[PPT] - Fate of axial U(1) symmetry at two flavor chiral limit of QCD in PowerPoint Presentation

SLIDE 1

Fate of axial U(1) symmetry at two flavor chiral limit of QCD in finite temperature

Yasumichi Aoki & XQCD 2018 @ Frankfurt am Main May 21, 2018

SLIDE 2

Thanks to

Those who gave me useful

information for this talk

Phillipe de Forcrand
Christian Lang
Gian Carlo Rossi
Peter Petreczky
Sayantan Sharma
Vicente Azcoiti
Bastian Brandt
for useful discussion
Ryuichiro Kitano
Norikazu Yamada
JLQCD members
Sinya Aoki
Guido Cossu
Shoji Hasihmoto
Hidenori Fukaya
Kei Suzuki ……

SLIDE 3

U(1) axial

violated by quantum anomaly

up to contact terms

at T=0, responsible for η’ mass
non-trivial topology of gauge field
at high T, this Ward-Takahashi identity is still valid
however, if configurations that contribute to RHS is suppressed………

➡ the symmetry effectively recovers ๏ here Nf=2 (including Nf=2+1 with “2” driven to chiral limit)

∂µJµ

5 = Nf

32π2 F ˜ F

h∂µJµ

5 (x) · O(0)i = Nf

32π2 hF ˜ F(x) · O(0)i

SLIDE 4

Why bother ?

Because it is unsettled problem !
fate of U(1)A - analytic
Gross-Pisarski-Yaffe (1981) restores in high temperature limit
Dilute instanton gas
Cohen (1996)
measure zero instanton effect → restores
Lee-Hatsuda (1996)
zero mode does contributes → broken
Aoki-Fukaya-Tanigchi (2012)
QCD analysis (overlap) → restores w/ assumption (lattice)
Kanazawa-Yamamoto (2015)
EFT case study how restore / break
Azcoiti (2017)
case study how restore / break

SLIDE 5

Why bother ?

Because it is unsettled problem !
fate of U(1)A lattice
HotQCD (DW, 2012) broken
JLQCD (topology fixed overlap, 2013) restores
TWQCD (optimal DW, 2013) restores ?
LLNL/RBC (DW, 2014) broken
HotQCD (DW, 2014) broken
Dick et al. (overlap on HISQ, 2015) broken
Brandt et al. (O(a) improved Wilson 2016) restores
JLQCD (reweighted overlap from DW, 2016) restores
JLQCD (current: see Suzuki et al Lattice 2017) restores
Ishikawa et al (Wilson, 2017) at least Z4 restores

SLIDE 6

Why bother ?

it may provide useful information on the phase transition
if the U(1)A continue to be broken
SU(2)L x SU(2)R ≃ O(4) universality class for 2nd order
if the U(1)A recovers
U(2)L x U(2)R / U(2)V for 2nd order
provides crucial information on the universality class
1st order possible for both cases
though often discussed in context with U(1)A restoration

SLIDE 7

Why bother ?

it may provide useful information on the phase transition

➡ Columbia plot

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

Physical pt : crossover

Wuppertal 2006

Right upper corner : 1st order

pure gauge

ther parts are less known

[original Columbia plot: Brown et al 1990]

SLIDE 8

Columbia plot: direct search of PT / scaling

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

2nd order
improved Wilson
WHOT-QCD Lat2016 (O(4) scaling)
Ejiri et al PRD 2016 [heavy many flavor]
1st oder
imaginary μ → 0
staggered Bonati et al PRD 2014
Wilson Phillipsen et al PRD 2016

SLIDE 9

Columbia plot: direct search of PT / scaling

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

2nd order
improved Wilson
WHOT-QCD Lat2016 (O(4) scaling)
Ejiri et al PRD 2016 [heavy many flavor]
1st oder
imaginary μ → 0
staggered Bonati et al PRD 2014
Wilson Phillipsen et al PRD 2016

external parameter → phase boundary → point of interest

➡ detour the demanding region

SLIDE 10

Columbia plot: direct search of PT / scaling

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

2nd order
improved Wilson
WHOT-QCD Lat2016 (O(4) scaling)
Ejiri et al PRD 2016 [heavy many flavor]
1st oder
imaginary μ → 0
staggered Bonati et al PRD 2014
Wilson Phillipsen et al PRD 2016

