Eur. Phys. J. C (2017) 77:112
DOI 10.1140/epjc/s10052-016-4509-7
Review
Review of lattice results concerning low-energy particle physics
Flavour Lattice Averaging Group (FLAG)
S. Aoki
1
, Y. Aoki
2,3,17
,D.Beˇcirevi´c
4
, C. Bernard
5
,T.Blum
3,6
, G. Colangelo
7
,M.DellaMorte
8,9
, P. Dimopoulos
10,11
,
S. Dürr
12,13
, H. Fukaya
14
, M. Golterman
15
, Steven Gottlieb
16
, S. Hashimoto
17,18
,U.M.Heller
19
,R.Horsley
20
,
A. Jüttner
21,a
, T. Kaneko
17,18
, L. Lellouch
22
,H.Leutwyler
7
,C.-J.D.Lin
22,23
, V. Lubicz
24,25
, E. Lunghi
16
,
R. Mawhinney
26
, T. Onogi
14
,C.Pena
27
, C. T. Sachrajda
21
, S. R. Sharpe
28
,S.Simula
25
, R. Sommer
29
, A. Vladikas
30
,
U. Wenger
7
, H. Wittig
31
1
Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto
606-8502, Japan
2
Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya University, Nagoya 464-8602, Japan
3
Brookhaven National Laboratory, RIKEN BNL Research Center, Upton, NY 11973, USA
4
Laboratoire de Physique Théorique (UMR8627), CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
5
Department of Physics, Washington University, Saint Louis, MO 63130, USA
6
Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA
7
Albert Einstein Center for Fundamental Physics, Institut für Theoretische Physik, Universität Bern, Sidlerstr. 5, 3012 Bern, Switzerland
8
CP3-Origins and Danish IAS, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark
9
IFIC (CSIC), c/ Catedrático José Beltrán, 2, 46980 Paterna, Spain
10
Centro Fermi-Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi Compendio del Viminale, Piazza del Viminiale 1, 00184 Rome,
Italy
11
c/o Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy
12
University of Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany
13
Jülich Supercomputing Center, Forschungszentrum Jülich, 52425 Jülich, Germany
14
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
15
Department of Physics and Astronomy, San Francisco State University, San Francisco, CA 94132, USA
16
Department of Physics, Indiana University, Bloomington, IN 47405, USA
17
High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
18
School of High Energy Accelerator Science, The Graduate University for Advanced Studies (Sokendai), Tsukuba 305-0801, Japan
19
American Physical Society (APS), One Research Road, Ridge, NY 11961, USA
20
Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, UK
21
School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK
22
Centre de Physique Théorique, UMR 7332, CNRS, Aix-Marseille Université, Université de Toulon, 13288 Marseille, France
23
Institute of Physics, National Chiao-Tung University, Hsinchu 30010, Taiwan
24
Dipartimento di Matematica e Fisica, Università Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy
25
Sezione di Roma Tre, INFN, Via della Vasca Navale 84, 00146 Rome, Italy
26
Physics Department, Columbia University, New York, NY 10027, USA
27
Departamento de Física Teórica, Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
28
Physics Department, University of Washington, Seattle, WA 98195-1560, USA
29
John von Neumann Institute for Computing (NIC), DESY, Platanenallee 6, 15738 Zeuthen, Germany
30
Sezione di Tor Vergata, INFN, c/o Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy
31
PRISMA Cluster of Excellence, Institut für Kernphysik and Helmholtz Institute Mainz, University of Mainz, 55099 Mainz, Germany
Received: 6 October 2016 / Accepted: 11 November 2016 / Published online: 17 February 2017
© The Author(s) 2017. This article is published with open access at Springerlink.com
Abstract 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 specif-
ically, we report on the determination of the light-quark
masses, the form factor f
+
(0), arising in the semileptonic
K → π transition at zero momentum transfer, as well as
a
e-mail:
juettner@soton.ac.uk
the decay constant ratio f
K
/ f
π
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 determi-
nation 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
123
112 Page 2 of 228 Eur. Phys. J. C (2017) 77 :112
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 param-
eters. 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 α
s
.
