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FLAG Review 2021

Note that the latest version of the FLAG review from 2021 is available here.

Please note that is is mandatory to cite the original sources when citing a result from FLAG. We facilitate this by providing corresponding bibtex files for each FLAG value, see our paragraph on citation policy.

The latest version of the FLAG 21 review as of February 2024 is accessible here. It contains an update of the following section:

$B$-meson decay constants, mixing parameters, and form factors: updated February 2024

The version of the FLAG 21 review as of February 2023 is accessible here. It contains updated sections as follows:

Quark masses: updated February 2023
$|V_{ud}|$ and $|V_{us}|$: updated February 2023
$B$-meson decay constants, mixing parameters, and form factors: updated February 2023
Scale setting: updated February 2023

The version of the FLAG review as of November 2021 is accessible here and corresponds to the one accepted for publication in EPJC. The EPJC version can be obtained from here.

The separate sections can be downloaded as separate pdf-files following the links in the table of contents below, or via the menu in the sidebar. Clicking on the FLAG logo in the upper left corner links back to this main page.

The latest figures can be downloaded in pdf and png format, together with a bib-file containing the bibtex-entries for the calculations which contribute to the FLAG averages and estimates. The downloads are available via the menu in the sidebar.

In the notes we compile detailed information on the simulations used to calculate the quantities discussed in the review. In the current report we present this information only for results that are new w.r.t. FLAG 19. For all other results the information is available in the corresponding Appendices of the previous reports.

The previous version of the FLAG review was updated as of December 2020 and is accessible at flag.unibe.ch/2019 where also the original 2019 edition of the FLAG review can be downloaded. The original complete 2015/2016 review is still accessible from EPJC. The 2016/2017 updates are available from here. The 2013/2014 review is accessible here or from EPJC.

Contents

Introduction
Quality criteria
Quark masses
$\small{V_{ud}}$ and $\small{V_{us}}$
Low-energy constants
Kaon mixing
$\small{D}$-meson decay constants and form factors
$\small{B}$-meson decay constants, mixing parameters, and form factors
The strong coupling $\alpha_s$
Nucleon matrix elements
Scale setting
Appendix
Notes

Inquiries and suggestions can be sent to the FLAG Editorial Board.

Introduction

The introduction with the summary tables can be downloaded here.

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

FLAG composition, guidelines and rules

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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 role of the Advisory Board is to provide oversight of the content, procedures, schedule and membership of FLAG, to help resolve disputes, to serve as a source of advice to the EB and to FLAG as a whole, and to provide a critical assessment of drafts. They also give their approval of the final version of the preprint before it is rendered public. The Editorial Board coordinates the activities of FLAG, sets priorities and intermediate deadlines, organizes votes on FLAG procedures, writes the introductory sections, and takes care of the editorial work needed to amalgamate the sections written by the individual working groups into a uniform and coherent review. The working groups concentrate on writing the review of the physical quantities for which they are responsible, which is subsequently circulated to the whole collaboration for critical evaluation.

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

Advisory Board (AB)		G. Colangelo, M. Golterman, P. Hernandez, T. Onogi, R. Van de Water
Editorial Board (EB)		S. Gottlieb, A. Jüttner, S. Hashimoto, S. Sharpe, U. Wenger
Working Groups (coordinator listed first)
Quark masses		T. Blum, A. Portelli, A. Ramos
$V_{us}, V_{ud}$		T. Kaneko, J.N. Simone, S. Simula, N. Tantalo
LEC		S. Dürr, H. Fukaya, U.M. Heller
$B_K$		P. Dimopoulos, X. Feng, G. Herdoiza
$f_{B_{(s)}}$, $f_{D_{(s)}}$, $B_B$		Y. Aoki, M. Della Morte, C. Monahan
$B_{(s)}$, $D$ semileptonic and radiative decays		E. Lunghi, S. Meinel, C. Pena
$\alpha_s$		S. Sint, R. Horsley, P. Petreczky
Nucleon Matrix Elements		R. Gupta, S. Collins, A. Nicholson, H. Wittig
Scale setting		R. Sommer, N. Tantalo, U. Wenger

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 members must belong to different lattice collaborations;
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.

As for FLAG 19, for this 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.

