Studies of b quark decays using experiment plus lattice QCD Matthew - - PowerPoint PPT Presentation
Studies of b quark decays using experiment plus lattice QCD Matthew Wingate DAMTP, University of Cambridge Particle Physics Seminar, University of Birmingham, 7 February 2018 Outline Quark flavour & Lattice QCD DiRAC facility
Matthew Wingate DAMTP, University of Cambridge
Particle Physics Seminar, University of Birmingham, 7 February 2018
2
interactions
physics
precision, new modes
inferences about quark interactions
4
5 CKM Fitter
Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb CKM matrix
Bc → J/ `⌫ B0
(s) − ¯
B0
(s)
D → ⇡`⌫ K → ⇡`⌫ B(s) → D(∗)
(s)`⌫
B → ⇡`⌫ D → K`⌫
u d′ W + e+ νe
1 − λ2/2 λ Aλ3(ρ − iη) −λ 1 − λ2/2 Aλ2 Aλ3(1 − ρ − iη) −Aλ2 1 + O(λ4)
=
tree
6
Lquark = ¯ Qi
L i /
D Qi
L + ¯
ui
R i /
D ui
R + ¯
di
R i /
D di
R
Jµ,+
weak = ¯
ui
Lγµdi L
Qi
L =
di
ui
R
di
R
Jµ,+
weak = ¯
ui
LγµV ij CKMdj L
Lquark,φ = − √ 2
d ¯
Qi
L dj R + ij u ¯
Qi
La ab† b uj R + h.c.
−
d ¯
di
Ldi R + mi u¯
ui
Lui R + h.c.
RH SU(2) singlets Interact with gauge bosons in covariant derivative Gives rise to weak current The coupling to the Higgs field is not apparently diagonal in generation Fields may be transformed to mass basis Showing the weak current allows mixing between generations
6
Lquark = ¯ Qi
L i /
D Qi
L + ¯
ui
R i /
D ui
R + ¯
di
R i /
D di
R
Jµ,+
weak = ¯
ui
Lγµdi L
Qi
L =
di
ui
R
di
R
Jµ,+
weak = ¯
ui
LγµV ij CKMdj L
Lquark,φ = − √ 2
d ¯
Qi
L dj R + ij u ¯
Qi
La ab† b uj R + h.c.
−
d ¯
di
Ldi R + mi u¯
ui
Lui R + h.c.
RH SU(2) singlets Interact with gauge bosons in covariant derivative Gives rise to weak current The coupling to the Higgs field is not apparently diagonal in generation Fields may be transformed to mass basis Showing the weak current allows mixing between generations
6
Lquark = ¯ Qi
L i /
D Qi
L + ¯
ui
R i /
D ui
R + ¯
di
R i /
D di
R
Jµ,+
weak = ¯
ui
Lγµdi L
Qi
L =
di
ui
R
di
R
Jµ,+
weak = ¯
ui
LγµV ij CKMdj L
Lquark,φ = − √ 2
d ¯
Qi
L dj R + ij u ¯
Qi
La ab† b uj R + h.c.
−
d ¯
di
Ldi R + mi u¯
ui
Lui R + h.c.
RH SU(2) singlets Interact with gauge bosons in covariant derivative Gives rise to weak current The coupling to the Higgs field is not apparently diagonal in generation Fields may be transformed to mass basis Showing the weak current allows mixing between generations
6
Lquark = ¯ Qi
L i /
D Qi
L + ¯
ui
R i /
D ui
R + ¯
di
R i /
D di
R
Jµ,+
weak = ¯
ui
Lγµdi L
Qi
L =
di
ui
R
di
R
Jµ,+
weak = ¯
ui
LγµV ij CKMdj L
Lquark,φ = − √ 2
d ¯
Qi
L dj R + ij u ¯
Qi
La ab† b uj R + h.c.
−
d ¯
di
Ldi R + mi u¯
ui
Lui R + h.c.
RH SU(2) singlets Interact with gauge bosons in covariant derivative Gives rise to weak current The coupling to the Higgs field is not apparently diagonal in generation Fields may be transformed to mass basis Showing the weak current allows mixing between generations
6
Lquark = ¯ Qi
L i /
D Qi
L + ¯
ui
R i /
D ui
R + ¯
di
R i /
D di
R
Jµ,+
weak = ¯
ui
Lγµdi L
Qi
L =
di
ui
R
di
R
Jµ,+
weak = ¯
ui
LγµV ij CKMdj L
Lquark,φ = − √ 2
d ¯
Qi
L dj R + ij u ¯
Qi
La ab† b uj R + h.c.
