Analysis of topological structure of the QCD vacuum with overlap
Transcript of Analysis of topological structure of the QCD vacuum with overlap
Analysis of topological structure of the QCD vacuumwith overlap-Dirac operator eigenmode
T.Iritani
High Energy Accelerator Research Organization (KEK)
Jul 29, 2013
Coll. S. Hashimoto and G. Cossu
...1 Introduction
...2 Topological structure of QCD vacuum from overlap-Dirac eigenmodes
...3 Flux-tube Formation by Low-lying Dirac Eigenmodes
...4 Chiral Condensate in Flux-tube
...5 Summary
...1 Introduction
...2 Topological structure of QCD vacuum from overlap-Dirac eigenmodes
...3 Flux-tube Formation by Low-lying Dirac Eigenmodes
...4 Chiral Condensate in Flux-tube
...5 Summary
Vacuum Structure of QCD
QCD vacuum is filled with interesting objects.
instanton/anti-instanton
⇒ chiral symmetry breaking
flux-tube
⇒ quark confinement
Figs (left) JLQCD Coll. ’12 (right) www.physics.adelaide.edu.au/theory/staff/leinweber/VisualQCD/Nobel/
1 / 15
Overlap-Dirac Operator
Dov = m0 [1 + γ5sgn HW (−m0)]
HW (−m0): hermitian Wilson-Dirac operator with a mass −m0 (Neuberger ’98)
overlap-Dirac eigenmode is an ideal probe to study QCD vacuum
exact chiral symmetry on the lattice
Banks-Casher relation— chiral symmetry breaking
〈q̄q〉 = −πρ(0)
index theorem — topology of QCD
12tr [γ5Dov] = nL − nR
0
0.001
0.002
0.003
0.004
0.005
0 0.01 0.02 0.03 0.04 0.05 0.06π
ρ Q(λ
)
λ
latticeChPT LO
ChPT NLO
Dirac spectral density ρ(λ)JLQCD Coll. ’10
2 / 15
Contents
...1 topological structure of QCD vacuum from overlap-Dirac eigenmodes
...2 flux-tube formation from overlap-Dirac eigenmodes
...3 chiral condensate in flux-tube
...1 Introduction
...2 Topological structure of QCD vacuum from overlap-Dirac eigenmodes
...3 Flux-tube Formation by Low-lying Dirac Eigenmodes
...4 Chiral Condensate in Flux-tube
...5 Summary
Dirac Eigenmode Decomposition of Field Strength
Dirac eigenmodes ψλ(x) ⇒ Field strength tensor Fµν
Gattringer ’02
[ /D(x)]2 =∑
µ
D2µ(x) +
∑µ<ν
γµγνFµν(x)
∴ Fµν(x) = −14tr
[γµγν /D2(x)
]∝
∑λ
λ2fµν(x)λ
with Dirac-mode components fµν(x)λ
fµν(x)λ ≡ ψ†λ(x)γµγνψλ(x)
references Gattringer ’02, Ilgenfritz et al. ’07, ’08.In this talk, we usedynamical overlap-fermion configurations by JLQCD Coll.β = 2.3 (Iwasaki action), a ∼ 0.11 fm, Volume 163 × 48,overlap-Dirac operator Nf = 2 + 1, mud = 0.015, ms = 0.080
3 / 15
Duality of Field Strength Tensor by Overlap-eigenmodelowest eigenmode components of fµν
f12(x)0
20 22 24 26 28 30 32 34 0 2 4 6 8 10 12 14 16-3E-04
-2E-04
-1E-04
0E+00
1E-04
f012(x) 0 around (8, 15, 7, 27)
t x
⇒ negative peak
f34(x)0
20 22 24 26 28 30 32 34 0 2 4 6 8 10 12 14 16-1E-04
0E+00
1E-04
2E-04
3E-04
f034(x) 0 around (8, 15, 7, 27)
t x
⇒ positive peak
at this point, an anti-self-dual lump exists
f12(x)0 ' −f34(x)0
4 / 15
Action and Topological Charge from Dirac Eigenmodesaction density
ρ(N)(x) =N∑i,j
λ2iλ