FLAG composition for the FLAG 2024 edition

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As usual, after the completion of the FLAG 2021 edition the composition of FLAG has partially been restructured due to FLAG members stepping back or taking on different roles within FLAG. The work towards the FLAG 2024 edition, as well as the work for the webupdates, is currently undertaken with the following composition of FLAG:

Advisory Board (AB)		M. Golterman, P. Hernandez, T. Onogi, S. Sharpe, R. Van de Water
Editorial Board (EB)		S. Gottlieb, A. Jüttner, U. Wenger
Working Groups (coordinator underlined)
Quark masses		T. Blum, A. Portelli, A. Ramos
$V_{us}, V_{ud}$		T. Kaneko, J.N. Simone, N. Tantalo
$B_K$		P. Dimopoulos, X. Feng, G. Herdoiza
$f_{B_{(s)}}$, $f_{D_{(s)}}$, $B_B$		Y. Aoki, M. Della Morte, C. Monahan
$B_{(s)}$, $D$ semileptonic and radiative decays		E. Lunghi, S. Meinel, S. Simula, A. Vaquero Avilès-Casco
$\alpha_s$		L. Del Debbio, P. Petreczky, S. Sint
Nucleon Matrix Elements		S. Collins, R. Gupta, A. Nicholson, H. Wittig
Scale setting		R. Sommer, N. Tantalo, U. Wenger

We note that the LEC section is no longer maintained.

Citation policy

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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, 3, 4, and 5, besides summarizing the main results of the present review, we also provide here a bibtex file containing the original references from which the results have been obtained. In addition, for each figure, we make a corresponding bibtex file available 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.

Quality criteria

The section on the quality criteria can be downloaded here.

Quark masses

The section on the quark masses can be downloaded here.

$\small{V_{ud}}$ and $\small{V_{us}}$

The section on $V_{ud}$ and $V_{us}$ can be downloaded here.

Low-energy constants

The section on the low-energy constants can be downloaded here.

Kaon mixing

The section on the kaon mixing can be downloaded here.

$\small{D}$-meson decay constants and form factors

The section on the $D$-meson decay constants and form factors can be downloaded here.

$\small{B}$-meson decay constants, mixing parameters, and form factors

The section on the $B$-meson decay constants, mixing parameters, and form factors can be downloaded here.

The strong coupling $\alpha_s$

The section on the strong coupling $\alpha_s$ can be downloaded here.

Nucleon matrix elements

The section on nucleon matrix elements can be downloaded here.

Scale setting

The section on scale setting can be downloaded here.

Appendix

The Appendix can be downloaded here.

Notes

Notes to the various sections can be downloaded here in reduced form (containing notes to results that are new w.r.t. FLAG 19).

This page has been accessed 179322 times.

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-== Review of lattice results concerning low energy particle physics ==
+== FLAG Review 2021 ==
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-The latest version of the complete review as of January 2017 is accessible  [[attachment:Media/FLAG_webupdate.pdf|here|&do=get]]. It contains the new section updated in November 2016 on leptonic and semileptonic kaon and pion decay and $|Vud|$ and $|Vus|$, and the new section updated in December 2016 on kaon mixing.
+||<(#ffbbbb tablestyle="width: 100%;"> <<BR>> '''&nbsp; Note that the latest version of the FLAG review from 2021 is available [[http://flag.unibe.ch/|here]].''' <<BR>>&nbsp; ||
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-The original complete 2015/2016 review is still accessible [[attachment:Media/FLAG.pdf|here|&do=get]] or from [[http://link.springer.com/article/10.1140%2Fepjc%2Fs10052-016-4509-7| EPJC]]. The separate sections can be downloaded as separate pdf-files following the links in the table of contents below.
+{{{#!wiki caution
<<BR>> '''&nbsp;Please note that is is mandatory to cite the original sources when citing a result from FLAG. We facilitate this by providing corresponding bibtex files for each FLAG value, see our paragraph on [[MainPage#citation_policy|citation policy]].''' <<BR>>&nbsp;
}}}
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-The latest figures can be downloaded in eps, pdf and png format, together with a bib-file containing the bibtex-entries for the calculations which contribute to the FLAG averages and estimates. The downloads are available via the menu in the sidebar.
+The latest version of the FLAG 21 review as of February 2024 is accessible  [[attachment:Media/FLAG_webupdate_February2024.pdf|here|&do=get]]. It contains an update of the following section:
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-The 2013/2014 review is accessible [[attachment:Media/FLAG_2013.pdf|here|&do=get]] or from [[http://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2890-7|EPJC]].
+ * $B$-meson decay constants, mixing parameters, and form factors:     updated February 2024

The version of the FLAG 21 review as of February 2023 is accessible  [[attachment:Media/FLAG_2023_webupdate.pdf|here|&do=get]]. It contains updated sections as follows:

 * Quark masses:                                                       updated February 2023
 * $|V_{ud}|$ and $|V_{us}|$:                                          updated February 2023
 * $B$-meson decay constants, mixing parameters, and form factors:     updated February 2023
 * Scale setting:                                                      updated February 2023

The version of the FLAG review as of November 2021 is accessible  [[attachment:Media/FLAG_2021.pdf|here|&do=get]] and corresponds to the one accepted for publication in EPJC. The EPJC version can be obtained from [[https://doi.org/10.1140/epjc/s10052-022-10536-1|here|]].

