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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 $|V_{ud}|$ and $|V_{us}|$, and the new section updated in December 2016 on kaon mixing. |
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 $|V_{us}|$: updated November 2016 * Low-energy constants: updated July 2017 * Kaon mixing: updated December 2016 * $B$-meson mixing parameters: updated July/November 2017 |
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The introduction can be downloaded [[attachment:Media/FLAG_introduction.pdf|here|&do=get]]. | The introduction with the updated summary tables can be downloaded [[attachment:Media/FLAG_introduction_webupdate.pdf|here|&do=get]]. |
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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. | The first two editions of the FLAG review were published in 2011 <<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")>> and 2014 <<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")>>. 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. |
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|| $m_u$ [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 $m_{ud}$.)>> || |
|| $m_u$ [MeV] || [[Quark masses|3.1.5]] || 2.36(24) || 2.16(9)(7) || 2.40(23) || <<BetterFootNote("This is a FLAG estimate, based on $\chi$PT and the isospin averaged up- and down-quark mass $m_{ud}$.", refName="flag_estimate")>> || || $m_d$ [MeV] || [[Quark masses|3.1.5]] || 5.03(26) || 4.68(14)(7) || 4.80(23) || <<BetterFootNote(refName="flag_estimate")>> || || $m_u/m_d$ || [[Quark masses|3.1.5]] || 0.470(56) || 0.46(2)(2) || 0.50(4) || <<BetterFootNote(refName="flag_estimate")>> || |
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|| $f_+(0)$ || [[Quark masses|4.3]] || 0.9704(24)(22) || 0.9677(27) || 0.9560(57)(62) || [[attachment:Media/F+0.bib|bib|&do=get]] || || $f_{K^\pm}/f_{\pi^\pm}$ || [[Quark masses|4.3]] || 1.193(3) || 1.192(5) || 1.205(6)(17) || [[attachment:Media/RfKfpi.bib|bib|&do=get]] || || $f_{\pi^\pm}$ [MeV] || [[Quark masses|4.6]] || || 130.2(1.4) || || [[attachment:Media/Fpi.bib|bib|&do=get]] || || $f_{K^\pm}$ [MeV] || [[Quark masses|4.6]] || 155.6(4) || 155.9(9) || 157.5(2.4) || [[attachment:Media/FK.bib|bib|&do=get]] || || $\Sigma^{1/3}$ [MeV] || [[Quark masses|5.2.1]] || 280(8)(15) || 274(3) || 266(10) || [[attachment:Media/Su2_Sigma.bib|bib|&do=get]] || || $F_\pi/F$ [MeV] || [[Quark masses|5.2.1]] || 1.076(2)(2) || 1.064(7) || 1.073(15) || [[attachment:Media/RFpiF.bib|bib|&do=get]] || || $\bar{\ell}_3$ [MeV] || [[Quark masses|5.2.2]] || 3.70(7)(26) || 2.81(64) || 3.41(82) || [[attachment:Media/Su2_l3bar.bib|bib|&do=get]] || || $\bar{\ell}_4$ [MeV] || [[Quark masses|5.2.2]] || 4.67(3)(10) || 4.10(45) || 4.51(26) || [[attachment:Media/Su2_l4bar.bib|bib|&do=get]] || || $\bar{\ell}_6$ [MeV] || [[Quark masses|5.2.2]] || || || 15.1(1.2) || [[attachment:Media/Su2_l6bar.bib|bib|&do=get]] || || $\hat B_K$ [MeV] || [[Quark masses|6.1]] || 0.717(18)(16) || 0.7625(97) || 0.727(22)(12) || [[attachment:Media/BK.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]] || |
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|| $f_{B_d}\sqrt{\hat{B}_{B_d}}$ [MeV] || [[B-meson decay constants, mixing parameters, and form factors|8.2]] || || 219(14) || 216(10) ||[[attachment:Media/FBsqrtBB.bib|bib|&do=get]] || || $f_{B_s}\sqrt{\hat{B}_{B_s}}$ [MeV] || [[B-meson decay constants, mixing parameters, and form factors|8.2]] || || 270(16) || 262(10) ||[[attachment:Media/FBssqrtBBs.bib|bib|&do=get]] || || $\hat{B}_{B_d}$ || [[B-meson decay constants, mixing parameters, and form factors|8.2]] || || 1.26(9) || 1.30(6) ||[[attachment:Media/BB.bib|bib|&do=get]] || || $\hat{B}_{B_s}$ || [[B-meson decay constants, mixing parameters, and form factors|8.2]] || || 1.32(6) || 1.32(5) ||[[attachment:Media/BBs.bib|bib|&do=get]] || || $\xi$ || [[B-meson decay constants, mixing parameters, and form factors|8.2]] || || 1.239(46) || 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]] || || 1.039(63) || 1.007(21) ||[[attachment:Media/RBB.bib|bib|&do=get]] || || $\alpha^{(5)}_{\overline{\text{MS} } }(M_Z)$ ||[[The strong coupling|9.9]] ||<-2:>0.1182(12) || ||[[attachment:Media/AlphaMSbarZ.bib|bib|&do=get]] || || $\Lambda^{(5)}_{\overline{\text{MS} } }$ [MeV] ||[[The strong coupling|9.9]] ||<-2:>211(14) || ||[[attachment:Media/R0LambdaMSbar.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]] || |
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||<28%-2 tablestyle="width: 60%;" style="border:none;">'''Advisory Board (AB)'''||<( style="border:none;">S. Aoki, C. Bernard, M. Golterman, H. Leutwyler, C. Sachrajda|| ||<28%-2 style="border:none;">'''Editorial Board (EB)'''||<( style="border:none;">G. Colangelo, A. Jüttner, S. Hashimoto, S. Sharpe, T. Vladikas, U. Wenger || |
||<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 || |
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||<28%-2 style="border:none;">''Quark masses''||<( style="border:none;">L. Lellouch, T. Blum, V. Lubicz || ||<28%-2 style="border:none;">''$V_{us}, V_{ud}$''||<( style="border:none;">S. Simula, P. Boyle,<<FootNote(Peter Boyle had participated actively in the early stages of the current FLAG effort.<<BR>>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 || ||<28%-2 style="border:none;">''LEC''||<( style="border:none;">S. Dürr, H. Fukaya, U.M. Heller|| ||<28%-2 style="border:none;">''$B_K$''||<( style="border:none;">H. Wittig, P. Dimopoulos, R. Mawhinney|| ||<28%-2 style="border:none;">''$f_{B_{(s)}}$, $f_{D_{(s)}}$, $B_B$''||<( style="border:none;">M. Della Morte, Y. Aoki, D. Lin|| ||<28%-2 style="border:none;">''$B_{(s)}$, $D$<<BR>>~-semileptonic and radiative decays-~''||<( style="border:none;">E. Lunghi, D. Becirevic, S. Gottlieb, C. Pena|| ||<28%-2 style="border:none;">''$\alpha_s$''||<( style="border:none;">R. Sommer, R. Horsley, T. Onogi|| |