0.05 0.1 0.15 0.2 0.25

(am u,d)2/5

1
0.75
0.5
0.25

(µ/T)

2

first order second order B region region

Bonati et al

external parameter → phase boundary → point of interest

➡ detour the demanding region

SLIDE 11

Columbia plot: direct search of PT / scaling

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

2nd order
improved Wilson
WHOT-QCD Lat2016 (O(4) scaling)
Ejiri et al PRD 2016 [heavy many flavor]
1st oder
imaginary μ → 0
staggered Bonati et al PRD 2014
Wilson Phillipsen et al PRD 2016

0.05 0.1 0.15 0.2 0.25

(am u,d)2/5

1
0.75
0.5
0.25

(µ/T)

2

first order second order B region region

Bonati et al

external parameter → phase boundary → point of interest

➡ detour the demanding region

for all Nt = 1/(aT) =4 or 6 problem not settled yet

SLIDE 12

χt(mf) for Nf=2 T=220 MeV

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV- DW OV index on DW ensemble OV-OV OV index on OV ensemble

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW GL-OV OV-DW OV-OV

32

3x12, β=4.3

JLQCD: Lattice 2017

SLIDE 13

χt(mf) for Nf=2 T=220 MeV

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV- DW OV index on DW ensemble OV-OV OV index on OV ensemble

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW GL-OV OV-DW OV-OV

32

3x12, β=4.3

JLQCD: Lattice 2017

physical ud

SLIDE 14

χt(mf) for Nf=2 T=220 MeV

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV- DW OV index on DW ensemble OV-OV OV index on OV ensemble

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW GL-OV OV-DW OV-OV

32

3x12, β=4.3

1st order transition ? JLQCD: Lattice 2017

physical ud

SLIDE 15

χt(mf) for Nf=2 T=220 MeV

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV- DW OV index on DW ensemble OV-OV OV index on OV ensemble

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW GL-OV OV-DW OV-OV

32

3x12, β=4.3

1st order transition ? make sense al a Pisarski & Wilczek JLQCD: Lattice 2017

physical ud

SLIDE 16

χt(mf) for Nf=2 T=220 MeV

GL-DW gluonic charge on DW ensemble GL-OV gluonic charge on OV ensemble OV- DW OV index on DW ensemble OV-OV OV index on OV ensemble

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW GL-OV OV-DW OV-OV

32

3x12, β=4.3

1st order transition ? make sense al a Pisarski & Wilczek JLQCD: Lattice 2017 JLQCD: U(1)A restoration

physical ud

SLIDE 17

if upper left corer is 1st order

0 ≤ mf < mc : 1st oder
might affect the physics around physical point

mud ms ∞ ∞

physical pt.

mud ms ∞ ∞ ?

SLIDE 18

Columbia plot: direct search of PT / scaling

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

SLIDE 19

Columbia plot: direct search of PT / scaling

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

SLIDE 20

Columbia plot: direct search of PT / scaling

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

SLIDE 21

Columbia plot: direct search of PT / scaling

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

Nf=2+1 or 3

either
no PT found
1st order region
shrinks as a→0

with both staggered and Wilson

r even disappear ?
for more information see eg
Meyer Lattice 2015
Ding Lattice 2016
de Forcrand

“Surprises in the Columbia plot” (Lapland talk 2018)

SLIDE 22

Columbia plot: direct search of PT / scaling

mud ms ∞ ∞

physical pt.