2j
2fa
µν(x)ifaµν(x)j
topological charge density
q(N)(x) =N∑i,j
λ2iλ
2j
2fa
µν(x)if̃aµν(x)j
Figure: action and topological charge densities of lowest eigenstate
20 22 24 26 28 30 32 34 0 2 4 6 8 10 12 14 160E+00
2E-16
4E-16
6E-16
8E-16
Action Density #0 around (8, 15, 7, 27)
t x
20 22 24 26 28 30 32 34 0 2 4 6 8 10 12 14 16-8E-16
-6E-16
-4E-16
-2E-16
0E+00
Topological Charge Density #0 around (8, 15, 7, 27)
t x
there is an “anti-instanton” at this point5 / 15
...1 Introduction
...2 Topological structure of QCD vacuum from overlap-Dirac eigenmodes
...3 Flux-tube Formation by Low-lying Dirac Eigenmodes
...4 Chiral Condensate in Flux-tube
...5 Summary
Flux-tube Formation Between Quarksaction density is “expelled” between quarks
⇒ formation of flux-tube ⇒ confinement potential
we discuss flux-tube structure from low-lying eigenmodes
flux-tube betweenquark and antiquark
Bali-Schilling-Schlichter ’95
Y-type flux-tube for 3Q-system
Ichie et al. ’036 / 15
Measurement of Flux-tube Structuredifference of action density ρ(x) with or without Wilson loop W (R, T )
〈ρ(x)〉W ≡ 〈ρ(x)W (R, T )〉〈W (R, T )〉
− 〈ρ〉 < 0
N.B. action density is lowered in the flux-tube
XY
Tim
e
Wilson Loop
action density
R7 / 15
Flux-tube Formation by Low-lying Eigenmodesρ(N)(x): low-mode truncated action density
〈ρ(x)〉(N)W ≡ 〈ρ(N)(x)W (R, T )〉
〈W (R, T )〉− 〈ρ〉(N)
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
8
inter-quark separation R = 10
XY
(−) 〈ρ(x)〉(N)W N = 100 eigenmodes
low-modes reallycontribute toflux-tube
1002, 359, 296
∼ 0.00004
λmax ∼ 300 MeV
8 / 15
Flux-tube Formation by Low-lying Eigenmodesρ(N)(x): low-mode truncated action density
〈ρ(x)〉(N)W ≡ 〈ρ(N)(x)W (R, T )〉
〈W (R, T )〉− 〈ρ〉(N)
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
8
inter-quark separation R = 9
XY
(−) 〈ρ(x)〉(N)W N = 100 eigenmodes
low-modes reallycontribute toflux-tube
1002, 359, 296
∼ 0.00004
λmax ∼ 300 MeV
8 / 15
Flux-tube Formation by Low-lying Eigenmodesρ(N)(x): low-mode truncated action density
〈ρ(x)〉(N)W ≡ 〈ρ(N)(x)W (R, T )〉
〈W (R, T )〉− 〈ρ〉(N)
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
8
inter-quark separation R = 8
XY
(−) 〈ρ(x)〉(N)W N = 100 eigenmodes
low-modes reallycontribute toflux-tube
1002, 359, 296
∼ 0.00004
λmax ∼ 300 MeV
8 / 15
Flux-tube Formation by Low-lying Eigenmodesρ(N)(x): low-mode truncated action density
〈ρ(x)〉(N)W ≡ 〈ρ(N)(x)W (R, T )〉
〈W (R, T )〉− 〈ρ〉(N)
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
8
inter-quark separation R = 7
XY
(−) 〈ρ(x)〉(N)W N = 100 eigenmodes
low-modes reallycontribute toflux-tube
1002, 359, 296
∼ 0.00004
λmax ∼ 300 MeV
8 / 15
Flux-tube Formation by Low-lying Eigenmodesρ(N)(x): low-mode truncated action density
〈ρ(x)〉(N)W ≡ 〈ρ(N)(x)W (R, T )〉
〈W (R, T )〉− 〈ρ〉(N)
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
8
inter-quark separation R = 6
XY
(−) 〈ρ(x)〉(N)W N = 100 eigenmodes
low-modes reallycontribute toflux-tube