Contents
1 Introduction ..................... 3
1.1 FLAG composition, guidelines and rules .... 4
1.2 Citation policy .................. 6
1.3 General issues .................. 6
2 Quality criteria, averaging and error estimation ... 8
2.1 Systematic errors and colour code ....... 8
2.1.1 Systematic effects and rating criteria ... 9
2.1.2 Heavy-quark actions ........... 11
2.1.3 Conventions for the figures ........ 11
2.2 Averages and estimates ............. 12
2.3 Averaging procedure and error analysis .... 13
3 Quark masses ..................... 13
3.1 Masses of the light quarks ........... 15
3.1.1 Contributions from the electromagnetic
interaction
................. 15
3.1.2 Pion and kaon masses in the isospin limit 17
3.1.3 Lattice determination of m
s
and m
ud
... 18
3.1.4 Lattice determinations of m
s
/m
ud
.... 23
3.1.5 Lattice determination of m
u
and m
d
... 25
3.1.6 Estimates for R and Q .......... 28
3.2 Charm-quark mass ............... 30
3.2.1 N
f
= 2 +1 +1 results
.......... 30
3.2.2 N
f
= 2 +1 results
............ 32
3.2.3 N
f
= 2 results
.............. 32
3.2.4 Lattice determinations of the ratio m
c
/m
s
32
3.3 Bottom-quark mass ............... 34
3.3.1 N
f
= 2 +1 +1
............. 34
3.3.2 N
f
= 2 +1
................ 35
3.3.3 N
f
= 2
.................. 35
3.3.4 Averages for m
b
(m
b
) ........... 35
4 Leptonic and semileptonic kaon and pion decay
and |V
ud
| and |V
us
|
.................. 36
4.1 Experimental information concerning |V
ud
|, |V
us
|,
f
+
(0) and f
K
±
/ f
π
±
.............. 36
4.2 Lattice results for f
+
(0) and f
K
±
/ f
π
±
..... 37
4.3 Direct determination of f
+
(0) and f
K
±
/ f
π
±
.. 39
4.4 Tests of the Standard Model .......... 43
4.5 Analysis within the Standard Model ...... 44
4.6 Direct determination of f
K
±
and f
π
±
..... 45
5 Low-energy constants ................ 47
5.1 Chiral perturbation theory ............ 47
5.1.1 Quark-mass dependence of pseudoscalar
masses and decay constants
....... 48
5.1.2 Pion form factors and charge radii .... 49
5.1.3 Partially quenched and mixed action for-
mulations
................. 50
5.1.4 Correlation functions in the ǫ-regime .. 51
5.1.5 Energy levels of the QCD Hamiltonian in
a box and δ-regime
............ 52
5.1.6 Other methods for the extraction of the
low-energy constants
........... 52
5.2 Extraction of SU(2) low-energy constants ... 53
5.2.1 Results for the LO SU(2) LECs ..... 59
5.2.2 Results for the NLO SU(2) LECs .... 60
5.2.3 Epilogue ................. 61
5.3 Extraction of SU(3) low-energy constants ... 61
5.3.1 Epilogue ................. 62
6 Kaon mixing ..................... 64
6.1 Indirect CP violation and ǫ
K
in the SM
..... 64
6.2 Lattice computation of B
K
........... 67
6.3 Kaon BSM B parameters ............ 72
7 D-meson-decay constants and form factors ..... 74
7.1 Leptonic decay constants f
D
and f
D
s
..... 74
7.2 Semileptonic form factors for D → πℓν and
D → K ℓν
.................... 78
7.2.1 Results for f
+
(0)
............. 79
7.3 Determinations of |V
cd
| and |V
cs
| and test of
second-row CKM unitarity
........... 81
8 B-meson-decay constants, mixing parameters and
form factors
...................... 83
8.1 Leptonic decay constants f
B
and f
B
s
...... 85
8.2 Neutral B-meson mixing matrix elements ... 91
8.3 Semileptonic form factors for B decays to light
flavours
..................... 95
8.3.1 Parameterizations of semileptonic form
factors
................... 95
8.3.2 Form factors for B → πℓν ........ 98
8.3.3 Form factors for B
s
→ K ℓν
.......101
8.3.4 Form factors for rare and radiative B-
semileptonic decays to light flavours
...102
8.4 Semileptonic form factors for B → Dℓν, B →
D
∗
ℓν, and B → Dτν
..............104
8.4.1 B
(s)
→ D
(s)
decays
............105
8.4.2 Ratios of B → Dℓν form factors .....108
8.4.3 B → D
∗
decays
.............108
8.5 Semileptonic form factors for
b
→ pℓν and
b
→
c
ℓν
...................109
8.6 Determination of |V
ub
|
.............110
8.7 Determination of |V
cb
| .............111
9 The strong coupling α
s
................113
9.1 Introduction ...................113
9.1.1 Scheme and scale dependence of α
s
