Lattie QCD with two light quarks
Luigi Del Debbio
luigi.del.debbio@ed.ac.uk
University of Edinburgh
Light Wilson fermions, UKQCD meeting Jan 06 – p.1/21
η
Precision tests of the SM
1.2
∆md
∆ms
1
∆md
0.8
∗
∗
∗
=0
+ Vtd Vtb
+ Vcd Vcb
Vud Vub
sin2β
0.6
0.4
εK
0.2
0
Vub
Vcb
-1
-0.5
0
0.5
1
=
∗
∗
Re (Vud Vub
) /Vcd Vcb
η̄
=
∗
∗
) /Vcd Vcb
Im (Vud Vub
ρ
1.5
lud
exc
excluded area has CL > 0.95
∆md
ed
L>
at C
γ
0.9
1
5
sin 2β
∆ms & ∆md
0.5
α
α
εK
η
-0.2
ρ̄
γ
0
β
|Vub/Vcb|
-0.5
εK
α
sol. w/ cos 2β < 0
(excl. at CL > 0.95)
-1
γ
CKM
fitter
EPS 2005
-1.5
-1
-0.5
0
0.5
1
1.5
2
ρ
Light Wilson fermions, UKQCD meeting Jan 06 – p.2/21
Match experimental precision
CLEO (endpoint)
4.02 ± 0.47 ± 0.35
BELLE (endpoint)
4.82 ± 0.45 ± 0.31
BABAR (endpoint)
4.23 ± 0.27 ± 0.31
BABAR (E e, q2)
4.06 ± 0.27 ± 0.36
BELLE m X
4.08 ± 0.27 ± 0.25
BELLE sim. ann. (m X, q2 )
4.38 ± 0.46 ± 0.30
BABAR (m X, q2)
4.76 ± 0.34 ± 0.32
Average +/- exp +/- (mb,theory)
4.38 ± 0.19 ± 0.27
χ2/dof = 5.9/ 6 (CL = 43.0%)
HQ input from b→ c l ν and b → s γ moments
2
4
HFAG
EPS-2005
6
|V ub | [× 10 -3 ]
Ball-Zwicky full q2
3.37 ± 0.14 + 0.66 - 0.41
HPQCD full q2
3.93 ± 0.17 + 0.77 - 0.48
FNAL full q2
3.76 ± 0.16 + 0.87 - 0.51
Ball-Zwicky q2 < 16
3.27 ± 0.16 + 0.54 - 0.36
HPQCD q2 > 16
4.47 ± 0.30 + 0.67 - 0.46
FNAL q2 > 16
3.78 ± 0.25 + 0.65 - 0.43
HFAG
EPS-2005
2
4
|V ub| [ × 10 ]
-3
Light Wilson fermions, UKQCD meeting Jan 06 – p.3/21
Precision studies
“Convincing precision” for CKM studies
a = 0.06 ÷ 0.09fm
mπ = 300MeV
Quantity
precision
Fπ
1.8%
FK /Fπ
<1 %
K→π
<1 %
BK
√
FB BB
5%
ξ
3%
B→π
7%
B→D
L > 2.5fm
5%
2%
[Sharpe 04]
Light Wilson fermions, UKQCD meeting Jan 06 – p.4/21
Fermionic action
Dynamics is determined by the lattice action
1
SL (U, ψ, ψ̄) −→ − G2 + ψ̄(D + m)ψ + O(a)
4
Different discretizations:
•
#
• gauge symmetry
• flavour symmetry
• chiral symmetry
• locality
• unitarity
"
!
