== Review of lattice results concerning low energy particle physics == The latest version of the complete review as of December 2017 is accessible [[attachment:Media/FLAG_webupdate.pdf|here|&do=get]]. It contains updated sections as follows: +-----------+--------------------- | $|V_{ud}|$ and $|V_{us}|$: | +-----------+--------------------- | Low-energy constants: | +-----------+--------------------- * $|V_{ud}|$ and $|V_{us}|$: :: updated November 2016 * Low-energy constants: :: updated July 2017 * Kaon mixing: ::updated December 2016 * $B$-meson mixing parameters: ::updated July/November 2017 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. 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 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]]. <> === Introduction === The introduction with the updated summary tables can be downloaded [[attachment:Media/FLAG_introduction_webupdate.pdf|here|&do=get]]. 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 $\alpha_s$. Flavour physics provides an important opportunity for exploring the limits of the Standard Model of particle physics and for constraining possible extensions that go beyond it. As the LHC explores a new energy frontier and as experiments continue to extend the precision frontier, the importance of flavour physics will grow, both in terms of searches for signatures of new physics through precision measurements and in terms of attempts to construct the theoretical framework behind direct discoveries of new particles. A major theoretical limitation consists in the precision with which strong-interaction effects can be quantified. Large-scale numerical simulations of lattice QCD allow for the computation of these effects from first principles. The scope of the Flavour Lattice Averaging Group (FLAG) is to review the current status of lattice results for a variety of physical quantities in low-energy physics. Set up in November 2007 it comprises experts in Lattice Field Theory, Chiral Perturbation Theory and Standard Model phenomenology. Our aim is to provide an answer to the frequently posed question “What is currently the best lattice value for a particular quantity?” in a way that is readily accessible to nonlattice-experts. This is generally not an easy question to answer; different collaborations use different lattice actions (discretizations of QCD) with a variety of lattice spacings and volumes, and with a range of masses for the $u$− and $d$−quarks. Not only are the systematic errors different, but also the methodology used to estimate these uncertainties varies between collaborations. In the present work we summarize the main features of each of the calculations and provide a framework for judging and combining the different results. Sometimes it is a single result that provides the “best” value; more often it is a combination of results from different collaborations. Indeed, the consistency of values obtained using different formulations adds significantly to our confidence in the results. The first two editions of the FLAG review were published in 2011 <> and 2014 <>. 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. 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. <> {{{#!wiki seesaw table1 ||<-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|&do=get]] || || $m_{ud}$ [MeV] || [[Quark masses|3.1.3]] || 3.70(17) || 3.373(80) || 3.6(2) || [[attachment:Media/Mud.bib|bib|&do=get]] || || $m_s/m_{ud}$ || [[Quark masses|3.1.4]] || 27.30(34) || 27.43(31) || 27.3(9) || [[attachment:Media/Ratio_msmud.bib|bib|&do=get]] || || $m_u$ [MeV] || [[Quark masses|3.1.5]] || 2.36(24) || 2.16(9)(7) || 2.40(23) || <> || || $m_d$ [MeV] || [[Quark masses|3.1.5]] || 5.03(26) || 4.68(14)(7) || 4.80(23) || <> || || $m_u/m_d$ || [[Quark masses|3.1.5]] || 0.470(56) || 0.46(2)(2) || 0.50(4) || <> || || $\overline{m}_c$(3 GeV) [GeV] || [[Quark masses|3.2.3]] || 0.996(25) || 0.987(6) || 1.03(4) || [[attachment:Media/Mc.bib|bib|&do=get]] || || $m_c/m_s$ || [[Quark masses|3.2.4]] || 11.70(6) || 11.82(16) || 11.74(35) || [[attachment:Media/Ratio_mcms.bib|bib|&do=get]] || || $\overline{m}_b(\overline{m}_b)$ [GeV] || [[Quark masses|3.3]] || 4.190(21) || 4.164(23) || 4.256(81) || [[attachment:Media/Mb.bib|bib|&do=get]] || || $f_+(0)$ || [[V(ud) and V(us)|4.3]] || 0.9706(27) || 0.9677(27) || 0.9560(57)(62) || [[attachment:Media/F+0.bib|bib|&do=get]] || || $f_{K^\pm}/f_{\pi^\pm}$ || [[V(ud) and V(us)|4.3]] || 1.193(3) || 1.192(4) || 1.205(6)(17) || [[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|5.2.1]] || 280(8)(15) || 272(5) || 266(10) || [[attachment:Media/Su2_Sigma.bib|bib|&do=get]] || || $F_\pi/F$ [MeV] || [[Low-energy constants|5.2.1]] || 1.077(2)(2) || 1.062(7) || 1.073(15) || [[attachment:Media/RFpiF.bib|bib|&do=get]] || || $\bar{\ell}_3$ [MeV] || [[Low-energy constants|5.2.2]] || 3.53(5)(26) || 3.07(64) || 3.41(82) || [[attachment:Media/Su2_l3bar.bib|bib|&do=get]] || || $\bar{\ell}_4$ [MeV] || [[Low-energy constants|5.2.2]] || 4.73(2)(10) || 4.02(45) || 4.40(28) || [[attachment:Media/Su2_l4bar.bib|bib|&do=get]] || || $\bar{\ell}_6$ [MeV] || [[Low-energy constants|5.2.2]] || || || 15.1(1.2) || [[attachment:Media/Su2_l6bar.bib|bib|&do=get]] || || $\hat B_K$ || [[Kaon mixing|6.1]] || 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/BSMB2.bib|bib|&do=get]] || || $B_3$ || [[Kaon mixing|6.3]] || 0.79(2)(4) || 0.766(32) || 0.78(4)(2) || [[attachment:Media/BSMB3.bib|bib|&do=get]] || || $B_4$ || [[Kaon