and
QCD
. 114
9.1.2 Overview of the review of α
s
.......115
9.1.3 Differences compared to the FLAG 13 report . 115
123
Eur. Phys. J. C (2017) 77 :112 Page 3 of 228 112
9.2 Discussion of criteria for computations entering
the averages
...................115
9.3 α
s
from the Schrödinger functional
.......118
9.3.1 General considerations ..........118
9.3.2 Discussion of computations .......119
9.4 α
s
from the potential at short distances
.....120
9.4.1 General considerations ..........120
9.4.2 Discussion of computations .......120
9.5 α
s
from the vacuum polarization at short distances
122
9.5.1 General considerations ..........122
9.5.2 Discussion of computations .......122
9.6 α
s
from observables at the lattice-spacing scale
123
9.6.1 General considerations ..........123
9.6.2 Continuum limit .............123
9.6.3 Discussion of computations .......124
9.7 α
s
from current 2-point functions
........125
9.7.1 General considerations ..........125
9.7.2 Discussion of computations .......127
9.8 α
s
from QCD vertices
..............128
9.8.1 General considerations ..........128
9.8.2 Discussion of computations .......129
9.9 Summary ....................129
9.9.1 The present situation ...........129
9.9.2 Our range for α
(5)
MS
............131
9.9.3 Ranges for [r
0
]
(N
f
)
and
MS
......134
9.9.4 Conclusions ................135
Appendix A: Glossary ..................136
A.1 Lattice actions ..................136
A.1.1 Gauge actions ...............136
A.1.2 Light-quark actions ............136
A.1.3 Heavy-quark actions ...........140
A.2 Setting the scale .................146
A.3 Matching and running ..............147
A.4 Chiral extrapolation ...............148
A.5 Summary of simulated lattice actions ......149
Appendix B: Notes ...................151
B.1 Notes to Sect. 3 on quark masses ........151
B.2 Notes to Sect. 4 on |V
ud
| and |V
us
|
.......159
B.3 Notes to Sect. 5 on low-energy constants ....166
B.4 Notes to Sect. 6 on Kaon mixing ........171
B.4.1 Kaon B-parameter B
K
..........171
B.4.2 Kaon BSM B-parameters .........177
B.5 Notes to Sect. 7 on D-meson-decay constants
and form factors
.................179
B.5.1 D
(s)
-meson-decay constants
.......179
B.5.2 D → πℓν and D → K ℓν form factors . 185
B.6 Notes to Sect. 8 on B-meson-decay constants and
mixing parameters
...............186
B.6.1 B
(s)
-meson-decay constants
.......186
B.6.2 B
(s)
-meson mixing matrix elements
...193
B.6.3 Form factors entering determinations of
|V
ub
| (B → πlν, B
s
→ Klν,
b
→ plν)
197
B.6.4 Form factors for B → K ℓ
+
ℓ
−
......200
B.6.5 Form factors entering determinations of
|V
cb
| (B → D
∗
lν, B → Dlν, B
s
→
D
s
lν,
b
→
c
lν) and R(D))
......201
B.7 Notes to Sect. 9 on the strong coupling α
s
....204
B.7.1 Renormalization scale and perturbative
behaviour
.................204
B.7.2 Continuum limit .............207
References210
1 Introduction
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 sig-
natures of new physics through precision measurements and
in terms of attempts to construct the theoretical framework
behind direct discoveries of new particles. A major theoret-
ical 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 Lat-
tice Averaging Group (FLAG) is to review the current sta-
tus 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 cur-
rently 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 collabo-
rations 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 consis-
tency 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 semilep-
tonic K → π transitions (evaluated at zero momentum trans-
123
112 Page 4 of 228 Eur. Phys. J. C (2017) 77 :112
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
MS scheme at
running scale μ = 2 GeV or as indicated; the other quantities listed are
specified in the quoted sections. For each result we list the references
that entered the FLAG average or estimate. From the entries in this col-
umn one can also read off the number of results that enter our averages
for each quantity. 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
Quantity Sects. N
f
= 2 +1 +1Refs. N
f
= 2 +1Refs. N
f
= 2Refs.