1st order 1st order crossover

Nf=2+1 or 3

either
no PT found
1st order region
shrinks as a→0

with both staggered and Wilson

r even disappear ?
for more information see eg
Meyer Lattice 2015
Ding Lattice 2016
de Forcrand

“Surprises in the Columbia plot” (Lapland talk 2018)

Understanding of the diagram being changed a lot

SLIDE 23

Why bother ?

in relation with “extended symmetry”
spin-chiral symmetry for vector and scalar props. at high T
SU(4) ⊃ SU(2)L x SU(2)R x U(1)A
C. Rohrhofer et al., PRD17 [1707.01881]
C. Lang [1803.08693]
riginal discussion on this symmetry: Glozman et al
for the T=0 but low-mode subtracted Dirac operator

SLIDE 24

Why bother ?

axion cosmology scenario may fail for U(1)A restoration

due to vanishing / suppressed topological susceptivility

χt |m=0 = 0 & dnχt / dmn|m=0 = 0 Aoki-Fukaya-Tanigchi

➡ χt = 0 for small non-zero m OR ➡ exponential decay for T>Tc

axion mass and decay constant:

➡ axion window can possibly be closed

Kitano-Yamada JHEP [1506.00370]

see also for θ=π QCD non-standard case with rich implications

Di Vecchia et al. JHEP [1709.00731]

χt(T) ∼

mqΛ3

QCD,

T < Tc, m2

qΛ2 QCDe−2c(mq)T 2/T 2

c , T > Tc,

h c(mq) → ∞ as mq → 0,

s χt = m2

af2 a

SLIDE 25

U(1)A restoration or not

need to make sure if not comparing apples and oranges…
key points
systematics effects of lattice discretization under control ?
ud chiral limit of
Nf=2 QCD or
Nf=2+1 QCD → strange quark mass effect !
discussing mud→0 or just around physical ud mass
discussing X = 0 ? or X ≃ 0 ?

SLIDE 26

a U(1)A order parameter

symmetry in switching flavor non-singlet pseudoscalar and

scalar

rder parameter:

➡ 0 for U(1)A restoration

as a result, screening masses for these channel will degenerate
not a sufficient condition for U(1)A restoration

Δπ−δ ¼ Z d4x½hπaðxÞπað0Þi − hδaðxÞδað0Þi;

SLIDE 27

screening mass from O(a) improved Wilson f Nf=2

1200
1000
800
600
400
200

200 10 20 30 40 50 ∆MP S [MeV] mud [MeV] T = TC linear chiral extrapolation T = 0 physical point T = 0 chiral limit

Brandt et al JHEP [1608.06882]

SLIDE 28

screening mass from O(a) improved Wilson f Nf=2

mass difference between π and δ
1200
1000
800
600
400
200

200 10 20 30 40 50 ∆MP S [MeV] mud [MeV] T = TC linear chiral extrapolation T = 0 physical point T = 0 chiral limit

Nt = 1/(aT) = 16 - quite fine lattice
T=Tc - on top of transition temperature
nly one existing study for Nf=2
ΔMPS = 0 (with a sizable error) → consistent with U(1)A restoration

Brandt et al JHEP [1608.06882]

SLIDE 29

relation with Dirac eigenmode spectrum ρ(λ)

hqqi = lim

m→0

Z ∞ dλρ(λ) 2m λ2 + m2 = πρ(0)

∆πδ = Z 1 dλρ(λ) 2m2 (λ2 + m2)2 →∼ ρ0(0)

SLIDE 30

relation with Dirac eigenmode spectrum ρ(λ)

chiral condensate : order parameter of SU(2)A
U(1)A:

very roughly speaking

very sensitive to the spectrum near λ=0
overlap fermion, able to distinguish zero/nonzero modes, is ideal

hqqi = lim

m→0

Z ∞ dλρ(λ) 2m λ2 + m2 = πρ(0)

∆πδ = Z 1 dλρ(λ) 2m2 (λ2 + m2)2 →∼ ρ0(0)

SLIDE 31

JLQCD 16: Hov, HDW spectrum: above Tc Nf=2

[JLQCD 2016 Tomiya et al]

DW: Domain wall fermion sea
OV: Overlap valence
exact “chiral symmetry”
reweighting to OV

SLIDE 32

JLQCD 16: Hov, HDW spectrum: above Tc Nf=2

[JLQCD 2016 Tomiya et al]

OV on DW OV on OV DW on DW

DW: Domain wall fermion sea
OV: Overlap valence
exact “chiral symmetry”
reweighting to OV

