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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 <<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. 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. |
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. |
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||<-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. BSM bag parameters $B_{2,3,4,5}$ are given in the $\overline{\text{MS}}$ scheme at $\mu=3$ GeV; 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 emphasize that these numbers only give a very rough indication of how thoroughly the quantity in question has been explored on the lattice and 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.|| | ||<-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 are given in the RGI scheme, except for those for $N_f=2$ which are given in the $\overline{\rm MS}$ scheme at running scale $\mu=2\,$GeV. Results for the quark condensate are given in the $\overline{\rm MS}$ scheme with $\mu=2\,$GeV. 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. For each result we provide the list of references that entered the FLAG average or estimate in the bib-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.|| |
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|| $\overline{m}_c^{\rm RGI}$ [GeV] || [[Quark masses|3.2.2]] || 1.521(17)(14) || 1.529(9)(14) || || [[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]] || 4.190(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]] || |
|| ${m}_c^{\rm RGI}$ [GeV] || [[Quark masses|3.2.2]] || 1.521(17)(14) || 1.529(9)(14) || || [[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]] || || ${m}_b^{\rm RGI}$ [GeV] || [[Quark masses|3.3]] || 6.936(20)(54) || 6.874(38)(54) || || [[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]] || 1.1932(19) || 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]] || |
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||<-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.|| | ||<-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 bib-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.|| |
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|| $f_D$ [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]] || |
|| $f_D$ [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]] || |
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|| $\xi$ || [[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. |
|| $\xi$ || [[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:>0.11823(81) || ||[[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 bib-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 {\bf 30 September 2018}. <<BetterFootNote("Working groups were given the option of including papers submitted to {\tt 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. hree papers of this type were included: Ref. [5] in Secs. 7 and 8, and Refs. [6] and [7] in Sec. 10.")>> |
FLAG Review 2019
The 2019 edition of the FLAG review can be downloaded here.
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.
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 EPJC. The 2016/2017 updates are available from here.
The 2013/2014 review is accessible here or from EPJC.
Contents
- Introduction
- Quality criteria
- Quark masses
- $\small{V_{ud}}$ and $\small{V_{us}}$
- Low-energy constants
- Kaon mixing
- $\small{D}$-meson decay constants and form factors
- $\small{B}$-meson decay constants, mixing parameters, and form factors
- The strong coupling $\alpha_s$
- Nucleon matrix elements
- Glossary
- Notes
Introduction
The introduction with the updated summary tables can be downloaded here.
We review lattice results related to pion, kaon, $D$-meson, $B$-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $f_+(0)$ arising in the semileptonic $K \to \pi$ transition at zero momentum transfer, as well as the decay constant ratio $f_K/f_\pi$ and its consequences for the CKM matrix elements $V_{us}$ and $V_{ud}$. 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 [1], 2013 [2], and 2016 [3] (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.
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 {\bf 30 September 2018}. [4]
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)
S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C77 (2017) 112, arXiv:1607.00299. (1)
Working groups were given the option of including papers submitted to {\tt 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. hree papers of this type were included: Ref. [5] in Secs. 7 and 8, and Refs. [6] and [7] in Sec. 10. (1)
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 $V_{us}$ can be downloaded here.
Low-energy constants
The section on the Low-energy constants can be downloaded here.
Kaon mixing
The section on the Kaon mixing can be downloaded here.
$\small{D}$-meson decay constants and form factors
The section on the $D$-meson decay constants and form factors can be downloaded here.
$\small{B}$-meson decay constants, mixing parameters, and form factors
The section on the $B$-meson decay constants, mixing parameters, and form factors can be downloaded here.
The strong coupling $\alpha_s$
The section on the strong coupling $\alpha_s$ can be downloaded here.
Nucleon matrix elements
The section on nucleon matrix elements can be downloaded here.
Glossary
The glossary can be downloaded here.
Notes
Notes to the various sections can be downloaded here.