restoration of chiral symmetry with overlap fermionsdk-user/talks/denissenya_18122013.pdf ·...

Restoration of Chiral Symmetry withOverlap Fermions

M.Denissenya, L.Ya.Glozman, C.B. Lang

Inst. f. Physik, FB Theoretische PhysikUniversität Graz

PhD Seminar talk

Graz, December 18, 20131 / 29

Outline

...1 Motivation

...2 Introduction

...3 Stochastic all-to-all propagators

...4 Mesons in all-to-all approach

...5 Meson spectrum under low-mode truncation

...6 Conclusions

2 / 29

Motivation:L. Ya. Glozman, C.B. Lang, M.Schroeck Phys. Rev.D 86 (2012)

Restoration of SU(2)L × SU(2)R requires the meson states to fallinto multiplets of SU(2)L × SU(2)R × Ci.

(0,0) : ω(0, 1−−) f1(0, 1++)

(1/2,1/2)a : h1(0, 1+−) ρ(1, 1−−) ↑

U(1)A(1/2,1/2)b : ω(0, 1−−) b1(1, 1

+−) ↓(0,1)+(1,0) : a1(1, 1

++)← SU(2)A → ρ(1, 1−−)

Do ma1 −mρ → 0 , mρ −mb1 → 0 if one artificially restoreschiral symmetry by removing the quark condensate ?

3 / 29

Introduction: Shifting to Overlap Fermions

Chirally Improved DCI

nf = 2 dynamicalsimulations163x32 lattice size161 gauge configurations

a = 0.1440(12) fm,L ≈ 2.3 fmmπ = 322(5)MeVeigenvalues and eigenmodesof γ5DCI

one-to-allQ unfixed

Overlap Dov

nf = 2 dynamicalsimulations163x32 lattice size100 gauge configurations(JLQCD) S.Aoki et al (2008)a = 0.1184(30) fm,L ≈ 1.9 fmmπ = 289(2)MeVeigenvalues and eigenmodesof Dov

stochastic all-to-allQ = 0

4 / 29

Overlap Operator Neuberger(1998)

D(m) = (ρ+m

2) + (ρ− m

2)γ5sign(Hw)

m - is the lattice quark mass,0 < ρ < 2 - is the simulationparameter (ρ = 1.6),Hw = γ5Dw(−ρ) -Wilson-Dirac operatoreigenvalues appear in pairs(λk, λ

∗k) (Q = 0)

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Imλ

Reλ

ρ-m/2

ρ+m/2

0 0.015

Zoom

m

Satisfies Ginsparg-Wilson relation (1982)

γ5D +Dγ5 = aDγ5D

⇒ Fermionic action is invariant under chiral rotations:ψ′ = exp(iθbT bγ5(1−

a

2Dov))ψ(x), ψ̄′ = ψ̄(x) exp(iθbT b(1− a

2Dov)γ5)

5 / 29

Spectral density of eigenvalues:

0

1

2

3

4

5

6

50 100 150 200 250 300 350 400 450

H(|λ|)

|λ|, MeV

m

0 10 20 30 40 50 60 70 80 90

100

0 50 100 150 200 250 300 350 400 450

k

|λ|, MeV

∫0|λ| (H(|ν|)d|ν|

According toBank-Casher(1980).

...... ⟨0|qq|0⟩ = −πρ(0)

- in the sequence oflimits V →∞ andmq → 0

ρ(0) ̸= 0⇐⇒ DχSB L L

N a

k

32x163m

λ

λ=0

32x16 x3x4 q

6 / 29

Stochastic All-to-all propagatorsFoley et al (2005)

Full propagator via spectral representation

D−1(x, y) =12V∑k

1

λk

uk(x)u†k(y)

Low-mode contribution.

......D−1

low =Ne∑k

1

λk

uk(x)u†k(y)

High-mode contribution can be estimated by

Dxr = P1ηr for (r = 1, ..., Nr), P1 = 1−Nep∑k=1

uku†k

.

