|
Size: 1797
Comment:
|
Size: 20452
Comment:
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 1: | Line 1: |
| === Review of lattice results concerning low energy particle physics === | == Review of lattice results concerning low energy particle physics == |
| Line 3: | Line 3: |
| The latest version of the complete review as of January 2017 is accessible XXX here. It contains the new section updated in November 2016 on leptonic and semileptonic kaon and pion decay and $|Vud|$ and $|Vus|$ , and the new section updated in December 2016 on kaon mixing. 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. |
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. |
| Line 8: | Line 5: |
| 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. | 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]]. <<BetterTableOfContents(mindepth=3)>> === Introduction === The introduction can be downloaded [[attachment:Media/FLAG_introduction.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 <<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. In the present paper we provide updated results for all the above-mentioned quantities, but also extend the scope of the review in two ways. First, we now present results for the charm and bottom quark masses, in addition to those of the three lightest quarks. Second, we review results obtained for the kaon mixing matrix elements of new operators that arise in theories of physics beyond the Standard Model. Our main results are collected in Tabs. 1 and 2. <<BetterSeeSaw(section="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, 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) || <<FootNote(This is a FLAG estimate, based on $\chi$PT and the isospin averaged up- and down-quark mass $m_{ud}$.)>> || || $m_d$ [MeV] || [[Quark masses|3.1.5]] || 5.03(26) || 4.68(14)(7) || 4.80(23) || <<FootNote(This is a FLAG estimate, based on $\chi$PT and the isospin averaged up- and down-quark mass $m_{ud}$.)>> || || $m_u/m_d$ || [[Quark masses|3.1.5]] || 0.470(56) || 0.46(2)(2) || 0.50(4) || <<FootNote(This is a FLAG estimate, based on $\chi$PT and the isospin averaged up- and down-quark mass $m_{ud}$.)>> || || $\overline{m}_c$(3 GeV) [GeV] || [[Quark masses|3.2.3]] || 0.996(25) || 0.987(6) || 1.03(4) || [[attachment:Media/Mc.bib|bib|&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]] || || $f_+(0)$ || [[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]] || }}} |
| Line 11: | Line 52: |
| $S_\text{G} = \beta\sum_{x} \sum_{\mu<\nu}\Big(1-\frac{1}{3}\text{Re Tr}\,W_{\mu\nu}^{1\times1}(x)\Big)$ | <<BetterSeeSaw(section="table2", toshow="Show Table 2", tohide="Hide Table 2")>> {{{#!wiki seesaw table2 |
| Line 13: | Line 55: |
| `$W_{\mu\nu}^{1\times1}(x) \equiv U_\mu(x)U_\nu(x+a\hat{\mu})){U_\mu(x+a\hat{\nu})}^{-1}$` | ||<-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]] || || 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]] || |
| Line 15: | Line 75: |
| $W_{\mu\nu}^{1\times1}(x) \equiv U_\mu(x)U_\nu(x+a\hat{\mu})U_\mu(x+a\hat{\nu})^{-1} U_\nu(x)^{-1}.$ | }}} 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 ==== <<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 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: ||<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 || ||<-3 style="border:none;">'''Working Groups (WG)''' || ||<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. 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|| 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;<<FootNote(The WG on semileptonic D and B decays has currently four members, but only three of them belong to 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 ==== ==== General issues ==== ==== References ==== <<FootNote()>> === Quality Criteria === === Quark Masses === === $V_{ud}$ and $V_{us}$ === === Low-energy constants === === Kaon mixing === === D-meson decay constants and form factors === === B-meson decay constants, mixing parameters, and form factors === === The strong coupling $\alpha_s$ === === Glossary === === Notes === |
Review of lattice results concerning low energy particle physics
The latest version of the complete review as of January 2017 is accessible here. 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 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 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.