−
d ¯
di
Ldi R + mi u¯
ui
Lui R + h.c.
RH SU(2) singlets Interact with gauge bosons in covariant derivative Gives rise to weak current The coupling to the Higgs field is not apparently diagonal in generation Fields may be transformed to mass basis Showing the weak current allows mixing between generations
7
Illustration from I. Shipsey, Nature 427, 591 (2004)
Models Experiment
7
Illustration from I. Shipsey, Nature 427, 591 (2004)
Models Experiment Lattice QCD
to turn a quantum physics problem into a statistical physics problem
glue + sea of quark-antiquark bubbles
methods: importance sampling & correlation functions
has a diverging condition number as amq ➙ 0 (vanishing lattice spacing × light quark mass)
8
9
⟨J(z′)J(z)⟩ = 1 Z
ψ][dU] J(z′)J(z) e−SE ⟨J(z′)J(z)⟩ = 1 Z Tr
QFT : Imaginary-time path integral SFT : Sum over all microstates
Monte Carlo Calculation : Find and use field “configurations” which dominate the integral/sum
10
= 1 Z
Gluonic expectation values
⟨Θ⟩ = 1 Z
ψ][dU] Θ[U] e−Sg[U]− ¯
ψQ[U]ψ
Fermionic expectation values
⟨ ¯ ψΓψ⟩ =
δ¯ ζ Γ δ δζ e−¯
ζQ−1[U]ζ det Q[U]e−Sg[U]
ζ → 0
10
Probability weight = 1 Z
Gluonic expectation values
⟨Θ⟩ = 1 Z
ψ][dU] Θ[U] e−Sg[U]− ¯
ψQ[U]ψ
Fermionic expectation values
⟨ ¯ ψΓψ⟩ =
δ¯ ζ Γ δ δζ e−¯
ζQ−1[U]ζ det Q[U]e−Sg[U]
ζ → 0
10
Probability weight = 1 Z
Gluonic expectation values
⟨Θ⟩ = 1 Z
ψ][dU] Θ[U] e−Sg[U]− ¯
ψQ[U]ψ
Fermionic expectation values
⟨ ¯ ψΓψ⟩ =
δ¯ ζ Γ δ δζ e−¯
ζQ−1[U]ζ det Q[U]e−Sg[U]
ζ → 0
Determinant in probability weight difficult 1) Requires nonlocal updating; 2) Matrix becomes singular
10
Partial quenching = different mass for valence than for sea Q−1 det Q
Probability weight = 1 Z
Gluonic expectation values
⟨Θ⟩ = 1 Z
ψ][dU] Θ[U] e−Sg[U]− ¯
ψQ[U]ψ
Fermionic expectation values
⟨ ¯ ψΓψ⟩ =
δ¯ ζ Γ δ δζ e−¯
ζQ−1[U]ζ det Q[U]e−Sg[U]
ζ → 0
Determinant in probability weight difficult 1) Requires nonlocal updating; 2) Matrix becomes singular
Lattice volume Lattice spacing Heavy quark mass Light quark mass
L 1/mπ
a 1/ΛQCD
mQ 1/a
mπ mρ, 4πfπ
Chiral pert. th. Brute force Chiral pert. th. Brute force Symanzik EFT NRQCD, HQET
Extra-fine, extra-improvement
mQ < 1/a mQ ≈ 1/a
Fermilab
Systematic error Controllable limit Theory
hΦπ(z)Vµ(y)ΦB(x)i = 1 Z Z Z Z [dψ][d ¯ ψ][dU] Φπ(z)Vµ(y)ΦB(x) e−S[ψ, ¯
ψ,U]
CTH Davies, [HPQCD Collaboration website]
12
CTH Davies, [HPQCD Collaboration website]
13
14
s W W t ν ℓ ℓ b
t W γ, Z s b
Flavour changing neutral decays
B → K∗`+`− Bs → `+`−
Horgan et al., (HPQCD) arXiv:1310.3722, arXiv:1310.3887
penguin box
Image credit: CIA World Factbook
international collaborations (e.g. HPQCD, RBC-UKQCD, HadSpec, QCDSF, FastSum)
flavour, hadron spectrum, hot/ dense QCD; BSM theories of EWSB, dark matter
Belle, JLab, J-PARC, FAIR, RHIC, NA62
16
facility for particle & nuclear physics, cosmology, & theoretical astrophysics. Recurrent costs funded by STFC