SLIDE 33

JLQCD 16: Hov, HDW spectrum: above Tc Nf=2

[JLQCD 2016 Tomiya et al]

Δ~0 OV on DW OV on OV DW on DW

DW: Domain wall fermion sea
OV: Overlap valence
exact “chiral symmetry”
reweighting to OV

SLIDE 34

JLQCD 16: Hov, HDW spectrum: above Tc Nf=2

Lowest bin→0 consistent with SUA(2) restoration

[JLQCD 2016 Tomiya et al]

Δ~0 OV on DW OV on OV DW on DW

DW: Domain wall fermion sea
OV: Overlap valence
exact “chiral symmetry”
reweighting to OV

SLIDE 35

Comparison: unitary <-> partially quench

Δ>0 Δ~0

range of JLQCD

Dick et al PRD [1502.06190]

SLIDE 36

Comparison: unitary <-> partially quench

Δ>0 Δ~0

range of JLQCD

OV on DW OV on HISQ OV on OV

Dick et al PRD [1502.06190]

SLIDE 37

Comparison: unitary <-> partially quench

Δ>0 Δ~0

range of JLQCD

OV on DW OV on HISQ OV on OV

Dick et al PRD [1502.06190]

Partially quench effect needs to be investigated

SLIDE 38

U(1)A residual chiral symmetry br. of DWF

SLIDE 39

U(1)A residual chiral symmetry br. of DWF

fraction of Δ from residual chiral symmetry breaking [JLQCD]
residual breaking, which is small in terms of mres

dominates the U(1)A br.

SLIDE 40

JLQCD 16: UA(1) susceptibility: T=190-220 MeV

¯ Δov

π−δ ≡ Δov π−δ − 2N0

Vm2 : ð

zero mode effect

ð Þ

SLIDE 41

JLQCD 16: UA(1) susceptibility: T=190-220 MeV

¯ Δov

π−δ ≡ Δov π−δ − 2N0

Vm2 : ð

zero mode effect

ð Þ

seemingly Δ→0

SLIDE 42

HotQCD 2014: DWF Nf=2+1

50 100 150 200 250 130 140 150 160 170 180 190 200 T [MeV] (χMS

π - χMS σ )/T2

mπ=135 MeV mπ=200 MeV 50 100 150 200 250 130 140 150 160 170 180 190 200 T [MeV] (χMS

π - χMS δ )/T2

mπ=135 MeV mπ=200 MeV

0.2

0.0 0.2 0.4 0.6 0.8 1.0

χπ − χσ χη− χδ χπ − χδ χσ− χη

SU(2)A U(1)A

[figures from Ding Lattice 2016] [figures from Ding Lattice 2016]

SLIDE 43

JLQCD 16: UA(1) susceptibility

is this showing really, exactly Δ→0 ?

ð Þ

update available closer to continuum limit

SLIDE 44

JLQCD 16: UA(1) susceptibility

is this showing really, exactly Δ→0 ?

ð Þ

seemingly Δ→0 update available closer to continuum limit

SLIDE 45

U(1)A susceptibility Nf=2 [JLQCD preliminary]

48

32 24

seemingly vanishing as m→0

Suzuki, JPS 2018.3

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 50 100 150 200 250 ρ(|λ|) [GeV3] Overlap Dirac eigenvalue |λ| [MeV] 323x12, β=4.3, T=220MeV, m=0.001(2.64MeV)

n DW
n OV

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 50 100 150 200 250 ρ(|λ|) [GeV3] Overlap Dirac eigenvalue |λ| [MeV] 323x12, β=4.3, T=220MeV, m=0.01(26.4MeV)

n DW
n OV

SLIDE 46

U(1)A susceptibility Nf=2 [JLQCD preliminary]

