== FLAG Review 2019 ==

||<(#ffbbbb tablestyle="width: 100%;"> <<BR>> '''&nbsp; Note that the latest version of the FLAG review from 2021 is available [[http://flag.unibe.ch/2021/|here]].''' <<BR>>&nbsp; ||

The version of the FLAG 19 review as of December 2020 is accessible  [[attachment:Media/FLAG_2020_webupdate.pdf|here|&do=get]]. It contains updated sections as follows:

 * $|V_{ud}|$ and $|V_{us}|$:                                          updated December 2020
 * $B$-meson decay constants, mixing parameters, and form factors:     updated December 2020
  
The original 2019 edition of the FLAG review can be downloaded [[attachment:Media/FLAG_2019.pdf|here|&do=get]]. The version dated March 4, 2020 is the one accepted for publication in EPJC. The EPJC version can be obtained from [[http://link.springer.com/article/10.1140%2Fepjc%2Fs10052-019-7354-7|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 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.
 
In the notes we   compile detailed information on the simulations used to calculate the quantities discussed in the review. Here we provide the complete tables, in contrast to the paper version of the review which contains this information only for results that have appeared since FLAG 16.

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]].  

<<BetterTableOfContents(mindepth=3)>>

=== Introduction ===
The introduction with the updated summary tables can be downloaded [[attachment:Media/FLAG_introduction.pdf|here|&do=get]].

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 $D$- 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.

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 <<BetterFootNote("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]]", refName="colangelo_1")>>, 
2013 <<BetterFootNote("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]].", refName="aoki_1")>>,
and 2016 <<BetterFootNote("S. Aoki et al., ''Review of lattice results concerning low-energy particle physics'', Eur. Phys. J. C77 (2017) 112, [[http://arxiv.org/abs/1607.00299|arXiv:1607.00299]].", refName="aoki_2")>> 
(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.

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
non-singlet 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.

<<BetterSeeSaw(section="table1", toshow="Show Table 1", tohide="Hide Table 1")>>
{{{#!wiki seesaw table1