coupled clusters with various levels of interconnectivity, memory, and fast I/O (Cambridge, Durham, Leicester)
17
20/7/2017)
35,365 citations (as of 20/7/2017)
exoplanets, MHD, particle pheno, nuclear physics
18
DiRAC-2 x10
19
DiRAC&3((2016/17(–(TBC)(
Extreme Scaling Data Intensive
Memory Intensive
Data Management
Internet Analytics
Many-Core Coding Data Analytics Programming
Fine Tuning Parallel Management Multi-threading
Disaster Recovery
Data Handling Archiving Tightly(coupled( compute(&(storage:( confronta7on(of( complex(simula7ons( with(large(data(sets!! Maximal( computa7onal( effort(applied( to(a(problem(of( fixed(size( Larger(memory(footprint(per(node:(problem( size(grows(with(increasing(machine(power((
and electricity
research and HPC training impact with PDRA and PhD support (Big Data CDTs)
e‑Infrastructure roadmap
20
2011/12 2018/19
2016/17 DiRAC 2.5 2017 DiRAC 2.5x Stop-gap funding:
access to Sandybridge system
0.5 Pflop/s KNL service
cores
21
After £1.67M capital injection
nodes (1-6 TB); NVMe storage for data intensive workflows
22
June 2017: £9M capital funding (BEIS), lifeline to DiRAC3:
costs through STFC
Lattice QCD (and BSM) efforts
experiment
23
25
b d
e− νe
W −
c d
¯ B0 D∗+
Vcb
26
0.036 0.038 0.040 0.042 0.044 0.046 0.048
|Vcb|
B → D FNAL/MILC, 2015 B → D HPQCD, 2015 B → D* FNAL/MILC, 2014 Inclusive Alberti et al., 2015 combined fit FLAG, 2016 (incl. new expt data)
(before 2/2017)
27
Source f+(%) f0(%) Statistics+matching+χPT cont. extrap. 1.2 1.1 (Statistics) (0.7) (0.7) (Matching) (0.7) (0.7) (χPT/cont. extrap.) (0.6) (0.5) Heavy-quark discretization 0.4 0.4 Lattice scale r1 0.2 0.2 Total error 1.2 1.1
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1 1.1 1.2 1.3 1.4 1.5 1.6 w z-parameterization f0 z-parameterization f+ BaBar ’10 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1 1.1 1.2 1.3 1.4 1.5 1.6 w simulated extrapolated
q
2 [GeV 2] 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4f0 and f+
BaBar 2010TABLE V. Error budget table for |Vcb|. The first three rows are from experiments, and the rest are from lattice simula- tions. Type Partial errors [%] experimental statistics 1.55 experimental systematic 3.3 meson masses 0.01 lattice statistics 1.22 chiral extrapolation 1.14 discretization 2.59 kinematic 0.96 matching 2.11 electro-weak 0.48 finite size effect 0.1 total 5.34
Fermilab/ MILC HPQCD
Na et al. (HPQCD), arXiv:1505.03925 Bailey et al. (FNAL/MILC), arXiv:1503.07237
0.02 0.062 0.092 0.122 0.152
a2/fm2
1002 2002 3002 4002 5002
m2
π/MeV2
MILC nf = 2 + 1
28
0.1 0.2 0.3
mπ
2 (GeV 2)
0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 hA1(1) a ≅ 0.15 fm a ≅ 0.12 fm a ≅ 0.09 fm a ≅ 0.06 fm a ≅ 0.045 fm extrapolated value
TABLE X. Final error budget for hA1ð1Þ where each error is discussed in the text. Systematic errors are added in quadrature and combined in quadrature with the statistical error to obtain the total error. Uncertainty hA1ð1Þ Statistics 0.4% Scale (r1) error 0.1% χPT fits 0.5% gDDπ 0.3% Discretization errors 1.0% Perturbation theory 0.4% Isospin 0.1% Total 1.4%