48

32 24

seemingly vanishing as m→0

Suzuki, JPS 2018.3 physical ud

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 50 100 150 200 250 ρ(|λ|) [GeV3] Overlap Dirac eigenvalue |λ| [MeV] 323x12, β=4.3, T=220MeV, m=0.001(2.64MeV)

n DW
n OV

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 50 100 150 200 250 ρ(|λ|) [GeV3] Overlap Dirac eigenvalue |λ| [MeV] 323x12, β=4.3, T=220MeV, m=0.01(26.4MeV)

n DW
n OV

SLIDE 47

U(1)A susceptibility Nf=2 [JLQCD preliminary]

seemingly vanishing as m→0, more evident in log-log prot

SLIDE 48

Analytic works

Aoki-Fukaya-Taniguchi
QCD with OV regulator
assuming analyticity of ρ(0)
fA → 0 : U(1)A br. parameter
χtop= 0 for 0<m<mc
Kanazawa-Yamamoto
assuming fA≠ 0
expansing free energy in m
discussing
finite m and V effect
contributions of topological

sectors

SLIDE 49

Kanazawa - Yamamoto

assuming fA≠ 0
expansing free energy in m

Z(T, V3, M) = exp

−V3

T f(T, V3, M)

,

f(T, V3, M) = f0 − f2 tr M†M − fA(det M + det M†) + O(M4) ,

represents the effect of axial as det M → e4iθA det M

breaks U(1)A

ther terms are invariant under U(1)A

all invariant under SU(2)LxR

M → e−2iθAVLMV †

R

logical sectors. As is w

via M → M eiθ/Nf

to study topological sectors

ZQ(T, V3, M) ≡ dθ 2π e−iQθ Z(T, V3, Meiθ/2). = e−V4[f0−f2(m2

u+m2 d)]

dθ 2π e−iQθ e2V4fAmumd cos θ = e−V4[f0−f2(m2

u+m2 d)] IQ(2V4fAmumd) ,

∆π−δ =

∞

X

Q=−∞

ZQ Z PQ PQ = 8fA

I′

Q(2V4fAm2)

IQ(2V4fAm2)

SLIDE 50

Kanazawa - Yamamoto: U(1)A br. scenario

logical sectors. As is w

via M → M eiθ/Nf

to study topological sectors

ZQ(T, V3, M) ≡ dθ 2π e−iQθ Z(T, V3, Meiθ/2). = e−V4[f0−f2(m2

u+m2 d)]

dθ 2π e−iQθ e2V4fAmumd cos θ = e−V4[f0−f2(m2

u+m2 d)] IQ(2V4fAmumd) ,

∆π−δ =

∞

X

Q=−∞

ZQ Z PQ PQ = 8fA

I′

Q(2V4fAm2)

IQ(2V4fAm2)

x = 2V4fAm2

relative contribution of modes

x ) )

∆|Q=0

∆|full

KY tells

fixed topology gives wrong result at small V
adding all Q sector or large enough volume necessary

JLQCD

does not fix topology (DW)
zero-mode subtraction may have similar effect to fix Q=0
for smallest m: actually effectively fixed to Q=0

SLIDE 51

compare with JLQCD Δ with non-zero modes

x ) )

x = 2V4fAm2

fix V: Δ→0 as m2 for m→0 even for U(1)A br. case

→ NOT inconsistent with JLQCD results

fix m : Δ∝V

10000 20000 30000 40000 V3 0.1 0.2 0.3 ∆subt m=12 MeV m= 9 MeV m= 2 MeV

Nt=8, T=217 MeV

100 200 300 m

2 [MeV 2]

0.1 0.2 0.3 16

3x8

32

3x8

Nt=8, T=217 MeV

Δ V

[JLQCD 2016 Tomiya et al]

SLIDE 52

compare with JLQCD Δ with non-zero modes

x ) )

x = 2V4fAm2

fix V: Δ→0 as m2 for m→0 even for U(1)A br. case

→ NOT inconsistent with JLQCD results

fix m : Δ∝V

10000 20000 30000 40000 V3 0.1 0.2 0.3 ∆subt m=12 MeV m= 9 MeV m= 2 MeV

Nt=8, T=217 MeV

100 200 300 m

2 [MeV 2]

0.1 0.2 0.3 16

3x8

32

3x8

Nt=8, T=217 MeV

Consistent with U(1)A breaking ??? Δ V

[JLQCD 2016 Tomiya et al]

