カイラル感受率とU(1)量子異常ppp.ws/PPP2020/slides/Fukaya.pdfカイラル感受率とU(1)量子異常 Hidenori Fukaya (Osaka U.) for JLQCD collaboration S. Aoki, Y. Aoki,

カイラル感受率とU(1)量子異常

Hidenori Fukaya (Osaka U.) for JLQCD collaboration S. Aoki, Y. Aoki, HF, S. Hashimoto, T. Kaneko, C. Rohrhofer, K. Suzuki, in preparation.

JLQCD collaborationKEKを中心とした大規模格子QCD数値計算の研究グループ。元々KEKのスパコンを

使ってたが、、、 2017年 shutdown（涙）.

現在はHPCI (革新的ハイパフォーマンス・コンピューティング・インフラ)

筑波大CCSの学際共同利用プログラムなどに応募、元気にやっています。 Oakforest-PACS at JCAHPC

HITACHI SR16000 IBM BG/Q

JLQCD’s finite T QCD project今日お話しする有限温度QCDプロジェクトは 2019年度HPCI利用研究課題優秀成果賞に選ばれました。

https://www.hpci-office.jp/pages/project_report_meeting　より抜粋。

https://www.hpci-office.jp/pages/project_report_meeting

https://www.hpci-office.jp/pages/project_report_meeting

Chiral phase transition

150- 180MeV (10μs after Big-bang)

Temperature

Chiral symmetric, deconfined

Chiral SSB, confined

Cf. Talk by Yonekura-san（Tuesday）

What is chiral susceptibility?QCD partition function

chiral condensate

chiral susceptibility (this talk)

Z(m) =

Z[dA] det(D(A) +m)Nf e

−SG(A)

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�hq̄qi =1

NfV

∂

∂mlnZ(m)

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χ(m) =1

NfV

∂2

∂m2lnZ(m)

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What is chiral susceptibility?Broken phase Symmetric phase

Z(m)<latexit sha1_base64="PupVJemZSKQb4N1YVoU/w7dysu8=">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</latexit>

hq̄qi(m)<latexit sha1_base64="uAestzlMWMKd85tT0CQVFq0LVF0=">AAACn3ichVG7SgNBFD2u7/hI1Ea0iQYlNmESBcUqaKGVJNH4wEjYXSdxyezD3U0gBsHaH7CwUlAQC+38ABt/wMJPEEsFGwtvNguiYrzL7Jx75p47c7iKJTTHZeypRWpta+/o7OoO9PT29QdDA4Prjlm2VZ5VTWHam4rscKEZPOtqruCbls1lXRF8Qykt1s83Ktx2NNNYc6sW39HloqEVNFV2icqHRsI5IRtFwXOKbNf2D/dztpdG9al8KMJizIvwbxD3QQR+pMzQHXLYhQkVZejgMOASFpDh0LeNOBgs4nZQI84mpHnnHIcIkLZMVZwqZGJL9C9Stu2zBuX1no6nVukWQcsmZRgT7JFdsVf2wK7ZM/v4s1fN61F/S5V2paHlVj54PLz6/q9Kp93F3peqiUKh6uaeXBQw53nRyJvlMXWXaqN/5eDkdXU+M1GbZOfshfydsSd2Tw6Nypt6keaZUwRoQPGf4/gN1hOx+HQskZ6JJBf8UXVhFOOI0jxmkcQyUsjSvUe4xA1upTFpSVqRUo1SqcXXDOFbSFuflACakw==</latexit>

χ(m)<latexit sha1_base64="CAFQjK9661PFXJtohAGrpdQxieE=">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</latexit>

m<latexit sha1_base64="OJX0bJofVkGxZN+RbpUB20+Ul9I=">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</latexit><latexit sha1_base64="OJX0bJofVkGxZN+RbpUB20+Ul9I=">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</latexit><latexit sha1_base64="OJX0bJofVkGxZN+RbpUB20+Ul9I=">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</latexit><latexit sha1_base64="OJX0bJofVkGxZN+RbpUB20+Ul9I=">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</latexit>

