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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).