||<-6(#eeeeee tablestyle="width: 70%;">'''Table 1'''||
||<-6(>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 $\overline{\rm MS}$ scheme at running scale  $\mu=2\,$GeV, unless otherwise indicated.  BSM bag parameters $B_{2,3,4,5}$ are given in the $\overline{\rm MS}$ scheme at scale $\mu=3\,$GeV. Further specifications of the quantities are given in the quoted sections. Results for $N_f = 2$ quark masses are unchanged since FLAG 16<<BetterFootNote(refName="aoki_2")>>. For each result we provide the list of references that entered the FLAG average or estimate in the bibtex file for download.  We 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.||
||<:#eeeeee>'''Quantity'''  ||<:#eeeeee>'''Sec.'''  ||<:#eeeeee>'''$N_f=2+1+1$'''   ||<:#eeeeee>'''$N_f=2+1$'''   ||<:#eeeeee>'''$N_f=2$'''||<:#eeeeee>'''Refs.''' ||
||  $m_{ud}$ [MeV]                         || [[Quark masses|3.1.4]] ||  3.410(43)     || 3.364(41)  ||           || [[attachment:Media/mud.bib|bib|&do=get]] ||
||  $m_s$ [MeV]                    || [[Quark masses|3.1.4]]    ||  93.44(68)     || 92.03(88)     ||              || [[attachment:Media/ms.bib|bib|&do=get]] ||
||  $m_s/m_{ud}$                           || [[Quark masses|3.1.5]] ||  27.23(10)      || 27.42(12)   ||          || [[attachment:Media/ratio_msmud.bib|bib|&do=get]] ||
||  $m_u$ [MeV]                            || [[Quark masses|3.1.6]] ||  2.50(17)      || 2.27(9)  ||        ||[[attachment:Media/mu.bib|bib|&do=get]]   ||
||  $m_d$ [MeV]                            || [[Quark masses|3.1.6]] ||  4.88(20) || 4.67(9) ||  ||[[attachment:Media/md.bib|bib|&do=get]]   ||
||  $m_u/m_d$                              || [[Quark masses|3.1.6]] ||  0.513(31) || 0.485(19)  ||         || [[attachment:Media/ratio_mumd.bib|bib|&do=get]]  ||
||  $\overline{m}_c(3\;{\rm GeV})$ [GeV]          || [[Quark masses|3.2.2]] ||  0.988(7)     || 0.992(6) ||          || [[attachment:Media/mc.bib|bib|&do=get]] ||
||  $m_c/m_s$                              || [[Quark masses|3.2.3]] ||  11.768(33)  || 11.82(16)  ||        || [[attachment:Media/ratio_mcms.bib|bib|&do=get]] ||
||  $\overline{m}_b(\overline{m}_b)$ [GeV] || [[Quark masses|3.3]]   ||  4.198 (12)      || 4.164 (23)    ||        || [[attachment:Media/mb.bib|bib|&do=get]] ||
||  $f_+(0)$                               || [[V(ud) and V(us)|4.3]]   ||  0.9698(17)  || 0.9677(27)  || 0.9560(57)(62)  || [[attachment:Media/f+0.bib|bib|&do=get]] ||
||  $f_{K&#94;\pm}/f_{\pi&#94;\pm}$        || [[V(ud) and V(us)|4.3]]   ||  1.1932(21)    || 1.1917(37)    || 1.205(18)    || [[attachment:Media/RfKfpi.bib|bib|&do=get]] ||
||  $f_{\pi^\pm}$ [MeV]                    || [[V(ud) and V(us)|4.6]]   ||              || 130.2(8)    ||                 || [[attachment:Media/fpi.bib|bib|&do=get]] ||
||  $f_{K^\pm}$ [MeV]                      || [[V(ud) and V(us)|4.6]]   ||  155.7(3)       || 155.7(7)    || 157.5(2.4)      || [[attachment:Media/fK.bib|bib|&do=get]] ||
||  $\Sigma^{1/3}$ [MeV]                   || [[Low-energy constants|5.2.2]] ||  286(23)     || 272(5)      || 266(10)         || [[attachment:Media/Sigma13.bib|bib|&do=get]] ||
||  $F_\pi/F$ [MeV]                        || [[Low-energy constants|5.2.2]] || 1.077(3)     || 1.062(7)    || 1.073(15)       || [[attachment:Media/FpioF.bib|bib|&do=get]] ||
||  $\bar{\ell}_3$ [MeV]                   || [[Low-energy constants|5.2.3]] ||  3.53(26)    || 3.07(64)    || 3.41(82)        || [[attachment:Media/lbar3.bib|bib|&do=get]] ||
||  $\bar{\ell}_4$ [MeV]                   || [[Low-energy constants|5.2.3]] ||  4.73(10)    || 4.02(45)    || 4.40(28)        || [[attachment:Media/lbar4.bib|bib|&do=get]] ||
||  $\bar{\ell}_6$ [MeV]                   || [[Low-energy constants|5.2.3]] ||                 ||             || 15.1(1.2)       || [[attachment:Media/lbar6.bib|bib|&do=get]] ||
||  $\hat B_K$                        || [[Kaon mixing|6.2]]   ||  0.717(18)(16)  ||  0.7625(97) || 0.727(22)(12)   || [[attachment:Media/BK.bib|bib|&do=get]] ||
||  $B_2$                        || [[Kaon mixing|6.3]]   ||  0.46(1)(3)  ||  0.502(14) || 0.47(2)(1)   || [[attachment:Media/BSM_all.bib|bib|&do=get]] ||
||  $B_3$                        || [[Kaon mixing|6.3]]   ||  0.79(2)(4)  ||  0.766(32) || 0.78(4)(2)   || [[attachment:Media/BSM_all.bib|bib|&do=get]] ||
||  $B_4$                        || [[Kaon mixing|6.3]]   ||  0.78(2)(4)  ||  0.926(19) || 0.76(2)(2)   || [[attachment:Media/BSM_all.bib|bib|&do=get]] ||
||  $B_5$                        || [[Kaon mixing|6.3]]   ||  0.49(3)(3)  ||  0.720(38) || 0.58(2)(2)   || [[attachment:Media/BSM_all.bib|bib|&do=get]] ||
}}}


<<BetterSeeSaw(section="table2", toshow="Show Table 2", tohide="Hide Table 2")>>
{{{#!wiki seesaw table2