The separate sections can be downloaded as separate pdf-files following the links in the table of contents below, or via the menu in the sidebar. Clicking on the FLAG logo in the upper left corner links back to this main page.

The latest figures can be downloaded in pdf and png format, together with a bib-file containing the bibtex-entries for the calculations which contribute to the FLAG averages and estimates. The downloads are available via the menu in the sidebar.
 
In the notes we  compile detailed information on the simulations used to calculate the quantities discussed in the review. In the current report we present this information only for results that are new w.r.t. FLAG 19. For all other results the information is available in the corresponding Appendices of the previous reports.

The previous version of the FLAG review was updated as of December 2020 and is accessible at [[http://flag.unibe.ch/2019/|flag.unibe.ch/2019|]] where also the original 2019 edition of the FLAG review can be downloaded.
The original complete 2015/2016 review is still accessible from [[http://link.springer.com/article/10.1140%2Fepjc%2Fs10052-016-4509-7| EPJC]]. The 2016/2017 updates are available from  [[http://flag.unibe.ch/2016|here]]. The 2013/2014 review is accessible [[attachment:Media/FLAG_2013.pdf|here|&do=get]] or from [[http://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2890-7|EPJC]].
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+Inquiries and suggestions can be sent to the [[mailto:flag-eb@itp.unibe.ch|FLAG Editorial Board]].
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-The introduction can be downloaded [[attachment:Media/FLAG_introduction.pdf|here]].
+The introduction with the summary tables can be downloaded [[attachment:Media/FLAG_introduction.pdf|here|&do=get]].
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-We review lattice results related to pion, kaon, $D$- and $B$-meson physics with the aim of making them easily accessible to the particle physics community. More specifically, we report on the determination of the light-quark masses, the form factor $f_+(0)$, arising in the semileptonic $K$→$\pi$ transition at zero momentum transfer, as well as the decay constant ratio $f_K/f_\pi$ 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)$_L$×SU(2)$_R$ and SU(3)$_L$×SU(3)$_R$ Chiral Perturbation Theory. We review the determination of the $B_K$ parameter of neutral kaon mixing as well as the additional four $B$ parameters that arise in theories of physics beyond the Standard Model. The latter quantities are an addition compared to the previous review. For the heavy-quark sector, we provide results for $m_c$ and $m_b$ (also new compared to the previous review), as well as those for $D$- and $B$-meson decay constants, form factors, and mixing parameters. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. Finally, we review the status of lattice determinations of the strong coupling constant $\alpha_s$.
+We review lattice results related to pion, kaon, $D$-meson, $B$-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $f_+(0)$ arising in the semileptonic $K \to \pi$ transition at zero momentum transfer, as well as the decay constant ratio $f_K/f_\pi$ and its   consequences for the CKM matrix elements $V_{us}$ and $V_{ud}$. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $SU(2)_L\times SU(2)_R$ and $SU(3)_L\times SU(3)_R$ Chiral Perturbation Theory. We review the determination of the $B_K$ parameter of neutral kaon mixing as well as the additional four $B$ parameters that arise in theories of physics beyond the Standard Model.         For the heavy-quark sector, we provide results for $m_c$ and $m_b$ as well as those for the decay constants, form factors, and mixing parameters of charmed and bottom mesons and baryons. 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 $\alpha_s$. We consider nucleon matrix elements, and review the determinations of the axial,  scalar and tensor bilinears, both isovector and flavor diagonal. Finally, in this review we have added a new section reviewing determinations of scale-setting quantities.