FNAL/MILC
B → D∗`⌫
Bailey et al. (FNAL/MILC), PRD89 (2014)
nature
0.02 0.062 0.092 0.122 0.152
a2/fm2
1002 2002 3002 4002 5002
m2
π/MeV2
MILC(HISQ) nf = 2 + 1 + 1
B → D* l ν
nature
Glue-field ensembles used
lattice spacing light quark mass
Judd Harrison, Christine Davies, MBW (HPQCD), arXiv:1711.11013
calculations from Fermilab/ MILC
29
30
F(1) = hA1(1) = MB + MD∗ 2√MBMD∗ A1(q2
max)
χ(1) = 1 dΓ dw( ¯ B0 → D∗+l−¯ νl) = G2
F M 3 D∗|¯
ηEW Vcb|2 4π3 (MB − MD∗)2p w2 − 1 χ(w)|F(w)|2
hD⇤(p0, ✏)|¯ qµQ|B(p)i = 2iV (q2) MB + MD∗ ✏µνρσ✏⇤
νp0 ρpσ
hD⇤(p0, ✏)|¯ qµ5Q|B(p)i = 2MD∗A0(q2)✏⇤ · q q2 qµ + (MB + MD∗)A1(q2) h ✏⇤µ ✏⇤ · q q2 qµi A2(q2) ✏⇤ · q MB + MD∗ h pµ + p0µ M 2
B M 2 D∗
q2 qµi .
31
Continuum- physical mass curve
FB→D∗(1) = hA1(1) = 0.895(10)stat(24)sys
32
Continuum- physical mass curve
→ FBs→D∗
s(1) = hs
A1(1) = 0.883(12)stat(28)sys
33
Continuum- physical mass curve
34
FB→D∗(1) = hA1(1) = 0.895(10)stat(24)sys
→ FBs→D∗
s(1) = hs
A1(1) = 0.883(12)stat(28)sys
FB→D∗(1) FBs→D∗
s(1) = hA1(1)
hs
A1(1) = 1.013(14)stat(17)sys
Uncertainty hA1(1) hs
A1(1) hA1(1)/hs A1(1)
α2
s
2.1 2.5 0.4 αsΛQCD/mb 0.9 0.9 0.0 (ΛQCD/mb)2 0.8 0.8 0.0 a2 0.7 1.4 1.4 gD∗Dπ 0.2 0.03 0.2 Total systematic 2.7 3.2 1.7 Data 1.1 1.4 1.4 Total 2.9 3.5 2.2
36
1 2 3 4 5 6χ
2.75 3.00 3.25 3.50 3.75 4.00 4.25 dΓ dχ/10−15 GeV Belle −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00cos θ`
4 6 8 10 12 14 dΓ d cos θ`/10−15 GeV Belle −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00cos θv
8 10 12 14 16 dΓ d cos θv/10−15 GeV Belle 1.0 1.1 1.2 1.3 1.4 1.5w
25 30 35 40 45 50 55 dΓ dw/10−15 GeV BelleAbdesselam et al., arXiv:1702.01521
B− → D∗(→ D⇡)`−¯ ⌫
37
hA1(w) = hA1(1)[1 − 8ρ2z + (rh2rρ2 + rh2)z2 + (rh3rρ2 + rh3)z3] R1(w) = R1(1) + r11(w − 1) + r12(w − 1)2 R2(w) = R2(1) + r21(w − 1) + r22(w − 1)2 rh2r = 53 , rh2 = −15 , rh3r = −231 , rh3 = 91 r11 = −0.12 , r12 = 0.05 , r21 = 0.11 , r22 = −0.06 Using this “tight” CLN parametrization Fixed: IHFLAV = 0.03561(11)(44) IBelle = 0.0348(12) (unfolded) I = |¯ ηEW Vcb|hA1(1) Form factors entering helicity amplitudes (massless leptons)
w = v · v0
37
hA1(w) = hA1(1)[1 − 8ρ2z + (rh2rρ2 + rh2)z2 + (rh3rρ2 + rh3)z3] R1(w) = R1(1) + r11(w − 1) + r12(w − 1)2 R2(w) = R2(1) + r21(w − 1) + r22(w − 1)2 rh2r = 53 , rh2 = −15 , rh3r = −231 , rh3 = 91 r11 = −0.12 , r12 = 0.05 , r21 = 0.11 , r22 = −0.06 Using this “tight” CLN parametrization Fixed: IHFLAV = 0.03561(11)(44) IBelle = 0.0348(12) (unfolded) I = |¯ ηEW Vcb|hA1(1) Form factors entering helicity amplitudes (massless leptons)
w = v · v0
38
hA1(w) = hA1(1)[1 − 8ρ2z + (rh2rρ2 + rh2)z2 + (rh3rρ2 + rh3)z3] R1(w) = R1(1) + r11(w − 1) + r12(w − 1)2 R2(w) = R2(1) + r21(w − 1) + r22(w − 1)2 rh2r = 53 , rh2 = −15 , rh3r = −231 , rh3 = 91 r11 = −0.12 , r12 = 0.05 , r21 = 0.11 , r22 = −0.06
BIG!
