HHIQCD at YITP
5th March (Thu.)

U(1) axial anomaly with chiralfermion at ﬁnite temperature

Akio Tomiya for JLQCD
time, and investigate how akio@het.phys.sci.osaka-u.ac.jp
the eﬀective potential for the Wilson line phase varies by running
an imaginary chemical potential. In flat space-time, the eﬀective potential of supersymmetric
Osaka
University(Ph.D
student)
theory is flat because of thecancellation
between thecontribution
from fermions and bosons.
2→ ???
for a post)
However, thanks to the curvatureof S(Looking, we obtain non-trivial
eﬀective potential. We findthat large R-charge is necessary in order to break the gauge symmetry. In addition, we
confirm a finite size eﬀect on our curved space taking smallish R-charge. This method has
several issues, which are discussed(Relatedfollowing
talk bysection.
G.Cossu today 17:00- )This thesis is based on following papers:

Reference:

1. A. Tomiya, G. Cossu, H. Fukaya, S. Hashimoto and J. Noaki, arXiv:1412.7306 [hep-lat].

Full paper is in preparation:

2. A. Tanaka, A. Tomiya and T. Shimotani, JHEP 1410, 136 (2014) [arXiv:1404.7639
15年3月5日木曜日[hep-th]].

1

PURPOSE:

We would like to judge the U(1)A symmetry
is restored above the QCD criticaltemperature or not

2
15年3月5日木曜日

PURPOSE:

We would like to judge the U(1)A symmetry
is restored above the QCD criticaltemperature or not

OUR RESULT:U(1)A sym. may be restored
2

15年3月5日木曜日

PURPOSE:

We would like to judge the U(1)A symmetry
is restored above the QCD criticaltemperature or not

OUR RESULT:U(1)A sym. may be restored
KEYWORD : Chiral symmetry
2

15年3月5日木曜日

1.Introduction

3
15年3月5日木曜日

Quarks in Nf = 2 QCD is described by the D
Sym. of Nf=2 QCD (m=0) !
SF =

d

4

L : SU (2)L ⇥ SU (2)R ⇥ U (1)V ⇥ U (1)A
where,Z¯
d x (D/)4

ψ(x) =

"

#

u(x)
,
d(x)

5

D
/ is the covariant derivative , which is defi
At zero temperature…
µ
µ
D
/
=
γ
D
=
γ
µ
SU(2) chiral symmetry is spontaneously broken
U(1)A Sym. is violated by the anomaly µ≡ γ (∂µ − ig
4

15年3月5日木曜日

Anomaly = Sym. Violation by Quantum Eﬀ.

Nf=2 massless QCD has chiral symmetriesZ=

Z

DAµ D ¯D exp

 Z

d x ¯(D
/)
4

✓ ◆
u
=
d

1. K. Fujikawa and H. Suzuki, Path Integrals and Quantum Anomalies (Oxford University Press, Oxford, 2004)
15年3月5日木曜日

5Anomaly = Sym. Violation by Quantum Eﬀ.

Nf=2 massless QCD has chiral symmetriesZ=

Z

DAµ D ¯D exp

!

0

i 5✓

 Z

=e
¯ ! ¯0 = ¯ei 5 ✓d x ¯(D
/)
4

✓ ◆
u
=
d

←U(1)ATrf(Chiral trf)

Rotate all R/L handed quark opposite dir.

/ =
5D

D
/ 5

1. K. Fujikawa and H. Suzuki, Path Integrals and Quantum Anomalies (Oxford University Press, Oxford, 2004)
15年3月5日木曜日

5Anomaly = Sym. Violation by Quantum Eﬀ.

Nf=2 massless QCD has chiral symmetriesZ=

Z

DAµ D ¯D exp

!

0

i 5✓

 Z

=e
¯ ! ¯0 = ¯ei 5 ✓

✓ ◆
u
=
d

d x ¯(D/)4←U(1)ATrf(Chiral trf)

Rotate all R/L handed quark opposite dir.