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-Flavour physics provides an important opportunity for exploring the limits of the Standard Model of particle physics and for constraining possible extensions that go beyond it. As the LHC explores a new energy frontier and as experiments continue to extend the precision frontier, the importance of flavour physics will grow, both in terms of searches for signatures of new physics through precision measurements and in terms of attempts to construct the theoretical framework behind direct discoveries of new particles. A major theoretical limitation consists in the precision with which strong-interaction effects can be quantified. Large-scale numerical simulations of lattice QCD allow for the computation of these effects from first principles. The scope of the Flavour Lattice Averaging Group (FLAG) is to review the current status of lattice results for a variety of physical quantities in low-energy physics. Set up in November 2007 it comprises experts in Lattice Field Theory, Chiral Perturbation Theory and Standard Model phenomenology. Our aim is to provide an answer to the frequently posed question “What is currently the best lattice value for a particular quantity?” in a way that is readily accessible to nonlattice-experts. This is generally not an easy question to answer; different collaborations use different lattice actions (discretizations of QCD) with a variety of lattice spacings and volumes, and with a range of masses for the $u$− and $d$−quarks. Not only are the systematic errors different, but also the methodology used to estimate these uncertainties varies between collaborations. In the present work we summarize the main features of each of the calculations and provide a framework for judging and combining the different results. Sometimes it is a single result that provides the “best” value; more often it is a combination of results from different collaborations. Indeed, the consistency of values obtained using different formulations adds significantly to our confidence in the results.
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-The first two editions of the FLAG review were published in 2011 <<FootNote(G. Colangelo et al., ''Review of lattice results concerning low energy particle physics'', Eur. Phys. J. C71 (2011) 1695, [[http://arxiv.org/abs/1011.4408|arXiv:1011.4408]])>>. and 2014 <<FootNote(S. Aoki et al., ''Review of lattice results concerning low-energy particle physics'', Eur. Phys. J. C74 (2014) 2890, [[http://arxiv.org/abs/1310.8555|arXiv:1310.8555]].)>>. The second edition reviewed results related to both light ($u$-, $d$- and $s$-), and heavy ($c$- and $b$-) flavours. The quantities related to pion and kaon physics were light-quark masses, the form factor $f_+(0)$ arising in semileptonic $K$→$\pi$ transitions (evaluated at zero momentum transfer), the decay constants $f_K$ and $f_\pi$, and the $B_K$ parameter from neutral kaon mixing. Their implications for the CKM matrix elements $V_{us}$ and $V_{ud}$ were also discussed. Furthermore, results were reported for some of the low-energy constants of SU(2)$_L$×SU(2)$_R$ and SU(3)$_L$×SU(3)$_R$ Chiral Perturbation Theory. The quantities related to $D$- and $B$-meson physics that were reviewed were the $B$- and $D$-meson decay constants, form factors, and mixing parameters. These are the heavy-light quantities most relevant to the determination of CKM matrix elements and the global CKM unitarity-triangle fit. Last but not least, the current status of lattice results on the QCD coupling $\alpha_s$ was reviewed.
+==== FLAG composition, guidelines and rules ====
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-In the present paper we provide updated results for all the above-mentioned quantities, but also extend the scope of the review in two ways. First, we now present results for the charm and bottom quark masses, in addition to those of the three lightest quarks. Second, we review results obtained for the kaon mixing matrix elements of new operators that arise in theories of physics beyond the Standard Model. Our main results are collected in Tabs. 1 and 2.
+<<BetterSeeSaw(section="flag_composition", toshow="Show Text", tohide="Hide Text")>>
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{{{#!wiki seesaw table1
+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.
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-||<-6(#eeeeee tablestyle="width: 70%;">'''Table 1'''||
||<-6(>Summary of the main results of this review, grouped in terms of $N_f$, the number of dynamical quark flavours in lattice simulations. Quark masses and the quark condensate are given in the $\overline{\text{MS}}$ scheme at running scale $\mu = 2$ GeV or as indicated; the other quantities listed are specified in the quoted sections. For each result we provide the list of references that entered the FLAG average or estimate in the bib-file for download. We recommend to consult the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors.||