small!
Coefficients calculated through Λ/m using HQET & sum rules
V (q2) = R1(w) r0 hA1(w) A2(q2) = R2(w) r0 hA1(w)
Ratios What are the uncertainties for the r ’s? 20%? 100%?
See papers by Bigi, Gambino, Schacht; Grinstein & Kobach; Bernlochner et al.; Jaiswal, et al.
Series (z) expansion z = √t+ − t − √t+ − t0 √t+ − t + √t+ − t0 t± = (mB ± mF )2 t = q2 Choose, e.g. z
branch cut t = t+
1
t = t− t = 0 t > t+
F (t) = 1 1 − t/m2
res
anzn
Simplified series expansion
t0 = t−
40
F(t) = QF (t)
KF −1
X
k=0
a(F )
k
zk(t, t0) QF (t) = 1 Bn(z)φF (z) Sg =
Kg−1
X
k=0
(a(g)
k )2 ≤ 1
SfF =
Kf −1
X
k=0
[(a(f)
k )2 + (a(F1) k
)2] ≤ 1 Bn(z) =
n
Y
i=1
z − zPi 1 − zzPi zPi = z(M 2
Pi, t−)
adopt the model estimates of Ref. [23], up to 3 digits. M1−/GeV method Ref. M1+/GeV method Ref. 6.335(6) lattice [77] 6.745(14) lattice [77] 6.926(19) lattice [77] 6.75 model [79, 80] 7.02 model [79] 7.15 model [79, 80] 7.28 model [81] 7.15 model [79, 80]
Blaschke factor Unitarity bounds
MB + MD∗ = 7.290 GeV
Predictions for Bc vector & axial vector resonances
41
F(t) = QF (t)
KF −1
X
k=0
a(F )
k
zk(t, t0) QF (t) = NF 1 −
t M 2
P
Using BGL as a guide, choose Nf = 300, NF1 = 7000, Ng = 5 Simple form which uses less theoretical information. Clean baseline, against which affects of theoretical input (HQET, unitarity bounds) can be measured
− − BCL – – 2 0.0367(15) 0.01496(19) −0.047(27) 0.002935(37) −0.0029(27) 0.027(13) 0.77(44) 0.0025(26) 0.60(69) BCL – – 3 0.0378(17) 0.01496(19) −0.065(40) 0.002935(37) −0.0135(82) 0.026(13) 0.82(46) 0.08(38) 0.67(75) BCL – – 4 0.0382(18) 0.01497(19) −0.310(42) 0.002936(37) −0.0151(83) 0.109(16) −0.29(37) 0.143(67) 0.10(22) BCL – – 5 0.0382(18) 0.01497(19) −0.310(42) 0.002936(37) −0.0151(83) 0.109(16) −0.29(37) 0.143(67) 0.10(22) fit n+
B n− B K
I a(f) a(f)
1
a(F1) a(F1)
1
a(g) a(g)
1
SfF Sg
42
1.0 1.1 1.2 1.3 1.4 1.5
w
20 25 30 35 40 45 50 55
dΓ dw/10−15 GeV
Belle CLN binned BGL binned BCL binned −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
cos θv
8 10 12 14 16
dΓ d cos θv/10−15 GeV
Belle CLN binned BGL binned BCL binned −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
cos θ`
4 6 8 10 12 14
dΓ d cos θ`/10−15 GeV
Belle CLN binned BGL binned BCL binned 1 2 3 4 5 6
χ
2.75 3.00 3.25 3.50 3.75 4.00 4.25
dΓ dχ/10−15 GeV
Belle CLN binned BGL binned BCL binned
42
1.0 1.1 1.2 1.3 1.4 1.5
w
20 25 30 35 40 45 50 55
dΓ dw/10−15 GeV
Belle CLN binned BGL binned BCL binned −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
cos θv
8 10 12 14 16
