FLAG Review 2024
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 average and the corresponding plot, see also our paragraph on citation policy.
The latest version of the FLAG 24 review as of June 2026 is accessible here. It contains updates of the following sections:
- Charm-hadron decay constants and form factors: updated June 2026
Leptonic decay constants $f_B$ and $f_{B_s}$: updated June 2026
The version of the FLAG review as of November 2024 is accessible here and corresponds to the one published in PRD. The PRD version can be obtained from PRD as an Editor's Suggestion.
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 21. 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 February 2024 and is accessible at flag.itp.unibe.ch/2021 where also the original 2021 edition of the FLAG review can be downloaded. The original complete 2019 review is still accessible from flag.itp.unibe.ch/2019 and the 2015/2016 review from EPJC. The 2016/2017 updates are available from here. The 2013/2014 review is accessible here or from EPJC.
Contents
- Introduction
- Quality criteria
- The low-energy limit of the Standard Model
- Quark masses
- $\small{V_{ud}}$ and $\small{V_{us}}$
- 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 $V_{ud}$. 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 flavour diagonal. We also review determinations of scale-setting quantities. Finally, in this review we have added a new section on the general definition of the low-energy limit of the Standard Model.
We note that the LEC section is omitted in the current review as progress in that area has slowed. FLAG will keep monitoring the situation and provide updates in future editions, should new results become available.
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 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) |
M. Golterman, P. Hernandez, T. Onogi, S.R. Sharpe, R. Van de Water |
|
Editorial Board (EB) |
S. Gottlieb, A. Jüttner, U. Wenger |
|
Working Groups (coordinator listed first) |
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Quark masses |
T. Blum, A. Portelli, A. Ramos |
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$V_{us}, V_{ud}$ |
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 |
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$B_{(s)}$, $D$ semileptonic and radiative decays |
E. Lunghi, S. Meinel, A. Vaquero |
|
$\alpha_s$ |
S. Sint, L. Del Debbio, P. Petreczky |
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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 21, 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.
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.
Quality criteria
The section on the quality criteria can be downloaded here.
The low-energy limit of the Standard Model
The section on the general definition of the low-energy limit of the Standard Model including a specific prescription for the separation of QCD and QED contributions to hadronic observables (Edinburgh consensus) can be downloaded here.
Edinburgh consensus |
||
|
QCD |
isoQCD |
$M_{\pi^+}$ |
135.0 MeV |
135.0 MeV |
$M_{K^+}$ |
491.6 MeV |
494.6 MeV |
$M_{K^0}$ |
497.6 MeV |
494.6 MeV |
$M_{D_s^+}$ |
1967$\,$ MeV |
1967$\,$ MeV |
$M_{B_s^0}$ |
5367$\,$ MeV |
5367$\,$ MeV |
$f_{\pi^+}$ |
130.5 MeV |
130.5 MeV |
|
QCD |
isoQCD |
$M_{\pi^+}/f_{\pi^+}$ |
1.034 |
1.034 |
$M_{K^+}/f_{\pi^+}$ |
3.767 |
3.790 |
$M_{K^0}/f_{\pi^+}$ |
3.813 |
3.790 |
$M_{D_s^+}/f_{\pi^+}$ |
15.07 |
15.07 |
$M_{B_s^0}/f_{\pi^+}$ |
41.13 |
41.13 |
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.
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 21).
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