||<:#eeeeee>'''Quantity'''  ||<:#eeeeee>'''Sec.'''  ||<:#eeeeee>'''$N_f=2+1+1$'''   ||<:#eeeeee>'''$N_f=2+1$'''   ||<:#eeeeee>'''$N_f=2$'''||<:#eeeeee>'''Refs.''' ||
||  $m_s$ [MeV]                    || [[Quark masses|3.1.3]]    ||  93.9(1.1)                || 92.0(2.1)               || 101(3)                || [[attachment:Media/Ms.bib|bib]] ||
||  $m_{ud}$ [MeV]                         || [[Quark masses|3.1.3]] ||  3.70(17)       || 3.373(80)   || 3.6(2)          || [[attachment:Media/Mud.bib|bib]] ||
||  $m_s/m_{ud}$                           || [[Quark masses|3.1.4]] ||  27.30(34)      || 27.43(31)   || 27.3(9)         || [[attachment:Media/Ratio_msmud.bib|bib]] ||
||  $m_u$ [MeV]                            || [[Quark masses|3.1.5]] ||  2.36(24)       || 2.16(9)(7)  || 2.40(23)        || <<FootNote(This is a FLAG estimate, based on $\chi$PT and the isospin averaged up- and down-quark mass $m_{ud}$.)>> ||
||  $m_d$ [MeV]                            || [[Quark masses|3.1.5]] ||  5.03(26)       || 4.68(14)(7) || 4.80(23)        || <<FootNote(This is a FLAG estimate, based on $\chi$PT and the isospin averaged up- and down-quark mass $m_{ud}$.)>> ||
||  $m_u/m_d$                              || [[Quark masses|3.1.5]] ||  0.470(56)      || 0.46(2)(2)  || 0.50(4)         || <<FootNote(This is a FLAG estimate, based on $\chi$PT and the isospin averaged up- and down-quark mass $m_{ud}$.)>> ||
||  $\overline{m}_c$(3 GeV) [GeV]          || [[Quark masses|3.2.3]] ||  0.996(25)      || 0.987(6)    || 1.03(4)         || [[attachment:Media/Mc.bib|bib]] ||
||  $m_c/m_s$                              || [[Quark masses|3.2.4]] ||  11.70(6)       || 11.82(16)   || 11.74(35)       || [[attachment:Media/Ratio_mcms.bib|bib]] ||
||  $\overline{m}_b(\overline{m}_b)$ [GeV] || [[Quark masses|3.3]]   ||  4.190(21)      || 4.164(23)   || 4.256(81)       || [[attachment:Media/Mb.bib|bib]] ||
||  $f_+(0)$                               || [[Quark masses|4.3]]   ||  0.9704(24)(22) || 0.9677(27)  || 0.9560(57)(62)  || [[attachment:Media/F+0.bib|bib]] ||
||  $f_{K^\pm}/f_{\pi^\pm}$                || [[Quark masses|4.3]]   ||  1.193(3)       || 1.192(5)    || 1.205(6)(17)    || [[attachment:Media/RfKfpi.bib|bib]] ||
||  $f_{\pi^\pm}$ [MeV]                    || [[Quark masses|4.6]]   ||                 || 130.2(1.4)  ||                 || [[attachment:Media/Fpi.bib|bib]] ||
||  $f_{K^\pm}$ [MeV]                      || [[Quark masses|4.6]]   ||  155.6(4)       || 155.9(9)    || 157.5(2.4)      || [[attachment:Media/FK.bib|bib]] ||
||  $\Sigma^{1/3}$ [MeV]                   || [[Quark masses|5.2.1]] ||  280(8)(15)     || 274(3)      || 266(10)         || [[attachment:Media/Su2_Sigma.bib|bib]] ||
||  $F_\pi/F$ [MeV]                        || [[Quark masses|5.2.1]] ||  1.076(2)(2)    || 1.064(7)    || 1.073(15)       || [[attachment:Media/RFpiF.bib|bib]] ||
||  $\bar{\ell}_3$ [MeV]                   || [[Quark masses|5.2.2]] ||  3.70(7)(26)    || 2.81(64)    || 3.41(82)        || [[attachment:Media/Su2_l3bar.bib|bib]] ||
||  $\bar{\ell}_4$ [MeV]                   || [[Quark masses|5.2.2]] ||  4.67(3)(10)    || 4.10(45)    || 4.51(26)        || [[attachment:Media/Su2_l4bar.bib|bib]] ||
||  $\bar{\ell}_6$ [MeV]                   || [[Quark masses|5.2.2]] ||                 ||             || 15.1(1.2)       || [[attachment:Media/Su2_l6bar.bib|bib]] ||
||  $\hat B_K$ [MeV]                       || [[Quark masses|6.1]]   ||  0.717(18)(16)  ||  0.7625(97) || 0.727(22)(12)   || [[attachment:Media/BK.bib|bib]] ||
+In order to work reliably and efficiently, we have adopted a formal structure and a set of rules by which all FLAG members abide.  The collaboration presently consists of an Advisory Board (AB), an Editorial Board (EB), and eight Working Groups (WG). The role of the Advisory Board is to provide oversight of the content, procedures, schedule and membership of FLAG, to help resolve disputes, to serve as a source of advice to the EB and to FLAG as a whole, and to provide a critical assessment of drafts. They also give their approval of the final version of the preprint before it is rendered public. The Editorial Board coordinates the activities of FLAG, sets priorities and intermediate deadlines, organizes votes on FLAG procedures, writes the introductory sections, and takes care of the editorial work needed to amalgamate the sections written by the individual working groups into a uniform and coherent review. The working groups concentrate on writing the review of the physical quantities for which they are responsible, which is subsequently circulated to the whole collaboration for critical evaluation.