||<-6(#eeeeee tablestyle="width: 70%;">'''Table 2'''||
||<-6(>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 provide the list of references that entered the FLAG average or estimate in the bibtex file for download.  We 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.||
||<:#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]]                     || 212.0(7) || 209.0(2.4) || 208(7)  ||[[attachment:Media/FD.bib|bib|&do=get]] ||
||  $f_{D_s}$ [MeV]                   || [[D-meson decay constants and form factors|7.1]]                     || 249.9(5) || 248.0(1.6) || 242.5(5.8)  ||[[attachment:Media/FDs.bib|bib|&do=get]] ||
||  $f_{D_s}/f_D$                     || [[D-meson decay constants and form factors|7.1]]                     || 1.1783(16)   || 1.174(7)  || 1.20(2) ||[[attachment:Media/fDsofD.bib|bib|&do=get]] ||
||  $f_+^{D\pi}(0)$ [MeV]             || [[D-meson decay constants and form factors|7.2]]                     || 0.612(35)     || 0.666(29)  ||          ||[[attachment:Media/DtoPiandDtoK.bib|bib|&do=get]] ||
||  $f_+^{DK}(0)$ [MeV]               || [[D-meson decay constants and form factors|7.2]]                     ||  0.765(31)    || 0.747(19)  ||          ||[[attachment:Media/DtoPiandDtoK.bib|bib|&do=get]] ||
||  $f_B$ [MeV]                       || [[B-meson decay constants, mixing parameters, and form factors|8.1]] || 190.0(1.3)       || 192.0(4.3) || 188(7)   ||[[attachment:Media/fB.bib|bib|&do=get]] ||
||  $f_{B_s}$ [MeV]                   || [[B-meson decay constants, mixing parameters, and form factors|8.1]] || 230.3(1.3)       || 228.4(3.7) || 227(7)   ||[[attachment:Media/fBs.bib|bib|&do=get]] ||
||  $f_{B_s}/f_B$                     || [[B-meson decay constants, mixing parameters, and form factors|8.1]] || 1.209(5)     || 1.201(16)|| 1.206(23) ||[[attachment:Media/fBofBs.bib|bib|&do=get]] ||
||  $f_{B_d}\sqrt{\hat{B}_{B_d}}$ [MeV] || [[B-meson decay constants, mixing parameters, and form factors|8.2]] ||              || 225(9)   || 216(10)   ||[[attachment:Media/fBsqrtBB.bib|bib|&do=get]] ||
||  $f_{B_s}\sqrt{\hat{B}_{B_s}}$ [MeV] || [[B-meson decay constants, mixing parameters, and form factors|8.2]] ||              || 274(8)   || 262(10)   ||[[attachment:Media/fBsqrtBB.bib|bib|&do=get]] ||
||  $\hat{B}_{B_d}$                    || [[B-meson decay constants, mixing parameters, and form factors|8.2]] ||              || 1.30(10)   ||  1.30(6)  ||[[attachment:Media/BBs.bib|bib|&do=get]] ||
||  $\hat{B}_{B_s}$                    || [[B-meson decay constants, mixing parameters, and form factors|8.2]] ||              || 1.35(6)   ||  1.32(5)   ||[[attachment:Media/BBs.bib|bib|&do=get]] ||
||  $\xi$                             || [[B-meson decay constants, mixing parameters, and form factors|8.2]] ||              || 1.206(17) ||   1.225(31)||[[attachment:Media/xiandRBB.bib|bib|&do=get]] ||
||  $B_{B_s}/B_{B_d}$                   || [[B-meson decay constants, mixing parameters, and form factors|8.2]] ||              || 1.032(38) || 1.007(21)  ||[[attachment:Media/xiandRBB.bib|bib|&do=get]] ||
|| $\alpha^{(5)}_{\overline{\text{MS} } }(M_Z)$   ||[[The strong coupling alpha_s|9.10]] ||<-2:>0.1182(8)                       ||             ||[[attachment:Media/alphaMSbarZ.bib|bib|&do=get]] ||
|| $\Lambda^{(5)}_{\overline{\text{MS} } }$ [MeV] ||[[The strong coupling alpha_s|9.10]] ||<-2:>211(10)                          ||             ||[[attachment:Media/alphaMSbarZ.bib|bib|&do=get]] ||
|| $\Lambda^{(4)}_{\overline{\text{MS} } }$ [MeV] ||[[The strong coupling alpha_s|9.10]] ||<-2:>294(12)                         ||             ||[[attachment:Media/alphaMSbarZ.bib|bib|&do=get]] ||
|| $\Lambda^{(3)}_{\overline{\text{MS} } }$ [MeV] ||[[The strong coupling alpha_s|9.10]] ||<-2:>343(12)                          ||             ||[[attachment:Media/alphaMSbarZ.bib|bib|&do=get]] ||