/ =
5D

D
/ 5

Action is inv., Measure is not invariant.
U(1)A Anomaly, Chiral anomaly

1. K. Fujikawa and H. Suzuki, Path Integrals and Quantum Anomalies (Oxford University Press, Oxford, 2004)
15年3月5日木曜日

5Anomaly = Sym. Violation by Quantum Eﬀ.

Nf=2 massless QCD has chiral symmetriesZ=

Z

DAµ D ¯D exp

!

0

 Z

d x ¯(D/)4←U(1)ATrf(Chiral trf)

i 5✓

=e
¯ ! ¯0 = ¯ei 5 ✓

✓ ◆
u
=
d

Rotate all R/L handed quark opposite dir.

/ =
5D

D
/ 5

Action is inv., Measure is not invariant.
U(1)A Anomaly, Chiral anomaly00¯¯
D D =D D e ,

⇠i

Z

4

d x[tr✏µ⌫↵Fµ⌫ F↵ ]1. K. Fujikawa and H. Suzuki, Path Integrals and Quantum Anomalies (Oxford University Press, Oxford, 2004)
15年3月5日木曜日

5Akio Tomiya(Osaka Univ.)

Chiral symmetry breaking in QCD (Nf=2, mud=0)
T =0

SU (2)LSU (2)RSSB

SU (2)VU (1)VU (1)A
Anomaly

U (1)V Residual symmetry

6
15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

Chiral symmetry breaking in QCD (Nf=2, mud=0)
T =0

SU (2)LSU (2)RU (1)VU (1)ASSB

SU (2)V

T > TcSU (2)VU (1)A

Anomaly

U (1)V Residual symmetry

SU (2)LSU (2)R

Restored??
6

15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

Chiral symmetry breaking in QCD (Nf=2, mud=0)
T =0

SU (2)LSU (2)RU (1)VU (1)ASSB

SU (2)V

T > TcSU (2)VU (1)A

Anomaly

U (1)V Residual symmetry

SU (2)LSU (2)R

Restored??

What happens on the anomaly above the Tc?
15年3月5日木曜日

6Akio Tomiya(Osaka Univ.)

U(1)A may be restored...
Two supporting evidences

1. Finite temperature = Theories on L3x(1/T)
T= ←→ Theory in D=3 (No anomaly)

Anomaly disappears at inﬁnite temperature(Anomaly could be disappeared at ﬁnite temperature?)
2. Cohenʼs argument (1996, Next page)SU(2) chiral symmetry restoration may be related toU(1)A restoration

7
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Cohen: SU(2) & U(1)A may be restored(1996)

Fact：T>Tc , when m→0, chiral symmetry is restored <=> h ¯ i = 0

8
15年3月5日木曜日

Cohen: SU(2) & U(1)A may be restored(1996)

Fact：T>Tc , when m→0, chiral symmetry is restored <=> h ¯ i = 0

Cohen：

If we assume h ¯

i = 0, gap in the Dirac spectrum, it may lead

Order parameterof SU(2) chiral sym.

U (1)A = 0

Order parameter ofU(1)A Sym. (Later)※Additional
condition is needed

8
15年3月5日木曜日

Cohen: SU(2) & U(1)A may be restored(1996)

Fact：T>Tc , when m→0, chiral symmetry is restored <=> h ¯ i = 0

Cohen：

If we assume h ¯

i = 0, gap in the Dirac spectrum, it may lead

Order parameterof SU(2) chiral sym.

U (1)A = 0

Order parameter ofU(1)A Sym. (Later)※Additional
condition is needed

Z 1
2m
¯
d ⇢( ) 2
h i = lim
m!0 0
+ m2
Z 1
4m2
d ⇢( ) 2
U (1)A = lim
m!0 0( + m2 )2Spectral rep :

lim

m!0

Banks-Casher rel.

