Quenched staggered eigenvalues
Eduardo Follana, Christine Davies, Alistair Hart
HPQCD and UKQCD
SUPA: Universities of Glasgow and Edinburgh
January 2006
Motivation
Summary of:
Phys. Rev. Lett. 93 (2004) 241601,
Phys. Rev. D 72 (2005) 054501,
Lat04, Lat05...
Study of low-lying (improved) staggered Dirac eigenmodes
quenched gauge backgrounds.
Results compared with:
continuum QCD expectations,
Random Matrix Theory.
(Constrains A. Hasenfratz’s musings on possible additive
quark mass renormalisation [hep-lat/0511021]).
Continuum QCD spectral expectations
D
/ anti-Hermitian ⇒ Df
/ s = i λs f s
{D,
/ γ5 } = 0
⇒
sp(D)
/ = {±i λs , λs ∈ R}.
(
±1 if λs = 0, with number n±
Chirality: χs = fs† γ5 fs =
0
otherwise
Topological charge:
Q=
R
F F̃ ∈ Z
Atiyah-Singer index theorem: Q = n+ − n−
The staggered spectrum
D
/ anti-Hermitian ⇒ Df
/ s = i λs f s
P
{D,
/ ǫ(x)} = 0 with ǫ(x) = (−1) µ xµ .
⇒ sp(D)
/ = {±i λs , λs ∈ R}.
Continuum limit:
SU(4)⊗SU(4) chiral symmetry
Expect (Nt = 4)-fold copy of continuum
Finite-a: O(a2 ) corrections:
Near zero modes: Q × Nt /2 = 2Q positive modes
Non-zero modes: Nt = 4-plets, slightly split
Do we see this continuum picture?
How large are the O(a2 ) corrections?
Measuring topological charge
No topological stability at finite a.
O(a2 ) ambiguity in Q
(Improved) staggered: no exact index
GW: index, but no exact index theorem
Us: 4 gluonic and 1 fermionic method
Each agrees with another to within 10%.
Simulation details
Glue:
SU(3), d = 3 + 1, quenched,
tadpole improved Symanzik action
Quarks: staggered actions
Naı̈ve,
Asqtad,
Hisq(-ish).
Lattices:
a/fm
0.125
0.093
0.077
size, aL/fm
1.1 1.5 1.9
×
× × ×
×
The staggered spectrum
Wilson Glue (a ≈ 0.1 fm, V = 163 x 32)
1
FAT7xASQ
ASQTAD
ONE-LINK
CHIRALITY
0.8
0.014
EIGENVALUES
0.012
0.01
0.6
0.008
0.4
0.006
0.2
0.004
0.002
0
0
2
4
6
8
10
12
14
16
2
4
6
8
10
12
14
16
10
12
14
16
10
12
14
16
Improved Glue (a ≈ 0.093 fm, V = 164)
1
0.09
FAT7xASQ
ASQTAD
ONE-LINK
CHIRALITY
0.8
EIGENVALUES
0.08
0.07
0.06
0.6
0.05
0.04
0.4
0.03
0.2
0.02
0.01
0
0
2
4
6
8
10
12
14
16
2
4
6
8
Improved Glue (a ≈ 0.077 fm, V = 204)
1
0.09
FAT7xASQ
ASQTAD
ONE-LINK
CHIRALITY
0.8
EIGENVALUES
0.08
0.07
0.06
0.6
0.05
0.04
0.4
0.03
0.2
0.02
0.01
0
0
2
4
6
8
10
12
14
16
2
4
6
8
Chirality gap
1
FAT7XASQ
0.8
0.6
0.4
fat7xasqtad, fine lattice
0.8
0.2
0
0.0001
0.7
0.001
0.01
0.1
1
0.6
ASQTAD
0.8
0.5
0.6
0.4
0.4
0.2
0
0.0001
0.3
0.001
0.01
0.1
0.2
1
ONE-LINK
0.1
0.8
0
1e-08
0.6
0.4
0.2
0
0.0001
0.001
0.01
0.1
1e-07
1e-06
1e-05
1e-04
0.001
0.01
0.1
Near zero modes and lattice spacing
|Q|=1
20
|Q|=2
ONE-LINK
ASQTAD
FAT7xASQ
aλ0 = 0 + O(a2 )
<Λ0>Q / MeV
15
10
5
0
0
0.01
0
0.01
2
(Lattice spacing, a / fm)
Near zero modes and volume
|Q|=1
|Q|=2
20
ONE-LINK
FAT7xASQ
√
aλ0 = O(1/ a4 V )
<Λ0>Q / MeV
15
10
5
0
0
0.3
0.6
0
4
0.5
1/(a V)
0.3
/ fm
-2
0.6
Non-zero modes and lattice spacing
Q=0
|Q|=1
|Q|=2
<Λ2>Q / MeV
125
ONE-LINK
ASQTAD
FAT7xASQ
100
75
50
100
<Λ1>Q / MeV
aλs = const. +
O(a2 )
75
50
25
0
0
0.005 0.01 0.015 0 0.005 0.01 0.015 0
2
(Lattice spacing, a / fm)
0.005 0.01 0.015
Non-zero modes and volume
|Q|=1
Q=0
|Q|=2
<Λ2>Q / MeV
ONE-LINK
FAT7xASQ
aλs = O(1/(a4 V ))
200
100
<Λ1>Q / MeV
0
200
100
0
0
0.2
0.4
0.6 0
0.2
0.4
0.6 0
4
-4
1/(a V) / fm
0.2
0.4
0.6
Intra-quartet splitting and lattice spacing
Q=0
|Q|=1
|Q|=2
30
ONE-LINK
ASQTAD
FAT7xASQ
<δΛ1>Q / MeV
20
10
0
0
0.01
0
0.01
0
2
(Lattice spacing, a / fm)
0.01
Conclusions
Features of continuum spectrum well reproduced
(Also predictions of Random Matrix Theory.)
Near-zero and non-zero modes clearly distinguished
O(a2 ) corrections small
Corrections → 0 as a → 0, V → ∞.
How small do we need? e.g. (a = 0.09, L = 20):
λ0 ∼ √
(3 + Q) MeV
Q̄ = χV = 2.6
⇒ λ¯0 ∼ 5 MeV ≃ mu,d
Suggests any additive quark mass renormalisation also small
(also volume dependent?)