1002, 359, 296
∼ 0.00004
λmax ∼ 300 MeV
8 / 15
Flux-tube Formation by Low-lying Eigenmodesρ(N)(x): low-mode truncated action density
〈ρ(x)〉(N)W ≡ 〈ρ(N)(x)W (R, T )〉
〈W (R, T )〉− 〈ρ〉(N)
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
8
inter-quark separation R = 5
XY
(−) 〈ρ(x)〉(N)W N = 100 eigenmodes
low-modes reallycontribute toflux-tube
1002, 359, 296
∼ 0.00004
λmax ∼ 300 MeV
8 / 15
Cross Section of Flux-tube
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
8
inter-quark separation R = 8
XY
-4e-10
-2e-10
0e+00
-8 -6 -4 -2 0 2 4 6 8
Y
σ = 2.42 ± 0.08R = 8
gaussian formρ(y) ∝ e−y2/σ2
thickness of flux∼ 0.6 fm
9 / 15
About Dirac Eigenmodes and Confinementconfinement remains without low or intermediate or higher eigenmodes
Refs. Gongyo-TI-Suganuma ’12, TI-Suganuma ’13
⇒ “seeds” of confinement seem to be widely distributed in eigenmodes
removing Dirac eigenmodes
0
200
400
600
800
1000
1200
1400
0 0.5 1 1.5 2 2.5
ρ (λ
) V
[a
]
λ [a-1]
interquark potential
0.5
1
1.5
0 1 2 3V
QQ–
(R
) [a
-1]
R [a]
interquark potential
Figs. Gongyo-TI-Suganuma ’12
flux-tube formation by low-lying modes
⇒ direct evidence of the seed of confinement in the low-lying modes6= low-lying modes are essential for confinement
10 / 15
About Dirac Eigenmodes and Confinementconfinement remains without low or intermediate or higher eigenmodes
Refs. Gongyo-TI-Suganuma ’12, TI-Suganuma ’13
⇒ “seeds” of confinement seem to be widely distributed in eigenmodes
removing Dirac eigenmodes
0
200
400
600
800
1000
1200
1400
0 0.5 1 1.5 2 2.5
ρ (λ
) V
[a
]
λ [a-1]
interquark potential
0.5
1
1.5
0 1 2 3V
QQ–
(R
) [a
-1]
R [a]
no cut
Dirac-mode cut
Figs. Gongyo-TI-Suganuma ’12
flux-tube formation by low-lying modes
⇒ direct evidence of the seed of confinement in the low-lying modes6= low-lying modes are essential for confinement
10 / 15
About Dirac Eigenmodes and Confinementconfinement remains without low or intermediate or higher eigenmodes
Refs. Gongyo-TI-Suganuma ’12, TI-Suganuma ’13
⇒ “seeds” of confinement seem to be widely distributed in eigenmodes
removing Dirac eigenmodes
0
200
400
600
800
1000
1200
1400
0 0.5 1 1.5 2 2.5
ρ (λ
) V
[a
]
λ [a-1]
interquark potential
0.5
1
1.5
0 1 2 3V
QQ–
(R
) [a
-1]
R [a]
no cut
Dirac-mode cut
Figs. Gongyo-TI-Suganuma ’12
flux-tube formation by low-lying modes
⇒ direct evidence of the seed of confinement in the low-lying modes6= low-lying modes are essential for confinement
10 / 15
About Dirac Eigenmodes and Confinementconfinement remains without low or intermediate or higher eigenmodes
Refs. Gongyo-TI-Suganuma ’12, TI-Suganuma ’13
⇒ “seeds” of confinement seem to be widely distributed in eigenmodes
removing Dirac eigenmodes
0
200
400
600
800
1000
1200
1400
0 0.5 1 1.5 2 2.5
ρ (λ
) V
[a
]
λ [a-1]
interquark potential
0.5
1
1.5
0 1 2 3V
QQ–
(R
) [a
-1]
R [a]
no cut
Dirac-mode cut
Figs. Gongyo-TI-Suganuma ’12