8
15年3月5日木曜日

Cohen: SU(2) & U(1)A may be restored(1996)

Fact：T>Tc , when m→0, chiral symmetry is restored <=> h ¯ i = 0

Cohen：

If we assume h ¯

i = 0, gap in the Dirac spectrum, it may lead

Order parameterof SU(2) chiral sym.

U (1)A = 0

Order parameter ofU(1)A Sym. (Later)※Additional
condition is needed

Z 1
2m
¯
d ⇢( ) 2
h i = lim
m!0 0
+ m2
Z 1
4m2
d ⇢( ) 2
U (1)A = lim
m!0 0( + m2 )2Spectral rep :

lim

m!0

→
15年3月5日木曜日

Banks-Casher rel.

Letʼs check this by Lattice QCD !

8

Akio Tomiya(Osaka Univ.)

Previous studies (DW type) are controversial !
Group

FermionSize

Gap in the
spectrum

JLQCD(2013)

Overlap(Top. ﬁxed)

2 fm

Gap

TWQCD
(2013)

Optimaldomain-wall

3 fm

UA(1)

Correlator

Degenerate

Restored

No gap Degenerate Restored？

LLNL/RBC
Domain-wall 2, 4 fm
(2013)Peak

No
Violateddegeneracy

What makes such diﬀerence?

Fermion, Volumes or Topology ?
9
15年3月5日木曜日

U(1)A

y
a
w
e
By th

If U(1)A symmetry is restored ?

Ref: http://personal.kent.edu/~mstrick6/
15年3月5日木曜日

10

y
a
w
e
By th

If U(1)A symmetry is restored ?

We are interested
in this point

Ref: http://personal.kent.edu/~mstrick6/
15年3月5日木曜日

10

y
a
w
e
By th

If U(1)A symmetry is restored ?

Our target
Nf = 22nd

1

ms 2nd1st

CrossoverPure Gauge
Quenched

Nf = 1

Physical point

1st
mu, md1
11

15年3月5日木曜日

y
a
w
e
By th

If U(1)A symmetry is restored ?

The order of the transition may be changed :2nd→1st (Pisarski&Wilczek1983)
Our target
Nf = 22nd

1

ms 2nd1st

CrossoverPure Gauge
Quenched

Nf = 1

Physical point

1st
mu, md1
11

15年3月5日木曜日

y
a
w
e
By th

If U(1)A symmetry is restored ?

The order of the transition may be changed :2nd→1st (Pisarski&Wilczek1983)
Our target
Nf = 22nd

1

ms 2nd1st

CrossoverPure Gauge
Quenched

Our target
Nf = 22nd

1

ms

Nf = 1

2nd

Physical point

CrossoverPure Gauge

Nf = 1

Physical point

1st

1st
mu, md1

mu, md1

Possibly realized...

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15年3月5日木曜日

1stTable of contents
1. Introduction
2. U(1)A symmetry and Correlators
3. The Dirac spectrum
4. Previous studies
5. Our setup
6. Our results
7. Summary
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2.U(1)ASym. & Correlators

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Sym. of QCD<=>Degeneracyh⇡(x)⇡(0)i

SU (2)L ⇥ SU (2)RU (1)Ah (x) (0)ih (x) (0)i
U (1)ASU (2)L ⇥ SU (2)Rh⌘(x)⌘(0)i

The relationship between meson correlators. Top and bottom correlators¯
chiral symmetry.
arepairedvia
U(1)⇡(x) Left= i ¯and(x) right
⌧
(x)
(x)
=
(x) chiral(x) symmetry. B5¯
mperature, thesearenotdegenerated.
¯
⌘(x)
=
i
(x) 5 (x)(x) = (x)⌧ (x)⇢( )Degeneracy of these channels<=>There are symmetries
14

