larger than 3.5, and this suggests that NLO contributions to \(a_0^0\) might be more relevant than NLO contributions to \(a_0^2\).

By means of \(M^2/(4\pi F)^2=M_{\pi }^2/(4\pi F_\pi )^2\{1+\frac{1}{2}\xi \ln (\Lambda _3^2/M_{\pi }^2)+2\xi \ln (\Lambda _4^2/M_{\pi }^2)+{\mathcal {O}}(\xi ^2)\}\) or equivalently through \(M^2/(4\pi F)^2=M_{\pi }^2/(4\pi F_\pi )^2\{1+\frac{1}{2}\xi {{\bar{\ell }}}_3+2\xi {{\bar{\ell }}}_4+{\mathcal {O}}(\xi ^2)\}\) Eqs. (104, 105) may be brought into the form

$$\begin{aligned} a_0^0M_{\pi }= & {} +\frac{7M_{\pi }^2}{32\pi F_\pi ^2} \bigg \{ 1 +\xi \frac{1}{2}{{\bar{\ell }}}_3 +\xi 2{{\bar{\ell }}}_4 \nonumber \\&+\,\xi \left[ \frac{20}{21}{{\bar{\ell }}}_1+\frac{40}{21}{{\bar{\ell }}}_2-\frac{18}{21}{{\bar{\ell }}}_3 +\frac{ 5}{ 2}\right] +{\mathcal {O}}(\xi ^2) \bigg \} \;, \end{aligned}$$
(106)
$$\begin{aligned} a_0^2M_{\pi }= & {} -\frac{ M_{\pi }^2}{16\pi F_\pi ^2} \bigg \{ 1 +\xi \frac{1}{2}{{\bar{\ell }}}_3 +\xi 2{{\bar{\ell }}}_4 \nonumber \\&-\,\xi \left[ \frac{ 4}{ 3}{{\bar{\ell }}}_1+\frac{ 8}{ 3}{{\bar{\ell }}}_2 +\frac{ 1}{ 2}\right] +{\mathcal {O}}(\xi ^2) \bigg \} \;. \end{aligned}$$
(107)

Finally, this expression can be summarized as

$$\begin{aligned} a_0^0M_{\pi }= & {} +\frac{7M_{\pi }^2}{32\pi F_\pi ^2} \bigg \{ 1+\frac{9M_{\pi }^2}{32\pi ^2F_\pi ^2}\ln \frac{(\lambda _0^0)^2}{M_{\pi }^2}+{\mathcal {O}}(\xi ^2) \bigg \} \;, \nonumber \\ \end{aligned}$$
(108)
$$\begin{aligned} a_0^2M_{\pi }= & {} -\frac{ M_{\pi }^2}{16\pi F_\pi ^2} \bigg \{ 1-\frac{3M_{\pi }^2}{32\pi ^2F_\pi ^2}\ln \frac{(\lambda _0^2)^2}{M_{\pi }^2}+{\mathcal {O}}(\xi ^2) \bigg \} \;, \nonumber \\ \end{aligned}$$
(109)

with the abbreviations

$$\begin{aligned} \frac{9}{2}\ln \frac{\left( \lambda _0^0\right) ^2}{M_{\pi ,\mathrm {phys}}^2}= & {} \frac{20}{21}{{\bar{\ell }}}_1 +\frac{40}{21}{{\bar{\ell }}}_2 -\frac{5}{14}{{\bar{\ell }}}_3 +2{{\bar{\ell }}}_4 +\frac{5}{2} \;, \end{aligned}$$
(110)
$$\begin{aligned} \frac{3}{2}\ln \frac{\left( \lambda _0^2\right) ^2}{M_{\pi ,\mathrm {phys}}^2}= & {} \frac{ 4}{ 3}{{\bar{\ell }}}_1 +\frac{ 8}{ 3}{{\bar{\ell }}}_2 -\frac{1}{ 2}{{\bar{\ell }}}_3 -2{{\bar{\ell }}}_4 +\frac{1}{2} \;, \end{aligned}$$
(111)

where \(\lambda _\ell ^I\) with \(\ell =0\) and \(I=0,2\) are scales like the \(\Lambda _i\) in \({{\bar{\ell }}}_i=\ln (\Lambda _i^2/M_{\pi ,\mathrm {phys}}^2)\) for \(i\in \{1,2,3,4\}\) (albeit they are not independent from the latter). Here we made use of the fact that \(M_{\pi }^2/M_{\pi ,\mathrm {phys}}^2=1+{\mathcal {O}}(\xi )\) and thus \(\xi \ln (M_{\pi }^2/M_{\pi ,\mathrm {phys}}^2)={\mathcal {O}}(\xi ^2)\). In the absence of any knowledge on the \({{\bar{\ell }}}_i\) one would assume \(\lambda _0^0\simeq \lambda _0^2\), and with this input Eqs. (108, 109) suggest that the NLO contribution to \(|a_0^0|\) is by a factor \(\sim 9\) larger than the NLO contribution to \(|a_0^2|\). The experimental numbers quoted before clearly support this view.

Given that all of this sounds like a complete success story for the determination of the scattering lengths \(a_0^0\) and \(a_0^2\), one may wonder whether lattice QCD is helpful at all. It is, because the “experimental” evaluation of these scattering lengths builds on a constraint between these two quantities that, in turn, is based on a (rather nontrivial) dispersive evaluation of scattering phase shifts [288, 303,304,305]. Hence, to overcome this possible loophole, an independent lattice determination of \(a_0^0\) and/or \(a_0^2\) is highly welcome.

On the lattice \(a_0^2\) is much easier to determine than \(a_0^0\), since the former quantity does not involve quark-line disconnected contributions. The main upshot of such activities (to be reviewed below) is that the lattice determination of \(a_0^2M_{\pi }\) at the physical mass point is in perfect agreement with the experimental numbers quoted before, thus supporting the view that the scalar condensate is – at least in the SU(2) case – the dominant order parameter, and the original estimate \({{\bar{\ell }}}_3=2.9\pm 2.4\) is correct (see below). Still, from a lattice perspective it is natural to see a determination of \(a_0^0M_{\pi }\) and/or \(a_0^2M_{\pi }\) as a means to access the specific linear combinations of \({{\bar{\ell }}}_i\) with \(i\in \{1,2,3,4\}\) defined in Eqs. (110, 111).

In passing we note that an alternative version of Eqs. (108, 109) is used in the literature, too. For instance Refs. [306,307,308,309,310] give their results in the form

$$\begin{aligned} a_0^0M_{\pi }= & {} +\frac{7M_{\pi }^2}{32\pi F_\pi ^2} \bigg \{ 1+\frac{M_{\pi }^2}{32\pi ^2F_\pi ^2}\left[ \ell ^{I=0}_{\pi \pi }+5-9\ln \frac{M_{\pi }^2}{2F_\pi ^2}\right] \nonumber \\&\quad \quad \quad \quad \quad +\,{\mathcal {O}}(\xi ^2) \bigg \} \;, \end{aligned}$$
(112)
$$\begin{aligned} a_0^2M_{\pi }= & {} -\frac{M_{\pi }^2}{16\pi F_\pi ^2} \bigg \{ 1-\frac{M_{\pi }^2}{32\pi ^2F_\pi ^2}\left[ \ell ^{I=2}_{\pi \pi }+1-3\ln \frac{M_{\pi }^2}{2F_\pi ^2}\right] \nonumber \\&\quad \quad \quad \qua