mixing|6.3]] || 0.78(2)(4) || 0.926(19) || 0.76(2)(2) || [[attachment:Media/BSMB4.bib|bib|&do=get]] || || $B_5$ || [[Kaon mixing|6.3]] || 0.49(3)(3) || 0.720(38) || 0.58(2)(2) || [[attachment:Media/BSMB5.bib|bib|&do=get]] || }}} <> {{{#!wiki seesaw table2 ||<-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|&do=get]] || || $f_{D_s}$ [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|&do=get]] || || $f_{D_s}/f_D$ || [[D-meson decay constants and form factors|7.1]] || 1.1716(32) || 1.187(12) || 1.20(2) ||[[attachment:Media/FDratio.bib|bib|&do=get]] || || $f_+^{D\pi}(0)$ [MeV] || [[D-meson decay constants and form factors|7.2]] || || 0.666(29) || ||[[attachment:Media/Dtopi.bib|bib|&do=get]] || || $f_+^{DK}(0)$ [MeV] || [[D-meson decay constants and form factors|7.2]] || || 0.747(19) || ||[[attachment:Media/DtoK.bib|bib|&do=get]] || || $f_B$ [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|&do=get]] || || $f_{B_s}$ [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|&do=get]] || || $f_{B_s}/f_B$ || [[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|&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/FBssqrtBBs.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/BB.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/Xi.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/RBB.bib|bib|&do=get]] || || $\alpha^{(5)}_{\overline{\text{MS} } }(M_Z)$ ||[[The strong coupling alpha_s|9.9]] ||<-2:>0.1182(12) || ||[[attachment:Media/AlphaMSbarZ.bib|bib|&do=get]] || || $\Lambda^{(5)}_{\overline{\text{MS} } }$ [MeV] ||[[The strong coupling alpha_s|9.9]] ||<-2:>211(14) || ||[[attachment:Media/R0LambdaMSbar.bib|bib|&do=get]] || }}} Our plan is to continue providing FLAG updates, in the form of a peer reviewed paper, roughly on a biennial basis. This effort is supplemented by our more frequently updated website here, where figures as well as pdf-files for the individual sections can be downloaded. The papers reviewed in the present edition have appeared before the closing date 30 November 2015. The section on leptonic and semileptonic kaon and pion decay and $|Vud|$ and $|Vus|$ has been updated in November 2016, while the section on kaon mixing has been updated in December 2016. ==== FLAG composition, guidelines and rules ==== <> {{{#!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 particle-physics community with a single source of reliable information on lattice results. In order to work reliably and efficiently, we have adopted a formal structure and a set of rules by which all FLAG members abide. The collaboration presently consists of an Advisory Board (AB), an Editorial Board (EB), and seven Working Groups (WG). The role of the Advisory Board is that of general supervision and consultation. Its members may interfere at any point in the process of drafting the paper, expressing their opinion and offering advice. They also give their approval of the final version of the preprint before it is rendered public. The Editorial Board coordinates the activities of FLAG, sets priorities and intermediate deadlines, and takes care of the editorial work needed to amalgamate the sections written by the individual working groups into a uniform and coherent review. The working groups concentrate on writing up the review of the physical quantities for which they are responsible, which is subsequently circulated to the whole collaboration for critical evaluation. The current list of FLAG members and their Working Group assignments is: ||<35%-2 tablestyle="width: 60%;" style="border:none;">'''Advisory Board (AB)'''||<( style="border:none;">S. Aoki, C. Bernard, M. Golterman, H. Leutwyler, C. Sachrajda|| ||<35%-2 style="border:none;">'''Editorial Board (EB)'''||<( style="border:none;">G. Colangelo, A. Jüttner, S. Hashimoto, S. Sharpe, T. Vladikas, U. Wenger || ||<-3 style="border:none;">'''Working Groups (WG)''' || ||<35%-2 style="border:none;">''Quark masses''||<( style="border:none;">L. Lellouch, T. Blum, V. Lubicz || ||<35%-2 style="border:none;">''$V_{us}, V_{ud}$''||<( style="border:none;">S. Simula, P. Boyle,<>Unfortunately, due to other commitments, it was impossible for him to contribute until the end, and he decided to withdraw from the collaboration.")>> T. Kaneko || ||<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;">H. Wittig, P. Dimopoulos, R. Mawhinney|| ||<35%-2 style="border:none;">''$f_{B_{(s)}}$, $f_{D_{(s)}}$, $B_B$''||<( style="border:none;">M. Della Morte, Y. Aoki, D. Lin|| ||<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|| As some members of the WG on quark masses were faced with unexpected hindrances, S. Simula has kindly assisted in the completion of the relevant section during the final phases of its composition. The most important FLAG guidelines and rules are the following: * the composition of the AB reflects the main geographical areas in which lattice collaborations are active, with members from America, Asia/Oceania and Europe; * the mandate of regular members is not limited in time, but we expect that a certain turnover will occur naturally; * whenever a replacement becomes necessary this has to keep, and possibly improve, the balance in FLAG, so that different collaborations, from different geographical areas are represented; * in all working groups the three members must belong to three different lattice collaborations;<> * a paper is in general not reviewed (nor colour-coded, as described in the next section) by any of its authors; * lattice collaborations not represented in FLAG will be consulted on the colour coding of their calculation; * there are also internal rules regulating our work, such as voting procedures. }}} ==== Citation policy ==== <> {{{#!