15年3月5日木曜日

Sym. of QCD<=>Degeneracyh⇡(x)⇡(0)i

SU (2)L ⇥ SU (2)RU (1)Ah (x) (0)ih (x) (0)i
U (1)ASU (2)L ⇥ SU (2)Rh⌘(x)⌘(0)i

The relationship between meson correlators. Top and bottom correlatorsZchiral symmetry. Left
and
right
are
pairedvia
U(1)chiralsymmetry.
B
4
Order parameter⌘ ared notx[h⇡(x)⇡(0)i
U (1)A
mperature,thesedegenerated. h (x) (0)i]of U(1)A⇢()
If this quantity 0 at V → , m→0,
U(1)A symmetry is restored.
15

15年3月5日木曜日

Letʼs check Cohenʼs argument
Z 1d ⇢( )h ¯ i = lim
m!0

lim

m!0

0

Z 1
d ⇢( )
U (1)A = lim
m!0

0

2m
2 + m224m( 2 + m2 )2Order parameter ofSU(2) Chiral symmetry

Order parameterU(1)A symmetry
What happens T>Tc

Checking U(1)A Sym. restoration at T>Tc ....
・Directly measure U (1)A・Measure ⇢( ) → Has a gap?
16
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3.Dirac spectrum
Z 1d ⇢( )h ¯ i = lim
m!0

0

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2m
2 + m2

Akio Tomiya(Osaka Univ.)⇢( ) :Dirac spectrum=Spectral density of the Dirac operator( 5D/) j ==µjj(@µ + Aµ )ρ(λ)：Distribution of λ→This reﬂects symmetry of quarks withthe gauge ﬁeld!
18
15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

Argument by Cohen(1996)

If there is a gap in the Dirac spectrum(and can be ignored exact zero-modes)

lim

m!0

Z

4

U(1)A violation

d x[h⇡(x)⇡(0)i

Z 1
h (x) (0)i] = lim
d
m!0

0

2

4m ⇢( )
(m2 + 2 )2=0

Cf : Aoki-Fukaya-Taniguchi (2012)19
15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

Argument by Cohen(1996)

If there is a gap in the Dirac spectrum(and can be ignored exact zero-modes)

lim

m!0

Z

4

U(1)A violation

d x[h⇡(x)⇡(0)i

Z 1
h (x) (0)i] = lim
d
m!0

0

2

4m ⇢( )
(m2 + 2 )2=0

Cf : Aoki-Fukaya-Taniguchi (2012)

λ〜0 modes are important
19
15年3月5日木曜日

4.Previous studies

20
15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

Previous studies (DW type) are controversial !
Group

FermionSize

Gap in the
spectrum

JLQCD(2013)

Overlap(Top. ﬁxed)

2 fm

Gap

TWQCD
(2013)

Optimaldomain-wall

3 fm

UA(1)

Correlator

Degenerate

Restored

No gap Degenerate Restored？

LLNL/RBC
Domain-wall 2, 4 fm
(2013)Peak

No
Violateddegeneracy

What makes such diﬀerence?
Fermion, Volume or Topology ?
21
15年3月5日木曜日

U(1)A

Akio Tomiya(Osaka Univ.)

Chiral symmetry on the Lat.
=Ginsparg-Wilson rel.

22
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Akio Tomiya(Osaka Univ.)

Chiral symmetry on the Lat.
=Ginsparg-Wilson rel.
S=Z

d x ¯D/
4

Cont.：

is chiral symmetric <=>

D
/ 5+

/ =0
5D

22
15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

Chiral symmetry on the Lat.
=Ginsparg-Wilson rel.
S=Z

d x ¯D/
4

Cont.：
Lat.：

is chiral symmetric <=>

D
/ 5+

D 5+

/ =0
5D
5 D = 2aD 5 D

Ginsparg-Wilson relation

a：lattice spacing

22
15年3月5日木曜日

Chiral sym. on the Lat.
Lat.： D 5 + 5 D = 2aD 5 D
Ginsparg-Wilson relation

a：lattice spacing

D (satisﬁes above) : 2 good things

1. Action has exact chiral symmetry!

0i 5 (1 aD)✓=e
¯ ! ¯0 = ¯ei 5 ✓→ The action has full SU(2) and U(1)A symmetries23

2. U(1)A sym. is violated by the quantizationas same as the continuum theory.

15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

The overlap fermion satisﬁes
the Ginsparg-Wilson relation exactly !