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

||<35%-2 tablestyle="width: 60%;" style="border:none;">'''Advisory Board (AB)'''||<( style="border:none;"> G. Colangelo, M. Golterman, P. Hernandez, T. Onogi, R. Van de Water ||
||<35%-2 style="border:none;">'''Editorial Board (EB)'''||<( style="border:none;">S. Gottlieb, A. Jüttner, S. Hashimoto, S. Sharpe, U. Wenger ||
||<-3 style="border:none;">'''Working Groups (coordinator listed first)'''                                               ||
||<35%-2 style="border:none;">''Quark masses''||<( style="border:none;">T. Blum, A. Portelli, A. Ramos ||
||<35%-2 style="border:none;">''$V_{us}, V_{ud}$''||<( style="border:none;">T. Kaneko, J.N. Simone, S. Simula, N. Tantalo ||
||<35%-2 style="border:none;">''LEC''||<( style="border:none;">S. Dürr, H. Fukaya, U.M. Heller||
||<35%-2 style="border:none;">''$B_K$''||<( style="border:none;">P. Dimopoulos, X. Feng, G. Herdoiza||
||<35%-2 style="border:none;">''$f_{B_{(s)}}$, $f_{D_{(s)}}$, $B_B$''||<( style="border:none;">Y. Aoki, M. Della Morte, C. Monahan||
||<35%-2 style="border:none;">''$B_{(s)}$, $D$ semileptonic and radiative decays''||<( style="border:none;">E. Lunghi, S. Meinel, C. Pena||
||<35%-2 style="border:none;">''$\alpha_s$''||<( style="border:none;">S. Sint, R. Horsley, P. Petreczky||
||<35%-2 style="border:none;">''Nucleon Matrix Elements''||<( style="border:none;">R. Gupta, S. Collins, A. Nicholson, H. Wittig||
||<35%-2 style="border:none;">''Scale setting''||<( style="border:none;">R. Sommer, N. Tantalo, U. Wenger||


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 members must belong to different lattice collaborations;
 * 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.

As for FLAG 19, for this 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.

}}}

==== FLAG composition for the FLAG 2024 edition ====

<<BetterSeeSaw(section="current_flag_composition", toshow="Show Text", tohide="Hide Text")>>
{{{#!wiki seesaw current_flag_composition

As usual, after the completion of the FLAG 2021 edition the composition of FLAG has partially been restructured due to FLAG members stepping back or taking on different roles within FLAG. The work towards the FLAG 2024 edition, as well as the work for the webupdates, is currently undertaken with the following composition of FLAG:

||<35%-2 tablestyle="width: 60%;" style="border:none;">'''Advisory Board (AB)'''||<( style="border:none;"> M. Golterman, P. Hernandez, T. Onogi, S. Sharpe, R. Van de Water ||
||<35%-2 style="border:none;">'''Editorial Board (EB)'''||<( style="border:none;">S. Gottlieb, A. Jüttner, U. Wenger ||
||<-3 style="border:none;">'''Working Groups (coordinator underlined)'''                                               ||
||<35%-2 style="border:none;">''Quark masses''||<( style="border:none;">__T. Blum__, A. Portelli, A. Ramos ||
||<35%-2 style="border:none;">''$V_{us}, V_{ud}$''||<( style="border:none;">__T. Kaneko__, J.N. Simone, N. Tantalo ||
||<35%-2 style="border:none;">''$B_K$''||<( style="border:none;">__P. Dimopoulos__, X. Feng, G. Herdoiza||
||<35%-2 style="border:none;">''$f_{B_{(s)}}$, $f_{D_{(s)}}$, $B_B$''||<( style="border:none;">Y. Aoki, M. Della Morte, __C. Monahan__||
||<35%-2 style="border:none;">''$B_{(s)}$, $D$ semileptonic and radiative decays''||<( style="border:none;">__E. Lunghi__, S. Meinel, S. Simula, A. Vaquero Avilès-Casco||
||<35%-2 style="border:none;">''$\alpha_s$''||<( style="border:none;">L. Del Debbio, P. Petreczky, __S. Sint__||
||<35%-2 style="border:none;">''Nucleon Matrix Elements''||<( style="border:none;">S. Collins, __R. Gupta__, A. Nicholson, H. Wittig||
||<35%-2 style="border:none;">''Scale setting''||<( style="border:none;">__R. Sommer__, N. Tantalo, U. Wenger||


We note that the LEC section is no longer maintained.