}}}

<<BetterSeeSaw(section="table3", toshow="Show Table 3", tohide="Hide Table 3")>>
{{{#!wiki seesaw table3

||<-6(#eeeeee tablestyle="width: 70%;">'''Table 3'''||
||<-6(>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 provide the list of references that entered the FLAG average or estimate in the bibtex file for download.  We 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.||
||<:#eeeeee>'''Quantity'''||<:#eeeeee>'''Sec.'''||<:#eeeeee>'''$N_f=2+1+1$'''||<:#eeeeee>'''$N_f=2+1$'''||<:#eeeeee>'''$N_f=2$'''||<:#eeeeee>'''Refs.'''||
||  $g_A^{u-d}$                        || [[Nucleon matrix elements|10.3.1]]       || 1.251(33) || 1.254(16)(30) || 1.278(86)  ||[[attachment:Media/gA.bib|bib|&do=get]] ||
||  $g_S^{u-d}$                        || [[Nucleon matrix elements|10.3.2]]       || 1.022(80)(60) ||  ||  ||[[attachment:Media/gS.bib|bib|&do=get]] ||
||  $g_T^{u-d}$                        || [[Nucleon matrix elements|10.3.3]]       || 0.989(32)(10) || ||   ||[[attachment:Media/gT.bib|bib|&do=get]] ||
||  $g_A^{u}$                        || [[Nucleon matrix elements|10.4.1]]       || 0.777(25)(30) || 0.847(18)(32)||   ||[[attachment:Media/gAu.bib|bib|&do=get]] ||
||  $g_A^{d}$                        || [[Nucleon matrix elements|10.4.1]]       || −0.438(18)(30) || −0.407(16)(18)||   ||[[attachment:Media/gAd.bib|bib|&do=get]] ||
||  $g_A^{s}$                        || [[Nucleon matrix elements|10.4.1]]       || −0.053(8) || −0.035(6)(7)(32)||   ||[[attachment:Media/gAs.bib|bib|&do=get]] ||
||  $\sigma_{\pi N}$ [MeV]                        || [[Nucleon matrix elements|10.4.4]]       || 64.9(1.5)(13.2) || 39.7(3.6)||  37(8)(6) ||[[attachment:Media/sigmapiN.bib|bib|&do=get]] ||
||  $\sigma_{s}$ [MeV]                        || [[Nucleon matrix elements|10.4.4]]       || 41.0(8.8) || 52.9(7.0)||   ||[[attachment:Media/sigmas.bib|bib|&do=get]] ||
||  $g_T^{u}$                        || [[Nucleon matrix elements|10.4.5]]       || 0.784(28)(10) ||  ||  ||[[attachment:Media/gTu.bib|bib|&do=get]] ||
||  $g_T^{d}$                        || [[Nucleon matrix elements|10.4.5]]       || −0.204(11)(10) ||  ||  ||[[attachment:Media/gTd.bib|bib|&do=get]] ||
||  $g_T^{s}$                        || [[Nucleon matrix elements|10.4.5]]       || −0.027(16) ||  ||  ||[[attachment:Media/gTs.bib|bib|&do=get]] ||

}}}

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 (this page),
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.
<<BetterFootNote("Working groups were given the option of including papers submitted to [[http://arxiv.org|arXiv.org]] before the closing date but published after this date. This flexibility allows this review to be up-to-date at the time of submission. Three papers of this type were included: Ref. [5] in Secs. 7 and 8, and Refs. [6] and [7] in Sec. 10.")>>

==== FLAG composition, guidelines and rules ====

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

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:

||<35%-2 tablestyle="width: 60%;" style="border:none;">'''Advisory Board (AB)'''||<( style="border:none;">S. Aoki, M. Golterman, R. Van de Water, T. Vladikas ||
||<35%-2 style="border:none;">'''Editorial Board (EB)'''||<( style="border:none;">G. Colangelo, A. Jüttner, S. Hashimoto, S. Sharpe, U. Wenger ||
||<-3 style="border:none;">'''Working Groups (WG)'''                                               ||
||<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;">S. Simula, T. Kaneko, J.N. Simone ||
||<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, G. Herdoiza, R. Mawhinney||
||<35%-2 style="border:none;">''$f_{B_{(s)}}$, $f_{D_{(s)}}$, $B_B$''||<( style="border:none;">D. Lin, Y. Aoki, M. Della Morte||
||<35%-2 style="border:none;">''$B_{(s)}$, $D$ semileptonic and radiative decays''||<( style="border:none;">E. Lunghi, D. Becirevic, S. Gottlieb, C. Pena||
||<35%-2 style="border:none;">''$\alpha_s$''||<( style="border:none;">R. Sommer, R. Horsley, T. Onogi||
||<35%-2 style="border:none;">''Nucleon Matrix Elements (NME)''||<( style="border:none;">R. Gupta, S. Collins, A. Nicholson, 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;<<BetterFootNote("The WG on semileptonic D and B decays has currently four members, but only three of them belong to lattice collaborations.")>> <<BetterFootNote("The NME WG, new in this addition of the FLAG review, has been formed with four members (all members of lattice collaborations) rather than three. This reflects the large amount of work needed to create a section for which some of the systematic errors are substantially different from those described in other sections, and to provide a better representation of relevant 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.