D 5+

1+m
Dov =
2

5 D = 2aD 5 D

1m
2

5 sgn(HT )HT is a hertimitan
Dirac operator◯ Exact chiral symmetry on the lattice Bad for the numerical simulation because of thesign function (needs special care).
24
15年3月5日木曜日

Previous result by JLQCD(2013)

Akio Tomiya(Osaka Univ.)

Red -> Yellow ( m→ light)

Gap

Simulation with the overlap (Exactly chiral)
FIG. 5: Spectral density ofthe masslessin two-flavor QCD. Top andVolume: L=2overlap-Dirac
fm & ﬁxingoperator
topology
Finitetemperature
simulation!
bottom panels are the data clearly
below
and above thecritical temperature, respectively. The→ As
a result,
is restored
the Tcerrors are shown for
middle panel corresponds
to thosearoundU(1)the transition
point.above
The jackknife

each bin of the histogram. When the histogram is terminated at the lower end, it implies that we

Objections: Do ﬁnite volume aﬀect near zero modes ?
find no eigenmode below that value. The statistical error in that case is also zero, because we use, Does topology-ﬁxing change the physics?

In this work, we change our set-up
argument about the power α and the point where gap opens would not be possible with the
and
Check
U(1)
restoration
25
currently available data. There is even a possibility that the gap develops right above thethe jackknife method. The lighter the color the lighter the mass.

15年3月5日木曜日

y
a
w
e
By th

Ideal simulation?

Ideal simulation• Overlap action:
Large volume,

several volume• Without topology
ﬁxing term

26
15年3月5日木曜日

y
a
w
e
By th

Ideal simulation?

Ideal simulation

Our set-up• Overlap action:• Overlap like action

several volume• Large volume、

ﬁxing term

• Without topology→reweighting to OV

Large volume,• Without topology

several volume
ﬁxing term

26
15年3月5日木曜日

5. Our set-up

27
15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

Mobius Domain-wall fermion= Better approximation of the overlap fermion
The overlap (Exactly chiral , used in JLQCD2013):

1+m
Dov =
2

1

m
2

Domain-wall fermion (used in RBC/LLNL)
1 digit precision chiral symmetry⇥

tanh Ls tanh

15年3月5日木曜日

1

(HT )

⇤

5 sgn(HT )HT is ahermitian Dirac op.

Akio Tomiya(Osaka Univ.)

Mobius Domain-wall fermion= Better approximation of the overlap fermion
The overlap (Exactly chiral , used in JLQCD2013):

1+m
Dov =
2

1

m
2

Domain-wall fermion (used in RBC/LLNL)
1 digit precision chiral symmetry⇥

tanh Ls tanh

1

(HT )

5 sgn(HT )⇤⇥tanh Ls tanh

15年3月5日木曜日

HT is ahermitian Dirac op.

1(2HT )⇤

Akio Tomiya(Osaka Univ.)

Mobius Domain-wall fermion= Better approximation of the overlap fermion
The overlap (Exactly chiral , used in JLQCD2013):

1+m
Dov =
2

1

m
2

Domain-wall fermion (used in RBC/LLNL)
1 digit precision chiral symmetry⇥

tanh Ls tanh

1

(HT )

5 sgn(HT )HT is ahermitian Dirac op.

⇤

Mobius Domain-wall fermion(This work)
3 digit chiral symmetry

Edwards-Heller (2000)⇥

tanh Ls tanh1(2HT )→ Better approximation of the OV(Still it violates the Ginsparg-Wilson rel. )15年3月5日木曜日⇤

Akio Tomiya(Osaka Univ.)