SLIDE 53

competing scenarios for χt and Δπ-δ (UA(1) oder parameter) @ T=~220 MeV

KY scenario [Kanazawa, Yamamoto 2016]
Δπ-δ: including zero mode cont. is proper
Δπ-δ = const >0
Δπ-δ ≃ 8 V fA2 m2 for Q=0 sector (for 2V fAm2 < 1)
Δπ-δ @ lightest point only from Q=0
χt = 2 fA m2
tension at m≥10 MeV χt sudden growth

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 5 10 15 20 25 323×12, β=4.3 T=220MeV U(1)A susceptibility ∆π-δ [GeV2] Quark mass m [MeV] ∆

ov on DW

∆

ov on OV
3
2
1

1 2 3 Qt 1 10 100 1000 histogram OV-DW OV-OV

OV index

32 3x12, β=4.3, m=0.001

5 10 15 20 25 30 m [MeV] 0.0 5.0×10

7

1.0×10

8

1.5×10

8

χt [MeV

4]

β=4.3 OV-OV T=0 (Nf=2+1) KY scenario from ∆π−δ (m=3MeV) AFT scenario

T=220 MeV

SLIDE 54

competing scenarios for χt and Δπ-δ (UA(1) oder parameter) @ T=~220 MeV

KY scenario [Kanazawa, Yamamoto 2016]
Δπ-δ: including zero mode cont. is proper
Δπ-δ = const >0
Δπ-δ ≃ 8 V fA2 m2 for Q=0 sector (for 2V fAm2 < 1)
Δπ-δ @ lightest point only from Q=0
χt = 2 fA m2
tension at m≥10 MeV χt sudden growth

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 5 10 15 20 25 323×12, β=4.3 T=220MeV U(1)A susceptibility ∆π-δ [GeV2] Quark mass m [MeV] ∆

ov on DW

∆

ov on OV
3
2
1

1 2 3 Qt 1 10 100 1000 histogram OV-DW OV-OV

OV index

32 3x12, β=4.3, m=0.001

Volume study would be useful to check this

5 10 15 20 25 30 m [MeV] 0.0 5.0×10

7

1.0×10

8

1.5×10

8

χt [MeV

4]

β=4.3 OV-OV T=0 (Nf=2+1) KY scenario from ∆π−δ (m=3MeV) AFT scenario

T=220 MeV

SLIDE 55

competing scenarios for χt and Δπ-δ (UA(1) oder parameter) @ T=~220 MeV

KY scenario [Kanazawa, Yamamoto 2016]
Δπ-δ: including zero mode cont. is proper
Δπ-δ = const >0
Δπ-δ ≃ 8 V fA2 m2 for Q=0 sector (for 2V fAm2 < 1)
Δπ-δ @ lightest point only from Q=0
χt = 2 fA m2
tension at m≥10 MeV χt sudden growth

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 5 10 15 20 25 323×12, β=4.3 T=220MeV U(1)A susceptibility ∆π-δ [GeV2] Quark mass m [MeV] ∆

ov on DW

∆

ov on OV
3
2
1

1 2 3 Qt 1 10 100 1000 histogram OV-DW OV-OV

OV index

32 3x12, β=4.3, m=0.001

Volume study would be useful to check this

5 10 15 20 25 30 m [MeV] 0.0 5.0×10

7

1.0×10

8

1.5×10

8

χt [MeV

4]

β=4.3 OV-OV T=0 (Nf=2+1) KY scenario from ∆π−δ (m=3MeV) AFT scenario

T=220 MeV

matching to KM not conclusive yet. JLQCD study along this line underway → Lattice 2018

SLIDE 56

Why bother ?