m
s
[MeV] 3.1.3 93.9(1.1) [4,5]92.0(2.1) [6–10] 101(3) [11,12]
m
ud
[MeV]
3.1.3 3.70(17) [4]3.373(80) [7–10,13]3.6(2) [11]
m
s
/m
ud
3.1.4 27.30(34) [4,14]27.43(31) [6–8,10]27.3(9) [11]
m
u
[MeV]
3.1.5 2.36(24) [4]2.16(9)(7)
a
2.40(23) [16]
m
d
[MeV]
3.1.5 5.03(26) [4]4.68(14)(7)
a
4.80(23) [16]
m
u
/m
d
3.1.5 0.470(56) [4]0.46(2)(2)
a
0.50(4) [16]
m
c
(3GeV) [GeV] 3.2 0.996(25) [4,5]0.987(6) [9,17]1.03(4) [11]
m
c
/m
s
3.2.4 11.70(6) [4,5,14]11.82(16) [17,18]11.74(35) [11,132]
m
b
(m
b
) [GeV] 3.3.4 4.190(21) [5,19]4.164(23) [9]4.256(81) [20,21]
f
+
(0)
4.3 0.9704(24)(22) [22]0.9677(27) [23,24]0.9560(57)(62) [25]
f
K
±
/ f
π
±
4.3 1.193(3) [14,26,27]1.192(5) [28–31]1.205(6)(17) [32]
f
π
±
[MeV]
4.6 130.2(1.4) [28,29,31]
f
K
±
[MeV]
4.6 155.6(4) [14,26,27] 155.9(9) [28,29,31] 157.5(2.4) [32]
1/3
[MeV]
5.2.1 280(8)(15) [33] 274(3) [10,13,34,35] 266(10) [33,36–38]
F
π
/F
5.2.1 1.076(2)(2) [39]1.064(7) [10,29,34,35,40]1.073(15) [36–38,41]
¯
ℓ
3
5.2.2 3.70(7)(26) [39]2.81(64) [10,29,34,35,40]3.41(82) [36,37,41]
¯
ℓ
4
5.2.2 4.67(3)(10) [39]4.10(45) [10,29,34,35,40]4.51(26) [36,37,41]
¯
ℓ
6
5.2.2 15.1(1.2) [37,41]
ˆ
B
K
6.1 0.717(18)(16) [42]0.7625(97) [10,43–45]0.727(22)(12) [46]
a
This is a FLAG estimate, based on χPT and the isospin averaged up- and down-quark mass m
ud
[
7–10,13]
fer), the decay constants f
K
and f
π
, and the B
K
parameter
from neutral kaon mixing. Their implications for the CKM
matrix elements V
us
and V
ud
were also discussed. Further-
more, results were reported for some of the low-energy con-
stants of SU(2)
L
× SU(2)
R
and SU(3)
L
× SU(3)
R
Chi-
ral 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 parame-
ters. 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 sta-
tus of lattice results on the QCD coupling α
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 Tables
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
http://itpwiki.unibe.ch/flag [3], 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.