Wilson fermions
SF
i
h
1
†
=
ψ̄(x) Uµ (x)γµ ψ(x + µ) − Uµ (x − µ)γµ ψ(x − µ)
2a
chiral symmetry-breaking term to eliminate doublers
irrelevant operators can be added: O(a2 ) effects only
Light Wilson fermions, UKQCD meeting Jan 06 – p.5/21
First principles calculation
Accuracy of the numerical evaluation is limited by:
•
•
statistical errors
systematic errors
To have a quantitative NP tool:
(a) large computers / good algorithms
—————————————————(b) finite-volume effects
(c) renormalization/improvement
—————————————————(e) continuum limit a → 0, at fixed La
(f) fermion masses (mq → 0)
(g) symmetry breaking due to regularization
taming systematic errors is crucial
Light Wilson fermions, UKQCD meeting Jan 06 – p.6/21
Finite-volume effects
RMπ = (Mπ (L) − Mπ )/Mπ
0.1
RM
π
LO, n = 1
NLO, n = 1
NNLO, n = 1
LO, all n
NLO, all n
NNLO, all n
Mπ L = 2
0.01
L = 2 fm
L = 3 fm
L = 4 fm
0.001
0.1
0.2
0.3
0.4
•
•
effect of pion loops
•
few % effects
computed in ChPT (no
new params)
0.5
Mπ (GeV)
[Lüscher 86 – Colangelo et al 04 – Becirevic 04]
Light Wilson fermions, UKQCD meeting Jan 06 – p.7/21
Hybrid Monte Carlo
•
cost of the inversion dictated by the small eigenvalues of D−1
⇒ more expensive as mq → 0, V → ∞
•
time-step in the classical evolution dictated by the magnitude of the forces
ǫk ||Fk || ≃ const
[Sexton & Weingarten 92]
•
chiral limit requires small a to have small lattice artefacts
•
total cost with Wilson/stag fermions:
TFlop × yrs
=
TFlop × yrs
=
«„
« „
«
«„
«6 „
Nconf
Ls × a 5 Lt
0.6
0.1fm 7
0.7
1000
3fm
2Ls
mπ /mρ
a
„
«„
« „
«
«„
«2.5 „
Nconf
Ls × a 4 Lt
0.2
0.1fm 7
1.31
1000
3fm
2Ls
m/ms
a
„
[Ukawa 02 - Gottlieb 02]
Light Wilson fermions, UKQCD meeting Jan 06 – p.8/21
SAP preconditioning
Design the algorithm using knowledge of the physical system
•
[Lüscher 04]
Domain decomposition
D = DΩ + DΩ∗ + D∂Ω + D∂Ω∗
yields:
det D =
Y
n
D̂Λ det 1 −
−1
−1
DΩ
D∂Ω DΩ
∗ D∂Ω∗
•
Block decoupling via active links; inner
integration can be performed in parallel
•
•
•
Block size ∼ 0.5fm
o
Schur complement only acts on ∂Ω∗
Hierarchical integration with different timesteps
Light Wilson fermions, UKQCD meeting Jan 06 – p.9/21
SAP preconditioning
Design the algorithm using knowledge of the physical system
•
[Lüscher 04]
Domain decomposition
D = DΩ + DΩ∗ + D∂Ω + D∂Ω∗
10
yields:
det D =
•
•
•
•
k=0
⟨||Fk(x,µ)||⟩
Y
n
D̂Λ det 1 −
−1
−1
DΩ
D∂Ω DΩ
∗ D∂Ω∗
Block decoupling via active links; inner
integration can be performed in parallel
Block size ∼ 0.5fm
Schur complement only acts on
o
k=1
1
0.1
k=2
1
∂Ω∗
2
3
4
d
5
Hierarchical integration with different timesteps
Light Wilson fermions, UKQCD meeting Jan 06 – p.10/21
SAP simulations
[Del Debbio, Giusti, Lüscher, Tantalo, Petronzio]
200
Lattice
κ
m/ms
mπ (MeV)
32 × 243
0.15750
0.93
676
0.15800
0.48
484
0.15825
0.30
381
0.15835
0.17
294
τ / ε2 =5
32 x243
150
6
100
8
10
50
64 x323
10
16
scale set by the sommer radius r0 : a = 0.08fm
0
0
0.01
0.02
am
0.03
Lattice
κ
m/ms
mπ (MeV)
Lattice
κ
τint [P ]
ν
64 × 323
0.15410
0.75
646
0.15750
53(22)
0.84(39)
0.15440
0.38
456
32 × 243
0.15800
33(11)
0.27(8)
0.15825
21(6)
0.40(11)
0.15835
17(6)
0.39(12)
scale set by the sommer radius r0 : a = 0.06fm
ν = 10−3 (2N2 + 3)τint [P ]
Light Wilson fermions, UKQCD meeting Jan 06 – p.11/21
Stability at large volume
1
∆H
0
Large lattice 64 × 323 , κ = 0.15440
−1
0.5940 P
•
•
∆H bounded btw -1 and 1
•
simulating large lattices pays off
0.5936
60
solver number NGCR has small
fluctuations
NGCR
50
2000
3000
4000
5000
6000
Light Wilson fermions, UKQCD meeting Jan 06 – p.12/21
Eigenvalues of the Dirac operator
0.2
A1
0.1
0
0.2
C1
A2
Property of the discretization (NOT of the algorithm)
A3
Small lattice 32 × 243 , 4 bare masses
0.1
0
0.2
0.1
0
0.2
•
•
•
hµi shifts linearly with m
stability is related to the width of the distribution
scaling with a and V
A4
0.1
0
0
20
40
60
80
µ[MeV]
Light Wilson fermions, UKQCD meeting Jan 06 – p.13/21
Scaling of the width
1.8
1.6
σV 1/2a −1
1.4
1.2
1
0.8
0.6
0.4
A1
A2
A3
A4
B1
B2
C1
D1
large V , small a do pay off!
