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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