This review is organized as follows. In the remainder of
Sect.
1 we summarize the composition and rules of FLAG
and discuss general issues that arise in modern lattice calcu-
lations. In Sect.
2 we explain our general methodology for
evaluating the robustness of lattice results. We also describe
the procedures followed for combining results from different
collaborations in a single average or estimate (see Sect.
2.2
for our definition of these terms). The rest of the paper con-
sists of sections, each dedicated to a single (or groups of
closely connected) physical quantity(ies). Each of these sec-
tions is accompanied by an Appendix with explicatory notes.
1.1 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.
123
Eur. Phys. J. C (2017) 77 :112 Page 5 of 228 112
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 list the references that entered the FLAG average or estimate. From
the entries in this column one can also read off the number of results that
enter our averages for each quantity. 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
Quantity Sects. N
f
= 2 +1 +1Refs. N
f
= 2 +1Refs. N
f
= 2Refs.
f
D
[MeV] 7.1 212.15(1.45) [14,27] 209.2(3.3) [47,48] 208(7) [20]
f
D
s
[MeV]
7.1 248.83(1.27) [14,27] 249.8(2.3) [17,48,49] 250(7) [20]
f
D
s
/ f
D
7.1 1.1716(32) [14,27]1.187(12) [47,48]1.20(2) [20]
f
Dπ
+
(0) 7.2 0.666(29) [50]
f
DK
+
(0)
7.2 0.747(19) [51]
f
B
[MeV]
8.1 186(4) [52] 192.0(4.3) [48,53–56] 188(7) [20,57,58]
f
B
s
[MeV]
8.1 224(5) [52] 228.4(3.7) [48,53–56] 227(7) [20,57,58]
f
B
s
/ f
B
8.1 1.205(7) [52]1.201(16) [48,53–55]1.206(23) [20,57,58]
f
B
d
ˆ
B
B
d
[MeV]
8.2 219(14) [54,59] 216(10) [20]
f
B
s
ˆ
B
B
s
[MeV]
8.2 270(16) [54,59] 262(10) [20]
ˆ
B
B
d
8.2 1.26(9) [54,59]1.30(6) [20]
ˆ
B
B
s
8.2 1.32(6) [54,59]1.32(5) [20]
ξ
8.2 1.239(46) [54,60]1.225(31) [20]
B
B
s
/B
B
d
8.2 1.039(63) [54,60]1.007(21) [20]
Quantity Sects. N
f
= 2 +1andN
f
= 2 +1 +1Refs.
α
(5)
MS
(M
Z
)
9.9 0.1182(12) [5,9,61–63]
(5)
MS
[MeV]
9.9 211(14) [5,9,61–63]
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 rôle 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 activi-
ties 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 uni-
form 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. Golter-
man, H. Leutwyler, and C. Sachrajda
• Editorial Board (EB): G. Colangelo, A. Jüttner,
S. Hashimoto, S. Sharpe, A. Vladikas, and U. Wenger
• Working Groups (coordinator listed first):
– Quark masses L. Lellouch, T. Blum, and V. Lubicz
– V
us
, V
ud
S. Simula, P. Boyle,
1
and T. Kaneko
– LEC S. Dürr, H. Fukaya, and U.M. Heller
– B
K
H. Wittig, P. Dimopoulos, and R. Mawhinney
– f
B
(s)
, f
D
(s)
, B
B
M. Della Morte, Y. Aoki, and D. Lin
– B
(s)
, D semileptonic and radiative decays E. Lunghi,
D. Becirevic, S. Gottlieb, and C. Pena
– α
s
R. Sommer, R. Horsley, and 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 geographi-
cal areas in which lattice collaborations are active, with
members from America, Asia/Oceania and Europe;
1
Peter Boyle had participated actively in the early stages of the current
FLAG effort. Unfortunately, due to other commitments, it was impos-
sible for him to contribute until the end, and he decided to withdraw
from the collaboration.
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