Gauge action:tree level Symanzik
Lattice set up Fermion :Mobius DW(b=2, c=1, Scaled Shamir + Tanh)
w/ Stout smearing(3)code:IroIro++(G.
Cossu et al.)Effects of near-zero Dirac eigenmodes
on axialU(1)Akio Tomiya
Resource :BG/Q(KEK)
L3 × Lt

L=2 fm

L=4 fm163 × 8163 × 8163 × 8163 × 8163 × 8163 × 8323 × 8323 × 8323 × 8323 × 8βmud (MeV)

Ls

mres (MeV)

Temp.(MeV)

4.07
4.07
4.07
4.10
4.10
4.10

30
15∗
3.0
32
16∗
3.2

12
12
24
12
12
24

2.5
2.4
1.4
1.2
1.2
0.8

180
180
180
200
200
200

4.07
4.10
4.10
4.10

3.0
32
16
3.2

24
12
24
24

5∗∗
1.7
1.7
0.7

180
200
200
200

Table 1: Our lattice set-up. Those with m∗ud are obtained by the stochastic reweighting of the Dirac operator
determinant from the ensemble with the higher quark mass. Residual mass with ∗∗ is estimated by weighted
average of gi with some threshold.

ｍres: Scale of violation of Ginsparg-Wilson relation

29
where
λ
is
the
i-th
eigenvalue
of
H
and
λ
is
a
certainthreshold,
gives
a
good
numerical
impleiMth
15年3月5日木曜日

6. Our results

30
15年3月5日木曜日

Result of the Mobius DW
T〜Tc

Akio Tomiya(Osaka Univ.)

T＞Tc(2 fm)

0.005

Small volume0.005 0.005

0

0
0

20

40

60

80

100

120

(2 fm)

0

140

0
0

20

20

40

40

60

λ(MeV)

0.005

0
20

40

60

80

100

140

120 140140

Large volume (4 fm)

0.005 0.005

00120 1400.015 0.015
am=0.01
am=0.001
am=0.005
am=0.001
0.01 0.01

ρ(λ) (GeV3)

Large volume (4 fm)

ρ(λ) (GeV3)

ρ(λ) (GeV3)

am=0.001

0.01

100 120

L=32 Domain-wall
Histogram(β=4.1)
T=200
MeV MeV
L=32 Overlap
Histogram(β=4.07)
T=184

L=32 Domain-wall Histogram(β=4.07) T=184 MeV
0.015

60 80
80 100λ(MeV)λ(MeV)120

ρ(λ) (GeV3)

Small volume0.015 0.015
am=0.01
am=0.01
am=0.005
am=0.001
am=0.001
0.01 0.01

ρ(λ) (GeV3)0.01

am=0.01
am=0.001

ρ(λ) (GeV3)

ρ(λ) (GeV3)

0.015

L=16 Overlap
Histogram(β=4.07)
T=184
L=16 Domain-wall
Histogram(β=4.1)
T=200
MeV MeVρ(λ) (GeV3)

L=16 Domain-wall Histogram(β=4.07) T=184 MeV

0

140

0
0

20

20

40

λ(MeV)

40

60

60 80
80 100λ(MeV)λ(MeV)

100 120

Figure5: Theeigenvalue
histogramsof the dFigure 4: The eigenvalue histograms of thedomainwall(left panels)
and reweighted
3
3The data for T ∼ 20
(right)
Diraconoperators.
(right)Dirac
operators.
The
data
for
T
∼
180MeVL=
16(toppanels)andL
No clear gap in the spectrum (bottom) lattices are presented.
(bottom) lattices are presented.
→UA(1) looks violated?
Even for the lightest mass (Red bar).

8.2

Ginsparg-Wilson relation violation
L=32x12 Domain-wall Histogram(β=4.24) T=192 MeV

15年3月5日木曜日

0.015

31

Akio Tomiya(Osaka Univ.)