What is chiral susceptibility?Broken phase Symmetric phase

Z(m)<latexit sha1_base64="PupVJemZSKQb4N1YVoU/w7dysu8=">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</latexit>

hq̄qi(m)<latexit sha1_base64="uAestzlMWMKd85tT0CQVFq0LVF0=">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</latexit>

χ(m)<latexit sha1_base64="CAFQjK9661PFXJtohAGrpdQxieE=">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</latexit>

m<latexit sha1_base64="OJX0bJofVkGxZN+RbpUB20+Ul9I=">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</latexit><latexit sha1_base64="OJX0bJofVkGxZN+RbpUB20+Ul9I=">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</latexit><latexit sha1_base64="OJX0bJofVkGxZN+RbpUB20+Ul9I=">AAACZHichVHLSsNAFD2N71q1KoIgSLEorsqNCIor0Y1L29oHaJEkjhqaF0la0OIP6FZx4UpBRPwMN/6Ai/6AIC4ruHHhTRoQFfUOM3PmzD13zsyojqF7PlEzJnV0dnX39PbF+xMDg0PJ4ZGiZ9dcTRQ027Ddsqp4wtAtUfB13xBlxxWKqRqipFZXg/1SXbieblsb/oEjKqayZ+m7uqb4TGXN7WSaMhRG6ieQI5BGFOt28gZb2IENDTWYELDgMzagwOO2CRkEh7kKGsy5jPRwX+AIcdbWOEtwhsJslcc9Xm1GrMXroKYXqjU+xeDusjKFaXqkW2rRA93RM73/WqsR1gi8HPCstrXC2R46Hs+//asyefax/6n607OPXSyGXnX27oRMcAutra8fnrfyS7npxgxd0Qv7v6Qm3fMNrPqrdp0VuQvE+QPk78/9ExTnMjJl5Ox8enkl+opeTGAKs/zeC1jGGtZR4HMFTnCKs9iTlJBGpbF2qhSLNKP4EtLkB+BMie0=</latexit><latexit sha1_base64="OJX0bJofVkGxZN+RbpUB20+Ul9I=">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</latexit>

finite V correction

SU(2) or U(1)?Chiral condensate or chiral susceptibility has been used as an order parameter of SU(2)LxSU(2)R chiral symmetry breaking. But the condensate also breaks axial U(1)… Naive answer : axial U(1) is anomalous given at a cut-off scale, and kept broken at all temperatures. We do not expect a strong dependence on T and m.

Naive expectation

m

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T

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Symmetry breaking at m=0

Symmetry breaking at T>Tc

U(1) anomalyU(1) anomaly

SU(2) SSB SU(2) SSB

T/m dependences of chiral condensate and its m-derivative should reflect SU(2) SSB rather than U(1) anomaly.

This talk = a counterintuitive result

In this work we show that the signal of chiral susceptibility is dominated by axial U(1) anomaly (at T>Tc), rather than SU(2)LxSU(2)R.

Contents✔ 1. Introduction

Chiral condensate and susceptibility are used as probe for SU(2)xSU(2) SSB but… 2. Dirac eigenmode decomposition 3. Lattice set-up 4. Numerical results 5. Summary

Dirac eigenmode decomposition

�hq̄qi =1

NfV

∂

∂mlnZ(m) =

1

V

*

X

λ

1

iλ(A) +m

+

,

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Z(m) =

Z[dA] det(D(A) +m)Nf e

−SG(A) =

Z[dA]

Yλ

(iλ(A) +m)Nf e−SG(A)

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χ(m) =1

NfV

∂2

∂m2lnZ(m) = χcon.(m) + χdis.(m),

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χcon.(m) = �∂

∂mvalence

hq̄qi

�

�

�

�

mvalence=m

χdis.(m) = �∂

∂msea

hq̄qi

�

�

�

�

msea=m

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chiral susceptibility

O(100)個ならLatticeで非摂動計算可能。

Connected partχcon.(m) = −

1

V

*

X

λ

1

(iλ(A) +m)2

+

= −

1

V

*

X

λ

2m2

(λ(A)2 +m2)2

+

+1

m

"

1

V

*

X

λ

m

λ(A)2 +m2

+#

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= �∆(m) +�hq̄qi

m,

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axial U(1) susceptibility!