}}}
<<Anchor(citation_policy)>>

==== Citation policy ====
<<BetterSeeSaw(section="citation_policy", toshow="Show Text", tohide="Hide Text")>>
{{{#!wiki seesaw 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, 3, 4, and 5, besides
summarizing the main results of the present review, we also provide here a bibtex file containing the
original references from which the results have been obtained. In addition, for each figure, we make a corresponding bibtex file available 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.
-Line 52:
+Line 128:
-<<BetterSeeSaw(section="table2", toshow="Show Table 2", tohide="Hide Table 2")>>
{{{#!wiki seesaw table2
+=== Quality criteria ===
-Line 55:
+Line 130:
-||<-6(#eeeeee tablestyle="width: 70%;">'''Table 2'''||
||<-6(>Summary of the main results of this review, grouped in terms of $N_f$, the number of dynamical quark flavours in lattice simulations. The quantities listed are specified in the quoted sections. For each result we provide the list of references that entered the FLAG average or estimate in the bib-file for download.  We recommend to consult the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors.||
||<:#eeeeee>'''Quantity'''||<:#eeeeee>'''Sec.'''||<:#eeeeee>'''$N_f=2+1+1$'''||<:#eeeeee>'''$N_f=2+1$'''||<:#eeeeee>'''$N_f=2$'''||<:#eeeeee>'''Refs.'''||
||  $f_D$ [MeV]                       || [[D-meson decay constants and form factors|7.1]]                     || 212.15(1.45) || 209.2(3.3) || 208(7)  ||[[attachment:Media/FD.bib|bib]] ||
||  $f_{D_s}$ [MeV]                   || [[D-meson decay constants and form factors|7.1]]                     || 248.83(1.27) || 249.8(2.3) || 250(7)  ||[[attachment:Media/FDs.bib|bib]] ||
||  $f_{D_s}/f_D$                     || [[D-meson decay constants and form factors|7.1]]                     || 1.1716(32)   || 1.187(12)  || 1.20(2) ||[[attachment:Media/FDratio.bib|bib]] ||
||  $f_+^{D\pi}(0)$ [MeV]             || [[D-meson decay constants and form factors|7.2]]                     ||              || 0.666(29)  ||          ||[[attachment:Media/Dtopi.bib|bib]] ||
||  $f_+^{DK}(0)$ [MeV]               || [[D-meson decay constants and form factors|7.2]]                     ||              || 0.747(19)  ||          ||[[attachment:Media/DtoK.bib|bib]] ||
||  $f_B$ [MeV]                       || [[B-meson decay constants, mixing parameters, and form factors|8.1]] || 186(4)       || 192.0(4.3) || 188(7)   ||[[attachment:Media/FB.bib|bib]] ||
||  $f_{B_s}$ [MeV]                   || [[B-meson decay constants, mixing parameters, and form factors|8.1]] || 224(5)       || 228.4(3.7) || 227(7)   ||[[attachment:Media/FBs.bib|bib]] ||
||  $f_{B_s}/f_B$                     || [[B-meson decay constants, mixing parameters, and form factors|8.1]] || 1.205(7)     || 1.201(16)  || 1.206(23) ||[[attachment:Media/FBratio.bib|bib]] ||
||  $f_{B_d}\sqrt{\hat{B}_{B_d}}$ [MeV] || [[B-meson decay constants, mixing parameters, and form factors|8.2]] ||              || 219(14)   || 216(10)   ||[[attachment:Media/FBsqrtBB.bib|bib]] ||
||  $f_{B_s}\sqrt{\hat{B}_{B_s}}$ [MeV] || [[B-meson decay constants, mixing parameters, and form factors|8.2]] ||              || 270(16)   || 262(10)   ||[[attachment:Media/FBssqrtBBs.bib|bib]] ||
||  $\hat{B}_{B_d}$                    || [[B-meson decay constants, mixing parameters, and form factors|8.2]] ||              || 1.26(9)   ||  1.30(6)  ||[[attachment:Media/BB.bib|bib]] ||
||  $\hat{B}_{B_s}$                    || [[B-meson decay constants, mixing parameters, and form factors|8.2]] ||              || 1.32(6)   ||  1.32(5)   ||[[attachment:Media/BBs.bib|bib]] ||
||  $\xi$                             || [[B-meson decay constants, mixing parameters, and form factors|8.2]] ||              || 1.239(46) ||   1.225(31)||[[attachment:Media/Xi.bib|bib]] ||
||  $B_{B_s}/B_{B_d}$                   || [[B-meson decay constants, mixing parameters, and form factors|8.2]] ||              || 1.039(63) || 1.007(21)  ||[[attachment:Media/RBB.bib|bib]] ||
|| $\alpha^{(5)}_{\overline{\text{MS} } }(M_Z)$   ||[[The strong coupling|9.9]] ||<-2:>0.1182(12)                       ||             ||[[attachment:Media/AlphaMSbarZ.bib|bib]] ||
|| $\Lambda^{(5)}_{\overline{\text{MS} } }$ [MeV] ||[[The strong coupling|9.9]] ||<-2:>211(14)                          ||             ||[[attachment:Media/R0LambdaMSbar.bib|bib]] ||
+The section on the [[Quality criteria|quality criteria]] can be downloaded [[attachment:Media/FLAG_criteria.pdf|here|&do=get]].
-Line 75:
+Line 132:
-}}}
+=== Quark masses ===
-Line 77:
+Line 134:
-==== FLAG composition, guidelines and rules ====
+The section on the [[Quark masses|quark masses]] can be downloaded [[attachment:Media/FLAG_qmass.pdf|here|&do=get]].
-Line 79:
+Line 136:
-==== Citation policy ====
+=== $\small{V_{ud}}$ and $\small{V_{us}}$ ===
-Line 81:
+Line 138:
-==== General issues ====
+The section on [[V(ud) and V(us)|$V_{ud}$ and $V_{us}$]] can be downloaded [[attachment:Media/FLAG_VudVus.pdf|here|&do=get]].
-Line 83:
+Line 140:
-==== References ====
<<FootNote()>>
+=== Low-energy constants ===
-Line 86:
+Line 142:
-=== Quality Criteria ===
=== Quark Masses ===
=== V(ud) and v(us) ===
=== Low-energy constants ===
+The section on the [[Low-energy constants|low-energy constants]] can be downloaded [[attachment:Media/FLAG_LECs.pdf|here|&do=get]].
-Line 91:
+Line 145:
-=== D-meson decay constants and form factors ===
=== B-meson decay constants, mixing parameters, and form factors ===
=== The strong coupling ===
=== Glossary ===
+The section on the [[Kaon mixing|kaon mixing]] can be downloaded [[attachment:Media/FLAG_BK.pdf|here|&do=get]].