3.Histogram for DW

No clear gap in the domain-wall spectrum
in large volume system→U(1)A looks violated.

Same result as the previous study by
LLNL/RBC 2013
What was wrong with previous JLQCD?

32
15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

3.Histogram for DW

No clear gap in the domain-wall spectrum
in large volume system→U(1)A looks violated.

Same result as the previous study by
LLNL/RBC 2013
What was wrong with previous JLQCD?
Possible causes

32
15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

3.Histogram for DW

No clear gap in the domain-wall spectrum
in large volume system→U(1)A looks violated.

Same result as the previous study by
LLNL/RBC 2013
What was wrong with previous JLQCD?
Possible causes

previous JLQCD：Finite volume eﬀect ?

32
15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

3.Histogram for DW

No clear gap in the domain-wall spectrum
in large volume system→U(1)A looks violated.

Same result as the previous study by
LLNL/RBC 2013
What was wrong with previous JLQCD?
Possible causes

previous JLQCD：Finite volume eﬀect ?
previous JLQCD：Topology ﬁxing changes physics?

32
15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

3.Histogram for DW

No clear gap in the domain-wall spectrum
in large volume system→U(1)A looks violated.

Same result as the previous study by
LLNL/RBC 2013
What was wrong with previous JLQCD?
Possible causes

previous JLQCD：Finite volume eﬀect ?
previous JLQCD：Topology ﬁxing changes physics?
Ginsparg-Wilson violation in DW ?
32

15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

3.Histogram for DW

No clear gap in the domain-wall spectrum
in large volume system→U(1)A looks violated.

Same result as the previous study by
LLNL/RBC 2013
What was wrong with previous JLQCD?
Possible causes

previous JLQCD：Finite volume eﬀect ?
previous JLQCD：Topology ﬁxing changes physics?
Ginsparg-Wilson violation in DW ?
32

15年3月5日木曜日

Akio Tomiya(Osaka Univ.)

Ginsparg-Wilson violation for each modeS=Z

d4 x ¯D

is chiral symmetric <=>Lattice： D 5 + 5 D = 2aD 5 D
Eigen-function of D↓

gi /

†
[D
+
5
5
i

5D

2aD 5 D] i

gi ＝0 for the chiral fermion
What happens on the Mobius domain-wall ?
33
15年3月5日木曜日

0

0
0

20

40

60

80

100

120

140

0

20

40

60

λ(MeV)

80

Akio Tomiya(Osaka Univ.)
100

120

140

λ(MeV)

Figure 6: The eigenvalue histograms of the domain wall (left panels) and reweighted(right) Dirac operators. The data for T ∼ 190MeV on L3 × Lt = 323 × 12 latpresented.

Ginsparg-Wilson relation is violated even forimproved domain-wall fermion
Violation of GW relation
DW,OV-16x8,T=180MeV,mud=3.0MeV|gi | 1.2

gi(DW)
gi(ov)

1

0.8

|gi|

|GW violation|

0.6
0.4
0.2
0
0

10

20

30

40
50
λ (MeV)

60

70

80

FigureNear
7: Thezero
violationof Ginsparg-Wilson
relation(g). Redissue...
symbol corresponds
modes
importantfor ithedomain-wall eigen vector and blue symbol corresponds to gi for overlap eigen vecto
34
15年3月5日木曜日

Observation

Eigenmodes of improved domain-wall :Ginsparg-Wilson is violated

35
15年3月5日木曜日

Observation

Eigenmodes of improved domain-wall :Ginsparg-Wilson is violated

Using reweighting technique,we switch the fermion determinant to the OV one(toward the ideal simulation...)35
15年3月5日木曜日

Akio Tomiya(Osaka Univ.)(skip)

Re-weighting tech. enables us to change
another fermion determinanthOiOverlap /
=
=
=

Z
Z

Z

Z

D ¯D DAµ O e Sgauge e
DAµ O eSgauge¯[DOV ]