flux-tube formation by low-lying modes
⇒ direct evidence of the seed of confinement in the low-lying modes6= low-lying modes are essential for confinement
10 / 15
About Dirac Eigenmodes and Confinementconfinement remains without low or intermediate or higher eigenmodes
Refs. Gongyo-TI-Suganuma ’12, TI-Suganuma ’13
⇒ “seeds” of confinement seem to be widely distributed in eigenmodes
removing Dirac eigenmodes
0
200
400
600
800
1000
1200
1400
0 0.5 1 1.5 2 2.5
ρ (λ
) V
[a
]
λ [a-1]
interquark potential
0.5
1
1.5
0 1 2 3V
QQ–
(R
) [a
-1]
R [a]
no cut
Dirac-mode cut
Figs. Gongyo-TI-Suganuma ’12
flux-tube formation by low-lying modes
⇒ direct evidence of the seed of confinement in the low-lying modes6= low-lying modes are essential for confinement
10 / 15
About Dirac Eigenmodes and Confinementconfinement remains without low or intermediate or higher eigenmodes
Refs. Gongyo-TI-Suganuma ’12, TI-Suganuma ’13
⇒ “seeds” of confinement seem to be widely distributed in eigenmodes
removing Dirac eigenmodes
0
200
400
600
800
1000
1200
1400
0 0.5 1 1.5 2 2.5
ρ (λ
) V
[a
]
λ [a-1]
interquark potential
0.5
1
1.5
0 1 2 3V
QQ–
(R
) [a
-1]
R [a]
no cut
Dirac-mode cut
Figs. Gongyo-TI-Suganuma ’12
flux-tube formation by low-lying modes
⇒ direct evidence of the seed of confinement in the low-lying modes6= low-lying modes are essential for confinement
10 / 15
About Dirac Eigenmodes and Confinementconfinement remains without low or intermediate or higher eigenmodes
Refs. Gongyo-TI-Suganuma ’12, TI-Suganuma ’13
⇒ “seeds” of confinement seem to be widely distributed in eigenmodes
removing Dirac eigenmodes
0
200
400
600
800
1000
1200
1400
0 0.5 1 1.5 2 2.5
ρ (λ
) V
[a
]
λ [a-1]
interquark potential
0.5
1
1.5
0 1 2 3V
QQ–
(R
) [a
-1]
R [a]
no cut
Dirac-mode cut
Figs. Gongyo-TI-Suganuma ’12
flux-tube formation by low-lying modes
⇒ direct evidence of the seed of confinement in the low-lying modes6= low-lying modes are essential for confinement
10 / 15
About Dirac Eigenmodes and Confinementconfinement remains without low or intermediate or higher eigenmodes
Refs. Gongyo-TI-Suganuma ’12, TI-Suganuma ’13
⇒ “seeds” of confinement seem to be widely distributed in eigenmodes
removing Dirac eigenmodes
0
200
400
600
800
1000
1200
1400
0 0.5 1 1.5 2 2.5
ρ (λ
) V
[a
]
λ [a-1]
interquark potential
0.5
1
1.5
0 1 2 3V
QQ–
(R
) [a
-1]
R [a]
no cut
Dirac-mode cut
Figs. Gongyo-TI-Suganuma ’12
flux-tube formation by low-lying modes
⇒ direct evidence of the seed of confinement in the low-lying modes6= low-lying modes are essential for confinement
10 / 15
About Dirac Eigenmodes and Confinementconfinement remains without low or intermediate or higher eigenmodes
Refs. Gongyo-TI-Suganuma ’12, TI-Suganuma ’13
⇒ “seeds” of confinement seem to be widely distributed in eigenmodes
removing Dirac eigenmodes
0
200
400
600
800
1000
1200
1400
0 0.5 1 1.5 2 2.5
ρ (λ
) V
[a
]
λ [a-1]