=== $\small{D}$-meson decay constants and form factors ===
The section on the [[D-meson decay constants and form factors|$D$-meson decay constants and form factors]] can be downloaded [[attachment:Media/FLAG_HQD.pdf|here|&do=get]].

=== $\small{B}$-meson decay constants, mixing parameters, and form factors ===
The section on the [[B-meson decay constants, mixing parameters, and form factors|$B$-meson decay constants, mixing parameters, and form factors]] can be downloaded [[attachment:Media/FLAG_HQB.pdf|here|&do=get]]. 

=== The strong coupling $\alpha_s$ ===
The section on [[The strong coupling alpha_s|the strong coupling $\alpha_s$]] can be downloaded [[attachment:Media/FLAG_Alpha_s.pdf|here|&do=get]].

=== Nucleon matrix elements ===
The section on [[Nucleon matrix elements|nucleon matrix elements]] can be downloaded [[attachment:Media/FLAG_NME.pdf|here|&do=get]].

=== Scale setting ===
The section on [[Scale setting|scale setting]] can be downloaded [[attachment:Media/FLAG_ScaleSetting.pdf|here|&do=get]].


=== Appendix ===
The Appendix can be downloaded [[attachment:Media/FLAG_glossary.pdf|here|&do=get]].
-Line 96:
+Line 168:
+Notes to the various sections can be downloaded [[attachment:Media/FLAG_notes.pdf|here|&do=get]] in reduced form (containing notes to results that are new w.r.t. FLAG 19). 

<<BR>>
<<BR>>
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FLAG 2023/24

Figures for download

Navigation

FLAG Review 2021

Introduction

FLAG composition, guidelines and rules

FLAG composition for the FLAG 2024 edition

Citation policy

Quality criteria

Quark masses

$\small{V_{ud}}$ and $\small{V_{us}}$

Low-energy constants

Kaon mixing

$\small{D}$-meson decay constants and form factors

$\small{B}$-meson decay constants, mixing parameters, and form factors

The strong coupling $\alpha_s$

Nucleon matrix elements

Scale setting

Appendix

Notes