Table 1 |
|||||
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. |
|||||
Quantity |
Sec. |
$N_f=2+1+1$ |
$N_f=2+1$ |
$N_f=2$ |
Refs. |
$m_s$ [MeV] |
93.9(1.1) |
92.0(2.1) |
101(3) |
||
$m_{ud}$ [MeV] |
3.70(17) |
3.373(80) |
3.6(2) |
||
$m_s/m_{ud}$ |
27.30(34) |
27.43(31) |
27.3(9) |
||
$m_u$ [MeV] |
2.36(24) |
2.16(9)(7) |
2.40(23) |
||
$m_d$ [MeV] |
5.03(26) |
4.68(14)(7) |
4.80(23) |
||
$m_u/m_d$ |
0.470(56) |
0.46(2)(2) |
0.50(4) |
||
$\overline{m}_c$(3 GeV) [GeV] |
0.996(25) |
0.987(6) |
1.03(4) |
||
$m_c/m_s$ |
11.70(6) |
11.82(16) |
11.74(35) |
||
$\overline{m}_b(\overline{m}_b)$ [GeV] |
4.190(21) |
4.164(23) |
4.256(81) |
||
$f_+(0)$ |
0.9704(24)(22) |
0.9677(27) |
0.9560(57)(62) |
||
$f_{K^\pm}/f_{\pi^\pm}$ |
1.193(3) |
1.192(5) |
1.205(6)(17) |
||
$f_{\pi^\pm}$ [MeV] |
|
130.2(1.4) |
|
||
$f_{K^\pm}$ [MeV] |
155.6(4) |
155.9(9) |
157.5(2.4) |
||
$\Sigma^{1/3}$ [MeV] |
280(8)(15) |
274(3) |
266(10) |
||
$F_\pi/F$ [MeV] |
1.076(2)(2) |
1.064(7) |
1.073(15) |
||
$\bar{\ell}_3$ [MeV] |
3.70(7)(26) |
2.81(64) |
3.41(82) |
||
$\bar{\ell}_4$ [MeV] |
4.67(3)(10) |
4.10(45) |
4.51(26) |
||
$\bar{\ell}_6$ [MeV] |
|
|
15.1(1.2) |
||
$\hat B_K$ [MeV] |
0.717(18)(16) |
0.7625(97) |
0.727(22)(12) |
||
Table 2 |
|||||
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. |
|||||
Quantity |
Sec. |
$N_f=2+1+1$ |
$N_f=2+1$ |
$N_f=2$ |
Refs. |
$f_D$ [MeV] |
212.15(1.45) |
209.2(3.3) |
208(7) |
||
$f_{D_s}$ [MeV] |
248.83(1.27) |
249.8(2.3) |
250(7) |
||
$f_{D_s}/f_D$ |
1.1716(32) |
1.187(12) |
1.20(2) |
||
$f_+^{D\pi}(0)$ [MeV] |
|
0.666(29) |
|
||
$f_+^{DK}(0)$ [MeV] |
|
0.747(19) |
|
||
$f_B$ [MeV] |
186(4) |
192.0(4.3) |
188(7) |
||
$f_{B_s}$ [MeV] |
224(5) |
228.4(3.7) |
227(7) |
||
$f_{B_s}/f_B$ |
1.205(7) |
1.201(16) |
1.206(23) |
||
$f_{B_d}\sqrt{\hat{B}_{B_d}}$ [MeV] |
|
219(14) |
216(10) |
||
$f_{B_s}\sqrt{\hat{B}_{B_s}}$ [MeV] |
|
270(16) |
262(10) |
||
$\hat{B}_{B_d}$ |
|
1.26(9) |
1.30(6) |
||
$\hat{B}_{B_s}$ |
|
1.32(6) |
1.32(5) |
||
$\xi$ |
|
1.239(46) |
1.225(31) |
||
$B_{B_s}/B_{B_d}$ |
|
1.039(63) |
1.007(21) |
||
$\alpha^{(5)}_{\overline{\text{MS} } }(M_Z)$ |
0.1182(12) |
|
|||
$\Lambda^{(5)}_{\overline{\text{MS} } }$ [MeV] |
211(14) |
|
|||
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
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:
Advisory Board (AB) |
S. Aoki, C. Bernard, M. Golterman, H. Leutwyler, C. Sachrajda |
|
Editorial Board (EB) |
G. Colangelo, A. Jüttner, S. Hashimoto, S. Sharpe, T. Vladikas, U. Wenger |
|
Working Groups (WG) |
||
Quark masses |
L. Lellouch, T. Blum, V. Lubicz |
|
$V_{us}, V_{ud}$ |
S. Simula, P. Boyle,4 T. Kaneko |
|
LEC |
S. Dürr, H. Fukaya, U.M. Heller |
|
$B_K$ |
H. Wittig, P. Dimopoulos, R. Mawhinney |
|
$f_{B_{(s)}}$, $f_{D_{(s)}}$, $B_B$ |
M. Della Morte, Y. Aoki, D. Lin |
|
$B_{(s)}$, $D$ |
E. Lunghi, D. Becirevic, S. Gottlieb, C. Pena |
|
$\alpha_s$ |
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;5
- 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
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)
S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C74 (2014) 2890, arXiv:1310.8555. (2)
This is a FLAG estimate, based on $\chi$PT and the isospin averaged up- and down-quark mass $m_{ud}$. (3 4 5)
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. (6)
The WG on semileptonic D and B decays has currently four members, but only three of them belong to lattice collaborations. (7)
Quality Criteria
Quark Masses
$V_{ud}$ and $V_{us}$
Low-energy constants
Kaon mixing
D-meson decay constants and form factors
B-meson decay constants, mixing parameters, and form factors
The strong coupling $\alpha_s$
Glossary