interquark potential
0.5
1
1.5
0 1 2 3V
QQ–
(R
) [a
-1]
R [a]
no cut
Dirac-mode cut
Figs. Gongyo-TI-Suganuma ’12
flux-tube formation by low-lying modes
⇒ direct evidence of the seed of confinement in the low-lying modes6= low-lying modes are essential for confinement
10 / 15
...1 Introduction
...2 Topological structure of QCD vacuum from overlap-Dirac eigenmodes
...3 Flux-tube Formation by Low-lying Dirac Eigenmodes
...4 Chiral Condensate in Flux-tube
...5 Summary
Dirac Eigenmodes and Chiral Condensatechiral condensate 〈q̄q〉
〈q̄q〉 = −Tr1
/D +mq= − 1
V
∑λ
1λ+mq
cf. Banks-Casher relation 〈q̄q〉 = −πρ(0)
“local chiral condensate” q̄q(x) is given by
q̄q(x) = −∑
λ
ψ†λ(x)ψλ(x)λ+mq
with Dirac eigenmodes ψλ(x) 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16-0.005
-0.004
-0.003
-0.002
-0.001
0
sample of a local condensate11 / 15
Chiral Condensate in Flux-tubedifference of chiral condensate q̄q(x) with or without Wilson loop W (R, T )
〈q̄q(x)〉W ≡ 〈q̄q(x)W (R, T )〉〈W (R, T )〉
− 〈q̄q〉
XY
Wilson Loop
local chiral condensate
R12 / 15
Reduction of Chiral Condensate in Fluxdifference of “truncated local chiral condensate”
〈q̄q(x)〉(N)W ≡ 〈q̄q(N)(x)W (R, T )〉
〈W (R, T )〉− 〈q̄q〉(N)
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
80.0000
0.0002
0.0004
0.0006 R = 10
XY
mq = 0.015 using N = 100 eigenmodes
〈q̄q(x)〉(N)W > 0
partially restored|〈q̄q〉|in flux < |〈q̄q〉|“Bag-model” like
cf. chiral condensate in color-electro/magnetic fields (Suganuma-Tatsumi ’93)
13 / 15
Reduction of Chiral Condensate in Fluxdifference of “truncated local chiral condensate”
〈q̄q(x)〉(N)W ≡ 〈q̄q(N)(x)W (R, T )〉
〈W (R, T )〉− 〈q̄q〉(N)
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
80.0000
0.0002
0.0004
0.0006 R = 9
XY
mq = 0.015 using N = 100 eigenmodes
〈q̄q(x)〉(N)W > 0
partially restored|〈q̄q〉|in flux < |〈q̄q〉|“Bag-model” like
cf. chiral condensate in color-electro/magnetic fields (Suganuma-Tatsumi ’93)
13 / 15
Reduction of Chiral Condensate in Fluxdifference of “truncated local chiral condensate”
〈q̄q(x)〉(N)W ≡ 〈q̄q(N)(x)W (R, T )〉
〈W (R, T )〉− 〈q̄q〉(N)
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
80.0000
0.0002
0.0004
0.0006 R = 8
XY
mq = 0.015 using N = 100 eigenmodes
〈q̄q(x)〉(N)W > 0
partially restored|〈q̄q〉|in flux < |〈q̄q〉|“Bag-model” like
cf. chiral condensate in color-electro/magnetic fields (Suganuma-Tatsumi ’93)
13 / 15
Reduction of Chiral Condensate in Fluxdifference of “truncated local chiral condensate”
〈q̄q(x)〉(N)W ≡ 〈q̄q(N)(x)W (R, T )〉
〈W (R, T )〉− 〈q̄q〉(N)
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
80.0000
0.0002
0.0004
0.0006 R = 7
XY
mq = 0.015 using N = 100 eigenmodes
〈q̄q(x)〉(N)W > 0
partially restored|〈q̄q〉|in flux < |〈q̄q〉|“Bag-model” like
cf. chiral condensate in color-electro/magnetic fields (Suganuma-Tatsumi ’93)
13 / 15
Reduction of Chiral Condensate in Fluxdifference of “truncated local chiral condensate”