dΓ d cos θv/10−15 GeV
Belle CLN binned BGL binned BCL binned −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
cos θ`
4 6 8 10 12 14
dΓ d cos θ`/10−15 GeV
Belle CLN binned BGL binned BCL binned 1 2 3 4 5 6
χ
2.75 3.00 3.25 3.50 3.75 4.00 4.25
dΓ dχ/10−15 GeV
Belle CLN binned BGL binned BCL binned
resolves inclusive/exclusive tension, at least in Belle data
non-zero recoil
43
0.032 0.034 0.036 0.038 0.040 0.042 0.044
I CLN 0% CLN h : 10%, R : 0(1) BGL 4 + 3 BCL
I = |Vcb ¯ ηEW| hA1(1)
0.0350 0.0375 0.0400 0.0425 0.0450 0.0475 0.0500 0.0525 0.0550
|Vcb| Inclusive B → D B → D∗, this work
Different fit Ansätze
“std” Our preferred
approach connecting hadronic observables and fundamental quark interactions
quark flavour
form factors, B mixing matrix elements, …
44
46
J(0)i
latt (x) = ¯
cγiγ5Q J(1)i
latt (x) = −
1 2amb ¯ cγiγ5γ · ∆Q Λ2
QCD
m2
b
1-loop coefficients η & τ from Monahan, Shigemitsu, Horgan, PRD87 (2013) Truncation errors enter at order: included as Gaussian noise
hJ ii = (1 + αs(η τ))hJ(0)i
latt i + hJ(1)i latt i + e4
Λ2
QCD
m2
b
46
J(0)i
latt (x) = ¯
cγiγ5Q J(1)i
latt (x) = −
1 2amb ¯ cγiγ5γ · ∆Q Λ2
QCD
m2
b
1-loop coefficients η & τ from Monahan, Shigemitsu, Horgan, PRD87 (2013) Truncation errors enter at order: included as Gaussian noise
hJ ii = (1 + αs(η τ))hJ(0)i
latt i + hJ(1)i latt i + e4
Λ2
QCD
m2
b
46
J(0)i
latt (x) = ¯
cγiγ5Q J(1)i
latt (x) = −
1 2amb ¯ cγiγ5γ · ∆Q Λ2
QCD
m2
b
1-loop coefficients η & τ from Monahan, Shigemitsu, Horgan, PRD87 (2013) Truncation errors enter at order: included as Gaussian noise
hJ ii = (1 + αs(η τ))hJ(0)i
latt i + hJ(1)i latt i + e4
Λ2
QCD
m2
b
46
J(0)i
latt (x) = ¯
cγiγ5Q J(1)i
latt (x) = −
1 2amb ¯ cγiγ5γ · ∆Q Λ2
QCD
m2
b
1-loop coefficients η & τ from Monahan, Shigemitsu, Horgan, PRD87 (2013) Truncation errors enter at order: included as Gaussian noise
hJ ii = (1 + αs(η τ))hJ(0)i
latt i + hJ(1)i latt i + e4
Λ2
QCD
m2
b
47
i αsτhJ(0)
latti
1) 0.00559(8) 4) 0.0064(1) 1) 0.0080(9)
hJ ii = (1 + αs(η τ))hJ(0)i
latt i + hJ(1)i latt i + e4
Λ2
QCD
m2
b
hJ(1)
latti
h 3 0.0078(66) 6 0.0055(48) 8 0.0048(6)
very coarse coarse fine & physical sea quark masses
Cancellation expected from Luke’s theorem
48
hA1(1) = (1 + δB
a )B + δg a
g2 48π2f 2 × chiral logs + C M 2
π
Λ2
χ
+ e1α2
s
h 1 + e5(amb − 2)/2 + e6((amb − 2)/2)2i J(0)
latt
Fit function:
Static limit
with g2 = 0.53(8)
δα : disc errors light quark mass 2-loop matching error
The αs2 uncertainty is the largest, by a factor of 2, compared to others