......D−1

high =1

Nr

Nr∑r=1

xr(P1ηr)†

7 / 29

Stochastic all-to-all Propagators

Stochastic all-to-all propagator is constructed as follows

D−1Full(x, y) =

Nev+Nr∑k=1

vk(x)wk(y)†

{vk} ={u1, u2, ..., uNev , x1, ..., xNr

}{wk} =

{u1

λ1

,u2

λ2

, ...,uNev

λNev

,Plη1Nr

, ...,PlηNr

Nr

}vk,wk are used in the construction of meson correlators

8 / 29

Mesons in all-to-all approachMeson two-point functions

CΓΓ′(t, t′;p = 0)

= ⟨(q̄2Γq1)(t′)(q̄1Γq2)(t)⟩

=∑x,x′

∑r,r′

ϕ(r)ϕ(r′)tr[Γ′D−1q1(x′, t′;x, t)Γ′D−1

q2(x + r, t;x′ + r′, t′)]

=

Nep+Nr∑n=1

Nep+Nr∑m=1

O(m,n)(t)O(n,m)(t′)

where.

......O(n,m)(t) =

∑r

ϕ(r)wm(x + r, t)Γvn(x, t)

wm, vn are reused for different smearing functions ϕ(r)

9 / 29

Smearing

Correlators are calculated with nine different choices of thesmearing functions (9× 9 combinations)

ϕ1(r) = δr,0, ϕ2(r) =const, ϕi(r) ∝ A|r|B e−C|r|D

with the normalization∑

r |ϕi(r)| = 1, (i = 1, 2, ..., 9).

We specify gamma matrices Γ:Γ = γ5 for pseudoscalar mesonsΓ = γi, γiγt for ρ mesonsΓ = γiγ5 for a1 mesonsΓ = σij for b1 mesons

where σij = i2[γi, γj]

10 / 29

Variational methodCross-correlation matrices Cij are computed with Oi’s involvingdifferent smearing functions at sink/source with an appropriate Γstructure

Cij(t) = ⟨0|Oi(t)O†j(0)|0⟩

Solving generalized eigenvalue problem:

C(t)υ⃗n = λ̃(n)(t)C(t0)υ⃗n,

meson ground and excited states are extracted from

λ̃(n)(t, t0) = e−En(t−t0)(1 +O

(e−∆En(t−t0)

))υ⃗n act as fingerprints of the corresponding states

11 / 29

Meson spectrum

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses, FULL

Mπ(285) MeV

Mπ(1683) MeV

Mπ(2625) MeV

0th1st

2nd3rd

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses, FULL

Mρ(866) MeV

Mρ(1693) MeV

Mρ(2615) MeV

0th1st

2nd

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses, FULL

Ma1(1172) MeV

Ma1(1805) MeV

0th1st

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective mass, FULL

Mb1(1217) MeV

0th

π - int:3,4,7,8,9; ρ - int: 3,4,5,6,7,9 (4,6,8,12,17,18)a1 - int:3,4,5,6,7,8; b1 - int:8,9

12 / 29

Meson spectrum

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses, FULL

Mπ(285) MeV

Mπ(1683) MeV

Mπ(2625) MeV

0th1st

2nd3rd

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses, b FULLMρ(863) MeVMρ(1339) MeVMρ(1840) MeV

0th1st

2nd3rd

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses, FULL

Ma1(1172) MeV

Ma1(1805) MeV

0th1st

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective mass, FULL

Mb1(1217) MeV

0th

π - int:3,4,7,8,9; ρ - int: 3,4,5,6,7,9 (4,6,8,12,17,18)a1 - int:3,4,5,6,7,8; b1 - int:8,9

13 / 29

Meson spectrum under low-mode truncationFormally

⟨OO†⟩ = ”D−1l D−1

l ” + ”D−1l D−1

h ” + ”D−1h D−1

l ” + ”D−1h D−1

h ”

Practically with{vk} =

{u1, u2, ..., uNev , x1, ..., xNr

}{wk} =

{u1

λ1

,u2

λ2

, ...,uNev

λNev

,Plη1Nr

, ...,PlηNr

Nr

}an arbitrary number k of the low modes can beincluded/excluded from the full propagator. This implies