Namely, connected part is dominated by the U(1) anomaly when condensate is zero.

∆(m) =X

x

[hπ(x)π(0)i � hδ(x)δ(0)i] ,

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U(1)A pair

Disconnected part

From the zero modes, we can separate the instanton number Q (U(1) anomaly effect) .

N0 = n+ + n−: number of zero modes

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χdis.(m) =Nf

V

2

4

*

X

λ

m

λ(A)2 +m2

!2+

−

*

X

λ

m

λ(A)2 +m2

+23

5 .

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=Nf

V

"

hN2

0i � hN0i

2

m2+

2

m

*

N0

X

λ>0

2m

λ(A)2 +m2

+

� hN0i

*

X

λ>0

2m

λ(A)2 +m2

+!

+

*

X

λ>0

2m

λ(A)2 +m2

!2+

�

*

X

λ>0

2m

λ(A)2 +m2

+23

5 .

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Q = n+ − n−= N0 − 2n

−= 2n+ −N0

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Disconnected partU(1) anomaly effect (topological susceptibility) χdis.(m)=

Nf

m2

hQ2i � hQi2

V

+2Nf

V

hn+n−i � hn+ihn−

i

m2

+Nf

V

"

2

m

*

N0

X

λ>0

2m

λ(A)2 +m2

+

� hN0i

*

X

λ>0

2m

λ(A)2 +m2

+!

+

*

X

λ>0

2m

λ(A)2 +m2

!2+

�

*

X

λ>0

2m

λ(A)2 +m2

+23

5 .

<latexit sha1_base64="WUromBA95d48lVYbMAe4bx7cSYA=">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</latexit>

Non-zero modes effect responsible for SU(2)LxSU(2)R.

empirically small

Lattice formulasWith the overlap Dirac operator, we have

where λm = eigenvalues of Hm = γ5[(1−m)Dov +m]

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χdis.lat(m) =Nf

V

2

4

1

(1�m2)2

*

X

allλm

m(1� λ2

m)

λ2m

!2+

� |hq̄qilat|2V 2

3

5 .

<latexit sha1_base64="MAa54fE/M9yZQfj7zvKPddKJPcs=">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</latexit>

χcon.lat(m) = �∆lat(m) +

�hq̄qilat

m,

∆lat(m) =

1

V (1�m2)2

*

X

allλm

2m2(1� λ2

m)2

λ4m

+

,

�hq̄qilat =1

V (1�m2)

*

X

allλm

m(1� λ2

m)

λ2m

+

.

<latexit sha1_base64="ncrudR/nb38wiuQgrKKniOwrX2w=">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</latexit>


Chiral condensate and susceptibility are used as a probe for SU(2)xSU(2) SSB but… 2. Dirac eigenmode decomposition Chiral susceptibility contains purely axial U(1) anomaly effects. 3. Lattice set-up 4. Numerical results 5. Summary

✔

Simulation setup厳密なカイラル対称性を保つDirac 演算子を用いた世界初の有限温度QCD大規模数値計算。 1/a = 2.6 GeV (0.074fm) L=24,32,40,48 [1.8-3.6fm] Symanzik gauge action Mobius domain-wall fermions with mres<1MeV and reweighted overlap fermion. Quark mass from 3MeV (< phys. pt. ~4MeV) to 30MeV. T=190, 220, 260, 330 MeV (Lt=8,10,12,14). Tc is estimated to be around 175MeV (from Polyakov loop).