√
Mπ L > C a (2.8, 2.3)
Light Wilson fermions, UKQCD meeting Jan 06 – p.14/21
Simulation cost
Simulation cost
a=0.080 fm L=2.0 fm
•
good preconditioning, wise
choice of simulation params
•
SAP alg opens new possibilities
to approach the chiral limit with
Wilson fermions
•
•
other candidates? why not?
Berlin ’01
no intrinsic difficulty
SAP
0
0.25
0.5
0.75
mπ/mρ
Light Wilson fermions, UKQCD meeting Jan 06 – p.15/21
Running SAP...
Light Wilson fermions, UKQCD meeting Jan 06 – p.16/21
Current simulations
mπ2 [GeV2]
SPQcdR ’04
0.3
CP-PACS ’01
TχL ’00
0.2
NF = 2
m ~ ms / 2
BNL ’04
UKQCD ’01
qq+q ’04
CERN-TOV ’05
0.1
CP-PACS ’04
m ~ ms / 6
Exp.
0
0
0.05
0.1
0.15
0.2
0.25 a [fm]
Light Wilson fermions, UKQCD meeting Jan 06 – p.17/21
Effective mass plateau
0.5
0.4
aWρeff
0.3
0.2
amπeff
0.1
am
0
5
10
15
t/a
20
25
30
[Del Debbio, Giusti, Lüscher, Petronzio, Tantalo]
Light Wilson fermions, UKQCD meeting Jan 06 – p.18/21
Pion mass
0.08
(amπ )
Mπ2
mπ ∼676 MeV
2
Rπ
0.06
484
0.04
0
0
M 2 Rπ
=
M2
1+
log(M 2 /Λ2π )
2
2
32π F
Rπ
1.00
381
0.02
=
M = 200 −500 MeV
0.95
294
0.01
0.02
0.03
0
am
0.1
0.2
M 2 [GeV 2]
[Del Debbio, Giusti, Lüscher, Petronzio, Tantalo]
Light Wilson fermions, UKQCD meeting Jan 06 – p.19/21
Pion decay constant
aFπ
0.06
»
Fπ = F 1 −
676
294
0.04
381
484
0.02
0
0
0.02
0.04
0.06
0.08 (am
π)
M2
2
2
log(M
/Λ
F)
16π 2 F 2
•
•
•
ZA : TI perturbation theory
•
•
Fπ |M =140Mev = 80(7)MeV
–
finite-volume corrections
˛
2
2
log(ΛF /M )˛M =140Mev ≃ 4.6 ± 0.9
Nf = 2, O(a) effects, ...
2
[Del Debbio, Giusti, Lüscher, Petronzio, Tantalo]
Light Wilson fermions, UKQCD meeting Jan 06 – p.20/21
Perspectives
•
•
unquenching is fundamental to eliminate systematic errors
•
•
dedicated algorithms + machines
•
•
establish QCD as the theory of strong interactions [chiral limit]
•
•
strong dynamics beyond the SM?
lattice QCD: robust tool for NP physics
renormalization/improvement?
precise determination of the SM parameters [Nf = 3]
large-N /susy theories - cfr with analytical results
Light Wilson fermions, UKQCD meeting Jan 06 – p.21/21