Because it is unsettled problem !
fate of U(1)A lattice
HotQCD (DW, 2012) broken
JLQCD (topology fixed overlap, 2013) restores
TWQCD (optimal DW, 2013) restores ?
LLNL/RBC (DW, 2014) broken
HotQCD (DW, 2014) broken
Dick et al. (overlap on HISQ, 2015) broken
Brandt et al. (O(a) improved Wilson 2016) restores
JLQCD (reweighted overlap from DW, 2016) restores
JLQCD (current: see Suzuki et al Lattice 2017) restores
Ishikawa et al (Wilson, 2017) at least Z4 restores

2+1 2+1 2+1 2+1

Nf

2 2 2 2 2 2

SLIDE 57

Summary

SLIDE 58

Summary

the status of the fate of U(1)A is still unclear at least to me
So far
Nf=2+1 studies suggest U(1)A breaking
Nf=2 studies suggest U(1)A restoration
needs to be carefully check these lattice technique / property
partially quenching
residual chiral symmetry breaking
ther possible source of systematic error
finite volume effect should be checked for zero-mode subtracted Δπ-δ

➡ JLQCD

More study needed !

SLIDE 59

Thank you very much for your attention !

SLIDE 60

Lattice framework

DWF ensemble → reweighted to overlap
Möbius DWF: almost exact chiral symmetry:

mres = 0.05(3) MeV (β=4.3, Ls=16)

Overlap: exact chiral symmetry
DW→OV reweighting

SLIDE 61

Lattice framework

DWF ensemble → reweighted to overlap
Möbius DWF: almost exact chiral symmetry:

mres = 0.05(3) MeV (β=4.3, Ls=16)

Overlap: exact chiral symmetry
DW→OV reweighting

hOiov ¼ hORiDW hRiDW ;

SLIDE 62

Lattice framework

DWF ensemble → reweighted to overlap
Möbius DWF: almost exact chiral symmetry:

mres = 0.05(3) MeV (β=4.3, Ls=16)

Overlap: exact chiral symmetry
DW→OV reweighting

hOiov ¼ hORiDW hRiDW ;

R ≡ det½HovðmÞ2 det½H4D

DWðmÞ2 × det½H4D DWð1=4aÞ2

det½Hovð1=4aÞ2 :

SLIDE 63

Lattice framework

DWF ensemble → reweighted to overlap
Möbius DWF: almost exact chiral symmetry:

mres = 0.05(3) MeV (β=4.3, Ls=16)

Overlap: exact chiral symmetry
DW→OV reweighting

Dov = 1 2

λi<λth

(1 + γ5sgnλi) |λi⇤⇥λi|

Exact low modes

+D4D

DW

1
λi<λth

|λi⇤⇥λi|

High modes

,

hOiov ¼ hORiDW hRiDW ;

R ≡ det½HovðmÞ2 det½H4D

DWðmÞ2 × det½H4D DWð1=4aÞ2

det½Hovð1=4aÞ2 :

SLIDE 64

Lattice framework

DWF ensemble → reweighted to overlap
Möbius DWF: almost exact chiral symmetry:

mres = 0.05(3) MeV (β=4.3, Ls=16)

Overlap: exact chiral symmetry
DW→OV reweighting

Dov = 1 2

λi<λth

(1 + γ5sgnλi) |λi⇤⇥λi|

Exact low modes

+D4D

DW

1
λi<λth

|λi⇤⇥λi|

High modes

,

hOiov ¼ hORiDW hRiDW ;

R ≡ det½HovðmÞ2 det½H4D

DWðmÞ2 × det½H4D DWð1=4aÞ2

det½Hovð1=4aÞ2 :

HM ¼ γ5 αDW 2 þ DW ;

λ for

SLIDE 65

resolution of susceptibility (ex: m=0.001)

null measurement of topological excitation after reweighting

does not readily mean χt=0: (this case <Q2>=4(4) x10-6)
there must be a resolution of χt under given statistics
[resolution of <Q2>] = 1/Neff
shall take the “statistical error” of <Q2> = max(Δ<Q2>, 1/Neff)

Effective number of statistics

decreases with reweighting
Neff=Nconf <R>/Rmax
Nconf=1326 → Neff = 32
3
2
1

1 2 3 Qt 1 10 100 1000 histogram OV-DW OV-OV

OV index

32

3x12, β=4.3, m=0.001

5 10 15 20 25 30 mf [MeV] 5e+07 1e+08 1.5e+08 2e+08 χ [MeV

4]

GL-DW GL-OV OV-DW OV-OV

32

3x12, β=4.3