t

z

Overlap & (Mobius) domain-wall

with Ls=16.

perfect chiral sym.

good chiral sym.

numerically violation ~ 1keV

numerically violation ~ 1MeV

(→a→0 Dcontinuum +m)

<latexit sha1_base64="/pWbZCOsR4d5mCOhDuB5nkTF1Pc=">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</latexit>

= Edge states of 5D topological insulator

Violation of chiral symmetry enhanced at finite TChecking chiral sym. for EACH eigenmode

Bad modes appear above Tc for a~0.1fm. [JLQCD (Cossu et al.) 2015, JLQCD(Tomiya et al.) 2016]

Note: residual mass is (weighted) average of them.

For T=0, gi are consistent with residual mass.

Overlap/domain-wall reweightingThe fermion action can be changed AFTER simulation.

can be numerically computable.


Chiral condensate and susceptibility are used as a probe for SU(2)xSU(2) SSB but… 2. Dirac eigenmode decomposition Chiral susceptibility contains purely axial U(1) anomaly effects. 3. Lattice set-up Nf=2 QCD w/ MDWF and rewegihted overlap. at T=190-330MeV near physical m~4MeV. 4. Numerical results 5. Summary

✔

✔

Low-mode approximationIn the eigenvalue summations,

where we truncate at 30-40th lowest mode ( ).

λm = eigenvalues of Hm = γ5[(1−m)Dov +m]

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λthreshold ∼ 150–300 MeV<latexit sha1_base64="OkxPQI5o3ZNVPs46yPfD1RHM1Zc=">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</latexit><latexit sha1_base64="OkxPQI5o3ZNVPs46yPfD1RHM1Zc=">AAAClnichVHLShxBFD22JuokxoluBDdNBkM2DrcTQ4ILESXoRhg1Mwq2DN09pVNY/aC7ZlCb+QF/wIWrCBKC3+BKSPIDLvwEydJANlnkTk9DSCTxFlV16tQ9t05VuZGSiSa67jP6Bx48HBwaLjx6PPJktPh0rJaErdgTVS9UYbzpOolQMhBVLbUSm1EsHN9VYsPdW+zub7RFnMgweK8PIrHtO7uB3JGeo5mqFy1bcXLDqad27Ju6GYukGapGx7QT6Zu274b7qfWapqdfEZkrotYx68USlSkL8y6wclBCHpWw+BE2GgjhoQUfAgE0YwUHCbctWCBEzG0jZS5mJLN9gQ4KrG1xluAMh9k9Hnd5tZWzAa+7NZNM7fEpinvMShNTdEWf6Ja+0jnd0M9/1kqzGl0vBzy7Pa2I6qNHE+s/7lX5PGs0f6v+61ljB28zr5K9RxnTvYXX07cPj2/XZ9em0ud0St/Y/we6pku+QdD+7p2tirUTFPgDrL+f+y6ovSxbVLZWZ0rzC/lXDGESz/CC3/sN5rGMCqp87jEu8BlfjAljznhnLPVSjb5cM44/wqj8Ah4Hm4g=</latexit><latexit sha1_base64="OkxPQI5o3ZNVPs46yPfD1RHM1Zc=">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</latexit><latexit sha1_base64="OkxPQI5o3ZNVPs46yPfD1RHM1Zc=">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</latexit>

χdis.lat(m) =Nf

V

2

4

1

(1�m2)2

*

X

allλm

m(1� λ2

m)

λ2m

!2+

� |hq̄qilat|2V 2

3

5 .

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χcon.lat(m) = �∆lat(m) +

�hq̄qilat

m,

∆lat(m) =

1

V (1�m2)2

*

X

allλm

2m2(1� λ2

m)2

λ4m

+

,

�hq̄qilat =1

V (1�m2)

*

X

allλm

m(1� λ2

m)

λ2m

+

.