2
Det[DOV ]2
Det[DSgauge2DW ]DAµ O eDet[DOV ]2 ]Det[DDW

D ¯D DAµ OR e Sgauge e/hORiDomain Wall¯[DDW ]

2
Det[DOV]R=
2 ]Det[DDW

Multiplying R and taking average, we obtain
the result with the overlap determinant
15年3月5日木曜日

36

Reweighting suppress near zero-modes!
Spectral density for the overlap Dirac operator
with diﬀerent fermion determinant (OV or DW)
T=180 MeV, L=2fm , m=3 MeV

ρa3λa
Red: Partially quenched = w/ det of DW
Blue: Reweighted =w/ det of OV
Cf: Microscopic Origin of UA(1) Symmetry Violation in the High Temperature Phaseof QCD - Dick, Viktor et al. arXiv:1502.06190 [hep-lat]37

15年3月5日木曜日

DW/OV Dirac spectrum has diﬀerent shapeOverlap

Domain-wall

2 fm L=16 Overlap Histogram(β=4.07) T=184 MeV

L=16 Domain-wall Histogram(β=4.07) T=184 MeV
0.015

am=0.01
am=0.001

0.01

3

ρ(λ) (GeV )ρ(λ) (GeV3)

0.015

0.005

0
0

20

40

60

80

100

120

am=0.01
am=0.005
am=0.001

0.01

0.005

0
140

0

λ(MeV)

3

ρ(λ) (GeV )ρ(λ) (GeV3)

0.005

0

20

40

60

40

60

80

100

120

140

120

140

4 fm L=32 Overlap Histogram(β=4.07) T=184 MeV

0.015

am=0.001

0.01

0

20

λ(MeV)

L=32 Domain-wall Histogram(β=4.07) T=184 MeV
0.015

T~Tc

80

100

120

0.01

0.005

0

140

am=0.001

020λ(MeV)

40

60

80

100

λ(MeV)

Figure 4: The eigenvalue histograms of the domain wall (left panels) and reweighted ov(right) Dirac
operators.
The data forT ∼ 180MeV on L3have= 163 (toppanels) and L3 =The
overlap
spectrums
gaps(bottom) lattices are presented.

on the spectruｍ(λ~20MeV)

8.2

38

Ginsparg-Wilson relation violation

15年3月5日木曜日

7.Summary1.

1.1. Thanks to the reweighting, we can perform the simulation
with the overlap in large volume without topology ﬁxing
1.2. We ﬁnd, the ov and DW spectrum have different shape
1.3. We ﬁnd Ginsparg-Wilson violation for DW each eigenmodes
1.4. We ﬁnd gaps in the spectrum for OV: T=180 MeV, T=200MeV ，2
fm and 4 fm.

time,
how
2. andWe investigate
are going
to,the eﬀective potential for the Wilson line phase varies by running
an imaginary chemical potential. In flat space-time, the eﬀective potential of supersymmetrictheory2.1.
is flatﬁnd
becauseof thebetweentheﬁner
contribution
from fermions and bosons.
a gapincancellation
the spectrum
forlatticeHowever, thanks to the curvature of S 2 , we obtain non-trivial eﬀective potential. We findthat large
is necessaryevaluationin order to break
the gap
gauge(Checksymmetry.
addition, we
2.2. R-charge
do quantitative
for thevol.Inscaling)
confirm a finite size eﬀect on our curved space taking smallish R-charge. This method has
several2.3.
issues,whichconsistency
are discussed following
section.
checkwith themeasurement for χU(1)This thesis is based on following papers:

Reference:

1. A. Tomiya, G. Cossu, H. Fukaya, S. Hashimoto and J. Noaki, arXiv:1412.7306 [hep-lat].

Full paper is in preparation:

2. A. Tanaka, A. Tomiya and T. Shimotani, JHEP 1410, 136 (2014) [arXiv:1404.7639

15年3月5日木曜日

39