⟨OO†⟩ = Ckll︸︷︷︸

CLM(k)

+Ck+1lh + Ck+1

hl + Chh︸ ︷︷ ︸CRD(k)

CLM(k) - contribution of k low modes onlyCRD(k) - contribution of all the eigenmodes except for k lowmodes

14 / 29

Saturating π with the low-modes

10-1

100

4 8 12 16 20 24 28 32

log

C(t

)

t

CorrelatorsLM006-LM090

LM100FULL

LM002LM004LM070

int:2 (wall smearing at source/sink)

15 / 29

π under the low-mode removal

0.173

0.5

1

0 2 4 6 8 10 12 14 16

mef

f

t

Effective massesFULL

RD004RD010RD030RD090RD100

int:2 (wall smearing at source/sink)

16 / 29

a1(1++)

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (FULL, int:3 4 5 6 7 8)0th1st

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (RD002, int:3 4 5 6 7 8)0th1st

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (RD002,State 0)345678

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (RD002,State 1)345678

17 / 29

a1(1++)

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (FULL, int:3 4 5 6 7 8)0th1st

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (RD020, int:3 4 5 6 7 8)0th1st

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (RD020,State 0)345678

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (RD020,State 1)345678

18 / 29

ρ(1−−)

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (FULL,int:4 6 8 12 17 18)0th1st

2nd3rd4th

10-4

10-3

10-2

10-1

100

0 2 4 6 8 10 12 14 16

C(t

)

t

Normalized Eigenvalues FULL (int:4 6 8 12 17 18)0th1st

2nd3rd4th5th

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (FULL, State 0)468

121718

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (FULL, State 1)468

121718

19 / 29

ρ(1−−)

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (FULL,int:4 6 8 12 17 18)0th1st

2nd3rd4th

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (RD002,int:4 6 8 12 17 18)0th1st

2nd3rd4th

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (FULL, State 0)468

121718

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (RD002, State 0)468

121718

20 / 29

ρ(1−−)

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (FULL,int:4 6 8 12 17 18)0th1st

2nd3rd4th

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (RD020,int:4 6 8 12 17 18)0th1st

2nd3rd4th

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (FULL, State 0)468

121718

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (RD020, State 0)468

121718

21 / 29

ρ(1−−)

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (FULL,int:4 6 8 12 17 18)0th1st

2nd3rd4th

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (RD002,int:4 6 8 12 17 18)0th1st

2nd3rd4th

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (FULL, State 1)468

121718

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (RD002, State 1)468

121718

22 / 29

ρ(1−−)

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (FULL,int:4 6 8 12 17 18)0th1st

2nd3rd4th

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (RD020,int:4 6 8 12 17 18)0th1st

2nd3rd4th

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (FULL, State 1)468

121718

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (RD020, State 1)468

121718

23 / 29

b1(1−−)

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (FULL ,int:8 9)0th

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (RD002 ,int:8 9)0th

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (FULL, State 0)89

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (RD002, State 0)89

24 / 29

b1(1−−)

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (FULL ,int:8 9)0th

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

mef

f

t

Effective masses (RD030 ,int:8 9)0th

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (FULL, State 0)89

-1

-0.5

0

0.5

1

2 4 6 8 10 12 14 16t

Eigenvector components (RD030, State 0)89

25 / 29

800

1000

1200

1400

1600

1800

0 5080 140 200 285 360

0 6 10 20 30 50 70

mef

f(k)

σ, MeV

kρ , 0thρ, 1st

a1, 0thb1, 0th

26 / 29

800

1100

1400

1700

2000

2300

0 5080 140 200 285 360

0 6 10 20 30 50 70

mef

f(k)

σ, MeV

kρ , 0thρ, 1st

a1, 0thb1, 0th

a1, 1stρ , 2nd

27 / 29

Conclusions

under the low-mode removalChiral symmetry gets restored (seen in the excited statestoo)U(1)A symmetry is restoredMultiple degeneracy of states indicates the presence of somehigher symmetry

28 / 29

Special Thanks to

S.Aoki, S. Hashimoto, T.Kaneko

and

for the collaboration29 / 29