<latexit sha1_base64="ncrudR/nb38wiuQgrKKniOwrX2w=">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</latexit>

Low mode approximation

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0 0.02 0.04 0.06 0.08 0.1 0.12

chir

al

sus.

connecte

d

λthre

L=24, m=0.01

L=24, m=0.005

L=24, m=0.00375

L=24, m=0.0025

L=24, m=0.001

L=32

L=40

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.02 0.04 0.06 0.08 0.1 0.12

chir

al

sus.

dis

connecte

d

λthre

L=24, m=0.01

L=24, m=0.005

L=24, m=0.00375

L=24, m=0.0025

L=24, m=0.001

L=32

L=40

connected (we do not use)

disconnected part is well described by 30-40 lowest modes.

Disconnected susceptibility at T=220MeV χdis.(m) =

Nf

V

2

4

*

X

λ

m

λ(A)2 +m2

!2+

−

*

X

λ

m

λ(A)2 +m2

+23

5 .

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0

0.02

0.04

0.06

0.08

0.1

0 0.002 0.004 0.006 0.008 0.01 0.012

chiral susceptibili

ty d

isconnecte

d

m

beta=4.30(T=220MeV) threshold=0.07

w/ nonzeromodes, L=24w/ nonzeromodes, L=32w/ nonzeromodes, L=40

Data on L=1.8-3.6fm lattices are consistent. Pseudo-peak at m=0.005 (14MeV)?

Disconnected susceptibility at T=220MeV

is dominated by chiral zero modes.

χdis.(m)

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⇠Nf

V

hN2

0i � hN0i

2

m2<latexit sha1_base64="GTK8PjPczRiv739TitJuBtL20TY=">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</latexit>

0

0.02

0.04

0.06

0.08

0.1

0 0.002 0.004 0.006 0.008 0.01 0.012

chiral susceptibili

ty d

isconnecte

d

m



zeromodes only

N0 :<latexit sha1_base64="lQnUs0hbmtDBXIc0SmiVKJYHxxY=">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</latexit>

number of zero modes

Disconnected susceptibility at T=220MeV

is dominated by U(1) anomaly !

χdis.(m)


⇠Nf

V

hN2

0i � hN0i

2

m2<latexit sha1_base64="GTK8PjPczRiv739TitJuBtL20TY=">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</latexit>

0

0.02

0.04

0.06

0.08

0.1

0 0.002 0.004 0.006 0.008 0.01 0.012

ch

ira

l su

sce

ptib

ility

dis

co

nn

ecte

d

m



zeromodes only2*top.sus./m2

⇠Nf

V

hQ2i � hQi2

m2<latexit sha1_base64="HCVM/Jzb5t8UYS975aLjSScoySs=">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</latexit> Q = n+ − n

−

<latexit sha1_base64="PFXXzCD2nY0yJfKEua838l2+msU=">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</latexit>

Disconnected part at different T

The dominance by topological susceptibility is seen at 4 different temperatures. Moreover, the chiral limit is consistent with zero.

0

200

400

600

800

1000

1200

1400

0 5 10 15 20 25 30 35 40

chiral susceptibili

ty1/2

(MeV

)

m(MeV)

chiral susceptibility1/2 (disconnected)

T=190 MeVT=220 MeVT=260 MeVT=330 MeV

2*top.sus./m2

χdis.(m)


⇠Nf

V

hQ2i � hQi2

m2<latexit sha1_base64="HCVM/Jzb5t8UYS975aLjSScoySs=">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</latexit>