||<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,<<BetterFootNote("Peter Boyle had participated actively in the early stages of the current FLAG effort.<<BR>>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|| |
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* in all working groups the three members must belong to three different lattice collaborations;<<FootNote(The WG on semileptonic D and B decays has currently four members, but only three of them belong to lattice collaborations.)>> | * 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.")>> |
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Several general issues concerning the present review are thoroughly discussed in Sec. 1.1 of our initial 2010 paper <<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 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 <<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]].)>>. | 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 essentially that of Sec. 1.2 of the 2013 review <<BetterFootNote(refName="aoki_1")>>. |
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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<<FootNote(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. | 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. |
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<<BetterSeeSaw(section="references", toshow="Show Text", tohide="Hide Text")>> {{{#!wiki seesaw references <<FootNote()>> }}} === Quality Criteria === The section on the quality criteria can be downloaded [[attachment:Media/FLAG_criteria.pdf|here|&do=get]]. === Quark Masses === The section on the quark masses can be downloaded [[attachment:Media/FLAG_qmass.pdf|here|&do=get]]. |
<<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]]. |
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The section on $V_{ud}$ and $V_{us}$ updated in November 2016 can be downloaded [[attachment:Media/FLAG_VudVus_webupdate.pdf|here|&do=get]]. | 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]]. |
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The section on the Low-energy constants can be downloaded [[attachment:Media/FLAG_LECs.pdf|here|&do=get]]. | 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]]. |
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The section on the Kaon mixing updated in December 2016 can be downloaded [[attachment:Media/FLAG_BK_webupdate.pdf|here|&do=get]]. | The section on the [[Kaon mixing]] updated in December 2016 can be downloaded [[attachment:Media/FLAG_BK_webupdate.pdf|here|&do=get]]. |
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The section on the $D$-meson decay constants and form factors can be downloaded [[attachment:Media/FLAG_HQD.pdf|here|&do=get]]. | 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]]. |
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The section on the $B$-meson decay constants, mixing parameters, and form factors can be downloaded [[attachment:Media/FLAG_HQB.pdf|here|&do=get]]. === The strong coupling === The section on the strong coupling $\alpha_s$ can be downloaded [[attachment:Media/FLAG_Alpha_s.pdf|here|&do=get]]. |
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]]. |
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The glossary can be downloaded [[attachment:Media/FLAG_glossary.pdf|here|&do=get]]. | The [[Glossary|glossary]] can be downloaded [[attachment:Media/FLAG_glossary.pdf|here|&do=get]]. |
Review of lattice results concerning low energy particle physics
The latest version of the complete review as of December 2017 is accessible here. It contains updated sections as follows:
$|V_{ud}|$ 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 here or from 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 here or from EPJC.
Contents
Introduction
The introduction with the updated summary tables can be downloaded here.
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 [1] and 2014 [2]. 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.
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
Citation policy
General issues
References
G. Colangelo et al., Review of lattice results concerning low energy particle physics, Eur. Phys. J. C71 (2011) 1695, arXiv:1011.4408 (1 2)
S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C74 (2014) 2890, arXiv:1310.8555. (1 2)
This is a FLAG estimate, based on $\chi$PT and the isospin averaged up- and down-quark mass $m_{ud}$. (1 2 3)
Peter Boyle had participated actively in the early stages of the current FLAG effort.
Unfortunately, due to other commitments, it was impossible for him to contribute until the end, and he decided to withdraw from the collaboration. (1)The WG on semileptonic D and B decays has currently four members, but only three of them belong to lattice collaborations. (1)
We also use terms like “quality criteria”, “rating”, “colour coding” etc. when referring to the classification of results, as described in Sec. 2. (1)
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 $V_{us}$ updated in November 2016 can be downloaded here. The original section is still available here.
Low-energy constants
The section on the Low-energy constants updated in Juy 2017 can be downloaded here. The original section is still available here.
Kaon mixing
The section on the Kaon mixing updated in December 2016 can be downloaded here. The original section is still available 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 updated in July 2017 can be downloaded here. The original section is still available here.
The strong coupling $\alpha_s$
The section on the strong coupling $\alpha_s$ can be downloaded here.
Glossary
The glossary can be downloaded here.
Notes
Notes to the various sections can be downloaded here.