〈q̄q(x)〉(N)W ≡ 〈q̄q(N)(x)W (R, T )〉
〈W (R, T )〉− 〈q̄q〉(N)
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
80.0000
0.0002
0.0004
0.0006 R = 6
XY
mq = 0.015 using N = 100 eigenmodes
〈q̄q(x)〉(N)W > 0
partially restored|〈q̄q〉|in flux < |〈q̄q〉|“Bag-model” like
cf. chiral condensate in color-electro/magnetic fields (Suganuma-Tatsumi ’93)
13 / 15
Reduction of Chiral Condensate in Fluxdifference of “truncated local chiral condensate”
〈q̄q(x)〉(N)W ≡ 〈q̄q(N)(x)W (R, T )〉
〈W (R, T )〉− 〈q̄q〉(N)
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
80.0000
0.0002
0.0004
0.0006 R = 5
XY
mq = 0.015 using N = 100 eigenmodes
〈q̄q(x)〉(N)W > 0
partially restored|〈q̄q〉|in flux < |〈q̄q〉|“Bag-model” like
cf. chiral condensate in color-electro/magnetic fields (Suganuma-Tatsumi ’93)
13 / 15
Reduction Ratio of Chiral Condensate in Flux-tube
r(x) ≡ 〈q̄q(subt)(x)W (R, T )〉〈q̄q(subt)〉〈W (R, T )〉
< 1
-8-6
-4-2
0 2
4 6
8
-8-6
-4-2
0 2
4 6
80.0000
0.0002
0.0004
0.0006 R = 8
XY
0.7
0.8
0.9
1.0
-8 -6 -4 -2 0 2 4 6 8
r (x), Y-axis with R = 8
at center of flux
reduction of |〈q̄q〉|⇒ about 20 %
cf. subtracted condensate 〈q̄q〉(N) = 〈q̄q(subt)〉 + c(N)1 mq/a2 + c
(N)2 m3
q
14 / 15
...1 Introduction
...2 Topological structure of QCD vacuum from overlap-Dirac eigenmodes
...3 Flux-tube Formation by Low-lying Dirac Eigenmodes
...4 Chiral Condensate in Flux-tube
...5 Summary
Summary
we discuss vacuum structure of QCD using overlap-Dirac eigenmodes
low-lying overlap-Dirac eigenmodes⇒ show “instanton”-like behavior
I (anti-)self-dual field strength fµν ' (−)f̃µν
I (anti-)self-dual lump of action density / topological charge density
flux-tube formation by low-lying Dirac eigenmodes
⇒ low-lying eigenmodes contribute to the formation of flux-tubei.e., confinement
chiral condensation in the flux-tube
⇒ chiral symmetry is partially restored in the flux-tubereduction of |〈q̄q〉| is about 20% at the center of flux
outlooks
chiral condensate in 3Q-system, quark-number densities in theflux-tube, light-quark content in quarkonia, topological susceptibilityfrom eigenmodes, and so on
15 / 15
...6 Appendix
Appendix
Lattice Setup
gauge configurationsI JLQCD dynamical overlap simulation Nf = 2 + 1I 163 × 48, β = 2.3, mud = 0.015, ms = 0.080, Q = 0I a−1 = 1.759(10) GeV (mπ ∼ 300 MeV)I 100 low-lying Dirac eigenmodes (λUV ∼ 400 MeV)I 50 configurations
Wilson loop W (R, T ) measurementI APE smearing for spatial link-variable Ui — 16 sweepsI T = 4 — ground-state component is dominantI measurement of action density/local chiral condensate at T = 2
Subtracted Chiral Condensate
〈q̄q〉(N) = 〈q̄q(subt)〉 + c(N)1
mq
a2+ c
(N)2 m3
q
0
0.001
0.002
0.003
0.004
0.005
0 0.05 0.1 0.15 0.2
- 〈 q
bar q
〉(N
)
mq
N = 30
N = 50
0
0.001
0.002
0.003
0.004
0.005
0 0.05 0.1 0.15 0.2- 〈 q
bar q〉(s
ubt)
mq
N = 30
N = 50
Figure: (a) 〈q̄q〉(N) (b) 〈q̄q〉(subt)
ref. J. Noaki et al., Phys. Rev. D81, 034502 (2010).