Q : instanton number<latexit sha1_base64="0/4IVhbyB0GA6BecsPHdHhpO3rM=">AAACfnichVFBSxtBGH1ZtdWoNerF4iUYIl6M30rB4EnspUdjjAYSCbvrqEN2Z5fdSVCDePcPePDUgogU2h/RS/+AB3+CeFTw4sEvm4VSxfYbZubNm+9982bGDlwZaaKblNHXP/Du/eBQenhk9MNYZnxiM/JboSMqju/6YdW2IuFKJSpaaldUg1BYnu2KLbv5ubu/1RZhJH21oQ8Dse1Ze0ruSsfSTDUyH0vLdc/2DzpSRdpS2ldZ1fJsER43MjkqUBzZ18BMQA5JrPmZS9SxAx8OWvAgoKAZu7AQcavBBCFgbhsd5kJGMt4XOEaatS3OEpxhMdvkcY9XtYRVvO7WjGK1w6e43ENWZpGna7qie/pN3+mWnt6s1YlrdL0c8mz3tCJojJ1OlR//q/J41tj/o/qnZ41dFGOvkr0HMdO9hdPTt4/O7svL6/nOLH2jO/b/lW7oF99AtR+ci5JYP0eaP8B8+dyvweZiwaSCWfqUW1lNvmIQ05jBHL/3ElbwBWuo8LknuMAP/DRgzBrzxkIv1Uglmkn8FUbxGZpCk74=</latexit><latexit sha1_base64="0/4IVhbyB0GA6BecsPHdHhpO3rM=">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</latexit><latexit sha1_base64="0/4IVhbyB0GA6BecsPHdHhpO3rM=">AAACfnichVFBSxtBGH1ZtdWoNerF4iUYIl6M30rB4EnspUdjjAYSCbvrqEN2Z5fdSVCDePcPePDUgogU2h/RS/+AB3+CeFTw4sEvm4VSxfYbZubNm+9982bGDlwZaaKblNHXP/Du/eBQenhk9MNYZnxiM/JboSMqju/6YdW2IuFKJSpaaldUg1BYnu2KLbv5ubu/1RZhJH21oQ8Dse1Ze0ruSsfSTDUyH0vLdc/2DzpSRdpS2ldZ1fJsER43MjkqUBzZ18BMQA5JrPmZS9SxAx8OWvAgoKAZu7AQcavBBCFgbhsd5kJGMt4XOEaatS3OEpxhMdvkcY9XtYRVvO7WjGK1w6e43ENWZpGna7qie/pN3+mWnt6s1YlrdL0c8mz3tCJojJ1OlR//q/J41tj/o/qnZ41dFGOvkr0HMdO9hdPTt4/O7svL6/nOLH2jO/b/lW7oF99AtR+ci5JYP0eaP8B8+dyvweZiwaSCWfqUW1lNvmIQ05jBHL/3ElbwBWuo8LknuMAP/DRgzBrzxkIv1Uglmkn8FUbxGZpCk74=</latexit><latexit sha1_base64="0/4IVhbyB0GA6BecsPHdHhpO3rM=">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</latexit>


Chiral condensate and susceptibility are used as a probe for SU(2)xSU(2) SSB but… 2. Dirac eigenmode decomposition Chiral susceptibility contains purely axial U(1) anomaly effects. 3. Lattice set-up Nf=2 QCD w/ MDWF and rewegihted overlap. at T=190-330MeV near physical m~4MeV. 4. Numerical results Disconnected chiral susceptibility is dominated by axial U(1) anomaly (topological susceptibility). 5. Summary (and discussion)

✔

✔

✔

Summary1. Chiral condensate/susceptibility are related to

both SU(2)xSU(2) and U(1)A. 2. In the Dirac eigenmode decomposition of

exactly chiral symmetric Dirac op, we can separate the purely U(1) anomaly effect.

3. Connected part = condensate/m + axial U(1) susceptibility.

4. Disconnected part 5. Axial U(1) anomaly may play more important

role in QCD phase transition than expected.

⇠2hQ2i

m2V

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May not be a surprise …* A long ago, people tried to explain the SU(2)xSU(2) breaking by instantons=U(1) anomaly [Polyakov,’t Hooft, Shuryak…] [Cf. talk by Hamada].