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 nonlattice-experts and we are well aware that it is likely that some readers will only consult the present paper and not the original lattice literature. It is very important that this paper be not the only one cited when our results are quoted. We strongly suggest that readers also cite the original sources. In order to facilitate this, in Tabs. 1 and 2, besides summarizing the main results of the present review, we also cite the original references from which they have been obtained. In addition, for each figure we make a bibtex-file available on this webpage which contains the bibtex-entries of all the calculations contributing to the FLAG average or estimate. The bibliography at the end of this paper should also make it easy to cite additional papers. Indeed we hope that the bibliography will be one of the most widely used elements of the whole paper. }}} ==== General issues ==== <> {{{#!wiki seesaw general_issues Several general issues concerning the present review are thoroughly discussed in Sec. 1.1 of our initial 2010 paper <> and we encourage the reader to consult the relevant pages. In the remainder of the present subsection, we focus on a few important points. Though the discussion has been duly updated, it is essentially that of Sec. 1.2 of the 2013 review <>. The present review aims to achieve two distinct goals: first, to provide a description of the work done on the lattice concerning low-energy particle physics; and, second, to draw conclusions on the basis of that work, summarizing the results obtained for the various quantities of physical interest. The core of the information about the work done on the lattice is presented in the form of tables, which not only list the various results, but also describe the quality of the data that underlie them. We consider it important that this part of the review represents a generally accepted description of the work done. For this reason, we explicitly specify the quality requirements<> used and provide sufficient details in appendices so that the reader can verify the information given in the tables. On the other hand, the conclusions drawn on the basis of the available lattice results are the responsibility of FLAG alone. Preferring to err on the side of caution, in several cases we draw conclusions that are more conservative than those resulting from a plain weighted average of the available lattice results. This cautious approach is usually adopted when the average is dominated by a single lattice result, or when only one lattice result is available for a given quantity. In such cases one does not have the same degree of confidence in results and errors as when there is agreement among several different calculations using different approaches. The reader should keep in mind that the degree of confidence cannot be quantified, and it is not reflected in the quoted errors. Each discretization has its merits, but also its shortcomings. For most topics covered in this review we have an increasingly broad database, and for most quantities lattice calculations based on totally different discretizations are now available. This is illustrated by the dense population of the tables and figures in most parts of this review. Those calculations that do satisfy our quality criteria indeed lead to consistent results, confirming universality within the accuracy reached. In our opinion, the consistency between independent lattice results, obtained with different discretizations, methods, and simulation parameters, is an important test of lattice QCD, and observing such consistency also provides further evidence that systematic errors are fully under control. In the sections dealing with heavy quarks and with $\alpha_s$, the situation is not the same. Since the $b$-quark mass cannot be resolved with current lattice spacings, all lattice methods for treating $b$ quarks use effective field theory at some level. This introduces additional complications not present in the light-quark sector. An overview of the issues specific to heavy-quark quantities is given in the introduction of [[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> === 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 $V_{us}$]] updated in November 2016 can be downloaded [[attachment:Media/FLAG_VudVus_webupdate.pdf|here|&do=get]]. The original section is still available [[attachment:Media/FLAG_VudVus.pdf|here|&do=get]]. === Low-energy constants === The section on the [[Low-energy constants]] updated in Juy 2017 can be downloaded [[attachment:Media/FLAG_LECs_webupdate.pdf|here|&do=get]]. The original section is still available [[attachment:Media/FLAG_LECs.pdf|here|&do=get]]. === Kaon mixing === The section on the [[Kaon mixing]] updated in December 2016 can be downloaded [[attachment:Media/FLAG_BK_webupdate.pdf|here|&do=get]]. The original section is still available [[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]] updated in July 2017 can be downloaded [[attachment:Media/FLAG_HQB_webupdate.pdf|here|&do=get]]. The original section is still available [[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]]. === 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]].