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 non-lattice 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 ====

<<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 and 3, 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. 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 ====

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

Several general issues concerning the present review are thoroughly discussed in Sec.~1.1 of our initial 2010 paper <<BetterFootNote(refName="colangelo_1")>>, and we encourage the reader to consult the relevant pages. In the remainder of the present subsection, we focus on a few important points. Though the discussion has been duly updated, it is similar to that of Sec.~1.2  in the previous two reviews <<BetterFootNote(refName="aoki_1")>><<BetterFootNote(refName="aoki_2")>>, 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<<BetterFootNote("We also use terms like “quality criteria”, “rating”, “colour coding” etc. when referring to the classification of results, as described in [[Quality criteria|Sec. 2]].")>> used and provide sufficient details in appendices so that the reader can verify the information given in the tables.

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

Each discretization has its merits, but also its shortcomings. For most topics covered in this review we have an increasingly broad database, and for most quantities lattice calculations based on totally different discretizations are now available. This is illustrated by the dense population of the tables and figures in most parts of this review. Those calculations that do satisfy our quality criteria indeed lead, 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, all lattice methods for treating $b$ quarks use effective field theory at some level. This introduces additional complications not present in the light-quark sector. An overview of the issues specific to heavy-quark quantities is given in the introduction of [[B-meson decay constants, mixing parameters, and form factors|Sec. 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 [[The strong coupling|Sec. 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 $r_0 \Lambda_{\overline{\rm MS}}$, 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 in the [[Glossary]]. 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 in the [[Glossary]].

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 in the [[Glossary]].

'''''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 in the [[Glossary]]. The conversion of the results obtained in these so-called intermediate schemes to more familiar regularization schemes, such as the $\overline{\rm MS}$-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 $\overline{\rm MS}$, 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; 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 in the [[Glossary]] 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 [[Nucleon matrix elements|Sec. 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. Many groups check for autocorrelations in a number of observables, including the topological charge, for which a rapid growth of the autocorrelation time is observed with decreasing lattice spacing. This is often referred to as topological freezing. A solution to the problem consists in using open boundary conditions in time, instead of the more common antiperiodic ones. More recently two other approaches have been proposed, one based on a multiscale thermalization algorithm and another based on defining QCD on a nonorientable manifold. The problem is also touched upon in [[The strong coupling $\alpha_s$|Sec. 9.2]], where it is stressed that attention must be paid to this issue. While large scale simulations with open boundary conditions are already far advanced, only one result reviewed here has been obtained with any of the above methods. It is usually assumed that the continuum limit can be reached by extrapolation from the existing simulations, and that potential systematic errors due to the long autocorrelation times have been adequately controlled. 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. 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.

'''''Simulation algorithms and numerical errors:'''''

Most of the modern lattice-QCD simulations use exact algorithms, 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.

}}}



==== References ====
<<BetterFootNote()>>


=== Quality criteria ===

The section on the [[Quality criteria|quality criteria]] can be downloaded [[attachment:Media/FLAG_criteria.pdf|here|&do=get]].

=== Quark masses ===

The section on the [[Quark masses|quark masses]] can be downloaded [[attachment:Media/FLAG_qmass.pdf|here|&do=get]].

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

The section on [[V(ud) and V(us)|$V_{ud}$ and $V_{us}$]] can be downloaded [[attachment:Media/FLAG_VudVus.pdf|here|&do=get]].

=== Low-energy constants ===

The section on the [[Low-energy constants|low-energy constants]] can be downloaded [[attachment:Media/FLAG_LECs.pdf|here|&do=get]].

=== Kaon mixing ===

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]].

=== Glossary ===
The [[Glossary|glossary]] can be downloaded [[attachment:Media/FLAG_glossary.pdf|here|&do=get]].

=== Notes ===
Notes to the various sections can be downloaded [[attachment:Media/FLAG_notes.pdf|here|&do=get]]. A reduced version containing only the notes on calculations which are new w.r.t. FLAG 13 is available [[attachment:Media/FLAG_notes_reduced.pdf|here|&do=get]].

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