But typical confs with Q≠0 is far from instanton semi-classical “solutions”. * Also, was also indicated from Ward-Takahashi identities [LLNL/RBC Collaboration 2013]

χdis.(m)


⇠Nf

V

hQ2i � hQi2

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χ(m = 0) = U(1) anomaly

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What’s still nontrivial?

* dominance by topological susceptibility, * seen at finite m up to 40 MeV, * suppression in the chiral limit, consistent with zero.

0

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chiral susceptibili

ty1/2

(MeV

)

m(MeV)

chiral susceptibility1/2 (disconnected)

T=190 MeVT=220 MeVT=260 MeVT=330 MeV

2*top.sus./m2

Disappearance of U(1) anomaly

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ΔU

Vsu

bt.

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eV

]m (MeV)

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T=220MeV, L=2.4fm

T=220MeV, L=3.0fm

T=220MeV, L=3.6fm

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en (

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L=40

L=48

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Δm

scre

en (

MeV

)

m (MeV)

T=220MeV, U(1)A

L=24

L=32

L=40

L=48

are seen in other observables.

U(1) susceptibility

meson screening mass baryon screening mass

What if axial U(1) “restored”? Not only SU(2)LxSU(2)R but also U(1)A may be restored at Tc. Then, the effective action = SU(2)xSU(2)[or O(4)] linear sigma model needs additional degrees of freedom. -> effective potential becomes complicated -> 1st-order transition is favored [Pisarski & Wilczek]

(the same suggestion as Yonekura-san’s but from different point of view.)

What if chiral phase transition is 1st order? * 1st order region may be spanned to finite quark mass.

* If physical point is not a crossover but 1st order, QCD may explain dark matter (Witten) -> ask Yonekura-san

* Gravitational waves due to QCD bubbles? * Axion dark matter scenario may be difficult (abundance is too big).

Interesting! But we have not detected its sign.

HotQCD 2012 (Domain-wall) Aoki-F-Taniguchi 2012 (theory) Ishikawa et al2013, 2014,2017. (Wilson) JLQCD 2013, 2016 (overlap) TWQCD 2013 (optimal DW) LLNL/RBC 2013 (Domain-wall) [may be at higher T] Pelisseto and Vicari 2013(theory) Nakayama-Ohtsuki 2015, 2016(CFT) Sato-Yamada 2015(theory), Kanazawa & Yamamoto 2015, 2016 (theory) Dick et al. 2015 (OV in HISQ sea) Sharma et al. 2015, 2016 (OV in DW sea) Glozman 2015, 2016 (theory) Borasnyi et al. 2015 (staggered & OV) Brandt et al. 2016 (Wilson) Ejiri et al. 2016 (Wilson) Azcoiti 2016,2017(theory) Gomez-Nicola & Ruiz de Elvira 2017 (theory) Rorhofer et al. 2017(Mobius DW) ……

CAN U(1)ANOMALY DISAPPEAR

AT FINITE T? → MANY ANSWERS.

Cohen 1996, 1998 (theory)

Bernard et al. 1996 (staggered)

Chandrasekharan et al. 1998 (staggered)

HotQCD 2011 (staggered)

Ohno et al. 2011 (staggered)

and many others

Before 2012 After 2012

Red: YES Blue: NO Green: Not (directly) answered but related

Take-home messageあなたが SU(2)LxSU(2)R の order parameterだと思っている

実は U(1)A 量子異常まみれかもしれません…。

∂

∂mhψ̄ψi

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Back-up slides

What we know at T=0

m

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meson mass2

η’ meson

pion

Temperature dependence

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1 1.2 1.4 1.6 1.8 2

χdis/ T

2

T/Tc

m=3 MeV

m=7 MeV

Nf=2 JLQCD 2020 preliminary Cf. Nf=2+1 result by HotQCD 2011

Documents

カイラル感受率とU(1)量子異常ppp.ws/PPP2020/slides/Fukaya.pdfカイラル感受率とU(1)量子異常 Hidenori Fukaya (Osaka U.) for JLQCD collaboration S. Aoki, Y. Aoki,