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### 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 $|Vud|$ and $|Vus|$, and the new section updated in December 2016 on kaon mixing.

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.

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### Introduction

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)

### References

1. 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. (2)

3. This is a FLAG estimate, based on $\chi$PT and the isospin averaged up- and down-quark mass $m_{ud}$. (3 4 5)

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